Purpose
To find a controllable realization for the linear time-invariant
single-input system
dX/dt = A * X + B * U,
where A is an N-by-N matrix and B is an N element vector which
are reduced by this routine to orthogonal canonical form using
(and optionally accumulating) orthogonal similarity
transformations.
Specification
SUBROUTINE AB01MD( JOBZ, N, A, LDA, B, NCONT, Z, LDZ, TAU, TOL,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBZ
INTEGER INFO, LDA, LDZ, LDWORK, N, NCONT
DOUBLE PRECISION TOL
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(*), DWORK(*), TAU(*), Z(LDZ,*)
Arguments
Mode Parameters
JOBZ CHARACTER*1
Indicates whether the user wishes to accumulate in a
matrix Z the orthogonal similarity transformations for
reducing the system, as follows:
= 'N': Do not form Z and do not store the orthogonal
transformations;
= 'F': Do not form Z, but store the orthogonal
transformations in the factored form;
= 'I': Z is initialized to the unit matrix and the
orthogonal transformation matrix Z is returned.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation,
i.e. the order of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state dynamics matrix A.
On exit, the leading NCONT-by-NCONT upper Hessenberg
part of this array contains the canonical form of the
state dynamics matrix, given by Z' * A * Z, of a
controllable realization for the original system. The
elements below the first subdiagonal are set to zero.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (N)
On entry, the original input/state vector B.
On exit, the leading NCONT elements of this array contain
canonical form of the input/state vector, given by Z' * B,
with all elements but B(1) set to zero.
NCONT (output) INTEGER
The order of the controllable state-space representation.
Z (output) DOUBLE PRECISION array, dimension (LDZ,N)
If JOBZ = 'I', then the leading N-by-N part of this array
contains the matrix of accumulated orthogonal similarity
transformations which reduces the given system to
orthogonal canonical form.
If JOBZ = 'F', the elements below the diagonal, with the
array TAU, represent the orthogonal transformation matrix
as a product of elementary reflectors. The transformation
matrix can then be obtained by calling the LAPACK Library
routine DORGQR.
If JOBZ = 'N', the array Z is not referenced and can be
supplied as a dummy array (i.e. set parameter LDZ = 1 and
declare this array to be Z(1,1) in the calling program).
LDZ INTEGER
The leading dimension of array Z. If JOBZ = 'I' or
JOBZ = 'F', LDZ >= MAX(1,N); if JOBZ = 'N', LDZ >= 1.
TAU (output) DOUBLE PRECISION array, dimension (N)
The elements of TAU contain the scalar factors of the
elementary reflectors used in the reduction of B and A.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in determining the
controllability of (A,B). If the user sets TOL > 0, then
the given value of TOL is used as an absolute tolerance;
elements with absolute value less than TOL are considered
neglijible. If the user sets TOL <= 0, then an implicitly
computed, default tolerance, defined by
TOLDEF = N*EPS*MAX( NORM(A), NORM(B) ) is used instead,
where EPS is the machine precision (see LAPACK Library
routine DLAMCH).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,N).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The Householder matrix which reduces all but the first element of vector B to zero is found and this orthogonal similarity transformation is applied to the matrix A. The resulting A is then reduced to upper Hessenberg form by a sequence of Householder transformations. Finally, the order of the controllable state- space representation (NCONT) is determined by finding the position of the first sub-diagonal element of A which is below an appropriate zero threshold, either TOL or TOLDEF (see parameter TOL); if NORM(B) is smaller than this threshold, NCONT is set to zero, and no computations for reducing the system to orthogonal canonical form are performed.References
[1] Konstantinov, M.M., Petkov, P.Hr. and Christov, N.D.
Orthogonal Invariants and Canonical Forms for Linear
Controllable Systems.
Proc. 8th IFAC World Congress, Kyoto, 1, pp. 49-54, 1981.
[2] Hammarling, S.J.
Notes on the use of orthogonal similarity transformations in
control.
NPL Report DITC 8/82, August 1982.
[3] Paige, C.C
Properties of numerical algorithms related to computing
controllability.
IEEE Trans. Auto. Contr., AC-26, pp. 130-138, 1981.
Numerical Aspects
3 The algorithm requires 0(N ) operations and is backward stable.Further Comments
NoneExample
Program Text
* AB01MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA, LDZ
PARAMETER ( LDA = NMAX, LDZ = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, J, N, NCONT
CHARACTER*1 JOBZ
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(NMAX), DWORK(LDWORK), TAU(NMAX),
$ Z(LDZ,NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL AB01MD, DORGQR
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read in the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, TOL, JOBZ
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( B(I), I = 1,N )
* Find a controllable realization for the given system.
CALL AB01MD( JOBZ, N, A, LDA, B, NCONT, Z, LDZ, TAU, TOL,
$ DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NCONT
DO 20 I = 1, NCONT
WRITE ( NOUT, FMT = 99994 ) ( A(I,J), J = 1,NCONT )
20 CONTINUE
WRITE ( NOUT, FMT = 99996 ) ( B(I), I = 1,NCONT )
IF ( LSAME( JOBZ, 'F' ) )
$ CALL DORGQR( N, N, N, Z, LDZ, TAU, DWORK, LDWORK, INFO )
IF ( LSAME( JOBZ, 'F' ).OR.LSAME( JOBZ, 'I' ) ) THEN
WRITE ( NOUT, FMT = 99995 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99994 ) ( Z(I,J), J = 1,N )
40 CONTINUE
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB01MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB01MD = ',I2)
99997 FORMAT (' The order of the controllable state-space representati',
$ 'on = ',I2,//' The state dynamics matrix A of a controlla',
$ 'ble realization is ')
99996 FORMAT (/' The input/state vector B of a controllable realizatio',
$ 'n is ',/(1X,F8.4))
99995 FORMAT (/' The similarity transformation matrix Z is ')
99994 FORMAT (20(1X,F8.4))
99993 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
AB01MD EXAMPLE PROGRAM DATA 3 0.0 I 1.0 2.0 0.0 4.0 -1.0 0.0 0.0 0.0 1.0 1.0 0.0 1.0Program Results
AB01MD EXAMPLE PROGRAM RESULTS The order of the controllable state-space representation = 3 The state dynamics matrix A of a controllable realization is 1.0000 1.4142 0.0000 2.8284 -1.0000 2.8284 0.0000 1.4142 1.0000 The input/state vector B of a controllable realization is -1.4142 0.0000 0.0000 The similarity transformation matrix Z is -0.7071 0.0000 -0.7071 0.0000 -1.0000 0.0000 -0.7071 0.0000 0.7071
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB01ND.html 0000664 0000000 0000000 00000032461 14560147231 0020325 0 ustar 00root root 0000000 0000000
Purpose
To find a controllable realization for the linear time-invariant
multi-input system
dX/dt = A * X + B * U,
where A and B are N-by-N and N-by-M matrices, respectively,
which are reduced by this routine to orthogonal canonical form
using (and optionally accumulating) orthogonal similarity
transformations. Specifically, the pair (A, B) is reduced to
the pair (Ac, Bc), Ac = Z' * A * Z, Bc = Z' * B, given by
[ Acont * ] [ Bcont ]
Ac = [ ], Bc = [ ],
[ 0 Auncont ] [ 0 ]
and
[ A11 A12 . . . A1,p-1 A1p ] [ B1 ]
[ A21 A22 . . . A2,p-1 A2p ] [ 0 ]
[ 0 A32 . . . A3,p-1 A3p ] [ 0 ]
Acont = [ . . . . . . . ], Bc = [ . ],
[ . . . . . . ] [ . ]
[ . . . . . ] [ . ]
[ 0 0 . . . Ap,p-1 App ] [ 0 ]
where the blocks B1, A21, ..., Ap,p-1 have full row ranks and
p is the controllability index of the pair. The size of the
block Auncont is equal to the dimension of the uncontrollable
subspace of the pair (A, B).
Specification
SUBROUTINE AB01ND( JOBZ, N, M, A, LDA, B, LDB, NCONT, INDCON,
$ NBLK, Z, LDZ, TAU, TOL, IWORK, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER JOBZ
INTEGER INDCON, INFO, LDA, LDB, LDWORK, LDZ, M, N, NCONT
DOUBLE PRECISION TOL
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), TAU(*), Z(LDZ,*)
INTEGER IWORK(*), NBLK(*)
Arguments
Mode Parameters
JOBZ CHARACTER*1
Indicates whether the user wishes to accumulate in a
matrix Z the orthogonal similarity transformations for
reducing the system, as follows:
= 'N': Do not form Z and do not store the orthogonal
transformations;
= 'F': Do not form Z, but store the orthogonal
transformations in the factored form;
= 'I': Z is initialized to the unit matrix and the
orthogonal transformation matrix Z is returned.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation,
i.e. the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs, or of columns of B. M >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state dynamics matrix A.
On exit, the leading NCONT-by-NCONT part contains the
upper block Hessenberg state dynamics matrix Acont in Ac,
given by Z' * A * Z, of a controllable realization for
the original system. The elements below the first block-
subdiagonal are set to zero.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input matrix B.
On exit, the leading NCONT-by-M part of this array
contains the transformed input matrix Bcont in Bc, given
by Z' * B, with all elements but the first block set to
zero.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
NCONT (output) INTEGER
The order of the controllable state-space representation.
INDCON (output) INTEGER
The controllability index of the controllable part of the
system representation.
NBLK (output) INTEGER array, dimension (N)
The leading INDCON elements of this array contain the
the orders of the diagonal blocks of Acont.
Z (output) DOUBLE PRECISION array, dimension (LDZ,N)
If JOBZ = 'I', then the leading N-by-N part of this
array contains the matrix of accumulated orthogonal
similarity transformations which reduces the given system
to orthogonal canonical form.
If JOBZ = 'F', the elements below the diagonal, with the
array TAU, represent the orthogonal transformation matrix
as a product of elementary reflectors. The transformation
matrix can then be obtained by calling the LAPACK Library
routine DORGQR.
If JOBZ = 'N', the array Z is not referenced and can be
supplied as a dummy array (i.e. set parameter LDZ = 1 and
declare this array to be Z(1,1) in the calling program).
LDZ INTEGER
The leading dimension of array Z. If JOBZ = 'I' or
JOBZ = 'F', LDZ >= MAX(1,N); if JOBZ = 'N', LDZ >= 1.
TAU (output) DOUBLE PRECISION array, dimension (N)
The elements of TAU contain the scalar factors of the
elementary reflectors used in the reduction of B and A.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in rank determination when
transforming (A, B). If the user sets TOL > 0, then
the given value of TOL is used as a lower bound for the
reciprocal condition number (see the description of the
argument RCOND in the SLICOT routine MB03OD); a
(sub)matrix whose estimated condition number is less than
1/TOL is considered to be of full rank. If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
defined by TOLDEF = N*N*EPS, is used instead, where EPS
is the machine precision (see LAPACK Library routine
DLAMCH).
Workspace
IWORK INTEGER array, dimension (M)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1, N, 3*M).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Matrix B is first QR-decomposed and the appropriate orthogonal similarity transformation applied to the matrix A. Leaving the first rank(B) states unchanged, the remaining lower left block of A is then QR-decomposed and the new orthogonal matrix, Q1, is also applied to the right of A to complete the similarity transformation. By continuing in this manner, a completely controllable state-space pair (Acont, Bcont) is found for the given (A, B), where Acont is upper block Hessenberg with each subdiagonal block of full row rank, and Bcont is zero apart from its (independent) first rank(B) rows. NOTE that the system controllability indices are easily calculated from the dimensions of the blocks of Acont.References
[1] Konstantinov, M.M., Petkov, P.Hr. and Christov, N.D.
Orthogonal Invariants and Canonical Forms for Linear
Controllable Systems.
Proc. 8th IFAC World Congress, Kyoto, 1, pp. 49-54, 1981.
[2] Paige, C.C.
Properties of numerical algorithms related to computing
controllablity.
IEEE Trans. Auto. Contr., AC-26, pp. 130-138, 1981.
[3] Petkov, P.Hr., Konstantinov, M.M., Gu, D.W. and
Postlethwaite, I.
Optimal Pole Assignment Design of Linear Multi-Input Systems.
Leicester University, Report 99-11, May 1996.
Numerical Aspects
3 The algorithm requires 0(N ) operations and is backward stable.Further Comments
If the system matrices A and B are badly scaled, it would be useful to scale them with SLICOT routine TB01ID, before calling the routine.Example
Program Text
* AB01ND EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX
PARAMETER ( NMAX = 20, MMAX = 20 )
INTEGER LDA, LDB, LDZ
PARAMETER ( LDA = NMAX, LDB = NMAX, LDZ = NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( NMAX, 3*MMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, INDCON, J, M, N, NCONT
CHARACTER*1 JOBZ
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), DWORK(LDWORK),
$ TAU(NMAX), Z(LDZ,NMAX)
INTEGER IWORK(LIWORK), NBLK(NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL AB01ND, DORGQR
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, TOL, JOBZ
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,N ), J = 1,M )
* Find a controllable ssr for the given system.
CALL AB01ND( JOBZ, N, M, A, LDA, B, LDB, NCONT, INDCON,
$ NBLK, Z, LDZ, TAU, TOL, IWORK, DWORK, LDWORK,
$ INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NCONT
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NCONT
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NCONT )
20 CONTINUE
WRITE ( NOUT, FMT = 99994 ) ( NBLK(I), I = 1,INDCON )
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NCONT
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
WRITE ( NOUT, FMT = 99992 ) INDCON
IF ( LSAME( JOBZ, 'F' ) )
$ CALL DORGQR( N, N, N, Z, LDZ, TAU, DWORK, LDWORK,
$ INFO )
IF ( LSAME( JOBZ, 'F' ).OR.LSAME( JOBZ, 'I' ) ) THEN
WRITE ( NOUT, FMT = 99991 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
60 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB01ND EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB01ND = ',I2)
99997 FORMAT (' The order of the controllable state-space representati',
$ 'on = ',I2)
99996 FORMAT (/' The transformed state dynamics matrix of a controllab',
$ 'le realization is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' and the dimensions of its diagonal blocks are ',
$ /20(1X,I2))
99993 FORMAT (/' The transformed input/state matrix B of a controllabl',
$ 'e realization is ')
99992 FORMAT (/' The controllability index of the transformed system r',
$ 'epresentation = ',I2)
99991 FORMAT (/' The similarity transformation matrix Z is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
END
Program Data
AB01ND EXAMPLE PROGRAM DATA 3 2 0.0 I -1.0 0.0 0.0 -2.0 -2.0 -2.0 -1.0 0.0 -3.0 1.0 0.0 0.0 0.0 2.0 1.0Program Results
AB01ND EXAMPLE PROGRAM RESULTS The order of the controllable state-space representation = 2 The transformed state dynamics matrix of a controllable realization is -3.0000 2.2361 0.0000 -1.0000 and the dimensions of its diagonal blocks are 2 The transformed input/state matrix B of a controllable realization is 0.0000 -2.2361 1.0000 0.0000 The controllability index of the transformed system representation = 1 The similarity transformation matrix Z is 0.0000 1.0000 0.0000 -0.8944 0.0000 -0.4472 -0.4472 0.0000 0.8944
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB01OD.html 0000664 0000000 0000000 00000036361 14560147231 0020331 0 ustar 00root root 0000000 0000000
Purpose
To reduce the matrices A and B using (and optionally accumulating)
state-space and input-space transformations U and V respectively,
such that the pair of matrices
Ac = U' * A * U, Bc = U' * B * V
are in upper "staircase" form. Specifically,
[ Acont * ] [ Bcont ]
Ac = [ ], Bc = [ ],
[ 0 Auncont ] [ 0 ]
and
[ A11 A12 . . . A1,p-1 A1p ] [ B1 ]
[ A21 A22 . . . A2,p-1 A2p ] [ 0 ]
[ 0 A32 . . . A3,p-1 A3p ] [ 0 ]
Acont = [ . . . . . . . ], Bc = [ . ],
[ . . . . . . ] [ . ]
[ . . . . . ] [ . ]
[ 0 0 . . . Ap,p-1 App ] [ 0 ]
where the blocks B1, A21, ..., Ap,p-1 have full row ranks and
p is the controllability index of the pair. The size of the
block Auncont is equal to the dimension of the uncontrollable
subspace of the pair (A, B). The first stage of the reduction,
the "forward" stage, accomplishes the reduction to the orthogonal
canonical form (see SLICOT library routine AB01ND). The blocks
B1, A21, ..., Ap,p-1 are further reduced in a second, "backward"
stage to upper triangular form using RQ factorization. Each of
these stages is optional.
Specification
SUBROUTINE AB01OD( STAGES, JOBU, JOBV, N, M, A, LDA, B, LDB, U,
$ LDU, V, LDV, NCONT, INDCON, KSTAIR, TOL, IWORK,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBU, JOBV, STAGES
INTEGER INDCON, INFO, LDA, LDB, LDU, LDV, LDWORK, M, N,
$ NCONT
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK(*), KSTAIR(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), U(LDU,*), V(LDV,*)
Arguments
Mode Parameters
STAGES CHARACTER*1
Specifies the reduction stages to be performed as follows:
= 'F': Perform the forward stage only;
= 'B': Perform the backward stage only;
= 'A': Perform both (all) stages.
JOBU CHARACTER*1
Indicates whether the user wishes to accumulate in a
matrix U the state-space transformations as follows:
= 'N': Do not form U;
= 'I': U is internally initialized to the unit matrix (if
STAGES <> 'B'), or updated (if STAGES = 'B'), and
the orthogonal transformation matrix U is
returned.
JOBV CHARACTER*1
Indicates whether the user wishes to accumulate in a
matrix V the input-space transformations as follows:
= 'N': Do not form V;
= 'I': V is initialized to the unit matrix and the
orthogonal transformation matrix V is returned.
JOBV is not referenced if STAGES = 'F'.
Input/Output Parameters
N (input) INTEGER
The actual state dimension, i.e. the order of the
matrix A. N >= 0.
M (input) INTEGER
The actual input dimension. M >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state transition matrix A to be transformed.
If STAGES = 'B', A should be in the orthogonal canonical
form, as returned by SLICOT library routine AB01ND.
On exit, the leading N-by-N part of this array contains
the transformed state transition matrix U' * A * U.
The leading NCONT-by-NCONT part contains the upper block
Hessenberg state matrix Acont in Ac, given by U' * A * U,
of a controllable realization for the original system.
The elements below the first block-subdiagonal are set to
zero. If STAGES <> 'F', the subdiagonal blocks of A are
triangularized by RQ factorization, and the annihilated
elements are explicitly zeroed.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input matrix B to be transformed.
If STAGES = 'B', B should be in the orthogonal canonical
form, as returned by SLICOT library routine AB01ND.
On exit with STAGES = 'F', the leading N-by-M part of
this array contains the transformed input matrix U' * B,
with all elements but the first block set to zero.
On exit with STAGES <> 'F', the leading N-by-M part of
this array contains the transformed input matrix
U' * B * V, with all elements but the first block set to
zero and the first block in upper triangular form.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
U (input/output) DOUBLE PRECISION array, dimension (LDU,N)
If STAGES <> 'B' or JOBU = 'N', then U need not be set
on entry.
If STAGES = 'B' and JOBU = 'I', then, on entry, the
leading N-by-N part of this array must contain the
transformation matrix U that reduced the pair to the
orthogonal canonical form.
On exit, if JOBU = 'I', the leading N-by-N part of this
array contains the transformation matrix U that performed
the specified reduction.
If JOBU = 'N', the array U is not referenced and can be
supplied as a dummy array (i.e. set parameter LDU = 1 and
declare this array to be U(1,1) in the calling program).
LDU INTEGER
The leading dimension of array U.
If JOBU = 'I', LDU >= MAX(1,N); if JOBU = 'N', LDU >= 1.
V (output) DOUBLE PRECISION array, dimension (LDV,M)
If JOBV = 'I', then the leading M-by-M part of this array
contains the transformation matrix V.
If STAGES = 'F', or JOBV = 'N', the array V is not
referenced and can be supplied as a dummy array (i.e. set
parameter LDV = 1 and declare this array to be V(1,1) in
the calling program).
LDV INTEGER
The leading dimension of array V.
If STAGES <> 'F' and JOBV = 'I', LDV >= MAX(1,M);
if STAGES = 'F' or JOBV = 'N', LDV >= 1.
NCONT (input/output) INTEGER
The order of the controllable state-space representation.
NCONT is input only if STAGES = 'B'.
INDCON (input/output) INTEGER
The number of stairs in the staircase form (also, the
controllability index of the controllable part of the
system representation).
INDCON is input only if STAGES = 'B'.
KSTAIR (input/output) INTEGER array, dimension (N)
The leading INDCON elements of this array contain the
dimensions of the stairs, or, also, the orders of the
diagonal blocks of Acont.
KSTAIR is input if STAGES = 'B', and output otherwise.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in rank determination when
transforming (A, B). If the user sets TOL > 0, then
the given value of TOL is used as a lower bound for the
reciprocal condition number (see the description of the
argument RCOND in the SLICOT routine MB03OD); a
(sub)matrix whose estimated condition number is less than
1/TOL is considered to be of full rank. If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
defined by TOLDEF = N*N*EPS, is used instead, where EPS
is the machine precision (see LAPACK Library routine
DLAMCH).
TOL is not referenced if STAGES = 'B'.
Workspace
IWORK INTEGER array, dimension (M)
IWORK is not referenced if STAGES = 'B'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
If STAGES <> 'B', LDWORK >= MAX(1, N + MAX(N,3*M));
If STAGES = 'B', LDWORK >= MAX(1, M + MAX(N,M)).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Staircase reduction of the pencil [B|sI - A] is used. Orthogonal
transformations U and V are constructed such that
|B |sI-A * . . . * * |
| 1| 11 . . . |
| | A sI-A . . . |
| | 21 22 . . . |
| | . . * * |
[U'BV|sI - U'AU] = |0 | 0 . . |
| | A sI-A * |
| | p,p-1 pp |
| | |
|0 | 0 0 sI-A |
| | p+1,p+1|
where the i-th diagonal block of U'AU has dimension KSTAIR(i),
for i = 1,...,p. The value of p is returned in INDCON. The last
block contains the uncontrollable modes of the (A,B)-pair which
are also the generalized eigenvalues of the above pencil.
The complete reduction is performed in two stages. The first,
forward stage accomplishes the reduction to the orthogonal
canonical form. The second, backward stage consists in further
reduction to triangular form by applying left and right orthogonal
transformations.
References
[1] Van Dooren, P.
The generalized eigenvalue problem in linear system theory.
IEEE Trans. Auto. Contr., AC-26, pp. 111-129, 1981.
[2] Miminis, G. and Paige, C.
An algorithm for pole assignment of time-invariant multi-input
linear systems.
Proc. 21st IEEE CDC, Orlando, Florida, 1, pp. 62-67, 1982.
Numerical Aspects
The algorithm requires O((N + M) x N**2) operations and is backward stable (see [1]).Further Comments
If the system matrices A and B are badly scaled, it would be useful to scale them with SLICOT routine TB01ID, before calling the routine.Example
Program Text
* AB01OD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX
PARAMETER ( NMAX = 20, MMAX = 20 )
INTEGER LDA, LDB, LDU, LDV
PARAMETER ( LDA = NMAX, LDB = NMAX, LDU = NMAX,
$ LDV = MMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX + MAX( NMAX, 3*MMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INDCON, INFO, J, M, N, NCONT
CHARACTER*1 JOBU, JOBV, STAGES
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), DWORK(LDWORK),
$ U(LDU,NMAX), V(LDV,MMAX)
INTEGER IWORK(LIWORK), KSTAIR(NMAX)
* .. External Subroutines ..
EXTERNAL AB01OD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, TOL, STAGES, JOBU, JOBV
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), I = 1,N ), J = 1,N )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
* Reduce the matrices A and B to upper "staircase" form.
CALL AB01OD( STAGES, JOBU, JOBV, N, M, A, LDA, B, LDB, U,
$ LDU, V, LDV, NCONT, INDCON, KSTAIR, TOL, IWORK,
$ DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
WRITE ( NOUT, FMT = 99994 ) INDCON
WRITE ( NOUT, FMT = 99993 ) ( KSTAIR(I), I = 1,INDCON )
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB01OD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB01OD = ',I2)
99997 FORMAT (' The transformed state transition matrix is ')
99996 FORMAT (/' The transformed input matrix is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' The number of stairs in the staircase form = ',I3,/)
99993 FORMAT (' The dimensions of the stairs are ',/(20(I3,2X)))
99992 FORMAT (/' N is out of range.',/' N = ',I5)
99991 FORMAT (/' M is out of range.',/' M = ',I5)
END
Program Data
AB01OD EXAMPLE PROGRAM DATA
5 2 0.0 F N N
17.0 24.0 1.0 8.0 15.0
23.0 5.0 7.0 14.0 16.0
4.0 6.0 13.0 20.0 22.0
10.0 12.0 19.0 21.0 3.0
11.0 18.0 25.0 2.0 9.0
-1.0 -4.0
4.0 9.0
-9.0 -16.0
16.0 25.0
-25.0 -36.0
Program Results
AB01OD EXAMPLE PROGRAM RESULTS The transformed state transition matrix is 12.8848 3.2345 11.8211 3.3758 -0.8982 4.4741 -12.5544 5.3509 5.9403 1.4360 14.4576 7.6855 23.1452 26.3872 -29.9557 0.0000 1.4805 27.4668 22.6564 -0.0072 0.0000 0.0000 -30.4822 0.6745 18.8680 The transformed input matrix is 31.1199 47.6865 3.2480 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 The number of stairs in the staircase form = 3 The dimensions of the stairs are 2 2 1
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB04MD.html 0000664 0000000 0000000 00000024404 14560147231 0020325 0 ustar 00root root 0000000 0000000
Purpose
To perform a transformation on the parameters (A,B,C,D) of a system, which is equivalent to a bilinear transformation of the corresponding transfer function matrix.Specification
SUBROUTINE AB04MD( TYPE, N, M, P, ALPHA, BETA, A, LDA, B, LDB, C,
$ LDC, D, LDD, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TYPE
INTEGER INFO, LDA, LDB, LDC, LDD, LDWORK, M, N, P
DOUBLE PRECISION ALPHA, BETA
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), DWORK(*)
Arguments
Mode Parameters
TYPE CHARACTER*1
Indicates the type of the original system and the
transformation to be performed as follows:
= 'D': discrete-time -> continuous-time;
= 'C': continuous-time -> discrete-time.
Input/Output Parameters
N (input) INTEGER
The order of the state matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
ALPHA, (input) DOUBLE PRECISION
BETA Parameters specifying the bilinear transformation.
Recommended values for stable systems: ALPHA = 1,
BETA = 1. ALPHA <> 0, BETA <> 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state matrix A of the original system.
On exit, the leading N-by-N part of this array contains
_
the state matrix A of the transformed system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input matrix B of the original system.
On exit, the leading N-by-M part of this array contains
_
the input matrix B of the transformed system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the output matrix C of the original system.
On exit, the leading P-by-N part of this array contains
_
the output matrix C of the transformed system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the input/output matrix D for the original system.
On exit, the leading P-by-M part of this array contains
_
the input/output matrix D of the transformed system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
Workspace
IWORK INTEGER array, dimension (N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,N).
For optimum performance LDWORK >= MAX(1,N*NB), where NB
is the optimal blocksize.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the matrix (ALPHA*I + A) is exactly singular;
= 2: if the matrix (BETA*I - A) is exactly singular.
Method
The parameters of the discrete-time system are transformed into
the parameters of the continuous-time system (TYPE = 'D'), or
vice-versa (TYPE = 'C') by the transformation:
1. Discrete -> continuous
_ -1
A = beta*(alpha*I + A) * (A - alpha*I)
_ -1
B = sqrt(2*alpha*beta) * (alpha*I + A) * B
_ -1
C = sqrt(2*alpha*beta) * C * (alpha*I + A)
_ -1
D = D - C * (alpha*I + A) * B
which is equivalent to the bilinear transformation
z - alpha
z -> s = beta --------- .
z + alpha
of one transfer matrix onto the other.
2. Continuous -> discrete
_ -1
A = alpha*(beta*I - A) * (beta*I + A)
_ -1
B = sqrt(2*alpha*beta) * (beta*I - A) * B
_ -1
C = sqrt(2*alpha*beta) * C * (beta*I - A)
_ -1
D = D + C * (beta*I - A) * B
which is equivalent to the bilinear transformation
beta + s
s -> z = alpha -------- .
beta - s
of one transfer matrix onto the other.
References
[1] Al-Saggaf, U.M. and Franklin, G.F.
Model reduction via balanced realizations: a extension and
frequency weighting techniques.
IEEE Trans. Autom. Contr., AC-33, pp. 687-692, 1988.
Numerical Aspects
3 The time taken is approximately proportional to N . The accuracy depends mainly on the condition number of the matrix to be inverted.Further Comments
NoneExample
Program Text
* AB04MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDD
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX )
* .. Local Scalars ..
DOUBLE PRECISION ALPHA, BETA
INTEGER I, INFO, J, M, N, P
CHARACTER*1 TYPE
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK)
INTEGER IWORK(NMAX)
* .. External Subroutines ..
EXTERNAL AB04MD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TYPE, ALPHA, BETA
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), I = 1,N ), J = 1,N )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,N ), J = 1,M )
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), I = 1,P ), J = 1,N )
READ ( NIN, FMT = * ) ( ( D(I,J), I = 1,P ), J = 1,M )
* Transform the parameters (A,B,C,D).
CALL AB04MD( TYPE, N, M, P, ALPHA, BETA, A, LDA, B, LDB,
$ C, LDC, D, LDD, IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( A(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( B(I,J), J = 1,M )
40 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99996 ) ( C(I,J), J = 1,N )
60 CONTINUE
WRITE ( NOUT, FMT = 99990 )
DO 80 I = 1, P
WRITE ( NOUT, FMT = 99996 ) ( D(I,J), J = 1,M )
80 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB04MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB04MD = ',I2)
99997 FORMAT (' The transformed state matrix is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The transformed input matrix is ')
99994 FORMAT (/' The transformed output matrix is ')
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' M is out of range.',/' M = ',I5)
99991 FORMAT (/' P is out of range.',/' P = ',I5)
99990 FORMAT (/' The transformed input/output matrix is ')
END
Program Data
AB04MD EXAMPLE PROGRAM DATA 2 2 2 C 1.0D0 1.0D0 1.0 0.5 0.5 1.0 0.0 -1.0 1.0 0.0 -1.0 0.0 0.0 1.0 1.0 0.0 0.0 -1.0Program Results
AB04MD EXAMPLE PROGRAM RESULTS The transformed state matrix is -1.0000 -4.0000 -4.0000 -1.0000 The transformed input matrix is 2.8284 0.0000 0.0000 -2.8284 The transformed output matrix is 0.0000 2.8284 -2.8284 0.0000 The transformed input/output matrix is -1.0000 0.0000 0.0000 -3.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB05MD.html 0000664 0000000 0000000 00000037470 14560147231 0020335 0 ustar 00root root 0000000 0000000
Purpose
To obtain the state-space model (A,B,C,D) for the cascaded inter-connection of two systems, each given in state-space form.Specification
SUBROUTINE AB05MD( UPLO, OVER, N1, M1, P1, N2, P2, A1, LDA1, B1,
$ LDB1, C1, LDC1, D1, LDD1, A2, LDA2, B2, LDB2,
$ C2, LDC2, D2, LDD2, N, A, LDA, B, LDB, C, LDC,
$ D, LDD, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER OVER, UPLO
INTEGER INFO, LDA, LDA1, LDA2, LDB, LDB1, LDB2, LDC,
$ LDC1, LDC2, LDD, LDD1, LDD2, LDWORK, M1, N, N1,
$ N2, P1, P2
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), A1(LDA1,*), A2(LDA2,*), B(LDB,*),
$ B1(LDB1,*), B2(LDB2,*), C(LDC,*), C1(LDC1,*),
$ C2(LDC2,*), D(LDD,*), D1(LDD1,*), D2(LDD2,*),
$ DWORK(*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Indicates whether the user wishes to obtain the matrix A
in the upper or lower block diagonal form, as follows:
= 'U': Obtain A in the upper block diagonal form;
= 'L': Obtain A in the lower block diagonal form.
OVER CHARACTER*1
Indicates whether the user wishes to overlap pairs of
arrays, as follows:
= 'N': Do not overlap;
= 'O': Overlap pairs of arrays: A1 and A, B1 and B,
C1 and C, and D1 and D (for UPLO = 'L'), or A2
and A, B2 and B, C2 and C, and D2 and D (for
UPLO = 'U'), i.e. the same name is effectively
used for each pair (for all pairs) in the routine
call. In this case, setting LDA1 = LDA,
LDB1 = LDB, LDC1 = LDC, and LDD1 = LDD, or
LDA2 = LDA, LDB2 = LDB, LDC2 = LDC, and LDD2 = LDD
will give maximum efficiency.
Input/Output Parameters
N1 (input) INTEGER
The number of state variables in the first system, i.e.
the order of the matrix A1. N1 >= 0.
M1 (input) INTEGER
The number of input variables for the first system.
M1 >= 0.
P1 (input) INTEGER
The number of output variables from the first system and
the number of input variables for the second system.
P1 >= 0.
N2 (input) INTEGER
The number of state variables in the second system, i.e.
the order of the matrix A2. N2 >= 0.
P2 (input) INTEGER
The number of output variables from the second system.
P2 >= 0.
A1 (input) DOUBLE PRECISION array, dimension (LDA1,N1)
The leading N1-by-N1 part of this array must contain the
state transition matrix A1 for the first system.
LDA1 INTEGER
The leading dimension of array A1. LDA1 >= MAX(1,N1).
B1 (input) DOUBLE PRECISION array, dimension (LDB1,M1)
The leading N1-by-M1 part of this array must contain the
input/state matrix B1 for the first system.
LDB1 INTEGER
The leading dimension of array B1. LDB1 >= MAX(1,N1).
C1 (input) DOUBLE PRECISION array, dimension (LDC1,N1)
The leading P1-by-N1 part of this array must contain the
state/output matrix C1 for the first system.
LDC1 INTEGER
The leading dimension of array C1.
LDC1 >= MAX(1,P1) if N1 > 0.
LDC1 >= 1 if N1 = 0.
D1 (input) DOUBLE PRECISION array, dimension (LDD1,M1)
The leading P1-by-M1 part of this array must contain the
input/output matrix D1 for the first system.
LDD1 INTEGER
The leading dimension of array D1. LDD1 >= MAX(1,P1).
A2 (input) DOUBLE PRECISION array, dimension (LDA2,N2)
The leading N2-by-N2 part of this array must contain the
state transition matrix A2 for the second system.
LDA2 INTEGER
The leading dimension of array A2. LDA2 >= MAX(1,N2).
B2 (input) DOUBLE PRECISION array, dimension (LDB2,P1)
The leading N2-by-P1 part of this array must contain the
input/state matrix B2 for the second system.
LDB2 INTEGER
The leading dimension of array B2. LDB2 >= MAX(1,N2).
C2 (input) DOUBLE PRECISION array, dimension (LDC2,N2)
The leading P2-by-N2 part of this array must contain the
state/output matrix C2 for the second system.
LDC2 INTEGER
The leading dimension of array C2.
LDC2 >= MAX(1,P2) if N2 > 0.
LDC2 >= 1 if N2 = 0.
D2 (input) DOUBLE PRECISION array, dimension (LDD2,P1)
The leading P2-by-P1 part of this array must contain the
input/output matrix D2 for the second system.
LDD2 INTEGER
The leading dimension of array D2. LDD2 >= MAX(1,P2).
N (output) INTEGER
The number of state variables (N1 + N2) in the resulting
system, i.e. the order of the matrix A, the number of rows
of B and the number of columns of C.
A (output) DOUBLE PRECISION array, dimension (LDA,N1+N2)
The leading N-by-N part of this array contains the state
transition matrix A for the cascaded system.
If OVER = 'O', the array A can overlap A1, if UPLO = 'L',
or A2, if UPLO = 'U'.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N1+N2).
B (output) DOUBLE PRECISION array, dimension (LDB,M1)
The leading N-by-M1 part of this array contains the
input/state matrix B for the cascaded system.
If OVER = 'O', the array B can overlap B1, if UPLO = 'L',
or B2, if UPLO = 'U'.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N1+N2).
C (output) DOUBLE PRECISION array, dimension (LDC,N1+N2)
The leading P2-by-N part of this array contains the
state/output matrix C for the cascaded system.
If OVER = 'O', the array C can overlap C1, if UPLO = 'L',
or C2, if UPLO = 'U'.
LDC INTEGER
The leading dimension of array C.
LDC >= MAX(1,P2) if N1+N2 > 0.
LDC >= 1 if N1+N2 = 0.
D (output) DOUBLE PRECISION array, dimension (LDD,M1)
The leading P2-by-M1 part of this array contains the
input/output matrix D for the cascaded system.
If OVER = 'O', the array D can overlap D1, if UPLO = 'L',
or D2, if UPLO = 'U'.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P2).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
The array DWORK is not referenced if OVER = 'N'.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( 1, P1*MAX(N1, M1, N2, P2) ) if OVER = 'O'.
LDWORK >= 1 if OVER = 'N'.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
After cascaded inter-connection of the two systems
X1' = A1*X1 + B1*U
V = C1*X1 + D1*U
X2' = A2*X2 + B2*V
Y = C2*X2 + D2*V
(where ' denotes differentiation with respect to time)
the following state-space model will be obtained:
X' = A*X + B*U
Y = C*X + D*U
where matrix A has the form ( A1 0 ),
( B2*C1 A2)
matrix B has the form ( B1 ),
( B2*D1 )
matrix C has the form ( D2*C1 C2 ) and
matrix D has the form ( D2*D1 ).
This form is returned by the routine when UPLO = 'L'. Note that
when A1 and A2 are block lower triangular, the resulting state
matrix is also block lower triangular.
By applying a similarity transformation to the system above,
using the matrix ( 0 I ), where I is the identity matrix of
( J 0 )
order N2, and J is the identity matrix of order N1, the
system matrices become
A = ( A2 B2*C1 ),
( 0 A1 )
B = ( B2*D1 ),
( B1 )
C = ( C2 D2*C1 ) and
D = ( D2*D1 ).
This form is returned by the routine when UPLO = 'U'. Note that
when A1 and A2 are block upper triangular (for instance, in the
real Schur form), the resulting state matrix is also block upper
triangular.
References
NoneNumerical Aspects
The algorithm requires P1*(N1+M1)*(N2+P2) operations.Further Comments
NoneExample
Program Text
* AB05MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER N1MAX, N2MAX, NMAX, M1MAX, P1MAX, P2MAX
PARAMETER ( N1MAX = 20, N2MAX = 20, NMAX = N1MAX+N2MAX,
$ M1MAX = 20, P1MAX = 20, P2MAX = 20 )
INTEGER LDA, LDA1, LDA2, LDB, LDB1, LDB2, LDC, LDC1,
$ LDC2, LDD, LDD1, LDD2, LDWORK
PARAMETER ( LDA = NMAX, LDA1 = N1MAX, LDA2 = N2MAX,
$ LDB = NMAX,LDB1 = N1MAX, LDB2 = N2MAX,
$ LDC = P2MAX, LDC1 = P1MAX, LDC2 = P2MAX,
$ LDD = P2MAX, LDD1 = P1MAX, LDD2 = P2MAX,
$ LDWORK = P1MAX*N1MAX )
* .. Local Scalars ..
CHARACTER*1 OVER, UPLO
INTEGER I, INFO, J, M1, N, N1, N2, P1, P2
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), A1(LDA1,N1MAX), A2(LDA2,N2MAX),
$ B(LDB,M1MAX), B1(LDB1,M1MAX), B2(LDB2,P1MAX),
$ C(LDC,NMAX), C1(LDC1,N1MAX), C2(LDC2,N2MAX),
$ D(LDD,M1MAX), D1(LDD1,M1MAX), D2(LDD2,P1MAX),
$ DWORK(LDWORK)
* .. External Subroutines ..
EXTERNAL AB05MD
* .. Executable Statements ..
*
UPLO = 'Lower'
OVER = 'N'
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N1, M1, P1, N2, P2
IF ( N1.LE.0 .OR. N1.GT.N1MAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N1
ELSE
READ ( NIN, FMT = * ) ( ( A1(I,J), J = 1,N1 ), I = 1,N1 )
IF ( M1.LE.0 .OR. M1.GT.M1MAX ) THEN
WRITE ( NOUT, FMT = 99991 ) M1
ELSE
READ ( NIN, FMT = * ) ( ( B1(I,J), I = 1,N1 ), J = 1,M1 )
IF ( P1.LE.0 .OR. P1.GT.P1MAX ) THEN
WRITE ( NOUT, FMT = 99990 ) P1
ELSE
READ ( NIN, FMT = * ) ( ( C1(I,J), J = 1,N1 ), I = 1,P1 )
READ ( NIN, FMT = * ) ( ( D1(I,J), J = 1,M1 ), I = 1,P1 )
IF ( N2.LE.0 .OR. N2.GT.N2MAX ) THEN
WRITE ( NOUT, FMT = 99989 ) N2
ELSE
READ ( NIN, FMT = * )
$ ( ( A2(I,J), J = 1,N2 ), I = 1,N2 )
READ ( NIN, FMT = * )
$ ( ( B2(I,J), I = 1,N2 ), J = 1,P1 )
IF ( P2.LE.0 .OR. P2.GT.P2MAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P2
ELSE
READ ( NIN, FMT = * )
$ ( ( C2(I,J), J = 1,N2 ), I = 1,P2 )
READ ( NIN, FMT = * )
$ ( ( D2(I,J), J = 1,P1 ), I = 1,P2 )
* Find the state-space model (A,B,C,D).
CALL AB05MD( UPLO, OVER, N1, M1, P1, N2, P2, A1,
$ LDA1, B1, LDB1, C1, LDC1, D1, LDD1,
$ A2, LDA2, B2, LDB2, C2, LDC2, D2,
$ LDD2, N, A, LDA, B, LDB, C, LDC, D,
$ LDD, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 )
$ ( A(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99996 )
$ ( B(I,J), J = 1,M1 )
40 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 60 I = 1, P2
WRITE ( NOUT, FMT = 99996 )
$ ( C(I,J), J = 1,N )
60 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 80 I = 1, P2
WRITE ( NOUT, FMT = 99996 )
$ ( D(I,J), J = 1,M1 )
80 CONTINUE
END IF
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB05MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB05MD = ',I2)
99997 FORMAT (' The state transition matrix of the cascaded system is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The input/state matrix of the cascaded system is ')
99994 FORMAT (/' The state/output matrix of the cascaded system is ')
99993 FORMAT (/' The input/output matrix of the cascaded system is ')
99992 FORMAT (/' N1 is out of range.',/' N1 = ',I5)
99991 FORMAT (/' M1 is out of range.',/' M1 = ',I5)
99990 FORMAT (/' P1 is out of range.',/' P1 = ',I5)
99989 FORMAT (/' N2 is out of range.',/' N2 = ',I5)
99988 FORMAT (/' P2 is out of range.',/' P2 = ',I5)
END
Program Data
AB05MD EXAMPLE PROGRAM DATA 3 2 2 3 2 1.0 0.0 -1.0 0.0 -1.0 1.0 1.0 1.0 2.0 1.0 1.0 0.0 2.0 0.0 1.0 3.0 -2.0 1.0 0.0 1.0 0.0 1.0 0.0 0.0 1.0 -3.0 0.0 0.0 1.0 0.0 1.0 0.0 -1.0 2.0 0.0 -1.0 0.0 1.0 0.0 2.0 1.0 1.0 0.0 1.0 1.0 -1.0 1.0 1.0 0.0 1.0Program Results
AB05MD EXAMPLE PROGRAM RESULTS The state transition matrix of the cascaded system is 1.0000 0.0000 -1.0000 0.0000 0.0000 0.0000 0.0000 -1.0000 1.0000 0.0000 0.0000 0.0000 1.0000 1.0000 2.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 -3.0000 0.0000 0.0000 -3.0000 2.0000 -1.0000 1.0000 0.0000 1.0000 0.0000 2.0000 0.0000 0.0000 -1.0000 2.0000 The input/state matrix of the cascaded system is 1.0000 2.0000 1.0000 0.0000 0.0000 1.0000 0.0000 1.0000 -1.0000 0.0000 0.0000 2.0000 The state/output matrix of the cascaded system is 3.0000 -1.0000 1.0000 1.0000 1.0000 0.0000 0.0000 1.0000 0.0000 1.0000 1.0000 -1.0000 The input/output matrix of the cascaded system is 1.0000 1.0000 0.0000 1.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB05ND.html 0000664 0000000 0000000 00000036422 14560147231 0020332 0 ustar 00root root 0000000 0000000
Purpose
To obtain the state-space model (A,B,C,D) for the feedback inter-connection of two systems, each given in state-space form.Specification
SUBROUTINE AB05ND( OVER, N1, M1, P1, N2, ALPHA, A1, LDA1, B1,
$ LDB1, C1, LDC1, D1, LDD1, A2, LDA2, B2, LDB2,
$ C2, LDC2, D2, LDD2, N, A, LDA, B, LDB, C, LDC,
$ D, LDD, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER OVER
INTEGER INFO, LDA, LDA1, LDA2, LDB, LDB1, LDB2, LDC,
$ LDC1, LDC2, LDD, LDD1, LDD2, LDWORK, M1, N, N1,
$ N2, P1
DOUBLE PRECISION ALPHA
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), A1(LDA1,*), A2(LDA2,*), B(LDB,*),
$ B1(LDB1,*), B2(LDB2,*), C(LDC,*), C1(LDC1,*),
$ C2(LDC2,*), D(LDD,*), D1(LDD1,*), D2(LDD2,*),
$ DWORK(*)
Arguments
Mode Parameters
OVER CHARACTER*1
Indicates whether the user wishes to overlap pairs of
arrays, as follows:
= 'N': Do not overlap;
= 'O': Overlap pairs of arrays: A1 and A, B1 and B,
C1 and C, and D1 and D, i.e. the same name is
effectively used for each pair (for all pairs)
in the routine call. In this case, setting
LDA1 = LDA, LDB1 = LDB, LDC1 = LDC, and LDD1 = LDD
will give maximum efficiency.
Input/Output Parameters
N1 (input) INTEGER
The number of state variables in the first system, i.e.
the order of the matrix A1. N1 >= 0.
M1 (input) INTEGER
The number of input variables for the first system and the
number of output variables from the second system.
M1 >= 0.
P1 (input) INTEGER
The number of output variables from the first system and
the number of input variables for the second system.
P1 >= 0.
N2 (input) INTEGER
The number of state variables in the second system, i.e.
the order of the matrix A2. N2 >= 0.
ALPHA (input) DOUBLE PRECISION
A coefficient multiplying the transfer-function matrix
(or the output equation) of the second system.
ALPHA = +1 corresponds to positive feedback, and
ALPHA = -1 corresponds to negative feedback.
A1 (input) DOUBLE PRECISION array, dimension (LDA1,N1)
The leading N1-by-N1 part of this array must contain the
state transition matrix A1 for the first system.
LDA1 INTEGER
The leading dimension of array A1. LDA1 >= MAX(1,N1).
B1 (input) DOUBLE PRECISION array, dimension (LDB1,M1)
The leading N1-by-M1 part of this array must contain the
input/state matrix B1 for the first system.
LDB1 INTEGER
The leading dimension of array B1. LDB1 >= MAX(1,N1).
C1 (input) DOUBLE PRECISION array, dimension (LDC1,N1)
The leading P1-by-N1 part of this array must contain the
state/output matrix C1 for the first system.
LDC1 INTEGER
The leading dimension of array C1.
LDC1 >= MAX(1,P1) if N1 > 0.
LDC1 >= 1 if N1 = 0.
D1 (input) DOUBLE PRECISION array, dimension (LDD1,M1)
The leading P1-by-M1 part of this array must contain the
input/output matrix D1 for the first system.
LDD1 INTEGER
The leading dimension of array D1. LDD1 >= MAX(1,P1).
A2 (input) DOUBLE PRECISION array, dimension (LDA2,N2)
The leading N2-by-N2 part of this array must contain the
state transition matrix A2 for the second system.
LDA2 INTEGER
The leading dimension of array A2. LDA2 >= MAX(1,N2).
B2 (input) DOUBLE PRECISION array, dimension (LDB2,P1)
The leading N2-by-P1 part of this array must contain the
input/state matrix B2 for the second system.
LDB2 INTEGER
The leading dimension of array B2. LDB2 >= MAX(1,N2).
C2 (input) DOUBLE PRECISION array, dimension (LDC2,N2)
The leading M1-by-N2 part of this array must contain the
state/output matrix C2 for the second system.
LDC2 INTEGER
The leading dimension of array C2.
LDC2 >= MAX(1,M1) if N2 > 0.
LDC2 >= 1 if N2 = 0.
D2 (input) DOUBLE PRECISION array, dimension (LDD2,P1)
The leading M1-by-P1 part of this array must contain the
input/output matrix D2 for the second system.
LDD2 INTEGER
The leading dimension of array D2. LDD2 >= MAX(1,M1).
N (output) INTEGER
The number of state variables (N1 + N2) in the connected
system, i.e. the order of the matrix A, the number of rows
of B and the number of columns of C.
A (output) DOUBLE PRECISION array, dimension (LDA,N1+N2)
The leading N-by-N part of this array contains the state
transition matrix A for the connected system.
The array A can overlap A1 if OVER = 'O'.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N1+N2).
B (output) DOUBLE PRECISION array, dimension (LDB,M1)
The leading N-by-M1 part of this array contains the
input/state matrix B for the connected system.
The array B can overlap B1 if OVER = 'O'.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N1+N2).
C (output) DOUBLE PRECISION array, dimension (LDC,N1+N2)
The leading P1-by-N part of this array contains the
state/output matrix C for the connected system.
The array C can overlap C1 if OVER = 'O'.
LDC INTEGER
The leading dimension of array C.
LDC >= MAX(1,P1) if N1+N2 > 0.
LDC >= 1 if N1+N2 = 0.
D (output) DOUBLE PRECISION array, dimension (LDD,M1)
The leading P1-by-M1 part of this array contains the
input/output matrix D for the connected system.
The array D can overlap D1 if OVER = 'O'.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P1).
Workspace
IWORK INTEGER array, dimension (P1)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK. If OVER = 'N',
LDWORK >= MAX(1, P1*P1, M1*M1, N1*P1), and if OVER = 'O',
LDWORK >= MAX(1, N1*P1 + MAX( P1*P1, M1*M1, N1*P1) ),
if M1 <= N*N2;
LDWORK >= MAX(1, N1*P1 + MAX( P1*P1, M1*(M1+1), N1*P1) ),
if M1 > N*N2.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
> 0: if INFO = i, 1 <= i <= P1, the system is not
completely controllable. That is, the matrix
(I + ALPHA*D1*D2) is exactly singular (the element
U(i,i) of the upper triangular factor of LU
factorization is exactly zero), possibly due to
rounding errors.
Method
After feedback inter-connection of the two systems,
X1' = A1*X1 + B1*U1
Y1 = C1*X1 + D1*U1
X2' = A2*X2 + B2*U2
Y2 = C2*X2 + D2*U2
(where ' denotes differentiation with respect to time)
the following state-space model will be obtained:
X' = A*X + B*U
Y = C*X + D*U
where U = U1 + alpha*Y2, X = ( X1 ),
Y = Y1 = U2, ( X2 )
matrix A has the form
( A1 - alpha*B1*E12*D2*C1 - alpha*B1*E12*C2 ),
( B2*E21*C1 A2 - alpha*B2*E21*D1*C2 )
matrix B has the form
( B1*E12 ),
( B2*E21*D1 )
matrix C has the form
( E21*C1 - alpha*E21*D1*C2 ),
matrix D has the form
( E21*D1 ),
E21 = ( I + alpha*D1*D2 )-INVERSE and
E12 = ( I + alpha*D2*D1 )-INVERSE = I - alpha*D2*E21*D1.
Taking N1 = 0 and/or N2 = 0 on the routine call will solve the
constant plant and/or constant feedback cases.
References
NoneNumerical Aspects
NoneFurther Comments
NoneExample
Program Text
* AB05ND EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER N1MAX, N2MAX, NMAX, M1MAX, P1MAX
PARAMETER ( N1MAX = 20, N2MAX = 20, NMAX = N1MAX+N2MAX,
$ M1MAX = 20, P1MAX = 20 )
INTEGER LDA, LDA1, LDA2, LDB, LDB1, LDB2, LDC, LDC1,
$ LDC2, LDD, LDD1, LDD2
PARAMETER ( LDA = NMAX, LDA1 = N1MAX, LDA2 = N2MAX,
$ LDB = NMAX, LDB1 = N1MAX, LDB2 = N2MAX,
$ LDC = P1MAX, LDC1 = P1MAX, LDC2 = M1MAX,
$ LDD = P1MAX, LDD1 = P1MAX, LDD2 = M1MAX )
INTEGER LDWORK
PARAMETER ( LDWORK = P1MAX*P1MAX )
DOUBLE PRECISION ONE
PARAMETER ( ONE=1.0D0 )
* .. Local Scalars ..
CHARACTER*1 OVER
INTEGER I, INFO, J, M1, N, N1, N2, P1
DOUBLE PRECISION ALPHA
* .. Local Arrays ..
INTEGER IWORK(P1MAX)
DOUBLE PRECISION A(LDA,NMAX), A1(LDA1,N1MAX), A2(LDA2,N2MAX),
$ B(LDB,M1MAX), B1(LDB1,M1MAX), B2(LDB2,P1MAX),
$ C(LDC,NMAX), C1(LDC1,N1MAX), C2(LDC2,N2MAX),
$ D(LDD,M1MAX), D1(LDD1,M1MAX), D2(LDD2,P1MAX),
$ DWORK(LDWORK)
* .. External Subroutines ..
EXTERNAL AB05ND
* .. Executable Statements ..
*
OVER = 'N'
ALPHA = ONE
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N1, M1, P1, N2
IF ( N1.LE.0 .OR. N1.GT.N1MAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N1
ELSE
READ ( NIN, FMT = * ) ( ( A1(I,J), J = 1,N1 ), I = 1,N1 )
IF ( M1.LE.0 .OR. M1.GT.M1MAX ) THEN
WRITE ( NOUT, FMT = 99991 ) M1
ELSE
READ ( NIN, FMT = * ) ( ( B1(I,J), I = 1,N1 ), J = 1,M1 )
IF ( P1.LE.0 .OR. P1.GT.P1MAX ) THEN
WRITE ( NOUT, FMT = 99990 ) P1
ELSE
READ ( NIN, FMT = * ) ( ( C1(I,J), J = 1,N1 ), I = 1,P1 )
READ ( NIN, FMT = * ) ( ( D1(I,J), J = 1,M1 ), I = 1,P1 )
IF ( N2.LE.0 .OR. N2.GT.N2MAX ) THEN
WRITE ( NOUT, FMT = 99989 ) N2
ELSE
READ ( NIN, FMT = * )
$ ( ( A2(I,J), J = 1,N2 ), I = 1,N2 )
READ ( NIN, FMT = * )
$ ( ( B2(I,J), I = 1,N2 ), J = 1,P1 )
READ ( NIN, FMT = * )
$ ( ( C2(I,J), J = 1,N2 ), I = 1,M1 )
READ ( NIN, FMT = * )
$ ( ( D2(I,J), J = 1,P1 ), I = 1,M1 )
* Find the state-space model (A,B,C,D).
CALL AB05ND( OVER, N1, M1, P1, N2, ALPHA, A1, LDA1,
$ B1, LDB1, C1, LDC1, D1, LDD1, A2, LDA2,
$ B2, LDB2, C2, LDC2, D2, LDD2, N, A, LDA,
$ B, LDB, C, LDC, D, LDD, IWORK, DWORK,
$ LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( A(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( B(I,J), J = 1,M1 )
40 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 60 I = 1, P1
WRITE ( NOUT, FMT = 99996 ) ( C(I,J), J = 1,N )
60 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 80 I = 1, P1
WRITE ( NOUT, FMT = 99996 ) ( D(I,J), J = 1,M1 )
80 CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB05ND EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB05ND = ',I2)
99997 FORMAT (' The state transition matrix of the connected system is')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The input/state matrix of the connected system is ')
99994 FORMAT (/' The state/output matrix of the connected system is ')
99993 FORMAT (/' The input/output matrix of the connected system is ')
99992 FORMAT (/' N1 is out of range.',/' N1 = ',I5)
99991 FORMAT (/' M1 is out of range.',/' M1 = ',I5)
99990 FORMAT (/' P1 is out of range.',/' P1 = ',I5)
99989 FORMAT (/' N2 is out of range.',/' N2 = ',I5)
END
Program Data
AB05ND EXAMPLE PROGRAM DATA 3 2 2 3 1.0 0.0 -1.0 0.0 -1.0 1.0 1.0 1.0 2.0 1.0 1.0 0.0 2.0 0.0 1.0 3.0 -2.0 1.0 0.0 1.0 0.0 1.0 0.0 0.0 1.0 -3.0 0.0 0.0 1.0 0.0 1.0 0.0 -1.0 2.0 0.0 -1.0 0.0 1.0 0.0 2.0 1.0 1.0 0.0 1.0 1.0 -1.0 1.0 1.0 0.0 1.0Program Results
AB05ND EXAMPLE PROGRAM RESULTS The state transition matrix of the connected system is -0.5000 -0.2500 -1.5000 -1.2500 -1.2500 0.7500 -1.5000 -0.2500 0.5000 -0.2500 -0.2500 -0.2500 1.0000 0.5000 2.0000 -0.5000 -0.5000 0.5000 0.0000 0.5000 0.0000 -3.5000 -0.5000 0.5000 -1.5000 1.2500 -0.5000 1.2500 0.2500 1.2500 0.0000 1.0000 0.0000 -1.0000 -2.0000 3.0000 The input/state matrix of the connected system is 0.5000 0.7500 0.5000 -0.2500 0.0000 0.5000 0.0000 0.5000 -0.5000 0.2500 0.0000 1.0000 The state/output matrix of the connected system is 1.5000 -1.2500 0.5000 -0.2500 -0.2500 -0.2500 0.0000 0.5000 0.0000 -0.5000 -0.5000 0.5000 The input/output matrix of the connected system is 0.5000 -0.2500 0.0000 0.5000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB05OD.html 0000664 0000000 0000000 00000035177 14560147231 0020341 0 ustar 00root root 0000000 0000000
Purpose
To obtain the state-space model (A,B,C,D) for rowwise concatenation (parallel inter-connection on outputs, with separate inputs) of two systems, each given in state-space form.Specification
SUBROUTINE AB05OD( OVER, N1, M1, P1, N2, M2, ALPHA, A1, LDA1, B1,
$ LDB1, C1, LDC1, D1, LDD1, A2, LDA2, B2, LDB2,
$ C2, LDC2, D2, LDD2, N, M, A, LDA, B, LDB, C,
$ LDC, D, LDD, INFO )
C .. Scalar Arguments ..
CHARACTER OVER
INTEGER INFO, LDA, LDA1, LDA2, LDB, LDB1, LDB2, LDC,
$ LDC1, LDC2, LDD, LDD1, LDD2, M, M1, M2, N, N1,
$ N2, P1
DOUBLE PRECISION ALPHA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), A1(LDA1,*), A2(LDA2,*), B(LDB,*),
$ B1(LDB1,*), B2(LDB2,*), C(LDC,*), C1(LDC1,*),
$ C2(LDC2,*), D(LDD,*), D1(LDD1,*), D2(LDD2,*)
Arguments
Mode Parameters
OVER CHARACTER*1
Indicates whether the user wishes to overlap pairs of
arrays, as follows:
= 'N': Do not overlap;
= 'O': Overlap pairs of arrays: A1 and A, B1 and B,
C1 and C, and D1 and D, i.e. the same name is
effectively used for each pair (for all pairs)
in the routine call. In this case, setting
LDA1 = LDA, LDB1 = LDB, LDC1 = LDC, and LDD1 = LDD
will give maximum efficiency.
Input/Output Parameters
N1 (input) INTEGER
The number of state variables in the first system, i.e.
the order of the matrix A1. N1 >= 0.
M1 (input) INTEGER
The number of input variables for the first system.
M1 >= 0.
P1 (input) INTEGER
The number of output variables from each system. P1 >= 0.
N2 (input) INTEGER
The number of state variables in the second system, i.e.
the order of the matrix A2. N2 >= 0.
M2 (input) INTEGER
The number of input variables for the second system.
M2 >= 0.
ALPHA (input) DOUBLE PRECISION
A coefficient multiplying the transfer-function matrix
(or the output equation) of the second system.
A1 (input) DOUBLE PRECISION array, dimension (LDA1,N1)
The leading N1-by-N1 part of this array must contain the
state transition matrix A1 for the first system.
LDA1 INTEGER
The leading dimension of array A1. LDA1 >= MAX(1,N1).
B1 (input) DOUBLE PRECISION array, dimension (LDB1,M1)
The leading N1-by-M1 part of this array must contain the
input/state matrix B1 for the first system.
LDB1 INTEGER
The leading dimension of array B1. LDB1 >= MAX(1,N1).
C1 (input) DOUBLE PRECISION array, dimension (LDC1,N1)
The leading P1-by-N1 part of this array must contain the
state/output matrix C1 for the first system.
LDC1 INTEGER
The leading dimension of array C1.
LDC1 >= MAX(1,P1) if N1 > 0.
LDC1 >= 1 if N1 = 0.
D1 (input) DOUBLE PRECISION array, dimension (LDD1,M1)
The leading P1-by-M1 part of this array must contain the
input/output matrix D1 for the first system.
LDD1 INTEGER
The leading dimension of array D1. LDD1 >= MAX(1,P1).
A2 (input) DOUBLE PRECISION array, dimension (LDA2,N2)
The leading N2-by-N2 part of this array must contain the
state transition matrix A2 for the second system.
LDA2 INTEGER
The leading dimension of array A2. LDA2 >= MAX(1,N2).
B2 (input) DOUBLE PRECISION array, dimension (LDB2,M2)
The leading N2-by-M2 part of this array must contain the
input/state matrix B2 for the second system.
LDB2 INTEGER
The leading dimension of array B2. LDB2 >= MAX(1,N2).
C2 (input) DOUBLE PRECISION array, dimension (LDC2,N2)
The leading P1-by-N2 part of this array must contain the
state/output matrix C2 for the second system.
LDC2 INTEGER
The leading dimension of array C2.
LDC2 >= MAX(1,P1) if N2 > 0.
LDC2 >= 1 if N2 = 0.
D2 (input) DOUBLE PRECISION array, dimension (LDD2,M2)
The leading P1-by-M2 part of this array must contain the
input/output matrix D2 for the second system.
LDD2 INTEGER
The leading dimension of array D2. LDD2 >= MAX(1,P1).
N (output) INTEGER
The number of state variables (N1 + N2) in the connected
system, i.e. the order of the matrix A, the number of rows
of B and the number of columns of C.
M (output) INTEGER
The number of input variables (M1 + M2) for the connected
system, i.e. the number of columns of B and D.
A (output) DOUBLE PRECISION array, dimension (LDA,N1+N2)
The leading N-by-N part of this array contains the state
transition matrix A for the connected system.
The array A can overlap A1 if OVER = 'O'.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N1+N2).
B (output) DOUBLE PRECISION array, dimension (LDB,M1+M2)
The leading N-by-M part of this array contains the
input/state matrix B for the connected system.
The array B can overlap B1 if OVER = 'O'.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N1+N2).
C (output) DOUBLE PRECISION array, dimension (LDC,N1+N2)
The leading P1-by-N part of this array contains the
state/output matrix C for the connected system.
The array C can overlap C1 if OVER = 'O'.
LDC INTEGER
The leading dimension of array C.
LDC >= MAX(1,P1) if N1+N2 > 0.
LDC >= 1 if N1+N2 = 0.
D (output) DOUBLE PRECISION array, dimension (LDD,M1+M2)
The leading P1-by-M part of this array contains the
input/output matrix D for the connected system.
The array D can overlap D1 if OVER = 'O'.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P1).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
After rowwise concatenation (parallel inter-connection with
separate inputs) of the two systems,
X1' = A1*X1 + B1*U
Y1 = C1*X1 + D1*U
X2' = A2*X2 + B2*V
Y2 = C2*X2 + D2*V
(where ' denotes differentiation with respect to time),
with the output equation for the second system multiplied by a
scalar alpha, the following state-space model will be obtained:
X' = A*X + B*(U)
(V)
Y = C*X + D*(U)
(V)
where matrix A has the form ( A1 0 ),
( 0 A2 )
matrix B has the form ( B1 0 ),
( 0 B2 )
matrix C has the form ( C1 alpha*C2 ) and
matrix D has the form ( D1 alpha*D2 ).
References
NoneNumerical Aspects
NoneFurther Comments
NoneExample
Program Text
* AB05OD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER N1MAX, N2MAX, NMAX, M1MAX, M2MAX, MMAX, P1MAX
PARAMETER ( N1MAX = 20, N2MAX = 20, NMAX = N1MAX+N2MAX,
$ M1MAX = 20, M2MAX = 20, MMAX = M1MAX+M2MAX,
$ P1MAX = 20 )
INTEGER LDA, LDA1, LDA2, LDB, LDB1, LDB2, LDC, LDC1,
$ LDC2, LDD, LDD1, LDD2
PARAMETER ( LDA = NMAX, LDA1 = N1MAX, LDA2 = N2MAX,
$ LDB = NMAX, LDB1 = N1MAX, LDB2 = N2MAX,
$ LDC = P1MAX, LDC1 = P1MAX, LDC2 = P1MAX,
$ LDD = P1MAX, LDD1 = P1MAX, LDD2 = P1MAX )
DOUBLE PRECISION ONE
PARAMETER ( ONE=1.0D0 )
* .. Local Scalars ..
CHARACTER*1 OVER
INTEGER I, INFO, J, M, M1, M2, N, N1, N2, P1
DOUBLE PRECISION ALPHA
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), A1(LDA1,N1MAX), A2(LDA2,N2MAX),
$ B(LDB,MMAX), B1(LDB1,M1MAX), B2(LDB2,M2MAX),
$ C(LDC,NMAX), C1(LDC1,N1MAX), C2(LDC2,N2MAX),
$ D(LDD,MMAX), D1(LDD1,M1MAX), D2(LDD2,M2MAX)
* .. External Subroutines ..
EXTERNAL AB05OD
* .. Executable Statements ..
*
OVER = 'N'
ALPHA = ONE
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N1, M1, P1, N2, M2
IF ( N1.LE.0 .OR. N1.GT.N1MAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N1
ELSE
READ ( NIN, FMT = * ) ( ( A1(I,J), J = 1,N1 ), I = 1,N1 )
IF ( M1.LE.0 .OR. M1.GT.M1MAX ) THEN
WRITE ( NOUT, FMT = 99992 ) M1
ELSE
READ ( NIN, FMT = * ) ( ( B1(I,J), I = 1,N1 ), J = 1,M1 )
IF ( P1.LE.0 .OR. P1.GT.P1MAX ) THEN
WRITE ( NOUT, FMT = 99991 ) P1
ELSE
READ ( NIN, FMT = * ) ( ( C1(I,J), J = 1,N1 ), I = 1,P1 )
READ ( NIN, FMT = * ) ( ( D1(I,J), J = 1,M1 ), I = 1,P1 )
IF ( N2.LE.0 .OR. N2.GT.N2MAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N2
ELSE
READ ( NIN, FMT = * )
$ ( ( A2(I,J), J = 1,N2 ), I = 1,N2 )
IF ( M2.LE.0 .OR. M2.GT.M2MAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M2
ELSE
READ ( NIN, FMT = * )
$ ( ( B2(I,J), I = 1,N2 ), J = 1,M2 )
READ ( NIN, FMT = * )
$ ( ( C2(I,J), J = 1,N2 ), I = 1,P1 )
READ ( NIN, FMT = * )
$ ( ( D2(I,J), J = 1,M2 ), I = 1,P1 )
* Find the state-space model (A,B,C,D).
CALL AB05OD( OVER, N1, M1, P1, N2, M2, ALPHA, A1,
$ LDA1, B1, LDB1, C1, LDC1, D1, LDD1,
$ A2, LDA2, B2, LDB2, C2, LDC2, D2,
$ LDD2, N, M, A, LDA, B, LDB, C, LDC,
$ D, LDD, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 )
$ ( A(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99996 )
$ ( B(I,J), J = 1,M )
40 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 60 I = 1, P1
WRITE ( NOUT, FMT = 99996 )
$ ( C(I,J), J = 1,N )
60 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 80 I = 1, P1
WRITE ( NOUT, FMT = 99996 )
$ ( D(I,J), J = 1,M )
80 CONTINUE
END IF
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB05OD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB05OD = ',I2)
99997 FORMAT (' The state transition matrix of the connected system is')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The input/state matrix of the connected system is ')
99994 FORMAT (/' The state/output matrix of the connected system is ')
99993 FORMAT (/' The input/output matrix of the connected system is ')
99992 FORMAT (/' N1 is out of range.',/' N1 = ',I5)
99991 FORMAT (/' M1 is out of range.',/' M1 = ',I5)
99990 FORMAT (/' P1 is out of range.',/' P1 = ',I5)
99989 FORMAT (/' N2 is out of range.',/' N2 = ',I5)
END
Program Data
AB05OD EXAMPLE PROGRAM DATA 3 2 2 3 2 1.0 0.0 -1.0 0.0 -1.0 1.0 1.0 1.0 2.0 1.0 1.0 0.0 2.0 0.0 1.0 3.0 -2.0 1.0 0.0 1.0 0.0 1.0 0.0 0.0 1.0 -3.0 0.0 0.0 1.0 0.0 1.0 0.0 -1.0 2.0 0.0 -1.0 0.0 1.0 0.0 2.0 1.0 1.0 0.0 1.0 1.0 -1.0 1.0 1.0 0.0 1.0Program Results
AB05OD EXAMPLE PROGRAM RESULTS The state transition matrix of the connected system is 1.0000 0.0000 -1.0000 0.0000 0.0000 0.0000 0.0000 -1.0000 1.0000 0.0000 0.0000 0.0000 1.0000 1.0000 2.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -3.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 -1.0000 2.0000 The input/state matrix of the connected system is 1.0000 2.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 -1.0000 0.0000 0.0000 0.0000 0.0000 2.0000 The state/output matrix of the connected system is 3.0000 -2.0000 1.0000 1.0000 1.0000 0.0000 0.0000 1.0000 0.0000 1.0000 1.0000 -1.0000 The input/output matrix of the connected system is 1.0000 0.0000 1.0000 1.0000 0.0000 1.0000 0.0000 1.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB05PD.html 0000664 0000000 0000000 00000032545 14560147231 0020336 0 ustar 00root root 0000000 0000000
Purpose
To compute the state-space model G = (A,B,C,D) corresponding to the sum G = G1 + alpha*G2, where G1 = (A1,B1,C1,D1) and G2 = (A2,B2,C2,D2). G, G1, and G2 are the transfer-function matrices of the corresponding state-space models.Specification
SUBROUTINE AB05PD( OVER, N1, M, P, N2, ALPHA, A1, LDA1, B1, LDB1,
$ C1, LDC1, D1, LDD1, A2, LDA2, B2, LDB2, C2,
$ LDC2, D2, LDD2, N, A, LDA, B, LDB, C, LDC, D,
$ LDD, INFO)
C .. Scalar Arguments ..
CHARACTER OVER
INTEGER INFO, LDA, LDA1, LDA2, LDB, LDB1, LDB2, LDC,
$ LDC1, LDC2, LDD, LDD1, LDD2, M, N, N1, N2, P
DOUBLE PRECISION ALPHA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), A1(LDA1,*), A2(LDA2,*), B(LDB,*),
$ B1(LDB1,*), B2(LDB2,*), C(LDC,*), C1(LDC1,*),
$ C2(LDC2,*), D(LDD,*), D1(LDD1,*), D2(LDD2,*)
Arguments
Mode Parameters
OVER CHARACTER*1
Indicates whether the user wishes to overlap pairs of
arrays, as follows:
= 'N': Do not overlap;
= 'O': Overlap pairs of arrays: A1 and A, B1 and B,
C1 and C, and D1 and D, i.e. the same name is
effectively used for each pair (for all pairs)
in the routine call. In this case, setting
LDA1 = LDA, LDB1 = LDB, LDC1 = LDC, and LDD1 = LDD
will give maximum efficiency.
Input/Output Parameters
N1 (input) INTEGER
The number of state variables in the first system, i.e.
the order of the matrix A1, the number of rows of B1 and
the number of columns of C1. N1 >= 0.
M (input) INTEGER
The number of input variables of the two systems, i.e. the
number of columns of matrices B1, D1, B2 and D2. M >= 0.
P (input) INTEGER
The number of output variables of the two systems, i.e.
the number of rows of matrices C1, D1, C2 and D2. P >= 0.
N2 (input) INTEGER
The number of state variables in the second system, i.e.
the order of the matrix A2, the number of rows of B2 and
the number of columns of C2. N2 >= 0.
ALPHA (input) DOUBLE PRECISION
The coefficient multiplying G2.
A1 (input) DOUBLE PRECISION array, dimension (LDA1,N1)
The leading N1-by-N1 part of this array must contain the
state transition matrix A1 for the first system.
LDA1 INTEGER
The leading dimension of array A1. LDA1 >= MAX(1,N1).
B1 (input) DOUBLE PRECISION array, dimension (LDB1,M)
The leading N1-by-M part of this array must contain the
input/state matrix B1 for the first system.
LDB1 INTEGER
The leading dimension of array B1. LDB1 >= MAX(1,N1).
C1 (input) DOUBLE PRECISION array, dimension (LDC1,N1)
The leading P-by-N1 part of this array must contain the
state/output matrix C1 for the first system.
LDC1 INTEGER
The leading dimension of array C1.
LDC1 >= MAX(1,P) if N1 > 0.
LDC1 >= 1 if N1 = 0.
D1 (input) DOUBLE PRECISION array, dimension (LDD1,M)
The leading P-by-M part of this array must contain the
input/output matrix D1 for the first system.
LDD1 INTEGER
The leading dimension of array D1. LDD1 >= MAX(1,P).
A2 (input) DOUBLE PRECISION array, dimension (LDA2,N2)
The leading N2-by-N2 part of this array must contain the
state transition matrix A2 for the second system.
LDA2 INTEGER
The leading dimension of array A2. LDA2 >= MAX(1,N2).
B2 (input) DOUBLE PRECISION array, dimension (LDB2,M)
The leading N2-by-M part of this array must contain the
input/state matrix B2 for the second system.
LDB2 INTEGER
The leading dimension of array B2. LDB2 >= MAX(1,N2).
C2 (input) DOUBLE PRECISION array, dimension (LDC2,N2)
The leading P-by-N2 part of this array must contain the
state/output matrix C2 for the second system.
LDC2 INTEGER
The leading dimension of array C2.
LDC2 >= MAX(1,P) if N2 > 0.
LDC2 >= 1 if N2 = 0.
D2 (input) DOUBLE PRECISION array, dimension (LDD2,M)
The leading P-by-M part of this array must contain the
input/output matrix D2 for the second system.
LDD2 INTEGER
The leading dimension of array D2. LDD2 >= MAX(1,P).
N (output) INTEGER
The number of state variables (N1 + N2) in the resulting
system, i.e. the order of the matrix A, the number of rows
of B and the number of columns of C.
A (output) DOUBLE PRECISION array, dimension (LDA,N1+N2)
The leading N-by-N part of this array contains the state
transition matrix A for the resulting system.
The array A can overlap A1 if OVER = 'O'.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N1+N2).
B (output) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array contains the
input/state matrix B for the resulting system.
The array B can overlap B1 if OVER = 'O'.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N1+N2).
C (output) DOUBLE PRECISION array, dimension (LDC,N1+N2)
The leading P-by-N part of this array contains the
state/output matrix C for the resulting system.
The array C can overlap C1 if OVER = 'O'.
LDC INTEGER
The leading dimension of array C.
LDC >= MAX(1,P) if N1+N2 > 0.
LDC >= 1 if N1+N2 = 0.
D (output) DOUBLE PRECISION array, dimension (LDD,M)
The leading P-by-M part of this array contains the
input/output matrix D for the resulting system.
The array D can overlap D1 if OVER = 'O'.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The matrices of the resulting systems are determined as:
( A1 0 ) ( B1 )
A = ( ) , B = ( ) ,
( 0 A2 ) ( B2 )
C = ( C1 alpha*C2 ) , D = D1 + alpha*D2 .
References
NoneNumerical Aspects
NoneFurther Comments
NoneExample
Program Text
* AB05PD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER N1MAX, N2MAX, NMAX, MMAX, PMAX
PARAMETER ( N1MAX = 20, N2MAX = 20, NMAX = N1MAX+N2MAX,
$ MMAX = 20, PMAX = 20 )
INTEGER LDA, LDA1, LDA2, LDB, LDB1, LDB2, LDC, LDC1,
$ LDC2, LDD, LDD1, LDD2
PARAMETER ( LDA = NMAX, LDA1 = N1MAX, LDA2 = N2MAX,
$ LDB = NMAX, LDB1 = N1MAX, LDB2 = N2MAX,
$ LDC = PMAX, LDC1 = PMAX, LDC2 = PMAX,
$ LDD = PMAX, LDD1 = PMAX, LDD2 = PMAX )
* .. Local Scalars ..
CHARACTER*1 OVER
INTEGER I, INFO, J, M, N, N1, N2, P
DOUBLE PRECISION ALPHA
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), A1(LDA1,N1MAX), A2(LDA2,N2MAX),
$ B(LDB,MMAX), B1(LDB1,MMAX), B2(LDB2,MMAX),
$ C(LDC,NMAX), C1(LDC1,N1MAX), C2(LDC2,N2MAX),
$ D(LDD,MMAX), D1(LDD1,MMAX), D2(LDD2,MMAX)
* .. External Subroutines ..
EXTERNAL AB05PD
* .. Executable Statements ..
*
OVER = 'N'
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N1, M, P, N2, ALPHA
IF ( N1.LE.0 .OR. N1.GT.N1MAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N1
ELSE
READ ( NIN, FMT = * ) ( ( A1(I,J), J = 1,N1 ), I = 1,N1 )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B1(I,J), I = 1,N1 ), J = 1,M )
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C1(I,J), J = 1,N1 ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D1(I,J), J = 1,M ), I = 1,P )
IF ( N2.LE.0 .OR. N2.GT.N2MAX ) THEN
WRITE ( NOUT, FMT = 99989 ) N2
ELSE
READ ( NIN, FMT = * )
$ ( ( A2(I,J), J = 1,N2 ), I = 1,N2 )
READ ( NIN, FMT = * )
$ ( ( B2(I,J), I = 1,N2 ), J = 1,M )
READ ( NIN, FMT = * )
$ ( ( C2(I,J), J = 1,N2 ), I = 1,P )
READ ( NIN, FMT = * )
$ ( ( D2(I,J), J = 1,M ), I = 1,P )
* Find the state-space model (A,B,C,D).
CALL AB05PD( OVER, N1, M, P, N2, ALPHA, A1, LDA1, B1,
$ LDB1, C1, LDC1, D1, LDD1, A2, LDA2, B2,
$ LDB2, C2, LDC2, D2, LDD2, N, A, LDA, B,
$ LDB, C, LDC, D, LDD, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 )
$ ( A(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99996 )
$ ( B(I,J), J = 1,M )
40 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99996 )
$ ( C(I,J), J = 1,N )
60 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 80 I = 1, P
WRITE ( NOUT, FMT = 99996 )
$ ( D(I,J), J = 1,M )
80 CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB05PD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB05PD = ',I2)
99997 FORMAT (' The state transition matrix of the connected system is')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The input/state matrix of the connected system is ')
99994 FORMAT (/' The state/output matrix of the connected system is ')
99993 FORMAT (/' The input/output matrix of the connected system is ')
99992 FORMAT (/' N1 is out of range.',/' N1 = ',I5)
99991 FORMAT (/' M is out of range.',/' M = ',I5)
99990 FORMAT (/' P is out of range.',/' P = ',I5)
99989 FORMAT (/' N2 is out of range.',/' N2 = ',I5)
END
Program Data
AB05PD EXAMPLE PROGRAM DATA 3 2 2 3 1.0D0 1.0 0.0 -1.0 0.0 -1.0 1.0 1.0 1.0 2.0 1.0 1.0 0.0 2.0 0.0 1.0 3.0 -2.0 1.0 0.0 1.0 0.0 1.0 0.0 0.0 1.0 -3.0 0.0 0.0 1.0 0.0 1.0 0.0 -1.0 2.0 0.0 -1.0 0.0 1.0 0.0 2.0 1.0 1.0 0.0 1.0 1.0 -1.0 1.0 1.0 0.0 1.0Program Results
AB05PD EXAMPLE PROGRAM RESULTS The state transition matrix of the connected system is 1.0000 0.0000 -1.0000 0.0000 0.0000 0.0000 0.0000 -1.0000 1.0000 0.0000 0.0000 0.0000 1.0000 1.0000 2.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -3.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 -1.0000 2.0000 The input/state matrix of the connected system is 1.0000 2.0000 1.0000 0.0000 0.0000 1.0000 0.0000 1.0000 -1.0000 0.0000 0.0000 2.0000 The state/output matrix of the connected system is 3.0000 -2.0000 1.0000 1.0000 1.0000 0.0000 0.0000 1.0000 0.0000 1.0000 1.0000 -1.0000 The input/output matrix of the connected system is 2.0000 1.0000 0.0000 2.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB05QD.html 0000664 0000000 0000000 00000036435 14560147231 0020341 0 ustar 00root root 0000000 0000000
Purpose
To append two systems G1 and G2 in state-space form together.
If G1 = (A1,B1,C1,D1) and G2 = (A2,B2,C2,D2) are the state-space
models of the given two systems having the transfer-function
matrices G1 and G2, respectively, this subroutine constructs the
state-space model G = (A,B,C,D) which corresponds to the
transfer-function matrix
( G1 0 )
G = ( )
( 0 G2 )
Specification
SUBROUTINE AB05QD( OVER, N1, M1, P1, N2, M2, P2, A1, LDA1, B1,
$ LDB1, C1, LDC1, D1, LDD1, A2, LDA2, B2, LDB2,
$ C2, LDC2, D2, LDD2, N, M, P, A, LDA, B, LDB,
$ C, LDC, D, LDD, INFO )
C .. Scalar Arguments ..
CHARACTER OVER
INTEGER INFO, LDA, LDA1, LDA2, LDB, LDB1, LDB2, LDC,
$ LDC1, LDC2, LDD, LDD1, LDD2, M, M1, M2, N, N1,
$ N2, P, P1, P2
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), A1(LDA1,*), A2(LDA2,*), B(LDB,*),
$ B1(LDB1,*), B2(LDB2,*), C(LDC,*), C1(LDC1,*),
$ C2(LDC2,*), D(LDD,*), D1(LDD1,*), D2(LDD2,*)
Arguments
Mode Parameters
OVER CHARACTER*1
Indicates whether the user wishes to overlap pairs of
arrays, as follows:
= 'N': Do not overlap;
= 'O': Overlap pairs of arrays: A1 and A, B1 and B,
C1 and C, and D1 and D, i.e. the same name is
effectively used for each pair (for all pairs)
in the routine call. In this case, setting
LDA1 = LDA, LDB1 = LDB, LDC1 = LDC, and LDD1 = LDD
will give maximum efficiency.
Input/Output Parameters
N1 (input) INTEGER
The number of state variables in the first system, i.e.
the order of the matrix A1, the number of rows of B1 and
the number of columns of C1. N1 >= 0.
M1 (input) INTEGER
The number of input variables in the first system, i.e.
the number of columns of matrices B1 and D1. M1 >= 0.
P1 (input) INTEGER
The number of output variables in the first system, i.e.
the number of rows of matrices C1 and D1. P1 >= 0.
N2 (input) INTEGER
The number of state variables in the second system, i.e.
the order of the matrix A2, the number of rows of B2 and
the number of columns of C2. N2 >= 0.
M2 (input) INTEGER
The number of input variables in the second system, i.e.
the number of columns of matrices B2 and D2. M2 >= 0.
P2 (input) INTEGER
The number of output variables in the second system, i.e.
the number of rows of matrices C2 and D2. P2 >= 0.
A1 (input) DOUBLE PRECISION array, dimension (LDA1,N1)
The leading N1-by-N1 part of this array must contain the
state transition matrix A1 for the first system.
LDA1 INTEGER
The leading dimension of array A1. LDA1 >= MAX(1,N1).
B1 (input) DOUBLE PRECISION array, dimension (LDB1,M1)
The leading N1-by-M1 part of this array must contain the
input/state matrix B1 for the first system.
LDB1 INTEGER
The leading dimension of array B1. LDB1 >= MAX(1,N1).
C1 (input) DOUBLE PRECISION array, dimension (LDC1,N1)
The leading P1-by-N1 part of this array must contain the
state/output matrix C1 for the first system.
LDC1 INTEGER
The leading dimension of array C1.
LDC1 >= MAX(1,P1) if N1 > 0.
LDC1 >= 1 if N1 = 0.
D1 (input) DOUBLE PRECISION array, dimension (LDD1,M1)
The leading P1-by-M1 part of this array must contain the
input/output matrix D1 for the first system.
LDD1 INTEGER
The leading dimension of array D1. LDD1 >= MAX(1,P1).
A2 (input) DOUBLE PRECISION array, dimension (LDA2,N2)
The leading N2-by-N2 part of this array must contain the
state transition matrix A2 for the second system.
LDA2 INTEGER
The leading dimension of array A2. LDA2 >= MAX(1,N2).
B2 (input) DOUBLE PRECISION array, dimension (LDB2,M2)
The leading N2-by-M2 part of this array must contain the
input/state matrix B2 for the second system.
LDB2 INTEGER
The leading dimension of array B2. LDB2 >= MAX(1,N2).
C2 (input) DOUBLE PRECISION array, dimension (LDC2,N2)
The leading P2-by-N2 part of this array must contain the
state/output matrix C2 for the second system.
LDC2 INTEGER
The leading dimension of array C2.
LDC2 >= MAX(1,P2) if N2 > 0.
LDC2 >= 1 if N2 = 0.
D2 (input) DOUBLE PRECISION array, dimension (LDD2,M2)
The leading P2-by-M2 part of this array must contain the
input/output matrix D2 for the second system.
LDD2 INTEGER
The leading dimension of array D2. LDD2 >= MAX(1,P2).
N (output) INTEGER
The number of state variables (N1 + N2) in the resulting
system, i.e. the order of the matrix A, the number of rows
of B and the number of columns of C.
M (output) INTEGER
The number of input variables (M1 + M2) in the resulting
system, i.e. the number of columns of B and D.
P (output) INTEGER
The number of output variables (P1 + P2) of the resulting
system, i.e. the number of rows of C and D.
A (output) DOUBLE PRECISION array, dimension (LDA,N1+N2)
The leading N-by-N part of this array contains the state
transition matrix A for the resulting system.
The array A can overlap A1 if OVER = 'O'.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N1+N2).
B (output) DOUBLE PRECISION array, dimension (LDB,M1+M2)
The leading N-by-M part of this array contains the
input/state matrix B for the resulting system.
The array B can overlap B1 if OVER = 'O'.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N1+N2).
C (output) DOUBLE PRECISION array, dimension (LDC,N1+N2)
The leading P-by-N part of this array contains the
state/output matrix C for the resulting system.
The array C can overlap C1 if OVER = 'O'.
LDC INTEGER
The leading dimension of array C.
LDC >= MAX(1,P1+P2) if N1+N2 > 0.
LDC >= 1 if N1+N2 = 0.
D (output) DOUBLE PRECISION array, dimension (LDD,M1+M2)
The leading P-by-M part of this array contains the
input/output matrix D for the resulting system.
The array D can overlap D1 if OVER = 'O'.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P1+P2).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The matrices of the resulting systems are determined as:
( A1 0 ) ( B1 0 )
A = ( ) , B = ( ) ,
( 0 A2 ) ( 0 B2 )
( C1 0 ) ( D1 0 )
C = ( ) , D = ( ) .
( 0 C2 ) ( 0 D2 )
References
NoneFurther Comments
NoneExample
Program Text
* AB05QD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER N1MAX, N2MAX, NMAX, M1MAX, M2MAX, MMAX, P1MAX,
$ P2MAX, PMAX
PARAMETER ( N1MAX = 20, N2MAX = 20, NMAX = N1MAX+N2MAX,
$ M1MAX = 20, M2MAX = 20, MMAX = M1MAX+M2MAX,
$ P1MAX = 20, P2MAX = 20, PMAX = P1MAX+P2MAX )
INTEGER LDA, LDA1, LDA2, LDB, LDB1, LDB2, LDC, LDC1,
$ LDC2, LDD, LDD1, LDD2
PARAMETER ( LDA = NMAX, LDA1 = N1MAX, LDA2 = N2MAX,
$ LDB = NMAX, LDB1 = N1MAX, LDB2 = N2MAX,
$ LDC = PMAX, LDC1 = P1MAX, LDC2 = P1MAX,
$ LDD = PMAX, LDD1 = P1MAX, LDD2 = P1MAX )
DOUBLE PRECISION ONE
PARAMETER ( ONE=1.0D0 )
* .. Local Scalars ..
CHARACTER*1 OVER
INTEGER I, INFO, J, M, M1, M2, N, N1, N2, P, P1, P2
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), A1(LDA1,N1MAX), A2(LDA2,N2MAX),
$ B(LDB,MMAX), B1(LDB1,M1MAX), B2(LDB2,M2MAX),
$ C(LDC,NMAX), C1(LDC1,N1MAX), C2(LDC2,N2MAX),
$ D(LDD,MMAX), D1(LDD1,M1MAX), D2(LDD2,M2MAX)
* .. External Subroutines ..
EXTERNAL AB05QD
* .. Executable Statements ..
*
OVER = 'N'
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N1, M1, P1, N2, M2, P2
IF ( N1.LE.0 .OR. N1.GT.N1MAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N1
ELSE
READ ( NIN, FMT = * ) ( ( A1(I,J), J = 1,N1 ), I = 1,N1 )
IF ( M1.LE.0 .OR. M1.GT.M1MAX ) THEN
WRITE ( NOUT, FMT = 99991 ) M1
ELSE
READ ( NIN, FMT = * ) ( ( B1(I,J), I = 1,N1 ), J = 1,M1 )
IF ( P1.LE.0 .OR. P1.GT.P1MAX ) THEN
WRITE ( NOUT, FMT = 99990 ) P1
ELSE
READ ( NIN, FMT = * ) ( ( C1(I,J), J = 1,N1 ), I = 1,P1 )
READ ( NIN, FMT = * ) ( ( D1(I,J), J = 1,M1 ), I = 1,P1 )
IF ( N2.LE.0 .OR. N2.GT.N2MAX ) THEN
WRITE ( NOUT, FMT = 99989 ) N2
ELSE
READ ( NIN, FMT = * )
$ ( ( A2(I,J), J = 1,N2 ), I = 1,N2 )
IF ( M2.LE.0 .OR. M2.GT.M2MAX ) THEN
WRITE ( NOUT, FMT = 99988 ) M2
ELSE
READ ( NIN, FMT = * )
$ ( ( B2(I,J), I = 1,N2 ), J = 1,M2 )
IF ( P2.LE.0 .OR. P2.GT.P2MAX ) THEN
WRITE ( NOUT, FMT = 99987 ) P2
ELSE
READ ( NIN, FMT = * )
$ ( ( C2(I,J), J = 1,N2 ), I = 1,P2 )
READ ( NIN, FMT = * )
$ ( ( D2(I,J), J = 1,M2 ), I = 1,P2 )
* Find the state-space model (A,B,C,D).
CALL AB05QD( OVER, N1, M1, P1, N2, M2, P2, A1,
$ LDA1, B1, LDB1, C1, LDC1, D1, LDD1,
$ A2, LDA2, B2, LDB2, C2, LDC2, D2,
$ LDD2, N, M, P, A, LDA, B, LDB, C,
$ LDC, D, LDD, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 )
$ ( A(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99996 )
$ ( B(I,J), J = 1,M )
40 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99996 )
$ ( C(I,J), J = 1,N )
60 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 80 I = 1, P
WRITE ( NOUT, FMT = 99996 )
$ ( D(I,J), J = 1,M )
80 CONTINUE
END IF
END IF
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB05QD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB05QD = ',I2)
99997 FORMAT (' The state transition matrix of the connected system is')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The input/state matrix of the connected system is ')
99994 FORMAT (/' The state/output matrix of the connected system is ')
99993 FORMAT (/' The input/output matrix of the connected system is ')
99992 FORMAT (/' N1 is out of range.',/' N1 = ',I5)
99991 FORMAT (/' M1 is out of range.',/' M1 = ',I5)
99990 FORMAT (/' P1 is out of range.',/' P1 = ',I5)
99989 FORMAT (/' N2 is out of range.',/' N2 = ',I5)
99988 FORMAT (/' M2 is out of range.',/' M2 = ',I5)
99987 FORMAT (/' P2 is out of range.',/' P2 = ',I5)
END
Program Data
AB05QD EXAMPLE PROGRAM DATA 3 2 2 3 2 2 1.0 0.0 -1.0 0.0 -1.0 1.0 1.0 1.0 2.0 1.0 1.0 0.0 2.0 0.0 1.0 3.0 -2.0 1.0 0.0 1.0 0.0 1.0 0.0 0.0 1.0 -3.0 0.0 0.0 1.0 0.0 1.0 0.0 -1.0 2.0 0.0 -1.0 0.0 1.0 0.0 2.0 1.0 1.0 0.0 1.0 1.0 -1.0 1.0 1.0 0.0 1.0Program Results
AB05QD EXAMPLE PROGRAM RESULTS The state transition matrix of the connected system is 1.0000 0.0000 -1.0000 0.0000 0.0000 0.0000 0.0000 -1.0000 1.0000 0.0000 0.0000 0.0000 1.0000 1.0000 2.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -3.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 -1.0000 2.0000 The input/state matrix of the connected system is 1.0000 2.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 -1.0000 0.0000 0.0000 0.0000 0.0000 2.0000 The state/output matrix of the connected system is 3.0000 -2.0000 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000 1.0000 -1.0000 The input/output matrix of the connected system is 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000 1.0000 0.0000 0.0000 0.0000 1.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB05RD.html 0000664 0000000 0000000 00000037203 14560147231 0020334 0 ustar 00root root 0000000 0000000
Purpose
To construct for a given state space system (A,B,C,D) the closed-
loop system (Ac,Bc,Cc,Dc) corresponding to the mixed output and
state feedback control law
u = alpha*F*y + beta*K*x + G*v
z = H*y.
Specification
SUBROUTINE AB05RD( FBTYPE, JOBD, N, M, P, MV, PZ, ALPHA, BETA, A,
$ LDA, B, LDB, C, LDC, D, LDD, F, LDF, K, LDK,
$ G, LDG, H, LDH, RCOND, BC, LDBC, CC, LDCC,
$ DC, LDDC, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER FBTYPE, JOBD
INTEGER INFO, LDA, LDB, LDBC, LDC, LDCC, LDD, LDDC,
$ LDF, LDG, LDH, LDK, LDWORK, M, MV, N, P, PZ
DOUBLE PRECISION ALPHA, BETA, RCOND
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), BC(LDBC,*), C(LDC,*),
$ CC(LDCC,*), D(LDD,*), DC(LDDC,*), DWORK(*),
$ F(LDF,*), G(LDG,*), H(LDH,*), K(LDK,*)
Arguments
Mode Parameters
FBTYPE CHARACTER*1
Specifies the type of the feedback law as follows:
= 'I': Unitary output feedback (F = I);
= 'O': General output feedback.
JOBD CHARACTER*1
Specifies whether or not a non-zero matrix D appears
in the given state space model:
= 'D': D is present;
= 'Z': D is assumed a zero matrix.
Input/Output Parameters
N (input) INTEGER
The dimension of state vector x, i.e. the order of the
matrix A, the number of rows of B and the number of
columns of C. N >= 0.
M (input) INTEGER
The dimension of input vector u, i.e. the number of
columns of matrices B and D, and the number of rows of F.
M >= 0.
P (input) INTEGER
The dimension of output vector y, i.e. the number of rows
of matrices C and D, and the number of columns of F.
P >= 0 and P = M if FBTYPE = 'I'.
MV (input) INTEGER
The dimension of the new input vector v, i.e. the number
of columns of matrix G. MV >= 0.
PZ (input) INTEGER.
The dimension of the new output vector z, i.e. the number
of rows of matrix H. PZ >= 0.
ALPHA (input) DOUBLE PRECISION
The coefficient alpha in the output feedback law.
BETA (input) DOUBLE PRECISION.
The coefficient beta in the state feedback law.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the system state transition matrix A.
On exit, the leading N-by-N part of this array contains
the state matrix Ac of the closed-loop system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the system input matrix B.
On exit, the leading N-by-M part of this array contains
the intermediary input matrix B1 (see METHOD).
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the system output matrix C.
On exit, the leading P-by-N part of this array contains
the intermediary output matrix C1+BETA*D1*K (see METHOD).
LDC INTEGER
The leading dimension of array C.
LDC >= MAX(1,P) if N > 0.
LDC >= 1 if N = 0.
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, if JOBD = 'D', the leading P-by-M part of this
array must contain the system direct input/output
transmission matrix D.
On exit, the leading P-by-M part of this array contains
the intermediary direct input/output transmission matrix
D1 (see METHOD).
The array D is not referenced if JOBD = 'Z'.
LDD INTEGER
The leading dimension of array D.
LDD >= MAX(1,P) if JOBD = 'D'.
LDD >= 1 if JOBD = 'Z'.
F (input) DOUBLE PRECISION array, dimension (LDF,P)
If FBTYPE = 'O', the leading M-by-P part of this array
must contain the output feedback matrix F.
If FBTYPE = 'I', then the feedback matrix is assumed to be
an M x M order identity matrix.
The array F is not referenced if FBTYPE = 'I' or
ALPHA = 0.
LDF INTEGER
The leading dimension of array F.
LDF >= MAX(1,M) if FBTYPE = 'O' and ALPHA <> 0.
LDF >= 1 if FBTYPE = 'I' or ALPHA = 0.
K (input) DOUBLE PRECISION array, dimension (LDK,N)
The leading M-by-N part of this array must contain the
state feedback matrix K.
The array K is not referenced if BETA = 0.
LDK INTEGER
The leading dimension of the array K.
LDK >= MAX(1,M) if BETA <> 0.
LDK >= 1 if BETA = 0.
G (input) DOUBLE PRECISION array, dimension (LDG,MV)
The leading M-by-MV part of this array must contain the
system input scaling matrix G.
LDG INTEGER
The leading dimension of the array G. LDG >= MAX(1,M).
H (input) DOUBLE PRECISION array, dimension (LDH,P)
The leading PZ-by-P part of this array must contain the
system output scaling matrix H.
LDH INTEGER
The leading dimension of the array H. LDH >= MAX(1,PZ).
RCOND (output) DOUBLE PRECISION
The reciprocal condition number of the matrix
I - alpha*D*F.
BC (output) DOUBLE PRECISION array, dimension (LDBC,MV)
The leading N-by-MV part of this array contains the input
matrix Bc of the closed-loop system.
LDBC INTEGER
The leading dimension of array BC. LDBC >= MAX(1,N).
CC (output) DOUBLE PRECISION array, dimension (LDCC,N)
The leading PZ-by-N part of this array contains the
system output matrix Cc of the closed-loop system.
LDCC INTEGER
The leading dimension of array CC.
LDCC >= MAX(1,PZ) if N > 0.
LDCC >= 1 if N = 0.
DC (output) DOUBLE PRECISION array, dimension (LDDC,MV)
If JOBD = 'D', the leading PZ-by-MV part of this array
contains the direct input/output transmission matrix Dc
of the closed-loop system.
The array DC is not referenced if JOBD = 'Z'.
LDDC INTEGER
The leading dimension of array DC.
LDDC >= MAX(1,PZ) if JOBD = 'D'.
LDDC >= 1 if JOBD = 'Z'.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK >= MAX(1,2*P) if JOBD = 'D'.
LIWORK >= 1 if JOBD = 'Z'.
IWORK is not referenced if JOBD = 'Z'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= wspace, where
wspace = MAX( 1, M, P*MV, P*P + 4*P ) if JOBD = 'D',
wspace = MAX( 1, M ) if JOBD = 'Z'.
For best performance, LDWORK >= MAX( wspace, N*M, N*P ).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the matrix I - alpha*D*F is numerically singular.
Method
The matrices of the closed-loop system have the expressions: Ac = A1 + beta*B1*K, Bc = B1*G, Cc = H*(C1 + beta*D1*K), Dc = H*D1*G, where A1 = A + alpha*B*F*E*C, B1 = B + alpha*B*F*E*D, C1 = E*C, D1 = E*D, with E = (I - alpha*D*F)**-1.Numerical Aspects
The accuracy of computations basically depends on the conditioning of the matrix I - alpha*D*F. If RCOND is very small, it is likely that the computed results are inaccurate.Further Comments
NoneExample
Program Text
* AB05RD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, MVMAX, PMAX, PZMAX
PARAMETER ( NMAX = 20, MMAX = 20, MVMAX = 20,
$ PMAX = 20, PZMAX = 20 )
INTEGER LDA, LDB, LDBC, LDC, LDCC, LDD, LDDC, LDF, LDG,
$ LDH, LDK, LDWORK, LIWORK
PARAMETER ( LDA = NMAX, LDB = NMAX, LDBC = NMAX,
$ LDC = PMAX, LDCC = PZMAX,
$ LDD = PMAX, LDDC = PZMAX, LDF = MMAX,
$ LDG = MMAX, LDH = PZMAX, LDK = MMAX,
$ LDWORK = MAX( MMAX, PMAX*MVMAX,
$ PMAX*PMAX + 4*PMAX ), LIWORK = 2*PMAX )
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
* .. Local Scalars ..
LOGICAL LJOBD, OUTPF
CHARACTER*1 FBTYPE, JOBD
INTEGER I, INFO, J, M, MV, N, P, PZ
DOUBLE PRECISION ALPHA, BETA, RCOND
* .. Local Arrays ..
INTEGER IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), BC(LDBC,MVMAX),
$ C(LDC,NMAX), CC(LDCC,NMAX),
$ D(LDD,MMAX), DC(LDDC,MVMAX), DWORK(LDWORK),
$ F(LDF,PMAX), G(LDG,MVMAX), H(LDH,PMAX),
$ K(LDK,NMAX)
* .. External functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL AB05RD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, MV, PZ, ALPHA, BETA, FBTYPE, JOBD
OUTPF = LSAME( FBTYPE, 'O' )
LJOBD = LSAME( JOBD, 'D' )
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,N ), J = 1,M )
IF ( BETA.NE.ZERO )
$ READ ( NIN, FMT = * ) ( ( K(I,J), J = 1,N ), I = 1,M )
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
IF ( LJOBD )
$ READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
IF ( OUTPF.AND.ALPHA.NE.ZERO )
$ READ ( NIN, FMT = * ) ( ( F(I,J), J = 1,P ), I = 1,M )
IF ( MV.LE.0 .OR. MV.GT.MVMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) MV
ELSE
READ ( NIN, FMT = * )
$ ( ( G(I,J), J = 1,MV ), I = 1,M )
IF ( PZ.LE.0 .OR. PZ.GT.PZMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) PZ
ELSE
READ ( NIN, FMT = * )
$ ( ( H(I,J), J = 1,P ), I = 1,PZ )
* Find the state-space model (A,B,C,D).
CALL AB05RD( FBTYPE, JOBD, N, M, P, MV, PZ, ALPHA,
$ BETA, A, LDA, B, LDB, C, LDC, D, LDD,
$ F, LDF, K, LDK, G, LDG, H, LDH, RCOND,
$ BC, LDBC, CC, LDCC, DC, LDDC, IWORK,
$ DWORK, LDWORK, INFO )
*
WRITE ( NOUT, FMT = 99987 ) RCOND
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 )
$ ( A(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99996 )
$ ( BC(I,J), J = 1,MV )
40 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 60 I = 1, PZ
WRITE ( NOUT, FMT = 99996 )
$ ( CC(I,J), J = 1,N )
60 CONTINUE
IF ( LJOBD ) THEN
WRITE ( NOUT, FMT = 99993 )
DO 80 I = 1, PZ
WRITE ( NOUT, FMT = 99996 )
$ ( DC(I,J), J = 1,MV )
80 CONTINUE
END IF
END IF
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB05RD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB05RD = ',I2)
99997 FORMAT (' The state transition matrix of the closed-loop system is
$')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The input/state matrix of the closed-loop system is ')
99994 FORMAT (/' The state/output matrix of the closed-loop system is ')
99993 FORMAT (/' The input/output matrix of the closed-loop system is ')
99992 FORMAT (/' N is out of range.',/' N = ',I5)
99991 FORMAT (/' M is out of range.',/' M = ',I5)
99990 FORMAT (/' P is out of range.',/' P = ',I5)
99989 FORMAT (/' MV is out of range.',/' MV = ',I5)
99988 FORMAT (/' PZ is out of range.',/' PZ = ',I5)
99987 FORMAT ( ' The reciprocal condition number of the matrix ',
$ ' I - alpha*D*F is',F8.4,/1X)
END
Program Data
AB05RD EXAMPLE PROGRAM DATA 3 2 2 2 2 1.0 1.0 O D 1.0 0.0 -1.0 0.0 -1.0 1.0 1.0 1.0 2.0 1.0 1.0 0.0 2.0 0.0 1.0 2.0 1.0 0.0 1.0 0.0 1.0 3.0 -2.0 1.0 0.0 1.0 0.0 1.0 0.0 0.0 1.0 1.0 2.0 3.0 4.0 1.0 1.0 0.0 1.0 4.0 3.0 2.0 1.0Program Results
AB05RD EXAMPLE PROGRAM RESULTS The reciprocal condition number of the matrix I - alpha*D*F is 0.2000 The state transition matrix of the closed-loop system is -4.8333 0.1667 -2.8333 -0.8333 0.1667 0.1667 -1.5000 0.5000 1.5000 The input/state matrix of the closed-loop system is -0.5000 -0.8333 0.5000 0.1667 -0.5000 -0.5000 The state/output matrix of the closed-loop system is 1.1667 -1.8333 -0.8333 1.8333 -1.1667 -0.1667 The input/output matrix of the closed-loop system is 0.5000 -0.8333 0.5000 -0.1667
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB05SD.html 0000664 0000000 0000000 00000014774 14560147231 0020345 0 ustar 00root root 0000000 0000000
Purpose
To construct for a given state space system (A,B,C,D) the closed-
loop system (Ac,Bc,Cc,Dc) corresponding to the output feedback
control law
u = alpha*F*y + v.
Specification
SUBROUTINE AB05SD( FBTYPE, JOBD, N, M, P, ALPHA, A, LDA, B, LDB,
$ C, LDC, D, LDD, F, LDF, RCOND, IWORK, DWORK,
$ LDWORK, INFO)
C .. Scalar Arguments ..
CHARACTER FBTYPE, JOBD
INTEGER INFO, LDA, LDB, LDC, LDD, LDF, LDWORK, M, N, P
DOUBLE PRECISION ALPHA, RCOND
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), F(LDF,*)
Arguments
Mode Parameters
FBTYPE CHARACTER*1
Specifies the type of the feedback law as follows:
= 'I': Unitary output feedback (F = I);
= 'O': General output feedback.
JOBD CHARACTER*1
Specifies whether or not a non-zero matrix D appears in
the given state space model:
= 'D': D is present;
= 'Z': D is assumed a zero matrix.
Input/Output Parameters
N (input) INTEGER
The number of state variables, i.e. the order of the
matrix A, the number of rows of B and the number of
columns of C. N >= 0.
M (input) INTEGER
The number of input variables, i.e. the number of columns
of matrices B and D, and the number of rows of F. M >= 0.
P (input) INTEGER
The number of output variables, i.e. the number of rows of
matrices C and D, and the number of columns of F. P >= 0
and P = M if FBTYPE = 'I'.
ALPHA (input) DOUBLE PRECISION
The coefficient alpha in the output feedback law.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the system state transition matrix A.
On exit, the leading N-by-N part of this array contains
the state matrix Ac of the closed-loop system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the system input matrix B.
On exit, the leading N-by-M part of this array contains
the input matrix Bc of the closed-loop system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the system output matrix C.
On exit, the leading P-by-N part of this array contains
the output matrix Cc of the closed-loop system.
LDC INTEGER
The leading dimension of array C.
LDC >= MAX(1,P) if N > 0.
LDC >= 1 if N = 0.
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the system direct input/output transmission
matrix D.
On exit, if JOBD = 'D', the leading P-by-M part of this
array contains the direct input/output transmission
matrix Dc of the closed-loop system.
The array D is not referenced if JOBD = 'Z'.
LDD INTEGER
The leading dimension of array D.
LDD >= MAX(1,P) if JOBD = 'D'.
LDD >= 1 if JOBD = 'Z'.
F (input) DOUBLE PRECISION array, dimension (LDF,P)
If FBTYPE = 'O', the leading M-by-P part of this array
must contain the output feedback matrix F.
If FBTYPE = 'I', then the feedback matrix is assumed to be
an M x M order identity matrix.
The array F is not referenced if FBTYPE = 'I' or
ALPHA = 0.
LDF INTEGER
The leading dimension of array F.
LDF >= MAX(1,M) if FBTYPE = 'O' and ALPHA <> 0.
LDF >= 1 if FBTYPE = 'I' or ALPHA = 0.
RCOND (output) DOUBLE PRECISION
The reciprocal condition number of the matrix
I - alpha*D*F.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK >= MAX(1,2*P) if JOBD = 'D'.
LIWORK >= 1 if JOBD = 'Z'.
IWORK is not referenced if JOBD = 'Z'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= wspace, where
wspace = MAX( 1, M, P*P + 4*P ) if JOBD = 'D',
wspace = MAX( 1, M ) if JOBD = 'Z'.
For best performance, LDWORK >= MAX( wspace, N*M, N*P ).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the matrix I - alpha*D*F is numerically singular.
Method
The matrices of the closed-loop system have the expressions: Ac = A + alpha*B*F*E*C, Bc = B + alpha*B*F*E*D, Cc = E*C, Dc = E*D, where E = (I - alpha*D*F)**-1.Numerical Aspects
The accuracy of computations basically depends on the conditioning of the matrix I - alpha*D*F. If RCOND is very small, it is likely that the computed results are inaccurate.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB07MD.html 0000664 0000000 0000000 00000017650 14560147231 0020335 0 ustar 00root root 0000000 0000000
Purpose
To find the dual of a given state-space representation.Specification
SUBROUTINE AB07MD( JOBD, N, M, P, A, LDA, B, LDB, C, LDC, D, LDD,
$ INFO )
C .. Scalar Arguments ..
CHARACTER JOBD
INTEGER INFO, LDA, LDB, LDC, LDD, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*)
Arguments
Mode Parameters
JOBD CHARACTER*1
Specifies whether or not a non-zero matrix D appears in
the given state space model:
= 'D': D is present;
= 'Z': D is assumed a zero matrix.
Input/Output Parameters
N (input) INTEGER
The order of the state-space representation. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state dynamics matrix A.
On exit, the leading N-by-N part of this array contains
the dual state dynamics matrix A'.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension
(LDB,MAX(M,P))
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, the leading N-by-P part of this array contains
the dual input/state matrix C'.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, the leading M-by-N part of this array contains
the dual state/output matrix B'.
LDC INTEGER
The leading dimension of array C.
LDC >= MAX(1,M,P) if N > 0.
LDC >= 1 if N = 0.
D (input/output) DOUBLE PRECISION array, dimension
(LDD,MAX(M,P))
On entry, if JOBD = 'D', the leading P-by-M part of this
array must contain the original direct transmission
matrix D.
On exit, if JOBD = 'D', the leading M-by-P part of this
array contains the dual direct transmission matrix D'.
The array D is not referenced if JOBD = 'Z'.
LDD INTEGER
The leading dimension of array D.
LDD >= MAX(1,M,P) if JOBD = 'D'.
LDD >= 1 if JOBD = 'Z'.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
If the given state-space representation is the M-input/P-output (A,B,C,D), its dual is simply the P-input/M-output (A',C',B',D').References
NoneNumerical Aspects
NoneFurther Comments
NoneExample
Program Text
* AB07MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER MAXMP
PARAMETER ( MAXMP = MAX( MMAX, PMAX ) )
INTEGER LDA, LDB, LDC, LDD
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = MAXMP,
$ LDD = MAXMP )
* .. Local Scalars ..
CHARACTER*1 JOBD
INTEGER I, INFO, J, M, N, P
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MAXMP), C(LDC,NMAX),
$ D(LDD,MAXMP)
* .. External functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL AB07MD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read in the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, JOBD
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
* Find the dual of the ssr (A,B,C,D).
CALL AB07MD( JOBD, N, M, P, A, LDA, B, LDB, C, LDC, D,
$ LDD, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( A(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( B(I,J), J = 1,P )
40 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 60 I = 1, M
WRITE ( NOUT, FMT = 99996 ) ( C(I,J), J = 1,N )
60 CONTINUE
IF ( LSAME( JOBD, 'D' ) ) THEN
WRITE ( NOUT, FMT = 99993 )
DO 80 I = 1, M
WRITE ( NOUT, FMT = 99996 ) ( D(I,J), J = 1,P )
80 CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB07MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB07MD = ',I2)
99997 FORMAT (' The dual state dynamics matrix is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The dual input/state matrix is ')
99994 FORMAT (/' The dual state/output matrix is ')
99993 FORMAT (/' The dual direct transmission matrix is ')
99992 FORMAT (/' N is out of range.',/' N = ',I5)
99991 FORMAT (/' M is out of range.',/' M = ',I5)
99990 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
AB07MD EXAMPLE PROGRAM DATA 3 1 2 D 1.0 2.0 0.0 4.0 -1.0 0.0 0.0 0.0 1.0 1.0 0.0 1.0 0.0 1.0 -1.0 0.0 0.0 1.0 0.0 1.0Program Results
AB07MD EXAMPLE PROGRAM RESULTS The dual state dynamics matrix is 1.0000 4.0000 0.0000 2.0000 -1.0000 0.0000 0.0000 0.0000 1.0000 The dual input/state matrix is 0.0000 0.0000 1.0000 0.0000 -1.0000 1.0000 The dual state/output matrix is 1.0000 0.0000 1.0000 The dual direct transmission matrix is 0.0000 1.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB07ND.html 0000664 0000000 0000000 00000021150 14560147231 0020324 0 ustar 00root root 0000000 0000000
Purpose
To compute the inverse (Ai,Bi,Ci,Di) of a given system (A,B,C,D).Specification
SUBROUTINE AB07ND( N, M, A, LDA, B, LDB, C, LDC, D, LDD, RCOND,
$ IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
DOUBLE PRECISION RCOND
INTEGER INFO, LDA, LDB, LDC, LDD, LDWORK, M, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*)
INTEGER IWORK(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the state matrix A. N >= 0.
M (input) INTEGER
The number of system inputs and outputs. M >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state matrix A of the original system.
On exit, the leading N-by-N part of this array contains
the state matrix Ai of the inverse system.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input matrix B of the original system.
On exit, the leading N-by-M part of this array contains
the input matrix Bi of the inverse system.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading M-by-N part of this array must
contain the output matrix C of the original system.
On exit, the leading M-by-N part of this array contains
the output matrix Ci of the inverse system.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,M).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading M-by-M part of this array must
contain the feedthrough matrix D of the original system.
On exit, the leading M-by-M part of this array contains
the feedthrough matrix Di of the inverse system.
LDD INTEGER
The leading dimension of the array D. LDD >= MAX(1,M).
RCOND (output) DOUBLE PRECISION
The estimated reciprocal condition number of the
feedthrough matrix D of the original system.
Workspace
IWORK INTEGER array, dimension (2*M)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or M+1, DWORK(1) returns the optimal
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,4*M).
For good performance, LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= i: the matrix D is exactly singular; the (i,i) diagonal
element is zero, i <= M; RCOND was set to zero;
= M+1: the matrix D is numerically singular, i.e., RCOND
is less than the relative machine precision, EPS
(see LAPACK Library routine DLAMCH). The
calculations have been completed, but the results
could be very inaccurate.
Method
The matrices of the inverse system are computed with the formulas:
-1 -1 -1 -1
Ai = A - B*D *C, Bi = -B*D , Ci = D *C, Di = D .
Numerical Aspects
The accuracy depends mainly on the condition number of the matrix D to be inverted. The estimated reciprocal condition number is returned in RCOND.Further Comments
NoneExample
Program Text
* AB07ND EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX
PARAMETER ( NMAX = 20, MMAX = 20 )
INTEGER LDA, LDB, LDC, LDD
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = MMAX,
$ LDD = MMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 4*MMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, M, N
DOUBLE PRECISION RCOND
* .. Local Arrays ..
INTEGER IWORK(2*MMAX)
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK)
* .. External Subroutines ..
EXTERNAL AB07ND
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read in the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,M )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,M )
* Find the inverse of the ssr (A,B,C,D).
CALL AB07ND( N, M, A, LDA, B, LDB, C, LDC, D, LDD, RCOND,
$ IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( A(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( B(I,J), J = 1,M )
40 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 60 I = 1, M
WRITE ( NOUT, FMT = 99996 ) ( C(I,J), J = 1,N )
60 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 80 I = 1, M
WRITE ( NOUT, FMT = 99996 ) ( D(I,J), J = 1,M )
80 CONTINUE
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB07ND EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB07ND = ',I2)
99997 FORMAT (' The state dynamics matrix of the inverse system is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The input/state matrix of the inverse system is ')
99994 FORMAT (/' The state/output matrix of the inverse system is ')
99993 FORMAT (/' The feedthrough matrix of the inverse system is ')
99992 FORMAT (/' N is out of range.',/' N = ',I5)
99991 FORMAT (/' M is out of range.',/' M = ',I5)
END
Program Data
AB07ND EXAMPLE PROGRAM DATA 3 2 1.0 2.0 0.0 4.0 -1.0 0.0 0.0 0.0 1.0 1.0 0.0 0.0 1.0 1.0 0.0 0.0 1.0 -1.0 0.0 0.0 1.0 4.0 0.0 0.0 1.0Program Results
AB07ND EXAMPLE PROGRAM RESULTS The state dynamics matrix of the inverse system is 1.0000 1.7500 0.2500 4.0000 -1.0000 -1.0000 0.0000 -0.2500 1.2500 The input/state matrix of the inverse system is -0.2500 0.0000 0.0000 -1.0000 -0.2500 0.0000 The state/output matrix of the inverse system is 0.0000 0.2500 -0.2500 0.0000 0.0000 1.0000 The feedthrough matrix of the inverse system is 0.2500 0.0000 0.0000 1.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB08MD.html 0000664 0000000 0000000 00000013172 14560147231 0020331 0 ustar 00root root 0000000 0000000
Purpose
To compute the normal rank of the transfer-function matrix of a state-space model (A,B,C,D).Specification
SUBROUTINE AB08MD( EQUIL, N, M, P, A, LDA, B, LDB, C, LDC, D, LDD,
$ RANK, TOL, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER EQUIL
INTEGER INFO, LDA, LDB, LDC, LDD, LDWORK, M, N, P, RANK
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), DWORK(*)
Arguments
Mode Parameters
EQUIL CHARACTER*1
Specifies whether the user wishes to balance the compound
matrix (see METHOD) as follows:
= 'S': Perform balancing (scaling);
= 'N': Do not perform balancing.
Input/Output Parameters
N (input) INTEGER
The number of state variables, i.e., the order of the
matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
state dynamics matrix A of the system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
input/state matrix B of the system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array must contain the
state/output matrix C of the system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading P-by-M part of this array must contain the
direct transmission matrix D of the system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
RANK (output) INTEGER
The normal rank of the transfer-function matrix.
Tolerances
TOL DOUBLE PRECISION
A tolerance used in rank decisions to determine the
effective rank, which is defined as the order of the
largest leading (or trailing) triangular submatrix in the
QR (or RQ) factorization with column (or row) pivoting
whose estimated condition number is less than 1/TOL.
If the user sets TOL to be less than SQRT((N+P)*(N+M))*EPS
then the tolerance is taken as SQRT((N+P)*(N+M))*EPS,
where EPS is the machine precision (see LAPACK Library
Routine DLAMCH).
Workspace
IWORK INTEGER array, dimension (2*N+MAX(M,P)+1)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= (N+P)*(N+M) +
MAX( MIN(P,M) + MAX(3*M-1,N), 1,
MIN(P,N) + MAX(3*P-1,N+P,N+M) )
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine reduces the (N+P)-by-(M+N) compound matrix (B A)
(D C)
to one with the same invariant zeros and with D of full row rank.
The normal rank of the transfer-function matrix is the rank of D.
References
[1] Svaricek, F.
Computation of the Structural Invariants of Linear
Multivariable Systems with an Extended Version of
the Program ZEROS.
System & Control Letters, 6, pp. 261-266, 1985.
[2] Emami-Naeini, A. and Van Dooren, P.
Computation of Zeros of Linear Multivariable Systems.
Automatica, 18, pp. 415-430, 1982.
Numerical Aspects
The algorithm is backward stable (see [2] and [1]).Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB08MZ.html 0000664 0000000 0000000 00000013304 14560147231 0020354 0 ustar 00root root 0000000 0000000
Purpose
To compute the normal rank of the transfer-function matrix of a state-space model (A,B,C,D).Specification
SUBROUTINE AB08MZ( EQUIL, N, M, P, A, LDA, B, LDB, C, LDC, D, LDD,
$ RANK, TOL, IWORK, DWORK, ZWORK, LZWORK, INFO )
C .. Scalar Arguments ..
CHARACTER EQUIL
INTEGER INFO, LDA, LDB, LDC, LDD, LZWORK, M, N, P, RANK
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK(*)
COMPLEX*16 A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), ZWORK(*)
DOUBLE PRECISION DWORK(*)
Arguments
Mode Parameters
EQUIL CHARACTER*1
Specifies whether the user wishes to balance the compound
matrix (see METHOD) as follows:
= 'S': Perform balancing (scaling);
= 'N': Do not perform balancing.
Input/Output Parameters
N (input) INTEGER
The number of state variables, i.e., the order of the
matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
A (input) COMPLEX*16 array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
state dynamics matrix A of the system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) COMPLEX*16 array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
input/state matrix B of the system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input) COMPLEX*16 array, dimension (LDC,N)
The leading P-by-N part of this array must contain the
state/output matrix C of the system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input) COMPLEX*16 array, dimension (LDD,M)
The leading P-by-M part of this array must contain the
direct transmission matrix D of the system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
RANK (output) INTEGER
The normal rank of the transfer-function matrix.
Tolerances
TOL DOUBLE PRECISION
A tolerance used in rank decisions to determine the
effective rank, which is defined as the order of the
largest leading (or trailing) triangular submatrix in the
QR (or RQ) factorization with column (or row) pivoting
whose estimated condition number is less than 1/TOL.
If the user sets TOL to be less than SQRT((N+P)*(N+M))*EPS
then the tolerance is taken as SQRT((N+P)*(N+M))*EPS,
where EPS is the machine precision (see LAPACK Library
Routine DLAMCH).
Workspace
IWORK INTEGER array, dimension (2*N+MAX(M,P)+1)
DWORK DOUBLE PRECISION array, dimension (2*MAX(M,P))
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) returns the optimal value
of LZWORK.
LZWORK INTEGER
The length of the array ZWORK.
LZWORK >= (N+P)*(N+M) + MAX(MIN(P,M) + MAX(3*M-1,N), 1,
MIN(P,N) + MAX(3*P-1,N+P,N+M))
For optimum performance LZWORK should be larger.
If LZWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
ZWORK array, returns this value as the first entry of
the ZWORK array, and no error message related to LZWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine reduces the (N+P)-by-(M+N) compound matrix (B A)
(D C)
to one with the same invariant zeros and with D of full row rank.
The normal rank of the transfer-function matrix is the rank of D.
References
[1] Svaricek, F.
Computation of the Structural Invariants of Linear
Multivariable Systems with an Extended Version of
the Program ZEROS.
System & Control Letters, 6, pp. 261-266, 1985.
[2] Emami-Naeini, A. and Van Dooren, P.
Computation of Zeros of Linear Multivariable Systems.
Automatica, 18, pp. 415-430, 1982.
Numerical Aspects
The algorithm is backward stable (see [2] and [1]).Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB08ND.html 0000664 0000000 0000000 00000046311 14560147231 0020333 0 ustar 00root root 0000000 0000000
Purpose
To construct for a linear multivariable system described by a
state-space model (A,B,C,D) a regular pencil (A - lambda*B ) which
f f
has the invariant zeros of the system as generalized eigenvalues.
The routine also computes the orders of the infinite zeros and the
right and left Kronecker indices of the system (A,B,C,D).
Specification
SUBROUTINE AB08ND( EQUIL, N, M, P, A, LDA, B, LDB, C, LDC, D, LDD,
$ NU, RANK, DINFZ, NKROR, NKROL, INFZ, KRONR,
$ KRONL, AF, LDAF, BF, LDBF, TOL, IWORK, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER EQUIL
INTEGER DINFZ, INFO, LDA, LDAF, LDB, LDBF, LDC, LDD,
$ LDWORK, M, N, NKROL, NKROR, NU, P, RANK
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER INFZ(*), IWORK(*), KRONL(*), KRONR(*)
DOUBLE PRECISION A(LDA,*), AF(LDAF,*), B(LDB,*), BF(LDBF,*),
$ C(LDC,*), D(LDD,*), DWORK(*)
Arguments
Mode Parameters
EQUIL CHARACTER*1
Specifies whether the user wishes to balance the compound
matrix (see METHOD) as follows:
= 'S': Perform balancing (scaling);
= 'N': Do not perform balancing.
Input/Output Parameters
N (input) INTEGER
The number of state variables, i.e., the order of the
matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
state dynamics matrix A of the system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
input/state matrix B of the system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array must contain the
state/output matrix C of the system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading P-by-M part of this array must contain the
direct transmission matrix D of the system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
NU (output) INTEGER
The number of (finite) invariant zeros.
RANK (output) INTEGER
The normal rank of the transfer function matrix.
DINFZ (output) INTEGER
The maximum degree of infinite elementary divisors.
NKROR (output) INTEGER
The number of right Kronecker indices.
NKROL (output) INTEGER
The number of left Kronecker indices.
INFZ (output) INTEGER array, dimension (N)
The leading DINFZ elements of INFZ contain information
on the infinite elementary divisors as follows:
the system has INFZ(i) infinite elementary divisors
of degree i, where i = 1,2,...,DINFZ.
KRONR (output) INTEGER array, dimension (MAX(N,M)+1)
The leading NKROR elements of this array contain the
right Kronecker (column) indices.
KRONL (output) INTEGER array, dimension (MAX(N,P)+1)
The leading NKROL elements of this array contain the
left Kronecker (row) indices.
AF (output) DOUBLE PRECISION array, dimension
(LDAF,N+MIN(P,M))
The leading NU-by-NU part of this array contains the
coefficient matrix A of the reduced pencil. The remainder
f
of the leading (N+M)-by-(N+MIN(P,M)) part is used as
internal workspace.
LDAF INTEGER
The leading dimension of array AF. LDAF >= MAX(1,N+M).
BF (output) DOUBLE PRECISION array, dimension (LDBF,N+M)
The leading NU-by-NU part of this array contains the
coefficient matrix B of the reduced pencil. The
f
remainder of the leading (N+P)-by-(N+M) part is used as
internal workspace.
LDBF INTEGER
The leading dimension of array BF. LDBF >= MAX(1,N+P).
Tolerances
TOL DOUBLE PRECISION
A tolerance used in rank decisions to determine the
effective rank, which is defined as the order of the
largest leading (or trailing) triangular submatrix in the
QR (or RQ) factorization with column (or row) pivoting
whose estimated condition number is less than 1/TOL.
If the user sets TOL to be less than SQRT((N+P)*(N+M))*EPS
then the tolerance is taken as SQRT((N+P)*(N+M))*EPS,
where EPS is the machine precision (see LAPACK Library
Routine DLAMCH).
Workspace
IWORK INTEGER array, dimension (MAX(M,P))
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( 1, MIN(P,M) + MAX(3*M-1,N),
MIN(P,N) + MAX(3*P-1,N+P,N+M),
MIN(M,N) + MAX(3*M-1,N+M) ).
An upper bound is MAX(s,N) + MAX(3*s-1,N+s), with
s = MAX(M,P).
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine extracts from the system matrix of a state-space
system (A,B,C,D) a regular pencil A - lambda*B which has the
f f
invariant zeros of the system as generalized eigenvalues as
follows:
(a) construct the (N+P)-by-(N+M) compound matrix (B A);
(D C)
(b) reduce the above system to one with the same invariant
zeros and with D of full row rank;
(c) pertranspose the system;
(d) reduce the system to one with the same invariant zeros and
with D square invertible;
(e) perform a unitary transformation on the columns of
(A - lambda*I B) in order to reduce it to
( C D)
(A - lambda*B X)
( f f ), with Y and B square invertible;
( 0 Y) f
(f) compute the right and left Kronecker indices of the system
(A,B,C,D), which together with the orders of the infinite
zeros (determined by steps (a) - (e)) constitute the
complete set of structural invariants under strict
equivalence transformations of a linear system.
References
[1] Svaricek, F.
Computation of the Structural Invariants of Linear
Multivariable Systems with an Extended Version of
the Program ZEROS.
System & Control Letters, 6, pp. 261-266, 1985.
[2] Emami-Naeini, A. and Van Dooren, P.
Computation of Zeros of Linear Multivariable Systems.
Automatica, 18, pp. 415-430, 1982.
Numerical Aspects
The algorithm is backward stable (see [2] and [1]).Further Comments
In order to compute the invariant zeros of the system explicitly,
a call to this routine may be followed by a call to the LAPACK
Library routine DGGEV with A = A , B = B and N = NU.
f f
If RANK = 0, the routine DGEEV can be used (since B = I).
f
Example
Program Text
* AB08ND EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER MPMAX
PARAMETER ( MPMAX = MAX( MMAX, PMAX ) )
INTEGER LDA, LDB, LDC, LDD, LDAF, LDBF, LDQ, LDZ
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX, LDAF = NMAX+MPMAX,
$ LDBF = NMAX+PMAX, LDQ = 1, LDZ = 1 )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( MAX( MPMAX+1, NMAX ) +
$ MAX( 3*(MPMAX+1), NMAX+MPMAX ),
$ 8*NMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER DINFZ, I, INFO, J, M, N, NINFZ, NKROL, NKROR,
$ NU, P, RANK
CHARACTER*1 EQUIL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), AF(LDAF,NMAX+PMAX), ALFI(NMAX),
$ ALFR(NMAX), B(LDB,MMAX), BETA(NMAX),
$ BF(LDBF,MMAX+NMAX), C(LDC,NMAX), D(LDD,MMAX),
$ DWORK(LDWORK), Q(LDQ,1), Z(LDZ,1)
INTEGER INFZ(NMAX), IWORK(MPMAX+1), KRONL(NMAX+1),
$ KRONR(NMAX+1)
* .. External Subroutines ..
EXTERNAL AB08ND, DGEGV
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL, EQUIL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99972 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99971 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99970 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
* Check the observability and compute the ordered set of
* the observability indices (call the routine with M = 0).
CALL AB08ND( EQUIL, N, 0, P, A, LDA, B, LDB, C, LDC, D,
$ LDD, NU, RANK, DINFZ, NKROR, NKROL, INFZ,
$ KRONR, KRONL, AF, LDAF, BF, LDBF, TOL,
$ IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) ( KRONL(I), I = 1,P )
IF ( NU.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99993 )
ELSE
WRITE ( NOUT, FMT = 99992 ) N - NU
WRITE ( NOUT, FMT = 99991 )
WRITE ( NOUT, FMT = 99990 )
DO 20 I = 1, NU
WRITE ( NOUT, FMT = 99989 )
$ ( AF(I,J), J = 1,NU )
20 CONTINUE
END IF
END IF
* Check the controllability and compute the ordered set of
* the controllability indices (call the routine with P = 0)
CALL AB08ND( EQUIL, N, M, 0, A, LDA, B, LDB, C, LDC, D,
$ LDD, NU, RANK, DINFZ, NKROR, NKROL, INFZ,
$ KRONR, KRONL, AF, LDAF, BF, LDBF, TOL,
$ IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99988 ) ( KRONR(I), I = 1,M )
IF ( NU.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99987 )
ELSE
WRITE ( NOUT, FMT = 99986 ) N - NU
WRITE ( NOUT, FMT = 99985 )
WRITE ( NOUT, FMT = 99990 )
DO 40 I = 1, NU
WRITE ( NOUT, FMT = 99989 )
$ ( AF(I,J), J = 1,NU )
40 CONTINUE
END IF
END IF
* Compute the structural invariants of the given system.
CALL AB08ND( EQUIL, N, M, P, A, LDA, B, LDB, C, LDC, D,
$ LDD, NU, RANK, DINFZ, NKROR, NKROL, INFZ,
$ KRONR, KRONL, AF, LDAF, BF, LDBF, TOL,
$ IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99984 ) NU
IF ( NU.GT.0 ) THEN
* Compute the invariant zeros of the given system.
* Workspace: need 8*NU.
WRITE ( NOUT, FMT = 99983 )
CALL DGEGV( 'No vectors', 'No vectors', NU, AF,
$ LDAF, BF, LDBF, ALFR, ALFI, BETA, Q,
$ LDQ, Z, LDZ, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99981 )
DO 60 I = 1, NU
IF ( ALFI(I).EQ.ZERO ) THEN
WRITE ( NOUT, FMT = 99980 )
$ ALFR(I)/BETA(I)
ELSE
WRITE ( NOUT, FMT = 99979 )
$ ALFR(I)/BETA(I),
$ ALFI(I)/BETA(I)
END IF
60 CONTINUE
WRITE ( NOUT, FMT = 99982 )
END IF
END IF
NINFZ = 0
DO 80 I = 1, DINFZ
IF ( INFZ(I).GT.0 ) THEN
NINFZ = NINFZ + INFZ(I)*I
END IF
80 CONTINUE
WRITE ( NOUT, FMT = 99978 ) NINFZ
IF ( NINFZ.GT.0 ) THEN
DO 100 I = 1, DINFZ
WRITE ( NOUT, FMT = 99977 ) INFZ(I), I
100 CONTINUE
END IF
WRITE ( NOUT, FMT = 99976 ) NKROR
IF ( NKROR.GT.0 ) WRITE ( NOUT, FMT = 99975 )
$ ( KRONR(I), I = 1,NKROR )
WRITE ( NOUT, FMT = 99974 ) NKROL
IF ( NKROL.GT.0 ) WRITE ( NOUT, FMT = 99973 )
$ ( KRONL(I), I = 1,NKROL )
END IF
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' AB08ND EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB08ND = ',I2)
99997 FORMAT (' INFO on exit from DGEGV = ',I2)
99994 FORMAT (' The left Kronecker indices of (A,C) are ',/(20(I3,2X)))
99993 FORMAT (/' The system (A,C) is completely observable ')
99992 FORMAT (/' The dimension of the observable subspace = ',I3)
99991 FORMAT (/' The output decoupling zeros are the eigenvalues of th',
$ 'e matrix AF. ')
99990 FORMAT (/' The matrix AF is ')
99989 FORMAT (20(1X,F8.4))
99988 FORMAT (//' The right Kronecker indices of (A,B) are ',/(20(I3,2X)
$ ))
99987 FORMAT (/' The system (A,B) is completely controllable ')
99986 FORMAT (/' The dimension of the controllable subspace = ',I3)
99985 FORMAT (/' The input decoupling zeros are the eigenvalues of the',
$ ' matrix AF. ')
99984 FORMAT (//' The number of finite invariant zeros = ',I3)
99983 FORMAT (/' The finite invariant zeros are ')
99982 FORMAT (/' which correspond to the generalized eigenvalues of (l',
$ 'ambda*BF - AF).')
99981 FORMAT (/' real part imag part ')
99980 FORMAT (1X,F9.4)
99979 FORMAT (1X,F9.4,6X,F9.4)
99978 FORMAT (//' The number of infinite zeros = ',I3)
99977 FORMAT ( I4,' infinite zero(s) of order ',I3)
99976 FORMAT (/' The number of right Kronecker indices = ',I3)
99975 FORMAT (/' Right Kronecker (column) indices of (A,B,C,D) are ',
$ /(20(I3,2X)))
99974 FORMAT (/' The number of left Kronecker indices = ',I3)
99973 FORMAT (/' The left Kronecker (row) indices of (A,B,C,D) are ',
$ /(20(I3,2X)))
99972 FORMAT (/' N is out of range.',/' N = ',I5)
99971 FORMAT (/' M is out of range.',/' M = ',I5)
99970 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
AB08ND EXAMPLE PROGRAM DATA 6 2 3 0.0 N 1.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0 -4.0 0.0 0.0 0.0 0.0 0.0 0.0 -1.0 0.0 0.0 0.0 0.0 0.0 0.0 3.0 0.0 -1.0 -1.0 0.0 1.0 -1.0 0.0 0.0 0.0 1.0 -1.0 -1.0 1.0 0.0 0.0 1.0 0.0 0.0 0.0 1.0 0.0 1.0 0.0 1.0 0.0 0.0 1.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0Program Results
AB08ND EXAMPLE PROGRAM RESULTS
The left Kronecker indices of (A,C) are
1 2 2
The dimension of the observable subspace = 5
The output decoupling zeros are the eigenvalues of the matrix AF.
The matrix AF is
-1.0000
The right Kronecker indices of (A,B) are
2 3
The dimension of the controllable subspace = 5
The input decoupling zeros are the eigenvalues of the matrix AF.
The matrix AF is
-4.0000
The number of finite invariant zeros = 2
The finite invariant zeros are
real part imag part
2.0000
-1.0000
which correspond to the generalized eigenvalues of (lambda*BF - AF).
The number of infinite zeros = 2
The orders of the infinite zeros are
1 1
The number of right Kronecker indices = 0
The number of left Kronecker indices = 1
The left Kronecker (row) indices of (A,B,C,D) are
2
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB08NW.html 0000664 0000000 0000000 00000050050 14560147231 0020351 0 ustar 00root root 0000000 0000000
Purpose
To extract from the system pencil
( A-lambda*I B )
S(lambda) = ( )
( C D )
a regular pencil Af-lambda*Ef which has the finite Smith zeros of
S(lambda) as generalized eigenvalues. The routine also computes
the orders of the infinite Smith zeros and determines the singular
and infinite Kronecker structure of the system pencil, i.e., the
right and left Kronecker indices, and the multiplicities of the
infinite eigenvalues.
Specification
SUBROUTINE AB08NW( EQUIL, N, M, P, A, LDA, B, LDB, C, LDC, D, LDD,
$ NFZ, NRANK, NIZ, DINFZ, NKROR, NINFE, NKROL,
$ INFZ, KRONR, INFE, KRONL, E, LDE, TOL, IWORK,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER EQUIL
INTEGER DINFZ, INFO, LDA, LDB, LDC, LDD, LDE, LDWORK,
$ M, N, NFZ, NINFE, NIZ, NKROL, NKROR, NRANK, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER INFE(*), INFZ(*), IWORK(*), KRONL(*), KRONR(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), E(LDE,*)
Arguments
Mode Parameters
EQUIL CHARACTER*1
Specifies whether the user wishes to balance the system
matrix as follows:
= 'S': Perform balancing (scaling);
= 'N': Do not perform balancing.
Input/Output Parameters
N (input) INTEGER.
The order of the square matrix A, the number of rows of
the matrix B, and number of columns of the matrix C.
N >= 0.
M (input) INTEGER.
The number of columns of the matrix B. M >= 0.
P (input) INTEGER.
The number of rows of the matrix C. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A of the system.
On exit, the leading NFZ-by-NFZ part of this array
contains the matrix Af of the reduced pencil.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input/state matrix B of the system.
On exit, this matrix does not contain useful information.
LDB INTEGER
The leading dimension of the array B. LDB >= 1, and
LDB >= MAX(1,N), if M > 0.
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C of the system.
On exit, this matrix does not contain useful information.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,P).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading P-by-M part of this array must contain the
direct transmission matrix D of the system.
LDD INTEGER
The leading dimension of the array D. LDD >= MAX(1,P).
NFZ (output) INTEGER
The number of finite zeros.
NRANK (output) INTEGER
The normal rank of the system pencil.
NIZ (output) INTEGER
The number of infinite zeros.
DINFZ (output) INTEGER
The maximal multiplicity of infinite Smith zeros.
NKROR (output) INTEGER
The number of right Kronecker indices.
NINFE (output) INTEGER
The number of elementary infinite blocks.
NKROL (output) INTEGER
The number of left Kronecker indices.
INFZ (output) INTEGER array, dimension (N+1)
The leading DINFZ elements of INFZ contain information
on the infinite elementary divisors as follows:
the system has INFZ(i) infinite elementary divisors in
the Smith form of degree i, where i = 1,2,...,DINFZ.
KRONR (output) INTEGER array, dimension (N+1)
The leading NKROR elements of this array contain the
right Kronecker (column) indices.
INFE (output) INTEGER array, dimension (N+1)
The leading NINFE elements of INFE contain the
multiplicities of infinite eigenvalues.
KRONL (output) INTEGER array, dimension (N+1)
The leading NKROL elements of this array contain the
left Kronecker (row) indices.
E (output) DOUBLE PRECISION array, dimension (LDE,N)
The leading NFZ-by-NFZ part of this array contains the
matrix Ef of the reduced pencil.
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,N).
Tolerances
TOL DOUBLE PRECISION
A tolerance used in rank decisions to determine the
effective rank, which is defined as the order of the
largest leading (or trailing) triangular submatrix in the
QR (or RQ) factorization with column (or row) pivoting
whose estimated condition number is less than 1/TOL.
If the user sets TOL <= 0, then an implicitly computed,
default tolerance TOLDEF = MAX(N+P,N+M)**2*EPS, is used
instead, where EPS is the machine precision (see LAPACK
Library routine DLAMCH). TOL < 1.
Workspace
IWORK INTEGER array, dimension (MAX(M,P))
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 1, if MAX(N,M,P) = 0; otherwise,
LDWORK >= MAX( MIN(P,M) + M + MAX(2*M,N) - 1,
MIN(P,N) + MAX(N + MAX(P,M), 3*P - 1 ) ) +
MAX(P+N,M+N)*MAX(P+N,M+N).
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine extracts from the system matrix of a state space
system, (A-lambda*I,B,C,D), a regular pencil Af-lambda*Ef, which
has the finite zeros of the system as generalized eigenvalues.
The procedure has the following main computational steps:
(a) construct the (N+P)-by-(M+N) system pencil
S(lambda) = (B A)-lambda*( 0 I );
(D C) ( 0 0 )
(b) reduce S(lambda) to S1(lambda) with the same finite zeros
and right Kronecker structure, but with D of full row rank;
(c) reduce the pencil S1(lambda) to S2(lambda) with the same
finite zeros and with D square invertible;
(d) perform a unitary transformation on the columns of
S2(lambda) = (A-lambda*I B), in order to reduce it to
( C D)
(Af-lambda*Ef X), with Y and Ef square invertible;
( 0 Y)
(e) compute the right and left Kronecker indices of the system
matrix, which, together with the multiplicities of the
finite and infinite eigenvalues, constitute the complete
set of structural invariants under strict equivalence
transformations of a linear system.
References
[1] Svaricek, F.
Computation of the Structural Invariants of Linear
Multivariable Systems with an Extended Version of the
Program ZEROS.
System & Control Letters, 6, pp. 261-266, 1985.
[2] Emami-Naeini, A. and Van Dooren, P.
Computation of Zeros of Linear Multivariable Systems.
Automatica, 18, pp. 415-430, 1982.
Numerical Aspects
The algorithm is backward stable (see [2] and [1]).Further Comments
In order to compute the finite Smith zeros of the system explicitly, a call to this routine may be followed by a call to the LAPACK Library routine DGGEV.Example
Program Text
* AB08NW EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER MPMAX
PARAMETER ( MPMAX = MAX( MMAX, PMAX ) )
INTEGER LDA, LDB, LDC, LDD, LDE, LDQ, LDZ
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX, LDE = NMAX, LDQ = 1, LDZ = 1 )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( MAX( MIN( PMAX, MMAX ) + MMAX +
$ MAX( 2*MMAX, NMAX ) - 1,
$ MIN( PMAX, NMAX ) +
$ MAX( NMAX + MPMAX,
$ 3*PMAX - 1 ) ) +
$ ( NMAX + MPMAX )**2, 8*NMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER DINFZ, I, INFO, J, M, N, NFZ, NINFE, NIZ, NKROL,
$ NKROR, NRANK, P
CHARACTER*1 EQUIL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), ALFI(NMAX), ALFR(NMAX),
$ AS(LDA,NMAX), B(LDB,MMAX), BS(LDB,MMAX),
$ BETA(NMAX), C(LDC,NMAX), CS(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK), E(LDE,NMAX),
$ Q(LDQ,1), Z(LDZ,1)
INTEGER INFE(NMAX+1), INFZ(NMAX+1), IWORK(MPMAX),
$ KRONL(NMAX+1), KRONR(NMAX+1)
* .. External Subroutines ..
EXTERNAL AB08NW, DLACPY, DGEGV
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL, EQUIL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99972 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99971 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99970 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
* Save the matrices A, B, and C.
CALL DLACPY( 'Full', N, N, A, LDA, AS, LDA )
CALL DLACPY( 'Full', N, M, B, LDB, BS, LDB )
CALL DLACPY( 'Full', P, N, C, LDC, CS, LDC )
* Check the observability and compute the ordered set of
* the observability indices (call the routine with M = 0).
CALL AB08NW( EQUIL, N, 0, P, A, LDA, B, LDB, C, LDC, D,
$ LDD, NFZ, NRANK, NIZ, DINFZ, NKROR, NINFE,
$ NKROL, INFZ, KRONR, INFE, KRONL, E, LDE,
$ TOL, IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) ( KRONL(I), I = 1,P )
IF ( NFZ.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99993 )
ELSE
WRITE ( NOUT, FMT = 99992 ) N - NFZ
WRITE ( NOUT, FMT = 99991 )
WRITE ( NOUT, FMT = 99990 )
DO 20 I = 1, NFZ
WRITE ( NOUT, FMT = 99989 )
$ ( A(I,J), J = 1,NFZ )
20 CONTINUE
END IF
END IF
* Restore the matrices A and C.
CALL DLACPY( 'Full', N, N, AS, LDA, A, LDA )
CALL DLACPY( 'Full', P, N, CS, LDC, C, LDC )
* Check the controllability and compute the ordered set of
* the controllability indices (call the routine with P = 0)
CALL AB08NW( EQUIL, N, M, 0, A, LDA, B, LDB, C, LDC, D,
$ LDD, NFZ, NRANK, NIZ, DINFZ, NKROR, NINFE,
$ NKROL, INFZ, KRONR, INFE, KRONL, E, LDE,
$ TOL, IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99988 ) ( KRONR(I), I = 1,M )
IF ( NFZ.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99987 )
ELSE
WRITE ( NOUT, FMT = 99986 ) N - NFZ
WRITE ( NOUT, FMT = 99985 )
WRITE ( NOUT, FMT = 99990 )
DO 40 I = 1, NFZ
WRITE ( NOUT, FMT = 99989 )
$ ( A(I,J), J = 1,NFZ )
40 CONTINUE
END IF
END IF
* Restore the matrices A and B.
CALL DLACPY( 'Full', N, N, AS, LDA, A, LDA )
CALL DLACPY( 'Full', N, M, BS, LDB, B, LDB )
* Compute the structural invariants of the given system.
CALL AB08NW( EQUIL, N, M, P, A, LDA, B, LDB, C, LDC, D,
$ LDD, NFZ, NRANK, NIZ, DINFZ, NKROR, NINFE,
$ NKROL, INFZ, KRONR, INFE, KRONL, E, LDE,
$ TOL, IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99984 ) NFZ
IF ( NFZ.GT.0 ) THEN
* Compute the invariant zeros of the given system.
* Workspace: need 8*NFZ.
WRITE ( NOUT, FMT = 99983 )
CALL DGEGV( 'No vectors', 'No vectors', NFZ, A,
$ LDA, E, LDE, ALFR, ALFI, BETA, Q,
$ LDQ, Z, LDZ, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99981 )
DO 60 I = 1, NFZ
IF ( ALFI(I).EQ.ZERO ) THEN
WRITE ( NOUT, FMT = 99980 )
$ ALFR(I)/BETA(I)
ELSE
WRITE ( NOUT, FMT = 99979 )
$ ALFR(I)/BETA(I),
$ ALFI(I)/BETA(I)
END IF
60 CONTINUE
WRITE ( NOUT, FMT = 99982 )
END IF
END IF
WRITE ( NOUT, FMT = 99978 ) NIZ
IF ( NIZ.GT.0 ) THEN
DO 100 I = 1, DINFZ
WRITE ( NOUT, FMT = 99977 ) INFZ(I), I
100 CONTINUE
END IF
WRITE ( NOUT, FMT = 99976 ) NKROR
IF ( NKROR.GT.0 ) WRITE ( NOUT, FMT = 99975 )
$ ( KRONR(I), I = 1,NKROR )
WRITE ( NOUT, FMT = 99974 ) NKROL
IF ( NKROL.GT.0 ) WRITE ( NOUT, FMT = 99973 )
$ ( KRONL(I), I = 1,NKROL )
IF ( NINFE.GT.0 ) WRITE ( NOUT, FMT = 99969 )
$ ( INFE(I), I = 1,NINFE )
END IF
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' AB08NW EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB08NW = ',I2)
99997 FORMAT (' INFO on exit from DGEGV = ',I2)
99994 FORMAT (' The left Kronecker indices of (A,C) are ',/(20(I3,2X)))
99993 FORMAT (/' The system (A,C) is completely observable ')
99992 FORMAT (/' The dimension of the observable subspace = ',I3)
99991 FORMAT (/' The output decoupling zeros are the eigenvalues of th',
$ 'e matrix AF. ')
99990 FORMAT (/' The matrix AF is ')
99989 FORMAT (20(1X,F8.4))
99988 FORMAT (//' The right Kronecker indices of (A,B) are ',/(20(I3,2X)
$ ))
99987 FORMAT (/' The system (A,B) is completely controllable ')
99986 FORMAT (/' The dimension of the controllable subspace = ',I3)
99985 FORMAT (/' The input decoupling zeros are the eigenvalues of the',
$ ' matrix AF. ')
99984 FORMAT (//' The number of finite invariant zeros = ',I3)
99983 FORMAT (/' The finite invariant zeros are ')
99982 FORMAT (/' which correspond to the generalized eigenvalues of (l',
$ 'ambda*EF - AF).')
99981 FORMAT (/' real part imag part ')
99980 FORMAT (1X,F9.4)
99979 FORMAT (1X,F9.4,6X,F9.4)
99978 FORMAT (//' The number of infinite zeros = ',I3)
99977 FORMAT ( I4,' infinite zero(s) of order ',I3)
99976 FORMAT (/' The number of right Kronecker indices = ',I3)
99975 FORMAT (/' Right Kronecker (column) indices of (A,B,C,D) are ',
$ /(20(I3,2X)))
99974 FORMAT (/' The number of left Kronecker indices = ',I3)
99973 FORMAT (/' The left Kronecker (row) indices of (A,B,C,D) are ',
$ /(20(I3,2X)))
99972 FORMAT (/' N is out of range.',/' N = ',I5)
99971 FORMAT (/' M is out of range.',/' M = ',I5)
99970 FORMAT (/' P is out of range.',/' P = ',I5)
99969 FORMAT (/' Multiplicities of infinite eigenvalues',/(20(I3,2X)))
END
Program Data
AB08NW EXAMPLE PROGRAM DATA 6 2 3 0.0 N 1.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0 -4.0 0.0 0.0 0.0 0.0 0.0 0.0 -1.0 0.0 0.0 0.0 0.0 0.0 0.0 3.0 0.0 -1.0 -1.0 0.0 1.0 -1.0 0.0 0.0 0.0 1.0 -1.0 -1.0 1.0 0.0 0.0 1.0 0.0 0.0 0.0 1.0 0.0 1.0 0.0 1.0 0.0 0.0 1.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0Program Results
AB08NW EXAMPLE PROGRAM RESULTS
The left Kronecker indices of (A,C) are
1 2 2
The dimension of the observable subspace = 5
The output decoupling zeros are the eigenvalues of the matrix AF.
The matrix AF is
-1.0000
The right Kronecker indices of (A,B) are
2 3
The dimension of the controllable subspace = 5
The input decoupling zeros are the eigenvalues of the matrix AF.
The matrix AF is
-4.0000
The number of finite invariant zeros = 2
The finite invariant zeros are
real part imag part
2.0000
-1.0000
which correspond to the generalized eigenvalues of (lambda*EF - AF).
The number of infinite zeros = 2
2 infinite zero(s) of order 1
The number of right Kronecker indices = 0
The number of left Kronecker indices = 1
The left Kronecker (row) indices of (A,B,C,D) are
2
Multiplicities of infinite eigenvalues
2 2
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB08NX.html 0000664 0000000 0000000 00000015073 14560147231 0020360 0 ustar 00root root 0000000 0000000
Purpose
To extract from the (N+P)-by-(M+N) system
( B A )
( D C )
an (NU+MU)-by-(M+NU) "reduced" system
( B' A')
( D' C')
having the same transmission zeros but with D' of full row rank.
Specification
SUBROUTINE AB08NX( N, M, P, RO, SIGMA, SVLMAX, ABCD, LDABCD,
$ NINFZ, INFZ, KRONL, MU, NU, NKROL, TOL, IWORK,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDABCD, LDWORK, M, MU, N, NINFZ, NKROL,
$ NU, P, RO, SIGMA
DOUBLE PRECISION SVLMAX, TOL
C .. Array Arguments ..
INTEGER INFZ(*), IWORK(*), KRONL(*)
DOUBLE PRECISION ABCD(LDABCD,*), DWORK(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The number of state variables. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
RO (input/output) INTEGER
On entry,
= P for the original system;
= MAX(P-M, 0) for the pertransposed system.
On exit, RO contains the last computed rank.
SIGMA (input/output) INTEGER
On entry,
= 0 for the original system;
= M for the pertransposed system.
On exit, SIGMA contains the last computed value sigma in
the algorithm.
SVLMAX (input) DOUBLE PRECISION
During each reduction step, the rank-revealing QR
factorization of a matrix stops when the estimated minimum
singular value is smaller than TOL * MAX(SVLMAX,EMSV),
where EMSV is the estimated maximum singular value.
SVLMAX >= 0.
ABCD (input/output) DOUBLE PRECISION array, dimension
(LDABCD,M+N)
On entry, the leading (N+P)-by-(M+N) part of this array
must contain the compound input matrix of the system.
On exit, the leading (NU+MU)-by-(M+NU) part of this array
contains the reduced compound input matrix of the system.
LDABCD INTEGER
The leading dimension of array ABCD.
LDABCD >= MAX(1,N+P).
NINFZ (input/output) INTEGER
On entry, the currently computed number of infinite zeros.
It should be initialized to zero on the first call.
NINFZ >= 0.
On exit, the number of infinite zeros.
INFZ (input/output) INTEGER array, dimension (N)
On entry, INFZ(i) must contain the current number of
infinite zeros of degree i, where i = 1,2,...,N, found in
the previous call(s) of the routine. It should be
initialized to zero on the first call.
On exit, INFZ(i) contains the number of infinite zeros of
degree i, where i = 1,2,...,N.
KRONL (input/output) INTEGER array, dimension (N+1)
On entry, this array must contain the currently computed
left Kronecker (row) indices found in the previous call(s)
of the routine. It should be initialized to zero on the
first call.
On exit, the leading NKROL elements of this array contain
the left Kronecker (row) indices.
MU (output) INTEGER
The normal rank of the transfer function matrix of the
original system.
NU (output) INTEGER
The dimension of the reduced system matrix and the number
of (finite) invariant zeros if D' is invertible.
NKROL (output) INTEGER
The number of left Kronecker indices.
Tolerances
TOL DOUBLE PRECISION
A tolerance used in rank decisions to determine the
effective rank, which is defined as the order of the
largest leading (or trailing) triangular submatrix in the
QR (or RQ) factorization with column (or row) pivoting
whose estimated condition number is less than 1/TOL.
NOTE that when SVLMAX > 0, the estimated ranks could be
less than those defined above (see SVLMAX).
Workspace
IWORK INTEGER array, dimension (MAX(M,P))
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( 1, MIN(P,M) + MAX(3*M-1,N),
MIN(P,N) + MAX(3*P-1,N+P,N+M) ).
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
References
[1] Svaricek, F.
Computation of the Structural Invariants of Linear
Multivariable Systems with an Extended Version of
the Program ZEROS.
System & Control Letters, 6, pp. 261-266, 1985.
[2] Emami-Naeini, A. and Van Dooren, P.
Computation of Zeros of Linear Multivariable Systems.
Automatica, 18, pp. 415-430, 1982.
Numerical Aspects
The algorithm is backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To extract from the (N+P)-by-(M+N) system pencil
( B A-lambda*I )
( D C )
an (NR+PR)-by-(M+NR) "reduced" system pencil,
( Br Ar-lambda*I ),
( Dr Cr )
having the same transmission zeros, but with Dr of full row rank.
Specification
SUBROUTINE AB08NY( FIRST, N, M, P, SVLMAX, ABCD, LDABCD, NINFZ,
$ NR, PR, DINFZ, NKRONL, INFZ, KRONL, TOL, IWORK,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
LOGICAL FIRST
INTEGER DINFZ, INFO, LDABCD, LDWORK, M, N, NINFZ,
$ NKRONL, NR, P, PR
DOUBLE PRECISION SVLMAX, TOL
C .. Array Arguments ..
INTEGER INFZ(*), IWORK(*), KRONL(*)
DOUBLE PRECISION ABCD(LDABCD,*), DWORK(*)
Arguments
Mode Parameters
FIRST LOGICAL
Specifies if AB08NY is called first time, or it is called
for an already reduced system, with D of full column rank,
with the last M rows in upper triangular form:
FIRST = .TRUE. : first time called;
FIRST = .FALSE. : not first time called.
Input/Output Parameters
N (input) INTEGER
The number of rows of the matrix B, the number of columns
of the matrix C, and the order of the square matrix A.
N >= 0.
M (input) INTEGER
The number of columns of the matrices B and D. M >= 0.
M <= P, if FIRST = .FALSE.
P (input) INTEGER
The number of rows of the matrices C and D. P >= 0.
SVLMAX (input) DOUBLE PRECISION
An estimate of the largest singular value of the original
matrix ABCD (for instance, the Frobenius norm of ABCD).
SVLMAX >= 0.
ABCD (input/output) DOUBLE PRECISION array, dimension
(LDABCD,M+N)
On entry, the leading (N+P)-by-(M+N) part of this array
must contain the compound matrix
( B A ),
( D C )
where A is an N-by-N matrix, B is an N-by-M matrix,
C is a P-by-N matrix, and D is a P-by-M matrix.
If FIRST = .FALSE., then D must be a full column rank
matrix, with the last M rows in an upper triangular form.
On exit, the leading (NR+PR)-by-(M+NR) part of this array
contains the reduced compound matrix
( Br Ar ),
( Dr Cr )
where Ar is an NR-by-NR matrix, Br is an NR-by-M matrix,
Cr is a PR-by-NR matrix, and Dr is a PR-by-M full row rank
left upper-trapezoidal matrix, with the first PR columns
in an upper triangular form.
LDABCD INTEGER
The leading dimension of the array ABCD.
LDABCD >= MAX(1,N+P).
NINFZ (input/output) INTEGER
On entry, the currently computed number of infinite zeros.
It should be initialized to zero on the first call.
NINFZ >= 0.
If FIRST = .FALSE., then NINFZ is not modified.
On exit, the number of infinite zeros.
NR (output) INTEGER
The order of the reduced matrix Ar; also, the number of
rows of the reduced matrix Br and the number of columns of
the reduced matrix Cr.
If Dr is invertible, NR is also the number of finite Smith
zeros.
PR (output) INTEGER
The normal rank of the transfer-function matrix of the
original system; also, the number of rows of the reduced
matrices Cr and Dr.
DINFZ (output) INTEGER
The maximal multiplicity of infinite zeros.
DINFZ = 0 if FIRST = .FALSE. .
NKRONL (output) INTEGER
The maximal dimension of left elementary Kronecker blocks.
INFZ (output) INTEGER array, dimension (N)
INFZ(i) contains the number of infinite zeros of degree i,
where i = 1,2,...,DINFZ.
INFZ is not referenced if FIRST = .FALSE. .
KRONL (output) INTEGER array, dimension (N+1)
KRONL(i) contains the number of left elementary Kronecker
blocks of dimension i-by-(i-1), where i = 1,2,...,NKRONL.
Tolerances
TOL DOUBLE PRECISION
A tolerance used in rank decisions to determine the
effective rank, which is defined as the order of the
largest leading (or trailing) triangular submatrix in the
QR (or RQ) factorization with column (or row) pivoting
whose estimated condition number is less than 1/TOL.
NOTE that when SVLMAX > 0, the estimated ranks could be
less than those defined above (see SVLMAX).
If the user sets TOL to be less than or equal to zero,
then the tolerance is taken as (N+P)*(N+M)*EPS, where EPS
is the machine precision (see LAPACK Library Routine
DLAMCH). TOL < 1.
Workspace
IWORK INTEGER array, dimension (MAX(M,P))
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 1, if MIN(P, MAX(N,M)) = 0; otherwise,
LDWORK >= MAX( MIN(P,M) + M + MAX(2*M,N) - 1,
MIN(P,N) + MAX(N + MAX( P, M), 3*P - 1 ) ).
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
References
[1] Svaricek, F.
Computation of the Structural Invariants of Linear
Multivariable Systems with an Extended Version of the
Program ZEROS.
System & Control Letters, 6, pp. 261-266, 1985.
[2] Emami-Naeini, A. and Van Dooren, P.
Computation of Zeros of Linear Multivariable Systems.
Automatica, 18, pp. 415-430, 1982.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( (P+N)*(M+N)*N ) floating point operations.Further Comments
The number of infinite zeros is computed (if FIRST = .TRUE.) as
DINFZ
NINFZ = Sum (INFZ(i)*i .
i=1
Note that each infinite zero of multiplicity k corresponds to an
infinite eigenvalue of multiplicity k+1.
The multiplicities of the infinite eigenvalues can be determined
from PR, DINFZ and INFZ(i), i = 1, ..., DINFZ, as follows:
DINFZ
- there are PR - Sum (INFZ(i)) simple infinite eigenvalues;
i=1
- there are INFZ(i) infinite eigenvalues with multiplicity i+1,
for i = 1, ..., DINFZ.
The left Kronecker indices are:
[ 0 0 ... 0 | 1 1 ... 1 | .... | NKRONL ... NKRONL ]
|<- KRONL(1) ->|<- KRONL(2) ->| |<- KRONL(NKRONL) ->|
Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To construct for a linear multivariable system described by a
state-space model (A,B,C,D) a regular pencil (A - lambda*B ) which
f f
has the invariant zeros of the system as generalized eigenvalues.
The routine also computes the orders of the infinite zeros and the
right and left Kronecker indices of the system (A,B,C,D).
Specification
SUBROUTINE AB08NZ( EQUIL, N, M, P, A, LDA, B, LDB, C, LDC, D, LDD,
$ NU, RANK, DINFZ, NKROR, NKROL, INFZ, KRONR,
$ KRONL, AF, LDAF, BF, LDBF, TOL, IWORK, DWORK,
$ ZWORK, LZWORK, INFO )
C .. Scalar Arguments ..
CHARACTER EQUIL
INTEGER DINFZ, INFO, LDA, LDAF, LDB, LDBF, LDC, LDD,
$ LZWORK, M, N, NKROL, NKROR, NU, P, RANK
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER INFZ(*), IWORK(*), KRONL(*), KRONR(*)
COMPLEX*16 A(LDA,*), AF(LDAF,*), B(LDB,*), BF(LDBF,*),
$ C(LDC,*), D(LDD,*), ZWORK(*)
DOUBLE PRECISION DWORK(*)
Arguments
Mode Parameters
EQUIL CHARACTER*1
Specifies whether the user wishes to balance the compound
matrix (see METHOD) as follows:
= 'S': Perform balancing (scaling);
= 'N': Do not perform balancing.
Input/Output Parameters
N (input) INTEGER
The number of state variables, i.e., the order of the
matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
A (input) COMPLEX*16 array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
state dynamics matrix A of the system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) COMPLEX*16 array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
input/state matrix B of the system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input) COMPLEX*16 array, dimension (LDC,N)
The leading P-by-N part of this array must contain the
state/output matrix C of the system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input) COMPLEX*16 array, dimension (LDD,M)
The leading P-by-M part of this array must contain the
direct transmission matrix D of the system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
NU (output) INTEGER
The number of (finite) invariant zeros.
RANK (output) INTEGER
The normal rank of the transfer function matrix.
DINFZ (output) INTEGER
The maximum degree of infinite elementary divisors.
NKROR (output) INTEGER
The number of right Kronecker indices.
NKROL (output) INTEGER
The number of left Kronecker indices.
INFZ (output) INTEGER array, dimension (N)
The leading DINFZ elements of INFZ contain information
on the infinite elementary divisors as follows:
the system has INFZ(i) infinite elementary divisors
of degree i, where i = 1,2,...,DINFZ.
KRONR (output) INTEGER array, dimension (MAX(N,M)+1)
The leading NKROR elements of this array contain the
right Kronecker (column) indices.
KRONL (output) INTEGER array, dimension (MAX(N,P)+1)
The leading NKROL elements of this array contain the
left Kronecker (row) indices.
AF (output) COMPLEX*16 array, dimension (LDAF,N+MIN(P,M))
The leading NU-by-NU part of this array contains the
coefficient matrix A of the reduced pencil. The remainder
f
of the leading (N+M)-by-(N+MIN(P,M)) part is used as
internal workspace.
LDAF INTEGER
The leading dimension of array AF. LDAF >= MAX(1,N+M).
BF (output) COMPLEX*16 array, dimension (LDBF,N+M)
The leading NU-by-NU part of this array contains the
coefficient matrix B of the reduced pencil. The
f
remainder of the leading (N+P)-by-(N+M) part is used as
internal workspace.
LDBF INTEGER
The leading dimension of array BF. LDBF >= MAX(1,N+P).
Tolerances
TOL DOUBLE PRECISION
A tolerance used in rank decisions to determine the
effective rank, which is defined as the order of the
largest leading (or trailing) triangular submatrix in the
QR (or RQ) factorization with column (or row) pivoting
whose estimated condition number is less than 1/TOL.
If the user sets TOL to be less than SQRT((N+P)*(N+M))*EPS
then the tolerance is taken as SQRT((N+P)*(N+M))*EPS,
where EPS is the machine precision (see LAPACK Library
Routine DLAMCH).
Workspace
IWORK INTEGER array, dimension (MAX(M,P))
DWORK DOUBLE PRECISION array, dimension (MAX(N,2*MAX(P,M)))
ZWORK DOUBLE PRECISION array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) returns the optimal value
of LZWORK.
LZWORK INTEGER
The length of the array ZWORK.
LZWORK >= MAX( 1, MIN(P,M) + MAX(3*M-1,N),
MIN(P,N) + MAX(3*P-1,N+P,N+M),
MIN(M,N) + MAX(3*M-1,N+M) ).
An upper bound is MAX(s,N) + MAX(3*s-1,N+s), with
s = MAX(M,P).
For optimum performance LZWORK should be larger.
If LZWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
ZWORK array, returns this value as the first entry of
the ZWORK array, and no error message related to LZWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine extracts from the system matrix of a state-space
system (A,B,C,D) a regular pencil A - lambda*B which has the
f f
invariant zeros of the system as generalized eigenvalues as
follows:
(a) construct the (N+P)-by-(N+M) compound matrix (B A);
(D C)
(b) reduce the above system to one with the same invariant
zeros and with D of full row rank;
(c) pertranspose the system;
(d) reduce the system to one with the same invariant zeros and
with D square invertible;
(e) perform a unitary transformation on the columns of
(A - lambda*I B) in order to reduce it to
( C D)
(A - lambda*B X)
( f f ), with Y and B square invertible;
( 0 Y) f
(f) compute the right and left Kronecker indices of the system
(A,B,C,D), which together with the orders of the infinite
zeros (determined by steps (a) - (e)) constitute the
complete set of structural invariants under strict
equivalence transformations of a linear system.
References
[1] Svaricek, F.
Computation of the Structural Invariants of Linear
Multivariable Systems with an Extended Version of
the Program ZEROS.
System & Control Letters, 6, pp. 261-266, 1985.
[2] Emami-Naeini, A. and Van Dooren, P.
Computation of Zeros of Linear Multivariable Systems.
Automatica, 18, pp. 415-430, 1982.
Numerical Aspects
The algorithm is backward stable (see [2] and [1]).Further Comments
In order to compute the invariant zeros of the system explicitly,
a call to this routine may be followed by a call to the LAPACK
Library routine ZGGEV with A = A , B = B and N = NU.
f f
If RANK = 0, the routine ZGEEV can be used (since B = I).
f
Example
Program Text
* AB08NZ EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 10, MMAX = 10, PMAX = 10 )
INTEGER MPMAX
PARAMETER ( MPMAX = MAX( MMAX, PMAX ) )
INTEGER LDA, LDB, LDC, LDD, LDAF, LDBF, LDQ, LDZ
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX, LDAF = NMAX+MPMAX,
$ LDBF = NMAX+PMAX, LDQ = 1, LDZ = 1 )
INTEGER LDWORK
PARAMETER ( LDWORK = 8*NMAX )
INTEGER LZWORK
PARAMETER ( LZWORK =
$ MAX( MIN( PMAX, MMAX ) +
$ MAX( 3*MMAX - 1, NMAX ),
$ MIN( PMAX, NMAX ) +
$ MAX( 3*PMAX, NMAX+PMAX, NMAX+MMAX ),
$ MIN( MMAX, NMAX ) +
$ MAX( 3*MMAX, NMAX+MMAX ), 1 ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER DINFZ, I, INFO, J, M, N, NINFZ, NKROL, NKROR,
$ NU, P, RANK
CHARACTER*1 EQUIL
* .. Local Arrays ..
COMPLEX*16 A(LDA,NMAX), AF(LDAF,NMAX+PMAX), ALPHA(NMAX),
$ B(LDB,MMAX), BETA(NMAX), BF(LDBF,MMAX+NMAX),
$ C(LDC,NMAX), D(LDD,MMAX), Q(LDQ,1), Z(LDZ,1),
$ ZWORK(LZWORK)
DOUBLE PRECISION DWORK(LDWORK)
INTEGER INFZ(NMAX), IWORK(MPMAX+1), KRONL(NMAX+1),
$ KRONR(NMAX+1)
* .. External Subroutines ..
EXTERNAL AB08NZ, ZGEGV
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL, EQUIL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99972 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99971 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99970 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
* Check the observability and compute the ordered set of
* the observability indices (call the routine with M = 0).
CALL AB08NZ( EQUIL, N, 0, P, A, LDA, B, LDB, C, LDC, D,
$ LDD, NU, RANK, DINFZ, NKROR, NKROL, INFZ,
$ KRONR, KRONL, AF, LDAF, BF, LDBF, TOL,
$ IWORK, DWORK, ZWORK, LZWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) ( KRONL(I), I = 1,P )
IF ( NU.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99993 )
ELSE
WRITE ( NOUT, FMT = 99992 ) N - NU
WRITE ( NOUT, FMT = 99991 )
WRITE ( NOUT, FMT = 99990 )
DO 20 I = 1, NU
WRITE ( NOUT, FMT = 99989 )
$ ( AF(I,J), J = 1,NU )
20 CONTINUE
END IF
END IF
* Check the controllability and compute the ordered set of
* the controllability indices (call the routine with P = 0)
CALL AB08NZ( EQUIL, N, M, 0, A, LDA, B, LDB, C, LDC, D,
$ LDD, NU, RANK, DINFZ, NKROR, NKROL, INFZ,
$ KRONR, KRONL, AF, LDAF, BF, LDBF, TOL,
$ IWORK, DWORK, ZWORK, LZWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99988 ) ( KRONR(I), I = 1,M )
IF ( NU.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99987 )
ELSE
WRITE ( NOUT, FMT = 99986 ) N - NU
WRITE ( NOUT, FMT = 99985 )
WRITE ( NOUT, FMT = 99990 )
DO 40 I = 1, NU
WRITE ( NOUT, FMT = 99989 )
$ ( AF(I,J), J = 1,NU )
40 CONTINUE
END IF
END IF
* Compute the structural invariants of the given system.
CALL AB08NZ( EQUIL, N, M, P, A, LDA, B, LDB, C, LDC, D,
$ LDD, NU, RANK, DINFZ, NKROR, NKROL, INFZ,
$ KRONR, KRONL, AF, LDAF, BF, LDBF, TOL,
$ IWORK, DWORK, ZWORK, LZWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99984 ) NU
IF ( NU.GT.0 ) THEN
* Compute the invariant zeros of the given system.
* Complex Workspace: need 2*NU.
* Real Workspace: need 8*NU.
WRITE ( NOUT, FMT = 99983 )
CALL ZGEGV( 'No vectors', 'No vectors', NU, AF,
$ LDAF, BF, LDBF, ALPHA, BETA, Q, LDQ,
$ Z, LDZ, ZWORK, LZWORK, DWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99981 )
DO 60 I = 1, NU
WRITE ( NOUT, FMT = 99980 ) ALPHA(I)/BETA(I)
60 CONTINUE
WRITE ( NOUT, FMT = 99982 )
END IF
END IF
NINFZ = 0
DO 80 I = 1, DINFZ
IF ( INFZ(I).GT.0 ) THEN
NINFZ = NINFZ + INFZ(I)*I
END IF
80 CONTINUE
WRITE ( NOUT, FMT = 99978 ) NINFZ
IF ( NINFZ.GT.0 ) THEN
DO 100 I = 1, DINFZ
WRITE ( NOUT, FMT = 99977 ) INFZ(I), I
100 CONTINUE
END IF
WRITE ( NOUT, FMT = 99976 ) NKROR
IF ( NKROR.GT.0 ) WRITE ( NOUT, FMT = 99975 )
$ ( KRONR(I), I = 1,NKROR )
WRITE ( NOUT, FMT = 99974 ) NKROL
IF ( NKROL.GT.0 ) WRITE ( NOUT, FMT = 99973 )
$ ( KRONL(I), I = 1,NKROL )
END IF
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' AB08NZ EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB08NZ = ',I2)
99997 FORMAT (' INFO on exit from ZGEGV = ',I2)
99994 FORMAT (' The left Kronecker indices of (A,C) are ',/(20(I3,2X)))
99993 FORMAT (/' The system (A,C) is completely observable ')
99992 FORMAT (/' The dimension of the observable subspace = ',I3)
99991 FORMAT (/' The output decoupling zeros are the eigenvalues of th',
$ 'e matrix AF. ')
99990 FORMAT (/' The matrix AF is ')
99989 FORMAT (20(1X,F9.4,SP,F9.4,S,'i '))
99988 FORMAT (//' The right Kronecker indices of (A,B) are ',/(20(I3,2X)
$ ))
99987 FORMAT (/' The system (A,B) is completely controllable ')
99986 FORMAT (/' The dimension of the controllable subspace = ',I3)
99985 FORMAT (/' The input decoupling zeros are the eigenvalues of the',
$ ' matrix AF. ')
99984 FORMAT (//' The number of finite invariant zeros = ',I3)
99983 FORMAT (/' The finite invariant zeros are ')
99982 FORMAT (/' which correspond to the generalized eigenvalues of (l',
$ 'ambda*BF - AF).')
99981 FORMAT (/' real part imag part ')
99980 FORMAT (1X,F9.4,SP,F9.4,S,'i ')
99978 FORMAT (//' The number of infinite zeros = ',I3)
99977 FORMAT ( I4,' infinite zero(s) of order ',I3)
99976 FORMAT (/' The number of right Kronecker indices = ',I3)
99975 FORMAT (/' Right Kronecker (column) indices of (A,B,C,D) are ',
$ /(20(I3,2X)))
99974 FORMAT (/' The number of left Kronecker indices = ',I3)
99973 FORMAT (/' The left Kronecker (row) indices of (A,B,C,D) are ',
$ /(20(I3,2X)))
99972 FORMAT (/' N is out of range.',/' N = ',I5)
99971 FORMAT (/' M is out of range.',/' M = ',I5)
99970 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
AB08NZ EXAMPLE PROGRAM DATA 6 2 3 0.0 N (1.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (1.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (3.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (-4.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (-1.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (3.0,0.0) (0.0,0.0) (-1.0,0.0) (-1.0,0.0) (0.0,0.0) (1.0,0.0) (-1.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (1.0,0.0) (-1.0,0.0) (-1.0,0.0) (1.0,0.0) (0.0,0.0) (0.0,0.0) (1.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (1.0,0.0) (0.0,0.0) (1.0,0.0) (0.0,0.0) (1.0,0.0) (0.0,0.0) (0.0,0.0) (1.0,0.0) (0.0,0.0) (0.0,0.0) (1.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0)Program Results
AB08NZ EXAMPLE PROGRAM RESULTS
The left Kronecker indices of (A,C) are
1 2 2
The dimension of the observable subspace = 5
The output decoupling zeros are the eigenvalues of the matrix AF.
The matrix AF is
-1.0000 +0.0000i
The right Kronecker indices of (A,B) are
2 3
The dimension of the controllable subspace = 5
The input decoupling zeros are the eigenvalues of the matrix AF.
The matrix AF is
-4.0000 +0.0000i
The number of finite invariant zeros = 2
The finite invariant zeros are
real part imag part
2.0000 +0.0000i
-1.0000 +0.0000i
which correspond to the generalized eigenvalues of (lambda*BF - AF).
The number of infinite zeros = 2
The orders of the infinite zeros are
1 1
The number of right Kronecker indices = 0
The number of left Kronecker indices = 1
The left Kronecker (row) indices of (A,B,C,D) are
2
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB09AD.html 0000664 0000000 0000000 00000034222 14560147231 0020315 0 ustar 00root root 0000000 0000000
Purpose
To compute a reduced order model (Ar,Br,Cr) for a stable original state-space representation (A,B,C) by using either the square-root or the balancing-free square-root Balance & Truncate (B & T) model reduction method.Specification
SUBROUTINE AB09AD( DICO, JOB, EQUIL, ORDSEL, N, M, P, NR, A, LDA,
$ B, LDB, C, LDC, HSV, TOL, IWORK, DWORK, LDWORK,
$ IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, EQUIL, JOB, ORDSEL
INTEGER INFO, IWARN, LDA, LDB, LDC, LDWORK, M, N, NR, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), HSV(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOB CHARACTER*1
Specifies the model reduction approach to be used
as follows:
= 'B': use the square-root Balance & Truncate method;
= 'N': use the balancing-free square-root
Balance & Truncate method.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily
equilibrate the triplet (A,B,C) as follows:
= 'S': perform equilibration (scaling);
= 'N': do not perform equilibration.
ORDSEL CHARACTER*1
Specifies the order selection method as follows:
= 'F': the resulting order NR is fixed;
= 'A': the resulting order NR is automatically determined
on basis of the given tolerance TOL.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation, i.e.
the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NR (input/output) INTEGER
On entry with ORDSEL = 'F', NR is the desired order of the
resulting reduced order system. 0 <= NR <= N.
On exit, if INFO = 0, NR is the order of the resulting
reduced order model. NR is set as follows:
if ORDSEL = 'F', NR is equal to MIN(NR,NMIN), where NR
is the desired order on entry and NMIN is the order of a
minimal realization of the given system; NMIN is
determined as the number of Hankel singular values greater
than N*EPS*HNORM(A,B,C), where EPS is the machine
precision (see LAPACK Library Routine DLAMCH) and
HNORM(A,B,C) is the Hankel norm of the system (computed
in HSV(1));
if ORDSEL = 'A', NR is equal to the number of Hankel
singular values greater than MAX(TOL,N*EPS*HNORM(A,B,C)).
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A.
On exit, if INFO = 0, the leading NR-by-NR part of this
array contains the state dynamics matrix Ar of the reduced
order system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, if INFO = 0, the leading NR-by-M part of this
array contains the input/state matrix Br of the reduced
order system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, if INFO = 0, the leading P-by-NR part of this
array contains the state/output matrix Cr of the reduced
order system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, it contains the Hankel singular values of
the original system ordered decreasingly. HSV(1) is the
Hankel norm of the system.
Tolerances
TOL DOUBLE PRECISION
If ORDSEL = 'A', TOL contains the tolerance for
determining the order of reduced system.
For model reduction, the recommended value is
TOL = c*HNORM(A,B,C), where c is a constant in the
interval [0.00001,0.001], and HNORM(A,B,C) is the
Hankel-norm of the given system (computed in HSV(1)).
For computing a minimal realization, the recommended
value is TOL = N*EPS*HNORM(A,B,C), where EPS is the
machine precision (see LAPACK Library Routine DLAMCH).
This value is used by default if TOL <= 0 on entry.
If ORDSEL = 'F', the value of TOL is ignored.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK = 0, if JOB = 'B';
LIWORK = N, if JOB = 'N'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,N*(2*N+MAX(N,M,P)+5)+N*(N+1)/2).
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: with ORDSEL = 'F', the selected order NR is greater
than the order of a minimal realization of the
given system. In this case, the resulting NR is
set automatically to a value corresponding to the
order of a minimal realization of the system.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction of A to the real Schur form failed;
= 2: the state matrix A is not stable (if DICO = 'C')
or not convergent (if DICO = 'D');
= 3: the computation of Hankel singular values failed.
Method
Let be the stable linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) (1)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system. The subroutine AB09AD determines for
the given system (1), the matrices of a reduced order system
d[z(t)] = Ar*z(t) + Br*u(t)
yr(t) = Cr*z(t) (2)
such that
HSV(NR) <= INFNORM(G-Gr) <= 2*[HSV(NR+1) + ... + HSV(N)],
where G and Gr are transfer-function matrices of the systems
(A,B,C) and (Ar,Br,Cr), respectively, and INFNORM(G) is the
infinity-norm of G.
If JOB = 'B', the square-root Balance & Truncate method of [1]
is used and, for DICO = 'C', the resulting model is balanced.
By setting TOL <= 0, the routine can be used to compute balanced
minimal state-space realizations of stable systems.
If JOB = 'N', the balancing-free square-root version of the
Balance & Truncate method [2] is used.
By setting TOL <= 0, the routine can be used to compute minimal
state-space realizations of stable systems.
References
[1] Tombs M.S. and Postlethwaite I.
Truncated balanced realization of stable, non-minimal
state-space systems.
Int. J. Control, Vol. 46, pp. 1319-1330, 1987.
[2] Varga A.
Efficient minimal realization procedure based on balancing.
Proc. of IMACS/IFAC Symp. MCTS, Lille, France, May 1991,
A. El Moudui, P. Borne, S. G. Tzafestas (Eds.),
Vol. 2, pp. 42-46.
Numerical Aspects
The implemented methods rely on accuracy enhancing square-root or
balancing-free square-root techniques.
3
The algorithms require less than 30N floating point operations.
Further Comments
NoneExample
Program Text
* AB09AD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX*( 2*NMAX + 5 +
$ MAX( NMAX, MMAX, PMAX ) ) +
$ ( NMAX*( NMAX + 1 ) )/2 )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, IWARN, J, M, N, NR, P
CHARACTER*1 DICO, EQUIL, JOB, ORDSEL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ DWORK(LDWORK), HSV(NMAX)
INTEGER IWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL AB09AD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, NR, TOL, DICO, JOB, EQUIL, ORDSEL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find a reduced ssr for (A,B,C).
CALL AB09AD( DICO, JOB, EQUIL, ORDSEL, N, M, P, NR,
$ A, LDA, B, LDB, C, LDC, HSV, TOL, IWORK,
$ DWORK, LDWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NR
WRITE ( NOUT, FMT = 99987 )
WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1,N )
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
60 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB09AD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB09AD = ',I2)
99997 FORMAT (' The order of reduced model = ',I2)
99996 FORMAT (/' The reduced state dynamics matrix Ar is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The reduced input/state matrix Br is ')
99992 FORMAT (/' The reduced state/output matrix Cr is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99987 FORMAT (/' The Hankel singular values HSV are')
END
Program Data
AB09AD EXAMPLE PROGRAM DATA (Continuous system) 7 2 3 0 1.E-1 C N N A -0.04165 0.0000 4.9200 -4.9200 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 0.0000 0.0000 0.5450 0.0000 0.0000 0.0000 -0.5450 0.0000 0.0000 0.0000 0.0000 0.0000 4.9200 -0.04165 0.0000 4.9200 0.0000 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 12.500 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000Program Results
AB09AD EXAMPLE PROGRAM RESULTS The order of reduced model = 5 The Hankel singular values HSV are 2.5139 2.0846 1.9178 0.7666 0.5473 0.0253 0.0246 The reduced state dynamics matrix Ar is 1.3451 5.0399 0.0000 0.0000 4.5315 -4.0214 -3.6604 0.0000 0.0000 -0.9056 0.0000 0.0000 0.5124 1.7910 0.0000 0.0000 0.0000 -4.2167 -2.9900 0.0000 1.2402 1.6416 0.0000 0.0000 -0.0586 The reduced input/state matrix Br is -0.3857 0.3857 -3.1753 3.1753 -0.7447 -0.7447 -3.6872 -3.6872 1.8197 -1.8197 The reduced state/output matrix Cr is -0.6704 0.1828 -0.6582 0.2222 -0.0104 0.1089 0.4867 0.0000 0.0000 0.8651 0.6704 -0.1828 -0.6582 0.2222 0.0104
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB09AX.html 0000664 0000000 0000000 00000023630 14560147231 0020342 0 ustar 00root root 0000000 0000000
Purpose
To compute a reduced order model (Ar,Br,Cr) for a stable original
state-space representation (A,B,C) by using either the square-root
or the balancing-free square-root Balance & Truncate model
reduction method. The state dynamics matrix A of the original
system is an upper quasi-triangular matrix in real Schur canonical
form. The matrices of the reduced order system are computed using
the truncation formulas:
Ar = TI * A * T , Br = TI * B , Cr = C * T .
Specification
SUBROUTINE AB09AX( DICO, JOB, ORDSEL, N, M, P, NR, A, LDA, B, LDB,
$ C, LDC, HSV, T, LDT, TI, LDTI, TOL, IWORK,
$ DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, JOB, ORDSEL
INTEGER INFO, IWARN, LDA, LDB, LDC, LDT, LDTI, LDWORK,
$ M, N, NR, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), HSV(*),
$ T(LDT,*), TI(LDTI,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOB CHARACTER*1
Specifies the model reduction approach to be used
as follows:
= 'B': use the square-root Balance & Truncate method;
= 'N': use the balancing-free square-root
Balance & Truncate method.
ORDSEL CHARACTER*1
Specifies the order selection method as follows:
= 'F': the resulting order NR is fixed;
= 'A': the resulting order NR is automatically determined
on basis of the given tolerance TOL.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation, i.e.
the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NR (input/output) INTEGER
On entry with ORDSEL = 'F', NR is the desired order of the
resulting reduced order system. 0 <= NR <= N.
On exit, if INFO = 0, NR is the order of the resulting
reduced order model. NR is set as follows:
if ORDSEL = 'F', NR is equal to MIN(NR,NMIN), where NR
is the desired order on entry and NMIN is the order of a
minimal realization of the given system; NMIN is
determined as the number of Hankel singular values greater
than N*EPS*HNORM(A,B,C), where EPS is the machine
precision (see LAPACK Library Routine DLAMCH) and
HNORM(A,B,C) is the Hankel norm of the system (computed
in HSV(1));
if ORDSEL = 'A', NR is equal to the number of Hankel
singular values greater than MAX(TOL,N*EPS*HNORM(A,B,C)).
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A in a real Schur
canonical form.
On exit, if INFO = 0, the leading NR-by-NR part of this
array contains the state dynamics matrix Ar of the
reduced order system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, if INFO = 0, the leading NR-by-M part of this
array contains the input/state matrix Br of the reduced
order system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, if INFO = 0, the leading P-by-NR part of this
array contains the state/output matrix Cr of the reduced
order system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, it contains the Hankel singular values of
the original system ordered decreasingly. HSV(1) is the
Hankel norm of the system.
T (output) DOUBLE PRECISION array, dimension (LDT,N)
If INFO = 0 and NR > 0, the leading N-by-NR part of this
array contains the right truncation matrix T.
LDT INTEGER
The leading dimension of array T. LDT >= MAX(1,N).
TI (output) DOUBLE PRECISION array, dimension (LDTI,N)
If INFO = 0 and NR > 0, the leading NR-by-N part of this
array contains the left truncation matrix TI.
LDTI INTEGER
The leading dimension of array TI. LDTI >= MAX(1,N).
Tolerances
TOL DOUBLE PRECISION
If ORDSEL = 'A', TOL contains the tolerance for
determining the order of reduced system.
For model reduction, the recommended value is
TOL = c*HNORM(A,B,C), where c is a constant in the
interval [0.00001,0.001], and HNORM(A,B,C) is the
Hankel-norm of the given system (computed in HSV(1)).
For computing a minimal realization, the recommended
value is TOL = N*EPS*HNORM(A,B,C), where EPS is the
machine precision (see LAPACK Library Routine DLAMCH).
This value is used by default if TOL <= 0 on entry.
If ORDSEL = 'F', the value of TOL is ignored.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK = 0, if JOB = 'B', or
LIWORK = N, if JOB = 'N'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,N*(MAX(N,M,P)+5) + N*(N+1)/2).
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: with ORDSEL = 'F', the selected order NR is greater
than the order of a minimal realization of the
given system. In this case, the resulting NR is
set automatically to a value corresponding to the
order of a minimal realization of the system.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the state matrix A is not stable (if DICO = 'C')
or not convergent (if DICO = 'D');
= 2: the computation of Hankel singular values failed.
Method
Let be the stable linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) (1)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system. The subroutine AB09AX determines for
the given system (1), the matrices of a reduced NR order system
d[z(t)] = Ar*z(t) + Br*u(t)
yr(t) = Cr*z(t) (2)
such that
HSV(NR) <= INFNORM(G-Gr) <= 2*[HSV(NR+1) + ... + HSV(N)],
where G and Gr are transfer-function matrices of the systems
(A,B,C) and (Ar,Br,Cr), respectively, and INFNORM(G) is the
infinity-norm of G.
If JOB = 'B', the square-root Balance & Truncate method of [1]
is used and, for DICO = 'C', the resulting model is balanced.
By setting TOL <= 0, the routine can be used to compute balanced
minimal state-space realizations of stable systems.
If JOB = 'N', the balancing-free square-root version of the
Balance & Truncate method [2] is used.
By setting TOL <= 0, the routine can be used to compute minimal
state-space realizations of stable systems.
References
[1] Tombs M.S. and Postlethwaite I.
Truncated balanced realization of stable, non-minimal
state-space systems.
Int. J. Control, Vol. 46, pp. 1319-1330, 1987.
[2] Varga A.
Efficient minimal realization procedure based on balancing.
Proc. of IMACS/IFAC Symp. MCTS, Lille, France, May 1991,
A. El Moudui, P. Borne, S. G. Tzafestas (Eds.),
Vol. 2, pp. 42-46.
Numerical Aspects
The implemented methods rely on accuracy enhancing square-root or
balancing-free square-root techniques.
3
The algorithms require less than 30N floating point operations.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute a reduced order model (Ar,Br,Cr,Dr) for a stable original state-space representation (A,B,C,D) by using either the square-root or the balancing-free square-root Singular Perturbation Approximation (SPA) model reduction method.Specification
SUBROUTINE AB09BD( DICO, JOB, EQUIL, ORDSEL, N, M, P, NR, A, LDA,
$ B, LDB, C, LDC, D, LDD, HSV, TOL1, TOL2, IWORK,
$ DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, EQUIL, JOB, ORDSEL
INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, LDWORK,
$ M, N, NR, P
DOUBLE PRECISION TOL1, TOL2
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), HSV(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOB CHARACTER*1
Specifies the model reduction approach to be used
as follows:
= 'B': use the square-root SPA method;
= 'N': use the balancing-free square-root SPA method.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily
equilibrate the triplet (A,B,C) as follows:
= 'S': perform equilibration (scaling);
= 'N': do not perform equilibration.
ORDSEL CHARACTER*1
Specifies the order selection method as follows:
= 'F': the resulting order NR is fixed;
= 'A': the resulting order NR is automatically determined
on basis of the given tolerance TOL1.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation, i.e.
the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NR (input/output) INTEGER
On entry with ORDSEL = 'F', NR is the desired order of
the resulting reduced order system. 0 <= NR <= N.
On exit, if INFO = 0, NR is the order of the resulting
reduced order model. NR is set as follows:
if ORDSEL = 'F', NR is equal to MIN(NR,NMIN), where NR
is the desired order on entry and NMIN is the order of a
minimal realization of the given system; NMIN is
determined as the number of Hankel singular values greater
than N*EPS*HNORM(A,B,C), where EPS is the machine
precision (see LAPACK Library Routine DLAMCH) and
HNORM(A,B,C) is the Hankel norm of the system (computed
in HSV(1));
if ORDSEL = 'A', NR is equal to the number of Hankel
singular values greater than MAX(TOL1,N*EPS*HNORM(A,B,C)).
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A.
On exit, if INFO = 0, the leading NR-by-NR part of this
array contains the state dynamics matrix Ar of the
reduced order system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, if INFO = 0, the leading NR-by-M part of this
array contains the input/state matrix Br of the reduced
order system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, if INFO = 0, the leading P-by-NR part of this
array contains the state/output matrix Cr of the reduced
order system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the original input/output matrix D.
On exit, if INFO = 0, the leading P-by-M part of this
array contains the input/output matrix Dr of the reduced
order system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, it contains the Hankel singular values of
the original system ordered decreasingly. HSV(1) is the
Hankel norm of the system.
Tolerances
TOL1 DOUBLE PRECISION
If ORDSEL = 'A', TOL1 contains the tolerance for
determining the order of reduced system.
For model reduction, the recommended value is
TOL1 = c*HNORM(A,B,C), where c is a constant in the
interval [0.00001,0.001], and HNORM(A,B,C) is the
Hankel-norm of the given system (computed in HSV(1)).
For computing a minimal realization, the recommended
value is TOL1 = N*EPS*HNORM(A,B,C), where EPS is the
machine precision (see LAPACK Library Routine DLAMCH).
This value is used by default if TOL1 <= 0 on entry.
If ORDSEL = 'F', the value of TOL1 is ignored.
TOL2 DOUBLE PRECISION
The tolerance for determining the order of a minimal
realization of the given system. The recommended value is
TOL2 = N*EPS*HNORM(A,B,C). This value is used by default
if TOL2 <= 0 on entry.
If TOL2 > 0, then TOL2 <= TOL1.
Workspace
IWORK INTEGER array, dimension (MAX(1,2*N))
On exit with INFO = 0, IWORK(1) contains the order of the
minimal realization of the system.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,N*(2*N+MAX(N,M,P)+5)+N*(N+1)/2).
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: with ORDSEL = 'F', the selected order NR is greater
than the order of a minimal realization of the
given system. In this case, the resulting NR is
set automatically to a value corresponding to the
order of a minimal realization of the system.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction of A to the real Schur form failed;
= 2: the state matrix A is not stable (if DICO = 'C')
or not convergent (if DICO = 'D');
= 3: the computation of Hankel singular values failed.
Method
Let be the stable linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t) (1)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system. The subroutine AB09BD determines for
the given system (1), the matrices of a reduced order system
d[z(t)] = Ar*z(t) + Br*u(t)
yr(t) = Cr*z(t) + Dr*u(t) (2)
such that
HSV(NR) <= INFNORM(G-Gr) <= 2*[HSV(NR+1) + ... + HSV(N)],
where G and Gr are transfer-function matrices of the systems
(A,B,C,D) and (Ar,Br,Cr,Dr), respectively, and INFNORM(G) is the
infinity-norm of G.
If JOB = 'B', the balancing-based square-root SPA method of [1]
is used and the resulting model is balanced.
If JOB = 'N', the balancing-free square-root SPA method of [2]
is used.
By setting TOL1 = TOL2, the routine can be used to compute
Balance & Truncate approximations.
References
[1] Liu Y. and Anderson B.D.O.
Singular Perturbation Approximation of Balanced Systems,
Int. J. Control, Vol. 50, pp. 1379-1405, 1989.
[2] Varga A.
Balancing-free square-root algorithm for computing singular
perturbation approximations.
Proc. 30-th IEEE CDC, Brighton, Dec. 11-13, 1991,
Vol. 2, pp. 1062-1065.
Numerical Aspects
The implemented methods rely on accuracy enhancing square-root or
balancing-free square-root techniques.
3
The algorithms require less than 30N floating point operations.
Further Comments
NoneExample
Program Text
* AB09BD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDD
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = 2*NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX*( 2*NMAX + 5 +
$ MAX( NMAX, MMAX, PMAX ) ) +
$ ( NMAX*( NMAX + 1 ) )/2 )
* .. Local Scalars ..
DOUBLE PRECISION TOL1, TOL2
INTEGER I, INFO, IWARN, J, M, N, NR, P
CHARACTER*1 DICO, EQUIL, JOB, ORDSEL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK), HSV(NMAX)
INTEGER IWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL AB09BD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, NR, TOL1, TOL2, DICO, JOB, EQUIL,
$ ORDSEL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
* Find a reduced ssr for (A,B,C).
CALL AB09BD( DICO, JOB, EQUIL, ORDSEL, N, M, P, NR,
$ A, LDA, B, LDB, C, LDC, D, LDD, HSV, TOL1,
$ TOL2, IWORK, DWORK, LDWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NR
WRITE ( NOUT, FMT = 99987 )
WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1,N )
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
20 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
60 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 70 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1,M )
70 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB09BD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB09BD = ',I2)
99997 FORMAT (' The order of reduced model = ',I2)
99996 FORMAT (/' The reduced state dynamics matrix Ar is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The reduced input/state matrix Br is ')
99992 FORMAT (/' The reduced state/output matrix Cr is ')
99991 FORMAT (/' The reduced input/output matrix Dr is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99987 FORMAT (/' The Hankel singular values are')
END
Program Data
AB09BD EXAMPLE PROGRAM DATA (Continuous system) 7 2 3 0 1.E-1 1.E-14 C N N A -0.04165 0.0000 4.9200 -4.9200 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 0.0000 0.0000 0.5450 0.0000 0.0000 0.0000 -0.5450 0.0000 0.0000 0.0000 0.0000 0.0000 4.9200 -0.04165 0.0000 4.9200 0.0000 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 12.500 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000Program Results
AB09BD EXAMPLE PROGRAM RESULTS The order of reduced model = 5 The Hankel singular values are 2.5139 2.0846 1.9178 0.7666 0.5473 0.0253 0.0246 The reduced state dynamics matrix Ar is 1.3960 5.1248 0.0000 0.0000 4.4331 -4.1411 -3.8605 0.0000 0.0000 -0.6738 0.0000 0.0000 0.5847 1.9230 0.0000 0.0000 0.0000 -4.3823 -3.2922 0.0000 1.3261 1.7851 0.0000 0.0000 -0.2249 The reduced input/state matrix Br is -0.2901 0.2901 -3.4004 3.4004 -0.6379 -0.6379 -3.9315 -3.9315 1.9813 -1.9813 The reduced state/output matrix Cr is -0.6570 0.2053 -0.6416 0.2526 -0.0364 0.1094 0.4875 0.0000 0.0000 0.8641 0.6570 -0.2053 -0.6416 0.2526 0.0364 The reduced input/output matrix Dr is 0.0498 -0.0007 0.0010 -0.0010 -0.0007 0.0498
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB09BX.html 0000664 0000000 0000000 00000025142 14560147231 0020343 0 ustar 00root root 0000000 0000000
Purpose
To compute a reduced order model (Ar,Br,Cr,Dr) for a stable
original state-space representation (A,B,C,D) by using either the
square-root or the balancing-free square-root
Singular Perturbation Approximation (SPA) model reduction method.
The state dynamics matrix A of the original system is an upper
quasi-triangular matrix in real Schur canonical form. The matrices
of a minimal realization are computed using the truncation
formulas:
Am = TI * A * T , Bm = TI * B , Cm = C * T . (1)
Am, Bm, Cm and D serve further for computing the SPA of the given
system.
Specification
SUBROUTINE AB09BX( DICO, JOB, ORDSEL, N, M, P, NR, A, LDA, B, LDB,
$ C, LDC, D, LDD, HSV, T, LDT, TI, LDTI, TOL1,
$ TOL2, IWORK, DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, JOB, ORDSEL
INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, LDT, LDTI,
$ LDWORK, M, N, NR, P
DOUBLE PRECISION TOL1, TOL2
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), HSV(*), T(LDT,*), TI(LDTI,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOB CHARACTER*1
Specifies the model reduction approach to be used
as follows:
= 'B': use the square-root SPA method;
= 'N': use the balancing-free square-root SPA method.
ORDSEL CHARACTER*1
Specifies the order selection method as follows:
= 'F': the resulting order NR is fixed;
= 'A': the resulting order NR is automatically determined
on basis of the given tolerance TOL1.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation, i.e.
the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NR (input/output) INTEGER
On entry with ORDSEL = 'F', NR is the desired order of
the resulting reduced order system. 0 <= NR <= N.
On exit, if INFO = 0, NR is the order of the resulting
reduced order model. NR is set as follows:
if ORDSEL = 'F', NR is equal to MIN(NR,NMIN), where NR
is the desired order on entry and NMIN is the order of a
minimal realization of the given system; NMIN is
determined as the number of Hankel singular values greater
than N*EPS*HNORM(A,B,C), where EPS is the machine
precision (see LAPACK Library Routine DLAMCH) and
HNORM(A,B,C) is the Hankel norm of the system (computed
in HSV(1));
if ORDSEL = 'A', NR is equal to the number of Hankel
singular values greater than MAX(TOL1,N*EPS*HNORM(A,B,C)).
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A in a real Schur
canonical form.
On exit, if INFO = 0, the leading NR-by-NR part of this
array contains the state dynamics matrix Ar of the
reduced order system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, if INFO = 0, the leading NR-by-M part of this
array contains the input/state matrix Br of the reduced
order system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, if INFO = 0, the leading P-by-NR part of this
array contains the state/output matrix Cr of the reduced
order system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the original input/output matrix D.
On exit, if INFO = 0, the leading P-by-M part of this
array contains the input/output matrix Dr of the reduced
order system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, it contains the Hankel singular values of
the original system ordered decreasingly. HSV(1) is the
Hankel norm of the system.
T (output) DOUBLE PRECISION array, dimension (LDT,N)
If INFO = 0 and NR > 0, the leading N-by-NR part of this
array contains the right truncation matrix T in (1).
LDT INTEGER
The leading dimension of array T. LDT >= MAX(1,N).
TI (output) DOUBLE PRECISION array, dimension (LDTI,N)
If INFO = 0 and NR > 0, the leading NR-by-N part of this
array contains the left truncation matrix TI in (1).
LDTI INTEGER
The leading dimension of array TI. LDTI >= MAX(1,N).
Tolerances
TOL1 DOUBLE PRECISION
If ORDSEL = 'A', TOL1 contains the tolerance for
determining the order of reduced system.
For model reduction, the recommended value is
TOL1 = c*HNORM(A,B,C), where c is a constant in the
interval [0.00001,0.001], and HNORM(A,B,C) is the
Hankel-norm of the given system (computed in HSV(1)).
For computing a minimal realization, the recommended
value is TOL1 = N*EPS*HNORM(A,B,C), where EPS is the
machine precision (see LAPACK Library Routine DLAMCH).
This value is used by default if TOL1 <= 0 on entry.
If ORDSEL = 'F', the value of TOL1 is ignored.
TOL2 DOUBLE PRECISION
The tolerance for determining the order of a minimal
realization of the given system. The recommended value is
TOL2 = N*EPS*HNORM(A,B,C). This value is used by default
if TOL2 <= 0 on entry.
If TOL2 > 0, then TOL2 <= TOL1.
Workspace
IWORK INTEGER array, dimension (MAX(1,2*N))
On exit with INFO = 0, IWORK(1) contains the order of the
minimal realization of the system.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,N*(MAX(N,M,P)+5) + N*(N+1)/2).
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: with ORDSEL = 'F', the selected order NR is greater
than the order of a minimal realization of the
given system. In this case, the resulting NR is
set automatically to a value corresponding to the
order of a minimal realization of the system.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the state matrix A is not stable (if DICO = 'C')
or not convergent (if DICO = 'D');
= 2: the computation of Hankel singular values failed.
Method
Let be the stable linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t) (2)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system. The subroutine AB09BX determines for
the given system (1), the matrices of a reduced NR order system
d[z(t)] = Ar*z(t) + Br*u(t)
yr(t) = Cr*z(t) + Dr*u(t) (3)
such that
HSV(NR) <= INFNORM(G-Gr) <= 2*[HSV(NR+1) + ... + HSV(N)],
where G and Gr are transfer-function matrices of the systems
(A,B,C,D) and (Ar,Br,Cr,Dr), respectively, and INFNORM(G) is the
infinity-norm of G.
If JOB = 'B', the balancing-based square-root SPA method of [1]
is used and the resulting model is balanced.
If JOB = 'N', the balancing-free square-root SPA method of [2]
is used.
By setting TOL1 = TOL2, the routine can be also used to compute
Balance & Truncate approximations.
References
[1] Liu Y. and Anderson B.D.O.
Singular Perturbation Approximation of Balanced Systems,
Int. J. Control, Vol. 50, pp. 1379-1405, 1989.
[2] Varga A.
Balancing-free square-root algorithm for computing singular
perturbation approximations.
Proc. 30-th IEEE CDC, Brighton, Dec. 11-13, 1991,
Vol. 2, pp. 1062-1065.
Numerical Aspects
The implemented methods rely on accuracy enhancing square-root or
balancing-free square-root techniques.
3
The algorithms require less than 30N floating point operations.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute a reduced order model (Ar,Br,Cr,Dr) for a stable original state-space representation (A,B,C,D) by using the optimal Hankel-norm approximation method in conjunction with square-root balancing.Specification
SUBROUTINE AB09CD( DICO, EQUIL, ORDSEL, N, M, P, NR, A, LDA, B,
$ LDB, C, LDC, D, LDD, HSV, TOL1, TOL2, IWORK,
$ DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, EQUIL, ORDSEL
INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, LDWORK,
$ M, N, NR, P
DOUBLE PRECISION TOL1, TOL2
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), HSV(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily
equilibrate the triplet (A,B,C) as follows:
= 'S': perform equilibration (scaling);
= 'N': do not perform equilibration.
ORDSEL CHARACTER*1
Specifies the order selection method as follows:
= 'F': the resulting order NR is fixed;
= 'A': the resulting order NR is automatically determined
on basis of the given tolerance TOL1.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation, i.e.
the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NR (input/output) INTEGER
On entry with ORDSEL = 'F', NR is the desired order of
the resulting reduced order system. 0 <= NR <= N.
On exit, if INFO = 0, NR is the order of the resulting
reduced order model. NR is set as follows:
if ORDSEL = 'F', NR is equal to MIN(MAX(0,NR-KR+1),NMIN),
where KR is the multiplicity of the Hankel singular value
HSV(NR+1), NR is the desired order on entry, and NMIN is
the order of a minimal realization of the given system;
NMIN is determined as the number of Hankel singular values
greater than N*EPS*HNORM(A,B,C), where EPS is the machine
precision (see LAPACK Library Routine DLAMCH) and
HNORM(A,B,C) is the Hankel norm of the system (computed
in HSV(1));
if ORDSEL = 'A', NR is equal to the number of Hankel
singular values greater than MAX(TOL1,N*EPS*HNORM(A,B,C)).
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A.
On exit, if INFO = 0, the leading NR-by-NR part of this
array contains the state dynamics matrix Ar of the
reduced order system in a real Schur form.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, if INFO = 0, the leading NR-by-M part of this
array contains the input/state matrix Br of the reduced
order system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, if INFO = 0, the leading P-by-NR part of this
array contains the state/output matrix Cr of the reduced
order system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the original input/output matrix D.
On exit, if INFO = 0, the leading P-by-M part of this
array contains the input/output matrix Dr of the reduced
order system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, it contains the Hankel singular values of
the original system ordered decreasingly. HSV(1) is the
Hankel norm of the system.
Tolerances
TOL1 DOUBLE PRECISION
If ORDSEL = 'A', TOL1 contains the tolerance for
determining the order of reduced system.
For model reduction, the recommended value is
TOL1 = c*HNORM(A,B,C), where c is a constant in the
interval [0.00001,0.001], and HNORM(A,B,C) is the
Hankel-norm of the given system (computed in HSV(1)).
For computing a minimal realization, the recommended
value is TOL1 = N*EPS*HNORM(A,B,C), where EPS is the
machine precision (see LAPACK Library Routine DLAMCH).
This value is used by default if TOL1 <= 0 on entry.
If ORDSEL = 'F', the value of TOL1 is ignored.
TOL2 DOUBLE PRECISION
The tolerance for determining the order of a minimal
realization of the given system. The recommended value is
TOL2 = N*EPS*HNORM(A,B,C). This value is used by default
if TOL2 <= 0 on entry.
If TOL2 > 0, then TOL2 <= TOL1.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK = MAX(1,M), if DICO = 'C';
LIWORK = MAX(1,N,M), if DICO = 'D'.
On exit, if INFO = 0, IWORK(1) contains NMIN, the order of
the computed minimal realization.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( LDW1, LDW2 ), where
LDW1 = N*(2*N+MAX(N,M,P)+5) + N*(N+1)/2,
LDW2 = N*(M+P+2) + 2*M*P + MIN(N,M) +
MAX( 3*M+1, MIN(N,M)+P ).
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: with ORDSEL = 'F', the selected order NR is greater
than the order of a minimal realization of the
given system. In this case, the resulting NR is set
automatically to a value corresponding to the order
of a minimal realization of the system.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction of A to the real Schur form failed;
= 2: the state matrix A is not stable (if DICO = 'C')
or not convergent (if DICO = 'D');
= 3: the computation of Hankel singular values failed;
= 4: the computation of stable projection failed;
= 5: the order of computed stable projection differs
from the order of Hankel-norm approximation.
Method
Let be the stable linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t) (1)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system. The subroutine AB09CD determines for
the given system (1), the matrices of a reduced order system
d[z(t)] = Ar*z(t) + Br*u(t)
yr(t) = Cr*z(t) + Dr*u(t) (2)
such that
HSV(NR) <= INFNORM(G-Gr) <= 2*[HSV(NR+1) + ... + HSV(N)],
where G and Gr are transfer-function matrices of the systems
(A,B,C,D) and (Ar,Br,Cr,Dr), respectively, and INFNORM(G) is the
infinity-norm of G.
The optimal Hankel-norm approximation method of [1], based on the
square-root balancing projection formulas of [2], is employed.
References
[1] Glover, K.
All optimal Hankel norm approximation of linear
multivariable systems and their L-infinity error bounds.
Int. J. Control, Vol. 36, pp. 1145-1193, 1984.
[2] Tombs M.S. and Postlethwaite I.
Truncated balanced realization of stable, non-minimal
state-space systems.
Int. J. Control, Vol. 46, pp. 1319-1330, 1987.
Numerical Aspects
The implemented methods rely on an accuracy enhancing square-root
technique.
3
The algorithms require less than 30N floating point operations.
Further Comments
NoneExample
Program Text
* AB09CD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDD
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MAX( NMAX, MMAX ) )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( NMAX*( 2*NMAX +
$ MAX( NMAX, MMAX, PMAX ) + 5 ) +
$ ( NMAX*( NMAX + 1 ) )/2,
$ NMAX*( MMAX + PMAX + 2 ) +
$ 2*MMAX*PMAX +
$ MIN( NMAX, MMAX ) + MAX( 3*MMAX + 1,
$ MIN( NMAX, MMAX ) + PMAX ) ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL1, TOL2
INTEGER I, INFO, IWARN, J, M, N, NR, P
CHARACTER*1 DICO, EQUIL, ORDSEL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK), HSV(NMAX)
INTEGER IWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL AB09CD
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, NR, TOL1, TOL2, DICO, EQUIL, ORDSEL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
* Find a reduced ssr for (A,B,C).
CALL AB09CD( DICO, EQUIL, ORDSEL, N, M, P, NR,
$ A, LDA, B, LDB, C, LDC, D, LDD, HSV, TOL1,
$ TOL2, IWORK, DWORK, LDWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NR
WRITE ( NOUT, FMT = 99987 )
WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1,N )
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
20 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
60 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 70 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1,M )
70 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB09CD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB09CD = ',I2)
99997 FORMAT (' The order of reduced model = ',I2)
99996 FORMAT (/' The reduced state dynamics matrix Ar is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The reduced input/state matrix Br is ')
99992 FORMAT (/' The reduced state/output matrix Cr is ')
99991 FORMAT (/' The reduced input/output matrix Dr is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99987 FORMAT (/' The Hankel singular values are')
END
Program Data
AB09CD EXAMPLE PROGRAM DATA (Continuous system) 7 2 3 0 1.E-1 1.E-14 C N A -0.04165 0.0000 4.9200 -4.9200 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 0.0000 0.0000 0.5450 0.0000 0.0000 0.0000 -0.5450 0.0000 0.0000 0.0000 0.0000 0.0000 4.9200 -0.04165 0.0000 4.9200 0.0000 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 12.500 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000Program Results
AB09CD EXAMPLE PROGRAM RESULTS The order of reduced model = 5 The Hankel singular values are 2.5139 2.0846 1.9178 0.7666 0.5473 0.0253 0.0246 The reduced state dynamics matrix Ar is -0.5038 -5.3070 -3.2250 0.0000 0.0000 1.8355 -0.5038 -2.6289 0.0000 0.0000 0.0000 0.0000 -1.5171 0.0000 0.0000 0.0000 0.0000 0.0000 -1.2925 -9.0718 0.0000 0.0000 0.0000 0.5047 -1.2925 The reduced input/state matrix Br is -1.5343 1.5343 -0.3614 0.3614 -1.1096 1.1096 -4.5325 -4.5325 -0.7396 -0.7396 The reduced state/output matrix Cr is 1.8971 -0.3055 -2.1124 0.4421 -2.1023 -0.0394 1.1112 -0.3119 0.0000 0.0000 -1.8971 0.3055 2.1124 0.4421 -2.1023 The reduced input/output matrix Dr is 0.0126 -0.0126 0.0005 -0.0005 -0.0126 0.0126
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB09CX.html 0000664 0000000 0000000 00000023523 14560147231 0020345 0 ustar 00root root 0000000 0000000
Purpose
To compute a reduced order model (Ar,Br,Cr,Dr) for a stable original state-space representation (A,B,C,D) by using the optimal Hankel-norm approximation method in conjunction with square-root balancing. The state dynamics matrix A of the original system is an upper quasi-triangular matrix in real Schur canonical form.Specification
SUBROUTINE AB09CX( DICO, ORDSEL, N, M, P, NR, A, LDA, B, LDB,
$ C, LDC, D, LDD, HSV, TOL1, TOL2, IWORK,
$ DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, ORDSEL
INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, LDWORK,
$ M, N, NR, P
DOUBLE PRECISION TOL1, TOL2
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), HSV(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
ORDSEL CHARACTER*1
Specifies the order selection method as follows:
= 'F': the resulting order NR is fixed;
= 'A': the resulting order NR is automatically determined
on basis of the given tolerance TOL1.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation, i.e.
the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NR (input/output) INTEGER
On entry with ORDSEL = 'F', NR is the desired order of
the resulting reduced order system. 0 <= NR <= N.
On exit, if INFO = 0, NR is the order of the resulting
reduced order model. NR is set as follows:
if ORDSEL = 'F', NR is equal to MIN(MAX(0,NR-KR+1),NMIN),
where KR is the multiplicity of the Hankel singular value
HSV(NR+1), NR is the desired order on entry, and NMIN is
the order of a minimal realization of the given system;
NMIN is determined as the number of Hankel singular values
greater than N*EPS*HNORM(A,B,C), where EPS is the machine
precision (see LAPACK Library Routine DLAMCH) and
HNORM(A,B,C) is the Hankel norm of the system (computed
in HSV(1));
if ORDSEL = 'A', NR is equal to the number of Hankel
singular values greater than MAX(TOL1,N*EPS*HNORM(A,B,C)).
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A in a real Schur
canonical form.
On exit, if INFO = 0, the leading NR-by-NR part of this
array contains the state dynamics matrix Ar of the
reduced order system in a real Schur form.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, if INFO = 0, the leading NR-by-M part of this
array contains the input/state matrix Br of the reduced
order system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, if INFO = 0, the leading P-by-NR part of this
array contains the state/output matrix Cr of the reduced
order system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the original input/output matrix D.
On exit, if INFO = 0, the leading P-by-M part of this
array contains the input/output matrix Dr of the reduced
order system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, it contains the Hankel singular values of
the original system ordered decreasingly. HSV(1) is the
Hankel norm of the system.
Tolerances
TOL1 DOUBLE PRECISION
If ORDSEL = 'A', TOL1 contains the tolerance for
determining the order of reduced system.
For model reduction, the recommended value is
TOL1 = c*HNORM(A,B,C), where c is a constant in the
interval [0.00001,0.001], and HNORM(A,B,C) is the
Hankel-norm of the given system (computed in HSV(1)).
For computing a minimal realization, the recommended
value is TOL1 = N*EPS*HNORM(A,B,C), where EPS is the
machine precision (see LAPACK Library Routine DLAMCH).
This value is used by default if TOL1 <= 0 on entry.
If ORDSEL = 'F', the value of TOL1 is ignored.
TOL2 DOUBLE PRECISION
The tolerance for determining the order of a minimal
realization of the given system. The recommended value is
TOL2 = N*EPS*HNORM(A,B,C). This value is used by default
if TOL2 <= 0 on entry.
If TOL2 > 0, then TOL2 <= TOL1.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK = MAX(1,M), if DICO = 'C';
LIWORK = MAX(1,N,M), if DICO = 'D'.
On exit, if INFO = 0, IWORK(1) contains NMIN, the order of
the computed minimal realization.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( LDW1,LDW2 ), where
LDW1 = N*(2*N+MAX(N,M,P)+5) + N*(N+1)/2,
LDW2 = N*(M+P+2) + 2*M*P + MIN(N,M) +
MAX( 3*M+1, MIN(N,M)+P ).
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: with ORDSEL = 'F', the selected order NR is greater
than the order of a minimal realization of the
given system. In this case, the resulting NR is set
automatically to a value corresponding to the order
of a minimal realization of the system.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the state matrix A is not stable (if DICO = 'C')
or not convergent (if DICO = 'D');
= 2: the computation of Hankel singular values failed;
= 3: the computation of stable projection failed;
= 4: the order of computed stable projection differs
from the order of Hankel-norm approximation.
Method
Let be the stable linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t) (1)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system. The subroutine AB09CX determines for
the given system (1), the matrices of a reduced order system
d[z(t)] = Ar*z(t) + Br*u(t)
yr(t) = Cr*z(t) + Dr*u(t) (2)
such that
HSV(NR) <= INFNORM(G-Gr) <= 2*[HSV(NR+1) + ... + HSV(N)],
where G and Gr are transfer-function matrices of the systems
(A,B,C,D) and (Ar,Br,Cr,Dr), respectively, and INFNORM(G) is the
infinity-norm of G.
The optimal Hankel-norm approximation method of [1], based on the
square-root balancing projection formulas of [2], is employed.
References
[1] Glover, K.
All optimal Hankel norm approximation of linear
multivariable systems and their L-infinity error bounds.
Int. J. Control, Vol. 36, pp. 1145-1193, 1984.
[2] Tombs M.S. and Postlethwaite I.
Truncated balanced realization of stable, non-minimal
state-space systems.
Int. J. Control, Vol. 46, pp. 1319-1330, 1987.
Numerical Aspects
The implemented methods rely on an accuracy enhancing square-root
technique.
3
The algorithms require less than 30N floating point operations.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute a reduced order model by using singular perturbation approximation formulas.Specification
SUBROUTINE AB09DD( DICO, N, M, P, NR, A, LDA, B, LDB, C, LDC,
$ D, LDD, RCOND, IWORK, DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO
INTEGER INFO, LDA, LDB, LDC, LDD, M, N, NR, P
DOUBLE PRECISION RCOND
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), DWORK(*)
INTEGER IWORK(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
Input/Output Parameters
N (input) INTEGER
The dimension of the state vector, i.e. the order of the
matrix A; also the number of rows of matrix B and the
number of columns of the matrix C. N >= 0.
M (input) INTEGER
The dimension of input vector, i.e. the number of columns
of matrices B and D. M >= 0.
P (input) INTEGER
The dimension of output vector, i.e. the number of rows of
matrices C and D. P >= 0.
NR (input) INTEGER
The order of the reduced order system. N >= NR >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix of the original system.
On exit, the leading NR-by-NR part of this array contains
the state dynamics matrix Ar of the reduced order system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input/state matrix of the original system.
On exit, the leading NR-by-M part of this array contains
the input/state matrix Br of the reduced order system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix of the original system.
On exit, the leading P-by-NR part of this array contains
the state/output matrix Cr of the reduced order system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the input/output matrix of the original system.
On exit, the leading P-by-M part of this array contains
the input/output matrix Dr of the reduced order system.
If NR = 0 and the given system is stable, then D contains
the steady state gain of the system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
RCOND (output) DOUBLE PRECISION
The reciprocal condition number of the matrix A22-g*I
(see METHOD).
Workspace
IWORK INTEGER array, dimension (2*(N-NR)) DWORK DOUBLE PRECISION array, dimension (4*(N-NR))Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the matrix A22-g*I (see METHOD) is numerically
singular.
Method
Given the system (A,B,C,D), partition the system matrices as
( A11 A12 ) ( B1 )
A = ( ) , B = ( ) , C = ( C1 C2 ),
( A21 A22 ) ( B2 )
where A11 is NR-by-NR, B1 is NR-by-M, C1 is P-by-NR, and the other
submatrices have appropriate dimensions.
The matrices of the reduced order system (Ar,Br,Cr,Dr) are
computed according to the following residualization formulas:
-1 -1
Ar = A11 + A12*(g*I-A22) *A21 , Br = B1 + A12*(g*I-A22) *B2
-1 -1
Cr = C1 + C2*(g*I-A22) *A21 , Dr = D + C2*(g*I-A22) *B2
where g = 0 if DICO = 'C' and g = 1 if DICO = 'D'.
Further Comments
NoneExample
Program Text
* AB09DD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDD
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = 2*NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 4*NMAX )
* .. Local Scalars ..
DOUBLE PRECISION RCOND
INTEGER I, INFO, J, M, N, NR, P
CHARACTER*1 DICO
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK)
INTEGER IWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL AB09DD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, NR, DICO
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
* Find a reduced ssr for (A,B,C).
CALL AB09DD( DICO, N, M, P, NR, A, LDA, B, LDB, C, LDC,
$ D, LDD, RCOND, IWORK, DWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) RCOND
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
20 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
60 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 70 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1,M )
70 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB09DD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB09DD = ',I2)
99997 FORMAT (' The computed reciprocal condition number = ',1PD12.5)
99996 FORMAT (/' The reduced state dynamics matrix Ar is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The reduced input/state matrix Br is ')
99992 FORMAT (/' The reduced state/output matrix Cr is ')
99991 FORMAT (/' The reduced input/output matrix Dr is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
AB09DD EXAMPLE PROGRAM DATA (Continuous system)
7 2 3 5 C
-0.04165 4.9200 -4.9200 0 0 0 0
0 -3.3300 0 0 0 3.3300 0
0.5450 0 0 -0.5450 0 0 0
0 0 4.9200 -0.04165 4.9200 0 0
0 0 0 0 -3.3300 0 3.3300
-5.2100 0 0 0 0 -12.5000 0
0 0 0 -5.2100 0 0 -12.5000
0 0
0 0
0 0
0 0
0 0
12.5000 0
0 12.5000
1 0 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 1 0 0 0
0 0
0 0
0 0
Program Results
AB09DD EXAMPLE PROGRAM RESULTS The computed reciprocal condition number = 1.00000D+00 The reduced state dynamics matrix Ar is -0.0416 4.9200 -4.9200 0.0000 0.0000 -1.3879 -3.3300 0.0000 0.0000 0.0000 0.5450 0.0000 0.0000 -0.5450 0.0000 0.0000 0.0000 4.9200 -0.0416 4.9200 0.0000 0.0000 0.0000 -1.3879 -3.3300 The reduced input/state matrix Br is 0.0000 0.0000 3.3300 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 The reduced state/output matrix Cr is 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 The reduced input/output matrix Dr is 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB09ED.html 0000664 0000000 0000000 00000044643 14560147231 0020331 0 ustar 00root root 0000000 0000000
Purpose
To compute a reduced order model (Ar,Br,Cr,Dr) for an original state-space representation (A,B,C,D) by using the optimal Hankel-norm approximation method in conjunction with square-root balancing for the ALPHA-stable part of the system.Specification
SUBROUTINE AB09ED( DICO, EQUIL, ORDSEL, N, M, P, NR, ALPHA,
$ A, LDA, B, LDB, C, LDC, D, LDD, NS, HSV, TOL1,
$ TOL2, IWORK, DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, EQUIL, ORDSEL
INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, LDWORK,
$ M, N, NR, NS, P
DOUBLE PRECISION ALPHA, TOL1, TOL2
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), HSV(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily
equilibrate the triplet (A,B,C) as follows:
= 'S': perform equilibration (scaling);
= 'N': do not perform equilibration.
ORDSEL CHARACTER*1
Specifies the order selection method as follows:
= 'F': the resulting order NR is fixed;
= 'A': the resulting order NR is automatically determined
on basis of the given tolerance TOL1.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation, i.e.
the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NR (input/output) INTEGER
On entry with ORDSEL = 'F', NR is the desired order of
the resulting reduced order system. 0 <= NR <= N.
On exit, if INFO = 0, NR is the order of the resulting
reduced order model. For a system with NU ALPHA-unstable
eigenvalues and NS ALPHA-stable eigenvalues (NU+NS = N),
NR is set as follows: if ORDSEL = 'F', NR is equal to
NU+MIN(MAX(0,NR-NU-KR+1),NMIN), where KR is the
multiplicity of the Hankel singular value HSV(NR-NU+1),
NR is the desired order on entry, and NMIN is the order
of a minimal realization of the ALPHA-stable part of the
given system; NMIN is determined as the number of Hankel
singular values greater than NS*EPS*HNORM(As,Bs,Cs), where
EPS is the machine precision (see LAPACK Library Routine
DLAMCH) and HNORM(As,Bs,Cs) is the Hankel norm of the
ALPHA-stable part of the given system (computed in
HSV(1));
if ORDSEL = 'A', NR is the sum of NU and the number of
Hankel singular values greater than
MAX(TOL1,NS*EPS*HNORM(As,Bs,Cs)).
ALPHA (input) DOUBLE PRECISION
Specifies the ALPHA-stability boundary for the eigenvalues
of the state dynamics matrix A. For a continuous-time
system (DICO = 'C'), ALPHA <= 0 is the boundary value for
the real parts of eigenvalues, while for a discrete-time
system (DICO = 'D'), 0 <= ALPHA <= 1 represents the
boundary value for the moduli of eigenvalues.
The ALPHA-stability domain does not include the boundary.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A.
On exit, if INFO = 0, the leading NR-by-NR part of this
array contains the state dynamics matrix Ar of the
reduced order system in a real Schur form.
The resulting A has a block-diagonal form with two blocks.
For a system with NU ALPHA-unstable eigenvalues and
NS ALPHA-stable eigenvalues (NU+NS = N), the leading
NU-by-NU block contains the unreduced part of A
corresponding to ALPHA-unstable eigenvalues.
The trailing (NR+NS-N)-by-(NR+NS-N) block contains
the reduced part of A corresponding to ALPHA-stable
eigenvalues.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, if INFO = 0, the leading NR-by-M part of this
array contains the input/state matrix Br of the reduced
order system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, if INFO = 0, the leading P-by-NR part of this
array contains the state/output matrix Cr of the reduced
order system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the original input/output matrix D.
On exit, if INFO = 0, the leading P-by-M part of this
array contains the input/output matrix Dr of the reduced
order system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
NS (output) INTEGER
The dimension of the ALPHA-stable subsystem.
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, the leading NS elements of HSV contain the
Hankel singular values of the ALPHA-stable part of the
original system ordered decreasingly.
HSV(1) is the Hankel norm of the ALPHA-stable subsystem.
Tolerances
TOL1 DOUBLE PRECISION
If ORDSEL = 'A', TOL1 contains the tolerance for
determining the order of reduced system.
For model reduction, the recommended value is
TOL1 = c*HNORM(As,Bs,Cs), where c is a constant in the
interval [0.00001,0.001], and HNORM(As,Bs,Cs) is the
Hankel-norm of the ALPHA-stable part of the given system
(computed in HSV(1)).
If TOL1 <= 0 on entry, the used default value is
TOL1 = NS*EPS*HNORM(As,Bs,Cs), where NS is the number of
ALPHA-stable eigenvalues of A and EPS is the machine
precision (see LAPACK Library Routine DLAMCH).
This value is appropriate to compute a minimal realization
of the ALPHA-stable part.
If ORDSEL = 'F', the value of TOL1 is ignored.
TOL2 DOUBLE PRECISION
The tolerance for determining the order of a minimal
realization of the ALPHA-stable part of the given system.
The recommended value is TOL2 = NS*EPS*HNORM(As,Bs,Cs).
This value is used by default if TOL2 <= 0 on entry.
If TOL2 > 0, then TOL2 <= TOL1.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK = MAX(1,M), if DICO = 'C';
LIWORK = MAX(1,N,M), if DICO = 'D'.
On exit, if INFO = 0, IWORK(1) contains NMIN, the order of
the computed minimal realization.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( LDW1, LDW2 ), where
LDW1 = N*(2*N + MAX(N,M,P) + 5) + N*(N+1)/2,
LDW2 = N*(M+P+2) + 2*M*P + MIN(N,M) +
MAX( 3*M+1, MIN(N,M)+P ).
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: with ORDSEL = 'F', the selected order NR is greater
than NSMIN, the sum of the order of the
ALPHA-unstable part and the order of a minimal
realization of the ALPHA-stable part of the given
system. In this case, the resulting NR is set equal
to NSMIN.
= 2: with ORDSEL = 'F', the selected order NR is less
than the order of the ALPHA-unstable part of the
given system. In this case NR is set equal to the
order of the ALPHA-unstable part.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the computation of the ordered real Schur form of A
failed;
= 2: the separation of the ALPHA-stable/unstable diagonal
blocks failed because of very close eigenvalues;
= 3: the computed ALPHA-stable part is just stable,
having stable eigenvalues very near to the imaginary
axis (if DICO = 'C') or to the unit circle
(if DICO = 'D');
= 4: the computation of Hankel singular values failed;
= 5: the computation of stable projection in the
Hankel-norm approximation algorithm failed;
= 6: the order of computed stable projection in the
Hankel-norm approximation algorithm differs
from the order of Hankel-norm approximation.
Method
Let be the following linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t) (1)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system. The subroutine AB09ED determines for
the given system (1), the matrices of a reduced order system
d[z(t)] = Ar*z(t) + Br*u(t)
yr(t) = Cr*z(t) + Dr*u(t) (2)
such that
HSV(NR+NS-N) <= INFNORM(G-Gr) <= 2*[HSV(NR+NS-N+1)+...+HSV(NS)],
where G and Gr are transfer-function matrices of the systems
(A,B,C,D) and (Ar,Br,Cr,Dr), respectively, and INFNORM(G) is the
infinity-norm of G.
The following procedure is used to reduce a given G:
1) Decompose additively G as
G = G1 + G2
such that G1 = (As,Bs,Cs,D) has only ALPHA-stable poles and
G2 = (Au,Bu,Cu,0) has only ALPHA-unstable poles.
2) Determine G1r, a reduced order approximation of the
ALPHA-stable part G1.
3) Assemble the reduced model Gr as
Gr = G1r + G2.
To reduce the ALPHA-stable part G1, the optimal Hankel-norm
approximation method of [1], based on the square-root
balancing projection formulas of [2], is employed.
References
[1] Glover, K.
All optimal Hankel norm approximation of linear
multivariable systems and their L-infinity error bounds.
Int. J. Control, Vol. 36, pp. 1145-1193, 1984.
[2] Tombs M.S. and Postlethwaite I.
Truncated balanced realization of stable, non-minimal
state-space systems.
Int. J. Control, Vol. 46, pp. 1319-1330, 1987.
Numerical Aspects
The implemented methods rely on an accuracy enhancing square-root
technique.
3
The algorithms require less than 30N floating point operations.
Further Comments
NoneExample
Program Text
* AB09ED EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDD
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MAX( NMAX, MMAX ) )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( NMAX*( 2*NMAX +
$ MAX( NMAX, MMAX, PMAX ) +
$ 5 ) + ( NMAX*( NMAX + 1 ) )/2,
$ NMAX*( MMAX + PMAX + 2 ) +
$ 2*MMAX*PMAX + MIN( NMAX, MMAX ) +
$ MAX( 3*MMAX + 1,
$ MIN( NMAX, MMAX ) +
$ PMAX ) ) )
* .. Local Scalars ..
DOUBLE PRECISION ALPHA, TOL1, TOL2
INTEGER I, INFO, IWARN, J, M, N, NR, NS, P
CHARACTER*1 DICO, EQUIL, ORDSEL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK), HSV(NMAX)
INTEGER IWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL AB09ED
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, NR, ALPHA, TOL1, TOL2,
$ DICO, EQUIL, ORDSEL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
* Find a reduced ssr for (A,B,C,D).
CALL AB09ED( DICO, EQUIL, ORDSEL, N, M, P, NR, ALPHA,
$ A, LDA, B, LDB, C, LDC, D, LDD, NS, HSV,
$ TOL1, TOL2, IWORK, DWORK, LDWORK,
$ IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NR
WRITE ( NOUT, FMT = 99987 )
WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1, NS )
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
20 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
60 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 70 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1,M )
70 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB09ED EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB09ED = ',I2)
99997 FORMAT (' The order of reduced model = ',I2)
99996 FORMAT (/' The reduced state dynamics matrix Ar is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The reduced input/state matrix Br is ')
99992 FORMAT (/' The reduced state/output matrix Cr is ')
99991 FORMAT (/' The reduced input/output matrix Dr is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99987 FORMAT (/' The Hankel singular values of ALPHA-stable part are')
END
Program Data
AB09ED EXAMPLE PROGRAM DATA (Continuous system) 7 2 3 0 -0.6D0 1.E-1 1.E-14 C N A -0.04165 0.0000 4.9200 -4.9200 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 0.0000 0.0000 0.5450 0.0000 0.0000 0.0000 -0.5450 0.0000 0.0000 0.0000 0.0000 0.0000 4.9200 -0.04165 0.0000 4.9200 0.0000 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 12.500 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000Program Results
AB09ED EXAMPLE PROGRAM RESULTS The order of reduced model = 5 The Hankel singular values of ALPHA-stable part are 1.9178 0.8621 0.7666 0.0336 0.0246 The reduced state dynamics matrix Ar is -0.5181 -1.1084 0.0000 0.0000 0.0000 8.8157 -0.5181 0.0000 0.0000 0.0000 0.0000 0.0000 -1.2769 7.3264 0.0000 0.0000 0.0000 -0.6203 -1.2769 0.0000 0.0000 0.0000 0.0000 0.0000 -1.5496 The reduced input/state matrix Br is -1.2837 1.2837 -0.7522 0.7522 3.2016 3.2016 -0.7640 -0.7640 1.3415 -1.3415 The reduced state/output matrix Cr is -0.1380 -0.6445 -0.6247 -2.0857 -0.8964 0.6246 0.0196 0.0000 0.0000 0.6131 0.1380 0.6445 -0.6247 -2.0857 0.8964 The reduced input/output matrix Dr is 0.0168 -0.0168 0.0008 -0.0008 -0.0168 0.0168
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB09FD.html 0000664 0000000 0000000 00000045240 14560147231 0020324 0 ustar 00root root 0000000 0000000
Purpose
To compute a reduced order model (Ar,Br,Cr) for an original state-space representation (A,B,C) by using either the square-root or the balancing-free square-root Balance & Truncate (B & T) model reduction method in conjunction with stable coprime factorization techniques.Specification
SUBROUTINE AB09FD( DICO, JOBCF, FACT, JOBMR, EQUIL, ORDSEL, N, M,
$ P, NR, ALPHA, A, LDA, B, LDB, C, LDC, NQ, HSV,
$ TOL1, TOL2, IWORK, DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, EQUIL, FACT, JOBCF, JOBMR, ORDSEL
INTEGER INFO, IWARN, LDA, LDB, LDC, LDWORK, M, N, NQ,
$ NR, P
DOUBLE PRECISION ALPHA, TOL1, TOL2
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), HSV(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOBCF CHARACTER*1
Specifies whether left or right coprime factorization is
to be used as follows:
= 'L': use left coprime factorization;
= 'R': use right coprime factorization.
FACT CHARACTER*1
Specifies the type of coprime factorization to be computed
as follows:
= 'S': compute a coprime factorization with prescribed
stability degree ALPHA;
= 'I': compute a coprime factorization with inner
denominator.
JOBMR CHARACTER*1
Specifies the model reduction approach to be used
as follows:
= 'B': use the square-root Balance & Truncate method;
= 'N': use the balancing-free square-root
Balance & Truncate method.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily
equilibrate the triplet (A,B,C) as follows:
= 'S': perform equilibration (scaling);
= 'N': do not perform equilibration.
ORDSEL CHARACTER*1
Specifies the order selection method as follows:
= 'F': the resulting order NR is fixed;
= 'A': the resulting order NR is automatically determined
on basis of the given tolerance TOL1.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation, i.e.
the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NR (input/output) INTEGER
On entry with ORDSEL = 'F', NR is the desired order of the
resulting reduced order system. 0 <= NR <= N.
On exit, if INFO = 0, NR is the order of the resulting
reduced order model. NR is set as follows:
if ORDSEL = 'F', NR is equal to MIN(NR,NQ,NMIN), where NR
is the desired order on entry, NQ is the order of the
computed coprime factorization of the given system, and
NMIN is the order of a minimal realization of the extended
system (see METHOD); NMIN is determined as the number of
Hankel singular values greater than NQ*EPS*HNORM(Ge),
where EPS is the machine precision (see LAPACK Library
Routine DLAMCH) and HNORM(Ge) is the Hankel norm of the
extended system (computed in HSV(1));
if ORDSEL = 'A', NR is equal to the number of Hankel
singular values greater than MAX(TOL1,NQ*EPS*HNORM(Ge)).
ALPHA (input) DOUBLE PRECISION
If FACT = 'S', the desired stability degree for the
factors of the coprime factorization (see SLICOT Library
routines SB08ED/SB08FD).
ALPHA < 0 for a continuous-time system (DICO = 'C'), and
0 <= ALPHA < 1 for a discrete-time system (DICO = 'D').
If FACT = 'I', ALPHA is not used.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state dynamics matrix A.
On exit, if INFO = 0, the leading NR-by-NR part of this
array contains the state dynamics matrix Ar of the reduced
order system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, if INFO = 0, the leading NR-by-M part of this
array contains the input/state matrix Br of the reduced
order system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, if INFO = 0, the leading P-by-NR part of this
array contains the state/output matrix Cr of the reduced
order system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
NQ (output) INTEGER
The order of the computed extended system Ge (see METHOD).
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, it contains the NQ Hankel singular values of
the extended system Ge ordered decreasingly (see METHOD).
Tolerances
TOL1 DOUBLE PRECISION
If ORDSEL = 'A', TOL1 contains the tolerance for
determining the order of reduced extended system.
For model reduction, the recommended value is
TOL1 = c*HNORM(Ge), where c is a constant in the
interval [0.00001,0.001], and HNORM(Ge) is the
Hankel-norm of the extended system (computed in HSV(1)).
The value TOL1 = NQ*EPS*HNORM(Ge) is used by default if
TOL1 <= 0 on entry, where EPS is the machine precision
(see LAPACK Library Routine DLAMCH).
If ORDSEL = 'F', the value of TOL1 is ignored.
TOL2 DOUBLE PRECISION
The absolute tolerance level below which the elements of
B or C are considered zero (used for controllability or
observability tests).
If the user sets TOL2 <= 0, then an implicitly computed,
default tolerance TOLDEF is used:
TOLDEF = N*EPS*NORM(C'), if JOBCF = 'L', or
TOLDEF = N*EPS*NORM(B), if JOBCF = 'R',
where EPS is the machine precision, and NORM(.) denotes
the 1-norm of a matrix.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK = PM, if JOBMR = 'B',
LIWORK = MAX(N,PM), if JOBMR = 'N', where
PM = P, if JOBCF = 'L',
PM = M, if JOBCF = 'R'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,LW1) if JOBCF = 'L' and FACT = 'S',
LDWORK >= MAX(1,LW2) if JOBCF = 'L' and FACT = 'I',
LDWORK >= MAX(1,LW3) if JOBCF = 'R' and FACT = 'S',
LDWORK >= MAX(1,LW4) if JOBCF = 'R' and FACT = 'I', where
LW1 = N*(2*MAX(M,P) + P) + MAX(M,P)*(MAX(M,P) + P) +
MAX( N*P+MAX(N*(N+5), 5*P, 4*M), LWR ),
LW2 = N*(2*MAX(M,P) + P) + MAX(M,P)*(MAX(M,P) + P) +
MAX( N*P+MAX(N*(N+5), P*(P+2), 4*P, 4*M), LWR ),
LW3 = (N+M)*(M+P) + MAX( 5*M, 4*P, LWR ),
LW4 = (N+M)*(M+P) + MAX( M*(M+2), 4*M, 4*P, LWR ), and
LWR = 2*N*N + N*(MAX(N,M+P)+5) + N*(N+1)/2.
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 10*K+I:
I = 1: with ORDSEL = 'F', the selected order NR is
greater than the order of the computed coprime
factorization of the given system. In this case,
the resulting NR is set automatically to a value
corresponding to the order of a minimal
realization of the system;
K > 0: K violations of the numerical stability
condition occured when computing the coprime
factorization using pole assignment (see SLICOT
Library routines SB08CD/SB08ED, SB08DD/SB08FD).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction of A to a real Schur form failed;
= 2: a failure was detected during the ordering of the
real Schur form of A, or in the iterative process
for reordering the eigenvalues of Z'*(A + H*C)*Z
(or Z'*(A + B*F)*Z) along the diagonal; see SLICOT
Library routines SB08CD/SB08ED (or SB08DD/SB08FD);
= 3: the matrix A has an observable or controllable
eigenvalue on the imaginary axis if DICO = 'C' or
on the unit circle if DICO = 'D';
= 4: the computation of Hankel singular values failed.
Method
Let be the linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) (1)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system, and let G be the corresponding
transfer-function matrix. The subroutine AB09FD determines
the matrices of a reduced order system
d[z(t)] = Ar*z(t) + Br*u(t)
yr(t) = Cr*z(t) (2)
with the transfer-function matrix Gr, by using the
balanced-truncation model reduction method in conjunction with
a left coprime factorization (LCF) or a right coprime
factorization (RCF) technique:
1. Compute the appropriate stable coprime factorization of G:
-1 -1
G = R *Q (LCF) or G = Q*R (RCF).
2. Perform the model reduction algorithm on the extended system
( Q )
Ge = ( Q R ) (LCF) or Ge = ( R ) (RCF)
to obtain a reduced extended system with reduced factors
( Qr )
Ger = ( Qr Rr ) (LCF) or Ger = ( Rr ) (RCF).
3. Recover the reduced system from the reduced factors as
-1 -1
Gr = Rr *Qr (LCF) or Gr = Qr*Rr (RCF).
The approximation error for the extended system satisfies
HSV(NR) <= INFNORM(Ge-Ger) <= 2*[HSV(NR+1) + ... + HSV(NQ)],
where INFNORM(G) is the infinity-norm of G.
If JOBMR = 'B', the square-root Balance & Truncate method of [1]
is used for model reduction.
If JOBMR = 'N', the balancing-free square-root version of the
Balance & Truncate method [2] is used for model reduction.
If FACT = 'S', the stable coprime factorization with prescribed
stability degree ALPHA is computed by using the algorithm of [3].
If FACT = 'I', the stable coprime factorization with inner
denominator is computed by using the algorithm of [4].
References
[1] Tombs M.S. and Postlethwaite I.
Truncated balanced realization of stable, non-minimal
state-space systems.
Int. J. Control, Vol. 46, pp. 1319-1330, 1987.
[2] Varga A.
Efficient minimal realization procedure based on balancing.
Proc. of IMACS/IFAC Symp. MCTS, Lille, France, May 1991,
A. El Moudui, P. Borne, S. G. Tzafestas (Eds.), Vol. 2,
pp. 42-46, 1991.
[3] Varga A.
Coprime factors model reduction method based on square-root
balancing-free techniques.
System Analysis, Modelling and Simulation, Vol. 11,
pp. 303-311, 1993.
[4] Varga A.
A Schur method for computing coprime factorizations with
inner denominators and applications in model reduction.
Proc. ACC'93, San Francisco, CA, pp. 2130-2131, 1993.
Numerical Aspects
The implemented methods rely on accuracy enhancing square-root or
balancing-free square-root techniques.
3
The algorithms require less than 30N floating point operations.
Further Comments
NoneExample
Program Text
* AB09FD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MAX( NMAX, MMAX, PMAX ) )
* The formula below uses that NMAX = MMAX = PMAX.
INTEGER LDWORK
PARAMETER ( LDWORK = 10*NMAX*NMAX + 5*NMAX )
* .. Local Scalars ..
DOUBLE PRECISION ALPHA, TOL1, TOL2
INTEGER I, INFO, IWARN, J, M, N, NQ, NR, P
CHARACTER*1 DICO, EQUIL, FACT, JOBCF, JOBMR, ORDSEL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ DWORK(LDWORK), HSV(NMAX)
INTEGER IWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL AB09FD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, NR, ALPHA, TOL1, TOL2,
$ DICO, JOBCF, FACT, JOBMR, EQUIL, ORDSEL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1, N ), I = 1, N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1, N ), I = 1, P )
* Find a reduced ssr for (A,B,C).
CALL AB09FD( DICO, JOBCF, FACT, JOBMR, EQUIL, ORDSEL,
$ N, M, P, NR, ALPHA, A, LDA, B, LDB, C, LDC,
$ NQ, HSV, TOL1, TOL2, IWORK, DWORK, LDWORK,
$ IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NR
WRITE ( NOUT, FMT = 99987 )
WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1, NQ )
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1, NR )
20 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1, M )
40 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1, NR )
60 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB09FD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB09FD = ',I2)
99997 FORMAT (' The order of reduced model = ',I2)
99996 FORMAT (/' The reduced state dynamics matrix Ar is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The reduced input/state matrix Br is ')
99992 FORMAT (/' The reduced state/output matrix Cr is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99987 FORMAT (/' The Hankel singular values of coprime factors are')
END
Program Data
AB08FD EXAMPLE PROGRAM DATA (Continuous system) 7 2 3 0 -1.e-1 .1 1.E-10 C L I B S A -0.04165 0.0000 4.9200 0.4920 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 0.0000 0.0000 0.5450 0.0000 0.0000 0.0000 0.0545 0.0000 0.0000 0.0000 0.0000 0.0000 -0.49200 0.004165 0.0000 4.9200 0.0000 0.0000 0.0000 0.0000 0.5210 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 12.500 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000Program Results
AB09FD EXAMPLE PROGRAM RESULTS The order of reduced model = 5 The Hankel singular values of coprime factors are 13.6047 9.4106 1.7684 0.7456 0.6891 0.0241 0.0230 The reduced state dynamics matrix Ar is 0.0520 -0.1491 0.0037 -0.0232 0.0168 0.2340 0.2618 0.0010 -0.0153 -0.0318 0.1197 0.0075 -0.5752 2.0119 -0.7779 0.1571 -0.2019 -2.1282 -2.1192 -0.3618 0.0368 -0.4810 0.8395 -0.2790 -2.8796 The reduced input/state matrix Br is 1.0454 0.5860 -0.0489 -1.9194 -1.4282 0.0541 -1.6144 -0.7533 0.5916 -1.9242 The reduced state/output matrix Cr is 0.4368 0.1122 -1.2917 1.5888 -0.6354 1.1170 0.3963 0.6115 0.1249 -0.0859 0.0756 -1.8904 0.0144 0.7964 1.9085
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB09GD.html 0000664 0000000 0000000 00000050230 14560147231 0020320 0 ustar 00root root 0000000 0000000
Purpose
To compute a reduced order model (Ar,Br,Cr,Dr) for an original state-space representation (A,B,C,D) by using either the square-root or the balancing-free square-root Singular Perturbation Approximation (SPA) model reduction method in conjunction with stable coprime factorization techniques.Specification
SUBROUTINE AB09GD( DICO, JOBCF, FACT, JOBMR, EQUIL, ORDSEL, N, M,
$ P, NR, ALPHA, A, LDA, B, LDB, C, LDC, D, LDD,
$ NQ, HSV, TOL1, TOL2, TOL3, IWORK, DWORK,
$ LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, EQUIL, FACT, JOBCF, JOBMR, ORDSEL
INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, LDWORK, M, N,
$ NQ, NR, P
DOUBLE PRECISION ALPHA, TOL1, TOL2, TOL3
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), HSV(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOBCF CHARACTER*1
Specifies whether left or right coprime factorization is
to be used as follows:
= 'L': use left coprime factorization;
= 'R': use right coprime factorization.
FACT CHARACTER*1
Specifies the type of coprime factorization to be computed
as follows:
= 'S': compute a coprime factorization with prescribed
stability degree ALPHA;
= 'I': compute a coprime factorization with inner
denominator.
JOBMR CHARACTER*1
Specifies the model reduction approach to be used
as follows:
= 'B': use the square-root Balance & Truncate method;
= 'N': use the balancing-free square-root
Balance & Truncate method.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily
equilibrate the triplet (A,B,C) as follows:
= 'S': perform equilibration (scaling);
= 'N': do not perform equilibration.
ORDSEL CHARACTER*1
Specifies the order selection method as follows:
= 'F': the resulting order NR is fixed;
= 'A': the resulting order NR is automatically determined
on basis of the given tolerance TOL1.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation, i.e.
the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NR (input/output) INTEGER
On entry with ORDSEL = 'F', NR is the desired order of the
resulting reduced order system. 0 <= NR <= N.
On exit, if INFO = 0, NR is the order of the resulting
reduced order model. NR is set as follows:
if ORDSEL = 'F', NR is equal to MIN(NR,NQ,NMIN), where NR
is the desired order on entry, NQ is the order of the
computed coprime factorization of the given system, and
NMIN is the order of a minimal realization of the extended
system (see METHOD); NMIN is determined as the number of
Hankel singular values greater than NQ*EPS*HNORM(Ge),
where EPS is the machine precision (see LAPACK Library
Routine DLAMCH) and HNORM(Ge) is the Hankel norm of the
extended system (computed in HSV(1));
if ORDSEL = 'A', NR is equal to the number of Hankel
singular values greater than MAX(TOL1,NQ*EPS*HNORM(Ge)).
ALPHA (input) DOUBLE PRECISION
If FACT = 'S', the desired stability degree for the
factors of the coprime factorization (see SLICOT Library
routines SB08ED/SB08FD).
ALPHA < 0 for a continuous-time system (DICO = 'C'), and
0 <= ALPHA < 1 for a discrete-time system (DICO = 'D').
If FACT = 'I', ALPHA is not used.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state dynamics matrix A.
On exit, if INFO = 0, the leading NR-by-NR part of this
array contains the state dynamics matrix Ar of the reduced
order system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, if INFO = 0, the leading NR-by-M part of this
array contains the input/state matrix Br of the reduced
order system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, if INFO = 0, the leading P-by-NR part of this
array contains the state/output matrix Cr of the reduced
order system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the original input/output matrix D.
On exit, if INFO = 0, the leading P-by-M part of this
array contains the input/output matrix Dr of the reduced
order system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
NQ (output) INTEGER
The order of the computed extended system Ge (see METHOD).
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, it contains the NQ Hankel singular values of
the extended system Ge ordered decreasingly (see METHOD).
Tolerances
TOL1 DOUBLE PRECISION
If ORDSEL = 'A', TOL1 contains the tolerance for
determining the order of reduced extended system.
For model reduction, the recommended value is
TOL1 = c*HNORM(Ge), where c is a constant in the
interval [0.00001,0.001], and HNORM(Ge) is the
Hankel-norm of the extended system (computed in HSV(1)).
The value TOL1 = NQ*EPS*HNORM(Ge) is used by default if
TOL1 <= 0 on entry, where EPS is the machine precision
(see LAPACK Library Routine DLAMCH).
If ORDSEL = 'F', the value of TOL1 is ignored.
TOL2 DOUBLE PRECISION
The tolerance for determining the order of a minimal
realization of the extended system Ge (see METHOD).
The recommended value is TOL2 = NQ*EPS*HNORM(Ge).
This value is used by default if TOL2 <= 0 on entry.
If TOL2 > 0, then TOL2 <= TOL1.
TOL3 DOUBLE PRECISION
The absolute tolerance level below which the elements of
B or C are considered zero (used for controllability or
observability tests by the coprime factorization method).
If the user sets TOL3 <= 0, then an implicitly computed,
default tolerance TOLDEF is used:
TOLDEF = N*EPS*NORM(C'), if JOBCF = 'L', or
TOLDEF = N*EPS*NORM(B), if JOBCF = 'R',
where EPS is the machine precision, and NORM(.) denotes
the 1-norm of a matrix.
Workspace
IWORK INTEGER array, dimension (MAX(1,2*N,PM))
where PM = P, if JOBCF = 'L',
PM = M, if JOBCF = 'R'.
On exit with INFO = 0, IWORK(1) contains the order of the
minimal realization of the system.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,LW1) if JOBCF = 'L' and FACT = 'S',
LDWORK >= MAX(1,LW2) if JOBCF = 'L' and FACT = 'I',
LDWORK >= MAX(1,LW3) if JOBCF = 'R' and FACT = 'S',
LDWORK >= MAX(1,LW4) if JOBCF = 'R' and FACT = 'I', where
LW1 = N*(2*MAX(M,P) + P) + MAX(M,P)*(MAX(M,P) + P) +
MAX( N*P+MAX(N*(N+5), 5*P, 4*M), LWR ),
LW2 = N*(2*MAX(M,P) + P) + MAX(M,P)*(MAX(M,P) + P) +
MAX( N*P+MAX(N*(N+5), P*(P+2), 4*P, 4*M), LWR ),
LW3 = (N+M)*(M+P) + MAX( 5*M, 4*P, LWR ),
LW4 = (N+M)*(M+P) + MAX( M*(M+2), 4*M, 4*P, LWR ), and
LWR = 2*N*N + N*(MAX(N,M+P)+5) + N*(N+1)/2.
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 10*K+I:
I = 1: with ORDSEL = 'F', the selected order NR is
greater than the order of the computed coprime
factorization of the given system. In this case,
the resulting NR is set automatically to a value
corresponding to the order of a minimal
realization of the system;
K > 0: K violations of the numerical stability
condition occured when computing the coprime
factorization using pole assignment (see SLICOT
Library routines SB08CD/SB08ED, SB08DD/SB08FD).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction of A to a real Schur form failed;
= 2: a failure was detected during the ordering of the
real Schur form of A, or in the iterative process
for reordering the eigenvalues of Z'*(A + H*C)*Z
(or Z'*(A + B*F)*Z) along the diagonal; see SLICOT
Library routines SB08CD/SB08ED (or SB08DD/SB08FD);
= 3: the matrix A has an observable or controllable
eigenvalue on the imaginary axis if DICO = 'C' or
on the unit circle if DICO = 'D';
= 4: the computation of Hankel singular values failed.
Method
Let be the linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t) (1)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system, and let G be the corresponding
transfer-function matrix. The subroutine AB09GD determines
the matrices of a reduced order system
d[z(t)] = Ar*z(t) + Br*u(t)
yr(t) = Cr*z(t) + Dr*u(t) (2)
with the transfer-function matrix Gr, by using the
singular perturbation approximation (SPA) method in conjunction
with a left coprime factorization (LCF) or a right coprime
factorization (RCF) technique:
1. Compute the appropriate stable coprime factorization of G:
-1 -1
G = R *Q (LCF) or G = Q*R (RCF).
2. Perform the model reduction algorithm on the extended system
( Q )
Ge = ( Q R ) (LCF) or Ge = ( R ) (RCF)
to obtain a reduced extended system with reduced factors
( Qr )
Ger = ( Qr Rr ) (LCF) or Ger = ( Rr ) (RCF).
3. Recover the reduced system from the reduced factors as
-1 -1
Gr = Rr *Qr (LCF) or Gr = Qr*Rr (RCF).
The approximation error for the extended system satisfies
HSV(NR) <= INFNORM(Ge-Ger) <= 2*[HSV(NR+1) + ... + HSV(NQ)],
where INFNORM(G) is the infinity-norm of G.
If JOBMR = 'B', the balancing-based square-root SPA method of [1]
is used for model reduction.
If JOBMR = 'N', the balancing-free square-root SPA method of [2]
is used for model reduction.
By setting TOL1 = TOL2, the routine can be used to compute
Balance & Truncate approximations.
If FACT = 'S', the stable coprime factorization with prescribed
stability degree ALPHA is computed by using the algorithm of [3].
If FACT = 'I', the stable coprime factorization with inner
denominator is computed by using the algorithm of [4].
References
[1] Liu Y. and Anderson B.D.O.
Singular Perturbation Approximation of Balanced Systems.
Int. J. Control, Vol. 50, pp. 1379-1405, 1989.
[2] Varga A.
Balancing-free square-root algorithm for computing singular
perturbation approximations.
Proc. 30-th IEEE CDC, Brighton, Dec. 11-13, 1991, Vol. 2,
pp. 1062-1065.
[3] Varga A.
Coprime factors model reduction method based on square-root
balancing-free techniques.
System Analysis, Modelling and Simulation, Vol. 11,
pp. 303-311, 1993.
[4] Varga A.
A Schur method for computing coprime factorizations with
inner denominators and applications in model reduction.
Proc. ACC'93, San Francisco, CA, pp. 2130-2131, 1993.
Numerical Aspects
The implemented methods rely on accuracy enhancing square-root or
balancing-free square-root techniques.
3
The algorithms require less than 30N floating point operations.
Further Comments
NoneExample
Program Text
* AB09GD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDD
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MAX( 2*NMAX, MMAX, PMAX ) )
* The formula below uses that NMAX = MMAX = PMAX.
INTEGER LDWORK
PARAMETER ( LDWORK = 10*NMAX*NMAX + 5*NMAX )
* .. Local Scalars ..
DOUBLE PRECISION ALPHA, TOL1, TOL2, TOL3
INTEGER I, INFO, IWARN, J, M, N, NQ, NR, P
CHARACTER*1 DICO, EQUIL, FACT, JOBCF, JOBMR, ORDSEL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK), HSV(NMAX)
INTEGER IWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL AB09GD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, NR, ALPHA, TOL1, TOL2, TOL3,
$ DICO, JOBCF, FACT, JOBMR, EQUIL, ORDSEL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1, N ), I = 1, N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1, N ), I = 1, P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1, M ), I = 1, P )
* Find a reduced ssr for (A,B,C,D).
CALL AB09GD( DICO, JOBCF, FACT, JOBMR, EQUIL, ORDSEL,
$ N, M, P, NR, ALPHA, A, LDA, B, LDB, C, LDC,
$ D, LDD, NQ, HSV, TOL1, TOL2, TOL3, IWORK,
$ DWORK, LDWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NR
WRITE ( NOUT, FMT = 99987 )
WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1, NQ )
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1, NR )
20 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1, M )
40 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1, NR )
60 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 80 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1, M )
80 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB09GD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB09GD = ',I2)
99997 FORMAT (' The order of reduced model = ',I2)
99996 FORMAT (/' The reduced state dynamics matrix Ar is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The reduced input/state matrix Br is ')
99992 FORMAT (/' The reduced state/output matrix Cr is ')
99991 FORMAT (/' The reduced input/output matrix Dr is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99987 FORMAT (/' The Hankel singular values of coprime factors are')
END
Program Data
AB08GD EXAMPLE PROGRAM DATA (Continuous system) 7 2 3 0 -1.e-1 .1 1.E-10 1.E-10 C L I B S A -0.04165 0.0000 4.9200 0.4920 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 0.0000 0.0000 0.5450 0.0000 0.0000 0.0000 0.0545 0.0000 0.0000 0.0000 0.0000 0.0000 -0.49200 0.004165 0.0000 4.9200 0.0000 0.0000 0.0000 0.0000 0.5210 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 12.500 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000Program Results
AB09GD EXAMPLE PROGRAM RESULTS The order of reduced model = 5 The Hankel singular values of coprime factors are 13.6047 9.4106 1.7684 0.7456 0.6891 0.0241 0.0230 The reduced state dynamics matrix Ar is 0.0521 -0.1491 0.0032 -0.0242 0.0181 0.2341 0.2615 0.0009 -0.0171 -0.0362 0.1170 0.0076 -0.5471 2.0904 -0.8098 0.1675 -0.2122 -2.2113 -2.4097 -0.4139 0.0390 -0.5061 0.8787 -0.3166 -3.2955 The reduced input/state matrix Br is 1.0449 0.5863 -0.0490 -1.9210 -1.3930 0.0540 -1.7206 -0.8039 0.6358 -2.0542 The reduced state/output matrix Cr is 0.4331 0.1125 -1.2534 1.6965 -0.6773 1.1171 0.3963 0.6102 0.1213 -0.0841 0.0736 -1.8815 0.0134 0.8457 2.0413 The reduced input/output matrix Dr is 0.0480 0.0003 -0.0017 0.0001 0.0005 0.0460
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB09HD.html 0000664 0000000 0000000 00000055047 14560147231 0020334 0 ustar 00root root 0000000 0000000
Purpose
To compute a reduced order model (Ar,Br,Cr,Dr) for an original state-space representation (A,B,C,D) by using the stochastic balancing approach in conjunction with the square-root or the balancing-free square-root Balance & Truncate (B&T) or Singular Perturbation Approximation (SPA) model reduction methods for the ALPHA-stable part of the system.Specification
SUBROUTINE AB09HD( DICO, JOB, EQUIL, ORDSEL, N, M, P, NR, ALPHA,
$ BETA, A, LDA, B, LDB, C, LDC, D, LDD, NS, HSV,
$ TOL1, TOL2, IWORK, DWORK, LDWORK, BWORK, IWARN,
$ INFO )
C .. Scalar Arguments ..
CHARACTER DICO, EQUIL, JOB, ORDSEL
INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, LDWORK,
$ M, N, NR, NS, P
DOUBLE PRECISION ALPHA, BETA, TOL1, TOL2
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), HSV(*)
LOGICAL BWORK(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOB CHARACTER*1
Specifies the model reduction approach to be used
as follows:
= 'B': use the square-root Balance & Truncate method;
= 'F': use the balancing-free square-root
Balance & Truncate method;
= 'S': use the square-root Singular Perturbation
Approximation method;
= 'P': use the balancing-free square-root
Singular Perturbation Approximation method.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily
equilibrate the triplet (A,B,C) as follows:
= 'S': perform equilibration (scaling);
= 'N': do not perform equilibration.
ORDSEL CHARACTER*1
Specifies the order selection method as follows:
= 'F': the resulting order NR is fixed;
= 'A': the resulting order NR is automatically determined
on basis of the given tolerance TOL1.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation,
i.e., the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
P <= M if BETA = 0.
NR (input/output) INTEGER
On entry with ORDSEL = 'F', NR is the desired order of the
resulting reduced order system. 0 <= NR <= N.
On exit, if INFO = 0, NR is the order of the resulting
reduced order model. For a system with NU ALPHA-unstable
eigenvalues and NS ALPHA-stable eigenvalues (NU+NS = N),
NR is set as follows: if ORDSEL = 'F', NR is equal to
NU+MIN(MAX(0,NR-NU),NMIN), where NR is the desired order
on entry, and NMIN is the order of a minimal realization
of the ALPHA-stable part of the given system; NMIN is
determined as the number of Hankel singular values greater
than NS*EPS, where EPS is the machine precision
(see LAPACK Library Routine DLAMCH);
if ORDSEL = 'A', NR is the sum of NU and the number of
Hankel singular values greater than MAX(TOL1,NS*EPS);
NR can be further reduced to ensure that
HSV(NR-NU) > HSV(NR+1-NU).
ALPHA (input) DOUBLE PRECISION
Specifies the ALPHA-stability boundary for the eigenvalues
of the state dynamics matrix A. For a continuous-time
system (DICO = 'C'), ALPHA <= 0 is the boundary value for
the real parts of eigenvalues, while for a discrete-time
system (DICO = 'D'), 0 <= ALPHA <= 1 represents the
boundary value for the moduli of eigenvalues.
The ALPHA-stability domain does not include the boundary.
BETA (input) DOUBLE PRECISION
BETA > 0 specifies the absolute/relative error weighting
parameter. A large positive value of BETA favours the
minimization of the absolute approximation error, while a
small value of BETA is appropriate for the minimization
of the relative error.
BETA = 0 means a pure relative error method and can be
used only if rank(D) = P.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A.
On exit, if INFO = 0, the leading NR-by-NR part of this
array contains the state dynamics matrix Ar of the reduced
order system.
The resulting A has a block-diagonal form with two blocks.
For a system with NU ALPHA-unstable eigenvalues and
NS ALPHA-stable eigenvalues (NU+NS = N), the leading
NU-by-NU block contains the unreduced part of A
corresponding to ALPHA-unstable eigenvalues in an
upper real Schur form.
The trailing (NR+NS-N)-by-(NR+NS-N) block contains
the reduced part of A corresponding to ALPHA-stable
eigenvalues.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, if INFO = 0, the leading NR-by-M part of this
array contains the input/state matrix Br of the reduced
order system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, if INFO = 0, the leading P-by-NR part of this
array contains the state/output matrix Cr of the reduced
order system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the original input/output matrix D.
On exit, if INFO = 0, the leading P-by-M part of this
array contains the input/output matrix Dr of the reduced
order system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
NS (output) INTEGER
The dimension of the ALPHA-stable subsystem.
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, the leading NS elements of HSV contain the
Hankel singular values of the phase system corresponding
to the ALPHA-stable part of the original system.
The Hankel singular values are ordered decreasingly.
Tolerances
TOL1 DOUBLE PRECISION
If ORDSEL = 'A', TOL1 contains the tolerance for
determining the order of reduced system.
For model reduction, the recommended value of TOL1 lies
in the interval [0.00001,0.001].
If TOL1 <= 0 on entry, the used default value is
TOL1 = NS*EPS, where NS is the number of
ALPHA-stable eigenvalues of A and EPS is the machine
precision (see LAPACK Library Routine DLAMCH).
If ORDSEL = 'F', the value of TOL1 is ignored.
TOL1 < 1.
TOL2 DOUBLE PRECISION
The tolerance for determining the order of a minimal
realization of the phase system (see METHOD) corresponding
to the ALPHA-stable part of the given system.
The recommended value is TOL2 = NS*EPS.
This value is used by default if TOL2 <= 0 on entry.
If TOL2 > 0 and ORDSEL = 'A', then TOL2 <= TOL1.
TOL2 < 1.
Workspace
IWORK INTEGER array, dimension (MAX(1,2*N))
On exit with INFO = 0, IWORK(1) contains the order of the
minimal realization of the system.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK and DWORK(2) contains RCOND, the reciprocal
condition number of the U11 matrix from the expression
used to compute the solution X = U21*inv(U11) of the
Riccati equation for spectral factorization.
A small value RCOND indicates possible ill-conditioning
of the respective Riccati equation.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 2*N*N + MB*(N+P) + MAX( 2, N*(MAX(N,MB,P)+5),
2*N*P+MAX(P*(MB+2),10*N*(N+1) ) ),
where MB = M if BETA = 0 and MB = M+P if BETA > 0.
For optimum performance LDWORK should be larger.
BWORK LOGICAL array, dimension 2*N
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: with ORDSEL = 'F', the selected order NR is greater
than NSMIN, the sum of the order of the
ALPHA-unstable part and the order of a minimal
realization of the ALPHA-stable part of the given
system; in this case, the resulting NR is set equal
to NSMIN;
= 2: with ORDSEL = 'F', the selected order NR corresponds
to repeated singular values for the ALPHA-stable
part, which are neither all included nor all
excluded from the reduced model; in this case, the
resulting NR is automatically decreased to exclude
all repeated singular values;
= 3: with ORDSEL = 'F', the selected order NR is less
than the order of the ALPHA-unstable part of the
given system; in this case NR is set equal to the
order of the ALPHA-unstable part.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the computation of the ordered real Schur form of A
failed;
= 2: the reduction of the Hamiltonian matrix to real
Schur form failed;
= 3: the reordering of the real Schur form of the
Hamiltonian matrix failed;
= 4: the Hamiltonian matrix has less than N stable
eigenvalues;
= 5: the coefficient matrix U11 in the linear system
X*U11 = U21 to determine X is singular to working
precision;
= 6: BETA = 0 and D has not a maximal row rank;
= 7: the computation of Hankel singular values failed;
= 8: the separation of the ALPHA-stable/unstable diagonal
blocks failed because of very close eigenvalues;
= 9: the resulting order of reduced stable part is less
than the number of unstable zeros of the stable
part.
Method
Let be the following linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t), (1)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system. The subroutine AB09HD determines for
the given system (1), the matrices of a reduced order system
d[z(t)] = Ar*z(t) + Br*u(t)
yr(t) = Cr*z(t) + Dr*u(t), (2)
such that
INFNORM[inv(conj(W))*(G-Gr)] <=
(1+HSV(NR+NS-N+1)) / (1-HSV(NR+NS-N+1)) + ...
+ (1+HSV(NS)) / (1-HSV(NS)) - 1,
where G and Gr are transfer-function matrices of the systems
(A,B,C,D) and (Ar,Br,Cr,Dr), respectively, W is the right, minimum
phase spectral factor satisfying
G1*conj(G1) = conj(W)* W, (3)
G1 is the NS-order ALPHA-stable part of G, and INFNORM(G) is the
infinity-norm of G. HSV(1), ... , HSV(NS) are the Hankel-singular
values of the stable part of the phase system (Ap,Bp,Cp)
with the transfer-function matrix
P = inv(conj(W))*G1.
If BETA > 0, then the model reduction is performed on [G BETA*I]
instead of G. This is the recommended approach to be used when D
has not a maximal row rank or when a certain balance between
relative and absolute approximation errors is desired. For
increasingly large values of BETA, the obtained reduced system
assymptotically approaches that computed by using the
Balance & Truncate or Singular Perturbation Approximation methods.
Note: conj(G) denotes either G'(-s) for a continuous-time system
or G'(1/z) for a discrete-time system.
inv(G) is the inverse of G.
The following procedure is used to reduce a given G:
1) Decompose additively G as
G = G1 + G2,
such that G1 = (As,Bs,Cs,D) has only ALPHA-stable poles and
G2 = (Au,Bu,Cu) has only ALPHA-unstable poles.
2) Determine G1r, a reduced order approximation of the
ALPHA-stable part G1 using the balancing stochastic method
in conjunction with either the B&T [1,2] or SPA methods [3].
3) Assemble the reduced model Gr as
Gr = G1r + G2.
Note: The employed stochastic truncation algorithm [2,3] has the
property that right half plane zeros of G1 remain as right half
plane zeros of G1r. Thus, the order can not be chosen smaller than
the sum of the number of unstable poles of G and the number of
unstable zeros of G1.
The reduction of the ALPHA-stable part G1 is done as follows.
If JOB = 'B', the square-root stochastic Balance & Truncate
method of [1] is used.
For an ALPHA-stable continuous-time system (DICO = 'C'),
the resulting reduced model is stochastically balanced.
If JOB = 'F', the balancing-free square-root version of the
stochastic Balance & Truncate method [1] is used to reduce
the ALPHA-stable part G1.
If JOB = 'S', the stochastic balancing method is used to reduce
the ALPHA-stable part G1, in conjunction with the square-root
version of the Singular Perturbation Approximation method [3,4].
If JOB = 'P', the stochastic balancing method is used to reduce
the ALPHA-stable part G1, in conjunction with the balancing-free
square-root version of the Singular Perturbation Approximation
method [3,4].
References
[1] Varga A. and Fasol K.H.
A new square-root balancing-free stochastic truncation model
reduction algorithm.
Proc. 12th IFAC World Congress, Sydney, 1993.
[2] Safonov M. G. and Chiang R. Y.
Model reduction for robust control: a Schur relative error
method.
Int. J. Adapt. Contr. Sign. Proc., vol. 2, pp. 259-272, 1988.
[3] Green M. and Anderson B. D. O.
Generalized balanced stochastic truncation.
Proc. 29-th CDC, Honolulu, Hawaii, pp. 476-481, 1990.
[4] Varga A.
Balancing-free square-root algorithm for computing
singular perturbation approximations.
Proc. 30-th IEEE CDC, Brighton, Dec. 11-13, 1991,
Vol. 2, pp. 1062-1065.
Numerical Aspects
The implemented methods rely on accuracy enhancing square-root or
balancing-free square-root techniques. The effectiveness of the
accuracy enhancing technique depends on the accuracy of the
solution of a Riccati equation. An ill-conditioned Riccati
solution typically results when [D BETA*I] is nearly
rank deficient.
3
The algorithm requires about 100N floating point operations.
Further Comments
NoneExample
Program Text
* AB09HD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDD
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX )
INTEGER LBWORK, LIWORK
PARAMETER ( LBWORK = 2*NMAX, LIWORK = 2*NMAX )
INTEGER LDWORK, MBMAX
PARAMETER ( MBMAX = MMAX + PMAX )
PARAMETER ( LDWORK = 2*NMAX*NMAX + MBMAX*(NMAX+PMAX) +
$ MAX( NMAX*(MAX( NMAX, MMAX, PMAX) + 5),
$ 2*NMAX*PMAX + MAX( PMAX*(MBMAX+2),
$ 10*NMAX*(NMAX+1) ) ) )
* .. Local Scalars ..
DOUBLE PRECISION ALPHA, BETA, TOL1, TOL2
INTEGER I, INFO, IWARN, J, M, N, NR, NS, P
CHARACTER*1 DICO, EQUIL, JOB, ORDSEL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK), HSV(NMAX)
LOGICAL BWORK(LBWORK)
INTEGER IWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL AB09HD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, NR, ALPHA, BETA, TOL1, TOL2,
$ DICO, JOB, EQUIL, ORDSEL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
* Find a reduced ssr for (A,B,C,D).
CALL AB09HD( DICO, JOB, EQUIL, ORDSEL, N, M, P, NR,
$ ALPHA, BETA, A, LDA, B, LDB, C, LDC, D, LDD,
$ NS, HSV, TOL1, TOL2, IWORK, DWORK, LDWORK,
$ BWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NR
WRITE ( NOUT, FMT = 99987 )
WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1,NS )
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
20 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
60 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 70 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1,M )
70 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB09HD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB09HD = ',I2)
99997 FORMAT (' The order of reduced model = ',I2)
99996 FORMAT (/' The reduced state dynamics matrix Ar is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The reduced input/state matrix Br is ')
99992 FORMAT (/' The reduced state/output matrix Cr is ')
99991 FORMAT (/' The reduced input/output matrix Dr is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99987 FORMAT (/' The stochastic Hankel singular values of ALPHA-stable'
$ ,' part are')
END
Program Data
AB09HD EXAMPLE PROGRAM DATA (Continuous system) 7 2 3 0 0.0 1.0 0.1E0 0.0 C F N A -0.04165 0.0000 4.9200 -4.9200 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 0.0000 0.0000 0.5450 0.0000 0.0000 0.0000 -0.5450 0.0000 0.0000 0.0000 0.0000 0.0000 4.9200 -0.04165 0.0000 4.9200 0.0000 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 12.500 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000Program Results
AB09HD EXAMPLE PROGRAM RESULTS The order of reduced model = 5 The stochastic Hankel singular values of ALPHA-stable part are 0.8803 0.8506 0.8038 0.4494 0.3973 0.0214 0.0209 The reduced state dynamics matrix Ar is 1.2729 0.0000 6.5947 0.0000 -3.4229 0.0000 0.8169 0.0000 2.4821 0.0000 -2.9889 0.0000 -2.9028 0.0000 -0.3692 0.0000 -3.3921 0.0000 -3.1126 0.0000 -1.4767 0.0000 -2.0339 0.0000 -0.6107 The reduced input/state matrix Br is 0.1331 -0.1331 -0.0862 -0.0862 -2.6777 2.6777 -3.5767 -3.5767 -2.3033 2.3033 The reduced state/output matrix Cr is -0.6907 -0.6882 0.0779 0.0958 -0.0038 0.0676 0.0000 0.6532 0.0000 -0.7522 0.6907 -0.6882 -0.0779 0.0958 0.0038 The reduced input/output matrix Dr is 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB09HX.html 0000664 0000000 0000000 00000031375 14560147231 0020356 0 ustar 00root root 0000000 0000000
Purpose
To compute a reduced order model (Ar,Br,Cr,Dr) for an original
stable state-space representation (A,B,C,D) by using the
stochastic balancing approach in conjunction with the square-root
or the balancing-free square-root Balance & Truncate (B&T) or
Singular Perturbation Approximation (SPA) model reduction methods.
The state dynamics matrix A of the original system is an upper
quasi-triangular matrix in real Schur canonical form and D must be
full row rank.
For the B&T approach, the matrices of the reduced order system
are computed using the truncation formulas:
Ar = TI * A * T , Br = TI * B , Cr = C * T . (1)
For the SPA approach, the matrices of a minimal realization
(Am,Bm,Cm) are computed using the truncation formulas:
Am = TI * A * T , Bm = TI * B , Cm = C * T . (2)
Am, Bm, Cm and D serve further for computing the SPA of the given
system.
Specification
SUBROUTINE AB09HX( DICO, JOB, ORDSEL, N, M, P, NR, A, LDA, B, LDB,
$ C, LDC, D, LDD, HSV, T, LDT, TI, LDTI, TOL1,
$ TOL2, IWORK, DWORK, LDWORK, BWORK, IWARN,
$ INFO )
C .. Scalar Arguments ..
CHARACTER DICO, JOB, ORDSEL
INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, LDT, LDTI,
$ LDWORK, M, N, NR, P
DOUBLE PRECISION TOL1, TOL2
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), HSV(*), T(LDT,*), TI(LDTI,*)
LOGICAL BWORK(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOB CHARACTER*1
Specifies the model reduction approach to be used
as follows:
= 'B': use the square-root Balance & Truncate method;
= 'F': use the balancing-free square-root
Balance & Truncate method;
= 'S': use the square-root Singular Perturbation
Approximation method;
= 'P': use the balancing-free square-root
Singular Perturbation Approximation method.
ORDSEL CHARACTER*1
Specifies the order selection method as follows:
= 'F': the resulting order NR is fixed;
= 'A': the resulting order NR is automatically determined
on basis of the given tolerance TOL1.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation,
i.e., the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. M >= P >= 0.
NR (input/output) INTEGER
On entry with ORDSEL = 'F', NR is the desired order of
the resulting reduced order system. 0 <= NR <= N.
On exit, if INFO = 0, NR is the order of the resulting
reduced order model. NR is set as follows:
if ORDSEL = 'F', NR is equal to MIN(NR,NMIN), where NR
is the desired order on entry and NMIN is the order of a
minimal realization of the given system; NMIN is
determined as the number of Hankel singular values greater
than N*EPS, where EPS is the machine precision
(see LAPACK Library Routine DLAMCH);
if ORDSEL = 'A', NR is equal to the number of Hankel
singular values greater than MAX(TOL1,N*EPS).
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A in a real Schur
canonical form.
On exit, if INFO = 0, the leading NR-by-NR part of this
array contains the state dynamics matrix Ar of the
reduced order system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, if INFO = 0, the leading NR-by-M part of this
array contains the input/state matrix Br of the reduced
order system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, if INFO = 0, the leading P-by-NR part of this
array contains the state/output matrix Cr of the reduced
order system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the original input/output matrix D.
On exit, if INFO = 0, the leading P-by-M part of this
array contains the input/output matrix Dr of the reduced
order system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, it contains the Hankel singular values,
ordered decreasingly, of the phase system. All singular
values are less than or equal to 1.
T (output) DOUBLE PRECISION array, dimension (LDT,N)
If INFO = 0 and NR > 0, the leading N-by-NR part of this
array contains the right truncation matrix T in (1), for
the B&T approach, or in (2), for the SPA approach.
LDT INTEGER
The leading dimension of array T. LDT >= MAX(1,N).
TI (output) DOUBLE PRECISION array, dimension (LDTI,N)
If INFO = 0 and NR > 0, the leading NR-by-N part of this
array contains the left truncation matrix TI in (1), for
the B&T approach, or in (2), for the SPA approach.
LDTI INTEGER
The leading dimension of array TI. LDTI >= MAX(1,N).
Tolerances
TOL1 DOUBLE PRECISION
If ORDSEL = 'A', TOL1 contains the tolerance for
determining the order of reduced system.
For model reduction, the recommended value lies in the
interval [0.00001,0.001].
If TOL1 <= 0 on entry, the used default value is
TOL1 = N*EPS, where EPS is the machine
precision (see LAPACK Library Routine DLAMCH).
If ORDSEL = 'F', the value of TOL1 is ignored.
TOL2 DOUBLE PRECISION
The tolerance for determining the order of a minimal
realization of the phase system (see METHOD) corresponding
to the given system.
The recommended value is TOL2 = N*EPS.
This value is used by default if TOL2 <= 0 on entry.
If TOL2 > 0 and ORDSEL = 'A', then TOL2 <= TOL1.
Workspace
IWORK INTEGER array, dimension (MAX(1,2*N))
On exit with INFO = 0, IWORK(1) contains the order of the
minimal realization of the system.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK and DWORK(2) contains RCOND, the reciprocal
condition number of the U11 matrix from the expression
used to compute the solution X = U21*inv(U11) of the
Riccati equation for spectral factorization.
A small value RCOND indicates possible ill-conditioning
of the respective Riccati equation.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( 2, N*(MAX(N,M,P)+5),
2*N*P+MAX(P*(M+2),10*N*(N+1) ) ).
For optimum performance LDWORK should be larger.
BWORK LOGICAL array, dimension 2*N
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: with ORDSEL = 'F', the selected order NR is greater
than the order of a minimal realization of the
given system. In this case, the resulting NR is
set automatically to a value corresponding to the
order of a minimal realization of the system.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the state matrix A is not stable (if DICO = 'C')
or not convergent (if DICO = 'D'), or it is not in
a real Schur form;
= 2: the reduction of Hamiltonian matrix to real
Schur form failed;
= 3: the reordering of the real Schur form of the
Hamiltonian matrix failed;
= 4: the Hamiltonian matrix has less than N stable
eigenvalues;
= 5: the coefficient matrix U11 in the linear system
X*U11 = U21, used to determine X, is singular to
working precision;
= 6: the feedthrough matrix D has not a full row rank P;
= 7: the computation of Hankel singular values failed.
Method
Let be the stable linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t), (3)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system. The subroutine AB09HX determines for
the given system (3), the matrices of a reduced NR-rder system
d[z(t)] = Ar*z(t) + Br*u(t)
yr(t) = Cr*z(t) + Dr*u(t), (4)
such that
HSV(NR) <= INFNORM(G-Gr) <= 2*[HSV(NR+1) + ... + HSV(N)],
where G and Gr are transfer-function matrices of the systems
(A,B,C,D) and (Ar,Br,Cr,Dr), respectively, and INFNORM(G) is the
infinity-norm of G.
If JOB = 'B', the square-root stochastic Balance & Truncate
method of [1] is used and the resulting model is balanced.
If JOB = 'F', the balancing-free square-root version of the
stochastic Balance & Truncate method [1] is used.
If JOB = 'S', the stochastic balancing method, in conjunction
with the square-root version of the Singular Perturbation
Approximation method [2,3] is used.
If JOB = 'P', the stochastic balancing method, in conjunction
with the balancing-free square-root version of the Singular
Perturbation Approximation method [2,3] is used.
By setting TOL1 = TOL2, the routine can be also used to compute
Balance & Truncate approximations.
References
[1] Varga A. and Fasol K.H.
A new square-root balancing-free stochastic truncation
model reduction algorithm.
Proc. of 12th IFAC World Congress, Sydney, 1993.
[2] Liu Y. and Anderson B.D.O.
Singular Perturbation Approximation of balanced systems.
Int. J. Control, Vol. 50, pp. 1379-1405, 1989.
[3] Varga A.
Balancing-free square-root algorithm for computing singular
perturbation approximations.
Proc. 30-th IEEE CDC, Brighton, Dec. 11-13, 1991,
Vol. 2, pp. 1062-1065.
Numerical Aspects
The implemented method relies on accuracy enhancing square-root
or balancing-free square-root methods. The effectiveness of the
accuracy enhancing technique depends on the accuracy of the
solution of a Riccati equation. Ill-conditioned Riccati solution
typically results when D is nearly rank deficient.
3
The algorithm requires about 100N floating point operations.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the Cholesky factors Su and Ru of the controllability
Grammian P = Su*Su' and observability Grammian Q = Ru'*Ru,
respectively, satisfying
A*P + P*A' + scalec^2*B*B' = 0, (1)
A'*Q + Q*A + scaleo^2*Cw'*Cw = 0, (2)
where
Cw = Hw - Bw'*X,
Hw = inv(Dw)*C,
Bw = (B*D' + P*C')*inv(Dw'),
D*D' = Dw*Dw' (Dw upper triangular),
and, with Aw = A - Bw*Hw, X is the stabilizing solution of the
Riccati equation
Aw'*X + X*Aw + Hw'*Hw + X*Bw*Bw'*X = 0. (3)
The P-by-M matrix D must have full row rank. Matrix A must be
stable and in a real Schur form.
Specification
SUBROUTINE AB09HY( N, M, P, A, LDA, B, LDB, C, LDC, D, LDD,
$ SCALEC, SCALEO, S, LDS, R, LDR, IWORK,
$ DWORK, LDWORK, BWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDD, LDR, LDS, LDWORK, M, N,
$ P
DOUBLE PRECISION SCALEC, SCALEO
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), R(LDR,*), S(LDS,*)
LOGICAL BWORK(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of state-space representation, i.e.,
the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. M >= P >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
stable state dynamics matrix A in a real Schur canonical
form.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
input/state matrix B, corresponding to the Schur matrix A.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array must contain the
state/output matrix C, corresponding to the Schur
matrix A.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading P-by-M part of this array must
contain the full row rank input/output matrix D.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
SCALEC (output) DOUBLE PRECISION
Scaling factor for the controllability Grammian in (1).
SCALEO (output) DOUBLE PRECISION
Scaling factor for the observability Grammian in (2).
S (output) DOUBLE PRECISION array, dimension (LDS,N)
The leading N-by-N upper triangular part of this array
contains the Cholesky factor Su of the cotrollability
Grammian P = Su*Su' satisfying (1).
LDS INTEGER
The leading dimension of array S. LDS >= MAX(1,N).
R (output) DOUBLE PRECISION array, dimension (LDR,N)
The leading N-by-N upper triangular part of this array
contains the Cholesky factor Ru of the observability
Grammian Q = Ru'*Ru satisfying (2).
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
Workspace
IWORK INTEGER array, dimension (2*N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK and DWORK(2) contains RCOND, the reciprocal
condition number of the U11 matrix from the expression
used to compute X = U21*inv(U11). A small value RCOND
indicates possible ill-conditioning of the Riccati
equation (3).
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( 2, N*(MAX(N,M,P)+5),
2*N*P+MAX(P*(M+2),10*N*(N+1) ) ).
For optimum performance LDWORK should be larger.
BWORK LOGICAL array, dimension 2*N
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the state matrix A is not stable or is not in a
real Schur form;
= 2: the reduction of Hamiltonian matrix to real Schur
form failed;
= 3: the reordering of the real Schur form of the
Hamiltonian matrix failed;
= 4: the Hamiltonian matrix has less than N stable
eigenvalues;
= 5: the coefficient matrix U11 in the linear system
X*U11 = U21, used to determine X, is singular to
working precision;
= 6: the feedthrough matrix D has not a full row rank P.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute a reduced order model (Ar,Br,Cr,Dr) for an original
state-space representation (A,B,C,D) by using the frequency
weighted square-root or balancing-free square-root
Balance & Truncate (B&T) or Singular Perturbation Approximation
(SPA) model reduction methods. The algorithm tries to minimize
the norm of the frequency-weighted error
||V*(G-Gr)*W||
where G and Gr are the transfer-function matrices of the original
and reduced order models, respectively, and V and W are
frequency-weighting transfer-function matrices. V and W must not
have poles on the imaginary axis for a continuous-time
system or on the unit circle for a discrete-time system.
If G is unstable, only the ALPHA-stable part of G is reduced.
In case of possible pole-zero cancellations in V*G and/or G*W,
the absolute values of parameters ALPHAO and/or ALPHAC must be
different from 1.
Specification
SUBROUTINE AB09ID( DICO, JOBC, JOBO, JOB, WEIGHT, EQUIL, ORDSEL,
$ N, M, P, NV, PV, NW, MW, NR, ALPHA, ALPHAC,
$ ALPHAO, A, LDA, B, LDB, C, LDC, D, LDD,
$ AV, LDAV, BV, LDBV, CV, LDCV, DV, LDDV,
$ AW, LDAW, BW, LDBW, CW, LDCW, DW, LDDW,
$ NS, HSV, TOL1, TOL2, IWORK, DWORK, LDWORK,
$ IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, EQUIL, JOB, JOBC, JOBO, ORDSEL, WEIGHT
INTEGER INFO, IWARN, LDA, LDAV, LDAW, LDB, LDBV, LDBW,
$ LDC, LDCV, LDCW, LDD, LDDV, LDDW, LDWORK, M, MW,
$ N, NR, NS, NV, NW, P, PV
DOUBLE PRECISION ALPHA, ALPHAC, ALPHAO, TOL1, TOL2
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), AV(LDAV,*), AW(LDAW,*),
$ B(LDB,*), BV(LDBV,*), BW(LDBW,*),
$ C(LDC,*), CV(LDCV,*), CW(LDCW,*),
$ D(LDD,*), DV(LDDV,*), DW(LDDW,*), DWORK(*),
$ HSV(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOBC CHARACTER*1
Specifies the choice of frequency-weighted controllability
Grammian as follows:
= 'S': choice corresponding to a combination method [4]
of the approaches of Enns [1] and Lin-Chiu [2,3];
= 'E': choice corresponding to the stability enhanced
modified combination method of [4].
JOBO CHARACTER*1
Specifies the choice of frequency-weighted observability
Grammian as follows:
= 'S': choice corresponding to a combination method [4]
of the approaches of Enns [1] and Lin-Chiu [2,3];
= 'E': choice corresponding to the stability enhanced
modified combination method of [4].
JOB CHARACTER*1
Specifies the model reduction approach to be used
as follows:
= 'B': use the square-root Balance & Truncate method;
= 'F': use the balancing-free square-root
Balance & Truncate method;
= 'S': use the square-root Singular Perturbation
Approximation method;
= 'P': use the balancing-free square-root
Singular Perturbation Approximation method.
WEIGHT CHARACTER*1
Specifies the type of frequency weighting, as follows:
= 'N': no weightings are used (V = I, W = I);
= 'L': only left weighting V is used (W = I);
= 'R': only right weighting W is used (V = I);
= 'B': both left and right weightings V and W are used.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily
equilibrate the triplet (A,B,C) as follows:
= 'S': perform equilibration (scaling);
= 'N': do not perform equilibration.
ORDSEL CHARACTER*1
Specifies the order selection method as follows:
= 'F': the resulting order NR is fixed;
= 'A': the resulting order NR is automatically determined
on basis of the given tolerance TOL1.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation,
i.e., the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NV (input) INTEGER
The order of the matrix AV. Also the number of rows of
the matrix BV and the number of columns of the matrix CV.
NV represents the dimension of the state vector of the
system with the transfer-function matrix V. NV >= 0.
PV (input) INTEGER
The number of rows of the matrices CV and DV. PV >= 0.
PV represents the dimension of the output vector of the
system with the transfer-function matrix V.
NW (input) INTEGER
The order of the matrix AW. Also the number of rows of
the matrix BW and the number of columns of the matrix CW.
NW represents the dimension of the state vector of the
system with the transfer-function matrix W. NW >= 0.
MW (input) INTEGER
The number of columns of the matrices BW and DW. MW >= 0.
MW represents the dimension of the input vector of the
system with the transfer-function matrix W.
NR (input/output) INTEGER
On entry with ORDSEL = 'F', NR is the desired order of the
resulting reduced order system. 0 <= NR <= N.
On exit, if INFO = 0, NR is the order of the resulting
reduced order model. For a system with NU ALPHA-unstable
eigenvalues and NS ALPHA-stable eigenvalues (NU+NS = N),
NR is set as follows: if ORDSEL = 'F', NR is equal to
NU+MIN(MAX(0,NR-NU),NMIN), where NR is the desired order
on entry, NMIN is the number of frequency-weighted Hankel
singular values greater than NS*EPS*S1, EPS is the
machine precision (see LAPACK Library Routine DLAMCH)
and S1 is the largest Hankel singular value (computed
in HSV(1)); NR can be further reduced to ensure
HSV(NR-NU) > HSV(NR+1-NU);
if ORDSEL = 'A', NR is the sum of NU and the number of
Hankel singular values greater than MAX(TOL1,NS*EPS*S1).
ALPHA (input) DOUBLE PRECISION
Specifies the ALPHA-stability boundary for the eigenvalues
of the state dynamics matrix A. For a continuous-time
system (DICO = 'C'), ALPHA <= 0 is the boundary value for
the real parts of eigenvalues, while for a discrete-time
system (DICO = 'D'), 0 <= ALPHA <= 1 represents the
boundary value for the moduli of eigenvalues.
The ALPHA-stability domain does not include the boundary.
ALPHAC (input) DOUBLE PRECISION
Combination method parameter for defining the
frequency-weighted controllability Grammian (see METHOD);
ABS(ALPHAC) <= 1.
ALPHAO (input) DOUBLE PRECISION
Combination method parameter for defining the
frequency-weighted observability Grammian (see METHOD);
ABS(ALPHAO) <= 1.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A.
On exit, if INFO = 0, the leading NR-by-NR part of this
array contains the state dynamics matrix Ar of the
reduced order system.
The resulting A has a block-diagonal form with two blocks.
For a system with NU ALPHA-unstable eigenvalues and
NS ALPHA-stable eigenvalues (NU+NS = N), the leading
NU-by-NU block contains the unreduced part of A
corresponding to ALPHA-unstable eigenvalues.
The trailing (NR+NS-N)-by-(NR+NS-N) block contains
the reduced part of A corresponding to ALPHA-stable
eigenvalues.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, if INFO = 0, the leading NR-by-M part of this
array contains the input/state matrix Br of the reduced
order system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, if INFO = 0, the leading P-by-NR part of this
array contains the state/output matrix Cr of the reduced
order system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the original input/output matrix D.
On exit, if INFO = 0, the leading P-by-M part of this
array contains the input/output matrix Dr of the reduced
order system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
AV (input/output) DOUBLE PRECISION array, dimension (LDAV,NV)
On entry, if WEIGHT = 'L' or 'B', the leading NV-by-NV
part of this array must contain the state matrix AV of
the system with the transfer-function matrix V.
On exit, if WEIGHT = 'L' or 'B', MIN(N,M,P) > 0 and
INFO = 0, the leading NVR-by-NVR part of this array
contains the state matrix of a minimal realization of V
in a real Schur form. NVR is returned in IWORK(2).
AV is not referenced if WEIGHT = 'R' or 'N',
or MIN(N,M,P) = 0.
LDAV INTEGER
The leading dimension of array AV.
LDAV >= MAX(1,NV), if WEIGHT = 'L' or 'B';
LDAV >= 1, if WEIGHT = 'R' or 'N'.
BV (input/output) DOUBLE PRECISION array, dimension (LDBV,P)
On entry, if WEIGHT = 'L' or 'B', the leading NV-by-P part
of this array must contain the input matrix BV of the
system with the transfer-function matrix V.
On exit, if WEIGHT = 'L' or 'B', MIN(N,M,P) > 0 and
INFO = 0, the leading NVR-by-P part of this array contains
the input matrix of a minimal realization of V.
BV is not referenced if WEIGHT = 'R' or 'N',
or MIN(N,M,P) = 0.
LDBV INTEGER
The leading dimension of array BV.
LDBV >= MAX(1,NV), if WEIGHT = 'L' or 'B';
LDBV >= 1, if WEIGHT = 'R' or 'N'.
CV (input/output) DOUBLE PRECISION array, dimension (LDCV,NV)
On entry, if WEIGHT = 'L' or 'B', the leading PV-by-NV
part of this array must contain the output matrix CV of
the system with the transfer-function matrix V.
On exit, if WEIGHT = 'L' or 'B', MIN(N,M,P) > 0 and
INFO = 0, the leading PV-by-NVR part of this array
contains the output matrix of a minimal realization of V.
CV is not referenced if WEIGHT = 'R' or 'N',
or MIN(N,M,P) = 0.
LDCV INTEGER
The leading dimension of array CV.
LDCV >= MAX(1,PV), if WEIGHT = 'L' or 'B';
LDCV >= 1, if WEIGHT = 'R' or 'N'.
DV (input) DOUBLE PRECISION array, dimension (LDDV,P)
If WEIGHT = 'L' or 'B', the leading PV-by-P part of this
array must contain the feedthrough matrix DV of the system
with the transfer-function matrix V.
DV is not referenced if WEIGHT = 'R' or 'N',
or MIN(N,M,P) = 0.
LDDV INTEGER
The leading dimension of array DV.
LDDV >= MAX(1,PV), if WEIGHT = 'L' or 'B';
LDDV >= 1, if WEIGHT = 'R' or 'N'.
AW (input/output) DOUBLE PRECISION array, dimension (LDAW,NW)
On entry, if WEIGHT = 'R' or 'B', the leading NW-by-NW
part of this array must contain the state matrix AW of
the system with the transfer-function matrix W.
On exit, if WEIGHT = 'R' or 'B', MIN(N,M,P) > 0 and
INFO = 0, the leading NWR-by-NWR part of this array
contains the state matrix of a minimal realization of W
in a real Schur form. NWR is returned in IWORK(3).
AW is not referenced if WEIGHT = 'L' or 'N',
or MIN(N,M,P) = 0.
LDAW INTEGER
The leading dimension of array AW.
LDAW >= MAX(1,NW), if WEIGHT = 'R' or 'B';
LDAW >= 1, if WEIGHT = 'L' or 'N'.
BW (input/output) DOUBLE PRECISION array, dimension (LDBW,MW)
On entry, if WEIGHT = 'R' or 'B', the leading NW-by-MW
part of this array must contain the input matrix BW of the
system with the transfer-function matrix W.
On exit, if WEIGHT = 'R' or 'B', MIN(N,M,P) > 0 and
INFO = 0, the leading NWR-by-MW part of this array
contains the input matrix of a minimal realization of W.
BW is not referenced if WEIGHT = 'L' or 'N',
or MIN(N,M,P) = 0.
LDBW INTEGER
The leading dimension of array BW.
LDBW >= MAX(1,NW), if WEIGHT = 'R' or 'B';
LDBW >= 1, if WEIGHT = 'L' or 'N'.
CW (input/output) DOUBLE PRECISION array, dimension (LDCW,NW)
On entry, if WEIGHT = 'R' or 'B', the leading M-by-NW part
of this array must contain the output matrix CW of the
system with the transfer-function matrix W.
On exit, if WEIGHT = 'R' or 'B', MIN(N,M,P) > 0 and
INFO = 0, the leading M-by-NWR part of this array contains
the output matrix of a minimal realization of W.
CW is not referenced if WEIGHT = 'L' or 'N',
or MIN(N,M,P) = 0.
LDCW INTEGER
The leading dimension of array CW.
LDCW >= MAX(1,M), if WEIGHT = 'R' or 'B';
LDCW >= 1, if WEIGHT = 'L' or 'N'.
DW (input) DOUBLE PRECISION array, dimension (LDDW,MW)
If WEIGHT = 'R' or 'B', the leading M-by-MW part of this
array must contain the feedthrough matrix DW of the system
with the transfer-function matrix W.
DW is not referenced if WEIGHT = 'L' or 'N',
or MIN(N,M,P) = 0.
LDDW INTEGER
The leading dimension of array DW.
LDDW >= MAX(1,M), if WEIGHT = 'R' or 'B';
LDDW >= 1, if WEIGHT = 'L' or 'N'.
NS (output) INTEGER
The dimension of the ALPHA-stable subsystem.
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, the leading NS elements of this array contain
the frequency-weighted Hankel singular values, ordered
decreasingly, of the ALPHA-stable part of the original
system.
Tolerances
TOL1 DOUBLE PRECISION
If ORDSEL = 'A', TOL1 contains the tolerance for
determining the order of reduced system.
For model reduction, the recommended value is
TOL1 = c*S1, where c is a constant in the
interval [0.00001,0.001], and S1 is the largest
frequency-weighted Hankel singular value of the
ALPHA-stable part of the original system (computed
in HSV(1)).
If TOL1 <= 0 on entry, the used default value is
TOL1 = NS*EPS*S1, where NS is the number of
ALPHA-stable eigenvalues of A and EPS is the machine
precision (see LAPACK Library Routine DLAMCH).
If ORDSEL = 'F', the value of TOL1 is ignored.
TOL2 DOUBLE PRECISION
The tolerance for determining the order of a minimal
realization of the ALPHA-stable part of the given system.
The recommended value is TOL2 = NS*EPS*S1.
This value is used by default if TOL2 <= 0 on entry.
If TOL2 > 0 and ORDSEL = 'A', then TOL2 <= TOL1.
Workspace
IWORK INTEGER array, dimension
( MAX( 3, LIWRK1, LIWRK2, LIWRK3 ) ), where
LIWRK1 = 0, if JOB = 'B';
LIWRK1 = N, if JOB = 'F';
LIWRK1 = 2*N, if JOB = 'S' or 'P';
LIWRK2 = 0, if WEIGHT = 'R' or 'N' or NV = 0;
LIWRK2 = NV+MAX(P,PV), if WEIGHT = 'L' or 'B' and NV > 0;
LIWRK3 = 0, if WEIGHT = 'L' or 'N' or NW = 0;
LIWRK3 = NW+MAX(M,MW), if WEIGHT = 'R' or 'B' and NW > 0.
On exit, if INFO = 0, IWORK(1) contains the order of a
minimal realization of the stable part of the system,
IWORK(2) and IWORK(3) contain the actual orders
of the state space realizations of V and W, respectively.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( LMINL, LMINR, LRCF,
2*N*N + MAX( 1, LLEFT, LRIGHT, 2*N*N+5*N,
N*MAX(M,P) ) ),
where
LMINL = 0, if WEIGHT = 'R' or 'N' or NV = 0; otherwise,
LMINL = MAX(LLCF,NV+MAX(NV,3*P)) if P = PV;
LMINL = MAX(P,PV)*(2*NV+MAX(P,PV))+
MAX(LLCF,NV+MAX(NV,3*P,3*PV)) if P <> PV;
LRCF = 0, and
LMINR = 0, if WEIGHT = 'L' or 'N' or NW = 0; otherwise,
LMINR = NW+MAX(NW,3*M) if M = MW;
LMINR = 2*NW*MAX(M,MW)+NW+MAX(NW,3*M,3*MW) if M <> MW;
LLCF = PV*(NV+PV)+PV*NV+MAX(NV*(NV+5), PV*(PV+2),
4*PV, 4*P);
LRCF = MW*(NW+MW)+MAX(NW*(NW+5),MW*(MW+2),4*MW,4*M)
LLEFT = (N+NV)*(N+NV+MAX(N+NV,PV)+5)
if WEIGHT = 'L' or 'B' and PV > 0;
LLEFT = N*(P+5) if WEIGHT = 'R' or 'N' or PV = 0;
LRIGHT = (N+NW)*(N+NW+MAX(N+NW,MW)+5)
if WEIGHT = 'R' or 'B' and MW > 0;
LRIGHT = N*(M+5) if WEIGHT = 'L' or 'N' or MW = 0.
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: with ORDSEL = 'F', the selected order NR is greater
than NSMIN, the sum of the order of the
ALPHA-unstable part and the order of a minimal
realization of the ALPHA-stable part of the given
system; in this case, the resulting NR is set equal
to NSMIN;
= 2: with ORDSEL = 'F', the selected order NR corresponds
to repeated singular values for the ALPHA-stable
part, which are neither all included nor all
excluded from the reduced model; in this case, the
resulting NR is automatically decreased to exclude
all repeated singular values;
= 3: with ORDSEL = 'F', the selected order NR is less
than the order of the ALPHA-unstable part of the
given system; in this case NR is set equal to the
order of the ALPHA-unstable part.
= 10+K: K violations of the numerical stability condition
occured during the assignment of eigenvalues in the
SLICOT Library routines SB08CD and/or SB08DD.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the computation of the ordered real Schur form of A
failed;
= 2: the separation of the ALPHA-stable/unstable
diagonal blocks failed because of very close
eigenvalues;
= 3: the reduction to a real Schur form of the state
matrix of a minimal realization of V failed;
= 4: a failure was detected during the ordering of the
real Schur form of the state matrix of a minimal
realization of V or in the iterative process to
compute a left coprime factorization with inner
denominator;
= 5: if DICO = 'C' and the matrix AV has an observable
eigenvalue on the imaginary axis, or DICO = 'D' and
AV has an observable eigenvalue on the unit circle;
= 6: the reduction to a real Schur form of the state
matrix of a minimal realization of W failed;
= 7: a failure was detected during the ordering of the
real Schur form of the state matrix of a minimal
realization of W or in the iterative process to
compute a right coprime factorization with inner
denominator;
= 8: if DICO = 'C' and the matrix AW has a controllable
eigenvalue on the imaginary axis, or DICO = 'D' and
AW has a controllable eigenvalue on the unit circle;
= 9: the computation of eigenvalues failed;
= 10: the computation of Hankel singular values failed.
Method
Let G be the transfer-function matrix of the original
linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t), (1)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system. The subroutine AB09ID determines
the matrices of a reduced order system
d[z(t)] = Ar*z(t) + Br*u(t)
yr(t) = Cr*z(t) + Dr*u(t), (2)
such that the corresponding transfer-function matrix Gr minimizes
the norm of the frequency-weighted error
V*(G-Gr)*W, (3)
where V and W are transfer-function matrices without poles on the
imaginary axis in continuous-time case or on the unit circle in
discrete-time case.
The following procedure is used to reduce G:
1) Decompose additively G, of order N, as
G = G1 + G2,
such that G1 = (A1,B1,C1,D) has only ALPHA-stable poles and
G2 = (A2,B2,C2,0), of order NU, has only ALPHA-unstable poles.
2) Compute for G1 a B&T or SPA frequency-weighted approximation
G1r of order NR-NU using the combination method or the
modified combination method of [4].
3) Assemble the reduced model Gr as
Gr = G1r + G2.
For the frequency-weighted reduction of the ALPHA-stable part,
several methods described in [4] can be employed in conjunction
with the combination method and modified combination method
proposed in [4].
If JOB = 'B', the square-root B&T method is used.
If JOB = 'F', the balancing-free square-root version of the
B&T method is used.
If JOB = 'S', the square-root version of the SPA method is used.
If JOB = 'P', the balancing-free square-root version of the
SPA method is used.
For each of these methods, left and right truncation matrices
are determined using the Cholesky factors of an input
frequency-weighted controllability Grammian P and an output
frequency-weighted observability Grammian Q.
P and Q are computed from the controllability Grammian Pi of G*W
and the observability Grammian Qo of V*G. Using special
realizations of G*W and V*G, Pi and Qo are computed in the
partitioned forms
Pi = ( P11 P12 ) and Qo = ( Q11 Q12 ) ,
( P12' P22 ) ( Q12' Q22 )
where P11 and Q11 are the leading N-by-N parts of Pi and Qo,
respectively. Let P0 and Q0 be non-negative definite matrices
defined below
-1
P0 = P11 - ALPHAC**2*P12*P22 *P21 ,
-1
Q0 = Q11 - ALPHAO**2*Q12*Q22 *Q21.
The frequency-weighted controllability and observability
Grammians, P and Q, respectively, are defined as follows:
P = P0 if JOBC = 'S' (standard combination method [4]);
P = P1 >= P0 if JOBC = 'E', where P1 is the controllability
Grammian defined to enforce stability for a modified combination
method of [4];
Q = Q0 if JOBO = 'S' (standard combination method [4]);
Q = Q1 >= Q0 if JOBO = 'E', where Q1 is the observability
Grammian defined to enforce stability for a modified combination
method of [4].
If JOBC = JOBO = 'S' and ALPHAC = ALPHAO = 0, the choice of
Grammians corresponds to the method of Enns [1], while if
ALPHAC = ALPHAO = 1, the choice of Grammians corresponds
to the method of Lin and Chiu [2,3].
If JOBC = 'S' and ALPHAC = 1, no pole-zero cancellations must
occur in G*W. If JOBO = 'S' and ALPHAO = 1, no pole-zero
cancellations must occur in V*G. The presence of pole-zero
cancellations leads to meaningless results and must be avoided.
The frequency-weighted Hankel singular values HSV(1), ....,
HSV(N) are computed as the square roots of the eigenvalues
of the product P*Q.
References
[1] Enns, D.
Model reduction with balanced realizations: An error bound
and a frequency weighted generalization.
Proc. 23-th CDC, Las Vegas, pp. 127-132, 1984.
[2] Lin, C.-A. and Chiu, T.-Y.
Model reduction via frequency-weighted balanced realization.
Control Theory and Advanced Technology, vol. 8,
pp. 341-351, 1992.
[3] Sreeram, V., Anderson, B.D.O and Madievski, A.G.
New results on frequency weighted balanced reduction
technique.
Proc. ACC, Seattle, Washington, pp. 4004-4009, 1995.
[4] Varga, A. and Anderson, B.D.O.
Square-root balancing-free methods for the frequency-weighted
balancing related model reduction.
(report in preparation)
Numerical Aspects
The implemented methods rely on accuracy enhancing square-root techniques.Further Comments
NoneExample
Program Text
* AB09ID EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, MWMAX, NMAX, NVMAX, NWMAX, PMAX, PVMAX
PARAMETER ( MMAX = 20, MWMAX = 20,
$ NMAX = 20, NVMAX = 20, NWMAX = 20,
$ PMAX = 20, PVMAX = 20 )
INTEGER LDA, LDAV, LDAW, LDB, LDBV, LDBW,
$ LDC, LDCV, LDCW, LDD, LDDV, LDDW
PARAMETER ( LDA = NMAX, LDAV = NVMAX, LDAW = NWMAX,
$ LDB = NMAX, LDBV = NVMAX, LDBW = NWMAX,
$ LDC = PMAX, LDCV = PVMAX, LDCW = MMAX,
$ LDD = PMAX, LDDV = PVMAX, LDDW = MMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MAX( 2*NMAX,
$ NVMAX + MAX( PMAX, PVMAX ),
$ NWMAX + MAX( MMAX, MWMAX ) ) )
INTEGER LDW1, LDW2, LDW3, LDW4, LDW5, LDW6, LDW7, LDW8,
$ LDWORK
PARAMETER ( LDW1 = NMAX + NVMAX, LDW2 = NMAX + NWMAX,
$ LDW3 = MAX( LDW1*( LDW1 + MAX( LDW1, PVMAX ) +
$ 5 ), NMAX*( PMAX + 5 ) ),
$ LDW4 = MAX( LDW2*( LDW2 + MAX( LDW2, MWMAX ) +
$ 5 ), NMAX*( MMAX + 5 ) ),
$ LDW5 = PVMAX*( NVMAX + PVMAX ) + PVMAX*NVMAX +
$ MAX( NVMAX*( NVMAX + 5 ), 4*PVMAX,
$ PVMAX*( PVMAX + 2 ), 4*PMAX ),
$ LDW6 = MAX( PMAX, PVMAX )*( 2*NVMAX +
$ MAX( PMAX, PVMAX ) ) +
$ MAX( LDW5, NVMAX +
$ MAX( NVMAX, 3*PMAX, 3*PVMAX )
$ ),
$ LDW7 = MAX( NWMAX + MAX( NWMAX, 3*MMAX ),
$ 2*NWMAX*MAX( MMAX, MWMAX ) +
$ NWMAX + MAX( NWMAX, 3*MMAX,
$ 3*MWMAX ) ),
$ LDW8 = MWMAX*( NWMAX + MWMAX ) +
$ MAX( NWMAX*( NWMAX + 5 ), 4*MWMAX,
$ MWMAX*( MWMAX + 2 ), 4*MMAX ) )
PARAMETER ( LDWORK = MAX( LDW6, LDW7, LDW8,
$ 2*NMAX*NMAX +
$ MAX( 1, LDW3, LDW4,
$ 2*NMAX*NMAX + 5*NMAX,
$ NMAX*MAX( MMAX, PMAX ) ) )
$ )
* .. Local Scalars ..
LOGICAL LEFTW, RIGHTW
DOUBLE PRECISION ALPHA, ALPHAC, ALPHAO, TOL1, TOL2
INTEGER I, INFO, IWARN, J, M, MW, N, NR, NS, NV, NW, P,
$ PV
CHARACTER*1 DICO, EQUIL, JOB, JOBC, JOBO, ORDSEL, WEIGHT
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), AV(LDAV,NVMAX), AW(LDAW,NWMAX),
$ B(LDB,MMAX), BV(LDBV,PMAX), BW(LDBW,MWMAX),
$ C(LDC,NMAX), CV(LDCV,NVMAX), CW(LDCW,NWMAX),
$ D(LDD,MMAX), DV(LDDV,PMAX), DW(LDDW,MWMAX),
$ DWORK(LDWORK), HSV(NMAX)
INTEGER IWORK(LIWORK)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL AB09ID
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, NV, PV, NW, MW, NR,
$ ALPHA, ALPHAC, ALPHAO, TOL1, TOL2,
$ DICO, JOBC, JOBO, JOB, WEIGHT,
$ EQUIL, ORDSEL
LEFTW = LSAME( WEIGHT, 'L' ) .OR. LSAME( WEIGHT, 'B' )
RIGHTW = LSAME( WEIGHT, 'R' ) .OR. LSAME( WEIGHT, 'B' )
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1, N )
IF( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
IF( LEFTW ) THEN
IF( NV.LT.0 .OR. NV.GT.NVMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) NV
ELSE
IF( NV.GT.0 ) THEN
READ ( NIN, FMT = * )
$ ( ( AV(I,J), J = 1,NV ), I = 1,NV )
READ ( NIN, FMT = * )
$ ( ( BV(I,J), J = 1,P ), I = 1,NV )
IF( PV.LE.0 .OR. PV.GT.PVMAX ) THEN
WRITE ( NOUT, FMT = 99985 ) PV
ELSE
READ ( NIN, FMT = * )
$ ( ( CV(I,J), J = 1,NV ), I = 1,PV )
END IF
END IF
IF( PV.LE.0 .OR. PV.GT.PVMAX ) THEN
WRITE ( NOUT, FMT = 99985 ) PV
ELSE
READ ( NIN, FMT = * )
$ ( ( DV(I,J), J = 1,P ), I = 1,PV )
END IF
END IF
END IF
IF( RIGHTW ) THEN
IF( NW.LT.0 .OR. NW.GT.NWMAX ) THEN
WRITE ( NOUT, FMT = 99984 ) NW
ELSE
IF( NW.GT.0 ) THEN
READ ( NIN, FMT = * )
$ ( ( AW(I,J), J = 1,NW ), I = 1,NW )
IF( MW.LE.0 .OR. MW.GT.MWMAX ) THEN
WRITE ( NOUT, FMT = 99983 ) MW
ELSE
READ ( NIN, FMT = * )
$ ( ( BW(I,J), J = 1,MW ), I = 1,NW )
END IF
READ ( NIN, FMT = * )
$ ( ( CW(I,J), J = 1,NW ), I = 1,M )
END IF
IF( MW.LE.0 .OR. MW.GT.MWMAX ) THEN
WRITE ( NOUT, FMT = 99983 ) MW
ELSE
READ ( NIN, FMT = * )
$ ( ( DW(I,J), J = 1,MW ), I = 1,M )
END IF
END IF
END IF
* Find a reduced ssr for (A,B,C,D).
CALL AB09ID( DICO, JOBC, JOBO, JOB, WEIGHT, EQUIL,
$ ORDSEL, N, M, P, NV, PV, NW, MW, NR, ALPHA,
$ ALPHAC, ALPHAO, A, LDA, B, LDB, C, LDC, D,
$ LDD, AV, LDAV, BV, LDBV, CV, LDCV, DV, LDDV,
$ AW, LDAW, BW, LDBW, CW, LDCW, DW, LDDW,
$ NS, HSV, TOL1, TOL2, IWORK, DWORK, LDWORK,
$ IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF( IWARN.NE.0) WRITE ( NOUT, FMT = 99982 ) IWARN
WRITE ( NOUT, FMT = 99997 ) NR
WRITE ( NOUT, FMT = 99987 )
WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1, NS )
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
20 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
60 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 70 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1,M )
70 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB09ID EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB09ID = ',I2)
99997 FORMAT (/' The order of reduced model = ',I2)
99996 FORMAT (/' The reduced state dynamics matrix Ar is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The reduced input/state matrix Br is ')
99992 FORMAT (/' The reduced state/output matrix Cr is ')
99991 FORMAT (/' The reduced input/output matrix Dr is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99987 FORMAT (/' The Hankel singular values of weighted ALPHA-stable',
$ ' part are')
99986 FORMAT (/' NV is out of range.',/' NV = ',I5)
99985 FORMAT (/' PV is out of range.',/' PV = ',I5)
99984 FORMAT (/' NW is out of range.',/' NW = ',I5)
99983 FORMAT (/' MW is out of range.',/' MW = ',I5)
99982 FORMAT (' IWARN on exit from AB09ID = ',I2)
END
Program Data
AB09ID EXAMPLE PROGRAM DATA (Continuous system)
3 1 1 6 1 0 0 2 0.0 0.0 0.0 0.1E0 0.0 C S S F L S F
-26.4000 6.4023 4.3868
32.0000 0 0
0 8.0000 0
16
0
0
9.2994 1.1624 0.1090
0
-1.0000 0 4.0000 -9.2994 -1.1624 -0.1090
0 2.0000 0 -9.2994 -1.1624 -0.1090
0 0 -3.0000 -9.2994 -1.1624 -0.1090
16.0000 16.0000 16.0000 -26.4000 6.4023 4.3868
0 0 0 32.0000 0 0
0 0 0 0 8.0000 0
1
1
1
0
0
0
1 1 1 0 0 0
0
Program Results
AB09ID EXAMPLE PROGRAM RESULTS The order of reduced model = 2 The Hankel singular values of weighted ALPHA-stable part are 3.8253 0.2005 The reduced state dynamics matrix Ar is 9.1900 0.0000 0.0000 -34.5297 The reduced input/state matrix Br is 11.9593 16.9329 The reduced state/output matrix Cr is 2.8955 6.9152 The reduced input/output matrix Dr is 0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB09IX.html 0000664 0000000 0000000 00000032551 14560147231 0020354 0 ustar 00root root 0000000 0000000
Purpose
To compute a reduced order model (Ar,Br,Cr,Dr) for an original
state-space representation (A,B,C,D) by using the square-root or
balancing-free square-root Balance & Truncate (B&T) or
Singular Perturbation Approximation (SPA) model reduction methods.
The computation of truncation matrices TI and T is based on
the Cholesky factor S of a controllability Grammian P = S*S'
and the Cholesky factor R of an observability Grammian Q = R'*R,
where S and R are given upper triangular matrices.
For the B&T approach, the matrices of the reduced order system
are computed using the truncation formulas:
Ar = TI * A * T , Br = TI * B , Cr = C * T . (1)
For the SPA approach, the matrices of a minimal realization
(Am,Bm,Cm) are computed using the truncation formulas:
Am = TI * A * T , Bm = TI * B , Cm = C * T . (2)
Am, Bm, Cm and D serve further for computing the SPA of the given
system.
Specification
SUBROUTINE AB09IX( DICO, JOB, FACT, ORDSEL, N, M, P, NR,
$ SCALEC, SCALEO, A, LDA, B, LDB, C, LDC, D, LDD,
$ TI, LDTI, T, LDT, NMINR, HSV, TOL1, TOL2,
$ IWORK, DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, FACT, JOB, ORDSEL
INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, LDT, LDTI,
$ LDWORK, M, N, NMINR, NR, P
DOUBLE PRECISION SCALEC, SCALEO, TOL1, TOL2
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), HSV(*), T(LDT,*), TI(LDTI,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOB CHARACTER*1
Specifies the model reduction approach to be used
as follows:
= 'B': use the square-root B&T method;
= 'F': use the balancing-free square-root B&T method;
= 'S': use the square-root SPA method;
= 'P': use the balancing-free square-root SPA method.
FACT CHARACTER*1
Specifies whether or not, on entry, the matrix A is in a
real Schur form, as follows:
= 'S': A is in a real Schur form;
= 'N': A is a general dense square matrix.
ORDSEL CHARACTER*1
Specifies the order selection method as follows:
= 'F': the resulting order NR is fixed;
= 'A': the resulting order NR is automatically determined
on basis of the given tolerance TOL1.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation,
i.e., the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NR (input/output) INTEGER
On entry with ORDSEL = 'F', NR is the desired order of
the resulting reduced order system. 0 <= NR <= N.
On exit, if INFO = 0, NR is the order of the resulting
reduced order model. NR is set as follows:
if ORDSEL = 'F', NR is equal to MIN(NR,NMINR), where NR
is the desired order on entry and NMINR is the number of
the Hankel singular values greater than N*EPS*S1, where
EPS is the machine precision (see LAPACK Library Routine
DLAMCH) and S1 is the largest Hankel singular value
(computed in HSV(1));
NR can be further reduced to ensure HSV(NR) > HSV(NR+1);
if ORDSEL = 'A', NR is equal to the number of Hankel
singular values greater than MAX(TOL1,N*EPS*S1).
SCALEC (input) DOUBLE PRECISION
Scaling factor for the Cholesky factor S of the
controllability Grammian, i.e., S/SCALEC is used to
compute the Hankel singular values. SCALEC > 0.
SCALEO (input) DOUBLE PRECISION
Scaling factor for the Cholesky factor R of the
observability Grammian, i.e., R/SCALEO is used to
compute the Hankel singular values. SCALEO > 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A. If FACT = 'S',
A is in a real Schur form.
On exit, if INFO = 0, the leading NR-by-NR part of this
array contains the state dynamics matrix Ar of the
reduced order system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, if INFO = 0, the leading NR-by-M part of this
array contains the input/state matrix Br of the reduced
order system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, if INFO = 0, the leading P-by-NR part of this
array contains the state/output matrix Cr of the reduced
order system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, if JOB = 'S' or JOB = 'P', the leading P-by-M
part of this array must contain the original input/output
matrix D.
On exit, if INFO = 0 and JOB = 'S' or JOB = 'P', the
leading P-by-M part of this array contains the
input/output matrix Dr of the reduced order system.
If JOB = 'B' or JOB = 'F', this array is not referenced.
LDD INTEGER
The leading dimension of array D.
LDD >= 1, if JOB = 'B' or JOB = 'F';
LDD >= MAX(1,P), if JOB = 'S' or JOB = 'P'.
TI (input/output) DOUBLE PRECISION array, dimension (LDTI,N)
On entry, the leading N-by-N upper triangular part of
this array must contain the Cholesky factor S of a
controllability Grammian P = S*S'.
On exit, if INFO = 0, and NR > 0, the leading NMINR-by-N
part of this array contains the left truncation matrix
TI in (1), for the B&T approach, or in (2), for the
SPA approach.
LDTI INTEGER
The leading dimension of array TI. LDTI >= MAX(1,N).
T (input/output) DOUBLE PRECISION array, dimension (LDT,N)
On entry, the leading N-by-N upper triangular part of
this array must contain the Cholesky factor R of an
observability Grammian Q = R'*R.
On exit, if INFO = 0, and NR > 0, the leading N-by-NMINR
part of this array contains the right truncation matrix
T in (1), for the B&T approach, or in (2), for the
SPA approach.
LDT INTEGER
The leading dimension of array T. LDT >= MAX(1,N).
NMINR (output) INTEGER
The number of Hankel singular values greater than
MAX(TOL2,N*EPS*S1).
Note: If S and R are the Cholesky factors of the
controllability and observability Grammians of the
original system (A,B,C,D), respectively, then NMINR is
the order of a minimal realization of the original system.
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, it contains the Hankel singular values,
ordered decreasingly. The Hankel singular values are
singular values of the product R*S.
Tolerances
TOL1 DOUBLE PRECISION
If ORDSEL = 'A', TOL1 contains the tolerance for
determining the order of the reduced system.
For model reduction, the recommended value lies in the
interval [0.00001,0.001].
If TOL1 <= 0 on entry, the used default value is
TOL1 = N*EPS*S1, where EPS is the machine precision
(see LAPACK Library Routine DLAMCH) and S1 is the largest
Hankel singular value (computed in HSV(1)).
If ORDSEL = 'F', the value of TOL1 is ignored.
TOL2 DOUBLE PRECISION
The tolerance for determining the order of a minimal
realization of the system.
The recommended value is TOL2 = N*EPS*S1.
This value is used by default if TOL2 <= 0 on entry.
If TOL2 > 0, and ORDSEL = 'A', then TOL2 <= TOL1.
Workspace
IWORK INTEGER array, dimension (LIWORK), where
LIWORK = 0, if JOB = 'B';
LIWORK = N, if JOB = 'F';
LIWORK = 2*N, if JOB = 'S' or 'P'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( 1, 2*N*N + 5*N, N*MAX(M,P) ).
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: with ORDSEL = 'F', the selected order NR is greater
than NMINR, the order of a minimal realization of
the given system; in this case, the resulting NR is
set automatically to NMINR;
= 2: with ORDSEL = 'F', the selected order NR corresponds
to repeated singular values, which are neither all
included nor all excluded from the reduced model;
in this case, the resulting NR is set automatically
to the largest value such that HSV(NR) > HSV(NR+1).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the computation of Hankel singular values failed.
Method
Let be the stable linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t), (3)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system. The subroutine AB09IX determines for
the given system (3), the matrices of a reduced NR order system
d[z(t)] = Ar*z(t) + Br*u(t)
yr(t) = Cr*z(t) + Dr*u(t), (4)
by using the square-root or balancing-free square-root
Balance & Truncate (B&T) or Singular Perturbation Approximation
(SPA) model reduction methods.
The projection matrices TI and T are determined using the
Cholesky factors S and R of a controllability Grammian P and an
observability Grammian Q.
The Hankel singular values HSV(1), ...., HSV(N) are computed as
singular values of the product R*S.
If JOB = 'B', the square-root Balance & Truncate technique
of [1] is used.
If JOB = 'F', the balancing-free square-root version of the
Balance & Truncate technique [2] is used.
If JOB = 'S', the square-root version of the Singular Perturbation
Approximation method [3,4] is used.
If JOB = 'P', the balancing-free square-root version of the
Singular Perturbation Approximation method [3,4] is used.
References
[1] Tombs M.S. and Postlethwaite I.
Truncated balanced realization of stable, non-minimal
state-space systems.
Int. J. Control, Vol. 46, pp. 1319-1330, 1987.
[2] Varga A.
Efficient minimal realization procedure based on balancing.
Proc. of IMACS/IFAC Symp. MCTS, Lille, France, May 1991,
A. El Moudni, P. Borne, S. G. Tzafestas (Eds.),
Vol. 2, pp. 42-46.
[3] Liu Y. and Anderson B.D.O.
Singular Perturbation Approximation of balanced systems.
Int. J. Control, Vol. 50, pp. 1379-1405, 1989.
[4] Varga A.
Balancing-free square-root algorithm for computing singular
perturbation approximations.
Proc. 30-th CDC, Brighton, Dec. 11-13, 1991,
Vol. 2, pp. 1062-1065.
Numerical Aspects
The implemented method relies on accuracy enhancing square-root or balancing-free square-root methods.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute for given state-space representations (A,B,C,0), (AV,BV,CV,DV), and (AW,BW,CW,DW) of the transfer-function matrices G, V and W, respectively, the Cholesky factors of the frequency-weighted controllability and observability Grammians corresponding to a frequency-weighted model reduction problem. G, V and W must be stable transfer-function matrices with the state matrices A, AV, and AW in real Schur form. It is assumed that the state space realizations (AV,BV,CV,DV) and (AW,BW,CW,DW) are minimal. In case of possible pole-zero cancellations in forming V*G and/or G*W, the parameters for the choice of frequency-weighted Grammians ALPHAO and/or ALPHAC, respectively, must be different from 1.Specification
SUBROUTINE AB09IY( DICO, JOBC, JOBO, WEIGHT, N, M, P, NV, PV,
$ NW, MW, ALPHAC, ALPHAO, A, LDA, B, LDB, C, LDC,
$ AV, LDAV, BV, LDBV, CV, LDCV, DV, LDDV,
$ AW, LDAW, BW, LDBW, CW, LDCW, DW, LDDW,
$ SCALEC, SCALEO, S, LDS, R, LDR,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, JOBC, JOBO, WEIGHT
INTEGER INFO, LDA, LDAV, LDAW, LDB, LDBV, LDBW,
$ LDC, LDCV, LDCW, LDDV, LDDW, LDR, LDS, LDWORK,
$ M, MW, N, NV, NW, P, PV
DOUBLE PRECISION ALPHAC, ALPHAO, SCALEC, SCALEO
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), AV(LDAV,*), AW(LDAW,*),
$ B(LDB,*), BV(LDBV,*), BW(LDBW,*),
$ C(LDC,*), CV(LDCV,*), CW(LDCW,*),
$ DV(LDDV,*), DW(LDDW,*),
$ DWORK(*), R(LDR,*), S(LDS,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the systems as follows:
= 'C': G, V and W are continuous-time systems;
= 'D': G, V and W are discrete-time systems.
JOBC CHARACTER*1
Specifies the choice of frequency-weighted controllability
Grammian as follows:
= 'S': choice corresponding to a combination method [4]
of the approaches of Enns [1] and Lin-Chiu [2,3];
= 'E': choice corresponding to the stability enhanced
modified combination method of [4].
JOBO CHARACTER*1
Specifies the choice of frequency-weighted observability
Grammian as follows:
= 'S': choice corresponding to a combination method [4]
of the approaches of Enns [1] and Lin-Chiu [2,3];
= 'E': choice corresponding to the stability enhanced
modified combination method of [4].
WEIGHT CHARACTER*1
Specifies the type of frequency weighting, as follows:
= 'N': no weightings are used (V = I, W = I);
= 'L': only left weighting V is used (W = I);
= 'R': only right weighting W is used (V = I);
= 'B': both left and right weightings V and W are used.
Input/Output Parameters
N (input) INTEGER
The order of the state-space representation of G, i.e.,
the order of the matrix A. N >= 0.
M (input) INTEGER
The number of columns of the matrix B and
the number of rows of the matrices CW and DW. M >= 0.
M represents the dimension of the input vector of the
system with the transfer-function matrix G and
also the dimension of the output vector of the system
with the transfer-function matrix W.
P (input) INTEGER
The number of rows of the matrix C and the
number of columns of the matrices BV and DV. P >= 0.
P represents the dimension of the output vector of the
system with the transfer-function matrix G and
also the dimension of the input vector of the system
with the transfer-function matrix V.
NV (input) INTEGER
The order of the matrix AV. Also the number of rows of
the matrix BV and the number of columns of the matrix CV.
NV represents the dimension of the state vector of the
system with the transfer-function matrix V. NV >= 0.
PV (input) INTEGER
The number of rows of the matrices CV and DV. PV >= 0.
PV represents the dimension of the output vector of the
system with the transfer-function matrix V.
NW (input) INTEGER
The order of the matrix AW. Also the number of rows of
the matrix BW and the number of columns of the matrix CW.
NW represents the dimension of the state vector of the
system with the transfer-function matrix W. NW >= 0.
MW (input) INTEGER
The number of columns of the matrices BW and DW. MW >= 0.
MW represents the dimension of the input vector of the
system with the transfer-function matrix W.
ALPHAC (input) DOUBLE PRECISION
Combination method parameter for defining the
frequency-weighted controllability Grammian (see METHOD);
ABS(ALPHAC) <= 1.
ALPHAO (input) DOUBLE PRECISION
Combination method parameter for defining the
frequency-weighted observability Grammian (see METHOD);
ABS(ALPHAO) <= 1.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must
contain the state matrix A (of the system with the
transfer-function matrix G) in a real Schur form.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
input/state matrix B.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array must contain the
state/output matrix C.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
AV (input) DOUBLE PRECISION array, dimension (LDAV,NV)
If WEIGHT = 'L' or 'B', the leading NV-by-NV part of this
array must contain the state matrix AV (of the system with
the transfer-function matrix V) in a real Schur form.
AV is not referenced if WEIGHT = 'R' or 'N'.
LDAV INTEGER
The leading dimension of array AV.
LDAV >= MAX(1,NV), if WEIGHT = 'L' or 'B';
LDAV >= 1, if WEIGHT = 'R' or 'N'.
BV (input) DOUBLE PRECISION array, dimension (LDBV,P)
If WEIGHT = 'L' or 'B', the leading NV-by-P part of this
array must contain the input matrix BV of the system with
the transfer-function matrix V.
BV is not referenced if WEIGHT = 'R' or 'N'.
LDBV INTEGER
The leading dimension of array BV.
LDBV >= MAX(1,NV), if WEIGHT = 'L' or 'B';
LDBV >= 1, if WEIGHT = 'R' or 'N'.
CV (input) DOUBLE PRECISION array, dimension (LDCV,NV)
If WEIGHT = 'L' or 'B', the leading PV-by-NV part of this
array must contain the output matrix CV of the system with
the transfer-function matrix V.
CV is not referenced if WEIGHT = 'R' or 'N'.
LDCV INTEGER
The leading dimension of array CV.
LDCV >= MAX(1,PV), if WEIGHT = 'L' or 'B';
LDCV >= 1, if WEIGHT = 'R' or 'N'.
DV (input) DOUBLE PRECISION array, dimension (LDDV,P)
If WEIGHT = 'L' or 'B', the leading PV-by-P part of this
array must contain the feedthrough matrix DV of the system
with the transfer-function matrix V.
DV is not referenced if WEIGHT = 'R' or 'N'.
LDDV INTEGER
The leading dimension of array DV.
LDDV >= MAX(1,PV), if WEIGHT = 'L' or 'B';
LDDV >= 1, if WEIGHT = 'R' or 'N'.
AW (input) DOUBLE PRECISION array, dimension (LDAW,NW)
If WEIGHT = 'R' or 'B', the leading NW-by-NW part of this
array must contain the state matrix AW (of the system with
the transfer-function matrix W) in a real Schur form.
AW is not referenced if WEIGHT = 'L' or 'N'.
LDAW INTEGER
The leading dimension of array AW.
LDAW >= MAX(1,NW), if WEIGHT = 'R' or 'B';
LDAW >= 1, if WEIGHT = 'L' or 'N'.
BW (input) DOUBLE PRECISION array, dimension (LDBW,MW)
If WEIGHT = 'R' or 'B', the leading NW-by-MW part of this
array must contain the input matrix BW of the system with
the transfer-function matrix W.
BW is not referenced if WEIGHT = 'L' or 'N'.
LDBW INTEGER
The leading dimension of array BW.
LDBW >= MAX(1,NW), if WEIGHT = 'R' or 'B';
LDBW >= 1, if WEIGHT = 'L' or 'N'.
CW (input) DOUBLE PRECISION array, dimension (LDCW,NW)
If WEIGHT = 'R' or 'B', the leading M-by-NW part of this
array must contain the output matrix CW of the system with
the transfer-function matrix W.
CW is not referenced if WEIGHT = 'L' or 'N'.
LDCW INTEGER
The leading dimension of array CW.
LDCW >= MAX(1,M), if WEIGHT = 'R' or 'B';
LDCW >= 1, if WEIGHT = 'L' or 'N'.
DW (input) DOUBLE PRECISION array, dimension (LDDW,MW)
If WEIGHT = 'R' or 'B', the leading M-by-MW part of this
array must contain the feedthrough matrix DW of the system
with the transfer-function matrix W.
DW is not referenced if WEIGHT = 'L' or 'N'.
LDDW INTEGER
The leading dimension of array DW.
LDDW >= MAX(1,M), if WEIGHT = 'R' or 'B';
LDDW >= 1, if WEIGHT = 'L' or 'N'.
SCALEC (output) DOUBLE PRECISION
Scaling factor for the controllability Grammian in (1)
or (3). See METHOD.
SCALEO (output) DOUBLE PRECISION
Scaling factor for the observability Grammian in (2)
or (4). See METHOD.
S (output) DOUBLE PRECISION array, dimension (LDS,N)
The leading N-by-N upper triangular part of this array
contains the Cholesky factor S of the frequency-weighted
cotrollability Grammian P = S*S'. See METHOD.
LDS INTEGER
The leading dimension of array S. LDS >= MAX(1,N).
R (output) DOUBLE PRECISION array, dimension (LDR,N)
The leading N-by-N upper triangular part of this array
contains the Cholesky factor R of the frequency-weighted
observability Grammian Q = R'*R. See METHOD.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( 1, LLEFT, LRIGHT ),
where
LLEFT = (N+NV)*(N+NV+MAX(N+NV,PV)+5)
if WEIGHT = 'L' or 'B' and PV > 0;
LLEFT = N*(P+5) if WEIGHT = 'R' or 'N' or PV = 0;
LRIGHT = (N+NW)*(N+NW+MAX(N+NW,MW)+5)
if WEIGHT = 'R' or 'B' and MW > 0;
LRIGHT = N*(M+5) if WEIGHT = 'L' or 'N' or MW = 0.
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the state matrices A and/or AV are not stable or
not in a real Schur form;
= 2: if the state matrices A and/or AW are not stable or
not in a real Schur form;
= 3: eigenvalues computation failure.
Method
Let Pi = Si*Si' and Qo = Ro'*Ro be the Cholesky factored
controllability and observability Grammians satisfying
in the continuous-time case
Ai*Pi + Pi*Ai' + scalec^2*Bi*Bi' = 0, (1)
Ao'*Qo + Qo*Ao + scaleo^2*Co'*Co = 0, (2)
and in the discrete-time case
Ai*Pi*Ai' - Pi + scalec^2*Bi*Bi' = 0, (3)
Ao'*Qo*Ao - Qo + scaleo^2*Co'*Co = 0, (4)
where
Ai = ( A B*Cw ) , Bi = ( B*Dw ) ,
( 0 Aw ) ( Bw )
Ao = ( A 0 ) , Co = ( Dv*C Cv ) .
( Bv*C Av )
Consider the partitioned Grammians
Pi = ( P11 P12 ) and Qo = ( Q11 Q12 ) ,
( P12' P22 ) ( Q12' Q22 )
where P11 and Q11 are the leading N-by-N parts of Pi and Qo,
respectively, and let P0 and Q0 be non-negative definite matrices
defined in the combination method [4]
-1
P0 = P11 - ALPHAC**2*P12*P22 *P21 ,
-1
Q0 = Q11 - ALPHAO**2*Q12*Q22 *Q21.
The frequency-weighted controllability and observability
Grammians, P and Q, respectively, are defined as follows:
P = P0 if JOBC = 'S' (standard combination method [4]);
P = P1 >= P0 if JOBC = 'E', where P1 is the controllability
Grammian defined to enforce stability for a modified combination
method of [4];
Q = Q0 if JOBO = 'S' (standard combination method [4]);
Q = Q1 >= Q0 if JOBO = 'E', where Q1 is the observability
Grammian defined to enforce stability for a modified combination
method of [4].
If JOBC = JOBO = 'S' and ALPHAC = ALPHAO = 0, the choice of
Grammians corresponds to the method of Enns [1], while if
ALPHAC = ALPHAO = 1, the choice of Grammians corresponds to the
method of Lin and Chiu [2,3].
The routine computes directly the Cholesky factors S and R
such that P = S*S' and Q = R'*R according to formulas
developed in [4]. No matrix inversions are involved.
References
[1] Enns, D.
Model reduction with balanced realizations: An error bound
and a frequency weighted generalization.
Proc. CDC, Las Vegas, pp. 127-132, 1984.
[2] Lin, C.-A. and Chiu, T.-Y.
Model reduction via frequency-weighted balanced realization.
Control Theory and Advanced Technology, vol. 8,
pp. 341-351, 1992.
[3] Sreeram, V., Anderson, B.D.O and Madievski, A.G.
New results on frequency weighted balanced reduction
technique.
Proc. ACC, Seattle, Washington, pp. 4004-4009, 1995.
[4] Varga, A. and Anderson, B.D.O.
Square-root balancing-free methods for the frequency-weighted
balancing related model reduction.
(report in preparation)
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute a reduced order model (Ar,Br,Cr,Dr) for an original
state-space representation (A,B,C,D) by using the frequency
weighted optimal Hankel-norm approximation method.
The Hankel norm of the weighted error
op(V)*(G-Gr)*op(W)
is minimized, where G and Gr are the transfer-function matrices
of the original and reduced systems, respectively, V and W are
invertible transfer-function matrices representing the left and
right frequency weights, and op(X) denotes X, inv(X), conj(X) or
conj(inv(X)). V and W are specified by their state space
realizations (AV,BV,CV,DV) and (AW,BW,CW,DW), respectively.
When minimizing ||V*(G-Gr)*W||, V and W must be antistable.
When minimizing inv(V)*(G-Gr)*inv(W), V and W must have only
antistable zeros.
When minimizing conj(V)*(G-Gr)*conj(W), V and W must be stable.
When minimizing conj(inv(V))*(G-Gr)*conj(inv(W)), V and W must
be minimum-phase.
If the original system is unstable, then the frequency weighted
Hankel-norm approximation is computed only for the
ALPHA-stable part of the system.
For a transfer-function matrix G, conj(G) denotes the conjugate
of G given by G'(-s) for a continuous-time system or G'(1/z)
for a discrete-time system.
Specification
SUBROUTINE AB09JD( JOBV, JOBW, JOBINV, DICO, EQUIL, ORDSEL,
$ N, NV, NW, M, P, NR, ALPHA, A, LDA, B, LDB,
$ C, LDC, D, LDD, AV, LDAV, BV, LDBV,
$ CV, LDCV, DV, LDDV, AW, LDAW, BW, LDBW,
$ CW, LDCW, DW, LDDW, NS, HSV, TOL1, TOL2,
$ IWORK, DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, EQUIL, JOBINV, JOBV, JOBW, ORDSEL
INTEGER INFO, IWARN, LDA, LDAV, LDAW, LDB, LDBV, LDBW,
$ LDC, LDCV, LDCW, LDD, LDDV, LDDW, LDWORK, M, N,
$ NR, NS, NV, NW, P
DOUBLE PRECISION ALPHA, TOL1, TOL2
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), AV(LDAV,*), AW(LDAW,*),
$ B(LDB,*), BV(LDBV,*), BW(LDBW,*),
$ C(LDC,*), CV(LDCV,*), CW(LDCW,*),
$ D(LDD,*), DV(LDDV,*), DW(LDDW,*), DWORK(*),
$ HSV(*)
Arguments
Mode Parameters
JOBV CHARACTER*1
Specifies the left frequency-weighting as follows:
= 'N': V = I;
= 'V': op(V) = V;
= 'I': op(V) = inv(V);
= 'C': op(V) = conj(V);
= 'R': op(V) = conj(inv(V)).
JOBW CHARACTER*1
Specifies the right frequency-weighting as follows:
= 'N': W = I;
= 'W': op(W) = W;
= 'I': op(W) = inv(W);
= 'C': op(W) = conj(W);
= 'R': op(W) = conj(inv(W)).
JOBINV CHARACTER*1
Specifies the computational approach to be used as
follows:
= 'N': use the inverse free descriptor system approach;
= 'I': use the inversion based standard approach;
= 'A': switch automatically to the inverse free
descriptor approach in case of badly conditioned
feedthrough matrices in V or W (see METHOD).
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily
equilibrate the triplet (A,B,C) as follows:
= 'S': perform equilibration (scaling);
= 'N': do not perform equilibration.
ORDSEL CHARACTER*1
Specifies the order selection method as follows:
= 'F': the resulting order NR is fixed;
= 'A': the resulting order NR is automatically determined
on basis of the given tolerance TOL1.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation,
i.e., the order of the matrix A. N >= 0.
NV (input) INTEGER
The order of the realization of the left frequency
weighting V, i.e., the order of the matrix AV. NV >= 0.
NW (input) INTEGER
The order of the realization of the right frequency
weighting W, i.e., the order of the matrix AW. NW >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NR (input/output) INTEGER
On entry with ORDSEL = 'F', NR is the desired order of
the resulting reduced order system. 0 <= NR <= N.
On exit, if INFO = 0, NR is the order of the resulting
reduced order model. For a system with NU ALPHA-unstable
eigenvalues and NS ALPHA-stable eigenvalues (NU+NS = N),
NR is set as follows: if ORDSEL = 'F', NR is equal to
NU+MIN(MAX(0,NR-NU-KR+1),NMIN), where KR is the
multiplicity of the Hankel singular value HSV(NR-NU+1),
NR is the desired order on entry, and NMIN is the order
of a minimal realization of the ALPHA-stable part of the
given system; NMIN is determined as the number of Hankel
singular values greater than NS*EPS*HNORM(As,Bs,Cs), where
EPS is the machine precision (see LAPACK Library Routine
DLAMCH) and HNORM(As,Bs,Cs) is the Hankel norm of the
ALPHA-stable part of the weighted system (computed in
HSV(1));
if ORDSEL = 'A', NR is the sum of NU and the number of
Hankel singular values greater than
MAX(TOL1,NS*EPS*HNORM(As,Bs,Cs)).
ALPHA (input) DOUBLE PRECISION
Specifies the ALPHA-stability boundary for the eigenvalues
of the state dynamics matrix A. For a continuous-time
system (DICO = 'C'), ALPHA <= 0 is the boundary value for
the real parts of eigenvalues, while for a discrete-time
system (DICO = 'D'), 0 <= ALPHA <= 1 represents the
boundary value for the moduli of eigenvalues.
The ALPHA-stability domain does not include the boundary.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A.
On exit, if INFO = 0, the leading NR-by-NR part of this
array contains the state dynamics matrix Ar of the
reduced order system in a real Schur form.
The resulting A has a block-diagonal form with two blocks.
For a system with NU ALPHA-unstable eigenvalues and
NS ALPHA-stable eigenvalues (NU+NS = N), the leading
NU-by-NU block contains the unreduced part of A
corresponding to ALPHA-unstable eigenvalues.
The trailing (NR+NS-N)-by-(NR+NS-N) block contains
the reduced part of A corresponding to ALPHA-stable
eigenvalues.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, if INFO = 0, the leading NR-by-M part of this
array contains the input/state matrix Br of the reduced
order system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, if INFO = 0, the leading P-by-NR part of this
array contains the state/output matrix Cr of the reduced
order system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the original input/output matrix D.
On exit, if INFO = 0, the leading P-by-M part of this
array contains the input/output matrix Dr of the reduced
order system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
AV (input/output) DOUBLE PRECISION array, dimension (LDAV,NV)
On entry, if JOBV <> 'N', the leading NV-by-NV part of
this array must contain the state matrix AV of a state
space realization of the left frequency weighting V.
On exit, if JOBV <> 'N', and INFO = 0, the leading
NV-by-NV part of this array contains the real Schur form
of AV.
AV is not referenced if JOBV = 'N'.
LDAV INTEGER
The leading dimension of the array AV.
LDAV >= MAX(1,NV), if JOBV <> 'N';
LDAV >= 1, if JOBV = 'N'.
BV (input/output) DOUBLE PRECISION array, dimension (LDBV,P)
On entry, if JOBV <> 'N', the leading NV-by-P part of
this array must contain the input matrix BV of a state
space realization of the left frequency weighting V.
On exit, if JOBV <> 'N', and INFO = 0, the leading
NV-by-P part of this array contains the transformed
input matrix BV corresponding to the transformed AV.
BV is not referenced if JOBV = 'N'.
LDBV INTEGER
The leading dimension of the array BV.
LDBV >= MAX(1,NV), if JOBV <> 'N';
LDBV >= 1, if JOBV = 'N'.
CV (input/output) DOUBLE PRECISION array, dimension (LDCV,NV)
On entry, if JOBV <> 'N', the leading P-by-NV part of
this array must contain the output matrix CV of a state
space realization of the left frequency weighting V.
On exit, if JOBV <> 'N', and INFO = 0, the leading
P-by-NV part of this array contains the transformed output
matrix CV corresponding to the transformed AV.
CV is not referenced if JOBV = 'N'.
LDCV INTEGER
The leading dimension of the array CV.
LDCV >= MAX(1,P), if JOBV <> 'N';
LDCV >= 1, if JOBV = 'N'.
DV (input) DOUBLE PRECISION array, dimension (LDDV,P)
If JOBV <> 'N', the leading P-by-P part of this array
must contain the feedthrough matrix DV of a state space
realization of the left frequency weighting V.
DV is not referenced if JOBV = 'N'.
LDDV INTEGER
The leading dimension of the array DV.
LDDV >= MAX(1,P), if JOBV <> 'N';
LDDV >= 1, if JOBV = 'N'.
AW (input/output) DOUBLE PRECISION array, dimension (LDAW,NW)
On entry, if JOBW <> 'N', the leading NW-by-NW part of
this array must contain the state matrix AW of a state
space realization of the right frequency weighting W.
On exit, if JOBW <> 'N', and INFO = 0, the leading
NW-by-NW part of this array contains the real Schur form
of AW.
AW is not referenced if JOBW = 'N'.
LDAW INTEGER
The leading dimension of the array AW.
LDAW >= MAX(1,NW), if JOBW <> 'N';
LDAW >= 1, if JOBW = 'N'.
BW (input/output) DOUBLE PRECISION array, dimension (LDBW,M)
On entry, if JOBW <> 'N', the leading NW-by-M part of
this array must contain the input matrix BW of a state
space realization of the right frequency weighting W.
On exit, if JOBW <> 'N', and INFO = 0, the leading
NW-by-M part of this array contains the transformed
input matrix BW corresponding to the transformed AW.
BW is not referenced if JOBW = 'N'.
LDBW INTEGER
The leading dimension of the array BW.
LDBW >= MAX(1,NW), if JOBW <> 'N';
LDBW >= 1, if JOBW = 'N'.
CW (input/output) DOUBLE PRECISION array, dimension (LDCW,NW)
On entry, if JOBW <> 'N', the leading M-by-NW part of
this array must contain the output matrix CW of a state
space realization of the right frequency weighting W.
On exit, if JOBW <> 'N', and INFO = 0, the leading
M-by-NW part of this array contains the transformed output
matrix CW corresponding to the transformed AW.
CW is not referenced if JOBW = 'N'.
LDCW INTEGER
The leading dimension of the array CW.
LDCW >= MAX(1,M), if JOBW <> 'N';
LDCW >= 1, if JOBW = 'N'.
DW (input) DOUBLE PRECISION array, dimension (LDDW,M)
If JOBW <> 'N', the leading M-by-M part of this array
must contain the feedthrough matrix DW of a state space
realization of the right frequency weighting W.
DW is not referenced if JOBW = 'N'.
LDDW INTEGER
The leading dimension of the array DW.
LDDW >= MAX(1,M), if JOBW <> 'N';
LDDW >= 1, if JOBW = 'N'.
NS (output) INTEGER
The dimension of the ALPHA-stable subsystem.
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, the leading NS elements of this array contain
the Hankel singular values, ordered decreasingly, of the
projection G1s of op(V)*G1*op(W) (see METHOD), where G1
is the ALPHA-stable part of the original system.
Tolerances
TOL1 DOUBLE PRECISION
If ORDSEL = 'A', TOL1 contains the tolerance for
determining the order of reduced system.
For model reduction, the recommended value is
TOL1 = c*HNORM(G1s), where c is a constant in the
interval [0.00001,0.001], and HNORM(G1s) is the
Hankel-norm of the projection G1s of op(V)*G1*op(W)
(see METHOD), computed in HSV(1).
If TOL1 <= 0 on entry, the used default value is
TOL1 = NS*EPS*HNORM(G1s), where NS is the number of
ALPHA-stable eigenvalues of A and EPS is the machine
precision (see LAPACK Library Routine DLAMCH).
If ORDSEL = 'F', the value of TOL1 is ignored.
TOL1 < 1.
TOL2 DOUBLE PRECISION
The tolerance for determining the order of a minimal
realization of the ALPHA-stable part of the given system.
The recommended value is TOL2 = NS*EPS*HNORM(G1s).
This value is used by default if TOL2 <= 0 on entry.
If TOL2 > 0 and ORDSEL = 'A', then TOL2 <= TOL1.
TOL2 < 1.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK = MAX(1,M,c,d), if DICO = 'C',
LIWORK = MAX(1,N,M,c,d), if DICO = 'D', where
c = 0, if JOBV = 'N',
c = MAX(2*P,NV+P+N+6,2*NV+P+2), if JOBV <> 'N',
d = 0, if JOBW = 'N',
d = MAX(2*M,NW+M+N+6,2*NW+M+2), if JOBW <> 'N'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( LDW1, LDW2, LDW3, LDW4 ), where
for NVP = NV+P and NWM = NW+M we have
LDW1 = 0 if JOBV = 'N' and
LDW1 = 2*NVP*(NVP+P) + P*P +
MAX( 2*NVP*NVP + MAX( 11*NVP+16, P*NVP ),
NVP*N + MAX( NVP*N+N*N, P*N, P*M ) )
if JOBV <> 'N',
LDW2 = 0 if JOBW = 'N' and
LDW2 = 2*NWM*(NWM+M) + M*M +
MAX( 2*NWM*NWM + MAX( 11*NWM+16, M*NWM ),
NWM*N + MAX( NWM*N+N*N, M*N, P*M ) )
if JOBW <> 'N',
LDW3 = N*(2*N + MAX(N,M,P) + 5) + N*(N+1)/2,
LDW4 = N*(M+P+2) + 2*M*P + MIN(N,M) +
MAX( 3*M+1, MIN(N,M)+P ).
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: with ORDSEL = 'F', the selected order NR is greater
than NSMIN, the sum of the order of the
ALPHA-unstable part and the order of a minimal
realization of the ALPHA-stable part of the given
system. In this case, the resulting NR is set equal
to NSMIN.
= 2: with ORDSEL = 'F', the selected order NR is less
than the order of the ALPHA-unstable part of the
given system. In this case NR is set equal to the
order of the ALPHA-unstable part.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the computation of the ordered real Schur form of A
failed;
= 2: the separation of the ALPHA-stable/unstable
diagonal blocks failed because of very close
eigenvalues;
= 3: the reduction of AV to a real Schur form failed;
= 4: the reduction of AW to a real Schur form failed;
= 5: the reduction to generalized Schur form of the
descriptor pair corresponding to the inverse of V
failed;
= 6: the reduction to generalized Schur form of the
descriptor pair corresponding to the inverse of W
failed;
= 7: the computation of Hankel singular values failed;
= 8: the computation of stable projection in the
Hankel-norm approximation algorithm failed;
= 9: the order of computed stable projection in the
Hankel-norm approximation algorithm differs
from the order of Hankel-norm approximation;
= 10: the reduction of AV-BV*inv(DV)*CV to a
real Schur form failed;
= 11: the reduction of AW-BW*inv(DW)*CW to a
real Schur form failed;
= 12: the solution of the Sylvester equation failed
because the poles of V (if JOBV = 'V') or of
conj(V) (if JOBV = 'C') are not distinct from
the poles of G1 (see METHOD);
= 13: the solution of the Sylvester equation failed
because the poles of W (if JOBW = 'W') or of
conj(W) (if JOBW = 'C') are not distinct from
the poles of G1 (see METHOD);
= 14: the solution of the Sylvester equation failed
because the zeros of V (if JOBV = 'I') or of
conj(V) (if JOBV = 'R') are not distinct from
the poles of G1sr (see METHOD);
= 15: the solution of the Sylvester equation failed
because the zeros of W (if JOBW = 'I') or of
conj(W) (if JOBW = 'R') are not distinct from
the poles of G1sr (see METHOD);
= 16: the solution of the generalized Sylvester system
failed because the zeros of V (if JOBV = 'I') or
of conj(V) (if JOBV = 'R') are not distinct from
the poles of G1sr (see METHOD);
= 17: the solution of the generalized Sylvester system
failed because the zeros of W (if JOBW = 'I') or
of conj(W) (if JOBW = 'R') are not distinct from
the poles of G1sr (see METHOD);
= 18: op(V) is not antistable;
= 19: op(W) is not antistable;
= 20: V is not invertible;
= 21: W is not invertible.
Method
Let G be the transfer-function matrix of the original
linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t), (1)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system. The subroutine AB09JD determines
the matrices of a reduced order system
d[z(t)] = Ar*z(t) + Br*u(t)
yr(t) = Cr*z(t) + Dr*u(t), (2)
such that the corresponding transfer-function matrix Gr minimizes
the Hankel-norm of the frequency-weighted error
op(V)*(G-Gr)*op(W). (3)
For minimizing (3) with op(V) = V and op(W) = W, V and W are
assumed to have poles distinct from those of G, while with
op(V) = conj(V) and op(W) = conj(W), conj(V) and conj(W) are
assumed to have poles distinct from those of G. For minimizing (3)
with op(V) = inv(V) and op(W) = inv(W), V and W are assumed to
have zeros distinct from the poles of G, while with
op(V) = conj(inv(V)) and op(W) = conj(inv(W)), conj(V) and conj(W)
are assumed to have zeros distinct from the poles of G.
Note: conj(G) = G'(-s) for a continuous-time system and
conj(G) = G'(1/z) for a discrete-time system.
The following procedure is used to reduce G (see [1]):
1) Decompose additively G as
G = G1 + G2,
such that G1 = (A1,B1,C1,D) has only ALPHA-stable poles and
G2 = (A2,B2,C2,0) has only ALPHA-unstable poles.
2) Compute G1s, the projection of op(V)*G1*op(W) containing the
poles of G1, using explicit formulas [4] or the inverse-free
descriptor system formulas of [5].
3) Determine G1sr, the optimal Hankel-norm approximation of G1s,
of order r.
4) Compute G1r, the projection of inv(op(V))*G1sr*inv(op(W))
containing the poles of G1sr, using explicit formulas [4]
or the inverse-free descriptor system formulas of [5].
5) Assemble the reduced model Gr as
Gr = G1r + G2.
To reduce the weighted ALPHA-stable part G1s at step 3, the
optimal Hankel-norm approximation method of [2], based on the
square-root balancing projection formulas of [3], is employed.
The optimal weighted approximation error satisfies
HNORM[op(V)*(G-Gr)*op(W)] >= S(r+1),
where S(r+1) is the (r+1)-th Hankel singular value of G1s, the
transfer-function matrix computed at step 2 of the above
procedure, and HNORM(.) denotes the Hankel-norm.
References
[1] Latham, G.A. and Anderson, B.D.O.
Frequency-weighted optimal Hankel-norm approximation of stable
transfer functions.
Systems & Control Letters, Vol. 5, pp. 229-236, 1985.
[2] Glover, K.
All optimal Hankel norm approximation of linear
multivariable systems and their L-infinity error bounds.
Int. J. Control, Vol. 36, pp. 1145-1193, 1984.
[3] Tombs, M.S. and Postlethwaite, I.
Truncated balanced realization of stable, non-minimal
state-space systems.
Int. J. Control, Vol. 46, pp. 1319-1330, 1987.
[4] Varga, A.
Explicit formulas for an efficient implementation
of the frequency-weighting model reduction approach.
Proc. 1993 European Control Conference, Groningen, NL,
pp. 693-696, 1993.
[5] Varga, A.
Efficient and numerically reliable implementation of the
frequency-weighted Hankel-norm approximation model reduction
approach.
Proc. 2001 ECC, Porto, Portugal, 2001.
Numerical Aspects
The implemented methods rely on an accuracy enhancing square-root technique.Further Comments
NoneExample
Program Text
* AB09JD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, NMAX, NVMAX, NVPMAX, NWMAX, NWMMAX, PMAX
PARAMETER ( MMAX = 20, NMAX = 20, NVMAX = 10, NWMAX = 10,
$ PMAX = 20, NVPMAX = NVMAX + PMAX,
$ NWMMAX = NWMAX + MMAX )
INTEGER LDA, LDAV, LDAW, LDB, LDBV, LDBW,
$ LDC, LDCV, LDCW, LDD, LDDV, LDDW
PARAMETER ( LDA = NMAX, LDAV = NVMAX, LDAW = NWMAX,
$ LDB = NMAX, LDBV = NVMAX, LDBW = NWMAX,
$ LDC = PMAX, LDCV = PMAX, LDCW = MMAX,
$ LDD = PMAX, LDDV = PMAX, LDDW = MMAX )
INTEGER LIW1, LIW2, LIW3, LIWORK
PARAMETER ( LIW1 = 2*MAX( MMAX, PMAX ),
$ LIW2 = MAX( NVPMAX, NWMMAX ) + NMAX + 6,
$ LIW3 = MAX( 2*NVMAX + PMAX + 2,
$ 2*NWMAX + MMAX + 2 ) )
PARAMETER ( LIWORK = MAX( LIW1, LIW2, LIW3 ) )
INTEGER LDW1, LDW2, LDW3, LDW4, LDWORK
PARAMETER ( LDW1 = 2*NVPMAX*( NVPMAX + PMAX ) + PMAX*PMAX +
$ MAX( 2*NVPMAX*NVPMAX +
$ MAX( 11*NVPMAX + 16, PMAX*NVPMAX ),
$ NVPMAX*NMAX +
$ MAX( NVPMAX*NMAX + NMAX*NMAX,
$ PMAX*NMAX, PMAX*MMAX ) ) )
PARAMETER ( LDW2 = 2*NWMMAX*( NWMMAX + MMAX ) + MMAX*MMAX +
$ MAX( 2*NWMMAX*NWMMAX +
$ MAX( 11*NWMMAX + 16, MMAX*NWMMAX ),
$ NWMMAX*NMAX +
$ MAX( NWMMAX*NMAX + NMAX*NMAX,
$ MMAX*NMAX, PMAX*MMAX ) ) )
PARAMETER ( LDW3 = NMAX*( 2*NMAX + MAX( NMAX, MMAX, PMAX )
$ + 5 ) + ( NMAX*( NMAX + 1 ) )/2 )
PARAMETER ( LDW4 = NMAX*( MMAX + PMAX + 2 ) + 2*MMAX*PMAX +
$ MIN( NMAX, MMAX ) +
$ MAX( 3*MMAX + 1,
$ MIN( NMAX, MMAX ) + PMAX ) )
PARAMETER ( LDWORK = MAX( LDW1, LDW2, LDW3, LDW4 ) )
* .. Local Scalars ..
LOGICAL LEFTW, RIGHTW
DOUBLE PRECISION ALPHA, TOL1, TOL2
INTEGER I, INFO, IWARN, J, M, N, NR, NS, NV, NW, P
CHARACTER*1 DICO, EQUIL, JOBINV, JOBV, JOBW, ORDSEL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), AV(LDAV,NVMAX), AW(LDAW,NWMAX),
$ B(LDB,MMAX), BV(LDBV,PMAX), BW(LDBW,MMAX),
$ C(LDC,NMAX), CV(LDCV,NVMAX), CW(LDCW,NWMAX),
$ D(LDD,MMAX), DV(LDDV,PMAX), DW(LDDW,MMAX),
$ DWORK(LDWORK), HSV(NMAX)
INTEGER IWORK(LIWORK)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL AB09JD
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, NV, NW, NR, ALPHA, TOL1, TOL2,
$ JOBV, JOBW, JOBINV, DICO, EQUIL, ORDSEL
LEFTW = .NOT.LSAME( JOBV, 'N' )
RIGHTW = .NOT.LSAME( JOBW, 'N' )
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
IF( LEFTW ) THEN
IF( NV.LT.0 .OR. NV.GT.NVMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) NV
ELSE
IF( NV.GT.0 ) THEN
READ ( NIN, FMT = * )
$ ( ( AV(I,J), J = 1,NV ), I = 1,NV )
READ ( NIN, FMT = * )
$ ( ( BV(I,J), J = 1,P ), I = 1,NV )
READ ( NIN, FMT = * )
$ ( ( CV(I,J), J = 1,NV ), I = 1,P )
END IF
IF( LEFTW )
$ READ ( NIN, FMT = * )
$ ( ( DV(I,J), J = 1,P ), I = 1,P )
END IF
END IF
IF( RIGHTW ) THEN
IF( NW.LT.0 .OR. NW.GT.NWMAX ) THEN
WRITE ( NOUT, FMT = 99985 ) NW
ELSE
IF( NW.GT.0 ) THEN
READ ( NIN, FMT = * )
$ ( ( AW(I,J), J = 1,NW ), I = 1,NW )
READ ( NIN, FMT = * )
$ ( ( BW(I,J), J = 1,M ), I = 1,NW )
READ ( NIN, FMT = * )
$ ( ( CW(I,J), J = 1,NW ), I = 1,M )
END IF
READ ( NIN, FMT = * )
$ ( ( DW(I,J), J = 1,M ), I = 1,M )
END IF
END IF
* Find a reduced ssr for (A,B,C,D).
CALL AB09JD( JOBV, JOBW, JOBINV, DICO, EQUIL, ORDSEL, N,
$ NV, NW, M, P, NR, ALPHA, A, LDA, B, LDB,
$ C, LDC, D, LDD, AV, LDAV, BV, LDBV,
$ CV, LDCV, DV, LDDV, AW, LDAW, BW, LDBW,
$ CW, LDCW, DW, LDDW, NS, HSV, TOL1, TOL2,
$ IWORK, DWORK, LDWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF( IWARN.NE.0 ) WRITE ( NOUT, FMT = 99994 ) IWARN
WRITE ( NOUT, FMT = 99997 ) NR
WRITE ( NOUT, FMT = 99987 )
WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1, NS )
IF( NR.GT.0 ) THEN
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
60 CONTINUE
END IF
WRITE ( NOUT, FMT = 99991 )
DO 70 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1,M )
70 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB09JD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB09JD = ',I2)
99997 FORMAT (/' The order of reduced model = ',I2)
99996 FORMAT (/' The reduced state dynamics matrix Ar is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (' IWARN on exit from AB09JD = ',I2)
99993 FORMAT (/' The reduced input/state matrix Br is ')
99992 FORMAT (/' The reduced state/output matrix Cr is ')
99991 FORMAT (/' The reduced input/output matrix Dr is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99987 FORMAT (/' The Hankel singular values of weighted ALPHA-stable',
$ ' part are')
99986 FORMAT (/' NV is out of range.',/' NV = ',I5)
99985 FORMAT (/' NW is out of range.',/' NW = ',I5)
END
Program Data
AB09JD EXAMPLE PROGRAM DATA (Continuous system)
6 1 1 2 0 0 0.0 1.E-1 1.E-14 V N I C S A
-3.8637 -7.4641 -9.1416 -7.4641 -3.8637 -1.0000
1.0000 0 0 0 0 0
0 1.0000 0 0 0 0
0 0 1.0000 0 0 0
0 0 0 1.0000 0 0
0 0 0 0 1.0000 0
1
0
0
0
0
0
0 0 0 0 0 1
0
0.2000 -1.0000
1.0000 0
1
0
-1.8000 0
1
Program Results
AB09JD EXAMPLE PROGRAM RESULTS The order of reduced model = 4 The Hankel singular values of weighted ALPHA-stable part are 2.6790 2.1589 0.8424 0.1929 0.0219 0.0011 The reduced state dynamics matrix Ar is -0.2391 0.3072 1.1630 1.1967 -2.9709 -0.2391 2.6270 3.1027 0.0000 0.0000 -0.5137 -1.2842 0.0000 0.0000 0.1519 -0.5137 The reduced input/state matrix Br is -1.0497 -3.7052 0.8223 0.7435 The reduced state/output matrix Cr is -0.4466 0.0143 -0.4780 -0.2013 The reduced input/output matrix Dr is 0.0219
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB09JV.html 0000664 0000000 0000000 00000034137 14560147231 0020355 0 ustar 00root root 0000000 0000000
Purpose
To construct a state-space representation (A,BS,CS,DS) of the projection of V*G or conj(V)*G containing the poles of G, from the state-space representations (A,B,C,D) and (AV-lambda*EV,BV,CV,DV), of the transfer-function matrices G and V, respectively. G is assumed to be a stable transfer-function matrix and the state matrix A must be in a real Schur form. When computing the stable projection of V*G, it is assumed that G and V have completely distinct poles. When computing the stable projection of conj(V)*G, it is assumed that G and conj(V) have completely distinct poles. Note: For a transfer-function matrix G, conj(G) denotes the conjugate of G given by G'(-s) for a continuous-time system or G'(1/z) for a discrete-time system.Specification
SUBROUTINE AB09JV( JOB, DICO, JOBEV, STBCHK, N, M, P, NV, PV,
$ A, LDA, B, LDB, C, LDC, D, LDD, AV, LDAV,
$ EV, LDEV, BV, LDBV, CV, LDCV, DV, LDDV, IWORK,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, JOB, JOBEV, STBCHK
INTEGER INFO, LDA, LDAV, LDB, LDBV, LDC, LDCV,
$ LDD, LDDV, LDEV, LDWORK, M, N, NV, P, PV
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), AV(LDAV,*), B(LDB,*), BV(LDBV,*),
$ C(LDC,*), CV(LDCV,*), D(LDD,*), DV(LDDV,*),
$ DWORK(*), EV(LDEV,*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the projection to be computed as follows:
= 'V': compute the projection of V*G containing
the poles of G;
= 'C': compute the projection of conj(V)*G containing
the poles of G.
DICO CHARACTER*1
Specifies the type of the systems as follows:
= 'C': G and V are continuous-time systems;
= 'D': G and V are discrete-time systems.
JOBEV CHARACTER*1
Specifies whether EV is a general square or an identity
matrix as follows:
= 'G': EV is a general square matrix;
= 'I': EV is the identity matrix.
STBCHK CHARACTER*1
Specifies whether stability/antistability of V is to be
checked as follows:
= 'C': check stability if JOB = 'C' or antistability if
JOB = 'V';
= 'N': do not check stability or antistability.
Input/Output Parameters
N (input) INTEGER
The dimension of the state vector of the system with
the transfer-function matrix G. N >= 0.
M (input) INTEGER
The dimension of the input vector of the system with
the transfer-function matrix G. M >= 0.
P (input) INTEGER
The dimension of the output vector of the system with the
transfer-function matrix G, and also the dimension of
the input vector if JOB = 'V', or of the output vector
if JOB = 'C', of the system with the transfer-function
matrix V. P >= 0.
NV (input) INTEGER
The dimension of the state vector of the system with
the transfer-function matrix V. NV >= 0.
PV (input) INTEGER
The dimension of the output vector, if JOB = 'V', or
of the input vector, if JOB = 'C', of the system with
the transfer-function matrix V. PV >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
state matrix A of the system with the transfer-function
matrix G in a real Schur form.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain
the input/state matrix B of the system with the
transfer-function matrix G. The matrix BS is equal to B.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the output matrix C of the system with the
transfer-function matrix G.
On exit, if INFO = 0, the leading PV-by-N part of this
array contains the output matrix CS of the projection of
V*G, if JOB = 'V', or of conj(V)*G, if JOB = 'C'.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,P,PV).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the feedthrough matrix D of the system with the
transfer-function matrix G.
On exit, if INFO = 0, the leading PV-by-M part of
this array contains the feedthrough matrix DS of the
projection of V*G, if JOB = 'V', or of conj(V)*G,
if JOB = 'C'.
LDD INTEGER
The leading dimension of the array D. LDD >= MAX(1,P,PV).
AV (input/output) DOUBLE PRECISION array, dimension (LDAV,NV)
On entry, the leading NV-by-NV part of this array must
contain the state matrix AV of the system with the
transfer-function matrix V.
On exit, if INFO = 0, the leading NV-by-NV part of this
array contains a condensed matrix as follows:
if JOBEV = 'I', it contains the real Schur form of AV;
if JOBEV = 'G' and JOB = 'V', it contains a quasi-upper
triangular matrix representing the real Schur matrix
in the real generalized Schur form of the pair (AV,EV);
if JOBEV = 'G', JOB = 'C' and DICO = 'C', it contains a
quasi-upper triangular matrix corresponding to the
generalized real Schur form of the pair (AV',EV');
if JOBEV = 'G', JOB = 'C' and DICO = 'D', it contains an
upper triangular matrix corresponding to the generalized
real Schur form of the pair (EV',AV').
LDAV INTEGER
The leading dimension of the array AV. LDAV >= MAX(1,NV).
EV (input/output) DOUBLE PRECISION array, dimension (LDEV,NV)
On entry, if JOBEV = 'G', the leading NV-by-NV part of
this array must contain the descriptor matrix EV of the
system with the transfer-function matrix V.
If JOBEV = 'I', EV is assumed to be an identity matrix
and is not referenced.
On exit, if INFO = 0 and JOBEV = 'G', the leading NV-by-NV
part of this array contains a condensed matrix as follows:
if JOB = 'V', it contains an upper triangular matrix
corresponding to the real generalized Schur form of the
pair (AV,EV);
if JOB = 'C' and DICO = 'C', it contains an upper
triangular matrix corresponding to the generalized real
Schur form of the pair (AV',EV');
if JOB = 'C' and DICO = 'D', it contains a quasi-upper
triangular matrix corresponding to the generalized
real Schur form of the pair (EV',AV').
LDEV INTEGER
The leading dimension of the array EV.
LDEV >= MAX(1,NV), if JOBEV = 'G';
LDEV >= 1, if JOBEV = 'I'.
BV (input/output) DOUBLE PRECISION array,
dimension (LDBV,MBV), where MBV = P, if JOB = 'V', and
MBV = PV, if JOB = 'C'.
On entry, the leading NV-by-MBV part of this array must
contain the input matrix BV of the system with the
transfer-function matrix V.
On exit, if INFO = 0, the leading NV-by-MBV part of this
array contains Q'*BV, where Q is the orthogonal matrix
that reduces AV to the real Schur form or the left
orthogonal matrix used to reduce the pair (AV,EV),
(AV',EV') or (EV',AV') to the generalized real Schur form.
LDBV INTEGER
The leading dimension of the array BV. LDBV >= MAX(1,NV).
CV (input/output) DOUBLE PRECISION array, dimension (LDCV,NV)
On entry, the leading PCV-by-NV part of this array must
contain the output matrix CV of the system with the
transfer-function matrix V, where PCV = PV, if JOB = 'V',
or PCV = P, if JOB = 'C'.
On exit, if INFO = 0, the leading PCV-by-NV part of this
array contains CV*Q, where Q is the orthogonal matrix that
reduces AV to the real Schur form, or CV*Z, where Z is the
right orthogonal matrix used to reduce the pair (AV,EV),
(AV',EV') or (EV',AV') to the generalized real Schur form.
LDCV INTEGER
The leading dimension of the array CV.
LDCV >= MAX(1,PV) if JOB = 'V';
LDCV >= MAX(1,P) if JOB = 'C'.
DV (input) DOUBLE PRECISION array,
dimension (LDDV,MBV), where MBV = P, if JOB = 'V', and
MBV = PV, if JOB = 'C'.
The leading PCV-by-MBV part of this array must contain
the feedthrough matrix DV of the system with the
transfer-function matrix V, where PCV = PV, if JOB = 'V',
or PCV = P, if JOB = 'C'.
LDDV INTEGER
The leading dimension of the array DV.
LDDV >= MAX(1,PV) if JOB = 'V';
LDDV >= MAX(1,P) if JOB = 'C'.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK = 0, if JOBEV = 'I';
LIWORK = NV+N+6, if JOBEV = 'G'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= LW1, if JOBEV = 'I',
LDWORK >= LW2, if JOBEV = 'G', where
LW1 = MAX( 1, NV*(NV+5), NV*N + MAX( a, PV*N, PV*M ) )
a = 0, if DICO = 'C' or JOB = 'V',
a = 2*NV, if DICO = 'D' and JOB = 'C';
LW2 = MAX( 2*NV*NV + MAX( 11*NV+16, P*NV, PV*NV ),
NV*N + MAX( NV*N+N*N, PV*N, PV*M ) ).
For good performance, LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction of the pair (AV,EV) to the real
generalized Schur form failed (JOBEV = 'G'),
or the reduction of the matrix AV to the real
Schur form failed (JOBEV = 'I);
= 2: the solution of the Sylvester equation failed
because the matrix A and the pencil AV-lambda*EV
have common eigenvalues (if JOB = 'V'), or the
pencil -AV-lambda*EV and A have common eigenvalues
(if JOB = 'C' and DICO = 'C'), or the pencil
AV-lambda*EV has an eigenvalue which is the
reciprocal of one of eigenvalues of A
(if JOB = 'C' and DICO = 'D');
= 3: the solution of the Sylvester equation failed
because the matrices A and AV have common
eigenvalues (if JOB = 'V'), or the matrices A
and -AV have common eigenvalues (if JOB = 'C' and
DICO = 'C'), or the matrix A has an eigenvalue
which is the reciprocal of one of eigenvalues of AV
(if JOB = 'C' and DICO = 'D');
= 4: JOB = 'V' and the pair (AV,EV) has not completely
unstable generalized eigenvalues, or JOB = 'C' and
the pair (AV,EV) has not completely stable
generalized eigenvalues.
Method
If JOB = 'V', the matrices of the stable projection of V*G are
computed as
BS = B, CS = CV*X + DV*C, DS = DV*D,
where X satisfies the generalized Sylvester equation
AV*X - EV*X*A + BV*C = 0.
If JOB = 'C', the matrices of the stable projection of conj(V)*G
are computed using the following formulas:
- for a continuous-time system, the matrices BS, CS and DS of
the stable projection are computed as
BS = B, CS = BV'*X + DV'*C, DS = DV'*D,
where X satisfies the generalized Sylvester equation
AV'*X + EV'*X*A + CV'*C = 0.
- for a discrete-time system, the matrices BS, CS and DS of
the stable projection are computed as
BS = B, CS = BV'*X*A + DV'*C, DS = DV'*D + BV'*X*B,
where X satisfies the generalized Sylvester equation
EV'*X - AV'*X*A = CV'*C.
References
[1] Varga, A.
Efficient and numerically reliable implementation of the
frequency-weighted Hankel-norm approximation model reduction
approach.
Proc. 2001 ECC, Porto, Portugal, 2001.
[2] Zhou, K.
Frequency-weighted H-infinity norm and optimal Hankel norm
model reduction.
IEEE Trans. Autom. Control, vol. 40, pp. 1687-1699, 1995.
Numerical Aspects
The implemented methods rely on numerically stable algorithms.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To construct a state-space representation (A,BS,CS,DS) of the projection of G*W or G*conj(W) containing the poles of G, from the state-space representations (A,B,C,D) and (AW-lambda*EW,BW,CW,DW), of the transfer-function matrices G and W, respectively. G is assumed to be a stable transfer-function matrix and the state matrix A must be in a real Schur form. When computing the stable projection of G*W, it is assumed that G and W have completely distinct poles. When computing the stable projection of G*conj(W), it is assumed that G and conj(W) have completely distinct poles. Note: For a transfer-function matrix G, conj(G) denotes the conjugate of G given by G'(-s) for a continuous-time system or G'(1/z) for a discrete-time system.Specification
SUBROUTINE AB09JW( JOB, DICO, JOBEW, STBCHK, N, M, P, NW, MW,
$ A, LDA, B, LDB, C, LDC, D, LDD, AW, LDAW,
$ EW, LDEW, BW, LDBW, CW, LDCW, DW, LDDW, IWORK,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, JOB, JOBEW, STBCHK
INTEGER INFO, LDA, LDAW, LDB, LDBW, LDC, LDCW,
$ LDD, LDDW, LDEW, LDWORK, M, MW, N, NW, P
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), AW(LDAW,*), B(LDB,*), BW(LDBW,*),
$ C(LDC,*), CW(LDCW,*), D(LDD,*), DW(LDDW,*),
$ DWORK(*), EW(LDEW,*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the projection to be computed as follows:
= 'W': compute the projection of G*W containing
the poles of G;
= 'C': compute the projection of G*conj(W) containing
the poles of G.
DICO CHARACTER*1
Specifies the type of the systems as follows:
= 'C': G and W are continuous-time systems;
= 'D': G and W are discrete-time systems.
JOBEW CHARACTER*1
Specifies whether EW is a general square or an identity
matrix as follows:
= 'G': EW is a general square matrix;
= 'I': EW is the identity matrix.
STBCHK CHARACTER*1
Specifies whether stability/antistability of W is to be
checked as follows:
= 'C': check stability if JOB = 'C' or antistability if
JOB = 'W';
= 'N': do not check stability or antistability.
Input/Output Parameters
N (input) INTEGER
The dimension of the state vector of the system with
the transfer-function matrix G. N >= 0.
M (input) INTEGER
The dimension of the input vector of the system with
the transfer-function matrix G, and also the dimension
of the output vector if JOB = 'W', or of the input vector
if JOB = 'C', of the system with the transfer-function
matrix W. M >= 0.
P (input) INTEGER
The dimension of the output vector of the system with the
transfer-function matrix G. P >= 0.
NW (input) INTEGER
The dimension of the state vector of the system with the
transfer-function matrix W. NW >= 0.
MW (input) INTEGER
The dimension of the input vector, if JOB = 'W', or of
the output vector, if JOB = 'C', of the system with the
transfer-function matrix W. MW >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
state matrix A of the system with the transfer-function
matrix G in a real Schur form.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array,
dimension (LDB,MAX(M,MW))
On entry, the leading N-by-M part of this array must
contain the input matrix B of the system with the
transfer-function matrix G.
On exit, if INFO = 0, the leading N-by-MW part of this
array contains the input matrix BS of the projection of
G*W, if JOB = 'W', or of G*conj(W), if JOB = 'C'.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array must contain
the output/state matrix C of the system with the
transfer-function matrix G. The matrix CS is equal to C.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array,
dimension (LDB,MAX(M,MW))
On entry, the leading P-by-M part of this array must
contain the feedthrough matrix D of the system with
the transfer-function matrix G.
On exit, if INFO = 0, the leading P-by-MW part of
this array contains the feedthrough matrix DS of the
projection of G*W, if JOB = 'W', or of G*conj(W),
if JOB = 'C'.
LDD INTEGER
The leading dimension of the array D. LDD >= MAX(1,P).
AW (input/output) DOUBLE PRECISION array, dimension (LDAW,NW)
On entry, the leading NW-by-NW part of this array must
contain the state matrix AW of the system with the
transfer-function matrix W.
On exit, if INFO = 0, the leading NW-by-NW part of this
array contains a condensed matrix as follows:
if JOBEW = 'I', it contains the real Schur form of AW;
if JOBEW = 'G' and JOB = 'W', it contains a quasi-upper
triangular matrix representing the real Schur matrix
in the real generalized Schur form of the pair (AW,EW);
if JOBEW = 'G', JOB = 'C' and DICO = 'C', it contains a
quasi-upper triangular matrix corresponding to the
generalized real Schur form of the pair (AW',EW');
if JOBEW = 'G', JOB = 'C' and DICO = 'D', it contains an
upper triangular matrix corresponding to the generalized
real Schur form of the pair (EW',AW').
LDAW INTEGER
The leading dimension of the array AW. LDAW >= MAX(1,NW).
EW (input/output) DOUBLE PRECISION array, dimension (LDEW,NW)
On entry, if JOBEW = 'G', the leading NW-by-NW part of
this array must contain the descriptor matrix EW of the
system with the transfer-function matrix W.
If JOBEW = 'I', EW is assumed to be an identity matrix
and is not referenced.
On exit, if INFO = 0 and JOBEW = 'G', the leading NW-by-NW
part of this array contains a condensed matrix as follows:
if JOB = 'W', it contains an upper triangular matrix
corresponding to the real generalized Schur form of the
pair (AW,EW);
if JOB = 'C' and DICO = 'C', it contains an upper
triangular matrix corresponding to the generalized real
Schur form of the pair (AW',EW');
if JOB = 'C' and DICO = 'D', it contains a quasi-upper
triangular matrix corresponding to the generalized
real Schur form of the pair (EW',AW').
LDEW INTEGER
The leading dimension of the array EW.
LDEW >= MAX(1,NW), if JOBEW = 'G';
LDEW >= 1, if JOBEW = 'I'.
BW (input/output) DOUBLE PRECISION array,
dimension (LDBW,MBW), where MBW = MW, if JOB = 'W', and
MBW = M, if JOB = 'C'.
On entry, the leading NW-by-MBW part of this array must
contain the input matrix BW of the system with the
transfer-function matrix W.
On exit, if INFO = 0, the leading NW-by-MBW part of this
array contains Q'*BW, where Q is the orthogonal matrix
that reduces AW to the real Schur form or the left
orthogonal matrix used to reduce the pair (AW,EW),
(AW',EW') or (EW',AW') to the generalized real Schur form.
LDBW INTEGER
The leading dimension of the array BW. LDBW >= MAX(1,NW).
CW (input/output) DOUBLE PRECISION array, dimension (LDCW,NW)
On entry, the leading PCW-by-NW part of this array must
contain the output matrix CW of the system with the
transfer-function matrix W, where PCW = M if JOB = 'W' or
PCW = MW if JOB = 'C'.
On exit, if INFO = 0, the leading PCW-by-NW part of this
array contains CW*Q, where Q is the orthogonal matrix that
reduces AW to the real Schur form, or CW*Z, where Z is the
right orthogonal matrix used to reduce the pair (AW,EW),
(AW',EW') or (EW',AW') to the generalized real Schur form.
LDCW INTEGER
The leading dimension of the array CW.
LDCW >= MAX(1,PCW), where PCW = M if JOB = 'W', or
PCW = MW if JOB = 'C'.
DW (input) DOUBLE PRECISION array,
dimension (LDDW,MBW), where MBW = MW if JOB = 'W', and
MBW = M if JOB = 'C'.
The leading PCW-by-MBW part of this array must contain
the feedthrough matrix DW of the system with the
transfer-function matrix W, where PCW = M if JOB = 'W',
or PCW = MW if JOB = 'C'.
LDDW INTEGER
LDDW >= MAX(1,PCW), where PCW = M if JOB = 'W', or
PCW = MW if JOB = 'C'.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK = 0, if JOBEW = 'I';
LIWORK = NW+N+6, if JOBEW = 'G'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= LW1, if JOBEW = 'I',
LDWORK >= LW2, if JOBEW = 'G', where
LW1 = MAX( 1, NW*(NW+5), NW*N + MAX( a, N*MW, P*MW ) )
a = 0, if DICO = 'C' or JOB = 'W',
a = 2*NW, if DICO = 'D' and JOB = 'C';
LW2 = MAX( 2*NW*NW + MAX( 11*NW+16, NW*M, MW*NW ),
NW*N + MAX( NW*N+N*N, MW*N, P*MW ) ).
For good performance, LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction of the pair (AW,EW) to the real
generalized Schur form failed (JOBEW = 'G'),
or the reduction of the matrix AW to the real
Schur form failed (JOBEW = 'I);
= 2: the solution of the Sylvester equation failed
because the matrix A and the pencil AW-lambda*EW
have common eigenvalues (if JOB = 'W'), or the
pencil -AW-lambda*EW and A have common eigenvalues
(if JOB = 'C' and DICO = 'C'), or the pencil
AW-lambda*EW has an eigenvalue which is the
reciprocal of one of eigenvalues of A
(if JOB = 'C' and DICO = 'D');
= 3: the solution of the Sylvester equation failed
because the matrices A and AW have common
eigenvalues (if JOB = 'W'), or the matrices A
and -AW have common eigenvalues (if JOB = 'C' and
DICO = 'C'), or the matrix A has an eigenvalue
which is the reciprocal of one of eigenvalues of AW
(if JOB = 'C' and DICO = 'D');
= 4: JOB = 'W' and the pair (AW,EW) has not completely
unstable generalized eigenvalues, or JOB = 'C' and
the pair (AW,EW) has not completely stable
generalized eigenvalues.
Method
If JOB = 'W', the matrices of the stable projection of G*W are
computed as
BS = B*DW + Y*BW, CS = C, DS = D*DW,
where Y satisfies the generalized Sylvester equation
-A*Y*EW + Y*AW + B*CW = 0.
If JOB = 'C', the matrices of the stable projection of G*conj(W)
are computed using the following formulas:
- for a continuous-time system, the matrices BS, CS and DS of
the stable projection are computed as
BS = B*DW' + Y*CW', CS = C, DS = D*DW',
where Y satisfies the generalized Sylvester equation
A*Y*EW' + Y*AW' + B*BW' = 0.
- for a discrete-time system, the matrices BS, CS and DS of
the stable projection are computed as
BS = B*DW' + A*Y*CW', CS = C, DS = D*DW' + C*Y*CW',
where Y satisfies the generalized Sylvester equation
Y*EW' - A*Y*AW' = B*BW'.
References
[1] Varga, A.
Efficient and numerically reliable implementation of the
frequency-weighted Hankel-norm approximation model reduction
approach.
Proc. 2001 ECC, Porto, Portugal, 2001.
[2] Zhou, K.
Frequency-weighted H-infinity norm and optimal Hankel norm
model reduction.
IEEE Trans. Autom. Control, vol. 40, pp. 1687-1699, 1995.
Numerical Aspects
The implemented methods rely on numerically stable algorithms.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To check stability/antistability of finite eigenvalues with respect to a given stability domain.Specification
SUBROUTINE AB09JX( DICO, STDOM, EVTYPE, N, ALPHA, ER, EI, ED,
$ TOLINF, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, EVTYPE, STDOM
INTEGER INFO, N
DOUBLE PRECISION ALPHA, TOLINF
C .. Array Arguments ..
DOUBLE PRECISION ED(*), EI(*), ER(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the stability domain as follows:
= 'C': for a continuous-time system;
= 'D': for a discrete-time system.
STDOM CHARACTER*1
Specifies whether the domain of interest is of stability
type (left part of complex plane or inside of a circle)
or of instability type (right part of complex plane or
outside of a circle) as follows:
= 'S': stability type domain;
= 'U': instability type domain.
EVTYPE CHARACTER*1
Specifies whether the eigenvalues arise from a standard
or a generalized eigenvalue problem as follows:
= 'S': standard eigenvalue problem;
= 'G': generalized eigenvalue problem;
= 'R': reciprocal generalized eigenvalue problem.
Input/Output Parameters
N (input) INTEGER
The dimension of vectors ER, EI and ED. N >= 0.
ALPHA (input) DOUBLE PRECISION
Specifies the boundary of the domain of interest for the
eigenvalues. For a continuous-time system
(DICO = 'C'), ALPHA is the boundary value for the real
parts of eigenvalues, while for a discrete-time system
(DICO = 'D'), ALPHA >= 0 represents the boundary value for
the moduli of eigenvalues.
ER, EI, (input) DOUBLE PRECISION arrays, dimension (N)
ED If EVTYPE = 'S', ER(j) + EI(j)*i, j = 1,...,N, are
the eigenvalues of a real matrix.
ED is not referenced and is implicitly considered as
a vector having all elements equal to one.
If EVTYPE = 'G' or EVTYPE = 'R', (ER(j) + EI(j)*i)/ED(j),
j = 1,...,N, are the generalized eigenvalues of a pair of
real matrices. If ED(j) is zero, then the j-th generalized
eigenvalue is infinite.
Complex conjugate pairs of eigenvalues must appear
consecutively.
Tolerances
TOLINF DOUBLE PRECISION
If EVTYPE = 'G' or 'R', TOLINF contains the tolerance for
detecting infinite generalized eigenvalues.
0 <= TOLINF < 1.
Error Indicator
INFO INTEGER
= 0: successful exit, i.e., all eigenvalues lie within
the domain of interest defined by DICO, STDOM
and ALPHA;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: some eigenvalues lie outside the domain of interest
defined by DICO, STDOM and ALPHA.
Method
The domain of interest for an eigenvalue lambda is defined by the
parameters ALPHA, DICO and STDOM as follows:
- for a continuous-time system (DICO = 'C'):
Real(lambda) < ALPHA if STDOM = 'S';
Real(lambda) > ALPHA if STDOM = 'U';
- for a discrete-time system (DICO = 'D'):
Abs(lambda) < ALPHA if STDOM = 'S';
Abs(lambda) > ALPHA if STDOM = 'U'.
If EVTYPE = 'R', the same conditions apply for 1/lambda.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute a reduced order model (Ar,Br,Cr,Dr) for an original
state-space representation (A,B,C,D) by using the frequency
weighted optimal Hankel-norm approximation method.
The Hankel norm of the weighted error
V*(G-Gr)*W or conj(V)*(G-Gr)*conj(W)
is minimized, where G and Gr are the transfer-function matrices
of the original and reduced systems, respectively, and V and W
are the transfer-function matrices of the left and right frequency
weights, specified by their state space realizations (AV,BV,CV,DV)
and (AW,BW,CW,DW), respectively. When minimizing the weighted
error V*(G-Gr)*W, V and W must be antistable transfer-function
matrices. When minimizing conj(V)*(G-Gr)*conj(W), V and W must be
stable transfer-function matrices.
Additionally, V and W must be invertible transfer-function
matrices, with the feedthrough matrices DV and DW invertible.
If the original system is unstable, then the frequency weighted
Hankel-norm approximation is computed only for the
ALPHA-stable part of the system.
For a transfer-function matrix G, conj(G) denotes the conjugate
of G given by G'(-s) for a continuous-time system or G'(1/z)
for a discrete-time system.
Specification
SUBROUTINE AB09KD( JOB, DICO, WEIGHT, EQUIL, ORDSEL, N, NV, NW, M,
$ P, NR, ALPHA, A, LDA, B, LDB, C, LDC, D, LDD,
$ AV, LDAV, BV, LDBV, CV, LDCV, DV, LDDV,
$ AW, LDAW, BW, LDBW, CW, LDCW, DW, LDDW,
$ NS, HSV, TOL1, TOL2, IWORK, DWORK, LDWORK,
$ IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, EQUIL, JOB, ORDSEL, WEIGHT
INTEGER INFO, IWARN, LDA, LDAV, LDAW, LDB, LDBV, LDBW,
$ LDC, LDCV, LDCW, LDD, LDDV, LDDW, LDWORK, M, N,
$ NR, NS, NV, NW, P
DOUBLE PRECISION ALPHA, TOL1, TOL2
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), AV(LDAV,*), AW(LDAW,*),
$ B(LDB,*), BV(LDBV,*), BW(LDBW,*),
$ C(LDC,*), CV(LDCV,*), CW(LDCW,*),
$ D(LDD,*), DV(LDDV,*), DW(LDDW,*), DWORK(*),
$ HSV(*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the frequency-weighting problem as follows:
= 'N': solve min||V*(G-Gr)*W||_H;
= 'C': solve min||conj(V)*(G-Gr)*conj(W)||_H.
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
WEIGHT CHARACTER*1
Specifies the type of frequency weighting, as follows:
= 'N': no weightings are used (V = I, W = I);
= 'L': only left weighting V is used (W = I);
= 'R': only right weighting W is used (V = I);
= 'B': both left and right weightings V and W are used.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily
equilibrate the triplet (A,B,C) as follows:
= 'S': perform equilibration (scaling);
= 'N': do not perform equilibration.
ORDSEL CHARACTER*1
Specifies the order selection method as follows:
= 'F': the resulting order NR is fixed;
= 'A': the resulting order NR is automatically determined
on basis of the given tolerance TOL1.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation,
i.e., the order of the matrix A. N >= 0.
NV (input) INTEGER
The order of the realization of the left frequency
weighting V, i.e., the order of the matrix AV. NV >= 0.
NW (input) INTEGER
The order of the realization of the right frequency
weighting W, i.e., the order of the matrix AW. NW >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NR (input/output) INTEGER
On entry with ORDSEL = 'F', NR is the desired order of
the resulting reduced order system. 0 <= NR <= N.
On exit, if INFO = 0, NR is the order of the resulting
reduced order model. For a system with NU ALPHA-unstable
eigenvalues and NS ALPHA-stable eigenvalues (NU+NS = N),
NR is set as follows: if ORDSEL = 'F', NR is equal to
NU+MIN(MAX(0,NR-NU-KR+1),NMIN), where KR is the
multiplicity of the Hankel singular value HSV(NR-NU+1),
NR is the desired order on entry, and NMIN is the order
of a minimal realization of the ALPHA-stable part of the
given system; NMIN is determined as the number of Hankel
singular values greater than NS*EPS*HNORM(As,Bs,Cs), where
EPS is the machine precision (see LAPACK Library Routine
DLAMCH) and HNORM(As,Bs,Cs) is the Hankel norm of the
ALPHA-stable part of the weighted system (computed in
HSV(1));
if ORDSEL = 'A', NR is the sum of NU and the number of
Hankel singular values greater than
MAX(TOL1,NS*EPS*HNORM(As,Bs,Cs)).
ALPHA (input) DOUBLE PRECISION
Specifies the ALPHA-stability boundary for the eigenvalues
of the state dynamics matrix A. For a continuous-time
system (DICO = 'C'), ALPHA <= 0 is the boundary value for
the real parts of eigenvalues, while for a discrete-time
system (DICO = 'D'), 0 <= ALPHA <= 1 represents the
boundary value for the moduli of eigenvalues.
The ALPHA-stability domain does not include the boundary.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A.
On exit, if INFO = 0, the leading NR-by-NR part of this
array contains the state dynamics matrix Ar of the
reduced order system in a real Schur form.
The resulting A has a block-diagonal form with two blocks.
For a system with NU ALPHA-unstable eigenvalues and
NS ALPHA-stable eigenvalues (NU+NS = N), the leading
NU-by-NU block contains the unreduced part of A
corresponding to ALPHA-unstable eigenvalues.
The trailing (NR+NS-N)-by-(NR+NS-N) block contains
the reduced part of A corresponding to ALPHA-stable
eigenvalues.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, if INFO = 0, the leading NR-by-M part of this
array contains the input/state matrix Br of the reduced
order system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, if INFO = 0, the leading P-by-NR part of this
array contains the state/output matrix Cr of the reduced
order system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the original input/output matrix D.
On exit, if INFO = 0, the leading P-by-M part of this
array contains the input/output matrix Dr of the reduced
order system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
AV (input/output) DOUBLE PRECISION array, dimension (LDAV,NV)
On entry, if WEIGHT = 'L' or 'B', the leading NV-by-NV
part of this array must contain the state matrix AV of a
state space realization of the left frequency weighting V.
On exit, if WEIGHT = 'L' or 'B', and INFO = 0, the leading
NV-by-NV part of this array contains a real Schur form
of the state matrix of a state space realization of the
inverse of V.
AV is not referenced if WEIGHT = 'R' or 'N'.
LDAV INTEGER
The leading dimension of the array AV.
LDAV >= MAX(1,NV), if WEIGHT = 'L' or 'B';
LDAV >= 1, if WEIGHT = 'R' or 'N'.
BV (input/output) DOUBLE PRECISION array, dimension (LDBV,P)
On entry, if WEIGHT = 'L' or 'B', the leading NV-by-P part
of this array must contain the input matrix BV of a state
space realization of the left frequency weighting V.
On exit, if WEIGHT = 'L' or 'B', and INFO = 0, the leading
NV-by-P part of this array contains the input matrix of a
state space realization of the inverse of V.
BV is not referenced if WEIGHT = 'R' or 'N'.
LDBV INTEGER
The leading dimension of the array BV.
LDBV >= MAX(1,NV), if WEIGHT = 'L' or 'B';
LDBV >= 1, if WEIGHT = 'R' or 'N'.
CV (input/output) DOUBLE PRECISION array, dimension (LDCV,NV)
On entry, if WEIGHT = 'L' or 'B', the leading P-by-NV part
of this array must contain the output matrix CV of a state
space realization of the left frequency weighting V.
On exit, if WEIGHT = 'L' or 'B', and INFO = 0, the leading
P-by-NV part of this array contains the output matrix of a
state space realization of the inverse of V.
CV is not referenced if WEIGHT = 'R' or 'N'.
LDCV INTEGER
The leading dimension of the array CV.
LDCV >= MAX(1,P), if WEIGHT = 'L' or 'B';
LDCV >= 1, if WEIGHT = 'R' or 'N'.
DV (input/output) DOUBLE PRECISION array, dimension (LDDV,P)
On entry, if WEIGHT = 'L' or 'B', the leading P-by-P part
of this array must contain the feedthrough matrix DV of a
state space realization of the left frequency weighting V.
On exit, if WEIGHT = 'L' or 'B', and INFO = 0, the leading
P-by-P part of this array contains the feedthrough matrix
of a state space realization of the inverse of V.
DV is not referenced if WEIGHT = 'R' or 'N'.
LDDV INTEGER
The leading dimension of the array DV.
LDDV >= MAX(1,P), if WEIGHT = 'L' or 'B';
LDDV >= 1, if WEIGHT = 'R' or 'N'.
AW (input/output) DOUBLE PRECISION array, dimension (LDAW,NW)
On entry, if WEIGHT = 'R' or 'B', the leading NW-by-NW
part of this array must contain the state matrix AW of
a state space realization of the right frequency
weighting W.
On exit, if WEIGHT = 'R' or 'B', and INFO = 0, the leading
NW-by-NW part of this array contains a real Schur form of
the state matrix of a state space realization of the
inverse of W.
AW is not referenced if WEIGHT = 'L' or 'N'.
LDAW INTEGER
The leading dimension of the array AW.
LDAW >= MAX(1,NW), if WEIGHT = 'R' or 'B';
LDAW >= 1, if WEIGHT = 'L' or 'N'.
BW (input/output) DOUBLE PRECISION array, dimension (LDBW,M)
On entry, if WEIGHT = 'R' or 'B', the leading NW-by-M part
of this array must contain the input matrix BW of a state
space realization of the right frequency weighting W.
On exit, if WEIGHT = 'R' or 'B', and INFO = 0, the leading
NW-by-M part of this array contains the input matrix of a
state space realization of the inverse of W.
BW is not referenced if WEIGHT = 'L' or 'N'.
LDBW INTEGER
The leading dimension of the array BW.
LDBW >= MAX(1,NW), if WEIGHT = 'R' or 'B';
LDBW >= 1, if WEIGHT = 'L' or 'N'.
CW (input/output) DOUBLE PRECISION array, dimension (LDCW,NW)
On entry, if WEIGHT = 'R' or 'B', the leading M-by-NW part
of this array must contain the output matrix CW of a state
space realization of the right frequency weighting W.
On exit, if WEIGHT = 'R' or 'B', and INFO = 0, the leading
M-by-NW part of this array contains the output matrix of a
state space realization of the inverse of W.
CW is not referenced if WEIGHT = 'L' or 'N'.
LDCW INTEGER
The leading dimension of the array CW.
LDCW >= MAX(1,M), if WEIGHT = 'R' or 'B';
LDCW >= 1, if WEIGHT = 'L' or 'N'.
DW (input/output) DOUBLE PRECISION array, dimension (LDDW,M)
On entry, if WEIGHT = 'R' or 'B', the leading M-by-M part
of this array must contain the feedthrough matrix DW of
a state space realization of the right frequency
weighting W.
On exit, if WEIGHT = 'R' or 'B', and INFO = 0, the leading
M-by-M part of this array contains the feedthrough matrix
of a state space realization of the inverse of W.
DW is not referenced if WEIGHT = 'L' or 'N'.
LDDW INTEGER
The leading dimension of the array DW.
LDDW >= MAX(1,M), if WEIGHT = 'R' or 'B';
LDDW >= 1, if WEIGHT = 'L' or 'N'.
NS (output) INTEGER
The dimension of the ALPHA-stable subsystem.
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, the leading NS elements of this array contain
the Hankel singular values, ordered decreasingly, of the
ALPHA-stable part of the weighted original system.
HSV(1) is the Hankel norm of the ALPHA-stable weighted
subsystem.
Tolerances
TOL1 DOUBLE PRECISION
If ORDSEL = 'A', TOL1 contains the tolerance for
determining the order of reduced system.
For model reduction, the recommended value is
TOL1 = c*HNORM(As,Bs,Cs), where c is a constant in the
interval [0.00001,0.001], and HNORM(As,Bs,Cs) is the
Hankel-norm of the ALPHA-stable part of the weighted
original system (computed in HSV(1)).
If TOL1 <= 0 on entry, the used default value is
TOL1 = NS*EPS*HNORM(As,Bs,Cs), where NS is the number of
ALPHA-stable eigenvalues of A and EPS is the machine
precision (see LAPACK Library Routine DLAMCH).
If ORDSEL = 'F', the value of TOL1 is ignored.
TOL2 DOUBLE PRECISION
The tolerance for determining the order of a minimal
realization of the ALPHA-stable part of the given system.
The recommended value is TOL2 = NS*EPS*HNORM(As,Bs,Cs).
This value is used by default if TOL2 <= 0 on entry.
If TOL2 > 0 and ORDSEL = 'A', then TOL2 <= TOL1.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK = MAX(1,M,c), if DICO = 'C',
LIWORK = MAX(1,N,M,c), if DICO = 'D',
where c = 0, if WEIGHT = 'N',
c = 2*P, if WEIGHT = 'L',
c = 2*M, if WEIGHT = 'R',
c = MAX(2*M,2*P), if WEIGHT = 'B'.
On exit, if INFO = 0, IWORK(1) contains NMIN, the order of
the computed minimal realization.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( LDW1, LDW2, LDW3, LDW4 ), where
LDW1 = 0 if WEIGHT = 'R' or 'N' and
LDW1 = MAX( NV*(NV+5), NV*N + MAX( a, P*N, P*M ) )
if WEIGHT = 'L' or WEIGHT = 'B',
LDW2 = 0 if WEIGHT = 'L' or 'N' and
LDW2 = MAX( NW*(NW+5), NW*N + MAX( b, M*N, P*M ) )
if WEIGHT = 'R' or WEIGHT = 'B', with
a = 0, b = 0, if DICO = 'C' or JOB = 'N',
a = 2*NV, b = 2*NW, if DICO = 'D' and JOB = 'C';
LDW3 = N*(2*N + MAX(N,M,P) + 5) + N*(N+1)/2,
LDW4 = N*(M+P+2) + 2*M*P + MIN(N,M) +
MAX( 3*M+1, MIN(N,M)+P ).
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: with ORDSEL = 'F', the selected order NR is greater
than NSMIN, the sum of the order of the
ALPHA-unstable part and the order of a minimal
realization of the ALPHA-stable part of the given
system; in this case, the resulting NR is set equal
to NSMIN;
= 2: with ORDSEL = 'F', the selected order NR is less
than the order of the ALPHA-unstable part of the
given system; in this case NR is set equal to the
order of the ALPHA-unstable part.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the computation of the ordered real Schur form of A
failed;
= 2: the separation of the ALPHA-stable/unstable
diagonal blocks failed because of very close
eigenvalues;
= 3: the reduction of AV or AV-BV*inv(DV)*CV to a
real Schur form failed;
= 4: the reduction of AW or AW-BW*inv(DW)*CW to a
real Schur form failed;
= 5: JOB = 'N' and AV is not antistable, or
JOB = 'C' and AV is not stable;
= 6: JOB = 'N' and AW is not antistable, or
JOB = 'C' and AW is not stable;
= 7: the computation of Hankel singular values failed;
= 8: the computation of stable projection in the
Hankel-norm approximation algorithm failed;
= 9: the order of computed stable projection in the
Hankel-norm approximation algorithm differs
from the order of Hankel-norm approximation;
= 10: DV is singular;
= 11: DW is singular;
= 12: the solution of the Sylvester equation failed
because the zeros of V (if JOB = 'N') or of conj(V)
(if JOB = 'C') are not distinct from the poles
of G1sr (see METHOD);
= 13: the solution of the Sylvester equation failed
because the zeros of W (if JOB = 'N') or of conj(W)
(if JOB = 'C') are not distinct from the poles
of G1sr (see METHOD).
Method
Let G be the transfer-function matrix of the original
linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t), (1)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system. The subroutine AB09KD determines
the matrices of a reduced order system
d[z(t)] = Ar*z(t) + Br*u(t)
yr(t) = Cr*z(t) + Dr*u(t), (2)
such that the corresponding transfer-function matrix Gr minimizes
the Hankel-norm of the frequency-weighted error
V*(G-Gr)*W, (3)
or
conj(V)*(G-Gr)*conj(W). (4)
For minimizing (3), V and W are assumed to be antistable, while
for minimizing (4), V and W are assumed to be stable transfer-
function matrices.
Note: conj(G) = G'(-s) for a continuous-time system and
conj(G) = G'(1/z) for a discrete-time system.
The following procedure is used to reduce G (see [1]):
1) Decompose additively G as
G = G1 + G2,
such that G1 = (A1,B1,C1,D) has only ALPHA-stable poles and
G2 = (A2,B2,C2,0) has only ALPHA-unstable poles.
2) Compute G1s, the stable projection of V*G1*W or
conj(V)*G1*conj(W), using explicit formulas [4].
3) Determine G1sr, the optimal Hankel-norm approximation of G1s
of order r.
4) Compute G1r, the stable projection of either inv(V)*G1sr*inv(W)
or conj(inv(V))*G1sr*conj(inv(W)), using explicit formulas [4].
5) Assemble the reduced model Gr as
Gr = G1r + G2.
To reduce the weighted ALPHA-stable part G1s at step 3, the
optimal Hankel-norm approximation method of [2], based on the
square-root balancing projection formulas of [3], is employed.
The optimal weighted approximation error satisfies
HNORM[V*(G-Gr)*W] = S(r+1),
or
HNORM[conj(V)*(G-Gr)*conj(W)] = S(r+1),
where S(r+1) is the (r+1)-th Hankel singular value of G1s, the
transfer-function matrix computed at step 2 of the above
procedure, and HNORM(.) denotes the Hankel-norm.
References
[1] Latham, G.A. and Anderson, B.D.O.
Frequency-weighted optimal Hankel-norm approximation of stable
transfer functions.
Systems & Control Letters, Vol. 5, pp. 229-236, 1985.
[2] Glover, K.
All optimal Hankel norm approximation of linear
multivariable systems and their L-infinity error bounds.
Int. J. Control, Vol. 36, pp. 1145-1193, 1984.
[3] Tombs M.S. and Postlethwaite I.
Truncated balanced realization of stable, non-minimal
state-space systems.
Int. J. Control, Vol. 46, pp. 1319-1330, 1987.
[4] Varga A.
Explicit formulas for an efficient implementation
of the frequency-weighting model reduction approach.
Proc. 1993 European Control Conference, Groningen, NL,
pp. 693-696, 1993.
Numerical Aspects
The implemented methods rely on an accuracy enhancing square-root
technique.
3
The algorithms require less than 30N floating point operations.
Further Comments
NoneExample
Program Text
* AB09KD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, NMAX, NVMAX, NWMAX, PMAX
PARAMETER ( MMAX = 20, NMAX = 20, NVMAX = 10, NWMAX = 10,
$ PMAX = 20 )
INTEGER LDA, LDAV, LDAW, LDB, LDBV, LDBW,
$ LDC, LDCV, LDCW, LDD, LDDV, LDDW
PARAMETER ( LDA = NMAX, LDAV = NVMAX, LDAW = NWMAX,
$ LDB = NMAX, LDBV = NVMAX, LDBW = NWMAX,
$ LDC = PMAX, LDCV = PMAX, LDCW = MMAX,
$ LDD = PMAX, LDDV = PMAX, LDDW = MMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = 2*MAX( MMAX, PMAX ) )
INTEGER LDW1, LDW2, LDW3, LDW4, LDWORK
PARAMETER ( LDW1 = MAX( NVMAX*( NVMAX + 5 ), NVMAX*NMAX +
$ MAX( 2*NVMAX, PMAX*NMAX, PMAX*MMAX ) ))
PARAMETER ( LDW2 = MAX( NWMAX*( NWMAX + 5 ), NWMAX*NMAX +
$ MAX( 2*NWMAX, MMAX*NMAX, PMAX*MMAX ) ))
PARAMETER ( LDW3 = NMAX*( 2*NMAX + MAX( NMAX, MMAX, PMAX )
$ + 5 ) + ( NMAX*( NMAX + 1 ) )/2 )
PARAMETER ( LDW4 = NMAX*( MMAX + PMAX + 2 ) + 2*MMAX*PMAX +
$ MIN( NMAX, MMAX ) +
$ MAX( 3*MMAX + 1,
$ MIN( NMAX, MMAX ) + PMAX ) )
PARAMETER ( LDWORK = MAX( LDW1, LDW2, LDW3, LDW4 ) )
* .. Local Scalars ..
LOGICAL LEFTW, RIGHTW
DOUBLE PRECISION ALPHA, TOL1, TOL2
INTEGER I, INFO, IWARN, J, M, N, NR, NS, NV, NW, P
CHARACTER*1 DICO, EQUIL, JOB, ORDSEL, WEIGHT
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), AV(LDAV,NVMAX), AW(LDAW,NWMAX),
$ B(LDB,MMAX), BV(LDBV,PMAX), BW(LDBW,MMAX),
$ C(LDC,NMAX), CV(LDCV,NVMAX), CW(LDCW,NWMAX),
$ D(LDD,MMAX), DV(LDDV,PMAX), DW(LDDW,MMAX),
$ DWORK(LDWORK), HSV(NMAX)
INTEGER IWORK(LIWORK)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL AB09KD
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, NV, NW, NR, ALPHA, TOL1, TOL2,
$ JOB, DICO, WEIGHT, EQUIL, ORDSEL
LEFTW = LSAME( WEIGHT, 'L' ) .OR. LSAME( WEIGHT, 'B' )
RIGHTW = LSAME( WEIGHT, 'R' ) .OR. LSAME( WEIGHT, 'B' )
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1, N )
IF( P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
IF( LEFTW .OR. NV.GT.0 ) THEN
IF( NV.LT.0 .OR. NV.GT.NVMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) NV
ELSE
IF( NV.GT.0 ) THEN
READ ( NIN, FMT = * )
$ ( ( AV(I,J), J = 1,NV ), I = 1,NV )
READ ( NIN, FMT = * )
$ ( ( BV(I,J), J = 1,P ), I = 1, NV )
READ ( NIN, FMT = * )
$ ( ( CV(I,J), J = 1,NV ), I = 1,P )
END IF
IF( LEFTW ) READ ( NIN, FMT = * )
$ ( ( DV(I,J), J = 1,P ), I = 1,P )
END IF
END IF
IF( RIGHTW ) THEN
IF( NW.LT.0 .OR. NW.GT.NWMAX ) THEN
WRITE ( NOUT, FMT = 99985 ) NW
ELSE
IF( NW.GT.0 ) THEN
READ ( NIN, FMT = * )
$ ( ( AW(I,J), J = 1,NW ), I = 1,NW )
READ ( NIN, FMT = * )
$ ( ( BW(I,J), J = 1,M ), I = 1, NW )
READ ( NIN, FMT = * )
$ ( ( CW(I,J), J = 1,NW ), I = 1,M )
END IF
READ ( NIN, FMT = * )
$ ( ( DW(I,J), J = 1,M ), I = 1,M )
END IF
END IF
* Find a reduced ssr for (A,B,C,D).
CALL AB09KD( JOB, DICO, WEIGHT, EQUIL, ORDSEL, N, NV, NW,
$ M, P, NR, ALPHA, A, LDA, B, LDB, C, LDC,
$ D, LDD, AV, LDAV, BV, LDBV, CV, LDCV,
$ DV, LDDV, AW, LDAW, BW, LDBW, CW, LDCW,
$ DW, LDDW, NS, HSV, TOL1, TOL2, IWORK,
$ DWORK, LDWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF( IWARN.NE.0) WRITE ( NOUT, FMT = 99984 ) IWARN
WRITE ( NOUT, FMT = 99997 ) NR
WRITE ( NOUT, FMT = 99987 )
WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1, NS )
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
20 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
60 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 70 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1,M )
70 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB09KD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB09KD = ',I2)
99997 FORMAT (/' The order of reduced model = ',I2)
99996 FORMAT (/' The reduced state dynamics matrix Ar is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The reduced input/state matrix Br is ')
99992 FORMAT (/' The reduced state/output matrix Cr is ')
99991 FORMAT (/' The reduced input/output matrix Dr is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99987 FORMAT (/' The Hankel singular values of weighted ALPHA-stable',
$ ' part are')
99986 FORMAT (/' NV is out of range.',/' NV = ',I5)
99985 FORMAT (/' NW is out of range.',/' NW = ',I5)
99984 FORMAT (' IWARN on exit from AB09KD = ',I2)
END
Program Data
AB09KD EXAMPLE PROGRAM DATA (Continuous system)
6 1 1 2 0 0 0.0 1.E-1 1.E-14 N C L S A
-3.8637 -7.4641 -9.1416 -7.4641 -3.8637 -1.0000
1.0000 0 0 0 0 0
0 1.0000 0 0 0 0
0 0 1.0000 0 0 0
0 0 0 1.0000 0 0
0 0 0 0 1.0000 0
1
0
0
0
0
0
0 0 0 0 0 1
0
0.2000 -1.0000
1.0000 0
1
0
-1.8000 0
1
Program Results
AB09KD EXAMPLE PROGRAM RESULTS The order of reduced model = 4 The Hankel singular values of weighted ALPHA-stable part are 2.6790 2.1589 0.8424 0.1929 0.0219 0.0011 The reduced state dynamics matrix Ar is -0.2391 0.3072 1.1630 1.1967 -2.9709 -0.2391 2.6270 3.1027 0.0000 0.0000 -0.5137 -1.2842 0.0000 0.0000 0.1519 -0.5137 The reduced input/state matrix Br is -1.0497 -3.7052 0.8223 0.7435 The reduced state/output matrix Cr is -0.4466 0.0143 -0.4780 -0.2013 The reduced input/output matrix Dr is 0.0219
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB09KX.html 0000664 0000000 0000000 00000036664 14560147231 0020367 0 ustar 00root root 0000000 0000000
Purpose
To construct a state-space representation (A,BS,CS,DS) of the stable projection of V*G*W or conj(V)*G*conj(W) from the state-space representations (A,B,C,D), (AV,BV,CV,DV), and (AW,BW,CW,DW) of the transfer-function matrices G, V and W, respectively. G is assumed to be a stable transfer-function matrix and the state matrix A must be in a real Schur form. When computing the stable projection of V*G*W, V and W are assumed to be completely unstable transfer-function matrices. When computing the stable projection of conj(V)*G*conj(W), V and W are assumed to be stable transfer-function matrices. For a transfer-function matrix G, conj(G) denotes the conjugate of G given by G'(-s) for a continuous-time system or G'(1/z) for a discrete-time system.Specification
SUBROUTINE AB09KX( JOB, DICO, WEIGHT, N, NV, NW, M, P,
$ A, LDA, B, LDB, C, LDC, D, LDD,
$ AV, LDAV, BV, LDBV, CV, LDCV, DV, LDDV,
$ AW, LDAW, BW, LDBW, CW, LDCW, DW, LDDW,
$ DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, JOB, WEIGHT
INTEGER INFO, IWARN, LDA, LDAV, LDAW, LDB, LDBV, LDBW,
$ LDC, LDCV, LDCW, LDD, LDDV, LDDW, LDWORK, M, N,
$ NV, NW, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ AV(LDAV,*), BV(LDBV,*), CV(LDCV,*), DV(LDDV,*),
$ AW(LDAW,*), BW(LDBW,*), CW(LDCW,*), DW(LDDW,*),
$ DWORK(*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies which projection to be computed as follows:
= 'N': compute the stable projection of V*G*W;
= 'C': compute the stable projection of
conj(V)*G*conj(W).
DICO CHARACTER*1
Specifies the type of the systems as follows:
= 'C': G, V and W are continuous-time systems;
= 'D': G, V and W are discrete-time systems.
WEIGHT CHARACTER*1
Specifies the type of frequency weighting, as follows:
= 'N': no weightings are used (V = I, W = I);
= 'L': only left weighting V is used (W = I);
= 'R': only right weighting W is used (V = I);
= 'B': both left and right weightings V and W are used.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. Also the number of rows of
the matrix B and the number of columns of the matrix C.
N represents the dimension of the state vector of the
system with the transfer-function matrix G. N >= 0.
NV (input) INTEGER
The order of the matrix AV. Also the number of rows of
the matrix BV and the number of columns of the matrix CV.
NV represents the dimension of the state vector of the
system with the transfer-function matrix V. NV >= 0.
NW (input) INTEGER
The order of the matrix AW. Also the number of rows of
the matrix BW and the number of columns of the matrix CW.
NW represents the dimension of the state vector of the
system with the transfer-function matrix W. NW >= 0.
M (input) INTEGER
The number of columns of the matrices B, D, BW and DW
and number of rows of the matrices CW and DW. M >= 0.
M represents the dimension of input vectors of the
systems with the transfer-function matrices G and W and
also the dimension of the output vector of the system
with the transfer-function matrix W.
P (input) INTEGER
The number of rows of the matrices C, D, CV and DV and the
number of columns of the matrices BV and DV. P >= 0.
P represents the dimension of output vectors of the
systems with the transfer-function matrices G and V and
also the dimension of the input vector of the system
with the transfer-function matrix V.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must
contain the state matrix A of the system with the
transfer-function matrix G in a real Schur form.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input matrix B of the system with the
transfer-function matrix G.
On exit, if INFO = 0, the leading N-by-M part of this
array contains the input matrix BS of the stable
projection of V*G*W if JOB = 'N', and of conj(V)*G*conj(W)
if JOB = 'C'.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the output matrix C of the system with the
transfer-function matrix G.
On exit, if INFO = 0, the leading P-by-N part of this
array contains the output matrix CS of the stable
projection of V*G*W if JOB = 'N', and of conj(V)*G*conj(W)
if JOB = 'C'.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the feedthrough matrix D of the system with the
transfer-function matrix G.
On exit, if INFO = 0, the leading P-by-M part of this
array contains the feedthrough matrix DS of the stable
projection of V*G*W if JOB = 'N', and of conj(V)*G*conj(W)
if JOB = 'C'.
LDD INTEGER
The leading dimension of the array D. LDD >= MAX(1,P).
AV (input/output) DOUBLE PRECISION array, dimension (LDAV,NV)
On entry, if WEIGHT = 'L' or 'B', the leading NV-by-NV
part of this array must contain the state matrix AV of
the system with the transfer-function matrix V.
On exit, if WEIGHT = 'L' or 'B', and INFO = 0, the leading
NV-by-NV part of this array contains a real Schur form
of AV.
AV is not referenced if WEIGHT = 'R' or 'N'.
LDAV INTEGER
The leading dimension of the array AV.
LDAV >= MAX(1,NV), if WEIGHT = 'L' or 'B';
LDAV >= 1, if WEIGHT = 'R' or 'N'.
BV (input/output) DOUBLE PRECISION array, dimension (LDBV,P)
On entry, if WEIGHT = 'L' or 'B', the leading NV-by-P part
of this array must contain the input matrix BV of the
system with the transfer-function matrix V.
On exit, if WEIGHT = 'L' or 'B', and INFO = 0, the leading
NV-by-P part of this array contains the transformed input
matrix BV.
BV is not referenced if WEIGHT = 'R' or 'N'.
LDBV INTEGER
The leading dimension of the array BV.
LDBV >= MAX(1,NV), if WEIGHT = 'L' or 'B';
LDBV >= 1, if WEIGHT = 'R' or 'N'.
CV (input/output) DOUBLE PRECISION array, dimension (LDCV,NV)
On entry, if WEIGHT = 'L' or 'B', the leading P-by-NV part
of this array must contain the output matrix CV of the
system with the transfer-function matrix V.
On exit, if WEIGHT = 'L' or 'B', and INFO = 0, the leading
P-by-NV part of this array contains the transformed output
matrix CV.
CV is not referenced if WEIGHT = 'R' or 'N'.
LDCV INTEGER
The leading dimension of the array CV.
LDCV >= MAX(1,P), if WEIGHT = 'L' or 'B';
LDCV >= 1, if WEIGHT = 'R' or 'N'.
DV (input) DOUBLE PRECISION array, dimension (LDDV,P)
If WEIGHT = 'L' or 'B', the leading P-by-P part of this
array must contain the feedthrough matrix DV of the system
with the transfer-function matrix V.
DV is not referenced if WEIGHT = 'R' or 'N'.
LDDV INTEGER
The leading dimension of the array DV.
LDDV >= MAX(1,P), if WEIGHT = 'L' or 'B';
LDDV >= 1, if WEIGHT = 'R' or 'N'.
AW (input/output) DOUBLE PRECISION array, dimension (LDAW,NW)
On entry, if WEIGHT = 'R' or 'B', the leading NW-by-NW
part of this array must contain the state matrix AW of
the system with the transfer-function matrix W.
On exit, if WEIGHT = 'R' or 'B', and INFO = 0, the leading
NW-by-NW part of this array contains a real Schur form
of AW.
AW is not referenced if WEIGHT = 'L' or 'N'.
LDAW INTEGER
The leading dimension of the array AW.
LDAW >= MAX(1,NW), if WEIGHT = 'R' or 'B';
LDAW >= 1, if WEIGHT = 'L' or 'N'.
BW (input/output) DOUBLE PRECISION array, dimension (LDBW,M)
On entry, if WEIGHT = 'R' or 'B', the leading NW-by-M part
of this array must contain the input matrix BW of the
system with the transfer-function matrix W.
On exit, if WEIGHT = 'R' or 'B', and INFO = 0, the leading
NW-by-M part of this array contains the transformed input
matrix BW.
BW is not referenced if WEIGHT = 'L' or 'N'.
LDBW INTEGER
The leading dimension of the array BW.
LDBW >= MAX(1,NW), if WEIGHT = 'R' or 'B';
LDBW >= 1, if WEIGHT = 'L' or 'N'.
CW (input/output) DOUBLE PRECISION array, dimension (LDCW,NW)
On entry, if WEIGHT = 'R' or 'B', the leading M-by-NW part
of this array must contain the output matrix CW of the
system with the transfer-function matrix W.
On exit, if WEIGHT = 'R' or 'B', and INFO = 0, the leading
M-by-NW part of this array contains the transformed output
matrix CW.
CW is not referenced if WEIGHT = 'L' or 'N'.
LDCW INTEGER
The leading dimension of the array CW.
LDCW >= MAX(1,M), if WEIGHT = 'R' or 'B';
LDCW >= 1, if WEIGHT = 'L' or 'N'.
DW (input) DOUBLE PRECISION array, dimension (LDDW,M)
If WEIGHT = 'R' or 'B', the leading M-by-M part of this
array must contain the feedthrough matrix DW of the system
with the transfer-function matrix W.
DW is not referenced if WEIGHT = 'L' or 'N'.
LDDW INTEGER
The leading dimension of the array DW.
LDDW >= MAX(1,M), if WEIGHT = 'R' or 'B';
LDDW >= 1, if WEIGHT = 'L' or 'N'.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( 1, LDW1, LDW2 ), where
LDW1 = 0 if WEIGHT = 'R' or 'N' and
LDW1 = MAX( NV*(NV+5), NV*N + MAX( a, P*N, P*M ) )
if WEIGHT = 'L' or WEIGHT = 'B',
LDW2 = 0 if WEIGHT = 'L' or 'N' and
LDW2 = MAX( NW*(NW+5), NW*N + MAX( b, M*N, P*M ) )
if WEIGHT = 'R' or WEIGHT = 'B',
a = 0, b = 0, if DICO = 'C' or JOB = 'N',
a = 2*NV, b = 2*NW, if DICO = 'D' and JOB = 'C'.
For good performance, LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: JOB = 'N' and AV is not completely unstable, or
JOB = 'C' and AV is not stable;
= 2: JOB = 'N' and AW is not completely unstable, or
JOB = 'C' and AW is not stable;
= 3: both above conditions appear.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction of AV to a real Schur form failed;
= 2: the reduction of AW to a real Schur form failed;
= 3: the solution of the Sylvester equation failed
because the matrices A and AV have common
eigenvalues (if JOB = 'N'), or -AV and A have
common eigenvalues (if JOB = 'C' and DICO = 'C'),
or AV has an eigenvalue which is the reciprocal of
one of the eigenvalues of A (if JOB = 'C' and
DICO = 'D');
= 4: the solution of the Sylvester equation failed
because the matrices A and AW have common
eigenvalues (if JOB = 'N'), or -AW and A have
common eigenvalues (if JOB = 'C' and DICO = 'C'),
or AW has an eigenvalue which is the reciprocal of
one of the eigenvalues of A (if JOB = 'C' and
DICO = 'D').
Method
The matrices of the stable projection of V*G*W are computed as
BS = B*DW + Y*BW, CS = CV*X + DV*C, DS = DV*D*DW,
where X and Y satisfy the continuous-time Sylvester equations
AV*X - X*A + BV*C = 0,
-A*Y + Y*AW + B*CW = 0.
The matrices of the stable projection of conj(V)*G*conj(W) are
computed using the explicit formulas established in [1].
For a continuous-time system, the matrices BS, CS and DS of
the stable projection are computed as
BS = B*DW' + Y*CW', CS = BV'*X + DV'*C, DS = DV'*D*DW',
where X and Y satisfy the continuous-time Sylvester equations
AV'*X + X*A + CV'*C = 0,
A*Y + Y*AW' + B*BW' = 0.
For a discrete-time system, the matrices BS, CS and DS of
the stable projection are computed as
BS = B*DW' + A*Y*CW', CS = BV'*X*A + DV'*C,
DS = DV'*D*DW' + BV'*X*B*DW' + DV'*C*Y*CW' + BV'*X*A*Y*CW',
where X and Y satisfy the discrete-time Sylvester equations
AV'*X*A + CV'*C = X,
A*Y*AW' + B*BW' = Y.
References
[1] Varga A.
Explicit formulas for an efficient implementation
of the frequency-weighting model reduction approach.
Proc. 1993 European Control Conference, Groningen, NL,
pp. 693-696, 1993.
Numerical Aspects
The implemented methods rely on numerically stable algorithms.Further Comments
The matrix A must be stable, but its stability is not checked by this routine.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute a reduced order model (Ar,Br,Cr) for an original state-space representation (A,B,C) by using either the square-root or the balancing-free square-root Balance & Truncate (B & T) model reduction method for the ALPHA-stable part of the system.Specification
SUBROUTINE AB09MD( DICO, JOB, EQUIL, ORDSEL, N, M, P, NR, ALPHA,
$ A, LDA, B, LDB, C, LDC, NS, HSV, TOL, IWORK,
$ DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, EQUIL, JOB, ORDSEL
INTEGER INFO, IWARN, LDA, LDB, LDC, LDWORK, M, N, NR,
$ NS, P
DOUBLE PRECISION ALPHA, TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), HSV(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOB CHARACTER*1
Specifies the model reduction approach to be used
as follows:
= 'B': use the square-root Balance & Truncate method;
= 'N': use the balancing-free square-root
Balance & Truncate method.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily
equilibrate the triplet (A,B,C) as follows:
= 'S': perform equilibration (scaling);
= 'N': do not perform equilibration.
ORDSEL CHARACTER*1
Specifies the order selection method as follows:
= 'F': the resulting order NR is fixed;
= 'A': the resulting order NR is automatically determined
on basis of the given tolerance TOL.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation, i.e.
the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NR (input/output) INTEGER
On entry with ORDSEL = 'F', NR is the desired order of the
resulting reduced order system. 0 <= NR <= N.
On exit, if INFO = 0, NR is the order of the resulting
reduced order model. For a system with NU ALPHA-unstable
eigenvalues and NS ALPHA-stable eigenvalues (NU+NS = N),
NR is set as follows: if ORDSEL = 'F', NR is equal to
NU+MIN(MAX(0,NR-NU),NMIN), where NR is the desired order
on entry, and NMIN is the order of a minimal realization
of the ALPHA-stable part of the given system; NMIN is
determined as the number of Hankel singular values greater
than NS*EPS*HNORM(As,Bs,Cs), where EPS is the machine
precision (see LAPACK Library Routine DLAMCH) and
HNORM(As,Bs,Cs) is the Hankel norm of the ALPHA-stable
part of the given system (computed in HSV(1));
if ORDSEL = 'A', NR is the sum of NU and the number of
Hankel singular values greater than
MAX(TOL,NS*EPS*HNORM(As,Bs,Cs)).
ALPHA (input) DOUBLE PRECISION
Specifies the ALPHA-stability boundary for the eigenvalues
of the state dynamics matrix A. For a continuous-time
system (DICO = 'C'), ALPHA <= 0 is the boundary value for
the real parts of eigenvalues, while for a discrete-time
system (DICO = 'D'), 0 <= ALPHA <= 1 represents the
boundary value for the moduli of eigenvalues.
The ALPHA-stability domain does not include the boundary.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A.
On exit, if INFO = 0, the leading NR-by-NR part of this
array contains the state dynamics matrix Ar of the reduced
order system.
The resulting A has a block-diagonal form with two blocks.
For a system with NU ALPHA-unstable eigenvalues and
NS ALPHA-stable eigenvalues (NU+NS = N), the leading
NU-by-NU block contains the unreduced part of A
corresponding to ALPHA-unstable eigenvalues in an
upper real Schur form.
The trailing (NR+NS-N)-by-(NR+NS-N) block contains
the reduced part of A corresponding to ALPHA-stable
eigenvalues.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, if INFO = 0, the leading NR-by-M part of this
array contains the input/state matrix Br of the reduced
order system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, if INFO = 0, the leading P-by-NR part of this
array contains the state/output matrix Cr of the reduced
order system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
NS (output) INTEGER
The dimension of the ALPHA-stable subsystem.
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, the leading NS elements of HSV contain the
Hankel singular values of the ALPHA-stable part of the
original system ordered decreasingly.
HSV(1) is the Hankel norm of the ALPHA-stable subsystem.
Tolerances
TOL DOUBLE PRECISION
If ORDSEL = 'A', TOL contains the tolerance for
determining the order of reduced system.
For model reduction, the recommended value is
TOL = c*HNORM(As,Bs,Cs), where c is a constant in the
interval [0.00001,0.001], and HNORM(As,Bs,Cs) is the
Hankel-norm of the ALPHA-stable part of the given system
(computed in HSV(1)).
If TOL <= 0 on entry, the used default value is
TOL = NS*EPS*HNORM(As,Bs,Cs), where NS is the number of
ALPHA-stable eigenvalues of A and EPS is the machine
precision (see LAPACK Library Routine DLAMCH).
This value is appropriate to compute a minimal realization
of the ALPHA-stable part.
If ORDSEL = 'F', the value of TOL is ignored.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK = 0, if JOB = 'B';
LIWORK = N, if JOB = 'N'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,N*(2*N+MAX(N,M,P)+5) + N*(N+1)/2).
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: with ORDSEL = 'F', the selected order NR is greater
than NSMIN, the sum of the order of the
ALPHA-unstable part and the order of a minimal
realization of the ALPHA-stable part of the given
system. In this case, the resulting NR is set equal
to NSMIN.
= 2: with ORDSEL = 'F', the selected order NR is less
than the order of the ALPHA-unstable part of the
given system. In this case NR is set equal to the
order of the ALPHA-unstable part.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the computation of the ordered real Schur form of A
failed;
= 2: the separation of the ALPHA-stable/unstable diagonal
blocks failed because of very close eigenvalues;
= 3: the computation of Hankel singular values failed.
Method
Let be the following linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) (1)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system. The subroutine AB09MD determines for
the given system (1), the matrices of a reduced order system
d[z(t)] = Ar*z(t) + Br*u(t)
yr(t) = Cr*z(t) (2)
such that
HSV(NR+NS-N) <= INFNORM(G-Gr) <= 2*[HSV(NR+NS-N+1)+...+HSV(NS)],
where G and Gr are transfer-function matrices of the systems
(A,B,C) and (Ar,Br,Cr), respectively, and INFNORM(G) is the
infinity-norm of G.
The following procedure is used to reduce a given G:
1) Decompose additively G as
G = G1 + G2
such that G1 = (As,Bs,Cs) has only ALPHA-stable poles and
G2 = (Au,Bu,Cu) has only ALPHA-unstable poles.
2) Determine G1r, a reduced order approximation of the
ALPHA-stable part G1.
3) Assemble the reduced model Gr as
Gr = G1r + G2.
To reduce the ALPHA-stable part G1, if JOB = 'B', the square-root
Balance & Truncate method of [1] is used, and for an ALPHA-stable
continuous-time system (DICO = 'C'), the resulting reduced model
is balanced. For ALPHA-stable systems, setting TOL < 0, the
routine can be used to compute balanced minimal state-space
realizations.
If JOB = 'N', the balancing-free square-root version of the
Balance & Truncate method [2] is used to reduce the ALPHA-stable
part G1.
References
[1] Tombs M.S. and Postlethwaite I.
Truncated balanced realization of stable, non-minimal
state-space systems.
Int. J. Control, Vol. 46, pp. 1319-1330, 1987.
[2] Varga A.
Efficient minimal realization procedure based on balancing.
Proc. of IMACS/IFAC Symp. MCTS, Lille, France, May 1991,
A. El Moudui, P. Borne, S. G. Tzafestas (Eds.),
Vol. 2, pp. 42-46.
Numerical Aspects
The implemented methods rely on accuracy enhancing square-root or
balancing-free square-root techniques.
3
The algorithms require less than 30N floating point operations.
Further Comments
NoneExample
Program Text
* AB09MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX*( 2*NMAX +
$ MAX( NMAX, MMAX, PMAX ) + 5 ) +
$ ( NMAX*( NMAX + 1 ) )/2 )
* .. Local Scalars ..
DOUBLE PRECISION ALPHA, TOL
INTEGER I, INFO, IWARN, J, M, N, NR, NS, P
CHARACTER*1 DICO, EQUIL, JOB, ORDSEL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ DWORK(LDWORK), HSV(NMAX)
INTEGER IWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL AB09MD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, NR, ALPHA, TOL, DICO, JOB, EQUIL,
$ ORDSEL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find a reduced ssr for (A,B,C).
CALL AB09MD( DICO, JOB, EQUIL, ORDSEL, N, M, P, NR,
$ ALPHA, A, LDA, B, LDB, C, LDC, NS, HSV,
$ TOL, IWORK, DWORK, LDWORK, IWARN, INFO)
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NR
WRITE ( NOUT, FMT = 99987 )
WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1,NS )
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
60 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB09MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB09MD = ',I2)
99997 FORMAT (' The order of reduced model = ',I2)
99996 FORMAT (/' The reduced state dynamics matrix Ar is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The reduced input/state matrix Br is ')
99992 FORMAT (/' The reduced state/output matrix Cr is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99987 FORMAT (/' The Hankel singular values of ALPHA-stable part are')
END
Program Data
AB09MD EXAMPLE PROGRAM DATA (Continuous system) 7 2 3 0 -.6D0 1.D-1 C N N A -0.04165 0.0000 4.9200 -4.9200 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 0.0000 0.0000 0.5450 0.0000 0.0000 0.0000 -0.5450 0.0000 0.0000 0.0000 0.0000 0.0000 4.9200 -0.04165 0.0000 4.9200 0.0000 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 12.500 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000Program Results
AB09MD EXAMPLE PROGRAM RESULTS The order of reduced model = 5 The Hankel singular values of ALPHA-stable part are 1.9178 0.8621 0.7666 0.0336 0.0246 The reduced state dynamics matrix Ar is -0.5181 -1.1084 0.0000 0.0000 0.0000 8.8157 -0.5181 0.0000 0.0000 0.0000 0.0000 0.0000 0.5124 0.0000 1.7910 0.0000 0.0000 0.0000 -1.4460 0.0000 0.0000 0.0000 -4.2167 0.0000 -2.9900 The reduced input/state matrix Br is -1.2837 1.2837 -0.7522 0.7522 -0.7447 -0.7447 1.9275 -1.9275 -3.6872 -3.6872 The reduced state/output matrix Cr is -0.1380 -0.6445 -0.6582 -0.5771 0.2222 0.6246 0.0196 0.0000 0.4131 0.0000 0.1380 0.6445 -0.6582 0.5771 0.2222
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB09ND.html 0000664 0000000 0000000 00000043531 14560147231 0020335 0 ustar 00root root 0000000 0000000
Purpose
To compute a reduced order model (Ar,Br,Cr,Dr) for an original state-space representation (A,B,C,D) by using either the square-root or the balancing-free square-root Singular Perturbation Approximation (SPA) model reduction method for the ALPHA-stable part of the system.Specification
SUBROUTINE AB09ND( DICO, JOB, EQUIL, ORDSEL, N, M, P, NR, ALPHA,
$ A, LDA, B, LDB, C, LDC, D, LDD, NS, HSV, TOL1,
$ TOL2, IWORK, DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, EQUIL, JOB, ORDSEL
INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, LDWORK,
$ M, N, NR, NS, P
DOUBLE PRECISION ALPHA, TOL1, TOL2
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), HSV(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOB CHARACTER*1
Specifies the model reduction approach to be used
as follows:
= 'B': use the square-root SPA method;
= 'N': use the balancing-free square-root SPA method.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily
equilibrate the triplet (A,B,C) as follows:
= 'S': perform equilibration (scaling);
= 'N': do not perform equilibration.
ORDSEL CHARACTER*1
Specifies the order selection method as follows:
= 'F': the resulting order NR is fixed;
= 'A': the resulting order NR is automatically determined
on basis of the given tolerance TOL1.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation, i.e.
the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NR (input/output) INTEGER
On entry with ORDSEL = 'F', NR is the desired order of the
resulting reduced order system. 0 <= NR <= N.
On exit, if INFO = 0, NR is the order of the resulting
reduced order model. For a system with NU ALPHA-unstable
eigenvalues and NS ALPHA-stable eigenvalues (NU+NS = N),
NR is set as follows: if ORDSEL = 'F', NR is equal to
NU+MIN(MAX(0,NR-NU),NMIN), where NR is the desired order
on entry, and NMIN is the order of a minimal realization
of the ALPHA-stable part of the given system; NMIN is
determined as the number of Hankel singular values greater
than NS*EPS*HNORM(As,Bs,Cs), where EPS is the machine
precision (see LAPACK Library Routine DLAMCH) and
HNORM(As,Bs,Cs) is the Hankel norm of the ALPHA-stable
part of the given system (computed in HSV(1));
if ORDSEL = 'A', NR is the sum of NU and the number of
Hankel singular values greater than
MAX(TOL1,NS*EPS*HNORM(As,Bs,Cs)).
ALPHA (input) DOUBLE PRECISION
Specifies the ALPHA-stability boundary for the eigenvalues
of the state dynamics matrix A. For a continuous-time
system (DICO = 'C'), ALPHA <= 0 is the boundary value for
the real parts of eigenvalues, while for a discrete-time
system (DICO = 'D'), 0 <= ALPHA <= 1 represents the
boundary value for the moduli of eigenvalues.
The ALPHA-stability domain does not include the boundary.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A.
On exit, if INFO = 0, the leading NR-by-NR part of this
array contains the state dynamics matrix Ar of the reduced
order system.
The resulting A has a block-diagonal form with two blocks.
For a system with NU ALPHA-unstable eigenvalues and
NS ALPHA-stable eigenvalues (NU+NS = N), the leading
NU-by-NU block contains the unreduced part of A
corresponding to ALPHA-unstable eigenvalues in an
upper real Schur form.
The trailing (NR+NS-N)-by-(NR+NS-N) block contains
the reduced part of A corresponding to ALPHA-stable
eigenvalues.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, if INFO = 0, the leading NR-by-M part of this
array contains the input/state matrix Br of the reduced
order system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, if INFO = 0, the leading P-by-NR part of this
array contains the state/output matrix Cr of the reduced
order system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the original input/output matrix D.
On exit, if INFO = 0, the leading P-by-M part of this
array contains the input/output matrix Dr of the reduced
order system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
NS (output) INTEGER
The dimension of the ALPHA-stable subsystem.
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, the leading NS elements of HSV contain the
Hankel singular values of the ALPHA-stable part of the
original system ordered decreasingly.
HSV(1) is the Hankel norm of the ALPHA-stable subsystem.
Tolerances
TOL1 DOUBLE PRECISION
If ORDSEL = 'A', TOL1 contains the tolerance for
determining the order of reduced system.
For model reduction, the recommended value is
TOL1 = c*HNORM(As,Bs,Cs), where c is a constant in the
interval [0.00001,0.001], and HNORM(As,Bs,Cs) is the
Hankel-norm of the ALPHA-stable part of the given system
(computed in HSV(1)).
If TOL1 <= 0 on entry, the used default value is
TOL1 = NS*EPS*HNORM(As,Bs,Cs), where NS is the number of
ALPHA-stable eigenvalues of A and EPS is the machine
precision (see LAPACK Library Routine DLAMCH).
This value is appropriate to compute a minimal realization
of the ALPHA-stable part.
If ORDSEL = 'F', the value of TOL1 is ignored.
TOL2 DOUBLE PRECISION
The tolerance for determining the order of a minimal
realization of the ALPHA-stable part of the given system.
The recommended value is TOL2 = NS*EPS*HNORM(As,Bs,Cs).
This value is used by default if TOL2 <= 0 on entry.
If TOL2 > 0, then TOL2 <= TOL1.
Workspace
IWORK INTEGER array, dimension (MAX(1,2*N))
On exit, if INFO = 0, IWORK(1) contains the order of the
minimal realization of the ALPHA-stable part of the
system.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,N*(2*N+MAX(N,M,P)+5) + N*(N+1)/2).
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: with ORDSEL = 'F', the selected order NR is greater
than NSMIN, the sum of the order of the
ALPHA-unstable part and the order of a minimal
realization of the ALPHA-stable part of the given
system. In this case, the resulting NR is set equal
to NSMIN.
= 2: with ORDSEL = 'F', the selected order NR is less
than the order of the ALPHA-unstable part of the
given system. In this case NR is set equal to the
order of the ALPHA-unstable part.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the computation of the ordered real Schur form of A
failed;
= 2: the separation of the ALPHA-stable/unstable diagonal
blocks failed because of very close eigenvalues;
= 3: the computation of Hankel singular values failed.
Method
Let be the following linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t) (1)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system. The subroutine AB09ND determines for
the given system (1), the matrices of a reduced order system
d[z(t)] = Ar*z(t) + Br*u(t)
yr(t) = Cr*z(t) + Dr*u(t) (2)
such that
HSV(NR+NS-N) <= INFNORM(G-Gr) <= 2*[HSV(NR+NS-N+1)+...+HSV(NS)],
where G and Gr are transfer-function matrices of the systems
(A,B,C,D) and (Ar,Br,Cr,Dr), respectively, and INFNORM(G) is the
infinity-norm of G.
The following procedure is used to reduce a given G:
1) Decompose additively G as
G = G1 + G2
such that G1 = (As,Bs,Cs,D) has only ALPHA-stable poles and
G2 = (Au,Bu,Cu,0) has only ALPHA-unstable poles.
2) Determine G1r, a reduced order approximation of the
ALPHA-stable part G1.
3) Assemble the reduced model Gr as
Gr = G1r + G2.
To reduce the ALPHA-stable part G1, if JOB = 'B', the square-root
balancing-based SPA method of [1] is used, and for an ALPHA-stable
system, the resulting reduced model is balanced.
If JOB = 'N', the balancing-free square-root SPA method of [2]
is used to reduce the ALPHA-stable part G1.
By setting TOL1 = TOL2, the routine can be used to compute
Balance & Truncate approximations as well.
References
[1] Liu Y. and Anderson B.D.O.
Singular Perturbation Approximation of Balanced Systems,
Int. J. Control, Vol. 50, pp. 1379-1405, 1989.
[2] Varga A.
Balancing-free square-root algorithm for computing
singular perturbation approximations.
Proc. 30-th IEEE CDC, Brighton, Dec. 11-13, 1991,
Vol. 2, pp. 1062-1065.
Numerical Aspects
The implemented methods rely on accuracy enhancing square-root or
balancing-free square-root techniques.
3
The algorithms require less than 30N floating point operations.
Further Comments
NoneExample
Program Text
* AB09ND EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDD
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = 2*NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX*( 2*NMAX +
$ MAX( NMAX, MMAX, PMAX ) + 5 ) +
$ ( NMAX*( NMAX + 1 ) )/2 )
* .. Local Scalars ..
DOUBLE PRECISION ALPHA, TOL1, TOL2
INTEGER I, INFO, IWARN, J, M, N, NR, NS, P
CHARACTER*1 DICO, EQUIL, JOB, ORDSEL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK), HSV(NMAX)
INTEGER IWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL AB09ND
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, NR, ALPHA, TOL1, TOL2,
$ DICO, JOB, EQUIL, ORDSEL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
* Find a reduced ssr for (A,B,C,D).
CALL AB09ND( DICO, JOB, EQUIL, ORDSEL, N, M, P, NR,
$ ALPHA, A, LDA, B, LDB, C, LDC, D, LDD,
$ NS, HSV, TOL1, TOL2, IWORK, DWORK, LDWORK,
$ IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NR
WRITE ( NOUT, FMT = 99987 )
WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1,NS )
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
20 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
60 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 70 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1,M )
70 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB09ND EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB09ND = ',I2)
99997 FORMAT (' The order of reduced model = ',I2)
99996 FORMAT (/' The reduced state dynamics matrix Ar is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The reduced input/state matrix Br is ')
99992 FORMAT (/' The reduced state/output matrix Cr is ')
99991 FORMAT (/' The reduced input/output matrix Dr is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99987 FORMAT (/' The Hankel singular values of ALPHA-stable part are')
END
Program Data
AB09ND EXAMPLE PROGRAM DATA (Continuous system) 7 2 3 0 -.6D0 1.D-1 1.E-14 C N N A -0.04165 0.0000 4.9200 -4.9200 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 0.0000 0.0000 0.5450 0.0000 0.0000 0.0000 -0.5450 0.0000 0.0000 0.0000 0.0000 0.0000 4.9200 -0.04165 0.0000 4.9200 0.0000 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 12.500 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000Program Results
AB09ND EXAMPLE PROGRAM RESULTS The order of reduced model = 5 The Hankel singular values of ALPHA-stable part are 1.9178 0.8621 0.7666 0.0336 0.0246 The reduced state dynamics matrix Ar is -0.5181 -1.1084 0.0000 0.0000 0.0000 8.8157 -0.5181 0.0000 0.0000 0.0000 0.0000 0.0000 0.5847 0.0000 1.9230 0.0000 0.0000 0.0000 -1.6606 0.0000 0.0000 0.0000 -4.3823 0.0000 -3.2922 The reduced input/state matrix Br is -1.2837 1.2837 -0.7522 0.7522 -0.6379 -0.6379 2.0656 -2.0656 -3.9315 -3.9315 The reduced state/output matrix Cr is -0.1380 -0.6445 -0.6416 -0.6293 0.2526 0.6246 0.0196 0.0000 0.4107 0.0000 0.1380 0.6445 -0.6416 0.6293 0.2526 The reduced input/output matrix Dr is 0.0582 -0.0090 0.0015 -0.0015 -0.0090 0.0582
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB13AD.html 0000664 0000000 0000000 00000026713 14560147231 0020316 0 ustar 00root root 0000000 0000000
Purpose
To compute the Hankel-norm of the ALPHA-stable projection of the transfer-function matrix G of the state-space system (A,B,C).Specification
DOUBLE PRECISION FUNCTION AB13AD( DICO, EQUIL, N, M, P, ALPHA, A,
$ LDA, B, LDB, C, LDC, NS, HSV,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, EQUIL
INTEGER INFO, LDA, LDB, LDC, LDWORK, M, N, NS, P
DOUBLE PRECISION ALPHA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), HSV(*)
Function Value
AB13AD DOUBLE PRECISION
The Hankel-norm of the ALPHA-stable projection of G
(if INFO = 0).
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily
equilibrate the triplet (A,B,C) as follows:
= 'S': perform equilibration (scaling);
= 'N': do not perform equilibration.
Input/Output Parameters
N (input) INTEGER
The order of the state-space representation, i.e. the
order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
ALPHA (input) DOUBLE PRECISION
Specifies the ALPHA-stability boundary for the eigenvalues
of the state dynamics matrix A. For a continuous-time
system (DICO = 'C'), ALPHA <= 0 is the boundary value for
the real parts of eigenvalues, while for a discrete-time
system (DICO = 'D'), 0 <= ALPHA <= 1 represents the
boundary value for the moduli of eigenvalues.
The ALPHA-stability domain does not include the boundary
(see the Note below).
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A.
On exit, if INFO = 0, the leading N-by-N part of this
array contains the state dynamics matrix A in a block
diagonal real Schur form with its eigenvalues reordered
and separated. The resulting A has two diagonal blocks.
The leading NS-by-NS part of A has eigenvalues in the
ALPHA-stability domain and the trailing (N-NS) x (N-NS)
part has eigenvalues outside the ALPHA-stability domain.
Note: The ALPHA-stability domain is defined either
as the open half complex plane left to ALPHA,
for a continous-time system (DICO = 'C'), or the
interior of the ALPHA-radius circle centered in the
origin, for a discrete-time system (DICO = 'D').
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, if INFO = 0, the leading N-by-M part of this
array contains the input/state matrix B of the transformed
system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, if INFO = 0, the leading P-by-N part of this
array contains the state/output matrix C of the
transformed system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
NS (output) INTEGER
The dimension of the ALPHA-stable subsystem.
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, the leading NS elements of HSV contain the
Hankel singular values of the ALPHA-stable part of the
original system ordered decreasingly.
HSV(1) is the Hankel norm of the ALPHA-stable subsystem.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,N*(MAX(N,M,P)+5)+N*(N+1)/2).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the computation of the ordered real Schur form of A
failed;
= 2: the separation of the ALPHA-stable/unstable diagonal
blocks failed because of very close eigenvalues;
= 3: the computed ALPHA-stable part is just stable,
having stable eigenvalues very near to the imaginary
axis (if DICO = 'C') or to the unit circle
(if DICO = 'D');
= 4: the computation of Hankel singular values failed.
Method
Let be the following linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) (1)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system, and let G be the corresponding
transfer-function matrix. The following procedure is used to
compute the Hankel-norm of the ALPHA-stable projection of G:
1) Decompose additively G as
G = G1 + G2
such that G1 = (As,Bs,Cs) has only ALPHA-stable poles and
G2 = (Au,Bu,Cu) has only ALPHA-unstable poles.
For the computation of the additive decomposition, the
algorithm presented in [1] is used.
2) Compute the Hankel-norm of ALPHA-stable projection G1 as the
the maximum Hankel singular value of the system (As,Bs,Cs).
The computation of the Hankel singular values is performed
by using the square-root method of [2].
References
[1] Safonov, M.G., Jonckheere, E.A., Verma, M. and Limebeer, D.J.
Synthesis of positive real multivariable feedback systems,
Int. J. Control, Vol. 45, pp. 817-842, 1987.
[2] Tombs, M.S. and Postlethwaite, I.
Truncated balanced realization of stable, non-minimal
state-space systems.
Int. J. Control, Vol. 46, pp. 1319-1330, 1987.
Numerical Aspects
The implemented method relies on a square-root technique.
3
The algorithms require about 17N floating point operations.
Further Comments
NoneExample
Program Text
* AB13AD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX*( MAX( NMAX, MMAX, PMAX ) + 5 )
$ + ( NMAX*( NMAX + 1 ) )/2 )
* .. Local Scalars ..
DOUBLE PRECISION ALPHA, SHNORM
INTEGER I, INFO, J, M, N, NS, P
CHARACTER*1 DICO, EQUIL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ DWORK(LDWORK), HSV(NMAX)
* .. External Functions ..
DOUBLE PRECISION AB13AD
EXTERNAL AB13AD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, ALPHA, DICO, EQUIL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Compute the Hankel-norm of the ALPHA-stable projection of
* (A,B,C).
SHNORM = AB13AD( DICO, EQUIL, N, M, P, ALPHA, A, LDA, B,
$ LDB, C, LDC, NS, HSV, DWORK, LDWORK,
$ INFO)
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) SHNORM
WRITE ( NOUT, FMT = 99987 )
WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1,NS )
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB13AD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB13AD = ',I2)
99997 FORMAT (' The Hankel-norm of the ALPHA-projection = ',1PD14.5)
99995 FORMAT (20(1X,F8.4))
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99987 FORMAT (/' The Hankel singular values of ALPHA-projection are')
END
Program Data
AB13AD EXAMPLE PROGRAM DATA (Continuous system) 7 2 3 0.0 C N -0.04165 0.0000 4.9200 -4.9200 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 0.0000 0.0000 0.5450 0.0000 0.0000 0.0000 -0.5450 0.0000 0.0000 0.0000 0.0000 0.0000 4.9200 -0.04165 0.0000 4.9200 0.0000 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 12.500 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000Program Results
AB13AD EXAMPLE PROGRAM RESULTS The Hankel-norm of the ALPHA-projection = 2.51388D+00 The Hankel singular values of ALPHA-projection are 2.5139 2.0846 1.9178 0.7666 0.5473 0.0253 0.0246
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB13AX.html 0000664 0000000 0000000 00000012001 14560147231 0020323 0 ustar 00root root 0000000 0000000
Purpose
To compute the Hankel-norm of the transfer-function matrix G of a stable state-space system (A,B,C). The state dynamics matrix A of the given system is an upper quasi-triangular matrix in real Schur form.Specification
DOUBLE PRECISION FUNCTION AB13AX( DICO, N, M, P, A, LDA, B, LDB,
$ C, LDC, HSV, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER DICO
INTEGER INFO, LDA, LDB, LDC, LDWORK, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), HSV(*)
Function Value
AB13AX DOUBLE PRECISION
The Hankel-norm of G (if INFO = 0).
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
Input/Output Parameters
N (input) INTEGER
The order of the state-space representation, i.e. the
order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
state dynamics matrix A in a real Schur canonical form.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
input/state matrix B.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array must contain the
state/output matrix C.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, this array contains the Hankel singular
values of the given system ordered decreasingly.
HSV(1) is the Hankel norm of the given system.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,N*(MAX(N,M,P)+5)+N*(N+1)/2).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the state matrix A is not stable (if DICO = 'C')
or not convergent (if DICO = 'D');
= 2: the computation of Hankel singular values failed.
Method
Let be the stable linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) (1)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system, and let G be the corresponding
transfer-function matrix. The Hankel-norm of G is computed as the
the maximum Hankel singular value of the system (A,B,C).
The computation of the Hankel singular values is performed
by using the square-root method of [1].
References
[1] Tombs M.S. and Postlethwaite I.
Truncated balanced realization of stable, non-minimal
state-space systems.
Int. J. Control, Vol. 46, pp. 1319-1330, 1987.
Numerical Aspects
The implemented method relies on a square-root technique.
3
The algorithms require about 17N floating point operations.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the H2 or L2 norm of the transfer-function matrix G of the system (A,B,C,D). G must not have poles on the imaginary axis, for a continuous-time system, or on the unit circle, for a discrete-time system. If the H2-norm is computed, the system must be stable.Specification
DOUBLE PRECISION FUNCTION AB13BD( DICO, JOBN, N, M, P, A, LDA,
$ B, LDB, C, LDC, D, LDD, NQ, TOL,
$ DWORK, LDWORK, IWARN, INFO)
C .. Scalar Arguments ..
CHARACTER DICO, JOBN
INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, LDWORK, M,
$ N, NQ, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), DWORK(*)
Function Value
AB13BD DOUBLE PRECISION
The H2-norm of G, if JOBN = 'H', or the L2-norm of G,
if JOBN = 'L' (if INFO = 0).
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOBN CHARACTER*1
Specifies the norm to be computed as follows:
= 'H': the H2-norm;
= 'L': the L2-norm.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A, the number of rows of the
matrix B, and the number of columns of the matrix C.
N represents the dimension of the state vector. N >= 0.
M (input) INTEGER
The number of columns of the matrices B and D.
M represents the dimension of input vector. M >= 0.
P (input) INTEGER
The number of rows of the matrices C and D.
P represents the dimension of output vector. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix of the system.
On exit, the leading NQ-by-NQ part of this array contains
the state dynamics matrix (in a real Schur form) of the
numerator factor Q of the right coprime factorization with
inner denominator of G (see METHOD).
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input/state matrix of the system.
On exit, the leading NQ-by-M part of this array contains
the input/state matrix of the numerator factor Q of the
right coprime factorization with inner denominator of G
(see METHOD).
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix of the system.
On exit, the leading P-by-NQ part of this array contains
the state/output matrix of the numerator factor Q of the
right coprime factorization with inner denominator of G
(see METHOD).
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the input/output matrix of the system.
If DICO = 'C', D must be a null matrix.
On exit, the leading P-by-M part of this array contains
the input/output matrix of the numerator factor Q of
the right coprime factorization with inner denominator
of G (see METHOD).
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
NQ (output) INTEGER
The order of the resulting numerator Q of the right
coprime factorization with inner denominator of G (see
METHOD).
Generally, NQ = N - NS, where NS is the number of
uncontrollable unstable eigenvalues.
Tolerances
TOL DOUBLE PRECISION
The absolute tolerance level below which the elements of
B are considered zero (used for controllability tests).
If the user sets TOL <= 0, then an implicitly computed,
default tolerance, defined by TOLDEF = N*EPS*NORM(B),
is used instead, where EPS is the machine precision
(see LAPACK Library routine DLAMCH) and NORM(B) denotes
the 1-norm of B.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The dimension of working array DWORK.
LDWORK >= MAX( 1, M*(N+M) + MAX( N*(N+5), M*(M+2), 4*P ),
N*( MAX( N, P ) + 4 ) + MIN( N, P ) ).
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= K: K violations of the numerical stability condition
occured during the assignment of eigenvalues in
computing the right coprime factorization with inner
denominator of G (see the SLICOT subroutine SB08DD).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction of A to a real Schur form failed;
= 2: a failure was detected during the reordering of the
real Schur form of A, or in the iterative process
for reordering the eigenvalues of Z'*(A + B*F)*Z
along the diagonal (see SLICOT routine SB08DD);
= 3: if DICO = 'C' and the matrix A has a controllable
eigenvalue on the imaginary axis, or DICO = 'D'
and A has a controllable eigenvalue on the unit
circle;
= 4: the solution of Lyapunov equation failed because
the equation is singular;
= 5: if DICO = 'C' and D is a nonzero matrix;
= 6: if JOBN = 'H' and the system is unstable.
Method
The subroutine is based on the algorithms proposed in [1] and [2].
If the given transfer-function matrix G is unstable, then a right
coprime factorization with inner denominator of G is first
computed
-1
G = Q*R ,
where Q and R are stable transfer-function matrices and R is
inner. If G is stable, then Q = G and R = I.
Let (AQ,BQ,CQ,DQ) be the state-space representation of Q.
If DICO = 'C', then the L2-norm of G is computed as
NORM2(G) = NORM2(Q) = SQRT(TRACE(BQ'*X*BQ)),
where X satisfies the continuous-time Lyapunov equation
AQ'*X + X*AQ + CQ'*CQ = 0.
If DICO = 'D', then the l2-norm of G is computed as
NORM2(G) = NORM2(Q) = SQRT(TRACE(BQ'*X*BQ+DQ'*DQ)),
where X satisfies the discrete-time Lyapunov equation
AQ'*X*AQ - X + CQ'*CQ = 0.
References
[1] Varga A.
On computing 2-norms of transfer-function matrices.
Proc. 1992 ACC, Chicago, June 1992.
[2] Varga A.
A Schur method for computing coprime factorizations with
inner denominators and applications in model reduction.
Proc. ACC'93, San Francisco, CA, pp. 2130-2131, 1993.
Numerical Aspects
3 The algorithm requires no more than 14N floating point operations.Further Comments
NoneExample
Program Text
* AB13BD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDD
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( MMAX*( NMAX + MMAX ) +
$ MAX( NMAX*( NMAX + 5 ),
$ MMAX*( MMAX + 2 ), 4*PMAX ),
$ NMAX*( MAX( NMAX, PMAX ) + 4 ) +
$ MIN( NMAX, PMAX ) ) )
* .. Local Scalars ..
DOUBLE PRECISION S2NORM, TOL
INTEGER I, INFO, IWARN, J, M, N, NQ, P
CHARACTER*1 DICO, JOBN
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK)
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION AB13BD
EXTERNAL AB13BD, LSAME
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL, DICO, JOBN
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1, N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1, M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1, N ), I = 1, P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1, M ), I = 1, P )
* Compute the H2 or L2 norm of (A,B,C,D).
S2NORM = AB13BD( DICO, JOBN, N, M, P, A, LDA, B, LDB,
* C, LDC, D, LDD, NQ, TOL, DWORK, LDWORK,
* IWARN, INFO)
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF( LSAME( JOBN, 'H' ) ) THEN
WRITE ( NOUT, FMT = 99997 ) S2NORM
ELSE
WRITE ( NOUT, FMT = 99996 ) S2NORM
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB13BD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB13BD = ',I2)
99997 FORMAT (' The H2-norm of the system = ',1PD14.5)
99996 FORMAT (' The L2-norm of the system = ',1PD14.5)
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
AB13BD EXAMPLE PROGRAM DATA (Continuous system) 7 2 3 1.E-10 C L -0.04165 0.0000 4.9200 0.4920 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 0.0000 0.0000 0.5450 0.0000 0.0000 0.0000 0.0545 0.0000 0.0000 0.0000 0.0000 0.0000 -0.49200 0.004165 0.0000 4.9200 0.0000 0.0000 0.0000 0.0000 0.5210 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 12.500 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000Program Results
AB13BD EXAMPLE PROGRAM RESULTS The L2-norm of the system = 7.93948D+00
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB13CD.html 0000664 0000000 0000000 00000022025 14560147231 0020310 0 ustar 00root root 0000000 0000000
Purpose
To compute the H-infinity norm of the continuous-time stable
system
| A | B |
G(s) = |---|---| .
| C | D |
Specification
DOUBLE PRECISION FUNCTION AB13CD( N, M, NP, A, LDA, B, LDB, C,
$ LDC, D, LDD, TOL, IWORK, DWORK,
$ LDWORK, CWORK, LCWORK, BWORK,
$ INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LCWORK, LDD, LDWORK, M, N,
$ NP
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK( * )
COMPLEX*16 CWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), DWORK( * )
LOGICAL BWORK( * )
Function Value
AB13CD DOUBLE PRECISION
If INFO = 0, the H-infinity norm of the system, HNORM,
i.e., the peak gain of the frequency response (as measured
by the largest singular value in the MIMO case).
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
NP (input) INTEGER
The row size of the matrix C. NP >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
system input matrix B.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading NP-by-N part of this array must contain the
system output matrix C.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,NP).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading NP-by-M part of this array must contain the
system input/output matrix D.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1,NP).
Tolerances
TOL DOUBLE PRECISION
Tolerance used to set the accuracy in determining the
norm.
Workspace
IWORK INTEGER array, dimension (N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal value
of LDWORK, and DWORK(2) contains the frequency where the
gain of the frequency response achieves its peak value
HNORM.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= max(2,4*N*N+2*M*M+3*M*N+M*NP+2*(N+NP)*NP+10*N+
6*max(M,NP)).
For good performance, LDWORK must generally be larger.
CWORK COMPLEX*16 array, dimension (LCWORK)
On exit, if INFO = 0, CWORK(1) contains the optimal value
of LCWORK.
LCWORK INTEGER
The dimension of the array CWORK.
LCWORK >= max(1,(N+M)*(N+NP)+3*max(M,NP)).
For good performance, LCWORK must generally be larger.
BWORK LOGICAL array, dimension (2*N)
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the system is unstable;
= 2: the tolerance is too small (the algorithm for
computing the H-infinity norm did not converge);
= 3: errors in computing the eigenvalues of A or of the
Hamiltonian matrix (the QR algorithm did not
converge);
= 4: errors in computing singular values.
Method
The routine implements the method presented in [1].References
[1] Bruinsma, N.A. and Steinbuch, M.
A fast algorithm to compute the Hinfinity-norm of a transfer
function matrix.
Systems & Control Letters, vol. 14, pp. 287-293, 1990.
Numerical Aspects
If the algorithm does not converge (INFO = 2), the tolerance must be increased.Further Comments
NoneExample
Program Text
* AB13CD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 10, MMAX = 10, PMAX = 10 )
INTEGER LDA, LDB, LDC, LDD
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX )
INTEGER LCWORK
PARAMETER ( LCWORK = ( NMAX + MMAX )*( NMAX + PMAX ) +
$ 3*MAX( MMAX, PMAX ) )
INTEGER LDWORK
PARAMETER ( LDWORK = 4*NMAX*NMAX + 2*MMAX*MMAX +
$ 2*PMAX*PMAX + 3*NMAX*MMAX +
$ 2*NMAX*PMAX + MMAX*PMAX + 10*NMAX +
$ 6*MAX( MMAX, PMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION FPEAK, HNORM, TOL
INTEGER I, INFO, J, M, N, NP
* .. Local Arrays ..
LOGICAL BWORK(2*NMAX)
INTEGER IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK)
COMPLEX*16 CWORK( LCWORK )
* .. External Functions ..
DOUBLE PRECISION AB13CD
EXTERNAL AB13CD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, NP
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE IF ( NP.LT.0 .OR. NP.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) NP
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,NP )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,NP )
READ ( NIN, FMT = * ) TOL
* Computing the Hinf norm
HNORM = AB13CD( N, M, NP, A, LDA, B, LDB, C, LDC, D, LDD, TOL,
$ IWORK, DWORK, LDWORK, CWORK, LCWORK, BWORK,
$ INFO )
*
IF ( INFO.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99997 )
WRITE ( NOUT, FMT = 99991 ) HNORM
FPEAK = DWORK(2)
WRITE ( NOUT, FMT = 99996 )
WRITE ( NOUT, FMT = 99991 ) FPEAK
ELSE
WRITE( NOUT, FMT = 99998 ) INFO
END IF
END IF
STOP
*
99999 FORMAT (' AB13CD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (/' INFO on exit from AB13CD =',I2)
99997 FORMAT (/' The H_infty norm of the system is'/)
99996 FORMAT (/' The peak frequency is'/)
99992 FORMAT (10(1X,F8.4))
99991 FORMAT (D17.10)
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' NP is out of range.',/' NP = ',I5)
END
Program Data
AB13CD EXAMPLE PROGRAM DATA 6 1 1 0.0 1.0 0.0 0.0 0.0 0.0 -0.5 -0.0002 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 -1.0 -0.00002 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 -2.0 -0.000002 1.0 0.0 1.0 0.0 1.0 0.0 1.0 0.0 1.0 0.0 1.0 0.0 0.0 0.000000001Program Results
AB13CD EXAMPLE PROGRAM RESULTS The H_infty norm of the system is 0.5000000006D+06 The peak frequency is 0.1414213562D+01
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB13DD.html 0000664 0000000 0000000 00000036671 14560147231 0020325 0 ustar 00root root 0000000 0000000
Purpose
To compute the L-infinity norm of a continuous-time or
discrete-time system, either standard or in the descriptor form,
-1
G(lambda) = C*( lambda*E - A ) *B + D .
The norm is finite if and only if the matrix pair (A,E) has no
eigenvalue on the boundary of the stability domain, i.e., the
imaginary axis, or the unit circle, respectively. It is assumed
that the matrix E is nonsingular.
Specification
SUBROUTINE AB13DD( DICO, JOBE, EQUIL, JOBD, N, M, P, FPEAK,
$ A, LDA, E, LDE, B, LDB, C, LDC, D, LDD, GPEAK,
$ TOL, IWORK, DWORK, LDWORK, CWORK, LCWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER DICO, EQUIL, JOBD, JOBE
INTEGER INFO, LCWORK, LDA, LDB, LDC, LDD, LDE, LDWORK,
$ M, N, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
COMPLEX*16 CWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), DWORK( * ), E( LDE, * ),
$ FPEAK( 2 ), GPEAK( 2 )
INTEGER IWORK( * )
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the system, as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOBE CHARACTER*1
Specifies whether E is a general square or an identity
matrix, as follows:
= 'G': E is a general square matrix;
= 'I': E is the identity matrix.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily
equilibrate the system (A,E,B,C) or (A,B,C), as follows:
= 'S': perform equilibration (scaling);
= 'N': do not perform equilibration.
JOBD CHARACTER*1
Specifies whether or not a non-zero matrix D appears in
the given state space model:
= 'D': D is present;
= 'Z': D is assumed a zero matrix.
Input/Output Parameters
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
P (input) INTEGER
The row size of the matrix C. P >= 0.
FPEAK (input/output) DOUBLE PRECISION array, dimension (2)
On entry, this parameter must contain an estimate of the
frequency where the gain of the frequency response would
achieve its peak value. Setting FPEAK(2) = 0 indicates an
infinite frequency. An accurate estimate could reduce the
number of iterations of the iterative algorithm. If no
estimate is available, set FPEAK(1) = 0, and FPEAK(2) = 1.
FPEAK(1) >= 0, FPEAK(2) >= 0.
On exit, if INFO = 0, this array contains the frequency
OMEGA, where the gain of the frequency response achieves
its peak value GPEAK, i.e.,
|| G ( j*OMEGA ) || = GPEAK , if DICO = 'C', or
j*OMEGA
|| G ( e ) || = GPEAK , if DICO = 'D',
where OMEGA = FPEAK(1), if FPEAK(2) > 0, and OMEGA is
infinite, if FPEAK(2) = 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
state dynamics matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
E (input) DOUBLE PRECISION array, dimension (LDE,N)
If JOBE = 'G', the leading N-by-N part of this array must
contain the descriptor matrix E of the system.
If JOBE = 'I', then E is assumed to be the identity
matrix and is not referenced.
LDE INTEGER
The leading dimension of the array E.
LDE >= MAX(1,N), if JOBE = 'G';
LDE >= 1, if JOBE = 'I'.
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
system input matrix B.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array must contain the
system output matrix C.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,P).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
If JOBD = 'D', the leading P-by-M part of this array must
contain the direct transmission matrix D.
The array D is not referenced if JOBD = 'Z'.
LDD INTEGER
The leading dimension of array D.
LDD >= MAX(1,P), if JOBD = 'D';
LDD >= 1, if JOBD = 'Z'.
GPEAK (output) DOUBLE PRECISION array, dimension (2)
The L-infinity norm of the system, i.e., the peak gain
of the frequency response (as measured by the largest
singular value in the MIMO case), coded in the same way
as FPEAK.
Tolerances
TOL DOUBLE PRECISION
Tolerance used to set the accuracy in determining the
norm. 0 <= TOL < 1.
Workspace
IWORK INTEGER array, dimension (N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= K, where K can be computed using the following
pseudo-code (or the Fortran code included in the routine)
d = 6*MIN(P,M);
c = MAX( 4*MIN(P,M) + MAX(P,M), d );
if ( MIN(P,M) = 0 ) then
K = 1;
else if( N = 0 or B = 0 or C = 0 ) then
if( JOBD = 'D' ) then
K = P*M + c;
else
K = 1;
end
else
if ( DICO = 'D' ) then
b = 0; e = d;
else
b = N*(N+M); e = c;
if ( JOBD = Z' ) then b = b + P*M; end
end
if ( JOBD = 'D' ) then
r = P*M;
if ( JOBE = 'I', DICO = 'C',
N > 0, B <> 0, C <> 0 ) then
K = P*P + M*M;
r = r + N*(P+M);
else
K = 0;
end
K = K + r + c; r = r + MIN(P,M);
else
r = 0; K = 0;
end
r = r + N*(N+P+M);
if ( JOBE = 'G' ) then
r = r + N*N;
if ( EQUIL = 'S' ) then
K = MAX( K, r + 9*N );
end
K = MAX( K, r + 4*N + MAX( M, 2*N*N, N+b+e ) );
else
K = MAX( K, r + N +
MAX( M, P, N*N+2*N, 3*N+b+e ) );
end
w = 0;
if ( JOBE = 'I', DICO = 'C' ) then
w = r + 4*N*N + 11*N;
if ( JOBD = 'D' ) then
w = w + MAX(M,P) + N*(P+M);
end
end
if ( JOBE = 'E' or DICO = 'D' or JOBD = 'D' ) then
w = MAX( w, r + 6*N + (2*N+P+M)*(2*N+P+M) +
MAX( 2*(N+P+M), 8*N*N + 16*N ) );
end
K = MAX( 1, K, w, r + 2*N + e );
end
For good performance, LDWORK must generally be larger.
An easily computable upper bound is
K = MAX( 1, 15*N*N + P*P + M*M + (6*N+3)*(P+M) + 4*P*M +
N*M + 22*N + 7*MIN(P,M) ).
The smallest workspace is obtained for DICO = 'C',
JOBE = 'I', and JOBD = 'Z', namely
K = MAX( 1, N*N + N*P + N*M + N +
MAX( N*N + N*M + P*M + 3*N + c,
4*N*N + 10*N ) ).
for which an upper bound is
K = MAX( 1, 6*N*N + N*P + 2*N*M + P*M + 11*N + MAX(P,M) +
6*MIN(P,M) ).
CWORK COMPLEX*16 array, dimension (LCWORK)
On exit, if INFO = 0, CWORK(1) contains the optimal
LCWORK.
LCWORK INTEGER
The dimension of the array CWORK.
LCWORK >= 1, if N = 0, or B = 0, or C = 0;
LCWORK >= MAX(1, (N+M)*(N+P) + 2*MIN(P,M) + MAX(P,M)),
otherwise.
For good performance, LCWORK must generally be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the matrix E is (numerically) singular;
= 2: the (periodic) QR (or QZ) algorithm for computing
eigenvalues did not converge;
= 3: the SVD algorithm for computing singular values did
not converge;
= 4: the tolerance is too small and the algorithm did
not converge.
Method
The routine implements the method presented in [1], with extensions and refinements for improving numerical robustness and efficiency. Structure-exploiting eigenvalue computations for Hamiltonian matrices are used if JOBE = 'I', DICO = 'C', and the symmetric matrices to be implicitly inverted are not too ill- conditioned. Otherwise, generalized eigenvalue computations are used in the iterative algorithm of [1].References
[1] Bruinsma, N.A. and Steinbuch, M.
A fast algorithm to compute the Hinfinity-norm of a transfer
function matrix.
Systems & Control Letters, vol. 14, pp. 287-293, 1990.
Numerical Aspects
If the algorithm does not converge in MAXIT = 30 iterations (INFO = 4), the tolerance must be increased.Further Comments
If the matrix E is singular, other SLICOT Library routines could be used before calling AB13DD, for removing the singular part of the system.Example
Program Text
* AB13DD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 10, MMAX = 10, PMAX = 10 )
INTEGER LDA, LDB, LDC, LDD, LDE
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX, LDE = NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX )
INTEGER LCWORK
PARAMETER ( LCWORK = ( NMAX + MMAX )*( NMAX + PMAX ) +
$ 2*MIN( PMAX, MMAX ) +
$ MAX( PMAX, MMAX ) )
INTEGER LDWORK
PARAMETER ( LDWORK = 15*NMAX*NMAX + PMAX*PMAX + MMAX*MMAX +
$ ( 6*NMAX + 3 )*( PMAX + MMAX ) +
$ 4*PMAX*MMAX + NMAX*MMAX + 22*NMAX +
$ 7*MIN( PMAX, MMAX ) )
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, J, M, N, P
CHARACTER DICO, EQUIL, JOBD, JOBE
* .. Local Arrays ..
INTEGER IWORK( LIWORK )
DOUBLE PRECISION A( LDA, NMAX ), B( LDB, MMAX ), C( LDC, NMAX ),
$ D( LDD, MMAX ), DWORK( LDWORK ), E( LDE, NMAX ),
$ FPEAK( 2 ), GPEAK( 2 )
COMPLEX*16 CWORK( LCWORK )
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL AB13DD
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, FPEAK, TOL, DICO, JOBE, EQUIL, JOBD
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) M
ELSE IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) P
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( JOBE, 'G' ) )
$ READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
IF ( LSAME( JOBD, 'D' ) )
$ READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
* Computing the Linf norm.
CALL AB13DD( DICO, JOBE, EQUIL, JOBD, N, M, P, FPEAK, A, LDA,
$ E, LDE, B, LDB, C, LDC, D, LDD, GPEAK, TOL, IWORK,
$ DWORK, LDWORK, CWORK, LCWORK, INFO )
*
IF ( INFO.EQ.0 ) THEN
IF ( GPEAK( 2 ).EQ.ZERO ) THEN
WRITE ( NOUT, FMT = 99991 )
ELSE
WRITE ( NOUT, FMT = 99997 )
WRITE ( NOUT, FMT = 99995 ) GPEAK( 1 )
END IF
IF ( FPEAK( 2 ).EQ.ZERO ) THEN
WRITE ( NOUT, FMT = 99990 )
ELSE
WRITE ( NOUT, FMT = 99996 )
WRITE ( NOUT, FMT = 99995 ) FPEAK( 1 )
END IF
ELSE
WRITE( NOUT, FMT = 99998 ) INFO
END IF
END IF
STOP
*
99999 FORMAT (' AB13DD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (/' INFO on exit from AB13DD =',I2)
99997 FORMAT (/' The L_infty norm of the system is'/)
99996 FORMAT (/' The peak frequency is'/)
99995 FORMAT (D17.10)
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' M is out of range.',/' M = ',I5)
99992 FORMAT (/' P is out of range.',/' P = ',I5)
99991 FORMAT (/' The L_infty norm of the system is infinite')
99990 FORMAT (/' The peak frequency is infinite'/)
END
Program Data
AB13CD EXAMPLE PROGRAM DATA 6 1 1 0.0 1.0 0.000000001 C I N D 0.0 1.0 0.0 0.0 0.0 0.0 -0.5 -0.0002 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 -1.0 -0.00002 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 -2.0 -0.000002 1.0 0.0 1.0 0.0 1.0 0.0 1.0 0.0 1.0 0.0 1.0 0.0 0.0Program Results
AB13DD EXAMPLE PROGRAM RESULTS The L_infty norm of the system is 0.5000000001D+06 The peak frequency is 0.1414213562D+01
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB13DX.html 0000664 0000000 0000000 00000022210 14560147231 0020331 0 ustar 00root root 0000000 0000000
Purpose
To compute the maximum singular value of a given continuous-time
or discrete-time transfer-function matrix, either standard or in
the descriptor form,
-1
G(lambda) = C*( lambda*E - A ) *B + D ,
for a given complex value lambda, where lambda = j*omega, in the
continuous-time case, and lambda = exp(j*omega), in the
discrete-time case. The matrices A, E, B, C, and D are real
matrices of appropriate dimensions. Matrix A must be in an upper
Hessenberg form, and if JOBE ='G', the matrix E must be upper
triangular. The matrices B and C must correspond to the system
in (generalized) Hessenberg form.
Specification
DOUBLE PRECISION FUNCTION AB13DX( DICO, JOBE, JOBD, N, M, P,
$ OMEGA, A, LDA, E, LDE, B, LDB,
$ C, LDC, D, LDD, IWORK, DWORK,
$ LDWORK, ZWORK, LZWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, JOBD, JOBE
INTEGER INFO, LDA, LDB, LDC, LDD, LDE, LDWORK, LZWORK,
$ M, N, P
DOUBLE PRECISION OMEGA
C .. Array Arguments ..
COMPLEX*16 ZWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), DWORK( * ), E( LDE, * )
INTEGER IWORK( * )
Function Value
AB13DX DOUBLE PRECISION
The maximum singular value of G(lambda).
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the system, as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOBE CHARACTER*1
Specifies whether E is an upper triangular or an identity
matrix, as follows:
= 'G': E is a general upper triangular matrix;
= 'I': E is the identity matrix.
JOBD CHARACTER*1
Specifies whether or not a non-zero matrix D appears in
the given state space model:
= 'D': D is present;
= 'Z': D is assumed a zero matrix.
Input/Output Parameters
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
P (input) INTEGER
The row size of the matrix C. P >= 0.
OMEGA (input) DOUBLE PRECISION
The frequency value for which the calculations should be
done.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N upper Hessenberg part of this
array must contain the state dynamics matrix A in upper
Hessenberg form. The elements below the subdiagonal are
not referenced.
On exit, if M > 0, P > 0, OMEGA = 0, DICO = 'C', B <> 0,
and C <> 0, the leading N-by-N upper Hessenberg part of
this array contains the factors L and U from the LU
factorization of A (A = P*L*U); the unit diagonal elements
of L are not stored, L is lower bidiagonal, and P is
stored in IWORK (see SLICOT Library routine MB02SD).
Otherwise, this array is unchanged on exit.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
E (input) DOUBLE PRECISION array, dimension (LDE,N)
If JOBE = 'G', the leading N-by-N upper triangular part of
this array must contain the upper triangular descriptor
matrix E of the system. The elements of the strict lower
triangular part of this array are not referenced.
If JOBE = 'I', then E is assumed to be the identity
matrix and is not referenced.
LDE INTEGER
The leading dimension of the array E.
LDE >= MAX(1,N), if JOBE = 'G';
LDE >= 1, if JOBE = 'I'.
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the system input matrix B.
On exit, if M > 0, P > 0, OMEGA = 0, DICO = 'C', B <> 0,
C <> 0, and INFO = 0 or N+1, the leading N-by-M part of
this array contains the solution of the system A*X = B.
Otherwise, this array is unchanged on exit.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array must contain the
system output matrix C.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, if JOBD = 'D', the leading P-by-M part of this
array must contain the direct transmission matrix D.
On exit, if (N = 0, or B = 0, or C = 0) and JOBD = 'D',
or (OMEGA = 0, DICO = 'C', JOBD = 'D', and INFO = 0 or
N+1), the contents of this array is destroyed.
Otherwise, this array is unchanged on exit.
This array is not referenced if JOBD = 'Z'.
LDD INTEGER
The leading dimension of array D.
LDD >= MAX(1,P), if JOBD = 'D';
LDD >= 1, if JOBD = 'Z'.
Workspace
IWORK INTEGER array, dimension (LIWORK), where
LIWORK = N, if N > 0, M > 0, P > 0, B <> 0, and C <> 0;
LIWORK = 0, otherwise.
This array contains the pivot indices in the LU
factorization of the matrix lambda*E - A; for 1 <= i <= N,
row i of the matrix was interchanged with row IWORK(i).
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal value
of LDWORK, and DWORK(2), ..., DWORK(MIN(P,M)) contain the
singular values of G(lambda), except for the first one,
which is returned in the function value AB13DX.
If (N = 0, or B = 0, or C = 0) and JOBD = 'Z', the last
MIN(P,M)-1 zero singular values of G(lambda) are not
stored in DWORK(2), ..., DWORK(MIN(P,M)).
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= MAX(1, LDW1 + LDW2 ),
LDW1 = P*M, if N > 0, B <> 0, C <> 0, OMEGA = 0,
DICO = 'C', and JOBD = 'Z';
LDW1 = 0, otherwise;
LDW2 = MIN(P,M) + MAX(3*MIN(P,M) + MAX(P,M), 5*MIN(P,M)),
if (N = 0, or B = 0, or C = 0) and JOBD = 'D',
or (N > 0, B <> 0, C <> 0, OMEGA = 0, and
DICO = 'C');
LDW2 = 0, if (N = 0, or B = 0, or C = 0) and JOBD = 'Z',
or MIN(P,M) = 0;
LDW2 = 6*MIN(P,M), otherwise.
For good performance, LDWORK must generally be larger.
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) contains the optimal
LZWORK.
LZWORK INTEGER
The dimension of the array ZWORK.
LZWORK >= 1, if N = 0, or B = 0, or C = 0, or (OMEGA = 0
and DICO = 'C') or MIN(P,M) = 0;
LZWORK >= MAX(1, (N+M)*(N+P) + 2*MIN(P,M) + MAX(P,M)),
otherwise.
For good performance, LZWORK must generally be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, U(i,i) is exactly zero; the LU
factorization of the matrix lambda*E - A has been
completed, but the factor U is exactly singular,
i.e., the matrix lambda*E - A is exactly singular;
= N+1: the SVD algorithm for computing singular values
did not converge.
Method
The routine implements standard linear algebra calculations, taking problem structure into account. LAPACK Library routines DGESVD and ZGESVD are used for finding the singular values.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To estimate beta(A), the 2-norm distance from a real matrix A to
the nearest complex matrix with an eigenvalue on the imaginary
axis. The estimate is given as
LOW <= beta(A) <= HIGH,
where either
(1 + TOL) * LOW >= HIGH,
or
LOW = 0 and HIGH = delta,
and delta is a small number approximately equal to the square root
of machine precision times the Frobenius norm (Euclidean norm)
of A. If A is stable in the sense that all eigenvalues of A lie
in the open left half complex plane, then beta(A) is the distance
to the nearest unstable complex matrix, i.e., the complex
stability radius.
Specification
SUBROUTINE AB13ED( N, A, LDA, LOW, HIGH, TOL, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
DOUBLE PRECISION HIGH, LOW, TOL
INTEGER INFO, LDA, LDWORK, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
matrix A.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
LOW (output) DOUBLE PRECISION
A lower bound for beta(A).
HIGH (output) DOUBLE PRECISION
An upper bound for beta(A).
Tolerances
TOL DOUBLE PRECISION
Specifies the accuracy with which LOW and HIGH approximate
beta(A). If the user sets TOL to be less than SQRT(EPS),
where EPS is the machine precision (see LAPACK Library
Routine DLAMCH), then the tolerance is taken to be
SQRT(EPS).
The recommended value is TOL = 9, which gives an estimate
of beta(A) correct to within an order of magnitude.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( 1, 3*N*(N+1) ).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the QR algorithm (LAPACK Library routine DHSEQR)
fails to converge; this error is very rare.
Method
Let beta(A) be the 2-norm distance from a real matrix A to the
nearest complex matrix with an eigenvalue on the imaginary axis.
It is known that beta(A) = minimum of the smallest singular
value of (A - jwI), where I is the identity matrix and j**2 = -1,
and the minimum is taken over all real w.
The algorithm computes a lower bound LOW and an upper bound HIGH
for beta(A) by a bisection method in the following way. Given a
non-negative real number sigma, the Hamiltonian matrix H(sigma)
is constructed:
| A -sigma*I | | A G |
H(sigma) = | | := | | .
| sigma*I -A' | | F -A' |
It can be shown [1] that H(sigma) has an eigenvalue whose real
part is zero if and only if sigma >= beta. Any lower and upper
bounds on beta(A) can be improved by choosing a number between
them and checking to see if H(sigma) has an eigenvalue with zero
real part. This decision is made by computing the eigenvalues of
H(sigma) using the square reduced algorithm of Van Loan [2].
References
[1] Byers, R.
A bisection method for measuring the distance of a stable
matrix to the unstable matrices.
SIAM J. Sci. Stat. Comput., Vol. 9, No. 5, pp. 875-880, 1988.
[2] Van Loan, C.F.
A symplectic method for approximating all the eigenvalues of a
Hamiltonian matrix.
Linear Algebra and its Applications, Vol 61, 233-251, 1984.
Numerical Aspects
Due to rounding errors the computed values of LOW and HIGH can be
proven to satisfy
LOW - p(n) * sqrt(e) * norm(A) <= beta(A)
and
beta(A) <= HIGH + p(n) * sqrt(e) * norm(A),
where p(n) is a modest polynomial of degree 3, e is the machine
precision and norm(A) is the Frobenius norm of A, see [1].
The recommended value for TOL is 9 which gives an estimate of
beta(A) correct to within an order of magnitude.
AB13ED requires approximately 38*N**3 flops for TOL = 9.
Further Comments
NoneExample
Program Text
* AB13ED EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA
PARAMETER ( LDA = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 3*NMAX*( NMAX + 1 ) )
* .. Local Scalars ..
INTEGER I, INFO, J, N
DOUBLE PRECISION HIGH, LOW, TOL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK)
* .. External Subroutines ..
EXTERNAL AB13ED, UD01MD
* ..
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
* Read N, TOL and next A (row wise).
READ ( NIN, FMT = * ) N, TOL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) N
ELSE
DO 10 I = 1, N
READ ( NIN, FMT = * ) ( A(I,J), J = 1, N )
10 CONTINUE
*
WRITE ( NOUT, FMT = 99998 ) N, TOL
CALL UD01MD( N, N, 5, NOUT, A, LDA, 'Matrix A', INFO )
*
CALL AB13ED( N, A, LDA, LOW, HIGH, TOL, DWORK, LDWORK, INFO )
IF ( INFO.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) LOW, HIGH
ELSE
WRITE ( NOUT, FMT = 99996 ) INFO
END IF
END IF
STOP
*
99999 FORMAT (' AB13ED EXAMPLE PROGRAM RESULTS', /1X)
99998 FORMAT (' N =', I4, 2X, 'TOL =', D10.3)
99997 FORMAT (' LOW =', D18.11, /' HIGH =', D18.11)
99996 FORMAT (' INFO on exit from AB13ED = ', I2)
99995 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
AB13ED EXAMPLE PROGRAM DATA 5, 9.0D0 1.0D-01 1.0D-00 0.0D-00 0.0D-00 0.0D-00 0.0D-00 1.0D-01 1.0D-00 0.0D-00 0.0D-00 0.0D-00 0.0D-00 1.0D-01 1.0D-00 0.0D-00 0.0D-00 0.0D-00 0.0D-00 1.0D-01 1.0D-00 0.0D-00 0.0D-00 0.0D-00 0.0D-00 1.0D-01Program Results
AB13ED EXAMPLE PROGRAM RESULTS
N = 5 TOL = 0.900D+01
Matrix A ( 5X 5)
1 2 3 4 5
1 0.1000000D+00 0.1000000D+01 0.0000000D+00 0.0000000D+00 0.0000000D+00
2 0.0000000D+00 0.1000000D+00 0.1000000D+01 0.0000000D+00 0.0000000D+00
3 0.0000000D+00 0.0000000D+00 0.1000000D+00 0.1000000D+01 0.0000000D+00
4 0.0000000D+00 0.0000000D+00 0.0000000D+00 0.1000000D+00 0.1000000D+01
5 0.0000000D+00 0.0000000D+00 0.0000000D+00 0.0000000D+00 0.1000000D+00
LOW = 0.20929379255D-05
HIGH = 0.20793050504D-04
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB13FD.html 0000664 0000000 0000000 00000020607 14560147231 0020317 0 ustar 00root root 0000000 0000000
Purpose
To compute beta(A), the 2-norm distance from a real matrix A to the nearest complex matrix with an eigenvalue on the imaginary axis. If A is stable in the sense that all eigenvalues of A lie in the open left half complex plane, then beta(A) is the complex stability radius, i.e., the distance to the nearest unstable complex matrix. The value of beta(A) is the minimum of the smallest singular value of (A - jwI), taken over all real w. The value of w corresponding to the minimum is also computed.Specification
SUBROUTINE AB13FD( N, A, LDA, BETA, OMEGA, TOL, DWORK, LDWORK,
$ CWORK, LCWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LCWORK, LDA, LDWORK, N
DOUBLE PRECISION BETA, OMEGA, TOL
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*)
COMPLEX*16 CWORK(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
matrix A.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
BETA (output) DOUBLE PRECISION
The computed value of beta(A), which actually is an upper
bound.
OMEGA (output) DOUBLE PRECISION
The value of w such that the smallest singular value of
(A - jwI) equals beta(A).
Tolerances
TOL DOUBLE PRECISION
Specifies the accuracy with which beta(A) is to be
calculated. (See the Numerical Aspects section below.)
If the user sets TOL to be less than EPS, where EPS is the
machine precision (see LAPACK Library Routine DLAMCH),
then the tolerance is taken to be EPS.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
If DWORK(1) is not needed, the first 2*N*N entries of
DWORK may overlay CWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( 1, 3*N*(N+2) ).
For optimum performance LDWORK should be larger.
CWORK COMPLEX*16 array, dimension (LCWORK)
On exit, if INFO = 0, CWORK(1) returns the optimal value
of LCWORK.
If CWORK(1) is not needed, the first N*N entries of
CWORK may overlay DWORK.
LCWORK INTEGER
The length of the array CWORK.
LCWORK >= MAX( 1, N*(N+3) ).
For optimum performance LCWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the routine fails to compute beta(A) within the
specified tolerance. Nevertheless, the returned
value is an upper bound on beta(A);
= 2: either the QR or SVD algorithm (LAPACK Library
routines DHSEQR, DGESVD or ZGESVD) fails to
converge; this error is very rare.
Method
AB13FD combines the methods of [1] and [2] into a provably reliable, quadratically convergent algorithm. It uses the simple bisection strategy of [1] to find an interval which contains beta(A), and then switches to the modified bisection strategy of [2] which converges quadratically to a minimizer. Note that the efficiency of the strategy degrades if there are several local minima that are near or equal the global minimum.References
[1] Byers, R.
A bisection method for measuring the distance of a stable
matrix to the unstable matrices.
SIAM J. Sci. Stat. Comput., Vol. 9, No. 5, pp. 875-880, 1988.
[2] Boyd, S. and Balakrishnan, K.
A regularity result for the singular values of a transfer
matrix and a quadratically convergent algorithm for computing
its L-infinity norm.
Systems and Control Letters, Vol. 15, pp. 1-7, 1990.
Numerical Aspects
In the presence of rounding errors, the computed function value
BETA satisfies
beta(A) <= BETA + epsilon,
BETA/(1+TOL) - delta <= MAX(beta(A), SQRT(2*N*EPS)*norm(A)),
where norm(A) is the Frobenius norm of A,
epsilon = p(N) * EPS * norm(A),
and
delta = p(N) * SQRT(EPS) * norm(A),
and p(N) is a low degree polynomial. It is recommended to choose
TOL greater than SQRT(EPS). Although rounding errors can cause
AB13FD to fail for smaller values of TOL, nevertheless, it usually
succeeds. Regardless of success or failure, the first inequality
holds.
Further Comments
NoneExample
Program Text
* AB13FD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA
PARAMETER ( LDA = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 3*NMAX*( NMAX + 2 ) )
INTEGER LCWORK
PARAMETER ( LCWORK = NMAX*( NMAX + 3 ) )
* .. Local Scalars ..
INTEGER I, INFO, J, N
DOUBLE PRECISION BETA, OMEGA, TOL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK)
COMPLEX*16 CWORK(LCWORK)
* .. External Subroutines ..
EXTERNAL AB13FD, UD01MD
* ..
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
* Read N, TOL and next A (row wise).
READ ( NIN, FMT = * ) N, TOL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) N
ELSE
DO 10 I = 1, N
READ ( NIN, FMT = * ) ( A(I,J), J = 1, N )
10 CONTINUE
*
WRITE ( NOUT, FMT = 99998 ) N, TOL
CALL UD01MD( N, N, 5, NOUT, A, LDA, 'A', INFO )
*
CALL AB13FD( N, A, LDA, BETA, OMEGA, TOL, DWORK, LDWORK, CWORK,
$ LCWORK, INFO )
*
IF ( INFO.NE.0 )
$ WRITE ( NOUT, FMT = 99996 ) INFO
WRITE ( NOUT, FMT = 99997 ) BETA, OMEGA
END IF
*
99999 FORMAT (' AB13FD EXAMPLE PROGRAM RESULTS', /1X)
99998 FORMAT (' N =', I2, 3X, 'TOL =', D10.3)
99997 FORMAT (' Stability radius :', D18.11, /
* ' Minimizing omega :', D18.11)
99996 FORMAT (' INFO on exit from AB13FD = ', I2)
99995 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
AB13FD EXAMPLE PROGRAM DATA
4 0.0D-00 0.0D-00
246.500 242.500 202.500 -197.500
-252.500 -248.500 -207.500 202.500
-302.500 -297.500 -248.500 242.500
-307.500 -302.500 -252.500 246.500
Program Results
AB13FD EXAMPLE PROGRAM RESULTS
N = 4 TOL = 0.000D+00
A ( 4X 4)
1 2 3 4
1 0.2465000D+03 0.2425000D+03 0.2025000D+03 -0.1975000D+03
2 -0.2525000D+03 -0.2485000D+03 -0.2075000D+03 0.2025000D+03
3 -0.3025000D+03 -0.2975000D+03 -0.2485000D+03 0.2425000D+03
4 -0.3075000D+03 -0.3025000D+03 -0.2525000D+03 0.2465000D+03
Stability radius : 0.39196472317D-02
Minimizing omega : 0.98966520430D+00
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB13HD.html 0000664 0000000 0000000 00000050672 14560147231 0020326 0 ustar 00root root 0000000 0000000
Purpose
To compute the L-infinity norm of a proper continuous-time or
causal discrete-time system, either standard or in the descriptor
form,
-1
G(lambda) = C*( lambda*E - A ) *B + D .
The norm is finite if and only if the matrix pair (A,E) has no
finite eigenvalue on the boundary of the stability domain, i.e.,
the imaginary axis, or the unit circle, respectively.
Specification
SUBROUTINE AB13HD( DICO, JOBE, EQUIL, JOBD, CKPROP, REDUCE, POLES,
$ N, M, P, RANKE, FPEAK, A, LDA, E, LDE, B, LDB,
$ C, LDC, D, LDD, NR, GPEAK, TOL, IWORK, DWORK,
$ LDWORK, ZWORK, LZWORK, BWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER CKPROP, DICO, EQUIL, JOBD, JOBE, POLES, REDUCE
INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, LDE, LDWORK,
$ LZWORK, M, N, NR, P, RANKE
C .. Array Arguments ..
COMPLEX*16 ZWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), DWORK( * ), E( LDE, * ),
$ FPEAK( 2 ), GPEAK( 2 ), TOL( * )
INTEGER IWORK( * )
LOGICAL BWORK( * )
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the system, as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOBE CHARACTER*1
Specifies whether E is an identity matrix, a general
square matrix, or a matrix in compressed form, as follows:
= 'I': E is the identity matrix;
= 'G': E is a general matrix;
= 'C': E is in compressed form, i.e., E = [ T 0 ],
[ 0 0 ]
with a square full-rank matrix T.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily
equilibrate the system (A,E,B,C) or (A,B,C), as follows:
= 'S': perform equilibration (scaling);
= 'N': do not perform equilibration.
JOBD CHARACTER*1
Specifies whether or not a non-zero matrix D appears in
the given state space model:
= 'D': D is present;
= 'Z': D is assumed a zero matrix;
= 'F': D is known to be well-conditioned (hence, to have
full rank), for DICO = 'C' and JOBE = 'I'.
The options JOBD = 'D' and JOBD = 'F' produce the same
results, but much less memory is needed for JOBD = 'F'.
CKPROP CHARACTER*1
If DICO = 'C' and JOBE <> 'I', specifies whether the user
wishes to check the properness of the transfer function of
the descriptor system, as follows:
= 'C': check the properness;
= 'N': do not check the properness.
If the test is requested and the system is found improper
then GPEAK and FPEAK are both set to infinity, i.e., their
second component is zero; in addition, IWARN is set to 2.
If the test is not requested, but the system is improper,
the resulted GPEAK and FPEAK may be wrong.
If DICO = 'D' or JOBE = 'I', this option is ineffective.
REDUCE CHARACTER*1
If CKPROP = 'C', specifies whether the user wishes to
reduce the system order, by removing all uncontrollable
and unobservable poles before computing the norm, as
follows:
= 'R': reduce the system order;
= 'N': compute the norm without reducing the order.
If CKPROP = 'N', this option is ineffective.
POLES CHARACTER*1
Specifies whether the user wishes to use all or part of
the poles to compute the test frequencies (in the non-
iterative part of the algorithm), or all or part of the
midpoints (in the iterative part of the algorithm), as
follows:
= 'A': use all poles with non-negative imaginary parts
and all midpoints;
= 'P': use part of the poles and midpoints.
Input/Output Parameters
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
P (input) INTEGER
The row size of the matrix C. P >= 0.
RANKE (input) INTEGER
If JOBE = 'C', RANKE denotes the rank of the descriptor
matrix E or the size of the full-rank block T.
0 <= RANKE <= N.
FPEAK (input/output) DOUBLE PRECISION array, dimension (2)
On entry, this parameter must contain an estimate of the
frequency where the gain of the frequency response would
achieve its peak value. Setting FPEAK(2) = 0 indicates an
infinite frequency. An accurate estimate could reduce the
number of iterations of the iterative algorithm. If no
estimate is available, set FPEAK(1) = 0, and FPEAK(2) = 1.
FPEAK(1) >= 0, FPEAK(2) >= 0.
On exit, if INFO = 0, this array contains the frequency
OMEGA, where the gain of the frequency response achieves
its peak value GPEAK, i.e.,
|| G ( j*OMEGA ) || = GPEAK , if DICO = 'C', or
j*OMEGA
|| G ( e ) || = GPEAK , if DICO = 'D',
where OMEGA = FPEAK(1), if FPEAK(2) > 0, and OMEGA is
infinite, if FPEAK(2) = 0. (If nonzero, FPEAK(2) = 1.)
For discrete-time systems, it is assumed that the sampling
period is Ts = 1. If Ts <> 1, the frequency corresponding
to the peak gain is OMEGA/Ts.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A.
On exit, if EQUIL = 'S' and CKPROP = 'N', the leading
N-by-N part of this array contains the state dynamics
matrix of an equivalent, scaled system.
On exit, if CKPROP = 'C', DICO = 'C', and JOBE <> 'I', the
leading NR-by-NR part of this array contains the state
dynamics matrix of an equivalent reduced, possibly scaled
(if EQUIL = 'S') system, used to check the properness.
Otherwise, the array A is unchanged.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
E (input/output) DOUBLE PRECISION array, dimension (LDE,K),
where K is N, RANKE, or 0, if JOBE = 'G', 'C', or 'I',
respectively.
On entry, if JOBE = 'G', the leading N-by-N part of this
array must contain the descriptor matrix E of the system.
If JOBE = 'C', the leading RANKE-by-RANKE part of this
array must contain the full-rank block T of the descriptor
matrix E.
If JOBE = 'I', then E is assumed to be the identity matrix
and is not referenced.
On exit, if EQUIL = 'S' and CKPROP = 'N', the leading
K-by-K part of this array contains the descriptor matrix
of an equivalent, scaled system.
On exit, if CKPROP = 'C', DICO = 'C', and JOBE <> 'I', the
leading MIN(K,NR)-by-MIN(K,NR) part of this array contains
the descriptor matrix of an equivalent reduced, possibly
scaled (if EQUIL = 'S') system, used to check the
properness.
Otherwise, the array E is unchanged.
LDE INTEGER
The leading dimension of the array E.
LDE >= MAX(1,N), if JOBE = 'G';
LDE >= MAX(1,RANKE), if JOBE = 'C';
LDE >= 1, if JOBE = 'I'.
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the system input matrix B.
On exit, if EQUIL = 'S' and CKPROP = 'N', the leading
NR-by-M part of this array contains the system input
matrix of an equivalent, scaled system.
On exit, if CKPROP = 'C', DICO = 'C', and JOBE <> 'I', the
leading NR-by-M part of this array contains the system
input matrix of an equivalent reduced, possibly scaled (if
EQUIL = 'S') system, used to check the properness.
Otherwise, the array B is unchanged.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the system output matrix C.
On exit, if EQUIL = 'S' and CKPROP = 'N', the leading
P-by-NR part of this array contains the system output
matrix of an equivalent, scaled system.
On exit, if CKPROP = 'C', DICO = 'C', and JOBE <> 'I', the
leading P-by-NR part of this array contains the system
output matrix of an equivalent reduced, possibly scaled
(if EQUIL = 'S') system, used to check the properness.
Otherwise, the array C is unchanged.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,P).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
If JOBD = 'D' or JOBD = 'F', the leading P-by-M part of
this array must contain the direct transmission matrix D.
The array D is not referenced if JOBD = 'Z'.
LDD INTEGER
The leading dimension of array D.
LDD >= MAX(1,P), if JOBD = 'D' or JOBD = 'F';
LDD >= 1, if JOBD = 'Z'.
NR (output) INTEGER
If CKPROP = 'C', DICO = 'C', and JOBE <> 'I', the order of
the reduced system. Otherwise, NR = N.
GPEAK (output) DOUBLE PRECISION array, dimension (2)
The L-infinity norm of the system, i.e., the peak gain
of the frequency response (as measured by the largest
singular value in the MIMO case), coded in the same way
as FPEAK.
Tolerances
TOL DOUBLE PRECISION array, dimension K, where K = 2, if
CKPROP = 'N' or DICO = 'D' or JOBE = 'I', and K = 4,
otherwise.
TOL(1) is the tolerance used to set the accuracy in
determining the norm. 0 <= TOL(1) < 1.
TOL(2) is the threshold value for magnitude of the matrix
elements, if EQUIL = 'S': elements with magnitude less
than or equal to TOL(2) are ignored for scaling. If the
user sets TOL(2) >= 0, then the given value of TOL(2) is
used. If the user sets TOL(2) < 0, then an implicitly
computed, default threshold, THRESH, is used instead,
defined by THRESH = 0.1, if MN/MX < EPS, and otherwise,
THRESH = MIN( 100*(MN/(EPS**0.25*MX))**0.5, 0.1 ), where
MX and MN are the maximum and the minimum nonzero absolute
value, respectively, of the elements of A and E, and EPS
is the machine precision (see LAPACK Library routine
DLAMCH). TOL(2) = 0 is not always a good choice.
TOL(2) < 1. TOL(2) is not used if EQUIL = 'N'.
TOL(3) is the tolerance to be used in rank determinations
when transforming (lambda*E-A,B,C), if CKPROP = 'C'. If
the user sets TOL(3) > 0, then the given value of TOL(3)
is used as a lower bound for reciprocal condition numbers
in rank determinations; a (sub)matrix whose estimated
condition number is less than 1/TOL(3) is considered to be
of full rank. If the user sets TOL(3) <= 0, then an
implicitly computed, default tolerance, defined by
TOLDEF1 = N*N*EPS, is used instead. TOL(3) < 1.
TOL(4) is the tolerance to be used for checking the
singularity of the matrices A and E when CKPROP = 'C'.
If the user sets TOL(4) > 0, then the given value of
TOL(4) is used. If the user sets TOL(4) <= 0, then an
implicitly computed, default tolerance, defined by
TOLDEF2 = N*EPS, is used instead. The 1-norms of A and E
are also taken into account. TOL(4) < 1.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK >= 1, if MIN(N,P,M) = 0, or B = 0, or C = 0; else
LIWORK >= MAX(1,N), if DICO = 'C', JOBE = 'I', and
JOBD <> 'D';
LIWORK >= 2*N + M + P + R + 12, otherwise, where
R = 0, if M + P is even,
R = 1, if M + P is odd.
On exit, if INFO = 0, IWORK(1) returns the number of
iterations performed by the iterative algorithm
(possibly 0).
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal value
of LDWORK.
On exit, if INFO = -28, DWORK(1) returns the minimum
value of LDWORK. These values are also set when LDWORK = 0
on entry, but no error message related to LDWORK is issued
by XERBLA.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= 1, if MIN(M,P) = 0 or ( JOBD = 'Z' and
( N = 0 or B = 0 or C = 0 ) );
LDWORK >= P*M + x, if ( ( N = 0 and MIN(M,P) > 0 )
or ( B = 0 or C = 0 ) ) and JOBD <> 'Z',
where
x = MAX( 4*MIN(M,P) + MAX(M,P), 6*MIN(M,P) ),
if DICO = 'C',
x = 6*MIN(M,P), if DICO = 'D';
LDWORK >= MAX( 1, N*(N+M+P+2) + MAX( N*(N+M+2) + P*M + x,
4*N*N + 9*N ) ),
if DICO = 'C', JOBE = 'I' and JOBD = 'Z'.
LDWORK >= MAX( 1, (N+M)*(M+P) + P*P + x,
2*N*(N+M+P+1) + N + MIN(P,M) +
MAX( M*(N+P) + N + x, N*N +
MAX( N*(P+M) + MAX(M,P),
2*N*N + 8*N ) ) ),
if DICO = 'C', JOBE = 'I' and JOBD = 'F'.
The formulas for other cases, e.g., for JOBE <> 'I' or
CKPROP = 'C', contain additional and/or other terms.
The minimum value of LDWORK for all cases can be obtained
in DWORK(1) when LDWORK is set to 0 on entry.
For good performance, LDWORK must generally be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) contains the optimal
LZWORK.
On exit, if INFO = -30, ZWORK(1) returns the minimum
value of LZWORK. These values are also set when LZWORK = 0
on entry, but no error message related to LZWORK is issued
by XERBLA.
If LDWORK = 0 and LZWORK = 0 are both set on entry, then
on exit, INFO = -30, but both DWORK(1) and ZWORK(1) are
set the minimum values of LDWORK and LZWORK, respectively.
LZWORK INTEGER
The dimension of the array ZWORK.
LZWORK >= 1, if MIN(N,M,P) = 0, or B = 0, or C = 0;
LZWORK >= MAX(1, (N+M)*(N+P) + 2*MIN(M,P) + MAX(M,P)),
otherwise.
For good performance, LZWORK must generally be larger.
If LZWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
ZWORK array, returns this value as the first entry of
the ZWORK array, and no error message related to LZWORK
is issued by XERBLA.
BWORK LOGICAL array, dimension (N)
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: the descriptor system is singular. GPEAK(1) and
GPEAK(2) are set to 0. FPEAK(1) and FPEAK(2) are
set to 0 and 1, respectively;
= 2: the descriptor system is improper. GPEAK(1) and
GPEAK(2) are set to 1 and 0, respectively,
corresponding to infinity. FPEAK(1) and FPEAK(2) are
set similarly. This warning can only appear if
CKPROP = 'C'.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: a matrix is (numerically) singular or the Sylvester
equation is very ill-conditioned, when computing the
largest singular value of G(infinity) (for
DICO = 'C'); the descriptor system is nearly
singular; the L-infinity norm could be infinite;
= 2: the (periodic) QR (or QZ) algorithm for computing
eigenvalues did not converge;
= 3: the SVD algorithm for computing singular values did
not converge;
= 4: the tolerance is too small and the algorithm did
not converge; this is a warning;
= 5: other computations than QZ iteration, or reordering
of eigenvalues, failed in the LAPACK Library
routines DHGEQZ or DTGSEN, respectively;
= 6: the numbers of "finite" eigenvalues before and after
reordering differ; the threshold used might be
unsuitable.
Method
The routine implements the method presented in [2], which is an extension of the method in [1] for descriptor systems. There are several improvements and refinements [3-5] to increase numerical robustness, accuracy and efficiency, such as the usage of structure-preserving eigenvalue computations for skew-Hamiltonian/ Hamiltonian eigenvalue problems in the iterative method in [2].References
[1] Bruinsma, N.A. and Steinbuch, M.
A fast algorithm to compute the H-infinity-norm of a transfer
function matrix.
Systems & Control Letters, vol. 14, pp. 287-293, 1990.
[2] Voigt, M.
L-infinity-Norm Computation for Descriptor Systems.
Diploma Thesis, Fakultaet fuer Mathematik, TU Chemnitz,
http://nbn-resolving.de/urn:nbn:de:bsz:ch1-201001050.
[3] Benner, P., Sima, V. and Voigt, M.
L-infinity-norm computation for continuous-time descriptor
systems using structured matrix pencils.
IEEE Trans. Auto. Contr., AC-57, pp.233-238, 2012.
[4] Benner, P., Sima, V. and Voigt, M.
Robust and efficient algorithms for L-infinity-norm
computations for descriptor systems.
7th IFAC Symposium on Robust Control Design (ROCOND'12),
pp. 189-194, 2012.
[5] Benner, P., Sima, V. and Voigt, M.
Algorithm 961: Fortran 77 subroutines for the solution of
skew-Hamiltonian/Hamiltonian eigenproblems.
ACM Trans. Math. Softw, 42, pp. 1-26, 2016.
Numerical Aspects
If the algorithm does not converge in MAXIT = 30 iterations (INFO = 4), the tolerance must be increased, or the system is improper.Further Comments
Setting POLES = 'P' usually saves some computational effort. The number of poles used is defined by the parameters BM, BNEICD, BNEICM, BNEICX, BNEIR and SWNEIC. Both real and complex optimal workspace sizes are computed if either LDWORK = -1 or LZWORK = -1.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB13ID.html 0000664 0000000 0000000 00000062763 14560147231 0020333 0 ustar 00root root 0000000 0000000
Purpose
To check whether the transfer function
-1
G(lambda) := C*( lambda*E - A ) *B
of a given linear time-invariant descriptor system with
generalized state space realization (lambda*E-A,B,C) is proper.
Optionally, if JOBEIG = 'A', the system (lambda*E-A,B,C) is
reduced to an equivalent one (lambda*Er-Ar,Br,Cr) with only
controllable and observable eigenvalues in order to use it for a
subsequent L_inf-norm computation; if JOBEIG = 'I', the system is
reduced to an equivalent one (lambda*Er-Ar,Br,Cr) without
uncontrollable and unobservable infinite eigenvalues. In this
case, intended mainly for checking the properness, the returned
system is not fully reduced, unless UPDATE = 'U'.
Specification
LOGICAL FUNCTION AB13ID( JOBSYS, JOBEIG, EQUIL, CKSING, RESTOR,
$ UPDATE, N, M, P, A, LDA, E, LDE, B, LDB,
$ C, LDC, NR, RANKE, TOL, IWORK, DWORK,
$ LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER CKSING, EQUIL, JOBEIG, JOBSYS, RESTOR, UPDATE
INTEGER INFO, IWARN, LDA, LDB, LDC, LDE, LDWORK, M, N,
$ NR, P, RANKE
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ DWORK( * ), E( LDE, * ), TOL( * )
Function Value
AB13ID LOGICAL
Indicates whether the transfer function is proper.
If AB13ID = .TRUE., the transfer function is proper;
otherwise, it is improper.
Arguments
Mode Parameters
JOBSYS CHARACTER*1
Indicates whether the system (lambda*E-A,B,C) is already
in the reduced form which is obtained as stated in
JOBEIG, as follows.
= 'R': The system is not in a reduced form, the reduction
step is performed;
= 'N': The system is in a reduced form; the reduction step
is omitted.
JOBEIG CHARACTER*1
Indicates which kind of eigenvalues of the matrix pencil
lambda*E-A should be removed if JOBSYS = 'R', as follows:
= 'A': All uncontrollable and unobservable eigenvalues
are removed; the reduced system is returned in
the arrays A, E, B, C;
= 'I': Only all uncontrollable and unobservable infinite
eigenvalues are removed; the returned system is not
fully reduced if UPDATE = 'N'.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily scale
the system (lambda*E-A,B,C) as follows:
= 'S': Perform scaling;
= 'N': Do not perform scaling.
CKSING CHARACTER*1
Specifies whether the user wishes to check if the pencil
(lambda*E-A) is singular as follows:
= 'C': Check singularity;
= 'N': Do not check singularity.
If the pencil is singular, the reduced system computed for
CKSING = 'N' may have completely different eigenvalues
than the given system.
The test is performed only if JOBSYS = 'R'.
RESTOR CHARACTER*1
Specifies whether the user wishes to save the system
matrices before each reduction phase (if JOBSYS = 'R') and
restore them if no order reduction took place as follows:
= 'R': Save and restore;
= 'N': Do not save the matrices.
This option is ineffective if JOBSYS = 'N'.
UPDATE CHARACTER*1
Specifies whether the user wishes to update the matrices
A, B, and C if JOBEIG = 'I' as follows:
= 'U': Update the matrices A, B and C;
= 'N': Do not update the matrices A, B and C when
performing URV decomposition of the matrix E
(see METHOD).
Input/Output Parameters
N (input) INTEGER
The dimension of the descriptor state vector; also the
order of square matrices A and E, the number of rows of
matrix B, and the number of columns of matrix C. N >= 0.
M (input) INTEGER
The dimension of the descriptor system input vector; also
the number of columns of matrix B. M >= 0.
P (input) INTEGER
The dimension of the descriptor system output vector; also
the number of rows of matrix C. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state matrix A.
On exit, if JOBSYS = 'R' and JOBEIG = 'A', the leading
NR-by-NR part of this array contains the reduced order
state matrix Ar of a fully controllable and observable
realization for the original system. If JOBSYS = 'R' and
JOBEIG = 'I', the leading NR-by-NR part of this array
contains the reduced order state matrix Ar of a
transformed system without uncontrollable and unobservable
infinite poles. In this case, the matrix Ar does not
correspond to the returned matrix Er (obtained after a
URV decomposition), unless UPDATE = 'U' or RANKE < NR.
On exit, if JOBSYS = 'N' and (JOBEIG = 'A' or UPDATE = 'U'
or RANKE < N), the leading N-by-N part of this array
contains the transformed matrix A corresponding to the
URV decomposition of E (see (2) in METHOD), and if
JOBEIG = 'I' and UPDATE = 'N', the submatrix A22 in (2) is
further transformed to estimate its rank.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain the descriptor matrix E.
On exit, if JOBSYS = 'R' and JOBEIG = 'A', the leading
NR-by-NR part of this array contains the reduced order
descriptor matrix Er of a completely controllable and
observable realization for the original system. The
reduced matrix Er is in upper triangular form.
If JOBSYS = 'R' and JOBEIG = 'I', the leading NR-by-NR
part of this array contains the reduced order descriptor
matrix Er of a transformed system without uncontrollable
and unobservable infinite poles. The reduced matrix Er is
upper triangular. In both cases, or if JOBSYS = 'N', the
matrix Er results from a URV decomposition of the matrix E
(see METHOD).
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension
(LDB,MAX(M,P))
On entry, the leading N-by-M part of this array must
contain the input matrix B; the remainder of the leading
N-by-MAX(M,P) part is used as internal workspace.
On exit, if JOBSYS = 'R' and JOBEIG = 'A', the leading
NR-by-M part of this array contains the reduced input
matrix Br of a completely controllable and observable
realization for the original system. If JOBSYS = 'R' and
JOBEIG = 'I', the leading NR-by-M part of this array
contains the transformed input matrix Br obtained after
removing the uncontrollable and unobservable infinite
poles; the transformations for the URV decomposition of
the matrix E are not applied if UPDATE = 'N'.
On exit, if JOBSYS = 'N' and (JOBEIG = 'A' or
UPDATE = 'U'), the leading N-by-M part of this array
contains the transformed matrix B corresponding to the
URV decomposition of E, but if JOBEIG = 'I', EQUIL = 'N'
and UPDATE = 'N', the array B is unchanged on exit.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the output matrix C; the remainder of the leading
MAX(M,P)-by-N part is used as internal workspace.
On exit, if JOBSYS = 'R' and JOBEIG = 'A', the leading
P-by-NR part of this array contains the transformed output
matrix Cr of a completely controllable and observable
realization for the original system. If JOBSYS = 'R' and
JOBEIG = 'I', the leading P-by-NR part of this array
contains the transformed output matrix Cr obtained after
removing the uncontrollable and unobservable infinite
poles; the transformations for the URV decomposition of
the matrix E are not applied if UPDATE = 'N'.
On exit, if JOBSYS = 'N' and (JOBEIG = 'A' or
UPDATE = 'U'), the leading P-by-N part of this array
contains the transformed matrix C corresponding to the
URV decomposition of E, but if JOBEIG = 'I', EQUIL = 'N'
and UPDATE = 'N', the array C is unchanged on exit.
LDC INTEGER
The leading dimension of the array C.
LDC >= MAX(1,M,P) if N > 0;
LDC >= 1 if N = 0.
NR (output) INTEGER
The order of the reduced generalized state space
representation (lambda*Er-Ar,Br,Cr) as stated in JOBEIG.
If JOBEIG = 'A', NR denotes the order of a reduced system
without any uncontrollable or unobservable eigenvalues; if
JOBEIG = 'I', NR denotes the order of the reduced system
without any uncontrollable or unobservable infinite
eigenvalues. If JOBSYS = 'N', then NR = N.
RANKE (output) INTEGER
The effective (estimated) rank of the reduced matrix Er.
Tolerances
TOL DOUBLE PRECISION array, dimension 3
TOL(1) is the tolerance to be used in rank determinations
when transforming (lambda*E-A,B,C). If the user sets
TOL(1) > 0, then the given value of TOL(1) is used as a
lower bound for reciprocal condition numbers in rank
determinations; a (sub)matrix whose estimated condition
number is less than 1/TOL(1) is considered to be of full
rank. If the user sets TOL(1) <= 0, then an implicitly
computed, default tolerance, defined by TOLDEF1 = N*N*EPS,
is used instead, where EPS is the machine precision (see
LAPACK Library routine DLAMCH). TOL(1) < 1.
TOL(2) is the tolerance to be used for checking pencil
singularity when CKSING = 'C', or singularity of the
matrices A and E when CKSING = 'N'. If the user sets
TOL(2) > 0, then the given value of TOL(2) is used.
If the user sets TOL(2) <= 0, then an implicitly
computed, default tolerance, defined by TOLDEF2 = 10*EPS,
is used instead. TOL(2) < 1.
TOL(3) is the threshold value for magnitude of the matrix
elements, if EQUIL = 'S': elements with magnitude less
than or equal to TOL(3) are ignored for scaling. If the
user sets TOL(3) >= 0, then the given value of TOL(3) is
used. If the user sets TOL(3) < 0, then an implicitly
computed, default threshold, defined by THRESH = c*EPS,
where c = MAX(||A||,||E||,||B||,||C||) is used instead and
1-norm is used. TOL(3) = 0 is not always a good choice.
TOL(3) < 1.
TOL(3) is not used if EQUIL = 'N'.
Workspace
IWORK INTEGER array, dimension (LIWORK)
If JOBSYS = 'R', LIWORK >= 2*N+MAX(M,P)+7;
If JOBSYS = 'N', LIWORK >= N.
If JOBSYS = 'R', the first 7 elements of IWORK contain
information on performed reduction and on structure of
resulting system matrices after removing the specified
eigenvalues (see the description of the parameter INFRED
of the SLICOT Library routine TG01JY).
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
If JOBSYS = 'R', and EQUIL = 'S',
LDWORK >= MAX(w+4*N+4,8*N,x,y),
where w = N*N, if JOBEIG = 'A',
w = 0, if JOBEIG = 'I',
x = MAX(2*(z+MAX(M,P)+N-1),N*N+4*N), if RESTOR = 'R'
x = MAX( 2*(MAX(M,P)+N-1),N*N+4*N), if RESTOR = 'N'
y = 2*N*N+10*N+MAX(N,23), if CKSING = 'C',
y = 0, if CKSING = 'N',
z = 2*N*N+N*M+N*P;
if JOBSYS = 'R', and EQUIL = 'N',
LDWORK >= MAX(w+4*N+4,x,y);
if JOBSYS = 'N', and JOBEIG = 'A' or UPDATE = 'U',
and EQUIL = 'S',
LDWORK >= MAX(N*N+4*N+4,8*N,N+M,N+P);
if JOBSYS = 'N', and JOBEIG = 'A' or UPDATE = 'U',
and EQUIL = 'N',
LDWORK >= MAX(N*N+4*N+4,N+M,N+P);
if JOBSYS = 'N', and JOBEIG = 'I' and UPDATE = 'N',
and EQUIL = 'S',
LDWORK >= MAX(4*N+4,8*N);
if JOBSYS = 'N', and JOBEIG = 'I' and UPDATE = 'N',
and EQUIL = 'N',
LDWORK >= 4*N+4.
If JOBSYS = 'R' and ( RESTOR = 'R' or
LDWORK >= MAX(1,2*N*N+N*M+N*P+2*(MAX(M,P)+N-1) ),
then more accurate results are to be expected by
considering only those reduction phases in the SLICOT
Library routine TG01JY, where effective order reduction
occurs. This is achieved by saving the system matrices
before each phase (after orthogonally triangularizing the
matrix A or the matrix E, if RESTOR = 'N') and restoring
them if no order reduction took place. However, higher
global accuracy is not guaranteed.
For good performance, LDWORK should be generally larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA. The optimal workspace includes the
extra space for improving the accuracy.
Warning Indicator
IWARN INTEGER
= 0: When determining the rank of a matrix, the rank can
be safely determined: a small decrease of TOL(1) will
not increase the rank.
= 1: The computed rank is possibly incorrect: a small
decrease of TOL(1) might increase the rank.
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: the given pencil A - lambda*E is numerically
singular and the reduced system is not computed.
However, the system is considered improper, and
AB13ID is set to .FALSE.
This error can be returned only if CKSING = 'C'.
Method
If JOBSYS = 'R', the routine first removes uncontrollable and
unobservable infinite eigenvalues of the pencil lambda*E-A. If, in
addition, JOBEIG = 'A', uncontrollable and unobservable zero
eigenvalues are also removed. Then, or if JOBSYS = 'N', a
URV decomposition of the matrix E is performed, i.e., orthogonal
matrices U and V are computed, such that
( T 0 )
U*E*V = ( ) with a full-rank matrix T. (1)
( 0 0 )
Then the matrix A (or a copy of A if JOBEIG = 'A' or UPDATE = 'U')
is updated and partioned as in (1), i.e.,
( A11 A12 )
U*A*V = ( ) , (2)
( A21 A22 )
and the rank of A22 is computed. If A22 is invertible, the
transfer function is proper, otherwise it is improper. If
required (i.e., JOBEIG = 'A' or UPDATE = 'U'), the matrices B and
C are updated as well in order to obtain an equivalent reduced
system with the same transfer function. See also Chapter 3 in [1],
[2] for more details.
References
[1] Voigt, M.
L_inf-Norm Computation for Descriptor Systems.
Diploma Thesis, Chemnitz University of Technology, Department
of Mathematics, Germany, July 2010.
[2] Benner, P., Sima, V., Voigt, M.
L_infinity-norm computation for continuous-time descriptor
systems using structured matrix pencils.
IEEE Trans. Automat. Contr., vol. 57, pp. 233-238, 2012.
Numerical Aspects
The algorithm requires O(N**3) floating point operations. During the algorithm it is necessary to determine the rank of certain matrices. Therefore it is crucial to use an appropriate tolerance TOL(1) to make correct rank decisions.Further Comments
NoneExample
Program Text
* AB13ID EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDE
PARAMETER ( LDA = NMAX, LDB = NMAX,
$ LDC = MAX( MMAX, PMAX ), LDE = NMAX )
INTEGER LDWORK, LIWORK
PARAMETER ( LDWORK = 2*NMAX*NMAX +
$ MAX( 2*( NMAX*( NMAX + MMAX + PMAX ) +
$ MAX( MMAX, PMAX ) + NMAX - 1 ),
$ 10*NMAX + MAX( NMAX, 23 ) ),
$ LIWORK = 2*NMAX + MAX( MMAX, PMAX ) + 7 )
* .. Local Scalars ..
LOGICAL LISPRP
CHARACTER CKSING, EQUIL, JOBEIG, JOBSYS, RESTOR, UPDATE
INTEGER I, INFO, IWARN, J, M, N, NO, NR, P, RANKE
* .. Local Arrays ..
INTEGER IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MAX(MMAX,PMAX)), C(LDC,NMAX),
$ DWORK(LDWORK), E(LDE,NMAX), TOL(3)
* .. External Functions ..
LOGICAL AB13ID, LSAME
EXTERNAL AB13ID, LSAME
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL(1), TOL(2), TOL(3), JOBSYS,
$ JOBEIG, EQUIL, CKSING, RESTOR, UPDATE
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Check whether the transfer function of the descriptor
* system is proper.
LISPRP = AB13ID( JOBSYS, JOBEIG, EQUIL, CKSING, RESTOR,
$ UPDATE, N, M, P, A, LDA, E, LDE, B, LDB,
$ C, LDC, NR, RANKE, TOL, IWORK, DWORK,
$ LDWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( LISPRP ) THEN
WRITE ( NOUT, FMT = 99991 )
ELSE
WRITE ( NOUT, FMT = 99990 )
END IF
WRITE ( NOUT, FMT = 99994 ) NR
WRITE ( NOUT, FMT = 99989 ) RANKE
IF ( LSAME( JOBSYS, 'N' ).AND.( LSAME( JOBEIG, 'A' )
$ .OR.LSAME( UPDATE, 'U' ) ) ) THEN
NO = N
ELSE
NO = NR
END IF
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, NO
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NO )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,NR )
20 CONTINUE
IF ( LSAME( JOBSYS, 'N' ).AND.LSAME( JOBEIG, 'I' )
$ .AND.LSAME( EQUIL, 'S' )
$ .AND.LSAME( UPDATE, 'N' ) )
$ NO = N
WRITE ( NOUT, FMT = 99993 )
DO 30 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
30 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 40 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
40 CONTINUE
IF ( IWARN.NE.0 )
$ WRITE ( NOUT, FMT = 99998 ) IWARN
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB13ID EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB13ID = ',I2)
99997 FORMAT (/' The reduced state dynamics matrix Ar is ')
99996 FORMAT (/' The reduced descriptor matrix Er is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' Order of reduced system =', I5 )
99993 FORMAT (/' The reduced input/state matrix Br is ')
99992 FORMAT (/' The reduced state/output matrix Cr is ')
99991 FORMAT ( ' The system is proper')
99990 FORMAT ( ' The system is improper')
99989 FORMAT (' Rank of matrix E =', I5 )
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
99986 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
AB13ID EXAMPLE PROGRAM DATA
9 2 2 0.0 0.0 0.0 R I N N N U
-2 -3 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0
0 0 -2 -3 0 0 0 0 0
0 0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1 0
1 0
0 0
0 1
0 0
-1 0
0 0
0 -1
0 0
0 0
1 0 1 -3 0 1 0 2 0
0 1 1 3 0 1 0 0 1
Program Results
AB13ID EXAMPLE PROGRAM RESULTS The system is improper Order of reduced system = 7 Rank of matrix E = 5 The reduced state dynamics matrix Ar is 0.2202 0.4554 -0.6171 -0.3695 -1.3751 0.8121 0.1953 0.6175 0.1352 -0.8444 -0.0262 -1.3999 -0.4013 0.3339 -0.6362 -1.1518 0.6708 -0.2221 0.8680 -0.0430 0.1827 0.1430 -0.0480 -0.3290 -0.1625 -0.6986 0.0008 -0.9031 0.0986 -0.8665 0.4541 0.6647 1.4067 0.4214 -0.0381 -0.6979 -0.0079 -0.1915 0.6588 -0.2054 0.0000 0.0000 -0.1861 -0.0021 -0.8313 -0.3063 0.4248 0.0000 0.0000 The reduced descriptor matrix Er is -1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 The reduced input/state matrix Br is -0.2140 -0.3805 -0.2558 -0.2539 0.1844 0.4001 -0.1115 -0.1176 0.1879 0.5325 -0.9954 -0.0961 0.0961 -0.9954 The reduced state/output matrix Cr is 0.0439 -3.8727 0.0000 0.0000 0.0000 0.6508 0.7593 0.1184 2.0671 -0.8971 -0.8236 -2.2869 1.4100 0.1085
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB13MD.html 0000664 0000000 0000000 00000022374 14560147231 0020331 0 ustar 00root root 0000000 0000000
Purpose
To compute an upper bound on the structured singular value for a given square complex matrix and a given block structure of the uncertainty.Specification
SUBROUTINE AB13MD( FACT, N, Z, LDZ, M, NBLOCK, ITYPE, X, BOUND, D,
$ G, IWORK, DWORK, LDWORK, ZWORK, LZWORK, INFO )
C .. Scalar Arguments ..
CHARACTER FACT
INTEGER INFO, LDWORK, LDZ, LZWORK, M, N
DOUBLE PRECISION BOUND
C .. Array Arguments ..
INTEGER ITYPE( * ), IWORK( * ), NBLOCK( * )
COMPLEX*16 Z( LDZ, * ), ZWORK( * )
DOUBLE PRECISION D( * ), DWORK( * ), G( * ), X( * )
Arguments
Mode Parameters
FACT CHARACTER*1
Specifies whether or not an information from the
previous call is supplied in the vector X.
= 'F': On entry, X contains information from the
previous call.
= 'N': On entry, X does not contain an information from
the previous call.
Input/Output Parameters
N (input) INTEGER
The order of the matrix Z. N >= 0.
Z (input) COMPLEX*16 array, dimension (LDZ,N)
The leading N-by-N part of this array must contain the
complex matrix Z for which the upper bound on the
structured singular value is to be computed.
LDZ INTEGER
The leading dimension of the array Z. LDZ >= max(1,N).
M (input) INTEGER
The number of diagonal blocks in the block structure of
the uncertainty. M >= 1.
NBLOCK (input) INTEGER array, dimension (M)
The vector of length M containing the block structure
of the uncertainty. NBLOCK(I), I = 1:M, is the size of
each block.
ITYPE (input) INTEGER array, dimension (M)
The vector of length M indicating the type of each block.
For I = 1:M,
ITYPE(I) = 1 indicates that the corresponding block is a
real block, and
ITYPE(I) = 2 indicates that the corresponding block is a
complex block.
NBLOCK(I) must be equal to 1 if ITYPE(I) is equal to 1.
X (input/output) DOUBLE PRECISION array, dimension
( M + MR - 1 ), where MR is the number of the real blocks.
On entry, if FACT = 'F' and NBLOCK(1) < N, this array
must contain information from the previous call to AB13MD.
If NBLOCK(1) = N, this array is not used.
On exit, if NBLOCK(1) < N, this array contains information
that can be used in the next call to AB13MD for a matrix
close to Z.
BOUND (output) DOUBLE PRECISION
The upper bound on the structured singular value.
D, G (output) DOUBLE PRECISION arrays, dimension (N)
The vectors of length N containing the diagonal entries
of the diagonal N-by-N matrices D and G, respectively,
such that the matrix
Z'*D^2*Z + sqrt(-1)*(G*Z-Z'*G) - BOUND^2*D^2
is negative semidefinite.
Workspace
IWORK INTEGER array, dimension (MAX(4*M-2,N))
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= 2*N*N*M - N*N + 9*M*M + N*M + 11*N + 33*M - 11.
For best performance
LDWORK >= 2*N*N*M - N*N + 9*M*M + N*M + 6*N + 33*M - 11 +
MAX( 5*N,2*N*NB )
where NB is the optimal blocksize returned by ILAENV.
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) contains the optimal value
of LZWORK.
LZWORK INTEGER
The dimension of the array ZWORK.
LZWORK >= 6*N*N*M + 12*N*N + 6*M + 6*N - 3.
For best performance
LZWORK >= 6*N*N*M + 12*N*N + 6*M + 3*N - 3 +
MAX( 3*N,N*NB )
where NB is the optimal blocksize returned by ILAENV.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the block sizes must be positive integers;
= 2: the sum of block sizes must be equal to N;
= 3: the size of a real block must be equal to 1;
= 4: the block type must be either 1 or 2;
= 5: errors in solving linear equations or in matrix
inversion;
= 6: errors in computing eigenvalues or singular values.
Method
The routine computes the upper bound proposed in [1].References
[1] Fan, M.K.H., Tits, A.L., and Doyle, J.C.
Robustness in the presence of mixed parametric uncertainty
and unmodeled dynamics.
IEEE Trans. Automatic Control, vol. AC-36, 1991, pp. 25-38.
Numerical Aspects
The accuracy and speed of computation depend on the value of the internal threshold TOL.Further Comments
NoneExample
Program Text
* AB13MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX
PARAMETER ( NMAX = 10, MMAX = 10 )
INTEGER LDZ
PARAMETER ( LDZ = NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MAX( 4*MMAX-2, NMAX ) )
INTEGER LDWORK
PARAMETER ( LDWORK = 2*NMAX*NMAX*MMAX - NMAX*NMAX +
$ 9*MMAX*MMAX + NMAX*MMAX + 11*NMAX +
$ 33*MMAX - 11 )
INTEGER LZWORK
PARAMETER ( LZWORK = 6*NMAX*NMAX*MMAX + 12*NMAX*NMAX +
$ 6*MMAX + 6*NMAX - 3 )
* .. Local Scalars ..
INTEGER I, INFO, J, M, N
DOUBLE PRECISION BOUND
* .. Local Arrays ..
INTEGER ITYPE(MMAX), IWORK(LIWORK), NBLOCK(MMAX)
DOUBLE PRECISION D(NMAX), DWORK(LDWORK), G(NMAX), X(2*MMAX-1)
COMPLEX*16 Z(LDZ,NMAX), ZWORK(LZWORK)
* .. External Subroutines ..
EXTERNAL AB13MD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( NBLOCK(I), I = 1, M )
READ ( NIN, FMT = * ) ( ITYPE(I), I = 1, M )
READ ( NIN, FMT = * ) ( ( Z(I,J), J = 1,N ), I = 1,N )
* Computing mu.
CALL AB13MD( 'N', N, Z, LDZ, M, NBLOCK, ITYPE, X, BOUND, D, G,
$ IWORK, DWORK, LDWORK, ZWORK, LZWORK, INFO )
*
IF ( INFO.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99997 )
WRITE ( NOUT, FMT = 99991 ) BOUND
ELSE
WRITE( NOUT, FMT = 99998 ) INFO
END IF
END IF
STOP
*
99999 FORMAT (' AB13MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB13MD =',I2)
99997 FORMAT (' The value of the structured singular value is'/)
99991 FORMAT (D17.10)
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
END
Program Data
AB13MD EXAMPLE PROGRAM DATA 6 5 1 1 2 1 1 1 1 2 2 2 (-1.0D0,6.0D0) (2.0D0,-3.0D0) (3.0D0,8.0D0) (3.0D0,8.0D0) (-5.0D0,-9.0D0) (-6.0D0,2.0D0) (4.0D0,2.0D0) (-2.0D0,5.0D0) (-6.0D0,-7.0D0) (-4.0D0,11.0D0) (8.0D0,-7.0D0) (12.0D0,-1.0D0) (5.0D0,-4.0D0) (-4.0D0,-8.0D0) (1.0D0,-3.0D0) (-6.0D0,14.0D0) (2.0D0,-5.0D0) (4.0D0,16.0D0) (-1.0D0,6.0D0) (2.0D0,-3.0D0) (3.0D0,8.0D0) (3.0D0,8.0D0) (-5.0D0,-9.0D0) (-6.0D0,2.0D0) (4.0D0,2.0D0) (-2.0D0,5.0D0) (-6.0D0,-7.0D0) (-4.0D0,11.0D0) (8.0D0,-7.0D0) (12.0D0,-1.0D0) (5.0D0,-4.0D0) (-4.0D0,-8.0D0) (1.0D0,-3.0D0) (-6.0D0,14.0D0) (2.0D0,-5.0D0) (4.0D0,16.0D0)Program Results
AB13MD EXAMPLE PROGRAM RESULTS The value of the structured singular value is 0.4174753408D+02
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AB8NXZ.html 0000664 0000000 0000000 00000015226 14560147231 0020432 0 ustar 00root root 0000000 0000000
Purpose
To extract from the (N+P)-by-(M+N) system
( B A )
( D C )
an (NU+MU)-by-(M+NU) "reduced" system
( B' A')
( D' C')
having the same transmission zeros but with D' of full row rank.
Specification
SUBROUTINE AB8NXZ( N, M, P, RO, SIGMA, SVLMAX, ABCD, LDABCD,
$ NINFZ, INFZ, KRONL, MU, NU, NKROL, TOL, IWORK,
$ DWORK, ZWORK, LZWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDABCD, LZWORK, M, MU, N, NINFZ, NKROL,
$ NU, P, RO, SIGMA
DOUBLE PRECISION SVLMAX, TOL
C .. Array Arguments ..
INTEGER INFZ(*), IWORK(*), KRONL(*)
COMPLEX*16 ABCD(LDABCD,*), ZWORK(*)
DOUBLE PRECISION DWORK(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The number of state variables. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
RO (input/output) INTEGER
On entry,
= P for the original system;
= MAX(P-M, 0) for the pertransposed system.
On exit, RO contains the last computed rank.
SIGMA (input/output) INTEGER
On entry,
= 0 for the original system;
= M for the pertransposed system.
On exit, SIGMA contains the last computed value sigma in
the algorithm.
SVLMAX (input) DOUBLE PRECISION
During each reduction step, the rank-revealing QR
factorization of a matrix stops when the estimated minimum
singular value is smaller than TOL * MAX(SVLMAX,EMSV),
where EMSV is the estimated maximum singular value.
SVLMAX >= 0.
ABCD (input/output) COMPLEX*16 array, dimension (LDABCD,M+N)
On entry, the leading (N+P)-by-(M+N) part of this array
must contain the compound input matrix of the system.
On exit, the leading (NU+MU)-by-(M+NU) part of this array
contains the reduced compound input matrix of the system.
LDABCD INTEGER
The leading dimension of array ABCD.
LDABCD >= MAX(1,N+P).
NINFZ (input/output) INTEGER
On entry, the currently computed number of infinite zeros.
It should be initialized to zero on the first call.
NINFZ >= 0.
On exit, the number of infinite zeros.
INFZ (input/output) INTEGER array, dimension (N)
On entry, INFZ(i) must contain the current number of
infinite zeros of degree i, where i = 1,2,...,N, found in
the previous call(s) of the routine. It should be
initialized to zero on the first call.
On exit, INFZ(i) contains the number of infinite zeros of
degree i, where i = 1,2,...,N.
KRONL (input/output) INTEGER array, dimension (N+1)
On entry, this array must contain the currently computed
left Kronecker (row) indices found in the previous call(s)
of the routine. It should be initialized to zero on the
first call.
On exit, the leading NKROL elements of this array contain
the left Kronecker (row) indices.
MU (output) INTEGER
The normal rank of the transfer function matrix of the
original system.
NU (output) INTEGER
The dimension of the reduced system matrix and the number
of (finite) invariant zeros if D' is invertible.
NKROL (output) INTEGER
The number of left Kronecker indices.
Tolerances
TOL DOUBLE PRECISION
A tolerance used in rank decisions to determine the
effective rank, which is defined as the order of the
largest leading (or trailing) triangular submatrix in the
QR (or RQ) factorization with column (or row) pivoting
whose estimated condition number is less than 1/TOL.
NOTE that when SVLMAX > 0, the estimated ranks could be
less than those defined above (see SVLMAX).
Workspace
IWORK INTEGER array, dimension (MAX(M,P))
DWORK DOUBLE PRECISION array, dimension (2*MAX(M,P))
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) returns the optimal value
of LZWORK.
LZWORK INTEGER
The length of the array ZWORK.
LZWORK >= MAX( 1, MIN(P,M) + MAX(3*M-1,N),
MIN(P,N) + MAX(3*P-1,N+P,N+M) ).
For optimum performance LZWORK should be larger.
If LZWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
ZWORK array, returns this value as the first entry of
the ZWORK array, and no error message related to LZWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
References
[1] Svaricek, F.
Computation of the Structural Invariants of Linear
Multivariable Systems with an Extended Version of
the Program ZEROS.
System & Control Letters, 6, pp. 261-266, 1985.
[2] Emami-Naeini, A. and Van Dooren, P.
Computation of Zeros of Linear Multivariable Systems.
Automatica, 18, pp. 415-430, 1982.
Numerical Aspects
The algorithm is backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the inverse (Ai-lambda*Ei,Bi,Ci,Di) of a given descriptor system (A-lambda*E,B,C,D).Specification
SUBROUTINE AG07BD( JOBE, N, M, A, LDA, E, LDE, B, LDB, C, LDC,
$ D, LDD, AI, LDAI, EI, LDEI, BI, LDBI, CI, LDCI,
$ DI, LDDI, INFO )
C .. Scalar Arguments ..
CHARACTER JOBE
INTEGER INFO, LDA, LDAI, LDB, LDBI, LDC, LDCI,
$ LDD, LDDI, LDE, LDEI, M, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), AI(LDAI,*), B(LDB,*), BI(LDBI,*),
$ C(LDC,*), CI(LDCI,*), D(LDD,*), DI(LDDI,*),
$ E(LDE,*), EI(LDEI,*)
Arguments
Mode Parameters
JOBE CHARACTER*1
Specifies whether E is a general square or an identity
matrix as follows:
= 'G': E is a general square matrix;
= 'I': E is the identity matrix.
Input/Output Parameters
N (input) INTEGER
The order of the square matrices A and E;
also the number of rows of matrix B and the number of
columns of matrix C. N >= 0.
M (input) INTEGER
The number of system inputs and outputs, i.e., the number
of columns of matrices B and D and the number of rows of
matrices C and D. M >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
state matrix A of the original system.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
E (input) DOUBLE PRECISION array, dimension (LDE,N)
If JOBE = 'G', the leading N-by-N part of this array must
contain the descriptor matrix E of the original system.
If JOBE = 'I', then E is assumed to be the identity
matrix and is not referenced.
LDE INTEGER
The leading dimension of the array E.
LDE >= MAX(1,N), if JOBE = 'G';
LDE >= 1, if JOBE = 'I'.
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
input matrix B of the original system.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading M-by-N part of this array must contain the
output matrix C of the original system.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,M).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading M-by-M part of this array must contain the
feedthrough matrix D of the original system.
LDD INTEGER
The leading dimension of the array D. LDD >= MAX(1,M).
AI (output) DOUBLE PRECISION array, dimension (LDAI,N+M)
The leading (N+M)-by-(N+M) part of this array contains
the state matrix Ai of the inverse system.
If LDAI = LDA >= N+M, then AI and A can share the same
storage locations.
LDAI INTEGER
The leading dimension of the array AI.
LDAI >= MAX(1,N+M).
EI (output) DOUBLE PRECISION array, dimension (LDEI,N+M)
The leading (N+M)-by-(N+M) part of this array contains
the descriptor matrix Ei of the inverse system.
If LDEI = LDE >= N+M, then EI and E can share the same
storage locations.
LDEI INTEGER
The leading dimension of the array EI.
LDEI >= MAX(1,N+M).
BI (output) DOUBLE PRECISION array, dimension (LDBI,M)
The leading (N+M)-by-M part of this array contains
the input matrix Bi of the inverse system.
If LDBI = LDB >= N+M, then BI and B can share the same
storage locations.
LDBI INTEGER
The leading dimension of the array BI.
LDBI >= MAX(1,N+M).
CI (output) DOUBLE PRECISION array, dimension (LDCI,N+M)
The leading M-by-(N+M) part of this array contains
the output matrix Ci of the inverse system.
If LDCI = LDC, CI and C can share the same storage
locations.
LDCI INTEGER
The leading dimension of the array CI. LDCI >= MAX(1,M).
DI (output) DOUBLE PRECISION array, dimension (LDDI,M)
The leading M-by-M part of this array contains
the feedthrough matrix Di = 0 of the inverse system.
DI and D can share the same storage locations.
LDDI INTEGER
The leading dimension of the array DI. LDDI >= MAX(1,M).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The matrices of the inverse system are computed with the formulas
( E 0 ) ( A B ) ( 0 )
Ei = ( ) , Ai = ( ) , Bi = ( ),
( 0 0 ) ( C D ) ( -I )
Ci = ( 0 I ), Di = 0.
Further Comments
The routine does not perform an invertibility test. This check can be performed by using the SLICOT routines AB08NX or AG08BY.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AG08BD.html 0000664 0000000 0000000 00000072465 14560147231 0020335 0 ustar 00root root 0000000 0000000
Purpose
To extract from the system pencil
( A-lambda*E B )
S(lambda) = ( )
( C D )
a regular pencil Af-lambda*Ef which has the finite Smith zeros of
S(lambda) as generalized eigenvalues. The routine also computes
the orders of the infinite Smith zeros and determines the singular
and infinite Kronecker structure of system pencil, i.e., the right
and left Kronecker indices, and the multiplicities of infinite
eigenvalues.
Specification
SUBROUTINE AG08BD( EQUIL, L, N, M, P, A, LDA, E, LDE, B, LDB,
$ C, LDC, D, LDD, NFZ, NRANK, NIZ, DINFZ, NKROR,
$ NINFE, NKROL, INFZ, KRONR, INFE, KRONL,
$ TOL, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER EQUIL
INTEGER DINFZ, INFO, L, LDA, LDB, LDC, LDD, LDE, LDWORK,
$ M, N, NFZ, NINFE, NIZ, NKROL, NKROR, NRANK, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER INFE(*), INFZ(*), IWORK(*), KRONL(*), KRONR(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), E(LDE,*)
Arguments
Mode Parameters
EQUIL CHARACTER*1
Specifies whether the user wishes to balance the system
matrix as follows:
= 'S': Perform balancing (scaling);
= 'N': Do not perform balancing.
Input/Output Parameters
L (input) INTEGER
The number of rows of matrices A, B, and E. L >= 0.
N (input) INTEGER
The number of columns of matrices A, E, and C. N >= 0.
M (input) INTEGER
The number of columns of matrix B. M >= 0.
P (input) INTEGER
The number of rows of matrix C. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading L-by-N part of this array must
contain the state dynamics matrix A of the system.
On exit, the leading NFZ-by-NFZ part of this array
contains the matrix Af of the reduced pencil.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,L).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading L-by-N part of this array must
contain the descriptor matrix E of the system.
On exit, the leading NFZ-by-NFZ part of this array
contains the matrix Ef of the reduced pencil.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,L).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading L-by-M part of this array must
contain the input/state matrix B of the system.
On exit, this matrix does not contain useful information.
LDB INTEGER
The leading dimension of array B.
LDB >= MAX(1,L) if M > 0;
LDB >= 1 if M = 0.
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C of the system.
On exit, this matrix does not contain useful information.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading P-by-M part of this array must contain the
direct transmission matrix D of the system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
NFZ (output) INTEGER
The number of finite zeros.
NRANK (output) INTEGER
The normal rank of the system pencil.
NIZ (output) INTEGER
The number of infinite zeros.
DINFZ (output) INTEGER
The maximal multiplicity of infinite Smith zeros.
NKROR (output) INTEGER
The number of right Kronecker indices.
NINFE (output) INTEGER
The number of elementary infinite blocks.
NKROL (output) INTEGER
The number of left Kronecker indices.
INFZ (output) INTEGER array, dimension (N+1)
The leading DINFZ elements of INFZ contain information
on the infinite elementary divisors as follows:
the system has INFZ(i) infinite elementary divisors of
degree i in the Smith form, where i = 1,2,...,DINFZ.
KRONR (output) INTEGER array, dimension (N+M+1)
The leading NKROR elements of this array contain the
right Kronecker (column) indices.
INFE (output) INTEGER array, dimension (1+MIN(L+P,N+M))
The leading NINFE elements of INFE contain the
multiplicities of infinite eigenvalues.
KRONL (output) INTEGER array, dimension (L+P+1)
The leading NKROL elements of this array contain the
left Kronecker (row) indices.
Tolerances
TOL DOUBLE PRECISION
A tolerance used in rank decisions to determine the
effective rank, which is defined as the order of the
largest leading (or trailing) triangular submatrix in the
QR (or RQ) factorization with column (or row) pivoting
whose estimated condition number is less than 1/TOL.
If the user sets TOL <= 0, then default tolerances are
used instead, as follows: TOLDEF = L*N*EPS in TG01FD
(to determine the rank of E) and TOLDEF = (L+P)*(N+M)*EPS
in the rest, where EPS is the machine precision
(see LAPACK Library routine DLAMCH). TOL < 1.
Workspace
IWORK INTEGER array, dimension (N+max(1,M))
On output, IWORK(1) contains the normal rank of the
transfer function matrix.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= max( 4*(L+N), LDW ), if EQUIL = 'S',
LDWORK >= LDW, if EQUIL = 'N', where
LDW = max(L+P,M+N)*(M+N) + max(1,5*max(L+P,M+N)).
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine extracts from the system matrix of a descriptor
system (A-lambda*E,B,C,D) a regular pencil Af-lambda*Ef which
has the finite zeros of the system as generalized eigenvalues.
The procedure has the following main computational steps:
(a) construct the (L+P)-by-(N+M) system pencil
S(lambda) = ( B A )-lambda*( 0 E );
( D C ) ( 0 0 )
(b) reduce S(lambda) to S1(lambda) with the same finite
zeros and right Kronecker structure but with E
upper triangular and nonsingular;
(c) reduce S1(lambda) to S2(lambda) with the same finite
zeros and right Kronecker structure but with D of
full row rank;
(d) reduce S2(lambda) to S3(lambda) with the same finite zeros
and with D square invertible;
(e) perform a unitary transformation on the columns of
S3(lambda) = (A-lambda*E B) in order to reduce it to
( C D)
(Af-lambda*Ef X), with Y and Ef square invertible;
( 0 Y)
(f) compute the right and left Kronecker indices of the system
matrix, which together with the multiplicities of the
finite and infinite eigenvalues constitute the
complete set of structural invariants under strict
equivalence transformations of a linear system.
References
[1] P. Misra, P. Van Dooren and A. Varga.
Computation of structural invariants of generalized
state-space systems.
Automatica, 30, pp. 1921-1936, 1994.
Numerical Aspects
The algorithm is backward stable (see [1]).Further Comments
In order to compute the finite Smith zeros of the system explicitly, a call to this routine may be followed by a call to the LAPACK Library routines DGEGV or DGGEV.Example
Program Text
* AG08BD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER LMAX, MMAX, NMAX, PMAX
PARAMETER ( LMAX = 20, MMAX = 20, NMAX = 20, PMAX = 20 )
INTEGER LDA, LDAEMX, LDB, LDC, LDD, LDE, LDQ, LDZ
PARAMETER ( LDA = LMAX, LDB = LMAX, LDC = PMAX,
$ LDD = PMAX, LDE = LMAX, LDQ = 1, LDZ = 1,
$ LDAEMX = MAX( PMAX + LMAX, NMAX + MMAX ) )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( 4*( LMAX + NMAX ), 8*NMAX,
$ LDAEMX*LDAEMX +
$ MAX( 1, 5*LDAEMX ) ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER DINFZ, I, INFO, J, L, M, N, NFZ, NINFE, NIZ,
$ NKROL, NKROR, NRANK, P
CHARACTER*1 EQUIL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), ALFI(NMAX), ALFR(NMAX),
$ ASAVE(LDA,NMAX), B(LDB,MMAX), BETA(NMAX),
$ BSAVE(LDB,MMAX), C(LDC,NMAX), CSAVE(LDC,NMAX),
$ D(LDD,MMAX), DSAVE(LDD,MMAX), DWORK(LDWORK),
$ E(LDE,NMAX), ESAVE(LDE,NMAX), Q(LDQ,1), Z(LDZ,1)
INTEGER INFE(1+LMAX+PMAX), INFZ(NMAX+1),
$ IWORK(NMAX+MMAX), KRONL(LMAX+PMAX+1),
$ KRONR(NMAX+MMAX+1)
* .. External Subroutines ..
EXTERNAL AG08BD, DGEGV, DLACPY
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) L, N, M, P, TOL, EQUIL
IF( ( L.LT.0 .OR. L.GT.LMAX ) .OR. ( N.LT.0 .OR. N.GT.NMAX ) )
$ THEN
WRITE ( NOUT, FMT = 99972 ) L, N
ELSE
IF( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99971 ) M
ELSE
IF( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99970 ) P
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,L )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,L )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,L )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
CALL DLACPY( 'F', L, N, A, LDA, ASAVE, LDA )
CALL DLACPY( 'F', L, N, E, LDE, ESAVE, LDE )
CALL DLACPY( 'F', L, M, B, LDB, BSAVE, LDB )
CALL DLACPY( 'F', P, N, C, LDC, CSAVE, LDC )
CALL DLACPY( 'F', P, M, D, LDD, DSAVE, LDD )
* Compute poles (call the routine with M = 0, P = 0).
CALL AG08BD( EQUIL, L, N, 0, 0, A, LDA, E, LDE, B, LDB,
$ C, LDC, D, LDD, NFZ, NRANK, NIZ, DINFZ,
$ NKROR, NINFE, NKROL, INFZ, KRONR, INFE,
$ KRONL, TOL, IWORK, DWORK, LDWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99968 ) NIZ
DO 10 I = 1, DINFZ
WRITE ( NOUT, FMT = 99967 ) INFZ(I), I
10 CONTINUE
WRITE ( NOUT, FMT = 99962 ) NINFE
IF( NINFE.GT.0 ) WRITE ( NOUT, FMT = 99958 )
$ ( INFE(I), I = 1,NINFE )
IF( NFZ.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99965 )
ELSE
WRITE ( NOUT, FMT = 99966 )
WRITE ( NOUT, FMT = 99990 )
DO 20 I = 1, NFZ
WRITE ( NOUT, FMT = 99989 )
$ ( A(I,J), J = 1,NFZ )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 30 I = 1, NFZ
WRITE ( NOUT, FMT = 99989 )
$ ( E(I,J), J = 1,NFZ )
30 CONTINUE
CALL DGEGV( 'No vectors', 'No vectors', NFZ, A,
$ LDA, E, LDE, ALFR, ALFI, BETA, Q,
$ LDQ, Z, LDZ, DWORK, LDWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99996 )
DO 40 I = 1, NFZ
IF( ALFI(I).EQ.ZERO ) THEN
WRITE ( NOUT, FMT = 99980 )
$ ALFR(I)/BETA(I)
ELSE
WRITE ( NOUT, FMT = 99979 )
$ ALFR(I)/BETA(I),
$ ALFI(I)/BETA(I)
END IF
40 CONTINUE
END IF
END IF
END IF
CALL DLACPY( 'F', L, N, ASAVE, LDA, A, LDA )
CALL DLACPY( 'F', L, N, ESAVE, LDE, E, LDE )
* Check the observability and compute the ordered set of
* the observability indices (call the routine with M = 0).
CALL AG08BD( EQUIL, L, N, 0, P, A, LDA, E, LDE, B, LDB,
$ C, LDC, D, LDD, NFZ, NRANK, NIZ, DINFZ,
$ NKROR, NINFE, NKROL, INFZ, KRONR, INFE,
$ KRONL, TOL, IWORK, DWORK, LDWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99964 ) NIZ
DO 50 I = 1, DINFZ
WRITE ( NOUT, FMT = 99967 ) INFZ(I), I
50 CONTINUE
WRITE ( NOUT, FMT = 99962 ) NINFE
IF( NINFE.GT.0 ) WRITE ( NOUT, FMT = 99960 )
$ ( INFE(I), I = 1,NINFE )
WRITE ( NOUT, FMT = 99994 ) ( KRONL(I), I = 1,NKROL )
IF( NFZ+NINFE.EQ.0 ) WRITE ( NOUT, FMT = 99993 )
IF( NFZ.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99957 )
ELSE
WRITE ( NOUT, FMT = 99991 )
WRITE ( NOUT, FMT = 99990 )
DO 60 I = 1, NFZ
WRITE ( NOUT, FMT = 99989 )
$ ( A(I,J), J = 1,NFZ )
60 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 70 I = 1, NFZ
WRITE ( NOUT, FMT = 99989 )
$ ( E(I,J), J = 1,NFZ )
70 CONTINUE
CALL DGEGV( 'No vectors', 'No vectors', NFZ, A,
$ LDA, E, LDE, ALFR, ALFI, BETA, Q,
$ LDQ, Z, LDZ, DWORK, LDWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99996 )
DO 80 I = 1, NFZ
IF( ALFI(I).EQ.ZERO ) THEN
WRITE ( NOUT, FMT = 99980 )
$ ALFR(I)/BETA(I)
ELSE
WRITE ( NOUT, FMT = 99979 )
$ ALFR(I)/BETA(I),
$ ALFI(I)/BETA(I)
END IF
80 CONTINUE
END IF
END IF
END IF
CALL DLACPY( 'F', L, N, ASAVE, LDA, A, LDA )
CALL DLACPY( 'F', L, N, ESAVE, LDE, E, LDE )
CALL DLACPY( 'F', P, N, CSAVE, LDC, C, LDC )
* Check the controllability and compute the ordered set of
* the controllability indices (call the routine with P = 0)
CALL AG08BD( EQUIL, L, N, M, 0, A, LDA, E, LDE, B, LDB,
$ C, LDC, D, LDD, NFZ, NRANK, NIZ, DINFZ,
$ NKROR, NINFE, NKROL, INFZ, KRONR, INFE,
$ KRONL, TOL, IWORK, DWORK, LDWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99963 ) NIZ
DO 90 I = 1, DINFZ
WRITE ( NOUT, FMT = 99967 ) INFZ(I), I
90 CONTINUE
WRITE ( NOUT, FMT = 99962 ) NINFE
IF( NINFE.GT.0 ) WRITE ( NOUT, FMT = 99959 )
$ ( INFE(I), I = 1,NINFE )
WRITE ( NOUT, FMT = 99988 ) ( KRONR(I), I = 1,NKROR )
IF( NFZ+NINFE.EQ.0 ) WRITE ( NOUT, FMT = 99987 )
IF( NFZ.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99956 )
ELSE
WRITE ( NOUT, FMT = 99985 )
WRITE ( NOUT, FMT = 99990 )
DO 100 I = 1, NFZ
WRITE ( NOUT, FMT = 99989 )
$ ( A(I,J), J = 1,NFZ )
100 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 110 I = 1, NFZ
WRITE ( NOUT, FMT = 99989 )
$ ( E(I,J), J = 1,NFZ )
110 CONTINUE
CALL DGEGV( 'No vectors', 'No vectors', NFZ, A,
$ LDA, E, LDE, ALFR, ALFI, BETA, Q,
$ LDQ, Z, LDZ, DWORK, LDWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99982 )
DO 120 I = 1, NFZ
IF( ALFI(I).EQ.ZERO ) THEN
WRITE ( NOUT, FMT = 99980 )
$ ALFR(I)/BETA(I)
ELSE
WRITE ( NOUT, FMT = 99979 )
$ ALFR(I)/BETA(I),
$ ALFI(I)/BETA(I)
END IF
120 CONTINUE
END IF
END IF
END IF
CALL DLACPY( 'F', L, N, ASAVE, LDA, A, LDA )
CALL DLACPY( 'F', L, N, ESAVE, LDE, E, LDE )
CALL DLACPY( 'F', L, M, BSAVE, LDB, B, LDB )
CALL DLACPY( 'F', P, N, CSAVE, LDC, C, LDC )
CALL DLACPY( 'F', P, M, DSAVE, LDD, D, LDD )
* Compute the structural invariants of the given system.
CALL AG08BD( EQUIL, L, N, M, P, A, LDA, E, LDE, B, LDB,
$ C, LDC, D, LDD, NFZ, NRANK, NIZ, DINFZ,
$ NKROR, NINFE, NKROL, INFZ, KRONR, INFE,
$ KRONL, TOL, IWORK, DWORK, LDWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF( L.EQ.N ) THEN
WRITE ( NOUT, FMT = 99969 ) NRANK - N
ELSE
WRITE ( NOUT, FMT = 99955 ) NRANK
END IF
WRITE ( NOUT, FMT = 99984 ) NFZ
IF( NFZ.GT.0 ) THEN
* Compute the finite zeros of the given system.
* Workspace: need 8*NFZ.
WRITE ( NOUT, FMT = 99983 )
WRITE ( NOUT, FMT = 99990 )
DO 130 I = 1, NFZ
WRITE ( NOUT, FMT = 99989 )
$ ( A(I,J), J = 1,NFZ )
130 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 140 I = 1, NFZ
WRITE ( NOUT, FMT = 99989 )
$ ( E(I,J), J = 1,NFZ )
140 CONTINUE
CALL DGEGV( 'No vectors', 'No vectors', NFZ, A,
$ LDA, E, LDE, ALFR, ALFI, BETA, Q,
$ LDQ, Z, LDZ, DWORK, LDWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99981 )
DO 150 I = 1, NFZ
IF( ALFI(I).EQ.ZERO ) THEN
WRITE ( NOUT, FMT = 99980 )
$ ALFR(I)/BETA(I)
ELSE
WRITE ( NOUT, FMT = 99979 )
$ ALFR(I)/BETA(I),
$ ALFI(I)/BETA(I)
END IF
150 CONTINUE
END IF
END IF
WRITE ( NOUT, FMT = 99978 ) NIZ
DO 160 I = 1, DINFZ
WRITE ( NOUT, FMT = 99977 ) INFZ(I), I
160 CONTINUE
WRITE ( NOUT, FMT = 99962 ) NINFE
IF( NINFE.GT.0 ) WRITE ( NOUT, FMT = 99961 )
$ ( INFE(I), I = 1,NINFE )
WRITE ( NOUT, FMT = 99976 ) NKROR
IF( NKROR.GT.0 ) WRITE ( NOUT, FMT = 99975 )
$ ( KRONR(I), I = 1,NKROR )
WRITE ( NOUT, FMT = 99974 ) NKROL
IF( NKROL.GT.0 ) WRITE ( NOUT, FMT = 99973 )
$ ( KRONL(I), I = 1,NKROL )
END IF
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' AG08BD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AG08BD = ',I2)
99997 FORMAT (' INFO on exit from DGEGV = ',I2)
99996 FORMAT (/'Unobservable finite eigenvalues'/
$ ' real part imag part ')
99995 FORMAT (/' The matrix Ef is ')
99994 FORMAT (/' The left Kronecker indices of [A-lambda*E;C] are ',
$ /(20(I3,2X)))
99993 FORMAT (/' The system (A-lambda*E,C) is completely observable ')
99991 FORMAT (/' The finite output decoupling zeros are the eigenvalues'
$ ,' of the pair (Af,Ef). ')
99990 FORMAT (/' The matrix Af is ')
99989 FORMAT (20(1X,F8.4))
99988 FORMAT (/' The right Kronecker indices of [A-lambda*E,B] are ',
$ /( 20(I3,2X) ) )
99987 FORMAT (/' The system (A-lambda*E,B) is completely controllable ')
99985 FORMAT (/' The input decoupling zeros are the eigenvalues of the',
$ ' pair (Af,Ef). ')
99984 FORMAT (/' The number of finite zeros = ',I3)
99983 FORMAT (/' The finite zeros are the eigenvalues ',
$ 'of the pair (Af,Ef)')
99982 FORMAT (/'Uncontrollable finite eigenvalues'/
$ ' real part imag part ')
99981 FORMAT (/'Finite zeros'/' real part imag part ')
99980 FORMAT (1X,F9.4)
99979 FORMAT (1X,F9.4,6X,F9.4)
99978 FORMAT (//' The number of infinite zeros = ',I3)
99977 FORMAT ( I4,' infinite zero(s) of order ',I3)
99976 FORMAT (/' The number of right Kronecker indices = ',I3)
99975 FORMAT (/' Right Kronecker indices of [A-lambda*E,B;C,D]'
$ ,' are ', /(20(I3,2X)))
99974 FORMAT (/' The number of left Kronecker indices = ',I3)
99973 FORMAT (/' The left Kronecker indices of [A-lambda*E,B;C,D]'
$ ,' are ', /(20(I3,2X)))
99972 FORMAT (/' L or N is out of range.',/' L = ', I5, ' N = ',I5)
99971 FORMAT (/' M is out of range.',/' M = ',I5)
99970 FORMAT (/' P is out of range.',/' P = ',I5)
99969 FORMAT (/' Normal rank of transfer function matrix = ',I3)
99968 FORMAT (//' The number of infinite poles = ',I3)
99967 FORMAT ( I4,' infinite pole(s) of order ',I3)
99966 FORMAT (/' The finite poles are the eigenvalues',
$ ' of the pair (Af,Ef). ')
99965 FORMAT (/' The system has no finite poles ')
99964 FORMAT (//' The number of unobservable infinite poles = ',I3)
99963 FORMAT (//' The number of uncontrollable infinite poles = ',I3)
99962 FORMAT (/' The number of infinite Kronecker blocks = ',I3)
99961 FORMAT (/' Multiplicities of infinite eigenvalues of '
$ ,'[A-lambda*E,B;C,D] are ', /(20(I3,2X)))
99960 FORMAT (/' Multiplicities of infinite eigenvalues of '
$ ,'[A-lambda*E;C] are ', /(20(I3,2X)))
99959 FORMAT (/' Multiplicities of infinite eigenvalues of '
$ ,'[A-lambda*E,B] are ', /(20(I3,2X)))
99958 FORMAT (/' Multiplicities of infinite eigenvalues of A-lambda*E'
$ ,' are ', /(20(I3,2X)))
99957 FORMAT (/' The system (A-lambda*E,C) has no finite output',
$ ' decoupling zeros ')
99956 FORMAT (/' The system (A-lambda*E,B) has no finite input',
$ ' decoupling zeros ')
99955 FORMAT (/' Normal rank of system pencil = ',I3)
END
Program Data
AG08BD EXAMPLE PROGRAM DATA
9 9 3 3 1.e-7 N
1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1 0
-1 0 0
0 0 0
0 0 0
0 -1 0
0 0 0
0 0 0
0 0 -1
0 0 0
0 0 0
0 1 1 0 3 4 0 0 2
0 1 0 0 4 0 0 2 0
0 0 1 0 -1 4 0 -2 2
1 2 -2
0 -1 -2
0 0 0
Program Results
AG08BD EXAMPLE PROGRAM RESULTS
The number of infinite poles = 6
0 infinite pole(s) of order 1
3 infinite pole(s) of order 2
The number of infinite Kronecker blocks = 3
Multiplicities of infinite eigenvalues of A-lambda*E are
3 3 3
The system has no finite poles
The number of unobservable infinite poles = 4
0 infinite pole(s) of order 1
2 infinite pole(s) of order 2
The number of infinite Kronecker blocks = 3
Multiplicities of infinite eigenvalues of [A-lambda*E;C] are
1 3 3
The left Kronecker indices of [A-lambda*E;C] are
0 1 1
The system (A-lambda*E,C) has no finite output decoupling zeros
The number of uncontrollable infinite poles = 0
The number of infinite Kronecker blocks = 3
Multiplicities of infinite eigenvalues of [A-lambda*E,B] are
1 1 1
The right Kronecker indices of [A-lambda*E,B] are
2 2 2
The system (A-lambda*E,B) has no finite input decoupling zeros
Normal rank of transfer function matrix = 2
The number of finite zeros = 1
The finite zeros are the eigenvalues of the pair (Af,Ef)
The matrix Af is
0.7705
The matrix Ef is
0.7705
Finite zeros
real part imag part
1.0000
The number of infinite zeros = 2
0 infinite zero(s) of order 1
1 infinite zero(s) of order 2
The number of infinite Kronecker blocks = 5
Multiplicities of infinite eigenvalues of [A-lambda*E,B;C,D] are
1 1 1 1 3
The number of right Kronecker indices = 1
Right Kronecker indices of [A-lambda*E,B;C,D] are
2
The number of left Kronecker indices = 1
The left Kronecker indices of [A-lambda*E,B;C,D] are
1
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AG08BY.html 0000664 0000000 0000000 00000021644 14560147231 0020353 0 ustar 00root root 0000000 0000000
Purpose
To extract from the (N+P)-by-(M+N) descriptor system pencil
S(lambda) = ( B A - lambda*E )
( D C )
with E nonsingular and upper triangular a
(NR+PR)-by-(M+NR) "reduced" descriptor system pencil
( Br Ar-lambda*Er )
Sr(lambda) = ( )
( Dr Cr )
having the same finite Smith zeros as the pencil
S(lambda) but with Dr, a PR-by-M full row rank
left upper trapezoidal matrix, and Er, an NR-by-NR
upper triangular nonsingular matrix.
Specification
SUBROUTINE AG08BY( FIRST, N, M, P, SVLMAX, ABCD, LDABCD, E, LDE,
$ NR, PR, NINFZ, DINFZ, NKRONL, INFZ, KRONL,
$ TOL, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER DINFZ, INFO, LDABCD, LDE, LDWORK, M, N, NINFZ,
$ NKRONL, NR, P, PR
DOUBLE PRECISION SVLMAX, TOL
LOGICAL FIRST
C .. Array Arguments ..
INTEGER INFZ( * ), IWORK(*), KRONL( * )
DOUBLE PRECISION ABCD( LDABCD, * ), DWORK( * ), E( LDE, * )
Arguments
Mode Parameters
FIRST LOGICAL
Specifies if AG08BY is called first time or it is called
for an already reduced system, with D full column rank
with the last M rows in upper triangular form:
FIRST = .TRUE., first time called;
FIRST = .FALSE., not first time called.
Input/Output Parameters
N (input) INTEGER
The number of rows of matrix B, the number of columns of
matrix C and the order of square matrices A and E.
N >= 0.
M (input) INTEGER
The number of columns of matrices B and D. M >= 0.
M <= P if FIRST = .FALSE. .
P (input) INTEGER
The number of rows of matrices C and D. P >= 0.
SVLMAX (input) DOUBLE PRECISION
During each reduction step, the rank-revealing QR
factorization of a matrix stops when the estimated minimum
singular value is smaller than TOL * MAX(SVLMAX,EMSV),
where EMSV is the estimated maximum singular value.
SVLMAX >= 0.
ABCD (input/output) DOUBLE PRECISION array, dimension
(LDABCD,M+N)
On entry, the leading (N+P)-by-(M+N) part of this array
must contain the compound matrix
( B A ) ,
( D C )
where A is an N-by-N matrix, B is an N-by-M matrix,
C is a P-by-N matrix and D is a P-by-M matrix.
If FIRST = .FALSE., then D must be a full column
rank matrix with the last M rows in upper triangular form.
On exit, the leading (NR+PR)-by-(M+NR) part of ABCD
contains the reduced compound matrix
( Br Ar ) ,
( Dr Cr )
where Ar is an NR-by-NR matrix, Br is an NR-by-M matrix,
Cr is a PR-by-NR matrix, Dr is a PR-by-M full row rank
left upper trapezoidal matrix with the first PR columns
in upper triangular form.
LDABCD INTEGER
The leading dimension of array ABCD.
LDABCD >= MAX(1,N+P).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain the upper triangular nonsingular matrix E.
On exit, the leading NR-by-NR part contains the reduced
upper triangular nonsingular matrix Er.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,N).
NR (output) INTEGER
The order of the reduced matrices Ar and Er; also the
number of rows of the reduced matrix Br and the number
of columns of the reduced matrix Cr.
If Dr is invertible, NR is also the number of finite
Smith zeros.
PR (output) INTEGER
The rank of the resulting matrix Dr; also the number of
rows of reduced matrices Cr and Dr.
NINFZ (output) INTEGER
Number of infinite zeros. NINFZ = 0 if FIRST = .FALSE. .
DINFZ (output) INTEGER
The maximal multiplicity of infinite zeros.
DINFZ = 0 if FIRST = .FALSE. .
NKRONL (output) INTEGER
The maximal dimension of left elementary Kronecker blocks.
INFZ (output) INTEGER array, dimension (N)
INFZ(i) contains the number of infinite zeros of
degree i, where i = 1,2,...,DINFZ.
INFZ is not referenced if FIRST = .FALSE. .
KRONL (output) INTEGER array, dimension (N+1)
KRONL(i) contains the number of left elementary Kronecker
blocks of dimension i-by-(i-1), where i = 1,2,...,NKRONL.
Tolerances
TOL DOUBLE PRECISION
A tolerance used in rank decisions to determine the
effective rank, which is defined as the order of the
largest leading (or trailing) triangular submatrix in the
QR (or RQ) factorization with column (or row) pivoting
whose estimated condition number is less than 1/TOL.
If the user sets TOL <= 0, then an implicitly computed,
default tolerance TOLDEF = (N+P)*(N+M)*EPS, is used
instead, where EPS is the machine precision
(see LAPACK Library routine DLAMCH).
NOTE that when SVLMAX > 0, the estimated ranks could be
less than those defined above (see SVLMAX). TOL <= 1.
Workspace
IWORK INTEGER array, dimension (M)
If FIRST = .FALSE., IWORK is not referenced.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 1, if P = 0; otherwise
LDWORK >= MAX( 1, N+M-1, MIN(P,M) + MAX(3*M-1,N), 5*P ),
if FIRST = .TRUE.;
LDWORK >= MAX( 1, N+M-1, 5*P ), if FIRST = .FALSE. .
The second term is not needed if M = 0.
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The subroutine is based on the reduction algorithm of [1].References
[1] P. Misra, P. Van Dooren and A. Varga.
Computation of structural invariants of generalized
state-space systems.
Automatica, 30, pp. 1921-1936, 1994.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( (P+N)*(M+N)*N ) floating point operations.Further Comments
The number of infinite zeros is computed as
DINFZ
NINFZ = Sum (INFZ(i)*i) .
i=1
Note that each infinite zero of multiplicity k corresponds to
an infinite eigenvalue of multiplicity k+1.
The multiplicities of the infinite eigenvalues can be determined
from PR, DINFZ and INFZ(i), i = 1, ..., DINFZ, as follows:
DINFZ
- there are PR - Sum (INFZ(i)) simple infinite eigenvalues;
i=1
- there are INFZ(i) infinite eigenvalues with multiplicity i+1,
for i = 1, ..., DINFZ.
The left Kronecker indices are:
[ 0 0 ... 0 | 1 1 ... 1 | .... | NKRONL ... NKRONL ]
|<- KRONL(1) ->|<- KRONL(2) ->| |<- KRONL(NKRONL) ->|
Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To extract from the system pencil
( A-lambda*E B )
S(lambda) = ( )
( C D )
a regular pencil Af-lambda*Ef which has the finite Smith zeros of
S(lambda) as generalized eigenvalues. The routine also computes
the orders of the infinite Smith zeros and determines the singular
and infinite Kronecker structure of system pencil, i.e., the right
and left Kronecker indices, and the multiplicities of infinite
eigenvalues.
Specification
SUBROUTINE AG08BZ( EQUIL, L, N, M, P, A, LDA, E, LDE, B, LDB,
$ C, LDC, D, LDD, NFZ, NRANK, NIZ, DINFZ, NKROR,
$ NINFE, NKROL, INFZ, KRONR, INFE, KRONL,
$ TOL, IWORK, DWORK, ZWORK, LZWORK, INFO )
C .. Scalar Arguments ..
CHARACTER EQUIL
INTEGER DINFZ, INFO, L, LDA, LDB, LDC, LDD, LDE, LZWORK,
$ M, N, NFZ, NINFE, NIZ, NKROL, NKROR, NRANK, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER INFE(*), INFZ(*), IWORK(*), KRONL(*), KRONR(*)
DOUBLE PRECISION DWORK(*)
COMPLEX*16 A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ E(LDE,*), ZWORK(*)
Arguments
Mode Parameters
EQUIL CHARACTER*1
Specifies whether the user wishes to balance the system
matrix as follows:
= 'S': Perform balancing (scaling);
= 'N': Do not perform balancing.
Input/Output Parameters
L (input) INTEGER
The number of rows of matrices A, B, and E. L >= 0.
N (input) INTEGER
The number of columns of matrices A, E, and C. N >= 0.
M (input) INTEGER
The number of columns of matrix B. M >= 0.
P (input) INTEGER
The number of rows of matrix C. P >= 0.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the leading L-by-N part of this array must
contain the state dynamics matrix A of the system.
On exit, the leading NFZ-by-NFZ part of this array
contains the matrix Af of the reduced pencil.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,L).
E (input/output) COMPLEX*16 array, dimension (LDE,N)
On entry, the leading L-by-N part of this array must
contain the descriptor matrix E of the system.
On exit, the leading NFZ-by-NFZ part of this array
contains the matrix Ef of the reduced pencil.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,L).
B (input/output) COMPLEX*16 array, dimension (LDB,M)
On entry, the leading L-by-M part of this array must
contain the input/state matrix B of the system.
On exit, this matrix does not contain useful information.
LDB INTEGER
The leading dimension of array B.
LDB >= MAX(1,L) if M > 0;
LDB >= 1 if M = 0.
C (input/output) COMPLEX*16 array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C of the system.
On exit, this matrix does not contain useful information.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input) COMPLEX*16 array, dimension (LDD,M)
The leading P-by-M part of this array must contain the
direct transmission matrix D of the system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
NFZ (output) INTEGER
The number of finite zeros.
NRANK (output) INTEGER
The normal rank of the system pencil.
NIZ (output) INTEGER
The number of infinite zeros.
DINFZ (output) INTEGER
The maximal multiplicity of infinite Smith zeros.
NKROR (output) INTEGER
The number of right Kronecker indices.
NINFE (output) INTEGER
The number of elementary infinite blocks.
NKROL (output) INTEGER
The number of left Kronecker indices.
INFZ (output) INTEGER array, dimension (N+1)
The leading DINFZ elements of INFZ contain information
on the infinite elementary divisors as follows:
the system has INFZ(i) infinite elementary divisors of
degree i in the Smith form, where i = 1,2,...,DINFZ.
KRONR (output) INTEGER array, dimension (N+M+1)
The leading NKROR elements of this array contain the
right Kronecker (column) indices.
INFE (output) INTEGER array, dimension (1+MIN(L+P,N+M))
The leading NINFE elements of INFE contain the
multiplicities of infinite eigenvalues.
KRONL (output) INTEGER array, dimension (L+P+1)
The leading NKROL elements of this array contain the
left Kronecker (row) indices.
Tolerances
TOL DOUBLE PRECISION
A tolerance used in rank decisions to determine the
effective rank, which is defined as the order of the
largest leading (or trailing) triangular submatrix in the
QR (or RQ) factorization with column (or row) pivoting
whose estimated condition number is less than 1/TOL.
If the user sets TOL <= 0, then default tolerances are
used instead, as follows: TOLDEF = L*N*EPS in TG01FZ
(to determine the rank of E) and TOLDEF = (L+P)*(N+M)*EPS
in the rest, where EPS is the machine precision
(see LAPACK Library routine DLAMCH). TOL < 1.
Workspace
IWORK INTEGER array, dimension (N+max(1,M))
On output, IWORK(1) contains the normal rank of the
transfer function matrix.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK >= max(4*(L+N), 2*max(L+P,M+N))), if EQUIL = 'S',
LDWORK >= 2*max(L+P,M+N)), if EQUIL = 'N'.
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) returns the optimal value
of LZWORK.
LZWORK INTEGER
The length of the array ZWORK.
LZWORK >= max( max(L+P,M+N)*(M+N) +
max(min(L+P,M+N) + max(min(L,N),3*(M+N)-1),
3*(L+P), 1))
For optimum performance LZWORK should be larger.
If LZWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
ZWORK array, returns this value as the first entry of
the ZWORK array, and no error message related to LZWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine extracts from the system matrix of a descriptor
system (A-lambda*E,B,C,D) a regular pencil Af-lambda*Ef which
has the finite zeros of the system as generalized eigenvalues.
The procedure has the following main computational steps:
(a) construct the (L+P)-by-(N+M) system pencil
S(lambda) = ( B A )-lambda*( 0 E );
( D C ) ( 0 0 )
(b) reduce S(lambda) to S1(lambda) with the same finite
zeros and right Kronecker structure but with E
upper triangular and nonsingular;
(c) reduce S1(lambda) to S2(lambda) with the same finite
zeros and right Kronecker structure but with D of
full row rank;
(d) reduce S2(lambda) to S3(lambda) with the same finite zeros
and with D square invertible;
(e) perform a unitary transformation on the columns of
S3(lambda) = (A-lambda*E B) in order to reduce it to
( C D)
(Af-lambda*Ef X), with Y and Ef square invertible;
( 0 Y)
(f) compute the right and left Kronecker indices of the system
matrix, which together with the multiplicities of the
finite and infinite eigenvalues constitute the
complete set of structural invariants under strict
equivalence transformations of a linear system.
References
[1] P. Misra, P. Van Dooren and A. Varga.
Computation of structural invariants of generalized
state-space systems.
Automatica, 30, pp. 1921-1936, 1994.
Numerical Aspects
The algorithm is backward stable (see [1]).Further Comments
In order to compute the finite Smith zeros of the system explicitly, a call to this routine may be followed by a call to the LAPACK Library routines ZGEGV or ZGGEV.Example
Program Text
* AG08BZ EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER LMAX, MMAX, NMAX, PMAX
PARAMETER ( LMAX = 20, MMAX = 20, NMAX = 20, PMAX = 20 )
INTEGER LDA, LDAEMX, LDB, LDC, LDD, LDE, LDQ, LDZ
PARAMETER ( LDA = LMAX, LDB = LMAX, LDC = PMAX,
$ LDD = PMAX, LDE = LMAX, LDQ = 1, LDZ = 1,
$ LDAEMX = MAX( PMAX + LMAX, NMAX + MMAX ) )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( 4*( LMAX + NMAX ), 2*LDAEMX,
$ 8*NMAX ) )
INTEGER LZWORK
PARAMETER ( LZWORK = MAX( 1, LDAEMX*LDAEMX +
$ MAX( MIN( LMAX+PMAX, MMAX+NMAX )+
$ MAX( MIN( LMAX, NMAX ),
$ 3*( MMAX+NMAX )-1 ),
$ 3*( LMAX+PMAX ) ) ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER DINFZ, I, INFO, J, L, M, N, NFZ, NINFE, NIZ,
$ NKROL, NKROR, NRANK, P
CHARACTER*1 EQUIL
* .. Local Arrays ..
COMPLEX*16 A(LDA,NMAX), ALPHA(NMAX), ASAVE(LDA,NMAX),
$ B(LDB,MMAX), BETA(NMAX), BSAVE(LDB,MMAX),
$ C(LDC,NMAX), CSAVE(LDC,NMAX),
$ D(LDD,MMAX), DSAVE(LDD,MMAX),
$ E(LDE,NMAX), ESAVE(LDE,NMAX), Q(LDQ,1), Z(LDZ,1),
$ ZWORK(LZWORK)
DOUBLE PRECISION DWORK(LDWORK)
INTEGER INFE(1+LMAX+PMAX), INFZ(NMAX+1),
$ IWORK(NMAX+MMAX), KRONL(LMAX+PMAX+1),
$ KRONR(NMAX+MMAX+1)
* .. External Subroutines ..
EXTERNAL AG08BZ, ZGEGV, ZLACPY
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) L, N, M, P, TOL, EQUIL
IF( ( L.LT.0 .OR. L.GT.LMAX ) .OR. ( N.LT.0 .OR. N.GT.NMAX ) )
$ THEN
WRITE ( NOUT, FMT = 99972 ) L, N
ELSE
IF( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99971 ) M
ELSE
IF( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99970 ) P
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,L )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,L )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,L )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
CALL ZLACPY( 'F', L, N, A, LDA, ASAVE, LDA )
CALL ZLACPY( 'F', L, N, E, LDE, ESAVE, LDE )
CALL ZLACPY( 'F', L, M, B, LDB, BSAVE, LDB )
CALL ZLACPY( 'F', P, N, C, LDC, CSAVE, LDC )
CALL ZLACPY( 'F', P, M, D, LDD, DSAVE, LDD )
* Compute poles (call the routine with M = 0, P = 0).
CALL AG08BZ( EQUIL, L, N, 0, 0, A, LDA, E, LDE, B, LDB,
$ C, LDC, D, LDD, NFZ, NRANK, NIZ, DINFZ,
$ NKROR, NINFE, NKROL, INFZ, KRONR, INFE,
$ KRONL, TOL, IWORK, DWORK, ZWORK, LZWORK,
$ INFO )
*
IF( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99968 ) NIZ
DO 10 I = 1, DINFZ
WRITE ( NOUT, FMT = 99967 ) INFZ(I), I
10 CONTINUE
WRITE ( NOUT, FMT = 99962 ) NINFE
IF( NINFE.GT.0 ) WRITE ( NOUT, FMT = 99958 )
$ ( INFE(I), I = 1,NINFE )
IF( NFZ.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99965 )
ELSE
WRITE ( NOUT, FMT = 99966 )
WRITE ( NOUT, FMT = 99990 )
DO 20 I = 1, NFZ
WRITE ( NOUT, FMT = 99989 )
$ ( A(I,J), J = 1,NFZ )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 30 I = 1, NFZ
WRITE ( NOUT, FMT = 99989 )
$ ( E(I,J), J = 1,NFZ )
30 CONTINUE
CALL ZGEGV( 'No vectors', 'No vectors', NFZ, A,
$ LDA, E, LDE, ALPHA, BETA, Q, LDQ,
$ Z, LDZ, ZWORK, LZWORK, DWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99996 )
DO 40 I = 1, NFZ
WRITE ( NOUT, FMT = 99979 ) ALPHA(I)/BETA(I)
40 CONTINUE
END IF
END IF
END IF
CALL ZLACPY( 'F', L, N, ASAVE, LDA, A, LDA )
CALL ZLACPY( 'F', L, N, ESAVE, LDE, E, LDE )
* Check the observability and compute the ordered set of
* the observability indices (call the routine with M = 0).
CALL AG08BZ( EQUIL, L, N, 0, P, A, LDA, E, LDE, B, LDB,
$ C, LDC, D, LDD, NFZ, NRANK, NIZ, DINFZ,
$ NKROR, NINFE, NKROL, INFZ, KRONR, INFE,
$ KRONL, TOL, IWORK, DWORK, ZWORK, LZWORK,
$ INFO )
*
IF( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99964 ) NIZ
DO 50 I = 1, DINFZ
WRITE ( NOUT, FMT = 99967 ) INFZ(I), I
50 CONTINUE
WRITE ( NOUT, FMT = 99962 ) NINFE
IF( NINFE.GT.0 ) WRITE ( NOUT, FMT = 99960 )
$ ( INFE(I), I = 1,NINFE )
WRITE ( NOUT, FMT = 99994 ) ( KRONL(I), I = 1,NKROL )
IF( NFZ+NINFE.EQ.0 ) WRITE ( NOUT, FMT = 99993 )
IF( NFZ.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99957 )
ELSE
WRITE ( NOUT, FMT = 99991 )
WRITE ( NOUT, FMT = 99990 )
DO 60 I = 1, NFZ
WRITE ( NOUT, FMT = 99989 )
$ ( A(I,J), J = 1,NFZ )
60 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 70 I = 1, NFZ
WRITE ( NOUT, FMT = 99989 )
$ ( E(I,J), J = 1,NFZ )
70 CONTINUE
CALL ZGEGV( 'No vectors', 'No vectors', NFZ, A,
$ LDA, E, LDE, ALPHA, BETA, Q, LDQ,
$ Z, LDZ, ZWORK, LZWORK, DWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99996 )
DO 80 I = 1, NFZ
WRITE ( NOUT, FMT = 99979 ) ALPHA(I)/BETA(I)
80 CONTINUE
END IF
END IF
END IF
CALL ZLACPY( 'F', L, N, ASAVE, LDA, A, LDA )
CALL ZLACPY( 'F', L, N, ESAVE, LDE, E, LDE )
CALL ZLACPY( 'F', P, N, CSAVE, LDC, C, LDC )
* Check the controllability and compute the ordered set of
* the controllability indices (call the routine with P = 0)
CALL AG08BZ( EQUIL, L, N, M, 0, A, LDA, E, LDE, B, LDB,
$ C, LDC, D, LDD, NFZ, NRANK, NIZ, DINFZ,
$ NKROR, NINFE, NKROL, INFZ, KRONR, INFE,
$ KRONL, TOL, IWORK, DWORK, ZWORK, LZWORK,
$ INFO )
*
IF( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99963 ) NIZ
DO 90 I = 1, DINFZ
WRITE ( NOUT, FMT = 99967 ) INFZ(I), I
90 CONTINUE
WRITE ( NOUT, FMT = 99962 ) NINFE
IF( NINFE.GT.0 ) WRITE ( NOUT, FMT = 99959 )
$ ( INFE(I), I = 1,NINFE )
WRITE ( NOUT, FMT = 99988 ) ( KRONR(I), I = 1,NKROR )
IF( NFZ+NINFE.EQ.0 ) WRITE ( NOUT, FMT = 99987 )
IF( NFZ.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99956 )
ELSE
WRITE ( NOUT, FMT = 99985 )
WRITE ( NOUT, FMT = 99990 )
DO 100 I = 1, NFZ
WRITE ( NOUT, FMT = 99989 )
$ ( A(I,J), J = 1,NFZ )
100 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 110 I = 1, NFZ
WRITE ( NOUT, FMT = 99989 )
$ ( E(I,J), J = 1,NFZ )
110 CONTINUE
CALL ZGEGV( 'No vectors', 'No vectors', NFZ, A,
$ LDA, E, LDE, ALPHA, BETA, Q, LDQ,
$ Z, LDZ, ZWORK, LZWORK, DWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99982 )
DO 120 I = 1, NFZ
WRITE ( NOUT, FMT = 99979 ) ALPHA(I)/BETA(I)
120 CONTINUE
END IF
END IF
END IF
CALL ZLACPY( 'F', L, N, ASAVE, LDA, A, LDA )
CALL ZLACPY( 'F', L, N, ESAVE, LDE, E, LDE )
CALL ZLACPY( 'F', L, M, BSAVE, LDB, B, LDB )
CALL ZLACPY( 'F', P, N, CSAVE, LDC, C, LDC )
CALL ZLACPY( 'F', P, M, DSAVE, LDD, D, LDD )
* Compute the structural invariants of the given system.
CALL AG08BZ( EQUIL, L, N, M, P, A, LDA, E, LDE, B, LDB,
$ C, LDC, D, LDD, NFZ, NRANK, NIZ, DINFZ,
$ NKROR, NINFE, NKROL, INFZ, KRONR, INFE,
$ KRONL, TOL, IWORK, DWORK, ZWORK, LZWORK,
$ INFO )
*
IF( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF( L.EQ.N ) THEN
WRITE ( NOUT, FMT = 99969 ) NRANK - N
ELSE
WRITE ( NOUT, FMT = 99955 ) NRANK
END IF
WRITE ( NOUT, FMT = 99984 ) NFZ
IF( NFZ.GT.0 ) THEN
* Compute the finite zeros of the given system.
* Workspace: need 8*NFZ.
WRITE ( NOUT, FMT = 99983 )
WRITE ( NOUT, FMT = 99990 )
DO 130 I = 1, NFZ
WRITE ( NOUT, FMT = 99989 )
$ ( A(I,J), J = 1,NFZ )
130 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 140 I = 1, NFZ
WRITE ( NOUT, FMT = 99989 )
$ ( E(I,J), J = 1,NFZ )
140 CONTINUE
CALL ZGEGV( 'No vectors', 'No vectors', NFZ, A,
$ LDA, E, LDE, ALPHA, BETA, Q, LDQ,
$ Z, LDZ, ZWORK, LZWORK, DWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99981 )
DO 150 I = 1, NFZ
WRITE ( NOUT, FMT = 99979 ) ALPHA(I)/BETA(I)
150 CONTINUE
END IF
END IF
WRITE ( NOUT, FMT = 99978 ) NIZ
DO 160 I = 1, DINFZ
WRITE ( NOUT, FMT = 99977 ) INFZ(I), I
160 CONTINUE
WRITE ( NOUT, FMT = 99962 ) NINFE
IF( NINFE.GT.0 ) WRITE ( NOUT, FMT = 99961 )
$ ( INFE(I), I = 1,NINFE )
WRITE ( NOUT, FMT = 99976 ) NKROR
IF( NKROR.GT.0 ) WRITE ( NOUT, FMT = 99975 )
$ ( KRONR(I), I = 1,NKROR )
WRITE ( NOUT, FMT = 99974 ) NKROL
IF( NKROL.GT.0 ) WRITE ( NOUT, FMT = 99973 )
$ ( KRONL(I), I = 1,NKROL )
END IF
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' AG08BZ EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AG08BZ = ',I2)
99997 FORMAT (' INFO on exit from ZGEGV = ',I2)
99996 FORMAT (/' Unobservable finite eigenvalues'/
$ ' real part imag part ')
99995 FORMAT (/' The matrix Ef is ')
99994 FORMAT (/' The left Kronecker indices of [A-lambda*E;C] are ',
$ /(20(I3,2X)))
99993 FORMAT (/' The system (A-lambda*E,C) is completely observable ')
99991 FORMAT (/' The finite output decoupling zeros are the eigenvalues'
$ , ' of the pair (Af,Ef). ')
99990 FORMAT (/' The matrix Af is ')
99989 FORMAT (20(1X,F9.4,SP,F9.4,S,'i '))
99988 FORMAT (/' The right Kronecker indices of [A-lambda*E,B] are ',
$ /( 20(I3,2X) ) )
99987 FORMAT (/' The system (A-lambda*E,B) is completely controllable ')
99985 FORMAT (/' The input decoupling zeros are the eigenvalues of the',
$ ' pair (Af,Ef). ')
99984 FORMAT (/' The number of finite zeros = ',I3)
99983 FORMAT (/' The finite zeros are the eigenvalues ',
$ 'of the pair (Af,Ef)')
99982 FORMAT (/' Uncontrollable finite eigenvalues'/
$ ' real part imag part ')
99981 FORMAT (/' Finite zeros'/' real part imag part ')
99979 FORMAT (1X,F9.4,SP,F9.4,S,'i ')
99978 FORMAT (//' The number of infinite zeros = ',I3)
99977 FORMAT ( I4,' infinite zero(s) of order ',I3)
99976 FORMAT (/' The number of right Kronecker indices = ',I3)
99975 FORMAT (/' Right Kronecker indices of [A-lambda*E,B;C,D]'
$ ,' are ', /(20(I3,2X)))
99974 FORMAT (/' The number of left Kronecker indices = ',I3)
99973 FORMAT (/' The left Kronecker indices of [A-lambda*E,B;C,D]'
$ ,' are ', /(20(I3,2X)))
99972 FORMAT (/' L or N is out of range.',/' L = ', I5, ' N = ',I5)
99971 FORMAT (/' M is out of range.',/' M = ',I5)
99970 FORMAT (/' P is out of range.',/' P = ',I5)
99969 FORMAT (/' Normal rank of transfer function matrix = ',I3)
99968 FORMAT (//' The number of infinite poles = ',I3)
99967 FORMAT ( I4,' infinite pole(s) of order ',I3)
99966 FORMAT (/' The finite poles are the eigenvalues',
$ ' of the pair (Af,Ef). ')
99965 FORMAT (/' The system has no finite poles ')
99964 FORMAT (//' The number of unobservable infinite poles = ',I3)
99963 FORMAT (//' The number of uncontrollable infinite poles = ',I3)
99962 FORMAT (/' The number of infinite Kronecker blocks = ',I3)
99961 FORMAT (/' Multiplicities of infinite eigenvalues of '
$ ,'[A-lambda*E,B;C,D] are ', /(20(I3,2X)))
99960 FORMAT (/' Multiplicities of infinite eigenvalues of '
$ ,'[A-lambda*E;C] are ', /(20(I3,2X)))
99959 FORMAT (/' Multiplicities of infinite eigenvalues of '
$ ,'[A-lambda*E,B] are ', /(20(I3,2X)))
99958 FORMAT (/' Multiplicities of infinite eigenvalues of A-lambda*E'
$ ,' are ', /(20(I3,2X)))
99957 FORMAT (/' The system (A-lambda*E,C) has no finite output',
$ ' decoupling zeros ')
99956 FORMAT (/' The system (A-lambda*E,B) has no finite input',
$ ' decoupling zeros ')
99955 FORMAT (/' Normal rank of system pencil = ',I3)
END
Program Data
AG08BZ EXAMPLE PROGRAM DATA
9 9 3 3 1.e-7 N
(1,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0)
(0,0) (1,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0)
(0,0) (0,0) (1,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0)
(0,0) (0,0) (0,0) (1,0) (0,0) (0,0) (0,0) (0,0) (0,0)
(0,0) (0,0) (0,0) (0,0) (1,0) (0,0) (0,0) (0,0) (0,0)
(0,0) (0,0) (0,0) (0,0) (0,0) (1,0) (0,0) (0,0) (0,0)
(0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (1,0) (0,0) (0,0)
(0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (1,0) (0,0)
(0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (1,0)
(0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0)
(1,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0)
(0,0) (1,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0)
(0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0)
(0,0) (0,0) (0,0) (1,0) (0,0) (0,0) (0,0) (0,0) (0,0)
(0,0) (0,0) (0,0) (0,0) (1,0) (0,0) (0,0) (0,0) (0,0)
(0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0)
(0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (1,0) (0,0) (0,0)
(0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (1,0) (0,0)
(-1,0) (0,0) (0,0)
(0,0) (0,0) (0,0)
(0,0) (0,0) (0,0)
(0,0) (-1,0) (0,0)
(0,0) (0,0) (0,0)
(0,0) (0,0) (0,0)
(0,0) (0,0) (-1,0)
(0,0) (0,0) (0,0)
(0,0) (0,0) (0,0)
(0,0) (1,0) (1,0) (0,0) (3,0) (4,0) (0,0) (0,0) (2,0)
(0,0) (1,0) (0,0) (0,0) (4,0) (0,0) (0,0) (2,0) (0,0)
(0,0) (0,0) (1,0) (0,0) (-1,0) (4,0) (0,0) (-2,0) (2,0)
(1,0) (2,0) (-2,0)
(0,0) (-1,0) (-2,0)
(0,0) (0,0) (0,0)
Program Results
AG08BZ EXAMPLE PROGRAM RESULTS
The number of infinite poles = 6
0 infinite pole(s) of order 1
3 infinite pole(s) of order 2
The number of infinite Kronecker blocks = 3
Multiplicities of infinite eigenvalues of A-lambda*E are
3 3 3
The system has no finite poles
The number of unobservable infinite poles = 4
0 infinite pole(s) of order 1
2 infinite pole(s) of order 2
The number of infinite Kronecker blocks = 3
Multiplicities of infinite eigenvalues of [A-lambda*E;C] are
1 3 3
The left Kronecker indices of [A-lambda*E;C] are
0 1 1
The system (A-lambda*E,C) has no finite output decoupling zeros
The number of uncontrollable infinite poles = 0
The number of infinite Kronecker blocks = 3
Multiplicities of infinite eigenvalues of [A-lambda*E,B] are
1 1 1
The right Kronecker indices of [A-lambda*E,B] are
2 2 2
The system (A-lambda*E,B) has no finite input decoupling zeros
Normal rank of transfer function matrix = 2
The number of finite zeros = 1
The finite zeros are the eigenvalues of the pair (Af,Ef)
The matrix Af is
-0.7705 +0.0000i
The matrix Ef is
-0.7705 +0.0000i
Finite zeros
real part imag part
1.0000 +0.0000i
The number of infinite zeros = 2
0 infinite zero(s) of order 1
1 infinite zero(s) of order 2
The number of infinite Kronecker blocks = 5
Multiplicities of infinite eigenvalues of [A-lambda*E,B;C,D] are
1 1 1 1 3
The number of right Kronecker indices = 1
Right Kronecker indices of [A-lambda*E,B;C,D] are
2
The number of left Kronecker indices = 1
The left Kronecker indices of [A-lambda*E,B;C,D] are
1
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/AG8BYZ.html 0000664 0000000 0000000 00000022146 14560147231 0020423 0 ustar 00root root 0000000 0000000
Purpose
To extract from the (N+P)-by-(M+N) descriptor system pencil
S(lambda) = ( B A - lambda*E )
( D C )
with E nonsingular and upper triangular a
(NR+PR)-by-(M+NR) "reduced" descriptor system pencil
( Br Ar-lambda*Er )
Sr(lambda) = ( )
( Dr Cr )
having the same finite Smith zeros as the pencil
S(lambda) but with Dr, a PR-by-M full row rank
left upper trapezoidal matrix, and Er, an NR-by-NR
upper triangular nonsingular matrix.
Specification
SUBROUTINE AG8BYZ( FIRST, N, M, P, SVLMAX, ABCD, LDABCD, E, LDE,
$ NR, PR, NINFZ, DINFZ, NKRONL, INFZ, KRONL,
$ TOL, IWORK, DWORK, ZWORK, LZWORK, INFO )
C .. Scalar Arguments ..
INTEGER DINFZ, INFO, LDABCD, LDE, LZWORK, M, N, NINFZ,
$ NKRONL, NR, P, PR
DOUBLE PRECISION SVLMAX, TOL
LOGICAL FIRST
C .. Array Arguments ..
INTEGER INFZ( * ), IWORK(*), KRONL( * )
DOUBLE PRECISION DWORK( * )
COMPLEX*16 ABCD( LDABCD, * ), E( LDE, * ), ZWORK( * )
Arguments
Mode Parameters
FIRST LOGICAL
Specifies if AG8BYZ is called first time or it is called
for an already reduced system, with D full column rank
with the last M rows in upper triangular form:
FIRST = .TRUE., first time called;
FIRST = .FALSE., not first time called.
Input/Output Parameters
N (input) INTEGER
The number of rows of matrix B, the number of columns of
matrix C and the order of square matrices A and E.
N >= 0.
M (input) INTEGER
The number of columns of matrices B and D. M >= 0.
M <= P if FIRST = .FALSE. .
P (input) INTEGER
The number of rows of matrices C and D. P >= 0.
SVLMAX (input) DOUBLE PRECISION
During each reduction step, the rank-revealing QR
factorization of a matrix stops when the estimated minimum
singular value is smaller than TOL * MAX(SVLMAX,EMSV),
where EMSV is the estimated maximum singular value.
SVLMAX >= 0.
ABCD (input/output) COMPLEX*16 array, dimension (LDABCD,M+N)
On entry, the leading (N+P)-by-(M+N) part of this array
must contain the compound matrix
( B A ) ,
( D C )
where A is an N-by-N matrix, B is an N-by-M matrix,
C is a P-by-N matrix and D is a P-by-M matrix.
If FIRST = .FALSE., then D must be a full column
rank matrix with the last M rows in upper triangular form.
On exit, the leading (NR+PR)-by-(M+NR) part of ABCD
contains the reduced compound matrix
( Br Ar ) ,
( Dr Cr )
where Ar is an NR-by-NR matrix, Br is an NR-by-M matrix,
Cr is a PR-by-NR matrix, Dr is a PR-by-M full row rank
left upper trapezoidal matrix with the first PR columns
in upper triangular form.
LDABCD INTEGER
The leading dimension of array ABCD.
LDABCD >= MAX(1,N+P).
E (input/output) COMPLEX*16 array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain the upper triangular nonsingular matrix E.
On exit, the leading NR-by-NR part contains the reduced
upper triangular nonsingular matrix Er.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,N).
NR (output) INTEGER
The order of the reduced matrices Ar and Er; also the
number of rows of the reduced matrix Br and the number
of columns of the reduced matrix Cr.
If Dr is invertible, NR is also the number of finite
Smith zeros.
PR (output) INTEGER
The rank of the resulting matrix Dr; also the number of
rows of reduced matrices Cr and Dr.
NINFZ (output) INTEGER
Number of infinite zeros. NINFZ = 0 if FIRST = .FALSE. .
DINFZ (output) INTEGER
The maximal multiplicity of infinite zeros.
DINFZ = 0 if FIRST = .FALSE. .
NKRONL (output) INTEGER
The maximal dimension of left elementary Kronecker blocks.
INFZ (output) INTEGER array, dimension (N)
INFZ(i) contains the number of infinite zeros of
degree i, where i = 1,2,...,DINFZ.
INFZ is not referenced if FIRST = .FALSE. .
KRONL (output) INTEGER array, dimension (N+1)
KRONL(i) contains the number of left elementary Kronecker
blocks of dimension i-by-(i-1), where i = 1,2,...,NKRONL.
Tolerances
TOL DOUBLE PRECISION
A tolerance used in rank decisions to determine the
effective rank, which is defined as the order of the
largest leading (or trailing) triangular submatrix in the
QR (or RQ) factorization with column (or row) pivoting
whose estimated condition number is less than 1/TOL.
If the user sets TOL <= 0, then an implicitly computed,
default tolerance TOLDEF = (N+P)*(N+M)*EPS, is used
instead, where EPS is the machine precision
(see LAPACK Library routine DLAMCH).
NOTE that when SVLMAX > 0, the estimated ranks could be
less than those defined above (see SVLMAX). TOL <= 1.
Workspace
IWORK INTEGER array, dimension (M)
If FIRST = .FALSE., IWORK is not referenced.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK >= 2*MAX(M,P), if FIRST = .TRUE.;
LDWORK >= 2*P, if FIRST = .FALSE. .
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) returns the optimal value
of LZWORK.
LZWORK INTEGER
The length of the array ZWORK.
LZWORK >= 1, if P = 0; otherwise
LZWORK >= MAX( 1, N+M-1, MIN(P,M) + MAX(3*M-1,N), 3*P ),
if FIRST = .TRUE.;
LZWORK >= MAX( 1, N+M-1, 3*P ), if FIRST = .FALSE. .
The second term is not needed if M = 0.
For optimum performance LZWORK should be larger.
If LZWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
ZWORK array, returns this value as the first entry of
the ZWORK array, and no error message related to LZWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The subroutine is based on the reduction algorithm of [1].References
[1] P. Misra, P. Van Dooren and A. Varga.
Computation of structural invariants of generalized
state-space systems.
Automatica, 30, pp. 1921-1936, 1994.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( (P+N)*(M+N)*N ) floating point operations.Further Comments
The number of infinite zeros is computed as
DINFZ
NINFZ = Sum (INFZ(i)*i) .
i=1
Note that each infinite zero of multiplicity k corresponds to
an infinite eigenvalue of multiplicity k+1.
The multiplicities of the infinite eigenvalues can be determined
from PR, DINFZ and INFZ(i), i = 1, ..., DINFZ, as follows:
DINFZ
- there are PR - Sum (INFZ(i)) simple infinite eigenvalues;
i=1
- there are INFZ(i) infinite eigenvalues with multiplicity i+1,
for i = 1, ..., DINFZ.
The left Kronecker indices are:
[ 0 0 ... 0 | 1 1 ... 1 | .... | NKRONL ... NKRONL ]
|<- KRONL(1) ->|<- KRONL(2) ->| |<- KRONL(NKRONL) ->|
Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To generate the benchmark examples for the numerical solution of
continuous-time algebraic Riccati equations (CAREs) of the form
0 = Q + A'X + XA - XGX
corresponding to the Hamiltonian matrix
( A G )
H = ( T ).
( Q -A )
A,G,Q,X are real N-by-N matrices, Q and G are symmetric and may
be given in factored form
-1 T T
(I) G = B R B , (II) Q = C W C .
Here, C is P-by-N, W P-by-P, B N-by-M, and R M-by-M, where W
and R are symmetric. In linear-quadratic optimal control problems,
usually W is positive semidefinite and R positive definite. The
factorized form can be used if the CARE is solved using the
deflating subspaces of the extended Hamiltonian pencil
( A 0 B ) ( I 0 0 )
( T ) ( )
H - s K = ( Q A 0 ) - s ( 0 -I 0 ) ,
( T ) ( )
( 0 B R ) ( 0 0 0 )
where I and 0 denote the identity and zero matrix, respectively,
of appropriate dimensions.
NOTE: the formulation of the CARE and the related matrix (pencils)
used here does not include CAREs as they arise in robust
control (H_infinity optimization).
Specification
SUBROUTINE BB01AD(DEF, NR, DPAR, IPAR, BPAR, CHPAR, VEC, N, M, P,
1 A, LDA, B, LDB, C, LDC, G, LDG, Q, LDQ, X, LDX,
2 DWORK, LDWORK, INFO)
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDG, LDQ, LDWORK, LDX, M, N,
$ P
CHARACTER DEF
C .. Array Arguments ..
INTEGER IPAR(4), NR(2)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DPAR(*), DWORK(*),
1 G(*), Q(*), X(LDX,*)
CHARACTER CHPAR*(*)
LOGICAL BPAR(6), VEC(9)
Arguments
Mode Parameters
DEF CHARACTER
This parameter specifies if the default parameters are
to be used or not.
= 'N' or 'n' : The parameters given in the input vectors
xPAR (x = 'D', 'I', 'B', 'CH') are used.
= 'D' or 'd' : The default parameters for the example
are used.
This parameter is not meaningful if NR(1) = 1.
Input/Output Parameters
NR (input) INTEGER array, dimension (2)
This array determines the example for which CAREX returns
data. NR(1) is the group of examples.
NR(1) = 1 : parameter-free problems of fixed size.
NR(1) = 2 : parameter-dependent problems of fixed size.
NR(1) = 3 : parameter-free problems of scalable size.
NR(1) = 4 : parameter-dependent problems of scalable size.
NR(2) is the number of the example in group NR(1).
Let NEXi be the number of examples in group i. Currently,
NEX1 = 6, NEX2 = 9, NEX3 = 2, NEX4 = 4.
1 <= NR(1) <= 4;
1 <= NR(2) <= NEXi , where i = NR(1).
DPAR (input/output) DOUBLE PRECISION array, dimension (7)
Double precision parameter vector. For explanation of the
parameters see [1].
DPAR(1) : defines the parameters
'delta' for NR(1) = 3,
'q' for NR(1).NR(2) = 4.1,
'a' for NR(1).NR(2) = 4.2, and
'mu' for NR(1).NR(2) = 4.3.
DPAR(2) : defines parameters
'r' for NR(1).NR(2) = 4.1,
'b' for NR(1).NR(2) = 4.2, and
'delta' for NR(1).NR(2) = 4.3.
DPAR(3) : defines parameters
'c' for NR(1).NR(2) = 4.2 and
'kappa' for NR(1).NR(2) = 4.3.
DPAR(j), j=4,5,6,7: These arguments are only used to
generate Example 4.2 and define in
consecutive order the intervals
['beta_1', 'beta_2'],
['gamma_1', 'gamma_2'].
NOTE that if DEF = 'D' or 'd', the values of DPAR entries
on input are ignored and, on output, they are overwritten
with the default parameters.
IPAR (input/output) INTEGER array, dimension (4)
On input, IPAR(1) determines the actual state dimension,
i.e., the order of the matrix A as follows, where
NO = NR(1).NR(2).
NR(1) = 1 or 2.1-2.8: IPAR(1) is ignored.
NO = 2.9 : IPAR(1) = 1 generates the CARE for
optimal state feedback (default);
IPAR(1) = 2 generates the Kalman
filter CARE.
NO = 3.1 : IPAR(1) is the number of vehicles
(parameter 'l' in the description
in [1]).
NO = 3.2, 4.1 or 4.2: IPAR(1) is the order of the matrix
A.
NO = 4.3 or 4.4 : IPAR(1) determines the dimension of
the second-order system, i.e., the
order of the stiffness matrix for
Examples 4.3 and 4.4 (parameter 'l'
in the description in [1]).
The order of the output matrix A is N = 2*IPAR(1) for
Example 4.3 and N = 2*IPAR(1)-1 for Examples 3.1 and 4.4.
NOTE that IPAR(1) is overwritten for Examples 1.1-2.8. For
the other examples, IPAR(1) is overwritten if the default
parameters are to be used.
On output, IPAR(1) contains the order of the matrix A.
On input, IPAR(2) is the number of colums in the matrix B
in (I) (in control problems, the number of inputs of the
system). Currently, IPAR(2) is fixed or determined by
IPAR(1) for all examples and thus is not referenced on
input.
On output, IPAR(2) is the number of columns of the
matrix B from (I).
NOTE that currently IPAR(2) is overwritten and that
rank(G) <= IPAR(2).
On input, IPAR(3) is the number of rows in the matrix C
in (II) (in control problems, the number of outputs of the
system). Currently, IPAR(3) is fixed or determined by
IPAR(1) for all examples and thus is not referenced on
input.
On output, IPAR(3) contains the number of rows of the
matrix C in (II).
NOTE that currently IPAR(3) is overwritten and that
rank(Q) <= IPAR(3).
On input, if NR(1) = NR(2) = 4, and other data file than
that used by default is desired, then IPAR(4) is the
length of the character string in CHPAR specifying the
file name.
BPAR (input) BOOLEAN array, dimension (6)
This array defines the form of the output of the examples
and the storage mode of the matrices G and Q.
BPAR(1) = .TRUE. : G is returned.
BPAR(1) = .FALSE. : G is returned in factored form, i.e.,
B and R from (I) are returned.
BPAR(2) = .TRUE. : The matrix returned in array G (i.e.,
G if BPAR(1) = .TRUE. and R if
BPAR(1) = .FALSE.) is stored as full
matrix.
BPAR(2) = .FALSE. : The matrix returned in array G is
provided in packed storage mode.
BPAR(3) = .TRUE. : If BPAR(2) = .FALSE., the matrix
returned in array G is stored in upper
packed mode, i.e., the upper triangle
of a symmetric n-by-n matrix is stored
by columns, e.g., the matrix entry
G(i,j) is stored in the array entry
G(i+j*(j-1)/2) for i <= j.
Otherwise, this entry is ignored.
BPAR(3) = .FALSE. : If BPAR(2) = .FALSE., the matrix
returned in array G is stored in lower
packed mode, i.e., the lower triangle
of a symmetric n-by-n matrix is stored
by columns, e.g., the matrix entry
G(i,j) is stored in the array entry
G(i+(2*n-j)*(j-1)/2) for j <= i.
Otherwise, this entry is ignored.
BPAR(4) = .TRUE. : Q is returned.
BPAR(4) = .FALSE. : Q is returned in factored form, i.e.,
C and W from (II) are returned.
BPAR(5) = .TRUE. : The matrix returned in array Q (i.e.,
Q if BPAR(4) = .TRUE. and W if
BPAR(4) = .FALSE.) is stored as full
matrix.
BPAR(5) = .FALSE. : The matrix returned in array Q is
provided in packed storage mode.
BPAR(6) = .TRUE. : If BPAR(5) = .FALSE., the matrix
returned in array Q is stored in upper
packed mode (see above).
Otherwise, this entry is ignored.
BPAR(6) = .FALSE. : If BPAR(5) = .FALSE., the matrix
returned in array Q is stored in lower
packed mode (see above).
Otherwise, this entry is ignored.
NOTE that there are no default values for BPAR. If all
entries are declared to be .TRUE., then matrices G and Q
are returned in conventional storage mode, i.e., as
N-by-N arrays where the array element Z(I,J) contains the
matrix entry Z_{i,j}.
CHPAR (input/output) CHARACTER*255
On input, this is the name of a data file supplied by the
user.
In the current version, only Example 4.4 allows a
user-defined data file. This file must contain
consecutively DOUBLE PRECISION vectors mu, delta, gamma,
and kappa. The length of these vectors is determined by
the input value for IPAR(1).
If on entry, IPAR(1) = L, then mu and delta must each
contain L DOUBLE PRECISION values, and gamma and kappa
must each contain L-1 DOUBLE PRECISION values.
On output, this string contains short information about
the chosen example.
VEC (output) LOGICAL array, dimension (9)
Flag vector which displays the availability of the output
data:
VEC(j), j=1,2,3, refer to N, M, and P, respectively, and
are always .TRUE.
VEC(4) refers to A and is always .TRUE.
VEC(5) is .TRUE. if BPAR(1) = .FALSE., i.e., the factors B
and R from (I) are returned.
VEC(6) is .TRUE. if BPAR(4) = .FALSE., i.e., the factors C
and W from (II) are returned.
VEC(7) refers to G and is always .TRUE.
VEC(8) refers to Q and is always .TRUE.
VEC(9) refers to X and is .TRUE. if the exact solution
matrix is available.
NOTE that VEC(i) = .FALSE. for i = 1 to 9 if on exit
INFO .NE. 0.
N (output) INTEGER
The order of the matrices A, X, G if BPAR(1) = .TRUE., and
Q if BPAR(4) = .TRUE.
M (output) INTEGER
The number of columns in the matrix B (or the dimension of
the control input space of the underlying dynamical
system).
P (output) INTEGER
The number of rows in the matrix C (or the dimension of
the output space of the underlying dynamical system).
A (output) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array contains the
coefficient matrix A of the CARE.
LDA INTEGER
The leading dimension of array A. LDA >= N.
B (output) DOUBLE PRECISION array, dimension (LDB,M)
If (BPAR(1) = .FALSE.), then the leading N-by-M part of
this array contains the matrix B of the factored form (I)
of G. Otherwise, B is used as workspace.
LDB INTEGER
The leading dimension of array B. LDB >= N.
C (output) DOUBLE PRECISION array, dimension (LDC,N)
If (BPAR(4) = .FALSE.), then the leading P-by-N part of
this array contains the matrix C of the factored form (II)
of Q. Otherwise, C is used as workspace.
LDC INTEGER
The leading dimension of array C.
LDC >= P, where P is the number of rows of the matrix C,
i.e., the output value of IPAR(3). (For all examples,
P <= N, where N equals the output value of the argument
IPAR(1), i.e., LDC >= LDA is always safe.)
G (output) DOUBLE PRECISION array, dimension (NG)
If (BPAR(2) = .TRUE.) then NG = LDG*N.
If (BPAR(2) = .FALSE.) then NG = N*(N+1)/2.
If (BPAR(1) = .TRUE.), then array G contains the
coefficient matrix G of the CARE.
If (BPAR(1) = .FALSE.), then array G contains the 'control
weighting matrix' R of G's factored form as in (I). (For
all examples, M <= N.) The symmetric matrix contained in
array G is stored according to BPAR(2) and BPAR(3).
LDG INTEGER
If conventional storage mode is used for G, i.e.,
BPAR(2) = .TRUE., then G is stored like a 2-dimensional
array with leading dimension LDG. If packed symmetric
storage mode is used, then LDG is not referenced.
LDG >= N if BPAR(2) = .TRUE..
Q (output) DOUBLE PRECISION array, dimension (NQ)
If (BPAR(5) = .TRUE.) then NQ = LDQ*N.
If (BPAR(5) = .FALSE.) then NQ = N*(N+1)/2.
If (BPAR(4) = .TRUE.), then array Q contains the
coefficient matrix Q of the CARE.
If (BPAR(4) = .FALSE.), then array Q contains the 'output
weighting matrix' W of Q's factored form as in (II).
The symmetric matrix contained in array Q is stored
according to BPAR(5) and BPAR(6).
LDQ INTEGER
If conventional storage mode is used for Q, i.e.,
BPAR(5) = .TRUE., then Q is stored like a 2-dimensional
array with leading dimension LDQ. If packed symmetric
storage mode is used, then LDQ is not referenced.
LDQ >= N if BPAR(5) = .TRUE..
X (output) DOUBLE PRECISION array, dimension (LDX,IPAR(1))
If an exact solution is available (NR = 1.1, 1.2, 2.1,
2.3-2.6, 3.2), then the leading N-by-N part of this array
contains the solution matrix X in conventional storage
mode. Otherwise, X is not referenced.
LDX INTEGER
The leading dimension of array X. LDX >= 1, and
LDX >= N if NR = 1.1, 1.2, 2.1, 2.3-2.6, 3.2.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= N*MAX(4,N).
Error Indicator
INFO INTEGER
= 0 : successful exit;
< 0 : if INFO = -i, the i-th argument had an illegal
value;
= 1 : data file could not be opened or had wrong format;
= 2 : division by zero;
= 3 : G can not be computed as in (I) due to a singular R
matrix.
References
[1] Abels, J. and Benner, P.
CAREX - A Collection of Benchmark Examples for Continuous-Time
Algebraic Riccati Equations (Version 2.0).
SLICOT Working Note 1999-14, November 1999. Available from
http://www.win.tue.nl/niconet/NIC2/reports.html.
This is an updated and extended version of
[2] Benner, P., Laub, A.J., and Mehrmann, V.
A Collection of Benchmark Examples for the Numerical Solution
of Algebraic Riccati Equations I: Continuous-Time Case.
Technical Report SPC 95_22, Fak. f. Mathematik,
TU Chemnitz-Zwickau (Germany), October 1995.
Numerical Aspects
If the original data as taken from the literature is given via matrices G and Q, but factored forms are requested as output, then these factors are obtained from Cholesky or LDL' decompositions of G and Q, i.e., the output data will be corrupted by roundoff errors.Further Comments
Some benchmark examples read data from the data files provided with the collection.Example
Program Text
* BB01AD EXAMPLE PROGRAM TEXT
*
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, NMAX, PMAX
PARAMETER ( MMAX = 100, NMAX = 100, PMAX = 100 )
INTEGER LDA, LDB, LDC, LDG, LDQ, LDX
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDG = NMAX, LDQ = NMAX, LDX = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX*MAX( 4, NMAX ) )
* .. Local Scalars ..
CHARACTER DEF
INTEGER I, INFO, ISYMM, J, LBPAR, LDPAR, LIPAR, M, N, P
* .. Local Arrays ..
DOUBLE PRECISION A(LDA, NMAX), B(LDB,MMAX), C(LDC, NMAX),
$ DPAR(7), DWORK(LDWORK), G(LDG, NMAX),
$ Q(LDQ, NMAX), X(LDX, NMAX)
INTEGER IPAR(4), NR(2)
LOGICAL BPAR(6), VEC(9)
CHARACTER CHPAR*255
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL BB01AD, MA02DD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
WRITE( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ( NIN, FMT = '()' )
READ( NIN, FMT = * ) DEF
READ( NIN, FMT = * ) ( NR(I), I = 1, 2 )
IF( LSAME( DEF, 'N' ) ) THEN
READ( NIN, FMT = * ) LBPAR
IF( LBPAR.GT.0 ) READ( NIN, FMT = * ) ( BPAR(I), I = 1, LBPAR )
READ( NIN, FMT = * ) LDPAR
IF( LDPAR.GT.0 ) READ( NIN, FMT = * ) ( DPAR(I), I = 1, LDPAR )
READ( NIN, FMT = * ) LIPAR
IF( LIPAR.GT.0 ) READ( NIN, FMT = * ) ( IPAR(I), I = 1, LIPAR )
END IF
* Generate benchmark example
CALL BB01AD( DEF, NR, DPAR, IPAR, BPAR, CHPAR, VEC, N, M, P, A,
$ LDA, B, LDB, C, LDC, G, LDG, Q, LDQ, X, LDX, DWORK,
$ LDWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99998 ) INFO
ELSE
WRITE( NOUT, FMT = * ) CHPAR(1:70)
WRITE( NOUT, FMT = 99997 ) N
WRITE( NOUT, FMT = 99996 ) M
WRITE( NOUT, FMT = 99995 ) P
WRITE( NOUT, FMT = 99994 )
DO 10 I = 1, N
WRITE( NOUT, FMT = 99979 ) ( A(I,J), J = 1, N )
10 CONTINUE
IF( VEC(5) ) THEN
WRITE( NOUT, FMT = 99993 )
DO 20 I = 1, N
WRITE( NOUT, FMT = 99979 ) ( B(I,J), J = 1, M )
20 CONTINUE
ELSE
WRITE( NOUT, FMT = 99992 )
END IF
IF( VEC(6) ) THEN
WRITE( NOUT,FMT = 99991 )
DO 30 I = 1, P
WRITE( NOUT, FMT = 99979 ) ( C(I,J), J = 1, N )
30 CONTINUE
ELSE
WRITE( NOUT, FMT = 99990 )
END IF
IF( .NOT.VEC(5) ) THEN
WRITE( NOUT, FMT = 99989 )
IF( .NOT.BPAR(2) ) THEN
ISYMM = ( N * ( N + 1 ) ) / 2
CALL DCOPY( ISYMM, G, 1, DWORK, 1 )
IF( BPAR(3) ) THEN
CALL MA02DD( 'Unpack', 'Upper', N, G, LDG, DWORK )
ELSE
CALL MA02DD( 'Unpack', 'Lower', N, G, LDG, DWORK )
END IF
END IF
DO 40 I = 1, N
WRITE( NOUT, FMT = 99979 ) ( G(I,J), J = 1, N )
40 CONTINUE
ELSE
WRITE( NOUT, FMT = 99988 )
END IF
IF( .NOT.VEC(6) ) THEN
IF( .NOT. BPAR(5) ) THEN
ISYMM = ( N * ( N + 1 ) ) / 2
CALL DCOPY( ISYMM, Q, 1, DWORK, 1 )
IF( BPAR(6) ) THEN
CALL MA02DD( 'Unpack', 'Upper', N, Q, LDQ, DWORK )
ELSE
CALL MA02DD( 'Unpack', 'Lower', N, Q, LDQ, DWORK )
END IF
END IF
WRITE( NOUT, FMT = 99987 )
DO 50 I = 1, N
WRITE( NOUT, FMT = 99979 ) ( Q(I,J), J = 1, N )
50 CONTINUE
ELSE
WRITE( NOUT, FMT = 99986 )
END IF
IF( VEC(6) ) THEN
IF( .NOT.BPAR(5) ) THEN
ISYMM = ( P * ( P + 1 ) ) / 2
CALL DCOPY( ISYMM, Q, 1, DWORK, 1 )
IF( BPAR(6) ) THEN
CALL MA02DD( 'Unpack', 'Upper', P, Q, LDQ, DWORK )
ELSE
CALL MA02DD( 'Unpack', 'Lower', P, Q, LDQ, DWORK )
END IF
END IF
WRITE( NOUT, FMT = 99985 )
DO 60 I = 1, N
WRITE( NOUT, FMT = 99979 ) ( Q(I,J), J = 1, N )
60 CONTINUE
ELSE
WRITE( NOUT, FMT = 99984 )
END IF
IF( VEC(5) ) THEN
IF( .NOT.BPAR(2) ) THEN
ISYMM = ( M * ( M + 1 ) ) / 2
CALL DCOPY( ISYMM, G, 1, DWORK, 1 )
IF( BPAR(3) ) THEN
CALL MA02DD( 'Unpack', 'Upper', M, G, LDG, DWORK )
ELSE
CALL MA02DD( 'Unpack', 'Lower', M, G, LDG, DWORK )
END IF
END IF
WRITE( NOUT, FMT = 99983 )
DO 70 I = 1, N
WRITE( NOUT, FMT = 99979 ) ( G(I,J), J = 1, N )
70 CONTINUE
ELSE
WRITE( NOUT, FMT = 99982 )
END IF
IF( VEC(9) ) THEN
WRITE( NOUT, FMT = 99981 )
DO 80 I = 1, N
WRITE( NOUT, FMT = 99979 ) ( X(I,J), J = 1, N )
80 CONTINUE
ELSE
WRITE( NOUT, FMT = 99980 )
END IF
END IF
STOP
*
99999 FORMAT (' BB01AD EXAMPLE PROGRAM RESULTS', /1X)
99998 FORMAT (' INFO on exit from BB03AD = ', I3)
99997 FORMAT (/' Order of matrix A: N = ', I3)
99996 FORMAT (' Number of columns in matrix B: M = ', I3)
99995 FORMAT (' Number of rows in matrix C: P = ', I3)
99994 FORMAT (' A = ')
99993 FORMAT (' B = ')
99992 FORMAT (' B is not provided.')
99991 FORMAT (' C = ')
99990 FORMAT (' C is not provided.')
99989 FORMAT (' G = ')
99988 FORMAT (' G is not provided.')
99987 FORMAT (' Q = ')
99986 FORMAT (' Q is not provided.')
99985 FORMAT (' W = ')
99984 FORMAT (' W is not provided.')
99983 FORMAT (' R = ')
99982 FORMAT (' R is not provided.')
99981 FORMAT (' X = ')
99980 FORMAT (' X is not provided.')
99979 FORMAT (20(1X,F8.4))
*
END
Program Data
BB01AD EXAMPLE PROGRAM DATA N 2 3 6 .T. .T. .T. .F. .F. .T. 1 .1234 0Program Results
BB01AD EXAMPLE PROGRAM RESULTS Kenney/Laub/Wette 1989, Ex.2: ARE ill conditioned for EPS -> oo Order of matrix A: N = 2 Number of columns in matrix B: M = 1 Number of rows in matrix C: P = 2 A = 0.0000 0.1234 0.0000 0.0000 B is not provided. C = 1.0000 0.0000 0.0000 1.0000 G = 0.0000 0.0000 0.0000 1.0000 Q is not provided. W = 1.0000 0.0000 0.0000 1.0000 R is not provided. X = 9.0486 1.0000 1.0000 1.1166
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/BB02AD.html 0000664 0000000 0000000 00000053100 14560147231 0020303 0 ustar 00root root 0000000 0000000
Purpose
To generate the benchmark examples for the numerical solution of
discrete-time algebraic Riccati equations (DAREs) of the form
T T T -1 T T
0 = A X A - X - (A X B + S) (R + B X B) (B X A + S ) + Q
as presented in [1]. Here, A,Q,X are real N-by-N matrices, B,S are
N-by-M, and R is M-by-M. The matrices Q and R are symmetric and Q
may be given in factored form
T
(I) Q = C Q0 C .
Here, C is P-by-N and Q0 is P-by-P. If R is nonsingular and S = 0,
the DARE can be rewritten equivalently as
T -1
0 = X - A X (I_n + G X) A - Q,
where I_n is the N-by-N identity matrix and
-1 T
(II) G = B R B .
Specification
SUBROUTINE BB02AD(DEF, NR, DPAR, IPAR, BPAR, CHPAR, VEC, N, M, P,
1 A, LDA, B, LDB, C, LDC, Q, LDQ, R, LDR, S, LDS,
2 X, LDX, DWORK, LDWORK, INFO)
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDQ, LDR, LDS, LDWORK, LDX,
$ M, N, P
CHARACTER DEF
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DPAR(*), DWORK(*),
1 Q(*), R(*), S(LDS,*), X(LDX,*)
INTEGER IPAR(3), NR(2)
CHARACTER CHPAR*255
LOGICAL BPAR(7), VEC(10)
Arguments
Mode Parameters
DEF CHARACTER
This parameter specifies if the default parameters are
to be used or not.
= 'N' or 'n' : The parameters given in the input vectors
xPAR (x = 'D', 'I', 'B', 'CH') are used.
= 'D' or 'd' : The default parameters for the example
are used.
This parameter is not meaningful if NR(1) = 1.
Input/Output Parameters
NR (input) INTEGER array, dimension (2)
This array determines the example for which DAREX returns
data. NR(1) is the group of examples.
NR(1) = 1 : parameter-free problems of fixed size.
NR(1) = 2 : parameter-dependent problems of fixed size.
NR(1) = 3 : parameter-free problems of scalable size.
NR(1) = 4 : parameter-dependent problems of scalable size.
NR(2) is the number of the example in group NR(1).
Let NEXi be the number of examples in group i. Currently,
NEX1 = 13, NEX2 = 5, NEX3 = 0, NEX4 = 1.
1 <= NR(1) <= 4;
0 <= NR(2) <= NEXi, where i = NR(1).
DPAR (input/output) DOUBLE PRECISION array, dimension (4)
Double precision parameter vector. For explanation of the
parameters see [1].
DPAR(1) defines the parameter 'epsilon' for
examples NR = 2.2,2.3,2.4, the parameter 'tau'
for NR = 2.5, and the 1-by-1 matrix R for NR = 2.1,4.1.
For Example 2.5, DPAR(2) - DPAR(4) define in
consecutive order 'D', 'K', and 'r'.
NOTE that DPAR is overwritten with default values
if DEF = 'D' or 'd'.
IPAR (input/output) INTEGER array, dimension (3)
On input, IPAR(1) determines the actual state dimension,
i.e., the order of the matrix A as follows:
NR(1) = 1, NR(1) = 2 : IPAR(1) is ignored.
NR = NR(1).NR(2) = 4.1 : IPAR(1) determines the order of
the output matrix A.
NOTE that IPAR(1) is overwritten for Examples 1.1-2.3. For
the other examples, IPAR(1) is overwritten if the default
parameters are to be used.
On output, IPAR(1) contains the order of the matrix A.
On input, IPAR(2) is the number of colums in the matrix B
and the order of the matrix R (in control problems, the
number of inputs of the system). Currently, IPAR(2) is
fixed for all examples and thus is not referenced on
input.
On output, IPAR(2) is the number of columns of the
matrix B from (I).
On input, IPAR(3) is the number of rows in the matrix C
(in control problems, the number of outputs of the
system). Currently, IPAR(3) is fixed for all examples
and thus is not referenced on input.
On output, IPAR(3) is the number of rows of the matrix C
from (I).
NOTE that IPAR(2) and IPAR(3) are overwritten and
IPAR(2) <= IPAR(1) and IPAR(3) <= IPAR(1) for all
examples.
BPAR (input) LOGICAL array, dimension (7)
This array defines the form of the output of the examples
and the storage mode of the matrices Q, G or R.
BPAR(1) = .TRUE. : Q is returned.
BPAR(1) = .FALSE. : Q is returned in factored form, i.e.,
Q0 and C from (I) are returned.
BPAR(2) = .TRUE. : The matrix returned in array Q (i.e.,
Q if BPAR(1) = .TRUE. and Q0 if
BPAR(1) = .FALSE.) is stored as full
matrix.
BPAR(2) = .FALSE. : The matrix returned in array Q is
provided in packed storage mode.
BPAR(3) = .TRUE. : If BPAR(2) = .FALSE., the matrix
returned in array Q is stored in upper
packed mode, i.e., the upper triangle
of a symmetric n-by-n matrix is stored
by columns, e.g., the matrix entry
Q(i,j) is stored in the array entry
Q(i+j*(j-1)/2) for i <= j.
Otherwise, this entry is ignored.
BPAR(3) = .FALSE. : If BPAR(2) = .FALSE., the matrix
returned in array Q is stored in lower
packed mode, i.e., the lower triangle
of a symmetric n-by-n matrix is stored
by columns, e.g., the matrix entry
Q(i,j) is stored in the array entry
Q(i+(2*n-j)*(j-1)/2) for j <= i.
Otherwise, this entry is ignored.
BPAR(4) = .TRUE. : The product G in (II) is returned.
BPAR(4) = .FALSE. : G is returned in factored form, i.e.,
B and R from (II) are returned.
BPAR(5) = .TRUE. : The matrix returned in array R (i.e.,
G if BPAR(4) = .TRUE. and R if
BPAR(4) = .FALSE.) is stored as full
matrix.
BPAR(5) = .FALSE. : The matrix returned in array R is
provided in packed storage mode.
BPAR(6) = .TRUE. : If BPAR(5) = .FALSE., the matrix
returned in array R is stored in upper
packed mode (see above).
Otherwise, this entry is ignored.
BPAR(6) = .FALSE. : If BPAR(5) = .FALSE., the matrix
returned in array R is stored in lower
packed mode (see above).
Otherwise, this entry is ignored.
BPAR(7) = .TRUE. : The coefficient matrix S of the DARE
is returned in array S.
BPAR(7) = .FALSE. : The coefficient matrix S of the DARE
is not returned.
NOTE that there are no default values for BPAR. If all
entries are declared to be .TRUE., then matrices Q, G or R
are returned in conventional storage mode, i.e., as
N-by-N or M-by-M arrays where the array element Z(I,J)
contains the matrix entry Z_{i,j}.
CHPAR (output) CHARACTER*255
On output, this string contains short information about
the chosen example.
VEC (output) LOGICAL array, dimension (10)
Flag vector which displays the availability of the output
data:
VEC(j), j=1,2,3, refer to N, M, and P, respectively, and
are always .TRUE.
VEC(4) refers to A and is always .TRUE.
VEC(5) is .TRUE. if BPAR(4) = .FALSE., i.e., the factors B
and R from (II) are returned.
VEC(6) is .TRUE. if BPAR(1) = .FALSE., i.e., the factors C
and Q0 from (I) are returned.
VEC(7) refers to Q and is always .TRUE.
VEC(8) refers to R and is always .TRUE.
VEC(9) is .TRUE. if BPAR(7) = .TRUE., i.e., the matrix S
is returned.
VEC(10) refers to X and is .TRUE. if the exact solution
matrix is available.
NOTE that VEC(i) = .FALSE. for i = 1 to 10 if on exit
INFO .NE. 0.
N (output) INTEGER
The order of the matrices A, X, G if BPAR(4) = .TRUE., and
Q if BPAR(1) = .TRUE.
M (output) INTEGER
The number of columns in the matrix B (or the dimension of
the control input space of the underlying dynamical
system).
P (output) INTEGER
The number of rows in the matrix C (or the dimension of
the output space of the underlying dynamical system).
A (output) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array contains the
coefficient matrix A of the DARE.
LDA INTEGER
The leading dimension of array A. LDA >= N.
B (output) DOUBLE PRECISION array, dimension (LDB,M)
If (BPAR(4) = .FALSE.), then the leading N-by-M part
of this array contains the coefficient matrix B of
the DARE. Otherwise, B is used as workspace.
LDB INTEGER
The leading dimension of array B. LDB >= N.
C (output) DOUBLE PRECISION array, dimension (LDC,N)
If (BPAR(1) = .FALSE.), then the leading P-by-N part
of this array contains the matrix C of the factored
form (I) of Q. Otherwise, C is used as workspace.
LDC INTEGER
The leading dimension of array C. LDC >= P.
Q (output) DOUBLE PRECISION array, dimension (NQ)
If (BPAR(1) = .TRUE.) and (BPAR(2) = .TRUE.), then
NQ = LDQ*N.
IF (BPAR(1) = .TRUE.) and (BPAR(2) = .FALSE.), then
NQ = N*(N+1)/2.
If (BPAR(1) = .FALSE.) and (BPAR(2) = .TRUE.), then
NQ = LDQ*P.
IF (BPAR(1) = .FALSE.) and (BPAR(2) = .FALSE.), then
NQ = P*(P+1)/2.
The symmetric matrix contained in array Q is stored
according to BPAR(2) and BPAR(3).
LDQ INTEGER
If conventional storage mode is used for Q, i.e.,
BPAR(2) = .TRUE., then Q is stored like a 2-dimensional
array with leading dimension LDQ. If packed symmetric
storage mode is used, then LDQ is irrelevant.
LDQ >= N if BPAR(1) = .TRUE.;
LDQ >= P if BPAR(1) = .FALSE..
R (output) DOUBLE PRECISION array, dimension (MR)
If (BPAR(4) = .TRUE.) and (BPAR(5) = .TRUE.), then
MR = LDR*N.
IF (BPAR(4) = .TRUE.) and (BPAR(5) = .FALSE.), then
MR = N*(N+1)/2.
If (BPAR(4) = .FALSE.) and (BPAR(5) = .TRUE.), then
MR = LDR*M.
IF (BPAR(4) = .FALSE.) and (BPAR(5) = .FALSE.), then
MR = M*(M+1)/2.
The symmetric matrix contained in array R is stored
according to BPAR(5) and BPAR(6).
LDR INTEGER
If conventional storage mode is used for R, i.e.,
BPAR(5) = .TRUE., then R is stored like a 2-dimensional
array with leading dimension LDR. If packed symmetric
storage mode is used, then LDR is irrelevant.
LDR >= N if BPAR(4) = .TRUE.;
LDR >= M if BPAR(4) = .FALSE..
S (output) DOUBLE PRECISION array, dimension (LDS,M)
If (BPAR(7) = .TRUE.), then the leading N-by-M part of
this array contains the coefficient matrix S of the DARE.
LDS INTEGER
The leading dimension of array S. LDS >= 1, and
LDS >= N if BPAR(7) = .TRUE..
X (output) DOUBLE PRECISION array, dimension (LDX,NX)
If an exact solution is available (NR = 1.1,1.3,1.4,2.1,
2.3,2.4,2.5,4.1), then NX = N and the leading N-by-N part
of this array contains the solution matrix X.
Otherwise, X is not referenced.
LDX INTEGER
The leading dimension of array X. LDX >= 1, and
LDX >= N if an exact solution is available.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK. LDWORK >= N*N.
Error Indicator
INFO INTEGER
= 0 : successful exit;
< 0 : if INFO = -i, the i-th argument had an illegal
value;
= 1 : data file could not be opened or had wrong format;
= 2 : division by zero;
= 3 : G can not be computed as in (II) due to a singular R
matrix. This error can only occur if
BPAR(4) = .TRUE..
References
[1] Abels, J. and Benner, P.
DAREX - A Collection of Benchmark Examples for Discrete-Time
Algebraic Riccati Equations (Version 2.0).
SLICOT Working Note 1999-16, November 1999. Available from
http://www.win.tue.nl/niconet/NIC2/reports.html.
This is an updated and extended version of
[2] Benner, P., Laub, A.J., and Mehrmann, V.
A Collection of Benchmark Examples for the Numerical Solution
of Algebraic Riccati Equations II: Discrete-Time Case.
Technical Report SPC 95_23, Fak. f. Mathematik,
TU Chemnitz-Zwickau (Germany), December 1995.
Further Comments
Some benchmark examples read data from the data files provided with the collection.Example
Program Text
* BB02AD EXAMPLE PROGRAM TEXT
*
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, NMAX, PMAX
PARAMETER ( MMAX = 100, NMAX = 100, PMAX = 100 )
INTEGER LDA, LDB, LDC, LDQ, LDR, LDS, LDX
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDQ = NMAX, LDR = NMAX, LDS = NMAX,
$ LDX = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX*NMAX )
* .. Local Scalars ..
CHARACTER DEF
INTEGER I, INFO, ISYMM, J, LBPAR, LDPAR, LIPAR, M, N, P
* .. Local Arrays ..
DOUBLE PRECISION A(LDA, NMAX), B(LDB,MMAX), C(LDC, NMAX),
$ DPAR(4), DWORK(LDWORK), Q(LDQ, NMAX),
$ R(LDR, NMAX), S(LDS, NMAX), X(LDX, NMAX)
INTEGER IPAR(3), NR(2)
LOGICAL BPAR(7), VEC(10)
CHARACTER CHPAR(255)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL BB02AD, MA02DD
* .. Executable Statements ..
WRITE( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ( NIN, FMT = '()' )
READ( NIN, FMT = * ) DEF
READ( NIN, FMT = * ) ( NR(I), I = 1, 2 )
IF( LSAME( DEF, 'N' ) ) THEN
READ( NIN, FMT = * ) LBPAR
IF( LBPAR.GT.0 ) READ( NIN, FMT = * ) ( BPAR(I), I = 1, LBPAR )
READ( NIN, FMT = * ) LDPAR
IF( LDPAR.GT.0 ) READ( NIN, FMT = * ) ( DPAR(I), I = 1, LDPAR )
READ( NIN, FMT = * ) LIPAR
IF( LIPAR.GT.0 ) READ( NIN, FMT = * ) ( IPAR(I), I = 1, LIPAR )
END IF
* Generate benchmark example
CALL BB02AD( DEF, NR, DPAR, IPAR, BPAR, CHPAR, VEC, N, M, P, A,
$ LDA, B, LDB, C, LDC, Q, LDQ, R, LDR, S, LDS, X, LDX,
$ DWORK, LDWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99998 ) INFO
ELSE
WRITE( NOUT, FMT = * ) CHPAR(1:70)
WRITE( NOUT, FMT = 99997 ) N
WRITE( NOUT, FMT = 99996 ) M
WRITE( NOUT, FMT = 99995 ) P
WRITE( NOUT, FMT = 99994 )
DO 10 I = 1, N
WRITE( NOUT, FMT = 99977 ) ( A(I,J), J = 1, N )
10 CONTINUE
IF( VEC(5) ) THEN
WRITE( NOUT, FMT = 99993 )
DO 20 I = 1, N
WRITE( NOUT, FMT = 99977 ) ( B(I,J), J = 1, M )
20 CONTINUE
ELSE
WRITE( NOUT, FMT = 99992 )
END IF
IF( VEC(6) ) THEN
WRITE( NOUT,FMT = 99991 )
DO 30 I = 1, P
WRITE( NOUT, FMT = 99977 ) ( C(I,J), J = 1, N )
30 CONTINUE
ELSE
WRITE( NOUT, FMT = 99990 )
END IF
IF( .NOT.VEC(5) ) THEN
WRITE( NOUT, FMT = 99989 )
IF( .NOT.BPAR(2) ) THEN
ISYMM = ( N * ( N + 1 ) ) / 2
CALL DCOPY( ISYMM, R, 1, DWORK, 1 )
IF( BPAR(3) ) THEN
CALL MA02DD( 'Unpack', 'Upper', N, R, LDR, DWORK )
ELSE
CALL MA02DD( 'Unpack', 'Lower', N, R, LDR, DWORK )
END IF
END IF
DO 40 I = 1, N
WRITE( NOUT, FMT = 99977 ) ( R(I,J), J = 1, N )
40 CONTINUE
ELSE
WRITE( NOUT, FMT = 99988 )
END IF
IF( .NOT.VEC(6) ) THEN
IF( .NOT.BPAR(5) ) THEN
ISYMM = ( N * ( N + 1 ) ) / 2
CALL DCOPY( ISYMM, Q, 1, DWORK, 1 )
IF( BPAR(6) ) THEN
CALL MA02DD( 'Unpack', 'Upper', N, Q, LDQ, DWORK )
ELSE
CALL MA02DD( 'Unpack', 'Lower', N, Q, LDQ, DWORK )
END IF
END IF
WRITE( NOUT, FMT = 99987 )
DO 50 I = 1, N
WRITE( NOUT, FMT = 99977 ) ( Q(I,J), J = 1, N )
50 CONTINUE
ELSE
WRITE( NOUT, FMT = 99986 )
END IF
IF( VEC(6) ) THEN
IF( .NOT.BPAR(5) ) THEN
ISYMM = ( P * ( P + 1 ) ) / 2
CALL DCOPY( ISYMM, Q, 1, DWORK, 1 )
IF( BPAR(6) ) THEN
CALL MA02DD( 'Unpack', 'Upper', P, Q, LDQ, DWORK )
ELSE
CALL MA02DD( 'Unpack', 'Lower', P, Q, LDQ, DWORK )
END IF
END IF
WRITE( NOUT, FMT = 99985 )
DO 60 I = 1, P
WRITE( NOUT, FMT = 99977 ) ( Q(I,J), J = 1, P )
60 CONTINUE
ELSE
WRITE( NOUT, FMT = 99984 )
END IF
IF( VEC(5) ) THEN
IF( .NOT.BPAR(2) ) THEN
ISYMM = ( M * ( M + 1 ) ) / 2
CALL DCOPY( ISYMM, R, 1, DWORK, 1 )
IF( BPAR(3) ) THEN
CALL MA02DD( 'Unpack', 'Upper', M, R, LDR, DWORK )
ELSE
CALL MA02DD( 'Unpack', 'Lower', M, R, LDR, DWORK )
END IF
END IF
WRITE( NOUT, FMT = 99983 )
DO 70 I = 1, M
WRITE( NOUT, FMT = 99977 ) ( R(I,J), J = 1, M )
70 CONTINUE
ELSE
WRITE( NOUT, FMT = 99982 )
END IF
IF( VEC(9) ) THEN
WRITE( NOUT, FMT = 99981 )
DO 80 I = 1, N
WRITE( NOUT, FMT = 99977 ) ( S(I,J), J = 1, M )
80 CONTINUE
ELSE
WRITE( NOUT, FMT = 99980 )
END IF
IF( VEC(10) ) THEN
WRITE( NOUT, FMT = 99979 )
DO 90 I = 1, N
WRITE( NOUT, FMT = 99977 ) ( X(I,J), J = 1, N )
90 CONTINUE
ELSE
WRITE( NOUT, FMT = 99978 )
END IF
END IF
STOP
*
99999 FORMAT (' BB02AD EXAMPLE PROGRAM RESULTS', /1X)
99998 FORMAT (' INFO on exit from BB02AD = ', I3)
99997 FORMAT (/' Order of matrix A: N = ', I3)
99996 FORMAT (' Number of columns in matrix B: M = ', I3)
99995 FORMAT (' Number of rows in matrix C: P = ', I3)
99994 FORMAT (' A = ')
99993 FORMAT (' B = ')
99992 FORMAT (' B is not provided.')
99991 FORMAT (' C = ')
99990 FORMAT (' C is not provided.')
99989 FORMAT (' G = ')
99988 FORMAT (' G is not provided.')
99987 FORMAT (' Q = ')
99986 FORMAT (' Q is not provided.')
99985 FORMAT (' Q0 = ')
99984 FORMAT (' Q0 is not provided.')
99983 FORMAT (' R = ')
99982 FORMAT (' R is not provided.')
99981 FORMAT (' S = ')
99980 FORMAT (' S is not provided.')
99979 FORMAT (' X = ')
99978 FORMAT (' X is not provided.')
99977 FORMAT (20(1X,F8.4))
*
END
Program Data
BB02AD EXAMPLE PROGRAM DATA N 2 3 7 .T. .T. .T. .F. .F. .T. .T. 1 .1234 0Program Results
BB02AD EXAMPLE PROGRAM RESULTS increasingly bad scaled system as eps -> oo Order of matrix A: N = 2 Number of columns in matrix B: M = 1 Number of rows in matrix C: P = 2 A = 0.0000 0.1234 0.0000 0.0000 B = 0.0000 1.0000 C is not provided. G is not provided. Q = 1.0000 0.0000 0.0000 1.0000 Q0 is not provided. R = 1.0000 S = 0.0000 0.0000 X = 1.0000 0.0000 0.0000 1.0152
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/BB03AD.html 0000664 0000000 0000000 00000030552 14560147231 0020312 0 ustar 00root root 0000000 0000000
Purpose
To generate benchmark examples of (generalized) continuous-time
Lyapunov equations
T T
A X E + E X A = Y .
In some examples, the right hand side has the form
T
Y = - B B
and the solution can be represented as a product of Cholesky
factors
T
X = U U .
E, A, Y, X, and U are real N-by-N matrices, and B is M-by-N. Note
that E can be the identity matrix. For some examples, B, X, or U
are not provided.
This routine is an implementation of the benchmark library
CTLEX (Version 1.0) described in [1].
Specification
SUBROUTINE BB03AD(DEF, NR, DPAR, IPAR, VEC, N, M, E, LDE, A, LDA,
1 Y, LDY, B, LDB, X, LDX, U, LDU, NOTE, DWORK,
2 LDWORK, INFO)
C .. Scalar Arguments ..
CHARACTER DEF
CHARACTER*70 NOTE
INTEGER INFO, LDA, LDB, LDE, LDU, LDWORK, LDX, LDY, M, N
C .. Array Arguments ..
LOGICAL VEC(8)
INTEGER IPAR(*), NR(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), DPAR(*), DWORK(LDWORK),
1 E(LDE,*), U(LDU,*), X(LDX,*), Y(LDY,*)
Arguments
Mode Parameters
DEF CHARACTER*1
Specifies the kind of values used as parameters when
generating parameter-dependent and scalable examples
(i.e., examples with NR(1) = 2, 3, or 4):
DEF = 'D' or 'd': Default values are used.
DEF = 'N' or 'n': Values set in DPAR and IPAR are used.
This parameter is not referenced if NR(1) = 1.
Note that the scaling parameter of examples with
NR(1) = 3 or 4 is considered as a regular parameter in
this context.
Input/Output Parameters
NR (input) INTEGER array, dimension 2
Specifies the index of the desired example according
to [1].
NR(1) defines the group:
1 : parameter-free problems of fixed size
2 : parameter-dependent problems of fixed size
3 : parameter-free problems of scalable size
4 : parameter-dependent problems of scalable size
NR(2) defines the number of the benchmark example
within a certain group according to [1].
DPAR (input/output) DOUBLE PRECISION array, dimension 2
On entry, if DEF = 'N' or 'n' and the desired example
depends on real parameters, then the array DPAR must
contain the values for these parameters.
For an explanation of the parameters see [1].
For Example 4.1, DPAR(1) and DPAR(2) define 'r' and 's',
respectively.
For Example 4.2, DPAR(1) and DPAR(2) define 'lambda' and
's', respectively.
For Examples 4.3 and 4.4, DPAR(1) defines the parameter
't'.
On exit, if DEF = 'D' or 'd' and the desired example
depends on real parameters, then the array DPAR is
overwritten by the default values given in [1].
IPAR (input/output) INTEGER array of DIMENSION at least 1
On entry, if DEF = 'N' or 'n' and the desired example
depends on integer parameters, then the array IPAR must
contain the values for these parameters.
For an explanation of the parameters see [1].
For Examples 4.1, 4.2, and 4.3, IPAR(1) defines 'n'.
For Example 4.4, IPAR(1) defines 'q'.
On exit, if DEF = 'D' or 'd' and the desired example
depends on integer parameters, then the array IPAR is
overwritten by the default values given in [1].
VEC (output) LOGICAL array, dimension 8
Flag vector which displays the availability of the output
data:
VEC(1) and VEC(2) refer to N and M, respectively, and are
always .TRUE.
VEC(3) is .TRUE. iff E is NOT the identity matrix.
VEC(4) and VEC(5) refer to A and Y, respectively, and are
always .TRUE.
VEC(6) is .TRUE. iff B is provided.
VEC(7) is .TRUE. iff the solution matrix X is provided.
VEC(8) is .TRUE. iff the Cholesky factor U is provided.
N (output) INTEGER
The actual state dimension, i.e., the order of the
matrices E and A.
M (output) INTEGER
The number of rows in the matrix B. If B is not provided
for the desired example, M = 0 is returned.
E (output) DOUBLE PRECISION array, dimension (LDE,N)
The leading N-by-N part of this array contains the
matrix E.
NOTE that this array is overwritten (by the identity
matrix), if VEC(3) = .FALSE.
LDE INTEGER
The leading dimension of array E. LDE >= N.
A (output) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array contains the
matrix A.
LDA INTEGER
The leading dimension of array A. LDA >= N.
Y (output) DOUBLE PRECISION array, dimension (LDY,N)
The leading N-by-N part of this array contains the
matrix Y.
LDY INTEGER
The leading dimension of array Y. LDY >= N.
B (output) DOUBLE PRECISION array, dimension (LDB,N)
The leading M-by-N part of this array contains the
matrix B.
LDB INTEGER
The leading dimension of array B. LDB >= M.
X (output) DOUBLE PRECISION array, dimension (LDX,N)
The leading N-by-N part of this array contains the
matrix X.
LDX INTEGER
The leading dimension of array X. LDX >= N.
U (output) DOUBLE PRECISION array, dimension (LDU,N)
The leading N-by-N part of this array contains the
matrix U.
LDU INTEGER
The leading dimension of array U. LDU >= N.
NOTE (output) CHARACTER*70
String containing short information about the chosen
example.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK.
For Examples 4.1 and 4.2., LDWORK >= 2*IPAR(1) is
required.
For the other examples, no workspace is needed, i.e.,
LDWORK >= 1.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value; in particular, INFO = -3 or -4 indicates
that at least one of the parameters in DPAR or
IPAR, respectively, has an illegal value.
References
[1] D. Kressner, V. Mehrmann, and T. Penzl.
CTLEX - a Collection of Benchmark Examples for Continuous-
Time Lyapunov Equations.
SLICOT Working Note 1999-6, 1999.
Numerical Aspects
NoneFurther Comments
NoneExample
Program Text
C BB03AD EXAMPLE PROGRAM TEXT
C
C .. Parameters ..
INTEGER NIN, NOUT
PARAMETER (NIN = 5, NOUT = 6)
INTEGER NMAX, MMAX
PARAMETER (NMAX = 100, MMAX = 100)
INTEGER LDE, LDA, LDY, LDB, LDX, LDU, LDWORK
PARAMETER (LDE = NMAX, LDA = NMAX, LDY = NMAX, LDB = MMAX,
1 LDX = NMAX, LDU = NMAX, LDWORK = 2*NMAX)
C .. Local Scalars ..
CHARACTER DEF
INTEGER INFO, N, M, I, J, LDPAR, LIPAR
CHARACTER*70 NOTE
C .. Local Arrays ..
DOUBLE PRECISION E(LDE,NMAX), A(LDA, NMAX), Y(LDY, NMAX),
1 B(LDB,NMAX), X(LDX, NMAX), U(LDU, NMAX),
2 DPAR(2), DWORK(LDWORK)
INTEGER NR(2), IPAR(1)
LOGICAL VEC(8)
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL BB03AD
C .. Executable Statements ..
WRITE (NOUT, FMT = 99999)
C Skip the heading in the data file and read the data.
READ (NIN, FMT = '()')
READ (NIN, FMT = *) DEF
READ (NIN, FMT = *) (NR(I), I = 1, 2)
IF (LSAME(DEF,'N')) THEN
READ (NIN, FMT = *) LDPAR
IF (LDPAR .GT. 0) READ (NIN, FMT = *) (DPAR(I), I = 1, LDPAR)
READ (NIN, FMT = *) LIPAR
IF (LIPAR .GT. 0) READ (NIN, FMT = *) (IPAR(I), I = 1, LIPAR)
END IF
C Generate benchmark example
CALL BB03AD(DEF, NR, DPAR, IPAR, VEC, N, M, E, LDE, A, LDA, Y,
1 LDY, B, LDB, X, LDX, U, LDU, NOTE, DWORK, LDWORK,
2 INFO)
C
IF (INFO .NE. 0) THEN
WRITE (NOUT, FMT = 99998) INFO
ELSE
WRITE (NOUT, FMT = *) NOTE
WRITE (NOUT, FMT = 99997) N
WRITE (NOUT, FMT = 99996) M
IF (VEC(3)) THEN
WRITE (NOUT, FMT = 99995)
DO 10 I = 1, N
WRITE (NOUT, FMT = 99985) (E(I,J), J = 1, N)
10 CONTINUE
ELSE
WRITE (NOUT, FMT = 99994)
END IF
WRITE (NOUT,FMT = 99993)
DO 20 I = 1, N
WRITE (NOUT, FMT = 99985) (A(I,J), J = 1, N)
20 CONTINUE
IF (VEC(6)) THEN
WRITE (NOUT,FMT = 99992)
DO 30 I = 1, M
WRITE (NOUT, FMT = 99985) (B(I,J), J = 1, N)
30 CONTINUE
ELSE
WRITE (NOUT, FMT = 99991)
END IF
WRITE (NOUT,FMT = 99990)
DO 40 I = 1, N
WRITE (NOUT, FMT = 99985) (Y(I,J), J = 1, N)
40 CONTINUE
IF (VEC(7)) THEN
WRITE (NOUT, FMT = 99989)
DO 50 I = 1, N
WRITE (NOUT, FMT = 99985) (X(I,J), J = 1, N)
50 CONTINUE
ELSE
WRITE (NOUT, FMT = 99988)
END IF
IF (VEC(8)) THEN
WRITE (NOUT, FMT = 99987)
DO 60 I = 1, N
WRITE (NOUT, FMT = 99985) (U(I,J), J = 1, N)
60 CONTINUE
ELSE
WRITE (NOUT, FMT = 99986)
END IF
END IF
C
99999 FORMAT (' BB03AD EXAMPLE PROGRAM RESULTS', /1X)
99998 FORMAT (' INFO on exit from BB03AD = ', I3)
99997 FORMAT (/' Order of matrix A: N = ', I3)
99996 FORMAT (' Number of rows in matrix B: M = ', I3)
99995 FORMAT (/' E = ')
99994 FORMAT (/' E is the identity matrix.')
99993 FORMAT (' A = ')
99992 FORMAT (' B = ')
99991 FORMAT (' B is not provided.')
99990 FORMAT (' Y = ')
99989 FORMAT (' X = ')
99988 FORMAT (' X is not provided.')
99987 FORMAT (' U = ')
99986 FORMAT (' U is not provided.')
99985 FORMAT (20(1X,F8.4))
C
END
Program Data
BB03AD EXAMPLE PROGRAM DATA N 4 1 2 .15D1 .15D1 1 5Program Results
BB03AD EXAMPLE PROGRAM RESULTS CTLEX: Example 4.1 Order of matrix A: N = 5 Number of rows in matrix B: M = 1 E is the identity matrix. A = -3.6360 -0.6921 -1.1933 -0.8137 0.3507 0.1406 -2.9375 0.9063 0.1562 0.3438 -2.5735 -1.4421 -2.8183 -1.1887 1.2257 -0.3779 0.0810 0.5544 -1.5891 0.0660 0.8961 1.1586 1.6279 0.5631 -2.2066 B = -3.6914 -3.9753 -0.0247 -1.9012 1.1111 Y = -13.6261 -14.6743 -0.0911 -7.0181 4.1015 -14.6743 -15.8031 -0.0982 -7.5580 4.4170 -0.0911 -0.0982 -0.0006 -0.0469 0.0274 -7.0181 -7.5580 -0.0469 -3.6147 2.1125 4.1015 4.4170 0.0274 2.1125 -1.2346 X = 1.7737 1.9307 -0.0703 1.0497 -0.4681 1.9307 2.1036 -0.0752 1.1489 -0.5069 -0.0703 -0.0752 0.0076 -0.0428 0.0178 1.0497 1.1489 -0.0428 0.6509 -0.2651 -0.4681 -0.5069 0.0178 -0.2651 0.1284 U is not provided.
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/BB04AD.html 0000664 0000000 0000000 00000030544 14560147231 0020314 0 ustar 00root root 0000000 0000000
Purpose
To generate benchmark examples of (generalized) discrete-time
Lyapunov equations
T T
A X A - E X E = Y .
In some examples, the right hand side has the form
T
Y = - B B
and the solution can be represented as a product of Cholesky
factors
T
X = U U .
E, A, Y, X, and U are real N-by-N matrices, and B is M-by-N. Note
that E can be the identity matrix. For some examples, B, X, or U
are not provided.
This routine is an implementation of the benchmark library
DTLEX (Version 1.0) described in [1].
Specification
SUBROUTINE BB04AD(DEF, NR, DPAR, IPAR, VEC, N, M, E, LDE, A, LDA,
1 Y, LDY, B, LDB, X, LDX, U, LDU, NOTE, DWORK,
2 LDWORK, INFO)
C .. Scalar Arguments ..
CHARACTER DEF
CHARACTER*70 NOTE
INTEGER INFO, LDA, LDB, LDE, LDU, LDWORK, LDX, LDY, M, N
C .. Array Arguments ..
LOGICAL VEC(8)
INTEGER IPAR(*), NR(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), DPAR(*), DWORK(LDWORK),
1 E(LDE,*), U(LDU,*), X(LDX,*), Y(LDY,*)
Arguments
Mode Parameters
DEF CHARACTER*1
Specifies the kind of values used as parameters when
generating parameter-dependent and scalable examples
(i.e., examples with NR(1) = 2, 3, or 4):
DEF = 'D' or 'd': Default values are used.
DEF = 'N' or 'n': Values set in DPAR and IPAR are used.
This parameter is not referenced if NR(1) = 1.
Note that the scaling parameter of examples with
NR(1) = 3 or 4 is considered as a regular parameter in
this context.
Input/Output Parameters
NR (input) INTEGER array, dimension 2
Specifies the index of the desired example according
to [1].
NR(1) defines the group:
1 : parameter-free problems of fixed size
2 : parameter-dependent problems of fixed size
3 : parameter-free problems of scalable size
4 : parameter-dependent problems of scalable size
NR(2) defines the number of the benchmark example
within a certain group according to [1].
DPAR (input/output) DOUBLE PRECISION array, dimension 2
On entry, if DEF = 'N' or 'n' and the desired example
depends on real parameters, then the array DPAR must
contain the values for these parameters.
For an explanation of the parameters see [1].
For Example 4.1, DPAR(1) and DPAR(2) define 'r' and 's',
respectively.
For Example 4.2, DPAR(1) and DPAR(2) define 'lambda' and
's', respectively.
For Examples 4.3 and 4.4, DPAR(1) defines the parameter
't'.
On exit, if DEF = 'D' or 'd' and the desired example
depends on real parameters, then the array DPAR is
overwritten by the default values given in [1].
IPAR (input/output) INTEGER array of DIMENSION at least 1
On entry, if DEF = 'N' or 'n' and the desired example
depends on integer parameters, then the array IPAR must
contain the values for these parameters.
For an explanation of the parameters see [1].
For Examples 4.1, 4.2, and 4.3, IPAR(1) defines 'n'.
For Example 4.4, IPAR(1) defines 'q'.
On exit, if DEF = 'D' or 'd' and the desired example
depends on integer parameters, then the array IPAR is
overwritten by the default values given in [1].
VEC (output) LOGICAL array, dimension 8
Flag vector which displays the availability of the output
data:
VEC(1) and VEC(2) refer to N and M, respectively, and are
always .TRUE.
VEC(3) is .TRUE. iff E is NOT the identity matrix.
VEC(4) and VEC(5) refer to A and Y, respectively, and are
always .TRUE.
VEC(6) is .TRUE. iff B is provided.
VEC(7) is .TRUE. iff the solution matrix X is provided.
VEC(8) is .TRUE. iff the Cholesky factor U is provided.
N (output) INTEGER
The actual state dimension, i.e., the order of the
matrices E and A.
M (output) INTEGER
The number of rows in the matrix B. If B is not provided
for the desired example, M = 0 is returned.
E (output) DOUBLE PRECISION array, dimension (LDE,N)
The leading N-by-N part of this array contains the
matrix E.
NOTE that this array is overwritten (by the identity
matrix), if VEC(3) = .FALSE.
LDE INTEGER
The leading dimension of array E. LDE >= N.
A (output) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array contains the
matrix A.
LDA INTEGER
The leading dimension of array A. LDA >= N.
Y (output) DOUBLE PRECISION array, dimension (LDY,N)
The leading N-by-N part of this array contains the
matrix Y.
LDY INTEGER
The leading dimension of array Y. LDY >= N.
B (output) DOUBLE PRECISION array, dimension (LDB,N)
The leading M-by-N part of this array contains the
matrix B.
LDB INTEGER
The leading dimension of array B. LDB >= M.
X (output) DOUBLE PRECISION array, dimension (LDX,N)
The leading N-by-N part of this array contains the
matrix X.
LDX INTEGER
The leading dimension of array X. LDX >= N.
U (output) DOUBLE PRECISION array, dimension (LDU,N)
The leading N-by-N part of this array contains the
matrix U.
LDU INTEGER
The leading dimension of array U. LDU >= N.
NOTE (output) CHARACTER*70
String containing short information about the chosen
example.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK.
For Examples 4.1 and 4.2., LDWORK >= 2*IPAR(1) is
required.
For the other examples, no workspace is needed, i.e.,
LDWORK >= 1.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value; in particular, INFO = -3 or -4 indicates
that at least one of the parameters in DPAR or
IPAR, respectively, has an illegal value.
References
[1] D. Kressner, V. Mehrmann, and T. Penzl.
DTLEX - a Collection of Benchmark Examples for Discrete-
Time Lyapunov Equations.
SLICOT Working Note 1999-7, 1999.
Numerical Aspects
NoneFurther Comments
NoneExample
Program Text
C BB04AD EXAMPLE PROGRAM TEXT
C
C .. Parameters ..
INTEGER NIN, NOUT
PARAMETER (NIN = 5, NOUT = 6)
INTEGER NMAX, MMAX
PARAMETER (NMAX = 100, MMAX = 100)
INTEGER LDE, LDA, LDY, LDB, LDX, LDU, LDWORK
PARAMETER (LDE = NMAX, LDA = NMAX, LDY = NMAX, LDB = MMAX,
1 LDX = NMAX, LDU = NMAX, LDWORK = 2*NMAX)
C .. Local Scalars ..
CHARACTER DEF
INTEGER INFO, N, M, I, J, LDPAR, LIPAR
CHARACTER*70 NOTE
C .. Local Arrays ..
DOUBLE PRECISION E(LDE,NMAX), A(LDA, NMAX), Y(LDY, NMAX),
1 B(LDB,NMAX), X(LDX, NMAX), U(LDU, NMAX),
2 DPAR(2), DWORK(LDWORK)
INTEGER NR(2), IPAR(1)
LOGICAL VEC(8)
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL BB04AD
C .. Executable Statements ..
WRITE (NOUT, FMT = 99999)
C Skip the heading in the data file and read the data.
READ (NIN, FMT = '()')
READ (NIN, FMT = *) DEF
READ (NIN, FMT = *) (NR(I), I = 1, 2)
IF (LSAME(DEF,'N')) THEN
READ (NIN, FMT = *) LDPAR
IF (LDPAR .GT. 0) READ (NIN, FMT = *) (DPAR(I), I = 1, LDPAR)
READ (NIN, FMT = *) LIPAR
IF (LIPAR .GT. 0) READ (NIN, FMT = *) (IPAR(I), I = 1, LIPAR)
END IF
C Generate benchmark example
CALL BB04AD(DEF, NR, DPAR, IPAR, VEC, N, M, E, LDE, A, LDA, Y,
1 LDY, B, LDB, X, LDX, U, LDU, NOTE, DWORK, LDWORK,
2 INFO)
C
IF (INFO .NE. 0) THEN
WRITE (NOUT, FMT = 99998) INFO
ELSE
WRITE (NOUT, FMT = *) NOTE
WRITE (NOUT, FMT = 99997) N
WRITE (NOUT, FMT = 99996) M
IF (VEC(3)) THEN
WRITE (NOUT, FMT = 99995)
DO 10 I = 1, N
WRITE (NOUT, FMT = 99985) (E(I,J), J = 1, N)
10 CONTINUE
ELSE
WRITE (NOUT, FMT = 99994)
END IF
WRITE (NOUT,FMT = 99993)
DO 20 I = 1, N
WRITE (NOUT, FMT = 99985) (A(I,J), J = 1, N)
20 CONTINUE
IF (VEC(6)) THEN
WRITE (NOUT,FMT = 99992)
DO 30 I = 1, M
WRITE (NOUT, FMT = 99985) (B(I,J), J = 1, N)
30 CONTINUE
ELSE
WRITE (NOUT, FMT = 99991)
END IF
WRITE (NOUT,FMT = 99990)
DO 40 I = 1, N
WRITE (NOUT, FMT = 99985) (Y(I,J), J = 1, N)
40 CONTINUE
IF (VEC(7)) THEN
WRITE (NOUT, FMT = 99989)
DO 50 I = 1, N
WRITE (NOUT, FMT = 99985) (X(I,J), J = 1, N)
50 CONTINUE
ELSE
WRITE (NOUT, FMT = 99988)
END IF
IF (VEC(8)) THEN
WRITE (NOUT, FMT = 99987)
DO 60 I = 1, N
WRITE (NOUT, FMT = 99985) (U(I,J), J = 1, N)
60 CONTINUE
ELSE
WRITE (NOUT, FMT = 99986)
END IF
END IF
C
99999 FORMAT (' BB04AD EXAMPLE PROGRAM RESULTS', /1X)
99998 FORMAT (' INFO on exit from BB04AD = ', I3)
99997 FORMAT (/' Order of matrix A: N = ', I3)
99996 FORMAT (' Number of rows in matrix B: M = ', I3)
99995 FORMAT (/' E = ')
99994 FORMAT (/' E is the identity matrix.')
99993 FORMAT (' A = ')
99992 FORMAT (' B = ')
99991 FORMAT (' B is not provided.')
99990 FORMAT (' Y = ')
99989 FORMAT (' X = ')
99988 FORMAT (' X is not provided.')
99987 FORMAT (' U = ')
99986 FORMAT (' U is not provided.')
99985 FORMAT (20(1X,F8.4))
C
END
Program Data
BB04AD EXAMPLE PROGRAM DATA N 4 1 2 .15D1 .15D1 1 5Program Results
BB04AD EXAMPLE PROGRAM RESULTS DTLEX: Example 4.1 Order of matrix A: N = 5 Number of rows in matrix B: M = 1 E is the identity matrix. A = 0.4562 0.0308 0.1990 0.0861 0.0217 0.0637 0.5142 -0.1828 0.0096 -0.1148 0.3139 0.1287 0.3484 0.1653 -0.1975 0.1500 0.0053 -0.1838 0.2501 -0.0687 0.0568 -0.1006 -0.3735 -0.0202 0.2285 B = 0.3086 0.0247 -0.4691 0.1728 -0.3704 Y = -0.0953 -0.0076 0.1448 -0.0533 0.1143 -0.0076 -0.0006 0.0116 -0.0043 0.0091 0.1448 0.0116 -0.2201 0.0811 -0.1738 -0.0533 -0.0043 0.0811 -0.0299 0.0640 0.1143 0.0091 -0.1738 0.0640 -0.1372 X = 0.0953 0.0076 -0.1448 0.0533 -0.1143 0.0076 0.0006 -0.0116 0.0043 -0.0091 -0.1448 -0.0116 0.2201 -0.0811 0.1738 0.0533 0.0043 -0.0811 0.0299 -0.0640 -0.1143 -0.0091 0.1738 -0.0640 0.1372 U is not provided.
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/BD01AD.html 0000664 0000000 0000000 00000027015 14560147231 0020312 0 ustar 00root root 0000000 0000000
Purpose
To generate benchmark examples for time-invariant,
continuous-time dynamical systems
.
E x(t) = A x(t) + B u(t)
y(t) = C x(t) + D u(t)
E, A are real N-by-N matrices, B is N-by-M, C is P-by-N, and
D is P-by-M. In many examples, E is the identity matrix and D is
the zero matrix.
This routine is an implementation of the benchmark library
CTDSX (Version 1.0) described in [1].
Specification
SUBROUTINE BD01AD( DEF, NR, DPAR, IPAR, VEC, N, M, P, E, LDE, A,
1 LDA, B, LDB, C, LDC, D, LDD, NOTE, DWORK,
2 LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DEF
CHARACTER*70 NOTE
INTEGER INFO, LDA, LDB, LDC, LDD, LDE, LDWORK, M, N, P
C .. Array Arguments ..
LOGICAL VEC(8)
INTEGER IPAR(*), NR(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), DPAR(*),
1 DWORK(*), E(LDE,*)
Arguments
Mode Parameters
DEF CHARACTER*1
Specifies the kind of values used as parameters when
generating parameter-dependent and scalable examples
(i.e., examples with NR(1) = 2, 3, or 4):
= 'D': Default values defined in [1] are used;
= 'N': Values set in DPAR and IPAR are used.
This parameter is not referenced if NR(1) = 1.
Note that the scaling parameter of examples with
NR(1) = 3 or 4 is considered as a regular parameter in
this context.
Input/Output Parameters
NR (input) INTEGER array, dimension (2)
Specifies the index of the desired example according
to [1].
NR(1) defines the group:
1 : parameter-free problems of fixed size
2 : parameter-dependent problems of fixed size
3 : parameter-free problems of scalable size
4 : parameter-dependent problems of scalable size
NR(2) defines the number of the benchmark example
within a certain group according to [1].
DPAR (input/output) DOUBLE PRECISION array, dimension (7)
On entry, if DEF = 'N' and the desired example depends on
real parameters, then the array DPAR must contain the
values for these parameters.
For an explanation of the parameters see [1].
For Examples 2.1 and 2.2, DPAR(1) defines the parameter
'epsilon'.
For Example 2.4, DPAR(1), ..., DPAR(7) define 'b', 'mu',
'r', 'r_c', 'k_l', 'sigma', 'a', respectively.
For Example 2.7, DPAR(1) and DPAR(2) define 'mu' and 'nu',
respectively.
For Example 4.1, DPAR(1), ..., DPAR(7) define 'a', 'b',
'c', 'beta_1', 'beta_2', 'gamma_1', 'gamma_2',
respectively.
For Example 4.2, DPAR(1), ..., DPAR(3) define 'mu',
'delta', 'kappa', respectively.
On exit, if DEF = 'D' and the desired example depends on
real parameters, then the array DPAR is overwritten by the
default values given in [1].
IPAR (input/output) INTEGER array, dimension (1)
On entry, if DEF = 'N' and the desired example depends on
integer parameters, then the array IPAR must contain the
values for these parameters.
For an explanation of the parameters see [1].
For Examples 2.3, 2.5, and 2.6, IPAR(1) defines the
parameter 's'.
For Example 3.1, IPAR(1) defines 'q'.
For Examples 3.2 and 3.3, IPAR(1) defines 'n'.
For Example 3.4, IPAR(1) defines 'l'.
For Example 4.1, IPAR(1) defines 'n'.
For Example 4.2, IPAR(1) defines 'l'.
On exit, if DEF = 'D' and the desired example depends on
integer parameters, then the array IPAR is overwritten by
the default values given in [1].
VEC (output) LOGICAL array, dimension (8)
Flag vector which displays the availabilty of the output
data:
VEC(1), ..., VEC(3) refer to N, M, and P, respectively,
and are always .TRUE..
VEC(4) is .TRUE. iff E is NOT the identity matrix.
VEC(5), ..., VEC(7) refer to A, B, and C, respectively,
and are always .TRUE..
VEC(8) is .TRUE. iff D is NOT the zero matrix.
N (output) INTEGER
The actual state dimension, i.e., the order of the
matrices E and A.
M (output) INTEGER
The number of columns in the matrices B and D.
P (output) INTEGER
The number of rows in the matrices C and D.
E (output) DOUBLE PRECISION array, dimension (LDE,N)
The leading N-by-N part of this array contains the
matrix E.
NOTE that this array is overwritten (by the identity
matrix), if VEC(4) = .FALSE..
LDE INTEGER
The leading dimension of array E. LDE >= N.
A (output) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array contains the
matrix A.
LDA INTEGER
The leading dimension of array A. LDA >= N.
B (output) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array contains the
matrix B.
LDB INTEGER
The leading dimension of array B. LDB >= N.
C (output) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array contains the
matrix C.
LDC INTEGER
The leading dimension of array C. LDC >= P.
D (output) DOUBLE PRECISION array, dimension (LDD,M)
The leading P-by-M part of this array contains the
matrix D.
NOTE that this array is overwritten (by the zero
matrix), if VEC(8) = .FALSE..
LDD INTEGER
The leading dimension of array D. LDD >= P.
NOTE (output) CHARACTER*70
String containing short information about the chosen
example.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK.
For Example 3.4, LDWORK >= 4*IPAR(1) is required.
For the other examples, no workspace is needed, i.e.,
LDWORK >= 1.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value; in particular, INFO = -3 or -4 indicates
that at least one of the parameters in DPAR or
IPAR, respectively, has an illegal value;
= 1: data file can not be opened or has wrong format.
References
[1] Kressner, D., Mehrmann, V. and Penzl, T.
CTDSX - a Collection of Benchmark Examples for State-Space
Realizations of Continuous-Time Dynamical Systems.
SLICOT Working Note 1998-9. 1998.
Numerical Aspects
NoneFurther Comments
NoneExample
Program Text
C BD01AD EXAMPLE PROGRAM TEXT
C
C .. Parameters ..
INTEGER NIN, NOUT
PARAMETER (NIN = 5, NOUT = 6)
INTEGER NMAX, MMAX, PMAX
PARAMETER (NMAX = 421, MMAX = 211, PMAX = 211)
INTEGER LDA, LDB, LDC, LDD, LDE, LDWORK
PARAMETER (LDA = NMAX, LDB = NMAX, LDC = PMAX, LDD = PMAX,
1 LDE = NMAX, LDWORK = 4*NMAX)
C .. Local Scalars ..
CHARACTER DEF
INTEGER I, INFO, J, LDPAR, LIPAR, M, N, P
CHARACTER*70 NOTE
C .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
1 D(LDD,MMAX), DPAR(7), DWORK(LDWORK), E(LDE,NMAX)
INTEGER NR(2), IPAR(7)
LOGICAL VEC(8)
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL BD01AD
C .. Executable Statements ..
WRITE (NOUT, FMT = 99999)
C Skip the heading in the data file and read the data.
READ (NIN, FMT = '()')
READ (NIN, FMT = *) DEF
READ (NIN, FMT = *) (NR(I), I = 1, 2)
IF (LSAME(DEF,'N')) THEN
READ (NIN, FMT = *) LDPAR
IF (LDPAR .GT. 0) READ (NIN, FMT = *) (DPAR(I), I = 1, LDPAR)
READ (NIN, FMT = *) LIPAR
IF (LIPAR .GT. 0) READ (NIN, FMT = *) (IPAR(I), I = 1, LIPAR)
END IF
C Generate benchmark example
CALL BD01AD(DEF, NR, DPAR, IPAR, VEC, N, M, P, E, LDE, A, LDA,
1 B, LDB, C, LDC, D, LDD, NOTE, DWORK, LDWORK, INFO)
C
IF (INFO .NE. 0) THEN
WRITE (NOUT, FMT = 99998) INFO
ELSE
WRITE (NOUT, FMT = *) NOTE
WRITE (NOUT, FMT = 99997) N
WRITE (NOUT, FMT = 99996) M
WRITE (NOUT, FMT = 99995) P
IF (VEC(4)) THEN
WRITE (NOUT, FMT = 99994)
DO 10 I = 1, N
WRITE (NOUT, FMT = 99987) (E(I,J), J = 1, N)
10 CONTINUE
ELSE
WRITE (NOUT, FMT = 99993)
END IF
WRITE (NOUT,FMT = 99992)
DO 20 I = 1, N
WRITE (NOUT, FMT = 99987) (A(I,J), J = 1, N)
20 CONTINUE
WRITE (NOUT,FMT = 99991)
DO 30 I = 1, N
WRITE (NOUT, FMT = 99987) (B(I,J), J = 1, M)
30 CONTINUE
WRITE (NOUT,FMT = 99990)
DO 40 I = 1, P
WRITE (NOUT, FMT = 99987) (C(I,J), J = 1, N)
40 CONTINUE
IF (VEC(8)) THEN
WRITE (NOUT,FMT = 99989)
DO 50 I = 1, P
WRITE (NOUT, FMT = 99987) (D(I,J), J = 1, M)
50 CONTINUE
ELSE
WRITE (NOUT, FMT = 99988)
END IF
END IF
C
99999 FORMAT (' BD01AD EXAMPLE PROGRAM RESULTS', /1X)
99998 FORMAT (' INFO on exit from BD01AD = ', I3)
99997 FORMAT (/' Order of matrix A: N = ', I3)
99996 FORMAT (' Number of columns in matrix B: M = ', I3)
99995 FORMAT (' Number of rows in matrix C: P = ', I3)
99994 FORMAT (/' E = ')
99993 FORMAT (/' E is the identity matrix.')
99992 FORMAT (' A = ')
99991 FORMAT (' B = ')
99990 FORMAT (' C = ')
99989 FORMAT (' D = ')
99988 FORMAT (' D is of zeros.')
99987 FORMAT (20(1X,F8.4))
C
END
Program Data
BD01AD EXAMPLE PROGRAM DATA D 1 1Program Results
BD01AD EXAMPLE PROGRAM RESULTS Laub 1979, Ex.1 Order of matrix A: N = 2 Number of columns in matrix B: M = 1 Number of rows in matrix C: P = 2 E is the identity matrix. A = 0.0000 1.0000 0.0000 0.0000 B = 0.0000 1.0000 C = 1.0000 0.0000 0.0000 1.0000 D is of zeros.
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/BD02AD.html 0000664 0000000 0000000 00000025307 14560147231 0020315 0 ustar 00root root 0000000 0000000
Purpose
To generate benchmark examples for time-invariant,
discrete-time dynamical systems
E x_k+1 = A x_k + B u_k
y_k = C x_k + D u_k
E, A are real N-by-N matrices, B is N-by-M, C is P-by-N, and
D is P-by-M. In many examples, E is the identity matrix and D is
the zero matrix.
This routine is an implementation of the benchmark library
DTDSX (Version 1.0) described in [1].
Specification
SUBROUTINE BD02AD( DEF, NR, DPAR, IPAR, VEC, N, M, P, E, LDE, A,
1 LDA, B, LDB, C, LDC, D, LDD, NOTE, DWORK,
2 LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DEF
CHARACTER*70 NOTE
INTEGER INFO, LDA, LDB, LDC, LDD, LDE, LDWORK, M, N, P
C .. Array Arguments ..
LOGICAL VEC(8)
INTEGER IPAR(*), NR(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), DPAR(*),
1 DWORK(*), E(LDE,*)
Arguments
Mode Parameters
DEF CHARACTER*1
Specifies the kind of values used as parameters when
generating parameter-dependent and scalable examples
(i.e., examples with NR(1) = 2, 3, or 4):
= 'D': Default values defined in [1] are used;
= 'N': Values set in DPAR and IPAR are used.
This parameter is not referenced if NR(1) = 1.
Note that the scaling parameter of examples with
NR(1) = 3 or 4 is considered as a regular parameter in
this context.
Input/Output Parameters
NR (input) INTEGER array, dimension (2)
Specifies the index of the desired example according
to [1].
NR(1) defines the group:
1 : parameter-free problems of fixed size
2 : parameter-dependent problems of fixed size
3 : parameter-free problems of scalable size
4 : parameter-dependent problems of scalable size
NR(2) defines the number of the benchmark example
within a certain group according to [1].
DPAR (input/output) DOUBLE PRECISION array, dimension (7)
On entry, if DEF = 'N' and the desired example depends on
real parameters, then the array DPAR must contain the
values for these parameters.
For an explanation of the parameters see [1].
For Example 2.1, DPAR(1), ..., DPAR(3) define the
parameters 'tau', 'delta', 'K', respectively.
On exit, if DEF = 'D' and the desired example depends on
real parameters, then the array DPAR is overwritten by the
default values given in [1].
IPAR (input/output) INTEGER array, dimension (1)
On entry, if DEF = 'N' and the desired example depends on
integer parameters, then the array IPAR must contain the
values for these parameters.
For an explanation of the parameters see [1].
For Example 3.1, IPAR(1) defines the parameter 'n'.
On exit, if DEF = 'D' and the desired example depends on
integer parameters, then the array IPAR is overwritten by
the default values given in [1].
VEC (output) LOGICAL array, dimension (8)
Flag vector which displays the availabilty of the output
data:
VEC(1), ..., VEC(3) refer to N, M, and P, respectively,
and are always .TRUE..
VEC(4) is .TRUE. iff E is NOT the identity matrix.
VEC(5), ..., VEC(7) refer to A, B, and C, respectively,
and are always .TRUE..
VEC(8) is .TRUE. iff D is NOT the zero matrix.
N (output) INTEGER
The actual state dimension, i.e., the order of the
matrices E and A.
M (output) INTEGER
The number of columns in the matrices B and D.
P (output) INTEGER
The number of rows in the matrices C and D.
E (output) DOUBLE PRECISION array, dimension (LDE,N)
The leading N-by-N part of this array contains the
matrix E.
NOTE that this array is overwritten (by the identity
matrix), if VEC(4) = .FALSE..
LDE INTEGER
The leading dimension of array E. LDE >= N.
A (output) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array contains the
matrix A.
LDA INTEGER
The leading dimension of array A. LDA >= N.
B (output) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array contains the
matrix B.
LDB INTEGER
The leading dimension of array B. LDB >= N.
C (output) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array contains the
matrix C.
LDC INTEGER
The leading dimension of array C. LDC >= P.
D (output) DOUBLE PRECISION array, dimension (LDD,M)
The leading P-by-M part of this array contains the
matrix D.
NOTE that this array is overwritten (by the zero
matrix), if VEC(8) = .FALSE..
LDD INTEGER
The leading dimension of array D. LDD >= P.
NOTE (output) CHARACTER*70
String containing short information about the chosen
example.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
NOTE that DWORK is not used in the current version
of BD02AD.
LDWORK INTEGER
LDWORK >= 1.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value; in particular, INFO = -3 or -4 indicates
that at least one of the parameters in DPAR or
IPAR, respectively, has an illegal value;
= 1: data file can not be opened or has wrong format.
References
[1] Kressner, D., Mehrmann, V. and Penzl, T.
DTDSX - a Collection of Benchmark Examples for State-Space
Realizations of Discrete-Time Dynamical Systems.
SLICOT Working Note 1998-10. 1998.
Numerical Aspects
NoneFurther Comments
NoneExample
Program Text
C BD02AD EXAMPLE PROGRAM TEXT
C
C .. Parameters ..
INTEGER NIN, NOUT
PARAMETER (NIN = 5, NOUT = 6)
INTEGER NMAX, MMAX, PMAX
PARAMETER (NMAX = 421, MMAX = 211, PMAX = 211)
INTEGER LDA, LDB, LDC, LDD, LDE, LDWORK
PARAMETER (LDA = NMAX, LDB = NMAX, LDC = PMAX, LDD = PMAX,
1 LDE = NMAX, LDWORK = 1)
C .. Local Scalars ..
CHARACTER DEF
INTEGER I, INFO, J, LDPAR, LIPAR, M, N, P
CHARACTER*70 NOTE
C .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
1 D(LDD,MMAX), DPAR(7), DWORK(LDWORK), E(LDE,NMAX)
INTEGER NR(2), IPAR(7)
LOGICAL VEC(8)
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL BD02AD
C .. Executable Statements ..
WRITE (NOUT, FMT = 99999)
C Skip the heading in the data file and read the data.
READ (NIN, FMT = '()')
READ (NIN, FMT = *) DEF
READ (NIN, FMT = *) (NR(I), I = 1, 2)
IF (LSAME(DEF,'N')) THEN
READ (NIN, FMT = *) LDPAR
IF (LDPAR .GT. 0) READ (NIN, FMT = *) (DPAR(I), I = 1, LDPAR)
READ (NIN, FMT = *) LIPAR
IF (LIPAR .GT. 0) READ (NIN, FMT = *) (IPAR(I), I = 1, LIPAR)
END IF
C Generate benchmark example
CALL BD02AD(DEF, NR, DPAR, IPAR, VEC, N, M, P, E, LDE, A, LDA,
1 B, LDB, C, LDC, D, LDD, NOTE, DWORK, LDWORK, INFO)
C
IF (INFO .NE. 0) THEN
WRITE (NOUT, FMT = 99998) INFO
ELSE
WRITE (NOUT, FMT = *) NOTE
WRITE (NOUT, FMT = 99997) N
WRITE (NOUT, FMT = 99996) M
WRITE (NOUT, FMT = 99995) P
IF (VEC(4)) THEN
WRITE (NOUT, FMT = 99994)
DO 10 I = 1, N
WRITE (NOUT, FMT = 99987) (E(I,J), J = 1, N)
10 CONTINUE
ELSE
WRITE (NOUT, FMT = 99993)
END IF
WRITE (NOUT,FMT = 99992)
DO 20 I = 1, N
WRITE (NOUT, FMT = 99987) (A(I,J), J = 1, N)
20 CONTINUE
WRITE (NOUT,FMT = 99991)
DO 30 I = 1, N
WRITE (NOUT, FMT = 99987) (B(I,J), J = 1, M)
30 CONTINUE
WRITE (NOUT,FMT = 99990)
DO 40 I = 1, P
WRITE (NOUT, FMT = 99987) (C(I,J), J = 1, N)
40 CONTINUE
IF (VEC(8)) THEN
WRITE (NOUT,FMT = 99989)
DO 50 I = 1, P
WRITE (NOUT, FMT = 99987) (D(I,J), J = 1, M)
50 CONTINUE
ELSE
WRITE (NOUT, FMT = 99988)
END IF
END IF
C
99999 FORMAT (' BD02AD EXAMPLE PROGRAM RESULTS', /1X)
99998 FORMAT (' INFO on exit from BD02AD = ', I3)
99997 FORMAT (/' Order of matrix A: N = ', I3)
99996 FORMAT (' Number of columns in matrix B: M = ', I3)
99995 FORMAT (' Number of rows in matrix C: P = ', I3)
99994 FORMAT (/' E = ')
99993 FORMAT (/' E is the identity matrix.')
99992 FORMAT (' A = ')
99991 FORMAT (' B = ')
99990 FORMAT (' C = ')
99989 FORMAT (' D = ')
99988 FORMAT (' D is of zeros.')
99987 FORMAT (20(1X,F8.4))
C
END
Program Data
BD02AD EXAMPLE PROGRAM DATA D 1 1Program Results
BD02AD EXAMPLE PROGRAM RESULTS Laub 1979, Ex. 2: uncontrollable-unobservable data Order of matrix A: N = 2 Number of columns in matrix B: M = 1 Number of rows in matrix C: P = 1 E is the identity matrix. A = 4.0000 3.0000 -4.5000 -3.5000 B = 1.0000 -1.0000 C = 3.0000 2.0000 D is of zeros.
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/DAESolver.html 0000664 0000000 0000000 00000340260 14560147231 0021243 0 ustar 00root root 0000000 0000000
Purpose
Interface for using a common entry point, DSblock compatible for defining Differential Algebraic Equations using several packages.
The equations follow the form (CASE A): F(dx(t)/dt, x(t), u(t), p, t) = 0 y(t) = g(dx(t)/dx, x(t), u(t), p, t) for the most general model which can only be solved by DASSL and DASSPK. A restricted case can be solved with RADAU5, LSODI, LSOIBT, if the system is expressed as (CASE B): F(x(t), u(t), p, t)*dx(t)/dt = A(x(t), u(t), p, t) y(t) = g(dx(t)/dx, x(t), u(t), p, t) And finally, the GELDA package is able to solve DAEs with the expression (CASE C): F(u(t), p, t)*dx(t)/dt = A(u(t), p, t)*x(t) + E(u(t), p, t) The user must define the subroutines: DAEDF: F(dx(t)/dt, x(t), u(t), p, t) for CASES: A, B and C DAEDA: A(x(t), u(t), p, t) for CASES: B and C DAEDE: E(u(t), p, t) for CASES: C DAEOUT: g(dx(t)/dx, x(t), u(t), p, t) and the Jacobians (JACFX, JACFU, JACFP) if used. The interface adapts the structure to fit all the codesSpecification
SUBROUTINE DAESolver(ISOLVER,CDAEDF_,CDAEDA_,CDAEDE_,CDAEOUT_, $ CJACFX_,CJACFU_,CJACFP_,CJACFXDOT_, $ NX, NY, NU, NP, TINI, TOUT, $ X, XDOTI, Y, U, P, $ IPAR, RPAR, RTOL, ATOL, $ IWORK, LIWORK, DWORK, LDWORK, $ IWARN, INFO) .. Scalar Arguments .. DOUBLE PRECISION TINI, TOUT INTEGER ISOLVER, IWARN, INFO, $ NX, NY, NU, NP, $ LDWORK, LIWORK CHARACTER*9 CDAEDF_, CDAEDA_,CDAEDE_, CDAEOUT_, $ CJACFX_, CJACFU_, CJACFP_, CJACFXDOT_, $ CDAEDF, CDAEDA,CDAEDE, CDAEOUT, $ CJACFX, CJACFU, CJACFP, CJACFXDOT .. Array Arguments .. DOUBLE PRECISION DWORK(LDWORK), RPAR(*), ATOL(*), RTOL(*) $ X(NX), XDOTI(NX), Y(NY), U(NU), P(NP) INTEGER IWORK(LIWORK), IPAR(*)Arguments
Mode Parameters
ISOLVER INTEGER Indicates the nonlinear solver packages to be used = 1: LSODI, = 2: LSOIBT, = 3: RADAU5, = 4: DASSL, = 5: DASPK, = 6: DGELDA.Input/Output Parameters
DAEDF (input) EXTERNAL Evaluates the F(dx(t)/dt, x(t), u(t), p, t). DAEDA (input) EXTERNAL Evaluates the A(x(t), u(t), p, t). DAEDE (input) EXTERNAL Evaluates the E(u(t), p, t). DAEOUT (input) EXTERNAL Evaluates the output signals function g. JACFX (input) EXTERNAL Evaluates the jacobian matrix with respect to X. JACFU (input) EXTERNAL Evaluates the jacobian matrix with respect to U. JACFP (input) EXTERNAL Evaluates the jacobian matrix with respect to P. NX (input) INTEGER Dimension of the state vector. NY (input) INTEGER Dimension of the output vector. NU (input) INTEGER Dimension of the input vector. NP (input) INTEGER Dimension of the parameter vector. TINI (input) DOUBLE PRECISION Initial value of time. TOUT (input) DOUBLE PRECISION Final value of time. X (input/output) DOUBLE PRECISION array, dimension (NX) On entry, array containing the initial state variables. On exit, it has the last value of the state variables. XDOTI (input) DOUBLE PRECISION array, dimension (NX) Array containing dx(t)/dt at initial point. Y (input/output) DOUBLE PRECISION array, dimension (NY) On entry, array containing the initial values of Y. On exit, it has the results of the system. U (input) DOUBLE PRECISION array, dimension (NU) Array containing the input initial values. P (input) DOUBLE PRECISION array, dimension (NP) Array containing the parameter variables. IPAR (input/output) INTEGER array, dimension (201) INPUT: 1..15 General 16..25 ODEPACK 26..35 RADAU5 36..50 DASSL/PK 51..60 GELDA 61..100 Reserved OUTPUT: 101..110 General 111..125 ODEPACK 126..135 RADAU5 136..145 DASSL/PK 146..155 GELDA 156..200 Reserved Any Mode: 201.. User Available Common integer parameters for SOLVERS: IPAR(1), Tolerance mode 0 : both rtol and atol are scalars 1 : rtol is a scalar and atol is a vector 2 : both rtol and atol are vectors IPAR(2), Compute Output Values only at TOUT (and not at the intermediate step). (1:Yes, 0:No) IPAR(3), mfjac, Method flag for jacobian 0 : No jacobian used (non-stiff method). 1 : User supplied full jacobian (stiff). 2 : User supplied banded jacobian (stiff). 3 : User supplied sparse jacobian (stiff). 10 : internally generated full jacobian (stiff). 11 : internally generated banded jacobian (stiff). 12 : internally generated sparse jacobian (stiff). IPAR(6), ml, lower half-bandwithds of the banded jacobian, excluding tne main diagonal. IPAR(7), mu, upper half-bandwithds of the banded jacobian, excluding the main diagonal. (Note: IPAR(6) and IPAR(7) are obligatories only if the jacobian matrix is banded) IPAR(101) = Number of steps taken for the problem. IPAR(102) = Number of residual evaluations. IPAR(103) = Number of jacobian evaluations. Common parameters for RADAU5, ODEPACK and DGELDA: IPAR(9), mfmass, Method flag for mass-matrix 0 : No mass-matrix used (non-stiff method). 1 : User supplied full mass-matrix (stiff). 2 : User supplied banded mass-matrix (stiff). 10 : Identity mass-matrix is used (stiff). IPAR(10), mlmass, lower half-bandwithds of the banded mass matrix, excluding the main diagonal. IPAR(11), mumass, upper half-bandwithds of the banded mass matrix, excluding the main diagonal. IPAR(12), Maximum number of steps allowed during one call to the solver. Common parameters for ODEPACK, DASSL, DASPK and DGELDA: IPAR(13), Maximum order to be allowed. default values : 12 if meth = 1 5 if meth = 2 If exceds the default value, it will be reduced to the default value. In DASSL, DASPK and DGELDA : (1 .LE. MAXORD .LE. 5) IPAR(111) = The method order last used(successfully). IPAR(112) = The order to be attempted on the next step. Common parameters for ODEPACK package: IPAR(16), Status Flag IPAR(17), Optional inputs, must be 0 IPAR(18), Maximum number of messages printed, default value is 10. IPAR(113) = Index of the component of largest in the weighted local error vector ( e(i)/ewt(i) ). IPAR(114) = Length of rwork actually required. IPAR(115) = Length of iwork actually required. - LSOIBIT IPAR(24), mb, block size. (mb .GE. 1) and mb*IPAR(28) = NX IPAR(25), nb, number of blocks in the main diagonal. (nb .ge. 4) and nb*IPAR(27) = NX - RADAU5 IPAR(26) Transforms the Jacobian matrix to Hessenberg form.(Only if IPAR(9)=1 and IPAR(3)=1 or 10) IPAR(27) Maximum number of Newton iterations in each step. IPAR(28) Starting values for Newton's method .EQ. 0 -> is taken the extrapolated collocation solution .NE. 0 -> zero values are used. IPAR(29) Dimension of the index 1 variables( >0 ). IPAR(30) Dimension of the index 2 variables. IPAR(31) Dimension of the index 3 variables. IPAR(32) Switch for step size strategy 0,1 Mod. Predictive controller(Gustafsson) 2 Classical step size control IPAR(33) Value of M1 (default 0). IPAR(34) Value of M2 (default(M2=M1). IPAR(126), Number of accepted steps. IPAR(127), Number of rejected steps. IPAR(128), Number of LU-Decompositions of both matrices IPAR(129), Number of forward-backward substitutions, of both systems. Common parameters for DASSL, DASPK and DGELDA solvers: IPAR(36), this parameter enables the code to initialize itself. Must set to 0 to indicate the start of every new problem. 0: Yes. (On each new problem) 1: No. (Allows 500 new steps) IPAR(38), Solver try to compute the initial T, X and XPRIME: 0: The initial T, X and XPRIME are consistent. 1: Given X_d calculate X_a and X'_d 2: Given X' calculate X. ( X_d differential variables in X X_a algebrac variables in X ) IPAR(136), Total number of error test failures so far. Common parameters for DASSL and DASPK solvers: IPAR(37), code solve the problem without invoking any special non negativity constraints: 0: Yes 1: To have constraint checking only in the initial condition calculation. 2: To enforze nonnegativity in X during the integration. 3: To enforce both options 1 and 2. IPAR(137), Total number of convergence test failures. - DASPK IPAR(39), DASPK use: 0: direct methods (dense or band) 1: Krylov method (iterative) 2: Krylov method + Jac (iterative) IPAR(41), Proceed to the integration after the initial condition calculation is done. Used when IPAR(38) > 0: 0: Yes 1: No IPAR(42), Errors are controled localy on all the variables: 0:Yes 1: No IPAR(8), Extra printing 0, no printing 1, for minimal printing 2, for full printing IPAR(44), maximum number of iterations in the SPIGMR algorithm. (.LE. NX) IPAR(45), number of vectors on which orthogonalization is done in the SPIGMR algorithm. (.LE. IPAR(44)) IPAR(46), maximum number of restarts of the SPIGMR algorithm per nonlinear iteration. (.GE. 0) IPAR(47), maximum number of Newton iterations per Jacobian or preconditioner evaluation. (> 0) IPAR(48), maximum number of Jacobian or preconditioner evaluations. (> 0) IPAR(49), maximum number of values of the artificial stepsize parameter H to be tried if IPAR(38) = 1. (> 0). IPAR(50), flag to turn off the linesearch algorithm. 0 : ON 1 : OFF (default) IPAR(138), number of convergence failures of the linear iteration IPAR(139), length of IWORK actually required. IPAR(140), length of RWORK actually required. IPAR(141), total number of nonlinear iterations. IPAR(142), total number of linear (Krylov) iterations IPAR(143), number of PSOL calls. - DGELDA IPAR(51), contains the strangeness index. IPAR(52), number of differential components IPAR(53), number of algebraic components IPAR(54), number of undetermined components IPAR(55), method used: if 1 then uses the BDF solver 2 then uses the Runge-Kutta solver IPAR(56), E(t) and A(t) are: 1 time dependent 0 constants IPAR(57), Maximum index of the problem. ( .GE. 0 ) IPAR(58), Step size strategy: 0, Mod. predictive controlled of Gustafsson(safer) 1, classical step size control(faster) RPAR (input/output) DOUBLE PRECISION array, dimension (201) INPUT: 1..15 General 16..25 ODEPACK 26..35 RADAU5 36..50 DASSL/PK 51..60 GELDA 61..100 Reserved OUTPUT: 101..110 General 111..125 ODEPACK 126..135 RADAU5 136..145 DASSL/PK 146..155 GELDA 156..200 Reserved Any Mode: 201.. User Available Common parameters for solvers: RPAR(1), Initial step size guess.Obligatory in RADAU5. RPAR(2), Maximum absolute step size allowed. Common parameters for ODEPACK, DASSL, DASPK and DGELDA: RPAR(111), Step size in t last used (successfully). RPAR(112), Step size to be attempted on the next step. RPAR(113), Current value of the independent variable which the solver has actually reached Common parameters for ODEPACK solver: RPAR(16), Critical value of t which the solver is not overshoot. RPAR(17), Minimum absolute step size allowed. RPAR(18), Tolerance scale factor, greater than 1.0. Parameters for RADAU5 solver: RPAR(26), The rounding unit, default 1E-16. RPAR(27), The safety factor in step size prediction, default 0.9D0. RPAR(28), Decides whether the jacobian should be recomputed, default 0.001D0. Increase when jacobian evaluations are costly For small systems should be smaller. RPAR(29), Stopping criterion for Newton's method, default MIN(0.03D0, RTOL(1)**0.5D0). RPAR(30), RPAR(31): This saves, together with a large RPAR(28), LU-decompositions and computing time for large systems. Small systems: RPAR(30)=1.D0, RPAR(31)=1.2D0 Large full systems: RPAR(30)=0.99D0, RPAR(31)=2.D0 might be good. RPAR(32), RPAR(33), Parameters for step size selection.Condition: RPAR(32)<=HNEW/HOLD<=RPAR(33) Parameters for DASSL, DASPK and DGELDA solvers: RPAR(36), Stopping point (Tstop) - DASPK RPAR(37), convergence test constant in SPIGMR algorithm. (0 .LT. RPAR(37) .LT. 1.0) RPAR(38), minimum scaled step in linesearch algorithm. The default is = (unit roundoff)**(2/3). (> 0) RPAR(39), swing factor in the Newton iteration convergence test. (default 0.1) (> 0) - DASPK RPAR(40), safety factor used in step size prediction. RPAR(41) and RPAR(42) restric the relation between the new and old stepsize in step size selection. 1/RPAR(41) .LE. Hnew/Hold .LE. 1/RPAR(42) RPAR(43), RPAR(44) QUOT1 and QUOT2 repectively. If QUOT1 < Hnew/Hold < QUOT2 and A and E are constants, the work can be saved by setting Hnew=Hold and using the system matrix of the previous step.Tolerances
RTOL DOUBLE PRECISION Relative Tolerance. ATOL DOUBLE PRECISION Absolute Tolerance.Workspace
IWORK INTEGER array, dimension (LIWORK) LIWORK INTEGER Minimum size of DWORK, depending on solver: - LSODI, LSOIBT, DASSL 20 + NX - RADAU5 3*N+20 DWORK DOUBLE PRECISION array, dimension (LDWORK) LDWORK INTEGER Size of DWORK, depending on solver: - LSODI 22 + 9*NX + NX**2 , IPAR(3) = 1 or 10 22 + 10*NX + (2*ML + MU)*NX , IPAR(3) = 2 or 11 - LSOIBT 20 + nyh*(maxord + 1) + 3*NX + lenw where nyh = Initial value of NX maxord = Maximum order allowed(default or IPAR(13) lenw = 3*mb*mb*nb + 2 - RADAU5 N*(LJAC+LMAS+3*LE+12)+20 where LJAC=N if (full jacobian) LJAC=MLJAC+MUJAC+1 if (banded jacobian) and LMAS=0 if (IPAR(9) = 10 or 11) LMAS=N if (IPAR(9) = 1) LMAS=MLMAS+MUMAS+1 if (IPAR(9) = 2) and LE=N if (IPAR(9) = 1 or 10) LE=2*MLJAC+MUJAC+1 if (IPAR(9) = 2 or 11) - DASSL >= 40 LRW .GE. 40+(MAXORD+4)*NEQ+NEQ**2, IPAR(3) = 1 or 10 >= 40+(MAXORD+4)*NEQ+(2*ML+MU+1)*NEQ, IPAR(3) = 2 >= 40+(MAXORD+4)*NEQ+(2*ML+MU+1)*NEQ +2*(NEQ/(ML+MU+1)+1), IPAR(3) = 11Warning Indicator
IWARN INTEGER = 0: no warning; = 1: LSODI/LSOIBT/RADAU5 do not use the input vector as argument; = 2: LSODI/LSOIBT do not use the param vector as argument; = 3: RTOL and ATOL are used as scalars;Error Indicator
INFO INTEGER = 0: Successful exit; < 0: If INFO = -i, the i-th argument had an illegal value; = 1: Wrong tolerance mode; = 2: Method (IPAR(9)) is not allowed for ODEPACK/RADAU5; = 3: Method (IPAR(3)) is not allowed for LSODE/RADAU5/DASSL; = 4: Option not allowed for IPAR(37); = 5: Option not allowed for IPAR(38); = 100+ERROR: RADAU5 returned -ERROR; = 200+ERROR: DASSL returned -ERROR; = 300+ERROR: DASPK returned -ERROR; = 400+ERROR: DGELDA returned -ERROR.Method
Since the package integrates 8 different solvers, it is possible to solve differential equations by means of Backward Differential Formulas, Runge-Kutta, using direct or iterative methods (including preconditioning) for the linear system associated, differential equations with time-varying coefficients or of order higher than one. The interface facilitates the user the work of changing the integrator and testing the results, thus leading a more robust and efficient integrated package.References
[1] A.C. Hindmarsh, Brief Description of ODEPACK: A Systematized Collection
of ODE Solvers, http://www.netlib.org/odepack/doc
[2] L.R. Petzold DASSL Library Documentation, http://www.netlib.org/ode/
[3] P.N. Brown, A.C. Hindmarsh, L.R. Petzold, DASPK Package 1995 Revision
[4] R.S. Maier, Using DASPK on the TMC CM5. Experiences with Two Programming
Models, Minesota Supercomputer Center, Technical Report.
[5] E. Hairer, G. Wanner, Solving Ordinary Dirential Equations II. Stiánd
Dirential- Algebraic Problems., Springer Seried in Computational
Mathermatics 14, Springer-Verlag 1991, Second Edition 1996.
[6] P. Kunkel, V. Mehrmann, W. Rath und J. Weickert, `GELDA: A Software
Package for the Solution of General Linear Dirential Algebraic
equations', SIAM Journal Scienti^Lc Computing, Vol. 18, 1997, pp.
115 - 138.
[7] M. Otter, DSblock: A neutral description of dynamic systems.
Version 3.3, http://www.netlib.org/odepack/doc
[8] M. Otter, H. Elmqvist, The DSblock model interface for exchanging model
components, Proceedings of EUROSIM 95, ed. F.Brenenecker, Vienna, Sep.
11-15, 1995
[9] M. Otter, The DSblock model interface, version 4.0, Incomplete Draft,
http://dv.op.dlr.de/~otter7dsblock/dsblock4.0a.html
[10] Ch. Lubich, U. Novak, U. Pohle, Ch. Engstler, MEXX - Numerical
Software for the Integration of Constrained Mechanical Multibody
Systems, http://www.netlib.org/odepack/doc
[11] Working Group on Software (WGS), SLICOT Implementation and Documentation
Standards (version 1.0), WGS-Report 90-1, Eindhoven University of
Technology, May 1990.
[12] P. Kunkel and V. Mehrmann, Canonical forms for linear differential-
algebraic equations with variable coeÆcients., J. Comput. Appl.
Math., 56:225{259, 1994.
[13] Working Group on Software (WGS), SLICOT Implementation and Documentation
Standards, WGS-Report 96-1, Eindhoven University of Technology, updated:
Feb. 1998, ../../REPORTS/rep96-1.ps.Z.
[14] A. Varga, Standarization of Interface for Nonlinear Systems Software
in SLICOT, Deutsches Zentrum ur Luft un Raumfahrt, DLR. SLICOT-Working
Note 1998-4, 1998, Available at
../../REPORTS/SLWN1998-4.ps.Z.
[15] D. Kirk, Optimal Control Theory: An Introduction, Prentice-Hall.
Englewood Cli, NJ, 1970.
[16] F.L. Lewis and V.L. Syrmos, Optimal Control, Addison-Wesley.
New York, 1995.
[17] W.M.Lioen, J.J.B de Swart, Test Set for Initial Value Problem Solvers,
Technical Report NM-R9615, CWI, Amsterdam, 1996.
http://www.cwi.nl/cwi/projects/IVPTestset/.
[18] V.Hernandez, I.Blanquer, E.Arias, and P.Ruiz,
Definition and Implementation of a SLICOT Standard Interface and the
associated MATLAB Gateway for the Solution of Nonlinear Control Systems
by using ODE and DAE Packages}, Universidad Politecnica de Valencia,
DSIC. SLICOT Working Note 2000-3: July 2000. Available at
../../REPORTS/SLWN2000-3.ps.Z.
[19] J.J.B. de Swart, W.M. Lioen, W.A. van der Veen, SIDE, November 25,
1998. Available at http://www.cwi.nl/cwi/projects/PSIDE/.
[20] Kim, H.Young, F.L.Lewis, D.M.Dawson, Intelligent optimal control of
robotic manipulators using neural networks.
[21] J.C.Fernandez, E.Arias, V.Hernandez, L.Penalver, High Performance
Algorithm for Tracking Trajectories of Robot Manipulators,
Preprints of the Proceedings of the 6th IFAC International Workshop on
Algorithms and Architectures for Real-Time Control (AARTC-2000),
pages 127-134.
Numerical Aspects
The numerical aspects of the routine lie on the features of the different packages integrated. Several packages are more robust than others, and other packages simply cannot deal with problems that others do. For a detailed description of the numerical aspects of each method is recommended to check the references above.Further Comments
Several packages (LSODES, LSOIBT) deal only with sparse matrices. The interface checks the suitability of the methods to the parameters and show a warning message if problems could arise.Example
Program Text
* DAESOLVER EXAMPLE PROGRAM TEXT FOR LSODIX
PROBLEM
*
* .. Parameters ..
INTEGER
NIN, NOUT
PARAMETER
( NIN = 5, NOUT = 6 )
INTEGER LSODI_, LSOIBT_, RADAU5_,
DASSL_, DASPK_, GELDA_
PARAMETER (LSODI_ = 1, LSOIBT_
= 2)
PARAMETER (RADAU5_ = 3, DASSL_
= 4, DASPK_ = 5)
PARAMETER (GELDA_ = 6)
* .. Executable Statements ..
*
EXTERNAL IARGC_
INTEGER IARGC_
INTEGER NUMARGS
CHARACTER*80 NAME
CHARACTER*80 SOLVER
*
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
*
NUMARGS = IARGC_()
*
CALL GETARG_(0, NAME)
IF (NUMARGS .NE. 1) THEN
WRITE (*,*) 'Syntax
Error: ',NAME(1:8),' <solver>'
WRITE (*,*) 'Solvers
: LSODI, LSOIBT, RADAU5, DASSL, DASPK, GELD
&A'
ELSE
*
CALL GETARG_(1, SOLVER)
*
WRITE (*,*) 'Problem:
LSODIX Solver: ',SOLVER(1:7)
*
IF (SOLVER(1:5) .EQ.
'LSODI') THEN
CALL TEST(LSODI_)
ELSEIF (SOLVER(1:6)
.EQ. 'LSOIBT') THEN
CALL TEST(LSOIBT_)
ELSEIF (SOLVER(1:6)
.EQ. 'RADAU5') THEN
CALL TEST(RADAU5_)
ELSEIF (SOLVER(1:5)
.EQ. 'GELDA') THEN
CALL TEST(GELDA_)
ELSEIF (SOLVER(1:5)
.EQ. 'DASSL') THEN
CALL TEST(DASSL_)
ELSEIF (SOLVER(1:5)
.EQ. 'DASPK') THEN
CALL TEST(DASPK_)
ELSE
WRITE (*,*)
'Error: Solver: ', SOLVER,' unknown'
ENDIF
ENDIF
*
99999 FORMAT (' DAESOLVER EXAMPLE PROGRAM RESULTS FOR LSODIX PROBLEM'
.
,/1X)
END
SUBROUTINE TEST( ISOLVER )
*
*
* PURPOSE
*
* Testing subroutine DAESolver
*
* ARGUMENTS
*
* Input/Output Parameters
*
* ISOLVER (input) INTEGER
*
Indicates the nonlinear solver package to be used:
*
= 1: LSODI,
*
= 2: LSOIBT,
*
= 3: RADAU5,
*
= 4: DASSL,
*
= 5: DASPK,
*
= 6: DGELDA.
*
* METHOD
*
* REFERENCES
*
* CONTRIBUTORS
*
* REVISIONS
*
* -
*
* KEYWORDS
*
*
* ******************************************************************
* .. Parameters ..
INTEGER LSODI_, LSOIBT_, RADAU5_,
DASSL_, DASPK_, GELDA_
PARAMETER (LSODI_ = 1, LSOIBT_
= 2)
PARAMETER (RADAU5_ = 3, DASSL_
= 4, DASPK_ = 5)
PARAMETER (GELDA_ = 6)
INTEGER
NIN, NOUT
PARAMETER
( NIN = 5, NOUT = 6 )
INTEGER
MD, ND, LPAR, LWORK
PARAMETER
( MD = 400, ND = 100, LPAR = 201,
$
LWORK = 10000 )
* .. Common variables ..
COMMON /TESTING/ ISOLVER2
INTEGER ISOLVER2
* .. Scalar Arguments ..
INTEGER ISOLVER
* .. Local Scalars ..
INTEGER
NEQN, NDISC, MLJAC, MUJAC, MLMAS, MUMAS
INTEGER
IWARN, INFO
DOUBLE PRECISION ATOL, RTOL, NORM
LOGICAL
NUMJAC, NUMMAS, CONSIS
* .. Local Arrays ..
CHARACTER FULLNM*40, PROBLM*8, TYPE*3
CHARACTER*9 CDAEDF,CDAEDA,CDAEDE,CDAEOUT,
$
CJACFX,CJACFU,CJACFP,CJACFXDOT
INTEGER
IND(MD), IPAR(LPAR), IWORK(LWORK)
DOUBLE PRECISION T(0:ND), RPAR(LPAR),
DWORK(LWORK)
DOUBLE PRECISION X(MD), XPRIME(MD),
Y(MD), U(MD), P(MD), SOLU(MD)
* .. External Functions ..
DOUBLE PRECISION DNRM2
EXTERNAL
DNRM2
* .. External Subroutines ..
EXTERNAL
PLSODIX, ILSODIX, SLSODIX
EXTERNAL
DAXPY
* .. Executable Statements ..
*
ISOLVER2 = ISOLVER
DO 20 I=1,NEQN
Y(I)=0D0
U(I)=0D0
P(I)=0D0
20 CONTINUE
DO 40 I=1,LPAR
IPAR(I)=0
RPAR(I)=0D0
40 CONTINUE
DO 60 I=1,LWORK
IWORK(I)=0
DWORK(I)=0D0
60 CONTINUE
* Get the problem dependent parameters.
RTOL=1D-4
ATOL=1D-6
IPAR(1)=0
IPAR(2)=1
IPAR(3)=1
IPAR(12)= 10000
IF (ISOLVER .EQ. LSODI_ .OR. ISOLVER
.EQ. RADAU5_) THEN
IPAR(9)=1
IPAR(16)=1
C IPAR(17)=0
RPAR(1)=1D-3
ELSE
C (ISOLVER
.EQ. DASSL_ .OR. ISOLVER .EQ. DASPK_)
C IPAR(36)=0
C IPAR(37)=0
C IPAR(38)=0
IPAR(39)=1
END IF
CALL PLSODIX(FULLNM,PROBLM,TYPE,NEQN,NDISC,T,NUMJAC,MLJAC,
$
MUJAC,NUMMAS,MLMAS,MUMAS,IND)
CALL ILSODIX(NEQN,T(0),X,XPRIME,CONSIS)
x(1) = 1.0d0
x(2) = 0.0d0
x(3) = 0.0d0
xprime(1) = -0.04D0
xprime(2) = 0.04D0
xprime(3) = 0.0D0
CALL SLSODIX(NEQN,T(1),SOLU)
IF ( TYPE.NE.'DAE' ) THEN
WRITE ( NOUT,
FMT = 99998 )
ELSE
WRITE ( NOUT,
FMT = 99997 ) FULLNM, PROBLM, TYPE, ISOLVER
CDAEDF=''
CDAEDA=''
CDAEDE=''
CDAEOUT=''
CJACFX=''
CJACFU=''
CJACFP=''
CJACFXDOT=''
CALL DAESolver(
ISOLVER, CDAEDF, CDAEDA, CDAEDE, CDAEOUT,
$
CJACFX, CJACFU, CJACFP, CJACFXDOT,
$
NEQN, NEQN, NEQN, NEQN, T(0), T(1),
$
X, XPRIME, Y, U, P,
$
IPAR, RPAR, RTOL, ATOL,
$
IWORK, LWORK, DWORK, LWORK, IWARN, INFO )
IF ( INFO.NE.0
) THEN
WRITE ( NOUT, FMT = 99996 ) INFO
ELSE
IF ( IWARN.NE.0 ) THEN
WRITE ( NOUT, FMT = 99995 ) IWARN
ENDIF
IF ( NEQN .LE. 30 ) THEN
WRITE ( NOUT, FMT = 99994 )
DO 80 I=1,NEQN
WRITE ( NOUT, FMT = 99993 ) I, X(I), SOLU(I)
80
CONTINUE
END IF
NORM=DNRM2(NEQN,SOLU,1)
IF ( NORM.EQ.0D0 ) THEN
NORM=1D0
END IF
CALL DAXPY(NEQN,-1D0,X,1,SOLU,1)
NORM=DNRM2(NEQN,SOLU,1)/NORM
WRITE ( NOUT, FMT = 99992 ) NORM
END IF
END IF
*
99998 FORMAT (' ERROR: This test is only intended for DAE problems')
99997 FORMAT (' ',A,' (',A,' , ',A,') with SOLVER ',I2)
99996 FORMAT (' INFO on exit from DAESolver = ',I3)
99995 FORMAT (' IWARN on exit from DAESolver = ',I3)
99994 FORMAT (' Solution: (calculated) (reference)')
99993 FORMAT (I,F,F)
99992 FORMAT (' Relative error comparing with the reference solution:'
$
,E,/1X)
* *** Last line of TEST ***
END
SUBROUTINE DAEDA_( RPAR, NRP, IPAR,
NIP, X, NX, U, NU, P, NP,
$
F, LDF, T, INFO )
*
*
* PURPOSE
*
* Interface routine between DAESolver and
the problem function FEVAL
*
* ARGUMENTS
*
* Input/Output Parameters
*
* RPAR (input/output)
DOUBLE PRECISION array, dimension (NRP)
*
Array for communication between the driver and FEVAL.
*
* NRP (input)
INTEGER
*
Dimension of RPAR array.
*
* IPAR (input/output)
INTEGER array, dimension (NIP)
*
Array for communication between the driver and FEVAL.
*
* NIP (input)
INTEGER
*
Dimension of IPAR array.
*
* X
(input) DOUBLE PRECISION array, dimension (NX)
*
Array containing the state variables.
*
* NX
(input) INTEGER
*
Dimension of the state vector.
*
* U
(input) DOUBLE PRECISION array, dimension (NU)
*
Array containing the input values.
*
* NU
(input) INTEGER
*
Dimension of the input vector.
*
* P
(input) DOUBLE PRECISION array, dimension (NP)
*
Array containing the parameter values.
*
* NP
(input) INTEGER
*
Dimension of the parameter vector.
*
* F
(output) DOUBLE PRECISION array, dimension (LDF,NX)
*
The resulting function value f(T,X).
*
* LDF (input)
INTEGER
*
The leading dimension of F.
*
* T
(input) INTEGER
*
The time point where the function is evaluated.
*
* Error Indicator
*
* INFO INTEGER
*
Returns values of error from FEVAL or 100 in case
*
a bad problem was choosen.
*
* METHOD
*
* REFERENCES
*
* CONTRIBUTORS
*
* REVISIONS
*
* -
*
* KEYWORDS
*
*
* ******************************************************************
*
* .. Common variables ..
COMMON /TESTING/ ISOLVER
INTEGER LSODI_, LSOIBT_, RADAU5_,
DASSL_, DASPK_, GELDA_
PARAMETER (LSODI_ = 1, LSOIBT_
= 2)
PARAMETER (RADAU5_ = 3, DASSL_
= 4, DASPK_ = 5)
PARAMETER (GELDA_ = 6)
* .. Scalar Arguments ..
INTEGER
NRP, NIP, NX, NU, NP, LDF, INFO
DOUBLE PRECISION T
* .. Array Arguments ..
INTEGER
IPAR(NIP)
DOUBLE PRECISION RPAR(NRP), X(NX),
U(NU), P(NP),
$ F(LDF,NX)
* .. External Subroutines ..
EXTERNAL
FLSODIX
* .. Executable Statements ..
CALL FLSODIX(NX,T,X,X,F,INFO,RPAR,IPAR)
* *** Last line of DAEDA_ ***
END
SUBROUTINE DAEDF_( RPAR, NRP, IPAR,
NIP, X, XPRIME, NX,
$
U, NU, P, NP, T, F, LDF, INFO )
*
*
* PURPOSE
*
* Interface routine between DAESolver and
the problem function MEVAL
*
* ARGUMENTS
*
* Input/Output Parameters
*
* RPAR (input/output)
DOUBLE PRECISION array, dimension (NRP)
*
Array for communication between the driver and MEVAL.
*
* NRP (input)
INTEGER
*
Dimension of RPAR array.
*
* IPAR (input/output)
INTEGER array, dimension (NIP)
*
Array for communication between the driver and MEVAL.
*
* NIP (input)
INTEGER
*
Dimension of IPAR array.
*
* X
(input) DOUBLE PRECISION array, dimension (NX)
*
Array containing the state variables.
*
* XPRIME (input) DOUBLE PRECISION
array, dimension (NX)
*
Array containing the state variables derivative.
*
* NX
(input) INTEGER
*
Dimension of the state vector.
*
* U
(input) DOUBLE PRECISION array, dimension (NU)
*
Array containing the input values.
*
* NU
(input) INTEGER
*
Dimension of the input vector.
*
* P
(input) DOUBLE PRECISION array, dimension (NP)
*
Array containing the parameter values.
*
* NP
(input) INTEGER
*
Dimension of the parameter vector.
*
* T
(input) INTEGER
*
The time point where the function is evaluated.
*
* F
(output) DOUBLE PRECISION array, dimension (LDF,NX)
*
The resulting function value f(T,X).
*
* LDF (input)
INTEGER
*
The leading dimension of F.
*
* Error Indicator
*
* INFO INTEGER
*
Returns values of error from MEVAL or 100 in case
*
a bad problem was choosen.
*
* METHOD
*
* REFERENCES
*
* CONTRIBUTORS
*
* REVISIONS
*
* -
*
* KEYWORDS
*
*
* ******************************************************************
*
* .. Common variables ..
COMMON /TESTING/ ISOLVER
INTEGER ISOLVER
INTEGER LSODI_, LSOIBT_, RADAU5_,
DASSL_, DASPK_, GELDA_
PARAMETER (LSODI_ = 1, LSOIBT_
= 2)
PARAMETER (RADAU5_ = 3, DASSL_
= 4, DASPK_ = 5)
PARAMETER (GELDA_ = 6)
* .. Scalar Arguments ..
INTEGER
NRP, NIP, NX, NU, NP, LDF, INFO
DOUBLE PRECISION T
* .. Array Arguments ..
INTEGER
IPAR(NIP)
DOUBLE PRECISION RPAR(NRP), X(NX),
XPRIME(NX), U(NU), P(NP),
$ F(LDF,NX)
* .. Local Scalars ..
INTEGER
I
* .. External Subroutines ..
EXTERNAL
MLSODIX, RLSODIX
* .. Executable Statements ..
IF (ISOLVER .EQ. DASSL_ .OR. ISOLVER
.EQ. DASPK_) THEN
CALL RLSODIX(LDF,NX,T,X,XPRIME,F,INFO,RPAR,IPAR)
ELSE
CALL MLSODIX(LDF,NX,T,X,XPRIME,F,INFO,RPAR,IPAR)
ENDIF
* *** Last line of DAEDF_ ***
END
SUBROUTINE JACFX_( NRP, NIP,
RPAR, IPAR, NX, NU,
$
NP, X, U, P, T, FX, LDFX,
$
INFO )
*
*
* PURPOSE
*
* Interface routine between DAESolver and
the problem function JEVAL
*
* ARGUMENTS
*
* Input/Output Parameters
*
* NRP (input)
INTEGER
*
Dimension of RPAR array.
*
* NIP (input)
INTEGER
*
Dimension of IPAR array.
*
* RPAR (input/output)
DOUBLE PRECISION array
*
Array for communication between the driver and JEVAL.
*
* IPAR (input/output)
INTEGER array
*
Array for communication between the driver and JEVAL.
*
* NX
(input) INTEGER
*
Dimension of the state vector.
*
* NU
(input) INTEGER
*
Dimension of the input vector.
*
* NP
(input) INTEGER
*
Dimension of the parameter vector.
*
* X
(input) DOUBLE PRECISION array, dimension (NX)
*
Array containing the state variables.
*
* U
(input) DOUBLE PRECISION array, dimension (NU)
*
Array containing the input values.
*
* P
(input) DOUBLE PRECISION array, dimension (NP)
*
Array containing the parameter values.
*
* T
(input) INTEGER
*
The time point where the derivative is evaluated.
*
* FX
(output) DOUBLE PRECISION array, dimension (LDFX,NX)
*
The array with the resulting Jacobian matrix.
*
* LDFX (input)
INTEGER
*
The leading dimension of the array FX.
*
* Error Indicator
*
* INFO INTEGER
*
Returns values of error from JEVAL or 100 in case
*
a bad problem was choosen.
*
* METHOD
*
* REFERENCES
*
* CONTRIBUTORS
*
* REVISIONS
*
* -
*
* KEYWORDS
*
*
* ******************************************************************
*
* .. Common variables ..
COMMON /TESTING/ ISOLVER
INTEGER LSODI_, LSOIBT_, RADAU5_,
DASSL_, DASPK_, GELDA_
PARAMETER (LSODI_ = 1, LSOIBT_
= 2)
PARAMETER (RADAU5_ = 3, DASSL_
= 4, DASPK_ = 5)
PARAMETER (GELDA_ = 6)
* .. Scalar Arguments ..
INTEGER
NRP, NIP, NX, NU, NP, LDFX, INFO
DOUBLE PRECISION T
* .. Array Arguments ..
INTEGER
IPAR(NIP)
DOUBLE PRECISION X(NX), U(NU), P(NP),
RPAR(NRP), FX(LDFX,NX)
* .. External Subroutines ..
EXTERNAL
JLSODIX
* .. Executable Statements ..
CALL JLSODIX(LDFX,NX,T,X,X,FX,INFO,RPAR,IPAR)
* *** Last line of JACFX_ ***
END
SUBROUTINE JACFXDOT_( NRP, NIP, RPAR,
IPAR,
$
NX, NU, NP, XPRIME, U, P, T, J, LDJ, INFO )
*
*
* PURPOSE
*
* MATJACFXDOT routine for TRANSAMP problem
*
* ARGUMENTS
*
* Input/Output Parameters
*
* NRP (input)
INTEGER
*
Dimension of RPAR array.
*
* NIP (input)
INTEGER
*
Dimension of IPAR array.
*
* RPAR (input/output)
DOUBLE PRECISION array
*
Array for communication with the driver.
*
* IPAR (input/output)
INTEGER array
*
Array for communication with the driver.
*
* NX
(input) INTEGER
*
Dimension of the state vector.
*
*
* NU
(input) INTEGER
*
Dimension of the input vector.
*
* NP
(input) INTEGER
*
Dimension of the parameter vector.
*
* XPRIME (input) DOUBLE PRECISION
array, dimension (NX)
*
Array containing the derivative of the state variables.
*
* U
(input) DOUBLE PRECISION array, dimension (NU)
*
Array containing the input values.
*
* P
(input) DOUBLE PRECISION array, dimension (NP)
*
Array containing the parameter values.
*
* T
(input) INTEGER
*
The time point where the derivative is evaluated.
*
* J
(output) DOUBLE PRECISION array, dimension (LDJ,NX)
*
The array with the resulting derivative matrix.
*
* LDJ (input)
INTEGER
*
The leading dimension of the array J.
*
* Error Indicator
*
* INFO INTEGER
*
Returns 1 in case a bad problem was choosen.
*
* METHOD
*
* REFERENCES
*
* CONTRIBUTORS
*
* REVISIONS
*
* -
*
* KEYWORDS
*
*
* ******************************************************************
*
* .. Common variables ..
COMMON /TESTING/ ISOLVER
INTEGER LSODI_, LSOIBT_, RADAU5_,
DASSL_, DASPK_, GELDA_
PARAMETER (LSODI_ = 1, LSOIBT_
= 2)
PARAMETER (RADAU5_ = 3, DASSL_
= 4, DASPK_ = 5)
PARAMETER (GELDA_ = 6)
* .. Scalar Arguments ..
INTEGER
NRP, NIP, NX, NU, NP, LDJ, INFO
DOUBLE PRECISION T
* .. Array Arguments ..
INTEGER
IPAR(NIP)
DOUBLE PRECISION XPRIME(NX), U(NU),
P(NP), RPAR(NRP), J(LDJ,NX)
* .. Executable Statements ..
*
CALL JDOTLSODIX(LDJ,NX,T,XPRIME,XPRIME,J,INFO,RPAR,IPAR)
ENDIF
* *** Last line of JACFXDOT_ ***
END
Program Data
No data requiredProgram Results
DAESOLVER EXAMPLE PROGRAM RESULTS Problem: LSODIX Solver: LSODI lsodix (lsodix , DAE) with SOLVER 1 IWARN on exit from DAESolver = 2 Solution: (calculated) (reference) 6.462112224297606E-07 1.255974374648338E-10 6.117680951077711E-07 Relative error comparing with the reference solution: .8898590685949503E-06
Return to index
SLICOT-SLICOT-Reference-a037f7e/doc/DE01OD.html 0000664 0000000 0000000 00000011260 14560147231 0020326 0 ustar 00root root 0000000 0000000
Purpose
To compute the convolution or deconvolution of two real signals A and B.Specification
SUBROUTINE DE01OD( CONV, N, A, B, INFO )
C .. Scalar Arguments ..
CHARACTER CONV
INTEGER INFO, N
C .. Array Arguments ..
DOUBLE PRECISION A(*), B(*)
Arguments
Mode Parameters
CONV CHARACTER*1
Indicates whether convolution or deconvolution is to be
performed as follows:
= 'C': Convolution;
= 'D': Deconvolution.
Input/Output Parameters
N (input) INTEGER
The number of samples. N must be a power of 2. N >= 2.
A (input/output) DOUBLE PRECISION array, dimension (N)
On entry, this array must contain the first signal.
On exit, this array contains the convolution (if
CONV = 'C') or deconvolution (if CONV = 'D') of the two
signals.
B (input) DOUBLE PRECISION array, dimension (N)
On entry, this array must contain the second signal.
NOTE that this array is overwritten.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
This routine computes the convolution or deconvolution of two real signals A and B using an FFT algorithm (SLICOT Library routine DG01MD).References
[1] Rabiner, L.R. and Rader, C.M.
Digital Signal Processing.
IEEE Press, 1972.
Numerical Aspects
The algorithm requires 0( N*log(N) ) operations.Further Comments
NoneExample
Program Text
* DE01OD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 128 )
* .. Local Scalars ..
INTEGER I, INFO, N
CHARACTER*1 CONV
* .. Local Arrays ..
DOUBLE PRECISION A(NMAX), B(NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL DE01OD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, CONV
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( A(I), B(I), I = 1,N )
* Perform convolution on A and B.
CALL DE01OD( CONV, N, A, B, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( LSAME( CONV, 'C' ) ) THEN
WRITE ( NOUT, FMT = 99997 )
ELSE
WRITE ( NOUT, FMT = 99996 )
END IF
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99995 ) I, A(I)
20 CONTINUE
END IF
END IF
STOP
*
99999 FORMAT (' DE01OD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from DE01OD = ',I2)
99997 FORMAT (' Convolution ',//' i A(i)',/)
99996 FORMAT (' Deconvolution ',//' i A(i)',/)
99995 FORMAT (I4,1X,F8.4)
99994 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
DE01OD EXAMPLE PROGRAM DATA 8 C 0.4862 0.2288 0.1948 0.3671 0.5788 0.6417 -0.5861 0.3875 0.8254 0.2380 0.1815 0.4682 0.2904 0.5312 -0.3599 0.6116Program Results
DE01OD EXAMPLE PROGRAM RESULTS Convolution i A(i) 1 0.5844 2 0.5769 3 0.6106 4 1.0433 5 0.6331 6 0.4531 7 0.7027 8 0.9929
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/DE01PD.html 0000664 0000000 0000000 00000013057 14560147231 0020335 0 ustar 00root root 0000000 0000000
Purpose
To compute the convolution or deconvolution of two real signals A and B using the Hartley transform.Specification
SUBROUTINE DE01PD( CONV, WGHT, N, A, B, W, INFO )
C .. Scalar Arguments ..
CHARACTER CONV, WGHT
INTEGER INFO, N
C .. Array Arguments ..
DOUBLE PRECISION A(*), B(*), W(*)
Arguments
Mode Parameters
CONV CHARACTER*1
Indicates whether convolution or deconvolution is to be
performed as follows:
= 'C': Convolution;
= 'D': Deconvolution.
WGHT CHARACTER*1
Indicates whether the precomputed weights are available
or not, as follows:
= 'A': available;
= 'N': not available.
Note that if N > 1 and WGHT = 'N' on entry, then WGHT is
set to 'A' on exit.
Input/Output Parameters
N (input) INTEGER
The number of samples. N must be a power of 2. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (N)
On entry, this array must contain the first signal.
On exit, this array contains the convolution (if
CONV = 'C') or deconvolution (if CONV = 'D') of the two
signals.
B (input) DOUBLE PRECISION array, dimension (N)
On entry, this array must contain the second signal.
NOTE that this array is overwritten.
W (input/output) DOUBLE PRECISION array,
dimension (N - LOG2(N))
On entry with WGHT = 'A', this array must contain the long
weight vector computed by a previous call of this routine
or of the SLICOT Library routine DG01OD.f, with the same
value of N. If WGHT = 'N', the contents of this array on
entry is ignored.
On exit, this array contains the long weight vector.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
This routine computes the convolution or deconvolution of two real signals A and B using three scrambled Hartley transforms (SLICOT Library routine DG01OD).References
[1] Van Loan, Charles.
Computational frameworks for the fast Fourier transform.
SIAM, 1992.
Numerical Aspects
The algorithm requires O(N log(N)) floating point operations.Further Comments
NoneExample
Program Text
* DE01PD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 128 )
* .. Local Scalars ..
INTEGER I, INFO, N
CHARACTER*1 CONV, WGHT
* .. Local Arrays ..
DOUBLE PRECISION A(NMAX), B(NMAX), W(NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL DE01PD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, CONV, WGHT
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( A(I), B(I), I = 1,N )
* Perform convolution on A and B.
CALL DE01PD( CONV, WGHT, N, A, B, W, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( LSAME( CONV, 'C' ) ) THEN
WRITE ( NOUT, FMT = 99997 )
ELSE
WRITE ( NOUT, FMT = 99996 )
END IF
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99995 ) I, A(I)
20 CONTINUE
END IF
END IF
STOP
*
99999 FORMAT (' DE01PD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from DE01PD = ',I2)
99997 FORMAT (' Convolution ',//' i A(i)',/)
99996 FORMAT (' Deconvolution ',//' i A(i)',/)
99995 FORMAT (I4,1X,F8.4)
99994 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
DE01PD EXAMPLE PROGRAM DATA 8 C N 0.4862 0.2288 0.1948 0.3671 0.5788 0.6417 -0.5861 0.3875 0.8254 0.2380 0.1815 0.4682 0.2904 0.5312 -0.3599 0.6116Program Results
DE01PD EXAMPLE PROGRAM RESULTS Convolution i A(i) 1 0.5844 2 0.5769 3 0.6106 4 1.0433 5 0.6331 6 0.4531 7 0.7027 8 0.9929
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/DF01MD.html 0000664 0000000 0000000 00000016501 14560147231 0020330 0 ustar 00root root 0000000 0000000
Purpose
To compute the sine transform or cosine transform of a real signal.Specification
SUBROUTINE DF01MD( SICO, N, DT, A, DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER SICO
INTEGER INFO, N
DOUBLE PRECISION DT
C .. Array Arguments ..
DOUBLE PRECISION A(*), DWORK(*)
Arguments
Mode Parameters
SICO CHARACTER*1
Indicates whether the sine transform or cosine transform
is to be computed as follows:
= 'S': The sine transform is computed;
= 'C': The cosine transform is computed.
Input/Output Parameters
N (input) INTEGER
The number of samples. N must be a power of 2 plus 1.
N >= 5.
DT (input) DOUBLE PRECISION
The sampling time of the signal.
A (input/output) DOUBLE PRECISION array, dimension (N)
On entry, this array must contain the signal to be
processed.
On exit, this array contains either the sine transform, if
SICO = 'S', or the cosine transform, if SICO = 'C', of the
given signal.
Workspace
DWORK DOUBLE PRECISION array, dimension (N+1)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Let A(1), A(2),..., A(N) be a real signal of N samples.
If SICO = 'S', the routine computes the sine transform of A as
follows. First, transform A(i), i = 1,2,...,N, into the complex
signal B(i), i = 1,2,...,(N+1)/2, where
B(1) = -2*A(2),
B(i) = {A(2i-2) - A(2i)} - j*A(2i-1) for i = 2,3,...,(N-1)/2,
B((N+1)/2) = 2*A(N-1) and j**2 = -1.
Next, perform a discrete inverse Fourier transform on B(i) by
calling SLICOT Library Routine DG01ND, to give the complex signal
Z(i), i = 1,2,...,(N-1)/2, from which the real signal C(i) may be
obtained as follows:
C(2i-1) = Re(Z(i)), C(2i) = Im(Z(i)) for i = 1,2,...,(N-1)/2.
Finally, compute the sine transform coefficients S ,S ,...,S
1 2 N
given by
S = 0,
1
{ [C(k) + C(N+1-k)] }
S = DT*{[C(k) - C(N+1-k)] - -----------------------},
k { [2*sin(pi*(k-1)/(N-1))]}
for k = 2,3,...,N-1, and
S = 0.
N
If SICO = 'C', the routine computes the cosine transform of A as
follows. First, transform A(i), i = 1,2,...,N, into the complex
signal B(i), i = 1,2,...,(N+1)/2, where
B(1) = 2*A(1),
B(i) = 2*A(2i-1) + 2*j*{[A(2i-2) - A(2i)]}
for i = 2,3,...,(N-1)/2 and B((N+1)/2) = 2*A(N).
Next, perform a discrete inverse Fourier transform on B(i) by
calling SLICOT Library Routine DG01ND, to give the complex signal
Z(i), i = 1,2,...,(N-1)/2, from which the real signal D(i) may be
obtained as follows:
D(2i-1) = Re(Z(i)), D(2i) = Im(Z(i)) for i = 1,2,...,(N-1)/2.
Finally, compute the cosine transform coefficients S ,S ,...,S
1 2 N
given by
S = 2*DT*[D(1) + A0],
1
{ [D(k) - D(N+1-k)] }
S = DT*{[D(k) + D(N+1-k)] - -----------------------},
k { [2*sin(pi*(k-1)/(N-1))]}
for k = 2,3,...,N-1, and
S = 2*DT*[D(1) - A0],
N
(N-1)/2
where A0 = 2*SUM A(2i).
i=1
References
[1] Rabiner, L.R. and Rader, C.M.
Digital Signal Processing.
IEEE Press, 1972.
[2] Oppenheim, A.V. and Schafer, R.W.
Discrete-Time Signal Processing.
Prentice-Hall Signal Processing Series, 1989.
Numerical Aspects
The algorithm requires 0( N*log(N) ) operations.Further Comments
NoneExample
Program Text
* DF01MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 129 )
* .. Local Scalars ..
DOUBLE PRECISION DT
INTEGER I, INFO, N
CHARACTER*1 SICO
* .. Local Arrays ..
DOUBLE PRECISION A(NMAX), DWORK(NMAX+1)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL DF01MD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, DT, SICO
IF ( N.LE.1 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( A(I), I = 1,N )
* Compute the sine/cosine transform of the given real signal.
CALL DF01MD( SICO, N, DT, A, DWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( LSAME( SICO, 'S' ) ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99995 ) I, A(I)
20 CONTINUE
ELSE
WRITE ( NOUT, FMT = 99996 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99995 ) I, A(I)
40 CONTINUE
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' DF01MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from DF01MD = ',I2)
99997 FORMAT (' Components of sine transform are',//' i',6X,'A(i)',/)
99996 FORMAT (' Components of cosine transform are',//' i',6X,'A(i)',
$ /)
99995 FORMAT (I4,3X,F8.4)
99994 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
DF01MD EXAMPLE PROGRAM DATA 17 1.0 C -0.1862 0.1288 0.3948 0.0671 0.6788 -0.2417 0.1861 0.8875 0.7254 0.9380 0.5815 -0.2682 0.4904 0.9312 -0.9599 -0.3116 0.8743Program Results
DF01MD EXAMPLE PROGRAM RESULTS Components of cosine transform are i A(i) 1 28.0536 2 3.3726 3 -20.8158 4 6.0566 5 5.7317 6 -3.9347 7 -12.8074 8 -6.8780 9 16.2892 10 -17.0788 11 21.7836 12 -20.8203 13 -7.3277 14 -2.5325 15 -0.3636 16 7.8792 17 11.0048
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/DG01MD.html 0000664 0000000 0000000 00000013516 14560147231 0020334 0 ustar 00root root 0000000 0000000
Purpose
To compute the discrete Fourier transform, or inverse transform, of a complex signal.Specification
SUBROUTINE DG01MD( INDI, N, XR, XI, INFO )
C .. Scalar Arguments ..
CHARACTER INDI
INTEGER INFO, N
C .. Array Arguments ..
DOUBLE PRECISION XI(*), XR(*)
Arguments
Mode Parameters
INDI CHARACTER*1
Indicates whether a Fourier transform or inverse Fourier
transform is to be performed as follows:
= 'D': (Direct) Fourier transform;
= 'I': Inverse Fourier transform.
Input/Output Parameters
N (input) INTEGER
The number of complex samples. N must be a power of 2.
N >= 2.
XR (input/output) DOUBLE PRECISION array, dimension (N)
On entry, this array must contain the real part of either
the complex signal z if INDI = 'D', or f(z) if INDI = 'I'.
On exit, this array contains either the real part of the
computed Fourier transform f(z) if INDI = 'D', or the
inverse Fourier transform z of f(z) if INDI = 'I'.
XI (input/output) DOUBLE PRECISION array, dimension (N)
On entry, this array must contain the imaginary part of
either z if INDI = 'D', or f(z) if INDI = 'I'.
On exit, this array contains either the imaginary part of
f(z) if INDI = 'D', or z if INDI = 'I'.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
If INDI = 'D', then the routine performs a discrete Fourier
transform on the complex signal Z(i), i = 1,2,...,N. If the result
is denoted by FZ(k), k = 1,2,...,N, then the relationship between
Z and FZ is given by the formula:
N ((k-1)*(i-1))
FZ(k) = SUM ( Z(i) * V ),
i=1
2
where V = exp( -2*pi*j/N ) and j = -1.
If INDI = 'I', then the routine performs an inverse discrete
Fourier transform on the complex signal FZ(k), k = 1,2,...,N. If
the result is denoted by Z(i), i = 1,2,...,N, then the
relationship between Z and FZ is given by the formula:
N ((k-1)*(i-1))
Z(i) = SUM ( FZ(k) * W ),
k=1
where W = exp( 2*pi*j/N ).
Note that a discrete Fourier transform, followed by an inverse
discrete Fourier transform, will result in a signal which is a
factor N larger than the original input signal.
References
[1] Rabiner, L.R. and Rader, C.M.
Digital Signal Processing.
IEEE Press, 1972.
Numerical Aspects
The algorithm requires 0( N*log(N) ) operations.Further Comments
NoneExample
Program Text
* DG01MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 128 )
* .. Local Scalars ..
INTEGER I, INFO, N
CHARACTER*1 INDI
* .. Local Arrays ..
DOUBLE PRECISION XI(NMAX), XR(NMAX)
* .. External Subroutines ..
EXTERNAL DG01MD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, INDI
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) N
ELSE
READ ( NIN, FMT = * ) ( XR(I), XI(I), I = 1,N )
* Find the Fourier transform of the given complex signal.
CALL DG01MD( INDI, N, XR, XI, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) I, XR(I), XI(I)
20 CONTINUE
END IF
END IF
STOP
*
99999 FORMAT (' DG01MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from DG01MD = ',I2)
99997 FORMAT (' Components of Fourier transform are',//' i',6X,
$ 'XR(i)',6X,'XI(i)',/)
99996 FORMAT (I4,3X,F8.4,3X,F8.4)
99995 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
DG01MD EXAMPLE PROGRAM DATA 8 D -0.1862 0.1288 0.3948 0.0671 0.6788 -0.2417 0.1861 0.8875 0.7254 0.9380 0.5815 -0.2682 0.4904 0.9312 -0.9599 -0.3116Program Results
DG01MD EXAMPLE PROGRAM RESULTS Components of Fourier transform are i XR(i) XI(i) 1 1.9109 2.1311 2 -1.9419 -2.2867 3 -1.4070 -1.3728 4 2.2886 -0.6883 5 1.5059 1.3815 6 -2.2271 0.2915 7 0.1470 2.1274 8 -1.7660 -0.5533
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/DG01ND.html 0000664 0000000 0000000 00000022104 14560147231 0020326 0 ustar 00root root 0000000 0000000
Purpose
To compute the discrete Fourier transform, or inverse Fourier transform, of a real signal.Specification
SUBROUTINE DG01ND( INDI, N, XR, XI, INFO )
C .. Scalar Arguments ..
CHARACTER INDI
INTEGER INFO, N
C .. Array Arguments ..
DOUBLE PRECISION XI(*), XR(*)
Arguments
Mode Parameters
INDI CHARACTER*1
Indicates whether a Fourier transform or inverse Fourier
transform is to be performed as follows:
= 'D': (Direct) Fourier transform;
= 'I': Inverse Fourier transform.
Input/Output Parameters
N (input) INTEGER
Half the number of real samples. N must be a power of 2.
N >= 2.
XR (input/output) DOUBLE PRECISION array, dimension (N+1)
On entry with INDI = 'D', the first N elements of this
array must contain the odd part of the input signal; for
example, XR(I) = A(2*I-1) for I = 1,2,...,N.
On entry with INDI = 'I', the first N+1 elements of this
array must contain the the real part of the input discrete
Fourier transform (computed, for instance, by a previous
call of the routine).
On exit with INDI = 'D', the first N+1 elements of this
array contain the real part of the output signal, that is
of the computed discrete Fourier transform.
On exit with INDI = 'I', the first N elements of this
array contain the odd part of the output signal, that is
of the computed inverse discrete Fourier transform.
XI (input/output) DOUBLE PRECISION array, dimension (N+1)
On entry with INDI = 'D', the first N elements of this
array must contain the even part of the input signal; for
example, XI(I) = A(2*I) for I = 1,2,...,N.
On entry with INDI = 'I', the first N+1 elements of this
array must contain the the imaginary part of the input
discrete Fourier transform (computed, for instance, by a
previous call of the routine).
On exit with INDI = 'D', the first N+1 elements of this
array contain the imaginary part of the output signal,
that is of the computed discrete Fourier transform.
On exit with INDI = 'I', the first N elements of this
array contain the even part of the output signal, that is
of the computed inverse discrete Fourier transform.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Let A(1),....,A(2*N) be a real signal of 2*N samples. Then the
first N+1 samples of the discrete Fourier transform of this signal
are given by the formula:
2*N ((m-1)*(i-1))
FA(m) = SUM ( A(i) * W ),
i=1
2
where m = 1,2,...,N+1, W = exp(-pi*j/N) and j = -1.
This transform can be computed as follows. First, transform A(i),
i = 1,2,...,2*N, into the complex signal Z(i) = (X(i),Y(i)),
i = 1,2,...,N. That is, X(i) = A(2*i-1) and Y(i) = A(2*i). Next,
perform a discrete Fourier transform on Z(i) by calling SLICOT
Library routine DG01MD. This gives a new complex signal FZ(k),
such that
N ((k-1)*(i-1))
FZ(k) = SUM ( Z(i) * V ),
i=1
where k = 1,2,...,N, V = exp(-2*pi*j/N). Using the values of
FZ(k), the components of the discrete Fourier transform FA can be
computed by simple linear relations, implemented in the DG01NY
subroutine.
Finally, let
XR(k) = Re(FZ(k)), XI(k) = Im(FZ(k)), k = 1,2,...,N,
be the contents of the arrays XR and XI on entry to DG01NY with
INDI = 'D', then on exit XR and XI contain the real and imaginary
parts of the Fourier transform of the original real signal A.
That is,
XR(m) = Re(FA(m)), XI(m) = Im(FA(m)),
where m = 1,2,...,N+1.
If INDI = 'I', then the routine evaluates the inverse Fourier
transform of a complex signal which may itself be the discrete
Fourier transform of a real signal.
Let FA(m), m = 1,2,...,2*N, denote the full discrete Fourier
transform of a real signal A(i), i=1,2,...,2*N. The relationship
between FA and A is given by the formula:
2*N ((m-1)*(i-1))
A(i) = SUM ( FA(m) * W ),
m=1
where W = exp(pi*j/N).
Let
XR(m) = Re(FA(m)) and XI(m) = Im(FA(m)) for m = 1,2,...,N+1,
be the contents of the arrays XR and XI on entry to the routine
DG01NY with INDI = 'I', then on exit the first N samples of the
complex signal FZ are returned in XR and XI such that
XR(k) = Re(FZ(k)), XI(k) = Im(FZ(k)) and k = 1,2,...,N.
Next, an inverse Fourier transform is performed on FZ (e.g. by
calling SLICOT Library routine DG01MD), to give the complex signal
Z, whose i-th component is given by the formula:
N ((k-1)*(i-1))
Z(i) = SUM ( FZ(k) * V ),
k=1
where i = 1,2,...,N and V = exp(2*pi*j/N).
Finally, the 2*N samples of the real signal A can then be obtained
directly from Z. That is,
A(2*i-1) = Re(Z(i)) and A(2*i) = Im(Z(i)), for i = 1,2,...N.
Note that a discrete Fourier transform, followed by an inverse
transform will result in a signal which is a factor 2*N larger
than the original input signal.
References
[1] Rabiner, L.R. and Rader, C.M.
Digital Signal Processing.
IEEE Press, 1972.
Numerical Aspects
The algorithm requires 0( N*log(N) ) operations.Further Comments
NoneExample
Program Text
* DG01ND EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 128 )
* .. Local Scalars ..
INTEGER I, IEND, INFO, N
CHARACTER*1 INDI
* .. Local Arrays ..
DOUBLE PRECISION A(2*NMAX), XI(NMAX+1), XR(NMAX+1)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL DG01ND
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, INDI
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) N
ELSE
READ ( NIN, FMT = * ) ( A(I), I = 1,2*N )
* Copy the odd and even parts of A into XR and XI respectively.
DO 20 I = 1, N
XR(I) = A(2*I-1)
XI(I) = A(2*I)
20 CONTINUE
* Find the Fourier transform of the given real signal.
CALL DG01ND( INDI, N, XR, XI, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
IEND = N
IF ( LSAME( INDI, 'D' ) ) IEND = N + 1
DO 40 I = 1, IEND
WRITE ( NOUT, FMT = 99996 ) I, XR(I), XI(I)
40 CONTINUE
END IF
END IF
STOP
*
99999 FORMAT (' DG01ND EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from DG01ND = ',I2)
99997 FORMAT (' Components of Fourier transform are',//' i',6X,
$ 'XR(i)',6X,'XI(i)',/)
99996 FORMAT (I4,3X,F8.4,3X,F8.4)
99995 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
DG01ND EXAMPLE PROGRAM DATA 8 D -0.1862 0.1288 0.3948 0.0671 0.6788 -0.2417 0.1861 0.8875 0.7254 0.9380 0.5815 -0.2682 0.4904 0.9312 -0.9599 -0.3116Program Results
DG01ND EXAMPLE PROGRAM RESULTS Components of Fourier transform are i XR(i) XI(i) 1 4.0420 0.0000 2 -3.1322 -0.2421 3 0.1862 -1.4675 4 -2.1312 -1.1707 5 1.5059 -1.3815 6 2.1927 -0.1908 7 -1.4462 2.0327 8 -0.5757 1.4914 9 -0.2202 0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/DG01OD.html 0000664 0000000 0000000 00000012433 14560147231 0020333 0 ustar 00root root 0000000 0000000
Purpose
To compute the (scrambled) discrete Hartley transform of a real signal.Specification
SUBROUTINE DG01OD( SCR, WGHT, N, A, W, INFO )
C .. Scalar Arguments ..
CHARACTER SCR, WGHT
INTEGER INFO, N
C .. Array Arguments ..
DOUBLE PRECISION A(*), W(*)
Arguments
Mode Parameters
SCR CHARACTER*1
Indicates whether the signal is scrambled on input or
on output as follows:
= 'N': the signal is not scrambled at all;
= 'I': the input signal is bit-reversed;
= 'O': the output transform is bit-reversed.
WGHT CHARACTER*1
Indicates whether the precomputed weights are available
or not, as follows:
= 'A': available;
= 'N': not available.
Note that if N > 1 and WGHT = 'N' on entry, then WGHT is
set to 'A' on exit.
Input/Output Parameters
N (input) INTEGER
Number of real samples. N must be a power of 2.
N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (N)
On entry with SCR = 'N' or SCR = 'O', this array must
contain the input signal.
On entry with SCR = 'I', this array must contain the
bit-reversed input signal.
On exit with SCR = 'N' or SCR = 'I', this array contains
the Hartley transform of the input signal.
On exit with SCR = 'O', this array contains the
bit-reversed Hartley transform.
W (input/output) DOUBLE PRECISION array,
dimension (N - LOG2(N))
On entry with WGHT = 'A', this array must contain the long
weight vector computed by a previous call of this routine
with the same value of N. If WGHT = 'N', the contents of
this array on entry is ignored.
On exit, this array contains the long weight vector.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
This routine uses a Hartley butterfly algorithm as described in [1].References
[1] Van Loan, Charles.
Computational frameworks for the fast Fourier transform.
SIAM, 1992.
Numerical Aspects
The algorithm is backward stable and requires O(N log(N)) floating point operations.Further Comments
NoneExample
Program Text
* DG01OD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 128 )
* .. Local Scalars ..
INTEGER I, INFO, N
CHARACTER*1 SCR, WGHT
* .. Local Arrays ..
DOUBLE PRECISION A(NMAX), W(NMAX)
* .. External Subroutines ..
EXTERNAL DG01OD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, SCR, WGHT
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) N
ELSE
READ ( NIN, FMT = * ) ( A(I), I = 1,N )
* Compute the Hartley transform.
CALL DG01OD( SCR, WGHT, N, A, W, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99996 ) I, A(I)
10 CONTINUE
END IF
END IF
STOP
*
99999 FORMAT (' DG01OD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from DG01OD = ',I2)
99997 FORMAT (' Hartley transform ',//' i A(i)',/)
99996 FORMAT (I4,1X,F8.4)
99995 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
DG01OD EXAMPLE 16 N N 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0Program Results
DG01OD EXAMPLE PROGRAM RESULTS Hartley transform i A(i) 1 136.0000 2 -48.2187 3 -27.3137 4 -19.9728 5 -16.0000 6 -13.3454 7 -11.3137 8 -9.5913 9 -8.0000 10 -6.4087 11 -4.6863 12 -2.6546 13 0.0000 14 3.9728 15 11.3137 16 32.2187
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/DK01MD.html 0000664 0000000 0000000 00000011140 14560147231 0020327 0 ustar 00root root 0000000 0000000
Purpose
To apply an anti-aliasing window to a real signal.Specification
SUBROUTINE DK01MD( TYPE, N, A, INFO )
C .. Scalar Arguments ..
CHARACTER TYPE
INTEGER INFO, N
C .. Array Arguments ..
DOUBLE PRECISION A(*)
Arguments
Mode Parameters
TYPE CHARACTER*1
Indicates the type of window to be applied to the signal
as follows:
= 'M': Hamming window;
= 'N': Hann window;
= 'Q': Quadratic window.
Input/Output Parameters
N (input) INTEGER
The number of samples. N >= 1.
A (input/output) DOUBLE PRECISION array, dimension (N)
On entry, this array must contain the signal to be
processed.
On exit, this array contains the windowing function.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
If TYPE = 'M', then a Hamming window is applied to A(1),...,A(N),
which yields
_
A(i) = (0.54 + 0.46*cos(pi*(i-1)/(N-1)))*A(i), i = 1,2,...,N.
If TYPE = 'N', then a Hann window is applied to A(1),...,A(N),
which yields
_
A(i) = 0.5*(1 + cos(pi*(i-1)/(N-1)))*A(i), i = 1,2,...,N.
If TYPE = 'Q', then a quadratic window is applied to A(1),...,
A(N), which yields
_
A(i) = (1 - 2*((i-1)/(N-1))**2)*(1 - (i-1)/(N-1))*A(i),
i = 1,2,...,(N-1)/2+1;
_
A(i) = 2*(1 - ((i-1)/(N-1))**3)*A(i), i = (N-1)/2+2,...,N.
References
[1] Rabiner, L.R. and Rader, C.M.
Digital Signal Processing.
IEEE Press, 1972.
Numerical Aspects
The algorithm requires 0( N ) operations.Further Comments
NoneExample
Program Text
* DK01MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
* .. Local Scalars ..
CHARACTER*1 TYPE
INTEGER I, INFO, N
* .. Local Arrays ..
DOUBLE PRECISION A(NMAX)
* .. External Subroutines ..
EXTERNAL DK01MD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, TYPE
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) N
ELSE
READ ( NIN, FMT = * ) ( A(I), I = 1,N )
* Apply a Hamming window to the given signal.
CALL DK01MD( TYPE, N, A, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) I, A(I)
20 CONTINUE
END IF
END IF
STOP
*
99999 FORMAT (' DK01MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from DK01MD = ',I2)
99997 FORMAT (' Components of the windowing function are',//' k ',
$ ' A(k)',/)
99996 FORMAT (I4,3X,F8.4)
99995 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
DK01MD EXAMPLE PROGRAM DATA 8 M 0.3262 0.8723 -0.7972 0.6673 -0.1722 0.3237 0.5263 -0.3275Program Results
DK01MD EXAMPLE PROGRAM RESULTS Components of the windowing function are k A(k) 1 0.3262 2 0.8326 3 -0.6591 4 0.4286 5 -0.0754 6 0.0820 7 0.0661 8 -0.0262
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/FB01QD.html 0000664 0000000 0000000 00000041364 14560147231 0020337 0 ustar 00root root 0000000 0000000
Purpose
To calculate a combined measurement and time update of one iteration of the time-varying Kalman filter. This update is given for the square root covariance filter, using dense matrices.Specification
SUBROUTINE FB01QD( JOBK, MULTBQ, N, M, P, S, LDS, A, LDA, B,
$ LDB, Q, LDQ, C, LDC, R, LDR, K, LDK, TOL,
$ IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBK, MULTBQ
INTEGER INFO, LDA, LDB, LDC, LDK, LDQ, LDR, LDS, LDWORK,
$ M, N, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
$ K(LDK,*), Q(LDQ,*), R(LDR,*), S(LDS,*)
Arguments
Mode Parameters
JOBK CHARACTER*1
Indicates whether the user wishes to compute the Kalman
filter gain matrix K as follows:
i
= 'K': K is computed and stored in array K;
i
= 'N': K is not required.
i
MULTBQ CHARACTER*1 1/2
Indicates how matrices B and Q are to be passed to
i i
the routine as follows:
= 'P': Array Q is not used and the array B must contain
1/2
the product B Q ;
i i
= 'N': Arrays B and Q must contain the matrices as
described below.
Input/Output Parameters
N (input) INTEGER
The actual state dimension, i.e., the order of the
matrices S and A . N >= 0.
i-1 i
M (input) INTEGER
The actual input dimension, i.e., the order of the matrix
1/2
Q . M >= 0.
i
P (input) INTEGER
The actual output dimension, i.e., the order of the matrix
1/2
R . P >= 0.
i
S (input/output) DOUBLE PRECISION array, dimension (LDS,N)
On entry, the leading N-by-N lower triangular part of this
array must contain S , the square root (left Cholesky
i-1
factor) of the state covariance matrix at instant (i-1).
On exit, the leading N-by-N lower triangular part of this
array contains S , the square root (left Cholesky factor)
i
of the state covariance matrix at instant i.
The strict upper triangular part of this array is not
referenced.
LDS INTEGER
The leading dimension of array S. LDS >= MAX(1,N).
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain A ,
i
the state transition matrix of the discrete system at
instant i.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain B ,
1/2 i
the input weight matrix (or the product B Q if
i i
MULTBQ = 'P') of the discrete system at instant i.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
Q (input) DOUBLE PRECISION array, dimension (LDQ,*)
If MULTBQ = 'N', then the leading M-by-M lower triangular
1/2
part of this array must contain Q , the square root
i
(left Cholesky factor) of the input (process) noise
covariance matrix at instant i.
The strict upper triangular part of this array is not
referenced.
If MULTBQ = 'P', Q is not referenced and can be supplied
as a dummy array (i.e., set parameter LDQ = 1 and declare
this array to be Q(1,1) in the calling program).
LDQ INTEGER
The leading dimension of array Q.
LDQ >= MAX(1,M) if MULTBQ = 'N';
LDQ >= 1 if MULTBQ = 'P'.
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array must contain C , the
i
output weight matrix of the discrete system at instant i.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
R (input/output) DOUBLE PRECISION array, dimension (LDR,P)
On entry, the leading P-by-P lower triangular part of this
1/2
array must contain R , the square root (left Cholesky
i
factor) of the output (measurement) noise covariance
matrix at instant i.
On exit, the leading P-by-P lower triangular part of this
1/2
array contains (RINOV ) , the square root (left Cholesky
i
factor) of the covariance matrix of the innovations at
instant i.
The strict upper triangular part of this array is not
referenced.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,P).
K (output) DOUBLE PRECISION array, dimension (LDK,P)
If JOBK = 'K', and INFO = 0, then the leading N-by-P part
of this array contains K , the Kalman filter gain matrix
i
at instant i.
If JOBK = 'N', or JOBK = 'K' and INFO = 1, then the
leading N-by-P part of this array contains AK , a matrix
i
related to the Kalman filter gain matrix at instant i (see
-1/2
METHOD). Specifically, AK = A P C'(RINOV') .
i i i|i-1 i i
LDK INTEGER
The leading dimension of array K. LDK >= MAX(1,N).
Tolerances
TOL DOUBLE PRECISION
If JOBK = 'K', then TOL is used to test for near
1/2
singularity of the matrix (RINOV ) . If the user sets
i
TOL > 0, then the given value of TOL is used as a
lower bound for the reciprocal condition number of that
matrix; a matrix whose estimated condition number is less
than 1/TOL is considered to be nonsingular. If the user
sets TOL <= 0, then an implicitly computed, default
tolerance, defined by TOLDEF = P*P*EPS, is used instead,
where EPS is the machine precision (see LAPACK Library
routine DLAMCH).
Otherwise, TOL is not referenced.
Workspace
IWORK INTEGER array, dimension (LIWORK),
where LIWORK = P if JOBK = 'K',
and LIWORK = 1 otherwise.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK. If INFO = 0 and JOBK = 'K', DWORK(2) returns
an estimate of the reciprocal of the condition number
1/2
(in the 1-norm) of (RINOV ) .
i
LDWORK The length of the array DWORK.
LDWORK >= MAX(1,N*(P+N)+2*P,N*(N+M+2)), if JOBK = 'N';
LDWORK >= MAX(2,N*(P+N)+2*P,N*(N+M+2),3*P), if JOBK = 'K'.
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
1/2
= 1: if JOBK = 'K' and the matrix (RINOV ) is singular,
i 1/2
i.e., the condition number estimate of (RINOV )
i
(in the 1-norm) exceeds 1/TOL. The matrices S, AK ,
1/2 i
and (RINOV ) have been computed.
i
Method
The routine performs one recursion of the square root covariance
filter algorithm, summarized as follows:
| 1/2 | | 1/2 |
| R C x S 0 | | (RINOV ) 0 0 |
| i i i-1 | | i |
| 1/2 | T = | |
| 0 A x S B x Q | | AK S 0 |
| i i-1 i i | | i i |
(Pre-array) (Post-array)
where T is an orthogonal transformation triangularizing the
pre-array.
The state covariance matrix P is factorized as
i|i-1
P = S S'
i|i-1 i i
and one combined time and measurement update for the state X
i|i-1
is given by
X = A X + K (Y - C X ),
i+1|i i i|i-1 i i i i|i-1
-1/2
where K = AK (RINOV ) is the Kalman filter gain matrix and Y
i i i i
is the observed output of the system.
The triangularization is done entirely via Householder
transformations exploiting the zero pattern of the pre-array.
References
[1] Anderson, B.D.O. and Moore, J.B.
Optimal Filtering.
Prentice Hall, Englewood Cliffs, New Jersey, 1979.
[2] Verhaegen, M.H.G. and Van Dooren, P.
Numerical Aspects of Different Kalman Filter Implementations.
IEEE Trans. Auto. Contr., AC-31, pp. 907-917, Oct. 1986.
[3] Vanbegin, M., Van Dooren, P., and Verhaegen, M.H.G.
Algorithm 675: FORTRAN Subroutines for Computing the Square
Root Covariance Filter and Square Root Information Filter in
Dense or Hessenberg Forms.
ACM Trans. Math. Software, 15, pp. 243-256, 1989.
Numerical Aspects
The algorithm requires
3 2 2 2
(7/6)N + N x (5/2 x P + M) + N x (1/2 x M + P )
operations and is backward stable (see [2]).
Further Comments
NoneExample
Program Text
* FB01QD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDK, LDQ, LDR, LDS
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDK = NMAX, LDQ = MMAX, LDR = PMAX,
$ LDS = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( NMAX*(PMAX+NMAX)+2*PMAX,
$ NMAX*(NMAX+MMAX+2), 3*PMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, ISTEP, J, M, N, P
CHARACTER*1 JOBK, MULTBQ
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ DIAG(PMAX), DWORK(LDWORK), K(LDK,PMAX),
$ Q(LDQ,MMAX), R(LDR,PMAX), S(LDS,NMAX)
INTEGER IWORK(PMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL DCOPY, FB01QD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, JOBK, TOL, MULTBQ
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( ( S(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( LSAME( MULTBQ, 'N' ) ) READ ( NIN, FMT = *)
$ ( ( Q(I,J), J = 1,M ), I = 1,M )
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( R(I,J), J = 1,P ), I = 1,P )
* Save the strict lower triangle of R in its strict upper
* triangle and the diagonal in the array DIAG.
DO 10 I = 2, P
CALL DCOPY( I, R(I,1), LDR, R(1,I), 1 )
10 CONTINUE
CALL DCOPY( P, R, LDR+1, DIAG, 1 )
* Perform three iterations of the (Kalman) filter recursion
* (in square root covariance form).
ISTEP = 1
20 CONTINUE
CALL FB01QD( JOBK, MULTBQ, N, M, P, S, LDS, A, LDA,
$ B, LDB, Q, LDQ, C, LDC, R, LDR, K, LDK,
$ TOL, IWORK, DWORK, LDWORK, INFO )
ISTEP = ISTEP + 1
IF ( INFO.EQ.0 .AND. ISTEP.LE.3 ) THEN
* Restore the lower triangle of R.
DO 30 I = 2, P
CALL DCOPY( I, R(1,I), 1, R(I,1), LDR )
30 CONTINUE
CALL DCOPY( P, DIAG, 1, R, LDR+1 )
GO TO 20
END IF
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( S(I,J), J = 1,N )
40 CONTINUE
IF ( LSAME( JOBK, 'K' ) ) THEN
WRITE ( NOUT, FMT = 99996 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( K(I,J), J = 1,P )
60 CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' FB01QD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from FB01QD = ',I2)
99997 FORMAT (' The square root of the state covariance matrix is ')
99996 FORMAT (/' The Kalman gain matrix is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' M is out of range.',/' M = ',I5)
99992 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
FB01QD EXAMPLE PROGRAM DATA 4 2 2 K 0.0 N 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2113 0.8497 0.7263 0.8833 0.7560 0.6857 0.1985 0.6525 0.0002 0.8782 0.5442 0.3076 0.3303 0.0683 0.2320 0.9329 0.5618 0.5042 0.5896 0.3493 0.6853 0.3873 0.8906 0.9222 1.0000 0.0000 0.0000 1.0000 0.3616 0.5664 0.5015 0.2693 0.2922 0.4826 0.4368 0.6325 0.9488 0.0000 0.3760 0.7340Program Results
FB01QD EXAMPLE PROGRAM RESULTS The square root of the state covariance matrix is -1.2936 0.0000 0.0000 0.0000 -1.1382 -0.2579 0.0000 0.0000 -0.9622 -0.1529 0.2974 0.0000 -1.3076 0.0936 0.4508 -0.4897 The Kalman gain matrix is 0.3638 0.9469 0.3532 0.8179 0.2471 0.5542 0.1982 0.6471
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/FB01RD.html 0000664 0000000 0000000 00000042754 14560147231 0020344 0 ustar 00root root 0000000 0000000
Purpose
To calculate a combined measurement and time update of one iteration of the time-invariant Kalman filter. This update is given for the square root covariance filter, using the condensed observer Hessenberg form.Specification
SUBROUTINE FB01RD( JOBK, MULTBQ, N, M, P, S, LDS, A, LDA, B,
$ LDB, Q, LDQ, C, LDC, R, LDR, K, LDK, TOL,
$ IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBK, MULTBQ
INTEGER INFO, LDA, LDB, LDC, LDK, LDQ, LDR, LDS, LDWORK,
$ M, N, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
$ K(LDK,*), Q(LDQ,*), R(LDR,*), S(LDS,*)
Arguments
Mode Parameters
JOBK CHARACTER*1
Indicates whether the user wishes to compute the Kalman
filter gain matrix K as follows:
i
= 'K': K is computed and stored in array K;
i
= 'N': K is not required.
i
MULTBQ CHARACTER*1 1/2
Indicates how matrices B and Q are to be passed to
i i
the routine as follows:
= 'P': Array Q is not used and the array B must contain
1/2
the product B Q ;
i i
= 'N': Arrays B and Q must contain the matrices as
described below.
Input/Output Parameters
N (input) INTEGER
The actual state dimension, i.e., the order of the
matrices S and A. N >= 0.
i-1
M (input) INTEGER
The actual input dimension, i.e., the order of the matrix
1/2
Q . M >= 0.
i
P (input) INTEGER
The actual output dimension, i.e., the order of the matrix
1/2
R . P >= 0.
i
S (input/output) DOUBLE PRECISION array, dimension (LDS,N)
On entry, the leading N-by-N lower triangular part of this
array must contain S , the square root (left Cholesky
i-1
factor) of the state covariance matrix at instant (i-1).
On exit, the leading N-by-N lower triangular part of this
array contains S , the square root (left Cholesky factor)
i
of the state covariance matrix at instant i.
The strict upper triangular part of this array is not
referenced.
LDS INTEGER
The leading dimension of array S. LDS >= MAX(1,N).
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain A,
the state transition matrix of the discrete system in
lower observer Hessenberg form (e.g., as produced by
SLICOT Library Routine TB01ND).
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain B ,
1/2 i
the input weight matrix (or the product B Q if
i i
MULTBQ = 'P') of the discrete system at instant i.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
Q (input) DOUBLE PRECISION array, dimension (LDQ,*)
If MULTBQ = 'N', then the leading M-by-M lower triangular
1/2
part of this array must contain Q , the square root
i
(left Cholesky factor) of the input (process) noise
covariance matrix at instant i.
The strict upper triangular part of this array is not
referenced.
Otherwise, Q is not referenced and can be supplied as a
dummy array (i.e., set parameter LDQ = 1 and declare this
array to be Q(1,1) in the calling program).
LDQ INTEGER
The leading dimension of array Q.
LDQ >= MAX(1,M) if MULTBQ = 'N';
LDQ >= 1 if MULTBQ = 'P'.
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array must contain C,
the output weight matrix of the discrete system in lower
observer Hessenberg form (e.g., as produced by SLICOT
Library routine TB01ND).
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
R (input/output) DOUBLE PRECISION array, dimension (LDR,P)
On entry, the leading P-by-P lower triangular part of this
1/2
array must contain R , the square root (left Cholesky
i
factor) of the output (measurement) noise covariance
matrix at instant i.
On exit, the leading P-by-P lower triangular part of this
1/2
array contains (RINOV ) , the square root (left Cholesky
i
factor) of the covariance matrix of the innovations at
instant i.
The strict upper triangular part of this array is not
referenced.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,P).
K (output) DOUBLE PRECISION array, dimension (LDK,P)
If JOBK = 'K', and INFO = 0, then the leading N-by-P part
of this array contains K , the Kalman filter gain matrix
i
at instant i.
If JOBK = 'N', or JOBK = 'K' and INFO = 1, then the
leading N-by-P part of this array contains AK , a matrix
i
related to the Kalman filter gain matrix at instant i (see
-1/2
METHOD). Specifically, AK = A P C'(RINOV') .
i i|i-1 i
LDK INTEGER
The leading dimension of array K. LDK >= MAX(1,N).
Tolerances
TOL DOUBLE PRECISION
If JOBK = 'K', then TOL is used to test for near
1/2
singularity of the matrix (RINOV ) . If the user sets
i
TOL > 0, then the given value of TOL is used as a
lower bound for the reciprocal condition number of that
matrix; a matrix whose estimated condition number is less
than 1/TOL is considered to be nonsingular. If the user
sets TOL <= 0, then an implicitly computed, default
tolerance, defined by TOLDEF = P*P*EPS, is used instead,
where EPS is the machine precision (see LAPACK Library
routine DLAMCH).
Otherwise, TOL is not referenced.
Workspace
IWORK INTEGER array, dimension (LIWORK)
where LIWORK = P if JOBK = 'K',
and LIWORK = 1 otherwise.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK. If INFO = 0 and JOBK = 'K', DWORK(2) returns
an estimate of the reciprocal of the condition number
1/2
(in the 1-norm) of (RINOV ) .
i
LDWORK The length of the array DWORK.
LDWORK >= MAX(1,N*(P+N+1),N*(P+N)+2*P,N*(N+M+2)),
if JOBK = 'N';
LDWORK >= MAX(2,N*(P+N+1),N*(P+N)+2*P,N*(N+M+2),3*P),
if JOBK = 'K'.
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
1/2
= 1: if JOBK = 'K' and the matrix (RINOV ) is singular,
i 1/2
i.e., the condition number estimate of (RINOV )
i
(in the 1-norm) exceeds 1/TOL. The matrices S, AK ,
1/2 i
and (RINOV ) have been computed.
i
Method
The routine performs one recursion of the square root covariance
filter algorithm, summarized as follows:
| 1/2 | | 1/2 |
| R 0 C x S | | (RINOV ) 0 0 |
| i i-1 | | i |
| 1/2 | T = | |
| 0 B x Q A x S | | AK S 0 |
| i i i-1 | | i i |
(Pre-array) (Post-array)
where T is unitary and (A,C) is in lower observer Hessenberg form.
An example of the pre-array is given below (where N = 6, P = 2
and M = 3):
|x | | x |
|x x | | x x |
|____|______|____________|
| | x x x| x x x |
| | x x x| x x x x |
| | x x x| x x x x x |
| | x x x| x x x x x x|
| | x x x| x x x x x x|
| | x x x| x x x x x x|
The corresponding state covariance matrix P is then
i|i-1
factorized as
P = S S'
i|i-1 i i
and one combined time and measurement update for the state X
i|i-1
is given by
X = A X + K (Y - C X )
i+1|i i|i-1 i i i|i-1
-1/2
where K = AK (RINOV ) is the Kalman filter gain matrix and Y
i i i i
is the observed output of the system.
The triangularization is done entirely via Householder
transformations exploiting the zero pattern of the pre-array.
References
[1] Anderson, B.D.O. and Moore, J.B.
Optimal Filtering.
Prentice Hall, Englewood Cliffs, New Jersey, 1979.
[2] Van Dooren, P. and Verhaegen, M.H.G.
Condensed Forms for Efficient Time-Invariant Kalman Filtering.
SIAM J. Sci. Stat. Comp., 9. pp. 516-530, 1988.
[3] Verhaegen, M.H.G. and Van Dooren, P.
Numerical Aspects of Different Kalman Filter Implementations.
IEEE Trans. Auto. Contr., AC-31, pp. 907-917, Oct. 1986.
[4] Vanbegin, M., Van Dooren, P., and Verhaegen, M.H.G.
Algorithm 675: FORTRAN Subroutines for Computing the Square
Root Covariance Filter and Square Root Information Filter in
Dense or Hessenberg Forms.
ACM Trans. Math. Software, 15, pp. 243-256, 1989.
Numerical Aspects
The algorithm requires
3 2 2 3
1/6 x N + N x (3/2 x P + M) + 2 x N x P + 2/3 x P
operations and is backward stable (see [3]).
Further Comments
NoneExample
Program Text
* FB01RD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDK, LDQ, LDR, LDS
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX, LDK = NMAX,
$ LDQ = MMAX, LDR = PMAX, LDS = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( NMAX*(PMAX+NMAX+1),
$ NMAX*(PMAX+NMAX)+2*PMAX,
$ NMAX*(NMAX+MMAX+2), 3*PMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, ISTEP, J, M, N, P
CHARACTER*1 JOBK, MULTBQ
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ DIAG(PMAX), DWORK(LDWORK), K(LDK,PMAX),
$ Q(LDQ,MMAX), R(LDR,PMAX), S(LDS,NMAX)
INTEGER IWORK(PMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL DCOPY, FB01RD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, JOBK, TOL, MULTBQ
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( ( S(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( LSAME( MULTBQ, 'N' ) ) READ ( NIN, FMT = * )
$ ( ( Q(I,J), J = 1,M ), I = 1,M )
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( R(I,J), J = 1,P ), I = 1,P )
* Save the strict lower triangle of R in its strict upper
* triangle and the diagonal in the array DIAG.
DO 10 I = 2, P
CALL DCOPY( I, R(I,1), LDR, R(1,I), 1 )
10 CONTINUE
CALL DCOPY( P, R, LDR+1, DIAG, 1 )
* Perform three iterations of the (Kalman) filter recursion
* (in square root covariance form).
ISTEP = 1
20 CONTINUE
CALL FB01RD( JOBK, MULTBQ, N, M, P, S, LDS, A, LDA,
$ B, LDB, Q, LDQ, C, LDC, R, LDR, K, LDK,
$ TOL, IWORK, DWORK, LDWORK, INFO )
ISTEP = ISTEP + 1
IF ( INFO.EQ.0 .AND. ISTEP.LE.3 ) THEN
* Restore the lower triangle of R.
DO 30 I = 2, P
CALL DCOPY( I, R(1,I), 1, R(I,1), LDR )
30 CONTINUE
CALL DCOPY( P, DIAG, 1, R, LDR+1 )
GO TO 20
END IF
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( S(I,J), J = 1,N )
40 CONTINUE
IF ( LSAME( JOBK, 'K' ) ) THEN
WRITE ( NOUT, FMT = 99996 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( K(I,J), J = 1,P )
60 CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' FB01RD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from FB01QD = ',I2)
99997 FORMAT (' The square root of the state covariance matrix is ')
99996 FORMAT (/' The Kalman gain matrix is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' M is out of range.',/' M = ',I5)
99992 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
FB01RD EXAMPLE PROGRAM DATA 4 2 2 K 0.0 N 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2113 0.8497 0.7263 0.0000 0.7560 0.6857 0.1985 0.6525 0.0002 0.8782 0.5442 0.3076 0.3303 0.0683 0.2320 0.9329 0.5618 0.5042 0.5896 0.3493 0.6853 0.3873 0.8906 0.9222 1.0000 0.0000 0.0000 1.0000 0.3616 0.0000 0.0000 0.0000 0.2922 0.4826 0.0000 0.0000 0.9488 0.0000 0.3760 0.7340Program Results
FB01RD EXAMPLE PROGRAM RESULTS The square root of the state covariance matrix is -1.7223 0.0000 0.0000 0.0000 -2.1073 0.5467 0.0000 0.0000 -1.7649 0.1412 -0.1710 0.0000 -1.8291 0.2058 -0.1497 0.7760 The Kalman gain matrix is -0.2135 1.6649 -0.2345 2.1442 -0.2147 1.7069 -0.1345 1.4777
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/FB01SD.html 0000664 0000000 0000000 00000047612 14560147231 0020343 0 ustar 00root root 0000000 0000000
Purpose
To calculate a combined measurement and time update of one iteration of the time-varying Kalman filter. This update is given for the square root information filter, using dense matrices.Specification
SUBROUTINE FB01SD( JOBX, MULTAB, MULTRC, N, M, P, SINV, LDSINV,
$ AINV, LDAINV, B, LDB, RINV, LDRINV, C, LDC,
$ QINV, LDQINV, X, RINVY, Z, E, TOL, IWORK,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBX, MULTAB, MULTRC
INTEGER INFO, LDAINV, LDB, LDC, LDQINV, LDRINV, LDSINV,
$ LDWORK, M, N, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION AINV(LDAINV,*), B(LDB,*), C(LDC,*), DWORK(*),
$ E(*), QINV(LDQINV,*), RINV(LDRINV,*), RINVY(*),
$ SINV(LDSINV,*), X(*), Z(*)
Arguments
Mode Parameters
JOBX CHARACTER*1
Indicates whether X is to be computed as follows:
i+1
= 'X': X is computed and stored in array X;
i+1
= 'N': X is not required.
i+1
MULTAB CHARACTER*1 -1
Indicates how matrices A and B are to be passed to
i i
the routine as follows: -1
= 'P': Array AINV must contain the matrix A and the
-1 i
array B must contain the product A B ;
i i
= 'N': Arrays AINV and B must contain the matrices
as described below.
MULTRC CHARACTER*1 -1/2
Indicates how matrices R and C are to be passed to
i+1 i+1
the routine as follows:
= 'P': Array RINV is not used and the array C must
-1/2
contain the product R C ;
i+1 i+1
= 'N': Arrays RINV and C must contain the matrices
as described below.
Input/Output Parameters
N (input) INTEGER
The actual state dimension, i.e., the order of the
-1 -1
matrices S and A . N >= 0.
i i
M (input) INTEGER
The actual input dimension, i.e., the order of the matrix
-1/2
Q . M >= 0.
i
P (input) INTEGER
The actual output dimension, i.e., the order of the matrix
-1/2
R . P >= 0.
i+1
SINV (input/output) DOUBLE PRECISION array, dimension
(LDSINV,N)
On entry, the leading N-by-N upper triangular part of this
-1
array must contain S , the inverse of the square root
i
(right Cholesky factor) of the state covariance matrix
P (hence the information square root) at instant i.
i|i
On exit, the leading N-by-N upper triangular part of this
-1
array contains S , the inverse of the square root (right
i+1
Cholesky factor) of the state covariance matrix P
i+1|i+1
(hence the information square root) at instant i+1.
The strict lower triangular part of this array is not
referenced.
LDSINV INTEGER
The leading dimension of array SINV. LDSINV >= MAX(1,N).
AINV (input) DOUBLE PRECISION array, dimension (LDAINV,N)
-1
The leading N-by-N part of this array must contain A ,
i
the inverse of the state transition matrix of the discrete
system at instant i.
LDAINV INTEGER
The leading dimension of array AINV. LDAINV >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain B ,
-1 i
the input weight matrix (or the product A B if
i i
MULTAB = 'P') of the discrete system at instant i.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
RINV (input) DOUBLE PRECISION array, dimension (LDRINV,*)
If MULTRC = 'N', then the leading P-by-P upper triangular
-1/2
part of this array must contain R , the inverse of the
i+1
covariance square root (right Cholesky factor) of the
output (measurement) noise (hence the information square
root) at instant i+1.
The strict lower triangular part of this array is not
referenced.
Otherwise, RINV is not referenced and can be supplied as a
dummy array (i.e., set parameter LDRINV = 1 and declare
this array to be RINV(1,1) in the calling program).
LDRINV INTEGER
The leading dimension of array RINV.
LDRINV >= MAX(1,P) if MULTRC = 'N';
LDRINV >= 1 if MULTRC = 'P'.
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array must contain C ,
-1/2 i+1
the output weight matrix (or the product R C if
i+1 i+1
MULTRC = 'P') of the discrete system at instant i+1.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
QINV (input/output) DOUBLE PRECISION array, dimension
(LDQINV,M)
On entry, the leading M-by-M upper triangular part of this
-1/2
array must contain Q , the inverse of the covariance
i
square root (right Cholesky factor) of the input (process)
noise (hence the information square root) at instant i.
On exit, the leading M-by-M upper triangular part of this
-1/2
array contains (QINOV ) , the inverse of the covariance
i
square root (right Cholesky factor) of the process noise
innovation (hence the information square root) at
instant i.
The strict lower triangular part of this array is not
referenced.
LDQINV INTEGER
The leading dimension of array QINV. LDQINV >= MAX(1,M).
X (input/output) DOUBLE PRECISION array, dimension (N)
On entry, this array must contain X , the estimated
i
filtered state at instant i.
On exit, if JOBX = 'X', and INFO = 0, then this array
contains X , the estimated filtered state at
i+1
instant i+1.
On exit, if JOBX = 'N', or JOBX = 'X' and INFO = 1, then
-1
this array contains S X .
i+1 i+1
RINVY (input) DOUBLE PRECISION array, dimension (P)
-1/2
This array must contain R Y , the product of the
i+1 i+1
-1/2
upper triangular matrix R and the measured output
i+1
vector Y at instant i+1.
i+1
Z (input) DOUBLE PRECISION array, dimension (M)
This array must contain Z , the mean value of the state
i
process noise at instant i.
E (output) DOUBLE PRECISION array, dimension (P)
This array contains E , the estimated error at instant
i+1
i+1.
Tolerances
TOL DOUBLE PRECISION
If JOBX = 'X', then TOL is used to test for near
-1
singularity of the matrix S . If the user sets
i+1
TOL > 0, then the given value of TOL is used as a
lower bound for the reciprocal condition number of that
matrix; a matrix whose estimated condition number is less
than 1/TOL is considered to be nonsingular. If the user
sets TOL <= 0, then an implicitly computed, default
tolerance, defined by TOLDEF = N*N*EPS, is used instead,
where EPS is the machine precision (see LAPACK Library
routine DLAMCH).
Otherwise, TOL is not referenced.
Workspace
IWORK INTEGER array, dimension (LIWORK)
where LIWORK = N if JOBX = 'X',
and LIWORK = 1 otherwise.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK. If INFO = 0 and JOBX = 'X', DWORK(2) returns
an estimate of the reciprocal of the condition number
-1
(in the 1-norm) of S .
i+1
LDWORK The length of the array DWORK.
LDWORK >= MAX(1,N*(N+2*M)+3*M,(N+P)*(N+1)+2*N),
if JOBX = 'N';
LDWORK >= MAX(2,N*(N+2*M)+3*M,(N+P)*(N+1)+2*N,3*N),
if JOBX = 'X'.
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value; -1
= 1: if JOBX = 'X' and the matrix S is singular,
i+1 -1
i.e., the condition number estimate of S (in the
i+1
-1 -1/2
1-norm) exceeds 1/TOL. The matrices S , Q
i+1 i
and E have been computed.
Method
The routine performs one recursion of the square root information
filter algorithm, summarized as follows:
| -1/2 -1/2 | | -1/2 |
| Q 0 Q Z | | (QINOV ) * * |
| i i i | | i |
| | | |
| -1 -1 -1 -1 -1 | | -1 -1 |
T | S A B S A S X | = | 0 S S X |
| i i i i i i i | | i+1 i+1 i+1|
| | | |
| -1/2 -1/2 | | |
| 0 R C R Y | | 0 0 E |
| i+1 i+1 i+1 i+1| | i+1 |
(Pre-array) (Post-array)
where T is an orthogonal transformation triangularizing the
-1/2
pre-array, (QINOV ) is the inverse of the covariance square
i
root (right Cholesky factor) of the process noise innovation
(hence the information square root) at instant i, and E is the
i+1
estimated error at instant i+1.
The inverse of the corresponding state covariance matrix P
i+1|i+1
(hence the information matrix I) is then factorized as
-1 -1 -1
I = P = (S )' S
i+1|i+1 i+1|i+1 i+1 i+1
and one combined time and measurement update for the state is
given by X .
i+1
The triangularization is done entirely via Householder
transformations exploiting the zero pattern of the pre-array.
References
[1] Anderson, B.D.O. and Moore, J.B.
Optimal Filtering.
Prentice Hall, Englewood Cliffs, New Jersey, 1979.
[2] Verhaegen, M.H.G. and Van Dooren, P.
Numerical Aspects of Different Kalman Filter Implementations.
IEEE Trans. Auto. Contr., AC-31, pp. 907-917, Oct. 1986.
[3] Vanbegin, M., Van Dooren, P., and Verhaegen, M.H.G.
Algorithm 675: FORTRAN Subroutines for Computing the Square
Root Covariance Filter and Square Root Information Filter in
Dense or Hessenberg Forms.
ACM Trans. Math. Software, 15, pp. 243-256, 1989.
Numerical Aspects
The algorithm requires approximately
3 2 2 2
(7/6)N + N x (7/2 x M + P) + N x (1/2 x P + M )
operations and is backward stable (see [2]).
Further Comments
NoneExample
Program Text
* FB01SD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDAINV, LDB, LDC, LDQINV, LDRINV, LDSINV
PARAMETER ( LDAINV = NMAX, LDB = NMAX, LDC = PMAX,
$ LDQINV = MMAX, LDRINV = PMAX, LDSINV = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( NMAX*(NMAX + 2*MMAX) + 3*MMAX,
$ (NMAX + PMAX)*(NMAX + 1) + 2*NMAX,
$ 3*NMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, ISTEP, J, M, N, P
CHARACTER*1 JOBX, MULTAB, MULTRC
* .. Local Arrays ..
DOUBLE PRECISION AINV(LDAINV,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ DIAG(MMAX), DWORK(LDWORK), E(PMAX),
$ QINV(LDQINV,MMAX), RINV(LDRINV,PMAX),
$ RINVY(PMAX), SINV(LDSINV,NMAX), X(NMAX), Z(MMAX)
INTEGER IWORK(NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL DCOPY, FB01SD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, JOBX, TOL, MULTAB, MULTRC
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( AINV(I,J), J = 1,N ), I = 1,N )
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
IF ( LSAME( MULTRC, 'N' ) ) READ ( NIN, FMT = * )
$ ( ( RINV(I,J), J = 1,P ), I = 1,P )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
READ ( NIN, FMT = * ) ( ( QINV(I,J), J = 1,M ), I = 1,M )
READ ( NIN, FMT = * ) ( ( SINV(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( Z(J), J = 1,M )
READ ( NIN, FMT = * ) ( X(J), J = 1,N )
READ ( NIN, FMT = * ) ( RINVY(J), J = 1,P )
* Save the strict upper triangle of QINV in its strict
* lower triangle and the diagonal in the array DIAG.
DO 10 I = 2, M
CALL DCOPY( I, QINV(1,I), 1, QINV(I,1), LDQINV )
10 CONTINUE
CALL DCOPY( M, QINV, LDQINV+1, DIAG, 1 )
* Perform three iterations of the (Kalman) filter recursion
* (in square root information form).
ISTEP = 1
20 CONTINUE
CALL FB01SD( JOBX, MULTAB, MULTRC, N, M, P, SINV,
$ LDSINV, AINV, LDAINV, B, LDB, RINV,
$ LDRINV, C, LDC, QINV, LDQINV, X, RINVY,
$ Z, E, TOL, IWORK, DWORK, LDWORK, INFO )
ISTEP = ISTEP + 1
IF ( INFO.EQ.0 .AND. ISTEP.LE.3 ) THEN
* Restore the upper triangle of QINV.
DO 30 I = 2, M
CALL DCOPY( I, QINV(I,1), LDQINV, QINV(1,I), 1 )
30 CONTINUE
CALL DCOPY( M, DIAG, 1, QINV, LDQINV+1 )
GO TO 20
END IF
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( SINV(I,J), J = 1,N )
40 CONTINUE
IF ( LSAME( JOBX, 'X' ) ) THEN
WRITE ( NOUT, FMT = 99995 )
DO 50 I = 1, N
WRITE ( NOUT, FMT = 99994 ) I, X(I)
50 CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' FB01SD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from FB01SD = ',I2)
99997 FORMAT (' The inverse of the square root of the state covariance',
$ ' matrix is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The components of the estimated filtered state are ',
$ //' k X(k)',/)
99994 FORMAT (I4,3X,F8.4)
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' M is out of range.',/' M = ',I5)
99991 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
FB01SD EXAMPLE PROGRAM DATA 4 2 2 X 0.0 P N 0.2113 0.7560 0.0002 0.3303 0.8497 0.6857 0.8782 0.0683 0.7263 0.1985 0.5442 0.2320 0.8833 0.6525 0.3076 0.9329 0.3616 0.5664 0.5015 0.2693 0.2922 0.4826 0.4368 0.6325 1.0000 0.0000 0.0000 1.0000 -0.8805 1.3257 2.1039 0.5207 -0.6075 1.0386 -0.8531 1.1688 1.1159 0.2305 0.0000 0.6597 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0019 0.5075 0.4076 0.8408 0.5017 0.9128 0.2129 0.5591Program Results
FB01SD EXAMPLE PROGRAM RESULTS The inverse of the square root of the state covariance matrix is 0.6897 0.7721 0.7079 0.6102 0.0000 -0.3363 -0.2252 -0.2642 0.0000 0.0000 -0.1650 0.0319 0.0000 0.0000 0.0000 0.3708 The components of the estimated filtered state are k X(k) 1 -0.7125 2 -1.8324 3 1.7500 4 1.5854
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/FB01TD.html 0000664 0000000 0000000 00000050343 14560147231 0020337 0 ustar 00root root 0000000 0000000
Purpose
To calculate a combined measurement and time update of one iteration of the time-invariant Kalman filter. This update is given for the square root information filter, using the condensed controller Hessenberg form.Specification
SUBROUTINE FB01TD( JOBX, MULTRC, N, M, P, SINV, LDSINV, AINV,
$ LDAINV, AINVB, LDAINB, RINV, LDRINV, C, LDC,
$ QINV, LDQINV, X, RINVY, Z, E, TOL, IWORK,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBX, MULTRC
INTEGER INFO, LDAINB, LDAINV, LDC, LDQINV, LDRINV,
$ LDSINV, LDWORK, M, N, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION AINV(LDAINV,*), AINVB(LDAINB,*), C(LDC,*),
$ DWORK(*), E(*), QINV(LDQINV,*), RINV(LDRINV,*),
$ RINVY(*), SINV(LDSINV,*), X(*), Z(*)
Arguments
Mode Parameters
JOBX CHARACTER*1
Indicates whether X is to be computed as follows:
i+1
= 'X': X is computed and stored in array X;
i+1
= 'N': X is not required.
i+1
MULTRC CHARACTER*1 -1/2
Indicates how matrices R and C are to be passed to
i+1 i+1
the routine as follows:
= 'P': Array RINV is not used and the array C must
-1/2
contain the product R C ;
i+1 i+1
= 'N': Arrays RINV and C must contain the matrices
as described below.
Input/Output Parameters
N (input) INTEGER
The actual state dimension, i.e., the order of the
-1 -1
matrices S and A . N >= 0.
i
M (input) INTEGER
The actual input dimension, i.e., the order of the matrix
-1/2
Q . M >= 0.
i
P (input) INTEGER
The actual output dimension, i.e., the order of the matrix
-1/2
R . P >= 0.
i+1
SINV (input/output) DOUBLE PRECISION array, dimension
(LDSINV,N)
On entry, the leading N-by-N upper triangular part of this
-1
array must contain S , the inverse of the square root
i
(right Cholesky factor) of the state covariance matrix
P (hence the information square root) at instant i.
i|i
On exit, the leading N-by-N upper triangular part of this
-1
array contains S , the inverse of the square root (right
i+1
Cholesky factor) of the state covariance matrix P
i+1|i+1
(hence the information square root) at instant i+1.
The strict lower triangular part of this array is not
referenced.
LDSINV INTEGER
The leading dimension of array SINV. LDSINV >= MAX(1,N).
AINV (input) DOUBLE PRECISION array, dimension (LDAINV,N)
-1
The leading N-by-N part of this array must contain A ,
the inverse of the state transition matrix of the discrete
system in controller Hessenberg form (e.g., as produced by
SLICOT Library Routine TB01MD).
LDAINV INTEGER
The leading dimension of array AINV. LDAINV >= MAX(1,N).
AINVB (input) DOUBLE PRECISION array, dimension (LDAINB,M)
-1
The leading N-by-M part of this array must contain A B,
-1
the product of A and the input weight matrix B of the
discrete system, in upper controller Hessenberg form
(e.g., as produced by SLICOT Library Routine TB01MD).
LDAINB INTEGER
The leading dimension of array AINVB. LDAINB >= MAX(1,N).
RINV (input) DOUBLE PRECISION array, dimension (LDRINV,*)
If MULTRC = 'N', then the leading P-by-P upper triangular
-1/2
part of this array must contain R , the inverse of the
i+1
covariance square root (right Cholesky factor) of the
output (measurement) noise (hence the information square
root) at instant i+1.
The strict lower triangular part of this array is not
referenced.
Otherwise, RINV is not referenced and can be supplied as a
dummy array (i.e., set parameter LDRINV = 1 and declare
this array to be RINV(1,1) in the calling program).
LDRINV INTEGER
The leading dimension of array RINV.
LDRINV >= MAX(1,P) if MULTRC = 'N';
LDRINV >= 1 if MULTRC = 'P'.
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array must contain C ,
-1/2 i+1
the output weight matrix (or the product R C if
i+1 i+1
MULTRC = 'P') of the discrete system at instant i+1.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
QINV (input/output) DOUBLE PRECISION array, dimension
(LDQINV,M)
On entry, the leading M-by-M upper triangular part of this
-1/2
array must contain Q , the inverse of the covariance
i
square root (right Cholesky factor) of the input (process)
noise (hence the information square root) at instant i.
On exit, the leading M-by-M upper triangular part of this
-1/2
array contains (QINOV ) , the inverse of the covariance
i
square root (right Cholesky factor) of the process noise
innovation (hence the information square root) at
instant i.
The strict lower triangular part of this array is not
referenced.
LDQINV INTEGER
The leading dimension of array QINV. LDQINV >= MAX(1,M).
X (input/output) DOUBLE PRECISION array, dimension (N)
On entry, this array must contain X , the estimated
i
filtered state at instant i.
On exit, if JOBX = 'X', and INFO = 0, then this array
contains X , the estimated filtered state at
i+1
instant i+1.
On exit, if JOBX = 'N', or JOBX = 'X' and INFO = 1, then
-1
this array contains S X .
i+1 i+1
RINVY (input) DOUBLE PRECISION array, dimension (P)
-1/2
This array must contain R Y , the product of the
i+1 i+1
-1/2
upper triangular matrix R and the measured output
i+1
vector Y at instant i+1.
i+1
Z (input) DOUBLE PRECISION array, dimension (M)
This array must contain Z , the mean value of the state
i
process noise at instant i.
E (output) DOUBLE PRECISION array, dimension (P)
This array contains E , the estimated error at instant
i+1
i+1.
Tolerances
TOL DOUBLE PRECISION
If JOBX = 'X', then TOL is used to test for near
-1
singularity of the matrix S . If the user sets
i+1
TOL > 0, then the given value of TOL is used as a
lower bound for the reciprocal condition number of that
matrix; a matrix whose estimated condition number is less
than 1/TOL is considered to be nonsingular. If the user
sets TOL <= 0, then an implicitly computed, default
tolerance, defined by TOLDEF = N*N*EPS, is used instead,
where EPS is the machine precision (see LAPACK Library
routine DLAMCH).
Otherwise, TOL is not referenced.
Workspace
IWORK INTEGER array, dimension (LIWORK)
where LIWORK = N if JOBX = 'X',
and LIWORK = 1 otherwise.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK. If INFO = 0 and JOBX = 'X', DWORK(2) returns
an estimate of the reciprocal of the condition number
-1
(in the 1-norm) of S .
i+1
LDWORK The length of the array DWORK.
LDWORK >= MAX(1,N*(N+2*M)+3*M,(N+P)*(N+1)+N+MAX(N-1,M+1)),
if JOBX = 'N';
LDWORK >= MAX(2,N*(N+2*M)+3*M,(N+P)*(N+1)+N+MAX(N-1,M+1),
3*N), if JOBX = 'X'.
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value; -1
= 1: if JOBX = 'X' and the matrix S is singular,
i+1 -1
i.e., the condition number estimate of S (in the
i+1
-1 -1/2
1-norm) exceeds 1/TOL. The matrices S , Q
i+1 i
and E have been computed.
Method
The routine performs one recursion of the square root information
filter algorithm, summarized as follows:
| -1/2 -1/2 | | -1/2 |
| Q 0 Q Z | | (QINOV ) * * |
| i i i | | i |
| | | |
| -1/2 -1/2 | | -1 -1 |
T | 0 R C R Y | = | 0 S S X |
| i+1 i+1 i+1 i+1| | i+1 i+1 i+1|
| | | |
| -1 -1 -1 -1 -1 | | |
| S A B S A S X | | 0 0 E |
| i i i i | | i+1 |
(Pre-array) (Post-array)
where T is an orthogonal transformation triangularizing the
-1/2
pre-array, (QINOV ) is the inverse of the covariance square
i
root (right Cholesky factor) of the process noise innovation
-1 -1
(hence the information square root) at instant i and (A ,A B) is
in upper controller Hessenberg form.
An example of the pre-array is given below (where N = 6, M = 2,
and P = 3):
|x x | | x|
| x | | x|
_______________________
| | x x x x x x | x|
| | x x x x x x | x|
| | x x x x x x | x|
_______________________
|x x | x x x x x x | x|
| x | x x x x x x | x|
| | x x x x x x | x|
| | x x x x x | x|
| | x x x x | x|
| | x x x | x|
The inverse of the corresponding state covariance matrix P
i+1|i+1
(hence the information matrix I) is then factorized as
-1 -1 -1
I = P = (S )' S
i+1|i+1 i+1|i+1 i+1 i+1
and one combined time and measurement update for the state is
given by X .
i+1
The triangularization is done entirely via Householder
transformations exploiting the zero pattern of the pre-array.
References
[1] Anderson, B.D.O. and Moore, J.B.
Optimal Filtering.
Prentice Hall, Englewood Cliffs, New Jersey, 1979.
[2] Van Dooren, P. and Verhaegen, M.H.G.
Condensed Forms for Efficient Time-Invariant Kalman Filtering.
SIAM J. Sci. Stat. Comp., 9. pp. 516-530, 1988.
[3] Verhaegen, M.H.G. and Van Dooren, P.
Numerical Aspects of Different Kalman Filter Implementations.
IEEE Trans. Auto. Contr., AC-31, pp. 907-917, Oct. 1986.
[4] Vanbegin, M., Van Dooren, P., and Verhaegen, M.H.G.
Algorithm 675: FORTRAN Subroutines for Computing the Square
Root Covariance Filter and Square Root Information Filter in
Dense or Hessenberg Forms.
ACM Trans. Math. Software, 15, pp. 243-256, 1989.
Numerical Aspects
The algorithm requires approximately
3 2 2 3
(1/6)N + N x (3/2 x M + P) + 2 x N x M + 2/3 x M
operations and is backward stable (see [3]).
Further Comments
NoneExample
Program Text
* FB01TD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDAINB, LDAINV, LDC, LDQINV, LDRINV, LDSINV
PARAMETER ( LDAINB = NMAX, LDAINV = NMAX, LDC = PMAX,
$ LDQINV = MMAX, LDRINV = PMAX, LDSINV = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( NMAX*(NMAX + 2*MMAX) + 3*MMAX,
$ (NMAX + PMAX)*(NMAX + 1) + NMAX +
$ MAX( NMAX - 1, MMAX + 1 ),
$ 3*NMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, ISTEP, J, M, N, P
CHARACTER*1 JOBX, MULTRC
* .. Local Arrays ..
DOUBLE PRECISION AINV(LDAINV,NMAX), AINVB(LDAINB,MMAX),
$ C(LDC,NMAX), DIAG(MMAX), DWORK(LDWORK), E(PMAX),
$ QINV(LDQINV,MMAX), RINV(LDRINV,PMAX),
$ RINVY(PMAX), SINV(LDSINV,NMAX), X(NMAX), Z(MMAX)
INTEGER IWORK(NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL DCOPY, FB01TD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, JOBX, TOL, MULTRC
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( AINV(I,J), J = 1,N ), I = 1,N )
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
IF ( LSAME( MULTRC, 'N' ) ) READ ( NIN, FMT = * )
$ ( ( RINV(I,J), J = 1,P ), I = 1,P )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) M
ELSE
READ ( NIN, FMT = * )
$ ( ( AINVB(I,J), J = 1,M ), I = 1,N )
READ ( NIN, FMT = * ) ( ( QINV(I,J), J = 1,M ), I = 1,M )
READ ( NIN, FMT = * ) ( ( SINV(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( Z(J), J = 1,M )
READ ( NIN, FMT = * ) ( X(J), J = 1,N )
READ ( NIN, FMT = * ) ( RINVY(J), J = 1,P )
* Save the strict upper triangle of QINV in its strict
* lower triangle and the diagonal in the array DIAG.
DO 10 I = 2, M
CALL DCOPY( I, QINV(1,I), 1, QINV(I,1), LDQINV )
10 CONTINUE
CALL DCOPY( M, QINV, LDQINV+1, DIAG, 1 )
* Perform three iterations of the (Kalman) filter
* recursion (in square root information form).
ISTEP = 1
20 CONTINUE
CALL FB01TD( JOBX, MULTRC, N, M, P, SINV, LDSINV,
$ AINV, LDAINV, AINVB, LDAINB, RINV,
$ LDRINV, C, LDC, QINV, LDQINV, X, RINVY,
$ Z, E, TOL, IWORK, DWORK, LDWORK, INFO )
ISTEP = ISTEP + 1
IF ( INFO.EQ.0 .AND. ISTEP.LE.3 ) THEN
* Restore the upper triangle of QINV.
DO 30 I = 2, M
CALL DCOPY( I, QINV(I,1), LDQINV, QINV(1,I), 1 )
30 CONTINUE
CALL DCOPY( M, DIAG, 1, QINV, LDQINV+1 )
GO TO 20
END IF
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( SINV(I,J), J = 1,N )
40 CONTINUE
IF ( LSAME( JOBX, 'X' ) ) THEN
WRITE ( NOUT, FMT = 99995 )
DO 50 I = 1, N
WRITE ( NOUT, FMT = 99994 ) I, X(I)
50 CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' FB01TD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from FB01TD = ',I2)
99997 FORMAT (' The inverse of the square root of the state covariance',
$ ' matrix is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The components of the estimated filtered state are ',
$ //' k X(k)',/)
99994 FORMAT (I4,3X,F8.4)
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' M is out of range.',/' M = ',I5)
99991 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
FB01TD EXAMPLE PROGRAM DATA 4 2 2 X 0.0 N 0.2113 0.7560 0.0002 0.3303 0.8497 0.6857 0.8782 0.0683 0.7263 0.1985 0.5442 0.2320 0.0000 0.6525 0.3076 0.9329 0.3616 0.5664 0.5015 0.2693 0.2922 0.4826 0.4368 0.6325 1.0000 0.0000 0.0000 1.0000 -0.8805 1.3257 0.0000 0.5207 0.0000 0.0000 0.0000 0.0000 1.1159 0.2305 0.0000 0.6597 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0019 0.5075 0.4076 0.8408 0.5017 0.9128 0.2129 0.5591Program Results
FB01TD EXAMPLE PROGRAM RESULTS The inverse of the square root of the state covariance matrix is -0.8731 -1.1461 -1.0260 -0.8901 0.0000 -0.2763 -0.1929 -0.3763 0.0000 0.0000 -0.1110 -0.1051 0.0000 0.0000 0.0000 0.3120 The components of the estimated filtered state are k X(k) 1 -2.0688 2 -0.7814 3 2.2181 4 0.9298
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/FB01VD.html 0000664 0000000 0000000 00000031644 14560147231 0020344 0 ustar 00root root 0000000 0000000
Purpose
To compute one recursion of the conventional Kalman filter equations. This is one update of the Riccati difference equation and the Kalman filter gain.Specification
SUBROUTINE FB01VD( N, M, L, P, LDP, A, LDA, B, LDB, C, LDC, Q,
$ LDQ, R, LDR, K, LDK, TOL, IWORK, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
INTEGER INFO, L, LDA, LDB, LDC, LDK, LDP, LDQ, LDR,
$ LDWORK, M, N
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
$ K(LDK,*), P(LDP,*), Q(LDQ,*), R(LDR,*)
Arguments
Input/Output Parameters
N (input) INTEGER
The actual state dimension, i.e., the order of the
matrices P and A . N >= 0.
i|i-1 i
M (input) INTEGER
The actual input dimension, i.e., the order of the matrix
Q . M >= 0.
i
L (input) INTEGER
The actual output dimension, i.e., the order of the matrix
R . L >= 0.
i
P (input/output) DOUBLE PRECISION array, dimension (LDP,N)
On entry, the leading N-by-N part of this array must
contain P , the state covariance matrix at instant
i|i-1
(i-1). The upper triangular part only is needed.
On exit, if INFO = 0, the leading N-by-N part of this
array contains P , the state covariance matrix at
i+1|i
instant i. The strictly lower triangular part is not set.
Otherwise, the leading N-by-N part of this array contains
P , its input value.
i|i-1
LDP INTEGER
The leading dimension of array P. LDP >= MAX(1,N).
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain A ,
i
the state transition matrix of the discrete system at
instant i.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain B ,
i
the input weight matrix of the discrete system at
instant i.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading L-by-N part of this array must contain C ,
i
the output weight matrix of the discrete system at
instant i.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,L).
Q (input) DOUBLE PRECISION array, dimension (LDQ,M)
The leading M-by-M part of this array must contain Q ,
i
the input (process) noise covariance matrix at instant i.
The diagonal elements of this array are modified by the
routine, but are restored on exit.
LDQ INTEGER
The leading dimension of array Q. LDQ >= MAX(1,M).
R (input/output) DOUBLE PRECISION array, dimension (LDR,L)
On entry, the leading L-by-L part of this array must
contain R , the output (measurement) noise covariance
i
matrix at instant i.
On exit, if INFO = 0, or INFO = L+1, the leading L-by-L
1/2
upper triangular part of this array contains (RINOV ) ,
i
the square root (left Cholesky factor) of the covariance
matrix of the innovations at instant i.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,L).
K (output) DOUBLE PRECISION array, dimension (LDK,L)
If INFO = 0, the leading N-by-L part of this array
contains K , the Kalman filter gain matrix at instant i.
i
If INFO > 0, the leading N-by-L part of this array
contains the matrix product P C'.
i|i-1 i
LDK INTEGER
The leading dimension of array K. LDK >= MAX(1,N).
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used to test for near singularity of
the matrix RINOV . If the user sets TOL > 0, then the
i
given value of TOL is used as a lower bound for the
reciprocal condition number of that matrix; a matrix whose
estimated condition number is less than 1/TOL is
considered to be nonsingular. If the user sets TOL <= 0,
then an implicitly computed, default tolerance, defined by
TOLDEF = L*L*EPS, is used instead, where EPS is the
machine precision (see LAPACK Library routine DLAMCH).
Workspace
IWORK INTEGER array, dimension (L)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, or INFO = L+1, DWORK(1) returns an
estimate of the reciprocal of the condition number (in the
1-norm) of the matrix RINOV .
i
LDWORK The length of the array DWORK.
LDWORK >= MAX(1,L*N+3*L,N*N,N*M).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -k, the k-th argument had an illegal
value;
= k: if INFO = k, 1 <= k <= L, the leading minor of order
k of the matrix RINOV is not positive-definite, and
i
its Cholesky factorization could not be completed;
= L+1: the matrix RINOV is singular, i.e., the condition
i
number estimate of RINOV (in the 1-norm) exceeds
i
1/TOL.
Method
The conventional Kalman filter gain used at the i-th recursion
step is of the form
-1
K = P C' RINOV ,
i i|i-1 i i
where RINOV = C P C' + R , and the state covariance matrix
i i i|i-1 i i
P is updated by the discrete-time difference Riccati equation
i|i-1
P = A (P - K C P ) A' + B Q B'.
i+1|i i i|i-1 i i i|i-1 i i i i
Using these two updates, the combined time and measurement update
of the state X is given by
i|i-1
X = A X + A K (Y - C X ),
i+1|i i i|i-1 i i i i i|i-1
where Y is the new observation at step i.
i
References
[1] Anderson, B.D.O. and Moore, J.B.
Optimal Filtering,
Prentice Hall, Englewood Cliffs, New Jersey, 1979.
[2] Verhaegen, M.H.G. and Van Dooren, P.
Numerical Aspects of Different Kalman Filter Implementations.
IEEE Trans. Auto. Contr., AC-31, pp. 907-917, 1986.
Numerical Aspects
The algorithm requires approximately
3 2
3/2 x N + N x (3 x L + M/2)
operations.
Further Comments
NoneExample
Program Text
* FB01VD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, LMAX
PARAMETER ( NMAX = 20, MMAX = 20, LMAX = 20 )
INTEGER LDA, LDB, LDC, LDK, LDP, LDQ, LDR
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = LMAX, LDK = NMAX,
$ LDP = NMAX, LDQ = MMAX, LDR = LMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( LMAX*NMAX + 3*LMAX, NMAX*NMAX,
$ MMAX*NMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, J, L, M, N
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ DWORK(LDWORK), K(LDK,LMAX), P(LDP,NMAX),
$ Q(LDQ,MMAX), R(LDR,LMAX)
INTEGER IWORK(LMAX)
* .. External Subroutines ..
EXTERNAL DCOPY, FB01VD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, L, TOL
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( P(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
READ ( NIN, FMT = * ) ( ( Q(I,J), J = 1,M ), I = 1,M )
IF ( L.LE.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) L
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,L )
READ ( NIN, FMT = * ) ( ( R(I,J), J = 1,L ), I = 1,L )
* Perform one iteration of the (Kalman) filter recursion.
CALL FB01VD( N, M, L, P, LDP, A, LDA, B, LDB, C, LDC,
$ Q, LDQ, R, LDR, K, LDK, TOL, IWORK, DWORK,
$ LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
CALL DCOPY( I-1, P(1,I), 1, P(I,1), LDP )
WRITE ( NOUT, FMT = 99994 ) ( P(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99994 ) ( K(I,J), J = 1,L )
40 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 60 I = 1, L
WRITE ( NOUT, FMT = 99994 ) ( R(I,J), J = 1,L )
60 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' FB01VD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from FB01VD = ',I3)
99997 FORMAT (' The state covariance matrix is ')
99996 FORMAT (/' The Kalman filter gain matrix is ')
99995 FORMAT (/' The square root of the covariance matrix of the innov',
$ 'ations is ')
99994 FORMAT (20(1X,F8.4))
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' M is out of range.',/' M = ',I5)
99991 FORMAT (/' L is out of range.',/' P = ',I5)
END
Program Data
FB01VD EXAMPLE PROGRAM DATA 4 3 2 0.0 0.5015 0.4368 0.2693 0.6325 0.4368 0.4818 0.2639 0.4148 0.2693 0.2639 0.1121 0.6856 0.6325 0.4148 0.6856 0.8906 0.2113 0.8497 0.7263 0.8833 0.7560 0.6857 0.1985 0.6525 0.0002 0.8782 0.5442 0.3076 0.3303 0.0683 0.2320 0.9329 0.0437 0.7783 0.5618 0.4818 0.2119 0.5896 0.2639 0.1121 0.6853 0.4148 0.6856 0.8906 0.9329 0.2146 0.3126 0.2146 0.2922 0.5664 0.3126 0.5664 0.5935 0.3873 0.9488 0.3760 0.0881 0.9222 0.3435 0.7340 0.4498 1.0000 0.0000 0.0000 1.0000Program Results
FB01VD EXAMPLE PROGRAM RESULTS The state covariance matrix is 1.6007 1.3283 1.1153 1.7177 1.3283 1.2763 1.0132 1.5137 1.1153 1.0132 0.8222 1.2722 1.7177 1.5137 1.2722 2.1562 The Kalman filter gain matrix is 0.1648 0.2241 0.2115 0.1610 0.0728 0.1673 0.1304 0.3892 The square root of the covariance matrix of the innovations is 1.5091 1.1543 0.0000 1.5072
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/FD01AD.html 0000664 0000000 0000000 00000034063 14560147231 0020317 0 ustar 00root root 0000000 0000000
Purpose
To solve the least-squares filtering problem recursively in time. Each subroutine call implements one time update of the solution. The algorithm uses a fast QR-decomposition based approach.Specification
SUBROUTINE FD01AD( JP, L, LAMBDA, XIN, YIN, EFOR, XF, EPSBCK,
$ CTETA, STETA, YQ, EPOS, EOUT, SALPH, IWARN,
$ INFO )
C .. Scalar Arguments ..
CHARACTER JP
INTEGER INFO, IWARN, L
DOUBLE PRECISION EFOR, EOUT, EPOS, LAMBDA, XIN, YIN
C .. Array Arguments ..
DOUBLE PRECISION CTETA(*), EPSBCK(*), SALPH(*), STETA(*), XF(*),
$ YQ(*)
Arguments
Mode Parameters
JP CHARACTER*1
Indicates whether the user wishes to apply both prediction
and filtering parts, as follows:
= 'B': Both prediction and filtering parts are to be
applied;
= 'P': Only the prediction section is to be applied.
Input/Output Parameters
L (input) INTEGER
The length of the impulse response of the equivalent
transversal filter model. L >= 1.
LAMBDA (input) DOUBLE PRECISION
Square root of the forgetting factor.
For tracking capabilities and exponentially stable error
propagation, LAMBDA < 1.0 (strict inequality) should
be used. 0.0 < LAMBDA <= 1.0.
XIN (input) DOUBLE PRECISION
The input sample at instant n.
(The situation just before and just after the call of
the routine are denoted by instant (n-1) and instant n,
respectively.)
YIN (input) DOUBLE PRECISION
If JP = 'B', then YIN must contain the reference sample
at instant n.
Otherwise, YIN is not referenced.
EFOR (input/output) DOUBLE PRECISION
On entry, this parameter must contain the square root of
exponentially weighted forward prediction error energy
at instant (n-1). EFOR >= 0.0.
On exit, this parameter contains the square root of the
exponentially weighted forward prediction error energy
at instant n.
XF (input/output) DOUBLE PRECISION array, dimension (L)
On entry, this array must contain the transformed forward
prediction variables at instant (n-1).
On exit, this array contains the transformed forward
prediction variables at instant n.
EPSBCK (input/output) DOUBLE PRECISION array, dimension (L+1)
On entry, the leading L elements of this array must
contain the normalized a posteriori backward prediction
error residuals of orders zero through L-1, respectively,
at instant (n-1), and EPSBCK(L+1) must contain the
square-root of the so-called "conversion factor" at
instant (n-1).
On exit, this array contains the normalized a posteriori
backward prediction error residuals, plus the square root
of the conversion factor at instant n.
CTETA (input/output) DOUBLE PRECISION array, dimension (L)
On entry, this array must contain the cosines of the
rotation angles used in time updates, at instant (n-1).
On exit, this array contains the cosines of the rotation
angles at instant n.
STETA (input/output) DOUBLE PRECISION array, dimension (L)
On entry, this array must contain the sines of the
rotation angles used in time updates, at instant (n-1).
On exit, this array contains the sines of the rotation
angles at instant n.
YQ (input/output) DOUBLE PRECISION array, dimension (L)
On entry, if JP = 'B', then this array must contain the
orthogonally transformed reference vector at instant
(n-1). These elements are also the tap multipliers of an
equivalent normalized lattice least-squares filter.
Otherwise, YQ is not referenced and can be supplied as
a dummy array (i.e., declare this array to be YQ(1) in
the calling program).
On exit, if JP = 'B', then this array contains the
orthogonally transformed reference vector at instant n.
EPOS (output) DOUBLE PRECISION
The a posteriori forward prediction error residual.
EOUT (output) DOUBLE PRECISION
If JP = 'B', then EOUT contains the a posteriori output
error residual from the least-squares filter at instant n.
SALPH (output) DOUBLE PRECISION array, dimension (L)
The element SALPH(i), i=1,...,L, contains the opposite of
the i-(th) reflection coefficient for the least-squares
normalized lattice predictor (whose value is -SALPH(i)).
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: an element to be annihilated by a rotation is less
than the machine precision (see LAPACK Library
routine DLAMCH).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The output error EOUT at instant n, denoted by EOUT(n), is the
reference sample minus a linear combination of L successive input
samples:
L-1
EOUT(n) = YIN(n) - SUM h_i * XIN(n-i),
i=0
where YIN(n) and XIN(n) are the scalar samples at instant n.
A least-squares filter uses those h_0,...,h_{L-1} which minimize
an exponentially weighted sum of successive output errors squared:
n
SUM [LAMBDA**(2(n-k)) * EOUT(k)**2].
k=1
Each subroutine call performs a time update of the least-squares
filter using a fast least-squares algorithm derived from a
QR decomposition, as described in references [1] and [2] (the
notation from [2] is followed in the naming of the arrays).
The algorithm does not compute the parameters h_0,...,h_{L-1} from
the above formula, but instead furnishes the parameters of an
equivalent normalized least-squares lattice filter, which are
available from the arrays SALPH (reflection coefficients) and YQ
(tap multipliers), as well as the exponentially weighted input
signal energy
n L
SUM [LAMBDA**(2(n-k)) * XIN(k)**2] = EFOR**2 + SUM XF(i)**2.
k=1 i=1
For more details on reflection coefficients and tap multipliers,
references [2] and [4] are recommended.
References
[1] Proudler, I. K., McWhirter, J. G., and Shepherd, T. J.
Fast QRD based algorithms for least-squares linear
prediction.
Proceedings IMA Conf. Mathematics in Signal Processing
Warwick, UK, December 1988.
[2] Regalia, P. A., and Bellanger, M. G.
On the duality between QR methods and lattice methods in
least-squares adaptive filtering.
IEEE Trans. Signal Processing, SP-39, pp. 879-891,
April 1991.
[3] Regalia, P. A.
Numerical stability properties of a QR-based fast
least-squares algorithm.
IEEE Trans. Signal Processing, SP-41, June 1993.
[4] Lev-Ari, H., Kailath, T., and Cioffi, J.
Least-squares adaptive lattice and transversal filters:
A unified geometric theory.
IEEE Trans. Information Theory, IT-30, pp. 222-236,
March 1984.
Numerical Aspects
The algorithm requires O(L) operations for each subroutine call. It is backward consistent for all input sequences XIN, and backward stable for persistently exciting input sequences, assuming LAMBDA < 1.0 (see [3]). If the condition of the signal is very poor (IWARN = 1), then the results are not guaranteed to be reliable.Further Comments
1. For tracking capabilities and exponentially stable error
propagation, LAMBDA < 1.0 should be used. LAMBDA is typically
chosen slightly less than 1.0 so that "past" data are
exponentially forgotten.
2. Prior to the first subroutine call, the variables must be
initialized. The following initial values are recommended:
XF(i) = 0.0, i=1,...,L
EPSBCK(i) = 0.0 i=1,...,L
EPSBCK(L+1) = 1.0
CTETA(i) = 1.0 i=1,...,L
STETA(i) = 0.0 i=1,...,L
YQ(i) = 0.0 i=1,...,L
EFOR = 0.0 (exact start)
EFOR = "small positive constant" (soft start).
Soft starts are numerically more reliable, but result in a
biased least-squares solution during the first few iterations.
This bias decays exponentially fast provided LAMBDA < 1.0.
If sigma is the standard deviation of the input sequence
XIN, then initializing EFOR = sigma*1.0E-02 usually works
well.
Example
Program Text
* FD01AD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT, NOUT1
PARAMETER ( NIN = 5, NOUT = 6, NOUT1 = 7 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
INTEGER IMAX, LMAX
PARAMETER ( IMAX = 500, LMAX = 10 )
DOUBLE PRECISION LAMBDA
PARAMETER ( LAMBDA = 0.99D0 )
* .. Local Scalars ..
CHARACTER JP
INTEGER I, INFO, IWARN, L
DOUBLE PRECISION DELTA, EFOR, EOUT, EPOS, XIN, YIN
* .. Local Arrays ..
DOUBLE PRECISION CTETA(LMAX), EPSBCK(LMAX+1), SALPH(LMAX),
$ STETA(LMAX), XF(LMAX), YQ(LMAX)
* .. External Functions ..
DOUBLE PRECISION XFCN, YFCN
EXTERNAL XFCN, YFCN
* NOTE: XFCN() generates at each iteration the next sample of the
* input sequence. YFCN() generates at each iteration the next
* sample of the reference sequence. These functions are user
* defined (obtained from data acquisition devices, for
* example).
* .. External Subroutines ..
EXTERNAL FD01AD
*
* .. File for the output error sequence ..
OPEN ( UNIT = NOUT1, FILE = 'ERR.OUT', STATUS = 'REPLACE' )
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) L, DELTA, JP
IF ( L.LE.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) L
ELSE
IF ( DELTA.LT.ZERO ) THEN
WRITE ( NOUT, FMT = 99991 )
ELSE
*
DO 10 I = 1, L
CTETA(I) = ONE
STETA(I) = ZERO
EPSBCK(I) = ZERO
XF(I) = ZERO
YQ(I) = ZERO
10 CONTINUE
EPSBCK(L+1) = ONE
EFOR = DELTA
* .. Run least squares filter.
DO 20 I = 1, IMAX
XIN = XFCN(I)
YIN = YFCN(I)
CALL FD01AD( JP, L, LAMBDA, XIN, YIN, EFOR, XF, EPSBCK,
$ CTETA, STETA, YQ, EPOS, EOUT, SALPH, IWARN,
$ INFO)
WRITE(NOUT1,*) EOUT
20 CONTINUE
CLOSE(NOUT1)
* NOTE: File 'ERR.OUT' now contains the output error
* sequence.
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 30 I = 1, L
WRITE ( NOUT, FMT = 99996 ) I, XF(I), YQ(I), EPSBCK(I)
30 CONTINUE
WRITE ( NOUT, FMT = 99995 ) L+1, EPSBCK(L+1)
WRITE ( NOUT, FMT = 99994 ) EFOR
IF ( IWARN.NE.0 ) THEN
WRITE ( NOUT, FMT = 99993 ) IWARN
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' FD01AD EXAMPLE PROGRAM RESULTS', /1X)
99998 FORMAT (' INFO on exit from FD01AD = ', I2)
99997 FORMAT (' i', 7X, 'XF(i)', 7X, 'YQ(i)', 6X, 'EPSBCK(i)', /1X)
99996 FORMAT ( I3, 2X, 3(2X, F10.6))
99995 FORMAT ( I3, 28X, F10.6, /1X)
99994 FORMAT (' EFOR = ', D10.3)
99993 FORMAT (' IWARN on exit from FD01AD = ', I2)
99992 FORMAT (/' L is out of range.',/' L = ',I5)
99991 FORMAT (/' The exponentially weighted forward prediction error',
$ ' energy must be non-negative.' )
*
END
*
* .. Example functions ..
*
DOUBLE PRECISION FUNCTION XFCN( I )
* .. Intrinsic Functions ..
INTRINSIC DBLE, SIN
* .. Local Scalar ..
INTEGER I
* .. Executable Statements ..
XFCN = SIN( 0.3D0*DBLE( I ) )
* *** Last line of XFCN ***
END
*
DOUBLE PRECISION FUNCTION YFCN( I )
* .. Intrinsic Functions ..
INTRINSIC DBLE, SIN
* .. Local Scalar ..
INTEGER I
* .. Executable Statements ..
YFCN = 0.5D0 * SIN( 0.3D0*DBLE( I ) ) +
$ 2.0D0 * SIN( 0.3D0*DBLE( I-1 ) )
* *** Last line of YFCN ***
END
Program Data
FD01AD EXAMPLE PROGRAM DATA 2 1.0D-2 BProgram Results
FD01AD EXAMPLE PROGRAM RESULTS i XF(i) YQ(i) EPSBCK(i) 1 4.880088 12.307615 -0.140367 2 -1.456881 2.914057 -0.140367 3 0.980099 EFOR = 0.197D-02
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/FSQP.html 0000664 0000000 0000000 00000040555 14560147231 0020234 0 ustar 00root root 0000000 0000000
Purpose
To perform the minimization of the maximum of a set of smooth
objective functions (possibly a single one or none at all) subject
to nonlinear equality and inequality constraints, linear equality
and inequality constraints, and simple bounds on the variables.
Specifically, the problem to solve is of the form
min max{f (x)}
i i
n
where 1 <= i <= NF, and x is a vector in R subject to the
following constraints
bl <= x <= bu
g (x)<=0, j = 1 ... NINEQN
j
g (x)<=0, j = NINEQN+1 ... NINEQ
j
h (x)=0, j = 1 ... NEQN
j
h (x)=0, j = NEQN+1 ... NEQ
j
n
where bl and bu are vectors in R , and
n
f : R -> R, j = 1 ... NF, smooth
j
n
g : R -> R, j = 1 ... NINEQN, nonlinear and smooth
j
n
g : R -> R, j = NINEQN+1 ... NINEQ, linear
j
n
h : R -> R, j = 1 ... NEQN, nonlinear and smooth
j
n
h : R -> R, j = NEQN+1 ... NEQ, linear
j
Specification
SUBROUTINE FSQP( MODE, IPRINT, OBJ, CONSTR, GRADOB, GRADCN,
& NPARAM, NF, NINEQN, NINEQ, NEQN, NEQ, MITER,
& UDELTA, BIGBND, BL, BU, X, F, G, TOL1, TOL2,
& IWORK, LIWORK, DWORK, LDWORK, INFO)
C .. Scalar Arguments ..
INTEGER NPARAM, NF, NINEQN, NINEQ, NEQN, NEQ, MODE,
& IPRINT, MITER, INFO, LIWORK, LDWORK
DOUBLE PRECISION BIGBND, TOL1, TOL2, UDELTA
C .. Array Arguments ..
INTEGER IWORK(LIWORK)
DOUBLE PRECISION BL(*), BU(*), X(*), F(*), G(*), DWORK(LDWORK)
Arguments
Mode Parameters
MODE (input) INTEGER
mode = CBA (a 3-digit number) with the following meaning
(see [1] for more details):
A specifies the problem to be solved:
A = 0: The problem is that described above.
A = 1: In the problem described above, the function to
minimize is replaced by
min max{ABS(f (x))}
i i
i.e., absolute values of the objective functions
are taken.
B indicates the method to be used:
B = 0: Algorithm FFSQP-AL is selected (see METHOD below).
B = 1: Algorithm FFSQP-NL is selected (see METHOD below).
C indicates the order of evaluation of objective
functions and constraints during the line search:
C = 1: The function that caused the previous value of t
to be rejected is checked first and all functions
of the same type ("objective" or "constraint") as
the latter will then be checked first (recommended
for most users).
C = 2: Constraints will be always checked first at each
trial point during the line search. If it is a
contraint that caused the previous value of t to
be rejected, that constraint will be checked first
(useful when objective functions are not defined or
are difficult to evaluate outside of the feasible
region).
IPRINT (input) INTEGER
Parameter indicating the desired output (see [1] for a
more complete description of the output).
IPRINT = 0: No information except for user-input errors
is displayed. This value is imposed during phase 1.
IPRINT = 1: Objective and constraint values at the
initial feasible point are displayed. At the end of
execution, status (INFO), iterate, objective values,
constraint values, number of evaluations of objectives
and nonlinear constraints, norm of the Kuhn-Tucker
vector, and sum of feasibility violation are
displayed.
IPRINT = 2: At the end of each iteration, the same
information as with IPRINT = 1 is displayed.
IPRINT = 3: At each iteration, the same information as
with IPRINT = 2,including detailed information on the
search direction computation, on the line search, and
on the update, is displayed.
IPRINT = 10*N +M: N any positive integer, M=2 or 3.
Information corresponding to IPRINT=M is displayed at
every (10*N)th iteration and at the last iteration.
User-supplied Subroutines
OBJ SUBROUTINE
Computes the value of the objective functions. If NF = 0,
a (dummy) subroutine must be provided anyway. The
specification of OBJ is
SUBROUTINE OBJ(NPARAM,J,X,FJ)
INTEGER NPARAM, J
DOUBLE PRECISION X(NPARAM),FJ
Arguments:
NPARAM (Input) Dimension of X.
J (Input) Number of the objective to be computed.
X (Input) Current iterate.
FJ (Output) Value of the jth objective function at X.
CONSTR SUBROUTINE
Computes the value of the constraints. If there are no
constraints, a (dummy) subroutine must be provided anyway.
The specification of CONSTR is as follows.
SUBROUTINE CONSTR(NPARAM,J,X,GJ)
INTEGER NPARAM,J
DOUBLE PRECISION X(NPARAM),GJ
Arguments:
NPARAM (Input) Dimension of X.
J (Input) Number of the constraint to be computed.
X (Input) Current iterate.
GJ (Output) Value of the jth constraint at X.
The order of the constraints must be as follows. First
the NINEQN (possibly zero) nonlinear inequality
constraints. Then the NINEQ-NINEQN (possibly zero) linear
inequality constraints. Finally, the NEQN (possibly zero)
nonlinear equality constraints followed by the NEQ-NEQN
(possibly zero) linear equality constraints.
GRADOB SUBROUTINE
Computes the gradients of the objective functions. The
user must pass the subroutine name GROBFD, if he/she
wishes that FSQP evaluate these gradients automatically,
by forward finite differences. The specification of GRADOB
is as follows.
SUBROUTINE GRADOB(NPARAM,J,X,GRADFJ,DUMMY)
INTEGER NPARAM,J
DOUBLE PRECISION X(NPARAM),GRADFJ(NPARAM)
DOUBLE PRECISION DUMMY
EXTERNAL DUMMY
Arguments:
NPARAM (Input) Dimension of X.
J (Input) Number of objective for which gradient is to be
computed.
X (Input) Current iterate.
GRADFJ (Output) Gradient of the jth objective function at
X.
DUMMY (Input) Used by GROBFD (internally assigned the
name of the objective function subroutine by FFSQP).
Note that DUMMY is passed as argument to GRADOB to allow
for forward finite difference computation of the gradient.
GRADCN SUBROUTINE
Computes the gradients of the constraints. The
user must pass the subroutine name GRCNFD, if he/she
wishes that FSQP evaluate these gradients automatically,
by forward finite differences. The specification of GRADCN
is as follows
SUBROUTINE GRADCN (NPARAM,J,X,GRADGJ,DUMMY)
INTEGER NPARAM,J
DOUBLE PRECISION X(NPARAM),GRADGJ(NPARAM)
DOUBLE PRECISION DUMMY
EXTERNAL DUMMY
Arguments:
NPARAM (Input) Dimension of X.
J (Input) Number of constraint for which gradient is to
be computed.
X (Input) Current iterate.
GRADGJ (Output) Gradient of the jth constraint evaluated
at X.
DUMMY (Input) Used by GRCNFD (internally assigned the
name of the constraint function subroutine by
FFSQP).
Note that DUMMY is passed as argument to GRADCN to allow
for forward finite difference computation of the gradients.
Input/Output Parameters
NPARAM (input) INTEGER
Number of free variables, i.e., the dimension of X.
NF (input) INTEGER
Number of objective functions (possibly zero).
NINEQN (input) INTEGER
Number (possibly zero) of nonlinear inequality
constraints.
NINEQ (input) INTEGER
Total number (possibly equal to nineqn) of inequality
constraints.
NEQN (input) INTEGER
Number (possibly zero) of nonlinear equality constraints.
NEQ (input) INTEGER
Total number (possibly equal to neqn) of equality
constraints.
MITER (input) INTEGER
Maximum number of iterations allowed by the user before
termination of execution.
UDELTA (input) DOUBLE PRECISION
The perturbation size the user suggests to use in
approximating gradients by finite difference. UDELTA
should be set to zero if the user has no idea how to
choose it. See [1] for details.
BIGBND (input) DOUBLE PRECISION
It plays the role of Infinite Bound (see also BL and BU
below).
BL (input) DOUBLE PRECISION array, dimension (NPARAM)
Lower bounds for the components of X. To specify a non-
existent lower bound for some j, the value used must
satisfy BL(j) <= -BIGBND.
BU (input) DOUBLE PRECISION array, dimension (NPARAM)
Upper bounds for the components of X. To specify a non-
existent upper bound for some j, the value used must
satisfy BU(j) >= BIGBND.
X (input/output) DOUBLE PRECISION array, dimension (NPARAM)
On entry, this is the initial guess.
On exit, this is the iterate at the end of execution.
F (output) DOUBLE PRECISION array, dimension ( MAX(1,NF) )
Value of functions f_i, i = 1, ..., NF, at X at the end
of execution.
G (output) DOUBLE PRECISION array, dimension
( MAX(1,NINEQ+NEQ) )
Value of constraint functions at X at the end of
execution.
Tolerances
TOL1 DOUBLE PRECISION
Corresponds to argument EPS in [1].
Final norm requirement for the Newton direction (see [1],
argument EPS). It must be bigger than the machine
precision epsmac (computed by FSQP). If the user does
not have a good feeling of what value should be chosen,
a very small number could be provided and IPRINT = 2 be
selected so that the user would be able to keep track of
the process of optimization and terminate FSQP at
appropriate time.
TOL2 DOUBLE PRECISION
Corresponds to argument EPSEQN in [1].
Maximum violation of nonlinear equality constraints
allowed by the user at an optimal point. It is in effect
only if NEQN > 0 and must be bigger than the machine
precision epsmac (computed by FSQP).
Workspace
IWORK INTEGER array, dimension (LIWORK)
Corresponds to argument IW in [1].
LIWORK INTEGER
Corresponds to argument IWSIZE in [1].
The length of array IWORK. It must be at least as big as
6*NPARAM + 8*max(1,NINEQ+NEQ) + 7*max(1,NF) + 30. This
estimate is usually very conservative and the smallest
suitable value will be displayed if the user-supplied
value is too small.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
Corresponds to argument W in [1].
On exit, it will contain estimates of Lagrange
multipliers at the end of execution. These multipliers
will be placed in the first NPARAM + NINEQ + NEQ + NFF
entries; where NFF = 0 if (in mode) A = 0 and NF = 1, and
NFF = NF otherwise. See [1] for details.
LDWORK INTEGER
Corresponds to argument NWSIZE in [1].
The length of array DWORK. It must be at least as big as
4*SQR(NPARAM) + 5*MAX(1,NINEQ+NEQ)*NPARAM +
3*MAX(1,NF)*NPARAM + 26*(NPARAM+MAX(1,NF)) +
45*MAX(1,NINEQ+NEQ) + 100.
This estimate is usually very conservative and the
smallest suitable value will be displayed if the user-
supplied value is too small.
Error Indicator
INFO (output) INTEGER
Corresponds to argument INFORM in [1].
INFO = 0: Normal termination of execution.
INFO = 1: The user-provided initial guess is infeasible
for linear constraints and FFSQP is unable to generate
a point satisfying all these constraints.
INFO = 2: The user-provided initial guess is infeasible
for nonlinear inequality constraints and linear
constraints; and FFSQP is unable to generate a point
satisfying all these constraints.
INFO = 3: The maximum number miter of iterations has
been reached before a solution is obtained.
INFO = 4: The line search fails to find a new iterate
(trial step size being smaller than the machine
precision epsmac computed by FFSQP).
INFO = 5: Failure of the QP solver in attempting to
construct d0. A more robust QP solver may succeed.
INFO = 6: Failure of the QP solver in attempting to
construct d1 . A more robust QP solver may succeed.
INFO = 7: Input data are not consistent (with printout
indicating the error).
INFO = 8: Two consecutive iterates are numerically
equivalent before a stopping criterion is satisfied.
INFO = 9: One of the penalty parameters exceeded
BIGBND. The algorithm is having trouble satisfying a
nonlinear equality constraint.
Method
If the initial guess provided by the user is infeasible for nonlinear inequality constraints and linear constraints, FFSQP first generates a point satisfying all these constraints by iterating on the problem of minimizing the maximum of these constraints. Then, using Mayne-Polak's scheme, nonlinear equality constraints are turned into nonlinear inequality constraints and the original objective function is replaced by a modified objective function. After obtaining feasibility, either (i) an Armijo-type line search may be used (algorithm FFSQP-AL), yielding a monotone decrease of the objective function at each iteration; or (ii) a nonmonotone line search may be selected (algorithm FFSQP-NL), forcing a decrease of the objective function within at most four iterations. See [1] for further details.References
[1] J. L. Zhou, A. L. Tits and C. T. Lawrence
User's Guide for FFSQP Version 3.7 : A Fortran Code for
Solving Optimization Programs, Possibly Minimax, with General
Inequality Constraints and Linear Equality Constraints,
Generating Feasible Iterates
Institute for Systems Research, University of Maryland,
Technical Report SRC-TR-92-107r5, 1997.
(Available at http://gachinese.com/aemdesign/FSQPframe.htm)
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/IB01AD.html 0000664 0000000 0000000 00000134603 14560147231 0020321 0 ustar 00root root 0000000 0000000
Purpose
To preprocess the input-output data for estimating the matrices of a linear time-invariant dynamical system and to find an estimate of the system order. The input-output data can, optionally, be processed sequentially.Specification
SUBROUTINE IB01AD( METH, ALG, JOBD, BATCH, CONCT, CTRL, NOBR, M,
$ L, NSMP, U, LDU, Y, LDY, N, R, LDR, SV, RCOND,
$ TOL, IWORK, DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
DOUBLE PRECISION RCOND, TOL
INTEGER INFO, IWARN, L, LDR, LDU, LDWORK, LDY, M, N,
$ NOBR, NSMP
CHARACTER ALG, BATCH, CONCT, CTRL, JOBD, METH
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION DWORK(*), R(LDR, *), SV(*), U(LDU, *),
$ Y(LDY, *)
Arguments
Mode Parameters
METH CHARACTER*1
Specifies the subspace identification method to be used,
as follows:
= 'M': MOESP algorithm with past inputs and outputs;
= 'N': N4SID algorithm.
ALG CHARACTER*1
Specifies the algorithm for computing the triangular
factor R, as follows:
= 'C': Cholesky algorithm applied to the correlation
matrix of the input-output data;
= 'F': Fast QR algorithm;
= 'Q': QR algorithm applied to the concatenated block
Hankel matrices.
JOBD CHARACTER*1
Specifies whether or not the matrices B and D should later
be computed using the MOESP approach, as follows:
= 'M': the matrices B and D should later be computed
using the MOESP approach;
= 'N': the matrices B and D should not be computed using
the MOESP approach.
This parameter is not relevant for METH = 'N'.
BATCH CHARACTER*1
Specifies whether or not sequential data processing is to
be used, and, for sequential processing, whether or not
the current data block is the first block, an intermediate
block, or the last block, as follows:
= 'F': the first block in sequential data processing;
= 'I': an intermediate block in sequential data
processing;
= 'L': the last block in sequential data processing;
= 'O': one block only (non-sequential data processing).
NOTE that when 100 cycles of sequential data processing
are completed for BATCH = 'I', a warning is
issued, to prevent for an infinite loop.
CONCT CHARACTER*1
Specifies whether or not the successive data blocks in
sequential data processing belong to a single experiment,
as follows:
= 'C': the current data block is a continuation of the
previous data block and/or it will be continued
by the next data block;
= 'N': there is no connection between the current data
block and the previous and/or the next ones.
This parameter is not used if BATCH = 'O'.
CTRL CHARACTER*1
Specifies whether or not the user's confirmation of the
system order estimate is desired, as follows:
= 'C': user's confirmation;
= 'N': no confirmation.
If CTRL = 'C', a reverse communication routine, IB01OY,
is indirectly called (by SLICOT Library routine IB01OD),
and, after inspecting the singular values and system order
estimate, n, the user may accept n or set a new value.
IB01OY is not called if CTRL = 'N'.
Input/Output Parameters
NOBR (input) INTEGER
The number of block rows, s, in the input and output
block Hankel matrices to be processed. NOBR > 0.
(In the MOESP theory, NOBR should be larger than n,
the estimated dimension of state vector.)
M (input) INTEGER
The number of system inputs. M >= 0.
When M = 0, no system inputs are processed.
L (input) INTEGER
The number of system outputs. L > 0.
NSMP (input) INTEGER
The number of rows of matrices U and Y (number of
samples, t). (When sequential data processing is used,
NSMP is the number of samples of the current data
block.)
NSMP >= 2*(M+L+1)*NOBR - 1, for non-sequential
processing;
NSMP >= 2*NOBR, for sequential processing.
The total number of samples when calling the routine with
BATCH = 'L' should be at least 2*(M+L+1)*NOBR - 1.
The NSMP argument may vary from a cycle to another in
sequential data processing, but NOBR, M, and L should
be kept constant. For efficiency, it is advisable to use
NSMP as large as possible.
U (input) DOUBLE PRECISION array, dimension (LDU,M)
The leading NSMP-by-M part of this array must contain the
t-by-m input-data sequence matrix U,
U = [u_1 u_2 ... u_m]. Column j of U contains the
NSMP values of the j-th input component for consecutive
time increments.
If M = 0, this array is not referenced.
LDU INTEGER
The leading dimension of the array U.
LDU >= NSMP, if M > 0;
LDU >= 1, if M = 0.
Y (input) DOUBLE PRECISION array, dimension (LDY,L)
The leading NSMP-by-L part of this array must contain the
t-by-l output-data sequence matrix Y,
Y = [y_1 y_2 ... y_l]. Column j of Y contains the
NSMP values of the j-th output component for consecutive
time increments.
LDY INTEGER
The leading dimension of the array Y. LDY >= NSMP.
N (output) INTEGER
The estimated order of the system.
If CTRL = 'C', the estimated order has been reset to a
value specified by the user.
R (output or input/output) DOUBLE PRECISION array, dimension
( LDR,2*(M+L)*NOBR )
On exit, if ALG = 'C' and BATCH = 'F' or 'I', the leading
2*(M+L)*NOBR-by-2*(M+L)*NOBR upper triangular part of this
array contains the current upper triangular part of the
correlation matrix in sequential data processing.
If ALG = 'F' and BATCH = 'F' or 'I', the array R is not
referenced.
On exit, if INFO = 0, ALG = 'Q', and BATCH = 'F' or 'I',
the leading 2*(M+L)*NOBR-by-2*(M+L)*NOBR upper triangular
part of this array contains the current upper triangular
factor R from the QR factorization of the concatenated
block Hankel matrices. Denote R_ij, i,j = 1:4, the
ij submatrix of R, partitioned by M*NOBR, M*NOBR,
L*NOBR, and L*NOBR rows and columns.
On exit, if INFO = 0 and BATCH = 'L' or 'O', the leading
2*(M+L)*NOBR-by-2*(M+L)*NOBR upper triangular part of
this array contains the matrix S, the processed upper
triangular factor R from the QR factorization of the
concatenated block Hankel matrices, as required by other
subroutines. Specifically, let S_ij, i,j = 1:4, be the
ij submatrix of S, partitioned by M*NOBR, L*NOBR,
M*NOBR, and L*NOBR rows and columns. The submatrix
S_22 contains the matrix of left singular vectors needed
subsequently. Useful information is stored in S_11 and
in the block-column S_14 : S_44. For METH = 'M' and
JOBD = 'M', the upper triangular part of S_31 contains
the upper triangular factor in the QR factorization of the
matrix R_1c = [ R_12' R_22' R_11' ]', and S_12
contains the corresponding leading part of the transformed
matrix R_2c = [ R_13' R_23' R_14' ]'. For METH = 'N',
the subarray S_41 : S_43 contains the transpose of the
matrix contained in S_14 : S_34.
The details of the contents of R need not be known if this
routine is followed by SLICOT Library routine IB01BD.
On entry, if ALG = 'C', or ALG = 'Q', and BATCH = 'I' or
'L', the leading 2*(M+L)*NOBR-by-2*(M+L)*NOBR upper
triangular part of this array must contain the upper
triangular matrix R computed at the previous call of this
routine in sequential data processing. The array R need
not be set on entry if ALG = 'F' or if BATCH = 'F' or 'O'.
LDR INTEGER
The leading dimension of the array R.
LDR >= MAX( 2*(M+L)*NOBR, 3*M*NOBR ),
for METH = 'M' and JOBD = 'M';
LDR >= 2*(M+L)*NOBR, for METH = 'M' and JOBD = 'N' or
for METH = 'N'.
SV (output) DOUBLE PRECISION array, dimension ( L*NOBR )
The singular values used to estimate the system order.
Tolerances
RCOND DOUBLE PRECISION
The tolerance to be used for estimating the rank of
matrices. If the user sets RCOND > 0, the given value
of RCOND is used as a lower bound for the reciprocal
condition number; an m-by-n matrix whose estimated
condition number is less than 1/RCOND is considered to
be of full rank. If the user sets RCOND <= 0, then an
implicitly computed, default tolerance, defined by
RCONDEF = m*n*EPS, is used instead, where EPS is the
relative machine precision (see LAPACK Library routine
DLAMCH).
This parameter is not used for METH = 'M'.
TOL DOUBLE PRECISION
Absolute tolerance used for determining an estimate of
the system order. If TOL >= 0, the estimate is
indicated by the index of the last singular value greater
than or equal to TOL. (Singular values less than TOL
are considered as zero.) When TOL = 0, an internally
computed default value, TOL = NOBR*EPS*SV(1), is used,
where SV(1) is the maximal singular value, and EPS is
the relative machine precision (see LAPACK Library routine
DLAMCH). When TOL < 0, the estimate is indicated by the
index of the singular value that has the largest
logarithmic gap to its successor.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK >= MAX(3,(M+L)*NOBR), if METH = 'N';
LIWORK >= MAX(3,M+L), if METH = 'M' and ALG = 'F';
LIWORK >= 3, if METH = 'M' and ALG = 'C' or 'Q'.
On entry with BATCH = 'I' or BATCH = 'L', IWORK(1:3)
must contain the values of ICYCLE, MAXWRK, and NSMPSM
set by the previous call of this routine.
On exit with BATCH = 'F' or BATCH = 'I', IWORK(1:3)
contains the values of ICYCLE, MAXWRK, and NSMPSM to be
used by the next call of the routine.
ICYCLE counts the cycles for BATCH = 'I'.
MAXWRK stores the current optimal workspace.
NSMPSM sums up the NSMP values for BATCH <> 'O'.
The first three elements of IWORK should be preserved
during successive calls of the routine with BATCH = 'F'
or BATCH = 'I', till the final call with BATCH = 'L'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK, and, for METH = 'N', and BATCH = 'L' or
'O', DWORK(2) and DWORK(3) contain the reciprocal
condition numbers of the triangular factors of the
matrices U_f and r_1 [6].
On exit, if INFO = -23, DWORK(1) returns the minimum
value of LDWORK.
Let
k = 0, if CONCT = 'N' and ALG = 'C' or 'Q';
k = 2*NOBR-1, if CONCT = 'C' and ALG = 'C' or 'Q';
k = 2*NOBR*(M+L+1), if CONCT = 'N' and ALG = 'F';
k = 2*NOBR*(M+L+2), if CONCT = 'C' and ALG = 'F'.
The first (M+L)*k elements of DWORK should be preserved
during successive calls of the routine with BATCH = 'F'
or 'I', till the final call with BATCH = 'L'.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= (4*NOBR-2)*(M+L), if ALG = 'C', BATCH = 'F' or
'I' and CONCT = 'C';
LDWORK >= 1, if ALG = 'C', BATCH = 'F' or 'I' and
CONCT = 'N';
LDWORK >= max((4*NOBR-2)*(M+L), 5*L*NOBR), if METH = 'M',
ALG = 'C', BATCH = 'L' and CONCT = 'C';
LDWORK >= max((2*M-1)*NOBR, (M+L)*NOBR, 5*L*NOBR),
if METH = 'M', JOBD = 'M', ALG = 'C',
BATCH = 'O', or
(BATCH = 'L' and CONCT = 'N');
LDWORK >= 5*L*NOBR, if METH = 'M', JOBD = 'N', ALG = 'C',
BATCH = 'O', or
(BATCH = 'L' and CONCT = 'N');
LDWORK >= 5*(M+L)*NOBR+1, if METH = 'N', ALG = 'C', and
BATCH = 'L' or 'O';
LDWORK >= (M+L)*2*NOBR*(M+L+3), if ALG = 'F',
BATCH <> 'O' and CONCT = 'C';
LDWORK >= (M+L)*2*NOBR*(M+L+1), if ALG = 'F',
BATCH = 'F', 'I' and CONCT = 'N';
LDWORK >= (M+L)*4*NOBR*(M+L+1)+(M+L)*2*NOBR, if ALG = 'F',
BATCH = 'L' and CONCT = 'N', or
BATCH = 'O';
LDWORK >= 4*(M+L)*NOBR, if ALG = 'Q', BATCH = 'F', and
LDR >= NS = NSMP - 2*NOBR + 1;
LDWORK >= max(4*(M+L)*NOBR, 5*L*NOBR), if METH = 'M',
ALG = 'Q', BATCH = 'O', and LDR >= NS;
LDWORK >= 5*(M+L)*NOBR+1, if METH = 'N', ALG = 'Q',
BATCH = 'O', and LDR >= NS;
LDWORK >= 6*(M+L)*NOBR, if ALG = 'Q', (BATCH = 'F' or 'O',
and LDR < NS), or (BATCH = 'I' or
'L' and CONCT = 'N');
LDWORK >= 4*(NOBR+1)*(M+L)*NOBR, if ALG = 'Q', BATCH = 'I'
or 'L' and CONCT = 'C'.
The workspace used for ALG = 'Q' is
LDRWRK*2*(M+L)*NOBR + 4*(M+L)*NOBR,
where LDRWRK = LDWORK/(2*(M+L)*NOBR) - 2; recommended
value LDRWRK = NS, assuming a large enough cache size.
For good performance, LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: the number of 100 cycles in sequential data
processing has been exhausted without signaling
that the last block of data was get; the cycle
counter was reinitialized;
= 2: a fast algorithm was requested (ALG = 'C' or 'F'),
but it failed, and the QR algorithm was then used
(non-sequential data processing);
= 3: all singular values were exactly zero, hence N = 0
(both input and output were identically zero);
= 4: the least squares problems with coefficient matrix
U_f, used for computing the weighted oblique
projection (for METH = 'N'), have a rank-deficient
coefficient matrix;
= 5: the least squares problem with coefficient matrix
r_1 [6], used for computing the weighted oblique
projection (for METH = 'N'), has a rank-deficient
coefficient matrix.
NOTE: the values 4 and 5 of IWARN have no significance
for the identification problem.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: a fast algorithm was requested (ALG = 'C', or 'F')
in sequential data processing, but it failed; the
routine can be repeatedly called again using the
standard QR algorithm;
= 2: the singular value decomposition (SVD) algorithm did
not converge.
Method
The procedure consists in three main steps, the first step being
performed by one of the three algorithms included.
1.a) For non-sequential data processing using QR algorithm, a
t x 2(m+l)s matrix H is constructed, where
H = [ Uf' Up' Y' ], for METH = 'M',
s+1,2s,t 1,s,t 1,2s,t
H = [ U' Y' ], for METH = 'N',
1,2s,t 1,2s,t
and Up , Uf , U , and Y are block Hankel
1,s,t s+1,2s,t 1,2s,t 1,2s,t
matrices defined in terms of the input and output data [3].
A QR factorization is used to compress the data.
The fast QR algorithm uses a QR factorization which exploits
the block-Hankel structure. Actually, the Cholesky factor of H'*H
is computed.
1.b) For sequential data processing using QR algorithm, the QR
decomposition is done sequentially, by updating the upper
triangular factor R. This is also performed internally if the
workspace is not large enough to accommodate an entire batch.
1.c) For non-sequential or sequential data processing using
Cholesky algorithm, the correlation matrix of input-output data is
computed (sequentially, if requested), taking advantage of the
block Hankel structure [7]. Then, the Cholesky factor of the
correlation matrix is found, if possible.
2) A singular value decomposition (SVD) of a certain matrix is
then computed, which reveals the order n of the system as the
number of "non-zero" singular values. For the MOESP approach, this
matrix is [ R_24' R_34' ]' := R(ms+1:(2m+l)s,(2m+l)s+1:2(m+l)s),
where R is the upper triangular factor R constructed by SLICOT
Library routine IB01MD. For the N4SID approach, a weighted
oblique projection is computed from the upper triangular factor R
and its SVD is then found.
3) The singular values are compared to the given, or default TOL,
and the estimated order n is returned, possibly after user's
confirmation.
References
[1] Verhaegen M., and Dewilde, P.
Subspace Model Identification. Part 1: The output-error
state-space model identification class of algorithms.
Int. J. Control, 56, pp. 1187-1210, 1992.
[2] Verhaegen M.
Subspace Model Identification. Part 3: Analysis of the
ordinary output-error state-space model identification
algorithm.
Int. J. Control, 58, pp. 555-586, 1993.
[3] Verhaegen M.
Identification of the deterministic part of MIMO state space
models given in innovations form from input-output data.
Automatica, Vol.30, No.1, pp.61-74, 1994.
[4] Van Overschee, P., and De Moor, B.
N4SID: Subspace Algorithms for the Identification of
Combined Deterministic-Stochastic Systems.
Automatica, Vol.30, No.1, pp. 75-93, 1994.
[5] Peternell, K., Scherrer, W. and Deistler, M.
Statistical Analysis of Novel Subspace Identification Methods.
Signal Processing, 52, pp. 161-177, 1996.
[6] Sima, V.
Subspace-based Algorithms for Multivariable System
Identification.
Studies in Informatics and Control, 5, pp. 335-344, 1996.
[7] Sima, V.
Cholesky or QR Factorization for Data Compression in
Subspace-based Identification ?
Proceedings of the Second NICONET Workshop on ``Numerical
Control Software: SLICOT, a Useful Tool in Industry'',
December 3, 1999, INRIA Rocquencourt, France, pp. 75-80, 1999.
Numerical Aspects
The implemented method is numerically stable (when QR algorithm is
used), reliable and efficient. The fast Cholesky or QR algorithms
are more efficient, but the accuracy could diminish by forming the
correlation matrix.
The most time-consuming computational step is step 1:
2
The QR algorithm needs 0(t(2(m+l)s) ) floating point operations.
2 3
The Cholesky algorithm needs 0(2t(m+l) s)+0((2(m+l)s) ) floating
point operations.
2 3 2
The fast QR algorithm needs 0(2t(m+l) s)+0(4(m+l) s ) floating
point operations.
3
Step 2 of the algorithm requires 0(((m+l)s) ) floating point
operations.
Further Comments
For ALG = 'Q', BATCH = 'O' and LDR < NS, or BATCH <> 'O', the calculations could be rather inefficient if only minimal workspace (see argument LDWORK) is provided. It is advisable to provide as much workspace as possible. Almost optimal efficiency can be obtained for LDWORK = (NS+2)*(2*(M+L)*NOBR), assuming that the cache size is large enough to accommodate R, U, Y, and DWORK.Example
Program Text
* IB01AD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER LDR, LDU, LDWORK, LDY, LIWORK, LMAX, MMAX,
$ NOBRMX, NSMPMX
PARAMETER ( LMAX = 5, MMAX = 5, NOBRMX = 20, NSMPMX = 2000,
$ LDR = MAX( 2*( MMAX + LMAX )*NOBRMX,
$ 3*MMAX*NOBRMX ), LDU = NSMPMX,
$ LDWORK = MAX( 6*( MMAX + LMAX )*NOBRMX,
$ ( MMAX + LMAX )*( 4*NOBRMX*
$ ( MMAX + LMAX + 1 ) + 2*NOBRMX ),
$ ( MMAX + LMAX )*4*NOBRMX*
$ ( NOBRMX + 1 ) ),
$ LDY = NSMPMX, LIWORK = ( MMAX + LMAX )*NOBRMX )
* .. Local Scalars ..
LOGICAL NGIVEN
CHARACTER ALG, BATCH, CONCT, CTRL, JOBD, METH
INTEGER I, ICYCLE, II, INFO, IWARN, J, L, M, N, NCYCLE,
$ NGIV, NOBR, NSAMPL, NSMP
DOUBLE PRECISION RCOND, TOL
* .. Local Arrays ..
DOUBLE PRECISION DWORK(LDWORK), R(LDR, 2*(MMAX+LMAX)*NOBRMX),
$ SV(LMAX*NOBRMX), U(LDU, MMAX), Y(LDY, LMAX)
INTEGER IWORK(LIWORK)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL IB01AD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
* If the value of N is positive, it will be taken as system order.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) NOBR, N, M, L, NSMP, RCOND, TOL, METH, ALG,
$ JOBD, BATCH, CONCT, CTRL
IF ( LSAME( BATCH, 'F' ) ) THEN
READ ( NIN, FMT = * ) NCYCLE
ELSE
NCYCLE = 1
END IF
NSAMPL = NCYCLE*NSMP
*
NGIVEN = N.GT.0
IF( NGIVEN )
$ NGIV = N
IF ( NOBR.LE.0 .OR. NOBR.GT.NOBRMX ) THEN
WRITE ( NOUT, FMT = 99997 ) NOBR
ELSE IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99996 ) M
ELSE IF ( L.LE.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) L
ELSE IF ( NSMP.LT.0 .OR. NSMP.GT.NSMPMX .OR.
$ ( NSMP.LT.2*( M + L + 1 )*NOBR - 1 .AND.
$ LSAME( BATCH, 'O' ) ) .OR.
$ ( NSAMPL.LT.2*( M + L + 1 )*NOBR - 1 .AND.
$ LSAME( BATCH, 'L' ) ) .OR.
$ NSMP.LT.2*NOBR .AND. ( LSAME( BATCH, 'F' ) .OR.
$ LSAME( BATCH, 'I' ) ) ) THEN
WRITE ( NOUT, FMT = 99994 ) NSMP
ELSE IF ( NCYCLE.LE.0 .OR. NSAMPL.GT.NSMPMX ) THEN
WRITE ( NOUT, FMT = 99993 ) NCYCLE
ELSE
* Read the matrices U and Y from the input file.
IF ( M.GT.0 )
$ READ ( NIN, FMT = * )
$ ( ( U(I,J), J = 1, M ), I = 1, NSAMPL )
READ ( NIN, FMT = * ) ( ( Y(I,J), J = 1, L ), I = 1, NSAMPL )
* Compute the R factor from a QR (or Cholesky) factorization
* of the Hankel-like matrix (or correlation matrix).
DO 10 ICYCLE = 1, NCYCLE
II = ( ICYCLE - 1 )*NSMP + 1
IF ( NCYCLE.GT.1 ) THEN
IF ( ICYCLE.GT.1 ) BATCH = 'I'
IF ( ICYCLE.EQ.NCYCLE ) BATCH = 'L'
END IF
CALL IB01AD( METH, ALG, JOBD, BATCH, CONCT, CTRL, NOBR, M,
$ L, NSMP, U(II,1), LDU, Y(II,1), LDY, N, R, LDR,
$ SV, RCOND, TOL, IWORK, DWORK, LDWORK, IWARN,
$ INFO )
10 CONTINUE
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( IWARN.NE.0 )
$ WRITE ( NOUT, FMT = 99990 ) IWARN
IF( NGIVEN )
$ N = NGIV
WRITE ( NOUT, FMT = 99992 ) N
WRITE ( NOUT, FMT = 99991 ) ( SV(I), I = 1,L*NOBR )
END IF
END IF
STOP
99999 FORMAT ( ' IB01AD EXAMPLE PROGRAM RESULTS', /1X)
99998 FORMAT ( ' INFO on exit from IB01AD = ',I2)
99997 FORMAT (/' NOBR is out of range.',/' NOBR = ', I5)
99996 FORMAT (/' M is out of range.',/' M = ', I5)
99995 FORMAT (/' L is out of range.',/' L = ', I5)
99994 FORMAT (/' NSMP is out of range.',/' NSMP = ', I5)
99993 FORMAT (/' NCYCLE is out of range.',/' NCYCLE = ', I5)
99992 FORMAT ( ' The order of the system is ', I5)
99991 FORMAT ( ' The singular values are ',/ (8(1X,F8.4)))
99990 FORMAT ( ' IWARN on exit from IB01AD = ',I2)
END
Program Data
IB01AD EXAMPLE PROGRAM DATA 15 0 1 1 1000 0.0 -1.0 M C N O N N 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 6.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 3.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 4.766099 4.763659 4.839359 5.002979 5.017629 5.056699 5.154379 5.361949 5.425439 5.569519 5.681849 5.742899 5.803949 5.918729 5.821049 5.447419 5.061589 4.629349 4.267939 4.011519 3.850349 3.711159 3.569519 3.518239 3.652549 3.818609 3.862559 4.011519 4.353409 4.705049 5.083559 5.344859 5.274039 5.127519 4.761219 4.451089 4.221539 4.045709 3.874769 3.730689 3.662319 3.576849 3.542659 3.479169 3.454749 3.359509 3.298459 3.225199 3.200779 3.225199 3.227639 3.274039 3.457189 3.867449 4.321659 4.492599 4.431549 4.243519 4.050599 3.857679 3.730689 3.791739 3.921169 3.955359 3.847909 3.725809 3.611039 3.716039 4.092109 4.480389 4.814939 5.054259 5.303339 5.486489 5.672089 5.779529 5.799069 5.664759 5.291129 4.880879 4.558529 4.184909 3.889419 3.708719 3.623249 3.569519 3.718479 4.033499 4.412009 4.629349 4.558529 4.394919 4.180019 4.197119 4.431549 4.714819 4.961459 5.300899 5.567079 5.681849 5.545099 5.188569 4.883319 4.600049 4.270379 4.038389 3.838139 3.711159 3.591499 3.535329 3.486489 3.476729 3.425439 3.381489 3.369279 3.364389 3.347299 3.381489 3.420559 3.413229 3.452309 3.635459 4.038389 4.375379 4.727029 5.056699 5.298459 5.532889 5.466959 5.195899 4.885759 4.763659 4.875989 5.042049 5.283809 5.491379 5.596379 5.672089 5.772209 5.830819 5.933379 5.899189 5.935819 5.894309 5.918729 5.994429 5.957799 6.031059 6.062809 6.040829 6.096999 6.123859 6.162929 6.040829 5.845469 5.772209 5.799069 5.923609 5.928499 6.001759 6.001759 6.060369 5.882099 5.510909 5.322879 5.371719 5.454749 5.437649 5.159269 4.902859 4.587839 4.502369 4.595159 4.824709 5.064029 5.271599 5.466959 5.615919 5.528009 5.254499 4.883319 4.517019 4.197119 4.001759 3.806399 3.904079 3.923609 3.869889 3.806399 3.720929 3.818609 4.140949 4.529229 4.805179 5.086009 5.339969 5.532889 5.576849 5.667199 5.791739 5.850349 5.923609 5.921169 5.977339 5.740459 5.388809 5.000539 4.849129 4.944369 5.173919 5.369279 5.447419 5.603709 5.730689 5.850349 5.979779 5.991989 6.084789 5.940709 5.803949 5.791739 5.603709 5.264269 4.946809 4.619579 4.514579 4.433989 4.285029 4.121419 3.945589 3.984659 4.219099 4.546319 4.873549 5.154379 5.388809 5.613479 5.835699 5.884539 5.955359 5.762439 5.459629 5.061589 4.707499 4.458409 4.267939 4.053039 3.943149 3.825929 3.967569 4.280149 4.480389 4.492599 4.390039 4.197119 4.111649 3.982219 3.867449 3.767319 3.872329 4.236189 4.663539 4.971229 5.066469 4.902859 4.675749 4.392479 4.099439 4.114089 4.326539 4.643999 4.971229 5.159269 5.388809 5.576849 5.652549 5.803949 5.913839 5.886979 5.799069 5.730689 5.762439 5.813719 5.821049 5.928499 6.013969 5.764879 5.413229 5.098219 4.678189 4.372939 4.392479 4.590279 4.919949 5.017629 4.858899 4.675749 4.619579 4.834479 5.090889 5.376599 5.681849 5.823489 5.952919 6.062809 6.089669 6.075019 6.026179 5.994429 6.077459 5.857679 5.701389 5.730689 5.784419 5.823489 5.894309 5.762439 5.415679 4.961459 4.595159 4.331429 4.297239 4.582949 4.861339 5.173919 5.166589 4.919949 4.607369 4.370499 4.182469 4.038389 4.145839 4.431549 4.556089 4.480389 4.375379 4.370499 4.558529 4.858899 4.895529 4.741679 4.744129 4.875989 5.105539 5.239849 5.518239 5.652549 5.723369 5.855239 5.962679 5.984659 5.984659 6.055479 6.062809 6.055479 6.070129 5.784419 5.440099 5.056699 4.941929 5.010299 5.134849 5.313109 5.479169 5.623249 5.562199 5.330209 5.010299 4.665979 4.414459 4.201999 4.048159 4.079899 4.189789 4.131179 4.004199 3.916289 3.960239 4.199559 4.624469 4.883319 5.137289 5.379049 5.623249 5.762439 5.833259 5.686739 5.366839 5.225199 5.239849 5.354629 5.508469 5.596379 5.752669 5.874769 5.906519 5.894309 5.742899 5.447419 5.024959 4.883319 4.885759 4.893089 4.714819 4.451089 4.233749 4.043269 3.864999 3.757559 3.669639 3.593939 3.547539 3.506029 3.454749 3.398579 3.361949 3.339969 3.374159 3.520679 3.713599 3.757559 3.779529 3.696509 3.777089 3.886979 3.904079 3.850349 3.965129 4.282589 4.521899 4.714819 4.971229 5.220319 5.532889 5.652549 5.781979 5.955359 6.035939 6.118969 6.133629 6.153159 6.192229 6.143389 6.167809 5.991989 5.652549 5.459629 5.437649 5.339969 5.098219 4.785639 4.492599 4.236189 4.067689 3.933379 3.823489 3.730689 3.611039 3.564639 3.549989 3.557309 3.513359 3.515799 3.694059 4.072579 4.480389 4.705049 4.612259 4.385149 4.201999 4.026179 3.904079 3.774649 3.691619 3.845469 4.201999 4.585399 4.902859 5.256949 5.510909 5.640339 5.843029 5.974889 5.935819 5.821049 5.528009 5.171479 4.810059 4.453529 4.380269 4.565859 4.805179 5.125079 5.354629 5.589059 5.764879 5.923609 5.940709 5.857679 5.694059 5.486489 5.149499 4.844249 4.541439 4.267939 4.060369 3.960239 3.789299 3.642779 3.525569 3.498699 3.454749 3.408349 3.379049 3.376599 3.361949 3.359509 3.369279 3.398579 3.579289 3.948029 4.412009 4.585399 4.514579 4.343639 4.155599 3.984659 4.043269 4.307009 4.421779 4.353409 4.223979 4.053039 3.940709 3.838139 3.730689 3.652549 3.611039 3.564639 3.496259 3.462069 3.454749 3.425439 3.379049 3.432769 3.623249 3.974889 4.380269 4.714819 5.073799 5.369279 5.603709 5.745349 5.652549 5.401019 5.015189 4.709939 4.416899 4.236189 4.236189 4.248399 4.221539 4.297239 4.590279 4.893089 5.134849 5.427889 5.379049 5.364389 5.452309 5.567079 5.672089 5.769769 5.830819 5.923609 5.965129 6.057919 6.050599 6.072579 6.111649 6.070129 5.896749 5.755109 5.718479 5.821049 6.001759 6.001759 5.901629 5.557309 5.173919 4.800289 4.431549 4.194679 4.006639 3.850349 3.747789 3.642779 3.591499 3.569519 3.528009 3.537779 3.554869 3.493819 3.447419 3.440099 3.408349 3.410789 3.452309 3.681849 4.060369 4.441319 4.854019 5.154379 5.425439 5.596379 5.586619 5.354629 5.027399 4.863779 4.761219 4.570739 4.368059 4.397359 4.573189 4.841809 5.203219 5.452309 5.652549 5.855239 5.906519 5.952919 5.828369 5.791739 5.799069 5.813719 5.877209 5.955359 5.781979 5.518239 5.127519 4.763659 4.492599 4.233749 4.011519 3.855239 3.691619 3.635459 3.818609 4.155599 4.590279 4.988329 5.076239 4.907739 4.648889 4.377829 4.216649 4.287469 4.590279 4.846689 5.139729 5.388809 5.689179 5.884539 6.043269 6.170259 6.211769 6.250839 6.209329 6.013969 5.701389 5.469399 5.479169 5.557309 5.728249 5.882099 5.984659 5.901629 5.581729 5.371719 5.418119 5.510909 5.667199 5.791739 5.698949 5.484049 5.154379 4.980999 5.061589 5.195899 5.359509 5.615919 5.762439 5.857679 5.948029 5.835699 5.706269 5.498699 5.188569 5.117749 5.191009 5.315549 5.532889 5.444979 5.396139 5.274039 5.027399 4.744129 4.668419 4.651329 4.514579 4.267939 4.260609 4.263049 4.189789 4.277699 4.600049 4.932159 5.283809 5.528009 5.740459 5.874769 5.955359 5.991989 5.845469 5.528009 5.061589 4.734359 4.534109 4.534109 4.697729 4.744129 4.619579 4.643999 4.832039 5.132399 5.410789 5.625689 5.603709 5.315549 4.961459 4.619579 4.358289 4.155599 4.033499 3.886979 3.772209 3.640339 3.532889 3.435209 3.427889 3.422999 3.398579 3.603709 4.023729 4.451089 4.792969 4.902859 4.780759 4.590279 4.336309 4.145839 4.216649 4.433989 4.714819 5.098219 5.359509 5.569519 5.772209 5.921169 6.055479 5.962679 5.642779 5.435209 5.388809 5.537779 5.681849 5.701389 5.615919 5.667199 5.740459 5.803949 5.882099 5.950469 6.072579 6.148279 6.116529 6.177579 6.201999 6.206889 5.991989 5.564639 5.178799 4.998089 5.051819 5.232529 5.484049 5.686739 5.899189 5.869889 5.977339 6.053039 6.079899 6.128739 6.079899 6.167809 6.194679 6.236189 6.053039 5.652549 5.274039 4.858899 4.534109 4.455969 4.619579 4.866229 5.117749 5.166589 5.056699 5.002979 5.098219 5.325319 5.567079 5.466959 5.252059 4.946809 4.880879 4.980999 5.225199 5.459629 5.723369 5.791739 5.906519 5.991989 5.835699 5.528009 5.142169 4.775869 4.490159 4.236189 4.023729 3.886979 3.752669 3.681849 3.806399 4.145839 4.600049 5.002979 5.303339 5.552429 5.615919 5.523119 5.611039 5.713599 5.845469 5.899189 5.994429 6.092109 6.092109 6.143389 6.153159 6.233749 6.187349 6.013969 5.835699 5.774649 5.686739 5.537779 5.327759 5.054259 4.700169 4.394919 4.180019 4.043269 3.877209 3.752669 3.728249 3.869889 4.206889 4.355849 4.426669 4.453529 4.521899 4.392479 4.155599 3.965129 3.877209 3.970009 4.258169 4.421779 4.336309 4.299679 4.392479 4.675749 4.761219 4.658659 4.490159 4.307009 4.126299 3.972449 4.077459 4.372939 4.741679 5.088449 5.186129 5.037169 4.785639 4.563419 4.534109 4.705049 4.741679 4.648889 4.431549 4.238629 4.065249 3.943149 3.811279 3.691619 3.652549 3.825929 4.223979 4.424219 4.429109 4.319219 4.138509 3.965129 3.886979 3.801509 3.701389 3.640339 3.767319 4.150719 4.648889 4.990769 5.088449 5.022509 4.783199 4.685519 4.665979 4.707499 4.912619 5.195899 5.415679 5.623249 5.740459 5.899189 5.928499 6.050599 6.153159 5.965129 5.586619 5.381489 5.371719 5.486489 5.567079 5.821049 5.913839 5.994429 6.011519 5.999309 6.018849 5.821049 5.728249 5.740459 5.764879 5.882099 5.926049 5.750229 5.415679 4.995649 4.861339 4.902859 5.103099 5.364389 5.596379 5.752669 5.845469 5.928499 6.006639 5.840579 5.518239 5.173919 4.739239 4.458409 4.426669 4.602489 4.822269 5.183689 5.430329 5.652549 5.821049 5.706269 5.369279 5.027399 4.705049 4.414459 4.145839 3.965129 4.033499 4.372939 4.683079Program Results
IB01AD EXAMPLE PROGRAM RESULTS The order of the system is 4 The singular values are 69.8841 14.9963 3.6675 1.9677 0.3000 0.2078 0.1651 0.1373 0.1133 0.1059 0.0856 0.0784 0.0733 0.0678 0.0571
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/IB01BD.html 0000664 0000000 0000000 00000143574 14560147231 0020331 0 ustar 00root root 0000000 0000000
Purpose
To estimate the system matrices A, C, B, and D, the noise covariance matrices Q, Ry, and S, and the Kalman gain matrix K of a linear time-invariant state space model, using the processed triangular factor R of the concatenated block Hankel matrices, provided by SLICOT Library routine IB01AD.Specification
SUBROUTINE IB01BD( METH, JOB, JOBCK, NOBR, N, M, L, NSMPL, R,
$ LDR, A, LDA, C, LDC, B, LDB, D, LDD, Q, LDQ,
$ RY, LDRY, S, LDS, K, LDK, TOL, IWORK, DWORK,
$ LDWORK, BWORK, IWARN, INFO )
C .. Scalar Arguments ..
DOUBLE PRECISION TOL
INTEGER INFO, IWARN, L, LDA, LDB, LDC, LDD, LDK, LDQ,
$ LDR, LDRY, LDS, LDWORK, M, N, NOBR, NSMPL
CHARACTER JOB, JOBCK, METH
C .. Array Arguments ..
DOUBLE PRECISION A(LDA, *), B(LDB, *), C(LDC, *), D(LDD, *),
$ DWORK(*), K(LDK, *), Q(LDQ, *), R(LDR, *),
$ RY(LDRY, *), S(LDS, *)
INTEGER IWORK( * )
LOGICAL BWORK( * )
Arguments
Mode Parameters
METH CHARACTER*1
Specifies the subspace identification method to be used,
as follows:
= 'M': MOESP algorithm with past inputs and outputs;
= 'N': N4SID algorithm;
= 'C': combined method: MOESP algorithm for finding the
matrices A and C, and N4SID algorithm for
finding the matrices B and D.
JOB CHARACTER*1
Specifies which matrices should be computed, as follows:
= 'A': compute all system matrices, A, B, C, and D;
= 'C': compute the matrices A and C only;
= 'B': compute the matrix B only;
= 'D': compute the matrices B and D only.
JOBCK CHARACTER*1
Specifies whether or not the covariance matrices and the
Kalman gain matrix are to be computed, as follows:
= 'C': the covariance matrices only should be computed;
= 'K': the covariance matrices and the Kalman gain
matrix should be computed;
= 'N': the covariance matrices and the Kalman gain matrix
should not be computed.
Input/Output Parameters
NOBR (input) INTEGER
The number of block rows, s, in the input and output
Hankel matrices processed by other routines. NOBR > 1.
N (input) INTEGER
The order of the system. NOBR > N > 0.
M (input) INTEGER
The number of system inputs. M >= 0.
L (input) INTEGER
The number of system outputs. L > 0.
NSMPL (input) INTEGER
If JOBCK = 'C' or 'K', the total number of samples used
for calculating the covariance matrices.
NSMPL >= 2*(M+L)*NOBR.
This parameter is not meaningful if JOBCK = 'N'.
R (input/workspace) DOUBLE PRECISION array, dimension
( LDR,2*(M+L)*NOBR )
On entry, the leading 2*(M+L)*NOBR-by-2*(M+L)*NOBR part
of this array must contain the relevant data for the MOESP
or N4SID algorithms, as constructed by SLICOT Library
routine IB01AD. Let R_ij, i,j = 1:4, be the
ij submatrix of R (denoted S in IB01AD), partitioned
by M*NOBR, L*NOBR, M*NOBR, and L*NOBR rows and
columns. The submatrix R_22 contains the matrix of left
singular vectors used. Also needed, for METH = 'N' or
JOBCK <> 'N', are the submatrices R_11, R_14 : R_44,
and, for METH = 'M' or 'C' and JOB <> 'C', the
submatrices R_31 and R_12, containing the processed
matrices R_1c and R_2c, respectively, as returned by
SLICOT Library routine IB01AD.
Moreover, if METH = 'N' and JOB = 'A' or 'C', the
block-row R_41 : R_43 must contain the transpose of the
block-column R_14 : R_34 as returned by SLICOT Library
routine IB01AD.
The remaining part of R is used as workspace.
On exit, part of this array is overwritten. Specifically,
if METH = 'M', R_22 and R_31 are overwritten if
JOB = 'B' or 'D', and R_12, R_22, R_14 : R_34,
and possibly R_11 are overwritten if JOBCK <> 'N';
if METH = 'N', all needed submatrices are overwritten.
The details of the contents of R need not be known if
this routine is called once just after calling the SLICOT
Library routine IB01AD.
LDR INTEGER
The leading dimension of the array R.
LDR >= 2*(M+L)*NOBR.
A (input or output) DOUBLE PRECISION array, dimension
(LDA,N)
On entry, if METH = 'N' or 'C' and JOB = 'B' or 'D',
the leading N-by-N part of this array must contain the
system state matrix.
If METH = 'M' or (METH = 'N' or 'C' and JOB = 'A'
or 'C'), this array need not be set on input.
On exit, if JOB = 'A' or 'C' and INFO = 0, the
leading N-by-N part of this array contains the system
state matrix.
LDA INTEGER
The leading dimension of the array A.
LDA >= N, if JOB = 'A' or 'C', or METH = 'N' or 'C'
and JOB = 'B' or 'D';
LDA >= 1, otherwise.
C (input or output) DOUBLE PRECISION array, dimension
(LDC,N)
On entry, if METH = 'N' or 'C' and JOB = 'B' or 'D',
the leading L-by-N part of this array must contain the
system output matrix.
If METH = 'M' or (METH = 'N' or 'C' and JOB = 'A'
or 'C'), this array need not be set on input.
On exit, if JOB = 'A' or 'C' and INFO = 0, or
INFO = 3 (or INFO >= 0, for METH = 'M'), the leading
L-by-N part of this array contains the system output
matrix.
LDC INTEGER
The leading dimension of the array C.
LDC >= L, if JOB = 'A' or 'C', or METH = 'N' or 'C'
and JOB = 'B' or 'D';
LDC >= 1, otherwise.
B (output) DOUBLE PRECISION array, dimension (LDB,M)
If M > 0, JOB = 'A', 'B', or 'D' and INFO = 0, the
leading N-by-M part of this array contains the system
input matrix. If M = 0 or JOB = 'C', this array is
not referenced.
LDB INTEGER
The leading dimension of the array B.
LDB >= N, if M > 0 and JOB = 'A', 'B', or 'D';
LDB >= 1, if M = 0 or JOB = 'C'.
D (output) DOUBLE PRECISION array, dimension (LDD,M)
If M > 0, JOB = 'A' or 'D' and INFO = 0, the leading
L-by-M part of this array contains the system input-output
matrix. If M = 0 or JOB = 'C' or 'B', this array is
not referenced.
LDD INTEGER
The leading dimension of the array D.
LDD >= L, if M > 0 and JOB = 'A' or 'D';
LDD >= 1, if M = 0 or JOB = 'C' or 'B'.
Q (output) DOUBLE PRECISION array, dimension (LDQ,N)
If JOBCK = 'C' or 'K', the leading N-by-N part of this
array contains the positive semidefinite state covariance
matrix. If JOBCK = 'K', this matrix has been used as
state weighting matrix for computing the Kalman gain.
This parameter is not referenced if JOBCK = 'N'.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= N, if JOBCK = 'C' or 'K';
LDQ >= 1, if JOBCK = 'N'.
RY (output) DOUBLE PRECISION array, dimension (LDRY,L)
If JOBCK = 'C' or 'K', the leading L-by-L part of this
array contains the positive (semi)definite output
covariance matrix. If JOBCK = 'K', this matrix has been
used as output weighting matrix for computing the Kalman
gain.
This parameter is not referenced if JOBCK = 'N'.
LDRY INTEGER
The leading dimension of the array RY.
LDRY >= L, if JOBCK = 'C' or 'K';
LDRY >= 1, if JOBCK = 'N'.
S (output) DOUBLE PRECISION array, dimension (LDS,L)
If JOBCK = 'C' or 'K', the leading N-by-L part of this
array contains the state-output cross-covariance matrix.
If JOBCK = 'K', this matrix has been used as state-
output weighting matrix for computing the Kalman gain.
This parameter is not referenced if JOBCK = 'N'.
LDS INTEGER
The leading dimension of the array S.
LDS >= N, if JOBCK = 'C' or 'K';
LDS >= 1, if JOBCK = 'N'.
K (output) DOUBLE PRECISION array, dimension ( LDK,L )
If JOBCK = 'K', the leading N-by-L part of this array
contains the estimated Kalman gain matrix.
If JOBCK = 'C' or 'N', this array is not referenced.
LDK INTEGER
The leading dimension of the array K.
LDK >= N, if JOBCK = 'K';
LDK >= 1, if JOBCK = 'C' or 'N'.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used for estimating the rank of
matrices. If the user sets TOL > 0, then the given value
of TOL is used as a lower bound for the reciprocal
condition number; an m-by-n matrix whose estimated
condition number is less than 1/TOL is considered to
be of full rank. If the user sets TOL <= 0, then an
implicitly computed, default tolerance, defined by
TOLDEF = m*n*EPS, is used instead, where EPS is the
relative machine precision (see LAPACK Library routine
DLAMCH).
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK >= max(LIW1,LIW2), where
LIW1 = N, if METH <> 'N' and M = 0
or JOB = 'C' and JOBCK = 'N';
LIW1 = M*NOBR+N, if METH <> 'N', JOB = 'C',
and JOBCK <> 'N';
LIW1 = max(L*NOBR,M*NOBR), if METH = 'M', JOB <> 'C',
and JOBCK = 'N';
LIW1 = max(L*NOBR,M*NOBR+N), if METH = 'M', JOB <> 'C',
and JOBCK = 'C' or 'K';
LIW1 = max(M*NOBR+N,M*(N+L)), if METH = 'N', or METH = 'C'
and JOB <> 'C';
LIW2 = 0, if JOBCK <> 'K';
LIW2 = N*N, if JOBCK = 'K'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK, and DWORK(2), DWORK(3), DWORK(4), and
DWORK(5) contain the reciprocal condition numbers of the
triangular factors of the following matrices (defined in
SLICOT Library routine IB01PD and in the lower level
routines):
GaL (GaL = Un(1:(s-1)*L,1:n)),
R_1c (if METH = 'M' or 'C'),
M (if JOBCK = 'C' or 'K' or METH = 'N'), and
Q or T (see SLICOT Library routine IB01PY or IB01PX),
respectively.
If METH = 'N', DWORK(3) is set to one without any
calculations. Similarly, if METH = 'M' and JOBCK = 'N',
DWORK(4) is set to one. If M = 0 or JOB = 'C',
DWORK(3) and DWORK(5) are set to one.
If JOBCK = 'K' and INFO = 0, DWORK(6) to DWORK(13)
contain information about the accuracy of the results when
computing the Kalman gain matrix, as follows:
DWORK(6) - reciprocal condition number of the matrix
U11 of the Nth order system of algebraic
equations from which the solution matrix X
of the Riccati equation is obtained;
DWORK(7) - reciprocal pivot growth factor for the LU
factorization of the matrix U11;
DWORK(8) - reciprocal condition number of the matrix
As = A - S*inv(Ry)*C, which is inverted by
the standard Riccati solver;
DWORK(9) - reciprocal pivot growth factor for the LU
factorization of the matrix As;
DWORK(10) - reciprocal condition number of the matrix
Ry;
DWORK(11) - reciprocal condition number of the matrix
Ry + C*X*C';
DWORK(12) - reciprocal condition number for the Riccati
equation solution;
DWORK(13) - forward error bound for the Riccati
equation solution.
On exit, if INFO = -30, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= max( LDW1,LDW2,LDW3 ), where, if METH = 'M',
LDW1 >= max( 2*(L*NOBR-L)*N+2*N, (L*NOBR-L)*N+N*N+7*N ),
if JOB = 'C' or JOB = 'A' and M = 0;
LDW1 >= max( 2*(L*NOBR-L)*N+N*N+7*N,
(L*NOBR-L)*N+N+6*M*NOBR, (L*NOBR-L)*N+N+
max( L+M*NOBR, L*NOBR +
max( 3*L*NOBR+1, M ) ) ),
if M > 0 and JOB = 'A', 'B', or 'D';
LDW2 >= 0, if JOBCK = 'N';
LDW2 >= L*NOBR*N+
max( (L*NOBR-L)*N+Aw+2*N+max(5*N,(2*M+L)*NOBR+L),
4*(M*NOBR+N)+1, M*NOBR+2*N+L ),
if JOBCK = 'C' or 'K',
where Aw = N+N*N, if M = 0 or JOB = 'C';
Aw = 0, otherwise;
if METH = 'N',
LDW1 >= L*NOBR*N+max( (L*NOBR-L)*N+2*N+(2*M+L)*NOBR+L,
2*(L*NOBR-L)*N+N*N+8*N,
N+4*(M*NOBR+N)+1, M*NOBR+3*N+L );
LDW2 >= 0, if M = 0 or JOB = 'C';
LDW2 >= L*NOBR*N+M*NOBR*(N+L)*(M*(N+L)+1)+
max( (N+L)**2, 4*M*(N+L)+1 ),
if M > 0 and JOB = 'A', 'B', or 'D';
and, if METH = 'C', LDW1 as
max( LDW1 for METH = 'M', JOB = 'C', LDW1 for METH = 'N'),
and LDW2 for METH = 'N' are used;
LDW3 >= 0, if JOBCK <> 'K';
LDW3 >= max( 4*N*N+2*N*L+L*L+max( 3*L,N*L ),
14*N*N+12*N+5 ), if JOBCK = 'K'.
For good performance, LDWORK should be larger.
BWORK LOGICAL array, dimension (LBWORK)
LBWORK = 2*N, if JOBCK = 'K';
LBWORK = 0, if JOBCK <> 'K'.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 4: a least squares problem to be solved has a
rank-deficient coefficient matrix;
= 5: the computed covariance matrices are too small.
The problem seems to be a deterministic one; the
gain matrix is set to zero.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 2: the singular value decomposition (SVD) algorithm did
not converge;
= 3: a singular upper triangular matrix was found;
= 3+i: if JOBCK = 'K' and the associated Riccati
equation could not be solved, where i = 1,...,6;
(see the description of the parameter INFO for the
SLICOT Library routine SB02RD for the meaning of
the i values);
= 10: the QR algorithm did not converge.
Method
In the MOESP approach, the matrices A and C are first computed from an estimated extended observability matrix [1], and then, the matrices B and D are obtained by solving an extended linear system in a least squares sense. In the N4SID approach, besides the estimated extended observability matrix, the solutions of two least squares problems are used to build another least squares problem, whose solution is needed to compute the system matrices A, C, B, and D. The solutions of the two least squares problems are also optionally used by both approaches to find the covariance matrices. The Kalman gain matrix is obtained by solving a discrete-time algebraic Riccati equation.References
[1] Verhaegen M., and Dewilde, P.
Subspace Model Identification. Part 1: The output-error
state-space model identification class of algorithms.
Int. J. Control, 56, pp. 1187-1210, 1992.
[2] Van Overschee, P., and De Moor, B.
N4SID: Two Subspace Algorithms for the Identification
of Combined Deterministic-Stochastic Systems.
Automatica, Vol.30, No.1, pp. 75-93, 1994.
[3] Van Overschee, P.
Subspace Identification : Theory - Implementation -
Applications.
Ph. D. Thesis, Department of Electrical Engineering,
Katholieke Universiteit Leuven, Belgium, Feb. 1995.
[4] Sima, V.
Subspace-based Algorithms for Multivariable System
Identification.
Studies in Informatics and Control, 5, pp. 335-344, 1996.
Numerical Aspects
The implemented method consists in numerically stable steps.Further Comments
The covariance matrices are computed using the N4SID approach. Therefore, for efficiency reasons, it is advisable to set METH = 'N', if the Kalman gain matrix or covariance matrices are needed (JOBCK = 'K', or 'C'). When JOBCK = 'N', it could be more efficient to use the combined method, METH = 'C'. Often, this combination will also provide better accuracy than MOESP algorithm. In some applications, it is useful to compute the system matrices using two calls to this routine, the first one with JOB = 'C', and the second one with JOB = 'B' or 'D'. This is slightly less efficient than using a single call with JOB = 'A', because some calculations are repeated. If METH = 'N', all the calculations at the first call are performed again at the second call; moreover, it is required to save the needed submatrices of R before the first call and restore them before the second call. If the covariance matrices and/or the Kalman gain are desired, JOBCK should be set to 'C' or 'K' at the second call. If B and D are both needed, they should be computed at once.Example
Program Text
* IB01BD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER LDA, LDB, LDC, LDD, LDK, LDQ, LDR, LDRY, LDS,
$ LDU, LDW1, LDW2, LDW3, LDWORK, LDY, LIWORK, LMAX,
$ MMAX, NMAX, NOBRMX, NSMPMX
PARAMETER ( LMAX = 5, MMAX = 5, NOBRMX = 20, NSMPMX = 2000,
$ NMAX = NOBRMX - 1, LDA = NMAX, LDB = NMAX,
$ LDC = LMAX, LDD = LMAX, LDK = NMAX,
$ LDQ = NMAX, LDRY = LMAX, LDS = NMAX,
$ LDR = MAX( 2*( MMAX + LMAX )*NOBRMX,
$ 3*MMAX*NOBRMX ), LDU = NSMPMX,
$ LDW1 = MAX( LMAX*( NOBRMX - 1 )*NMAX + NMAX +
$ MAX( 6*MMAX, 4*LMAX )*NOBRMX,
$ LMAX*NOBRMX*NMAX +
$ MAX( LMAX*( NOBRMX - 1 )*NMAX +
$ 3*NMAX + LMAX +
$ ( 2*MMAX + LMAX )*NOBRMX,
$ 2*LMAX*( NOBRMX - 1 )*NMAX +
$ NMAX*NMAX + 8*NMAX,
$ NMAX +
$ 4*( MMAX*NOBRMX + NMAX ) ) ),
$ LDW2 = LMAX*NOBRMX*NMAX +
$ MMAX*NOBRMX*( NMAX + LMAX )*
$ ( MMAX*( NMAX + LMAX ) + 1 ) +
$ MAX( ( NMAX + LMAX )**2,
$ 4*MMAX*( NMAX + LMAX ) + 1 ),
$ LDW3 = MAX( 4*NMAX*NMAX + 2*NMAX*LMAX +
$ LMAX*LMAX +
$ MAX( 3*LMAX, NMAX*LMAX ),
$ 14*NMAX*NMAX + 12*NMAX + 5 ),
$ LDWORK = MAX( 6*( MMAX + LMAX )*NOBRMX,
$ ( MMAX + LMAX )*( 4*NOBRMX*
$ ( MMAX + LMAX + 2 ) - 2 ),
$ ( MMAX + LMAX )*4*NOBRMX*
$ ( NOBRMX + 1 ), LDW1, LDW2,
$ LDW3 ),
$ LDY = NSMPMX,
$ LIWORK = MAX( ( MMAX + LMAX )*NOBRMX,
$ MMAX*NOBRMX + NMAX, LMAX*NOBRMX,
$ MMAX*( NMAX + LMAX ), NMAX*NMAX )
$ )
* .. Local Scalars ..
LOGICAL NGIVEN
CHARACTER ALG, BATCH, CONCT, CTRL, JOB, JOBCK, JOBD, JOBDA,
$ METH, METHA
INTEGER I, ICYCLE, II, INFO, IWARN, J, L, M, N, NCYCLE,
$ NGIV, NOBR, NSAMPL, NSMP
DOUBLE PRECISION RCOND, TOL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA, NMAX), B(LDB, MMAX), C(LDC, NMAX),
$ D(LDD, MMAX), DWORK(LDWORK), K(LDK, LMAX),
$ Q(LDQ, NMAX), R(LDR, 2*(MMAX+LMAX)*NOBRMX),
$ RY(LDRY, LMAX), S(LDS, LMAX), SV(LMAX*NOBRMX),
$ U(LDU, MMAX), Y(LDY, LMAX)
INTEGER IWORK(LIWORK)
LOGICAL BWORK(2*NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL IB01AD, IB01BD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
* If the value of N is positive, it will be taken as system order.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) NOBR, N, M, L, NSMP, RCOND, TOL
READ ( NIN, FMT = * ) METH, ALG, JOBD, BATCH, CONCT, CTRL, JOB,
$ JOBCK
IF ( LSAME( BATCH, 'F' ) ) THEN
READ ( NIN, FMT = * ) NCYCLE
ELSE
NCYCLE = 1
END IF
NSAMPL = NCYCLE*NSMP
*
NGIVEN = N.GT.0
IF( NGIVEN )
$ NGIV = N
IF ( NOBR.LE.0 .OR. NOBR.GT.NOBRMX ) THEN
WRITE ( NOUT, FMT = 99997 ) NOBR
ELSE IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99996 ) M
ELSE IF ( L.LE.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) L
ELSE IF ( NSMP.LT.0 .OR. NSMP.GT.NSMPMX .OR.
$ ( NSMP.LT.2*( M + L + 1 )*NOBR - 1 .AND.
$ LSAME( BATCH, 'O' ) ) .OR.
$ ( NSAMPL.LT.2*( M + L + 1 )*NOBR - 1 .AND.
$ LSAME( BATCH, 'L' ) ) .OR.
$ NSMP.LT.2*NOBR .AND. ( LSAME( BATCH, 'F' ) .OR.
$ LSAME( BATCH, 'I' ) ) ) THEN
WRITE ( NOUT, FMT = 99994 ) NSMP
ELSE IF ( NCYCLE.LE.0 .OR. NSAMPL.GT.NSMPMX ) THEN
WRITE ( NOUT, FMT = 99993 ) NCYCLE
ELSE IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99980 ) N
ELSE
* Read the matrices U and Y from the input file.
IF ( M.GT.0 )
$ READ ( NIN, FMT = * )
$ ( ( U(I,J), J = 1, M ), I = 1, NSAMPL )
READ ( NIN, FMT = * ) ( ( Y(I,J), J = 1, L ), I = 1, NSAMPL )
* Read A and C matrices, if METH <> 'M' and JOB = 'B' or 'D'.
IF ( .NOT.LSAME( METH, 'M' ) .AND.
$ ( LSAME( JOB, 'B' ) .OR. LSAME( JOB, 'D' ) ) ) THEN
DO 10 I = 1, N
READ ( NIN, FMT = * ) ( A(I,J), J = 1, N )
10 CONTINUE
DO 20 I = 1, L
READ ( NIN, FMT = * ) ( C(I,J), J = 1, N )
20 CONTINUE
END IF
* Force some options for IB01AD, depending on the specifications.
IF ( LSAME( METH, 'C' ) ) THEN
METHA = 'M'
JOBDA = 'N'
ELSE
METHA = METH
JOBDA = JOBD
END IF
* Compute the R factor from a QR (or Cholesky) factorization
* of the Hankel-like matrix (or correlation matrix).
DO 30 ICYCLE = 1, NCYCLE
II = ( ICYCLE - 1 )*NSMP + 1
IF ( NCYCLE.GT.1 ) THEN
IF ( ICYCLE.GT.1 ) BATCH = 'I'
IF ( ICYCLE.EQ.NCYCLE ) BATCH = 'L'
END IF
CALL IB01AD( METHA, ALG, JOBDA, BATCH, CONCT, CTRL, NOBR, M,
$ L, NSMP, U(II,1), LDU, Y(II,1), LDY, N, R, LDR,
$ SV, RCOND, TOL, IWORK, DWORK, LDWORK, IWARN,
$ INFO )
30 CONTINUE
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( IWARN.NE.0 )
$ WRITE ( NOUT, FMT = 99990 ) IWARN
IF( NGIVEN )
$ N = NGIV
* Compute the system matrices.
CALL IB01BD( METH, JOB, JOBCK, NOBR, N, M, L, NSMP, R,
$ LDR, A, LDA, C, LDC, B, LDB, D, LDD, Q, LDQ,
$ RY, LDRY, S, LDS, K, LDK, RCOND, IWORK, DWORK,
$ LDWORK, BWORK, IWARN, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99992 ) INFO
ELSE
IF ( IWARN.NE.0 )
$ WRITE ( NOUT, FMT = 99991 ) IWARN
IF ( LSAME( JOB, 'A' ) .OR. LSAME( JOB, 'C' ) ) THEN
WRITE ( NOUT, FMT = 99989 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99988 ) ( A(I,J), J = 1,N )
40 CONTINUE
WRITE ( NOUT, FMT = 99987 )
DO 50 I = 1, L
WRITE ( NOUT, FMT = 99988 ) ( C(I,J), J = 1,N )
50 CONTINUE
END IF
IF ( .NOT.LSAME( JOB, 'C' ) ) THEN
WRITE ( NOUT, FMT = 99986 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99988 ) ( B(I,J), J = 1,M )
60 CONTINUE
END IF
IF ( LSAME( JOB, 'A' ) .OR. LSAME( JOB, 'D' ) ) THEN
WRITE ( NOUT, FMT = 99985 )
DO 70 I = 1, L
WRITE ( NOUT, FMT = 99988 ) ( D(I,J), J = 1,M )
70 CONTINUE
END IF
IF ( LSAME( JOBCK, 'K' ) ) THEN
WRITE ( NOUT, FMT = 99984 )
DO 80 I = 1, N
WRITE ( NOUT, FMT = 99988 ) ( K(I,J), J = 1,L )
80 CONTINUE
END IF
IF ( .NOT.LSAME( JOBCK, 'N' ) ) THEN
WRITE ( NOUT, FMT = 99983 )
DO 90 I = 1, N
WRITE ( NOUT, FMT = 99988 ) ( Q(I,J), J = 1,N )
90 CONTINUE
WRITE ( NOUT, FMT = 99982 )
DO 100 I = 1, L
WRITE ( NOUT, FMT = 99988 ) ( RY(I,J), J = 1,L )
100 CONTINUE
WRITE ( NOUT, FMT = 99981 )
DO 110 I = 1, N
WRITE ( NOUT, FMT = 99988 ) ( S(I,J), J = 1,L )
110 CONTINUE
END IF
END IF
END IF
END IF
STOP
99999 FORMAT ( ' IB01BD EXAMPLE PROGRAM RESULTS', /1X)
99998 FORMAT ( ' INFO on exit from IB01AD = ',I2)
99997 FORMAT (/' NOBR is out of range.',/' NOBR = ', I5)
99996 FORMAT (/' M is out of range.',/' M = ', I5)
99995 FORMAT (/' L is out of range.',/' L = ', I5)
99994 FORMAT (/' NSMP is out of range.',/' NSMP = ', I5)
99993 FORMAT (/' NCYCLE is out of range.',/' NCYCLE = ', I5)
99992 FORMAT ( ' INFO on exit from IB01BD = ',I2)
99991 FORMAT ( ' IWARN on exit from IB01BD = ',I2)
99990 FORMAT ( ' IWARN on exit from IB01AD = ',I2)
99989 FORMAT (/' The system state matrix A is ')
99988 FORMAT (20(1X,F8.4))
99987 FORMAT (/' The system output matrix C is ')
99986 FORMAT (/' The system input matrix B is ')
99985 FORMAT (/' The system input-output matrix D is ')
99984 FORMAT (/' The Kalman gain matrix K is ')
99983 FORMAT (/' The state covariance matrix Q is ')
99982 FORMAT (/' The output covariance matrix Ry is ')
99981 FORMAT (/' The state-output cross-covariance matrix S is ')
99980 FORMAT (/' N is out of range.',/' N = ', I5)
END
Program Data
IB01BD EXAMPLE PROGRAM DATA 15 0 1 1 1000 0.0 -1.0 C C N O N N A K 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 6.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 3.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 4.766099 4.763659 4.839359 5.002979 5.017629 5.056699 5.154379 5.361949 5.425439 5.569519 5.681849 5.742899 5.803949 5.918729 5.821049 5.447419 5.061589 4.629349 4.267939 4.011519 3.850349 3.711159 3.569519 3.518239 3.652549 3.818609 3.862559 4.011519 4.353409 4.705049 5.083559 5.344859 5.274039 5.127519 4.761219 4.451089 4.221539 4.045709 3.874769 3.730689 3.662319 3.576849 3.542659 3.479169 3.454749 3.359509 3.298459 3.225199 3.200779 3.225199 3.227639 3.274039 3.457189 3.867449 4.321659 4.492599 4.431549 4.243519 4.050599 3.857679 3.730689 3.791739 3.921169 3.955359 3.847909 3.725809 3.611039 3.716039 4.092109 4.480389 4.814939 5.054259 5.303339 5.486489 5.672089 5.779529 5.799069 5.664759 5.291129 4.880879 4.558529 4.184909 3.889419 3.708719 3.623249 3.569519 3.718479 4.033499 4.412009 4.629349 4.558529 4.394919 4.180019 4.197119 4.431549 4.714819 4.961459 5.300899 5.567079 5.681849 5.545099 5.188569 4.883319 4.600049 4.270379 4.038389 3.838139 3.711159 3.591499 3.535329 3.486489 3.476729 3.425439 3.381489 3.369279 3.364389 3.347299 3.381489 3.420559 3.413229 3.452309 3.635459 4.038389 4.375379 4.727029 5.056699 5.298459 5.532889 5.466959 5.195899 4.885759 4.763659 4.875989 5.042049 5.283809 5.491379 5.596379 5.672089 5.772209 5.830819 5.933379 5.899189 5.935819 5.894309 5.918729 5.994429 5.957799 6.031059 6.062809 6.040829 6.096999 6.123859 6.162929 6.040829 5.845469 5.772209 5.799069 5.923609 5.928499 6.001759 6.001759 6.060369 5.882099 5.510909 5.322879 5.371719 5.454749 5.437649 5.159269 4.902859 4.587839 4.502369 4.595159 4.824709 5.064029 5.271599 5.466959 5.615919 5.528009 5.254499 4.883319 4.517019 4.197119 4.001759 3.806399 3.904079 3.923609 3.869889 3.806399 3.720929 3.818609 4.140949 4.529229 4.805179 5.086009 5.339969 5.532889 5.576849 5.667199 5.791739 5.850349 5.923609 5.921169 5.977339 5.740459 5.388809 5.000539 4.849129 4.944369 5.173919 5.369279 5.447419 5.603709 5.730689 5.850349 5.979779 5.991989 6.084789 5.940709 5.803949 5.791739 5.603709 5.264269 4.946809 4.619579 4.514579 4.433989 4.285029 4.121419 3.945589 3.984659 4.219099 4.546319 4.873549 5.154379 5.388809 5.613479 5.835699 5.884539 5.955359 5.762439 5.459629 5.061589 4.707499 4.458409 4.267939 4.053039 3.943149 3.825929 3.967569 4.280149 4.480389 4.492599 4.390039 4.197119 4.111649 3.982219 3.867449 3.767319 3.872329 4.236189 4.663539 4.971229 5.066469 4.902859 4.675749 4.392479 4.099439 4.114089 4.326539 4.643999 4.971229 5.159269 5.388809 5.576849 5.652549 5.803949 5.913839 5.886979 5.799069 5.730689 5.762439 5.813719 5.821049 5.928499 6.013969 5.764879 5.413229 5.098219 4.678189 4.372939 4.392479 4.590279 4.919949 5.017629 4.858899 4.675749 4.619579 4.834479 5.090889 5.376599 5.681849 5.823489 5.952919 6.062809 6.089669 6.075019 6.026179 5.994429 6.077459 5.857679 5.701389 5.730689 5.784419 5.823489 5.894309 5.762439 5.415679 4.961459 4.595159 4.331429 4.297239 4.582949 4.861339 5.173919 5.166589 4.919949 4.607369 4.370499 4.182469 4.038389 4.145839 4.431549 4.556089 4.480389 4.375379 4.370499 4.558529 4.858899 4.895529 4.741679 4.744129 4.875989 5.105539 5.239849 5.518239 5.652549 5.723369 5.855239 5.962679 5.984659 5.984659 6.055479 6.062809 6.055479 6.070129 5.784419 5.440099 5.056699 4.941929 5.010299 5.134849 5.313109 5.479169 5.623249 5.562199 5.330209 5.010299 4.665979 4.414459 4.201999 4.048159 4.079899 4.189789 4.131179 4.004199 3.916289 3.960239 4.199559 4.624469 4.883319 5.137289 5.379049 5.623249 5.762439 5.833259 5.686739 5.366839 5.225199 5.239849 5.354629 5.508469 5.596379 5.752669 5.874769 5.906519 5.894309 5.742899 5.447419 5.024959 4.883319 4.885759 4.893089 4.714819 4.451089 4.233749 4.043269 3.864999 3.757559 3.669639 3.593939 3.547539 3.506029 3.454749 3.398579 3.361949 3.339969 3.374159 3.520679 3.713599 3.757559 3.779529 3.696509 3.777089 3.886979 3.904079 3.850349 3.965129 4.282589 4.521899 4.714819 4.971229 5.220319 5.532889 5.652549 5.781979 5.955359 6.035939 6.118969 6.133629 6.153159 6.192229 6.143389 6.167809 5.991989 5.652549 5.459629 5.437649 5.339969 5.098219 4.785639 4.492599 4.236189 4.067689 3.933379 3.823489 3.730689 3.611039 3.564639 3.549989 3.557309 3.513359 3.515799 3.694059 4.072579 4.480389 4.705049 4.612259 4.385149 4.201999 4.026179 3.904079 3.774649 3.691619 3.845469 4.201999 4.585399 4.902859 5.256949 5.510909 5.640339 5.843029 5.974889 5.935819 5.821049 5.528009 5.171479 4.810059 4.453529 4.380269 4.565859 4.805179 5.125079 5.354629 5.589059 5.764879 5.923609 5.940709 5.857679 5.694059 5.486489 5.149499 4.844249 4.541439 4.267939 4.060369 3.960239 3.789299 3.642779 3.525569 3.498699 3.454749 3.408349 3.379049 3.376599 3.361949 3.359509 3.369279 3.398579 3.579289 3.948029 4.412009 4.585399 4.514579 4.343639 4.155599 3.984659 4.043269 4.307009 4.421779 4.353409 4.223979 4.053039 3.940709 3.838139 3.730689 3.652549 3.611039 3.564639 3.496259 3.462069 3.454749 3.425439 3.379049 3.432769 3.623249 3.974889 4.380269 4.714819 5.073799 5.369279 5.603709 5.745349 5.652549 5.401019 5.015189 4.709939 4.416899 4.236189 4.236189 4.248399 4.221539 4.297239 4.590279 4.893089 5.134849 5.427889 5.379049 5.364389 5.452309 5.567079 5.672089 5.769769 5.830819 5.923609 5.965129 6.057919 6.050599 6.072579 6.111649 6.070129 5.896749 5.755109 5.718479 5.821049 6.001759 6.001759 5.901629 5.557309 5.173919 4.800289 4.431549 4.194679 4.006639 3.850349 3.747789 3.642779 3.591499 3.569519 3.528009 3.537779 3.554869 3.493819 3.447419 3.440099 3.408349 3.410789 3.452309 3.681849 4.060369 4.441319 4.854019 5.154379 5.425439 5.596379 5.586619 5.354629 5.027399 4.863779 4.761219 4.570739 4.368059 4.397359 4.573189 4.841809 5.203219 5.452309 5.652549 5.855239 5.906519 5.952919 5.828369 5.791739 5.799069 5.813719 5.877209 5.955359 5.781979 5.518239 5.127519 4.763659 4.492599 4.233749 4.011519 3.855239 3.691619 3.635459 3.818609 4.155599 4.590279 4.988329 5.076239 4.907739 4.648889 4.377829 4.216649 4.287469 4.590279 4.846689 5.139729 5.388809 5.689179 5.884539 6.043269 6.170259 6.211769 6.250839 6.209329 6.013969 5.701389 5.469399 5.479169 5.557309 5.728249 5.882099 5.984659 5.901629 5.581729 5.371719 5.418119 5.510909 5.667199 5.791739 5.698949 5.484049 5.154379 4.980999 5.061589 5.195899 5.359509 5.615919 5.762439 5.857679 5.948029 5.835699 5.706269 5.498699 5.188569 5.117749 5.191009 5.315549 5.532889 5.444979 5.396139 5.274039 5.027399 4.744129 4.668419 4.651329 4.514579 4.267939 4.260609 4.263049 4.189789 4.277699 4.600049 4.932159 5.283809 5.528009 5.740459 5.874769 5.955359 5.991989 5.845469 5.528009 5.061589 4.734359 4.534109 4.534109 4.697729 4.744129 4.619579 4.643999 4.832039 5.132399 5.410789 5.625689 5.603709 5.315549 4.961459 4.619579 4.358289 4.155599 4.033499 3.886979 3.772209 3.640339 3.532889 3.435209 3.427889 3.422999 3.398579 3.603709 4.023729 4.451089 4.792969 4.902859 4.780759 4.590279 4.336309 4.145839 4.216649 4.433989 4.714819 5.098219 5.359509 5.569519 5.772209 5.921169 6.055479 5.962679 5.642779 5.435209 5.388809 5.537779 5.681849 5.701389 5.615919 5.667199 5.740459 5.803949 5.882099 5.950469 6.072579 6.148279 6.116529 6.177579 6.201999 6.206889 5.991989 5.564639 5.178799 4.998089 5.051819 5.232529 5.484049 5.686739 5.899189 5.869889 5.977339 6.053039 6.079899 6.128739 6.079899 6.167809 6.194679 6.236189 6.053039 5.652549 5.274039 4.858899 4.534109 4.455969 4.619579 4.866229 5.117749 5.166589 5.056699 5.002979 5.098219 5.325319 5.567079 5.466959 5.252059 4.946809 4.880879 4.980999 5.225199 5.459629 5.723369 5.791739 5.906519 5.991989 5.835699 5.528009 5.142169 4.775869 4.490159 4.236189 4.023729 3.886979 3.752669 3.681849 3.806399 4.145839 4.600049 5.002979 5.303339 5.552429 5.615919 5.523119 5.611039 5.713599 5.845469 5.899189 5.994429 6.092109 6.092109 6.143389 6.153159 6.233749 6.187349 6.013969 5.835699 5.774649 5.686739 5.537779 5.327759 5.054259 4.700169 4.394919 4.180019 4.043269 3.877209 3.752669 3.728249 3.869889 4.206889 4.355849 4.426669 4.453529 4.521899 4.392479 4.155599 3.965129 3.877209 3.970009 4.258169 4.421779 4.336309 4.299679 4.392479 4.675749 4.761219 4.658659 4.490159 4.307009 4.126299 3.972449 4.077459 4.372939 4.741679 5.088449 5.186129 5.037169 4.785639 4.563419 4.534109 4.705049 4.741679 4.648889 4.431549 4.238629 4.065249 3.943149 3.811279 3.691619 3.652549 3.825929 4.223979 4.424219 4.429109 4.319219 4.138509 3.965129 3.886979 3.801509 3.701389 3.640339 3.767319 4.150719 4.648889 4.990769 5.088449 5.022509 4.783199 4.685519 4.665979 4.707499 4.912619 5.195899 5.415679 5.623249 5.740459 5.899189 5.928499 6.050599 6.153159 5.965129 5.586619 5.381489 5.371719 5.486489 5.567079 5.821049 5.913839 5.994429 6.011519 5.999309 6.018849 5.821049 5.728249 5.740459 5.764879 5.882099 5.926049 5.750229 5.415679 4.995649 4.861339 4.902859 5.103099 5.364389 5.596379 5.752669 5.845469 5.928499 6.006639 5.840579 5.518239 5.173919 4.739239 4.458409 4.426669 4.602489 4.822269 5.183689 5.430329 5.652549 5.821049 5.706269 5.369279 5.027399 4.705049 4.414459 4.145839 3.965129 4.033499 4.372939 4.683079Program Results
IB01BD EXAMPLE PROGRAM RESULTS The system state matrix A is 0.8924 0.3887 0.1285 0.1716 -0.0837 0.6186 -0.6273 -0.4582 0.0052 0.1307 0.6685 -0.6755 0.0055 0.0734 -0.2148 0.4788 The system output matrix C is -0.4442 0.6663 0.3961 0.4102 The system input matrix B is -0.2142 -0.1968 0.0525 0.0361 The system input-output matrix D is -0.0041 The Kalman gain matrix K is -1.9513 -0.1867 0.6348 -0.3486 The state covariance matrix Q is 0.0052 0.0005 -0.0017 0.0009 0.0005 0.0000 -0.0002 0.0001 -0.0017 -0.0002 0.0006 -0.0003 0.0009 0.0001 -0.0003 0.0002 The output covariance matrix Ry is 0.0012 The state-output cross-covariance matrix S is -0.0025 -0.0002 0.0008 -0.0005
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/IB01CD.html 0000664 0000000 0000000 00000153044 14560147231 0020323 0 ustar 00root root 0000000 0000000
Purpose
To estimate the initial state and, optionally, the system matrices
B and D of a linear time-invariant (LTI) discrete-time system,
given the system matrices (A,B,C,D), or (when B and D are
estimated) only the matrix pair (A,C), and the input and output
trajectories of the system. The model structure is :
x(k+1) = Ax(k) + Bu(k), k >= 0,
y(k) = Cx(k) + Du(k),
where x(k) is the n-dimensional state vector (at time k),
u(k) is the m-dimensional input vector,
y(k) is the l-dimensional output vector,
and A, B, C, and D are real matrices of appropriate dimensions.
The input-output data can internally be processed sequentially.
Specification
SUBROUTINE IB01CD( JOBX0, COMUSE, JOB, N, M, L, NSMP, A, LDA, B,
$ LDB, C, LDC, D, LDD, U, LDU, Y, LDY, X0, V,
$ LDV, TOL, IWORK, DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
DOUBLE PRECISION TOL
INTEGER INFO, IWARN, L, LDA, LDB, LDC, LDD, LDU, LDV,
$ LDWORK, LDY, M, N, NSMP
CHARACTER COMUSE, JOB, JOBX0
C .. Array Arguments ..
DOUBLE PRECISION A(LDA, *), B(LDB, *), C(LDC, *), D(LDD, *),
$ DWORK(*), U(LDU, *), V(LDV, *), X0(*),
$ Y(LDY, *)
INTEGER IWORK(*)
Arguments
Mode Parameters
JOBX0 CHARACTER*1
Specifies whether or not the initial state should be
computed, as follows:
= 'X': compute the initial state x(0);
= 'N': do not compute the initial state (possibly,
because x(0) is known to be zero).
COMUSE CHARACTER*1
Specifies whether the system matrices B and D should be
computed or used, as follows:
= 'C': compute the system matrices B and D, as specified
by JOB;
= 'U': use the system matrices B and D, as specified by
JOB;
= 'N': do not compute/use the matrices B and D.
If JOBX0 = 'N' and COMUSE <> 'N', then x(0) is set
to zero.
If JOBX0 = 'N' and COMUSE = 'N', then x(0) is
neither computed nor set to zero.
JOB CHARACTER*1
If COMUSE = 'C' or 'U', specifies which of the system
matrices B and D should be computed or used, as follows:
= 'B': compute/use the matrix B only (D is known to be
zero);
= 'D': compute/use the matrices B and D.
The value of JOB is irrelevant if COMUSE = 'N' or if
JOBX0 = 'N' and COMUSE = 'U'.
The combinations of options, the data used, and the
returned results, are given in the table below, where
'*' denotes an irrelevant value.
JOBX0 COMUSE JOB Data used Returned results
----------------------------------------------------------
X C B A,C,u,y x,B
X C D A,C,u,y x,B,D
N C B A,C,u,y x=0,B
N C D A,C,u,y x=0,B,D
----------------------------------------------------------
X U B A,B,C,u,y x
X U D A,B,C,D,u,y x
N U * - x=0
----------------------------------------------------------
X N * A,C,y x
N N * - -
----------------------------------------------------------
For JOBX0 = 'N' and COMUSE = 'N', the routine just
sets DWORK(1) to 2 and DWORK(2) to 1, and returns
(see the parameter DWORK).
Input/Output Parameters
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
L (input) INTEGER
The number of system outputs. L > 0.
NSMP (input) INTEGER
The number of rows of matrices U and Y (number of
samples, t).
NSMP >= 0, if JOBX0 = 'N' and COMUSE <> 'C';
NSMP >= N, if JOBX0 = 'X' and COMUSE <> 'C';
NSMP >= N*M + a + e, if COMUSE = 'C',
where a = 0, if JOBX0 = 'N';
a = N, if JOBX0 = 'X';
e = 0, if JOBX0 = 'X' and JOB = 'B';
e = 1, if JOBX0 = 'N' and JOB = 'B';
e = M, if JOB = 'D'.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
If JOBX0 = 'X' or COMUSE = 'C', the leading N-by-N
part of this array must contain the system state matrix A.
If N = 0, or JOBX0 = 'N' and COMUSE <> 'C', this
array is not referenced.
LDA INTEGER
The leading dimension of the array A.
LDA >= MAX(1,N), if JOBX0 = 'X' or COMUSE = 'C';
LDA >= 1, if JOBX0 = 'N' and COMUSE <> 'C'.
B (input or output) DOUBLE PRECISION array, dimension
(LDB,M)
If JOBX0 = 'X' and COMUSE = 'U', B is an input
parameter and, on entry, the leading N-by-M part of this
array must contain the system input matrix B.
If COMUSE = 'C', B is an output parameter and, on exit,
if INFO = 0, the leading N-by-M part of this array
contains the estimated system input matrix B.
If min(N,M) = 0, or JOBX0 = 'N' and COMUSE = 'U',
or COMUSE = 'N', this array is not referenced.
LDB INTEGER
The leading dimension of the array B.
LDB >= MAX(1,N), if M > 0, COMUSE = 'U', JOBX0 = 'X',
or M > 0, COMUSE = 'C';
LDB >= 1, if min(N,M) = 0, or COMUSE = 'N',
or JOBX0 = 'N' and COMUSE = 'U'.
C (input) DOUBLE PRECISION array, dimension (LDC,N)
If JOBX0 = 'X' or COMUSE = 'C', the leading L-by-N
part of this array must contain the system output
matrix C.
If N = 0, or JOBX0 = 'N' and COMUSE <> 'C', this
array is not referenced.
LDC INTEGER
The leading dimension of the array C.
LDC >= L, if N > 0, and JOBX0 = 'X' or COMUSE = 'C';
LDC >= 1, if N = 0, or JOBX0 = 'N' and COMUSE <> 'C'.
D (input or output) DOUBLE PRECISION array, dimension
(LDD,M)
If JOBX0 = 'X', COMUSE = 'U', and JOB = 'D', D is an
input parameter and, on entry, the leading L-by-M part of
this array must contain the system input-output matrix D.
If COMUSE = 'C' and JOB = 'D', D is an output
parameter and, on exit, if INFO = 0, the leading
L-by-M part of this array contains the estimated system
input-output matrix D.
If M = 0, or JOBX0 = 'N' and COMUSE = 'U', or
COMUSE = 'N', or JOB = 'B', this array is not
referenced.
LDD INTEGER
The leading dimension of the array D.
LDD >= L, if M > 0, JOBX0 = 'X', COMUSE = 'U', and
JOB = 'D', or
if M > 0, COMUSE = 'C', and JOB = 'D';
LDD >= 1, if M = 0, or JOBX0 = 'N' and COMUSE = 'U',
or COMUSE = 'N', or JOB = 'B'.
U (input or input/output) DOUBLE PRECISION array, dimension
(LDU,M)
On entry, if COMUSE = 'C', or JOBX0 = 'X' and
COMUSE = 'U', the leading NSMP-by-M part of this array
must contain the t-by-m input-data sequence matrix U,
U = [u_1 u_2 ... u_m]. Column j of U contains the
NSMP values of the j-th input component for consecutive
time increments.
On exit, if COMUSE = 'C' and JOB = 'D', the leading
NSMP-by-M part of this array contains details of the
QR factorization of the t-by-m matrix U, possibly
computed sequentially (see METHOD).
If COMUSE = 'C' and JOB = 'B', or COMUSE = 'U', this
array is unchanged on exit.
If M = 0, or JOBX0 = 'N' and COMUSE = 'U', or
COMUSE = 'N', this array is not referenced.
LDU INTEGER
The leading dimension of the array U.
LDU >= MAX(1,NSMP), if M > 0 and COMUSE = 'C' or
JOBX0 = 'X' and COMUSE = 'U;
LDU >= 1, if M = 0, or COMUSE = 'N', or
JOBX0 = 'N' and COMUSE = 'U'.
Y (input) DOUBLE PRECISION array, dimension (LDY,L)
On entry, if JOBX0 = 'X' or COMUSE = 'C', the leading
NSMP-by-L part of this array must contain the t-by-l
output-data sequence matrix Y, Y = [y_1 y_2 ... y_l].
Column j of Y contains the NSMP values of the j-th
output component for consecutive time increments.
If JOBX0 = 'N' and COMUSE <> 'C', this array is not
referenced.
LDY INTEGER
The leading dimension of the array Y.
LDY >= MAX(1,NSMP), if JOBX0 = 'X' or COMUSE = 'C;
LDY >= 1, if JOBX0 = 'N' and COMUSE <> 'C'.
X0 (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0 and JOBX0 = 'X', this array contains the
estimated initial state of the system, x(0).
If JOBX0 = 'N' and COMUSE = 'C', this array is used as
workspace and finally it is set to zero.
If JOBX0 = 'N' and COMUSE = 'U', then x(0) is set to
zero without any calculations.
If JOBX0 = 'N' and COMUSE = 'N', this array is not
referenced.
V (output) DOUBLE PRECISION array, dimension (LDV,N)
On exit, if INFO = 0 or 2, JOBX0 = 'X' or
COMUSE = 'C', the leading N-by-N part of this array
contains the orthogonal matrix V of a real Schur
factorization of the matrix A.
If JOBX0 = 'N' and COMUSE <> 'C', this array is not
referenced.
LDV INTEGER
The leading dimension of the array V.
LDV >= MAX(1,N), if JOBX0 = 'X' or COMUSE = 'C;
LDV >= 1, if JOBX0 = 'N' and COMUSE <> 'C'.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used for estimating the rank of
matrices. If the user sets TOL > 0, then the given value
of TOL is used as a lower bound for the reciprocal
condition number; a matrix whose estimated condition
number is less than 1/TOL is considered to be of full
rank. If the user sets TOL <= 0, then EPS is used
instead, where EPS is the relative machine precision
(see LAPACK Library routine DLAMCH). TOL <= 1.
Workspace
IWORK INTEGER array, dimension (LIWORK), where
LIWORK >= 0, if JOBX0 = 'N' and COMUSE <> 'C';
LIWORK >= N, if JOBX0 = 'X' and COMUSE <> 'C';
LIWORK >= N*M + a, if COMUSE = 'C' and JOB = 'B',
LIWORK >= max(N*M + a,M), if COMUSE = 'C' and JOB = 'D',
with a = 0, if JOBX0 = 'N';
a = N, if JOBX0 = 'X'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK; DWORK(2) contains the reciprocal condition
number of the triangular factor of the QR factorization of
the matrix W2, if COMUSE = 'C', or of the matrix
Gamma, if COMUSE = 'U' (see METHOD); if JOBX0 = 'N'
and COMUSE <> 'C', DWORK(2) is set to one;
if COMUSE = 'C', M > 0, and JOB = 'D', DWORK(3)
contains the reciprocal condition number of the triangular
factor of the QR factorization of U; denoting
g = 2, if JOBX0 = 'X' and COMUSE <> 'C' or
COMUSE = 'C' and M = 0 or JOB = 'B',
g = 3, if COMUSE = 'C' and M > 0 and JOB = 'D',
then DWORK(i), i = g+1:g+N*N,
DWORK(j), j = g+1+N*N:g+N*N+L*N, and
DWORK(k), k = g+1+N*N+L*N:g+N*N+L*N+N*M,
contain the transformed system matrices At, Ct, and Bt,
respectively, corresponding to the real Schur form of the
given system state matrix A, i.e.,
At = V'*A*V, Bt = V'*B, Ct = C*V.
The matrices At, Ct, Bt are not computed if JOBX0 = 'N'
and COMUSE <> 'C'.
On exit, if INFO = -26, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 2, if JOBX0 = 'N' and COMUSE <> 'C', or
if max( N, M ) = 0.
Otherwise,
LDWORK >= LDW1 + N*( N + M + L ) +
max( 5*N, LDW1, min( LDW2, LDW3 ) ),
where, if COMUSE = 'C', then
LDW1 = 2, if M = 0 or JOB = 'B',
LDW1 = 3, if M > 0 and JOB = 'D',
LDWa = t*L*(r + 1) + max( N + max( d, f ), 6*r ),
LDW2 = LDWa, if M = 0 or JOB = 'B',
LDW2 = max( LDWa, t*L*(r + 1) + 2*M*M + 6*M ),
if M > 0 and JOB = 'D',
LDWb = (b + r)*(r + 1) +
max( q*(r + 1) + N*N*M + c + max( d, f ), 6*r ),
LDW3 = LDWb, if M = 0 or JOB = 'B',
LDW3 = max( LDWb, (b + r)*(r + 1) + 2*M*M + 6*M ),
if M > 0 and JOB = 'D',
r = N*M + a,
a = 0, if JOBX0 = 'N',
a = N, if JOBX0 = 'X';
b = 0, if JOB = 'B',
b = L*M, if JOB = 'D';
c = 0, if JOBX0 = 'N',
c = L*N, if JOBX0 = 'X';
d = 0, if JOBX0 = 'N',
d = 2*N*N + N, if JOBX0 = 'X';
f = 2*r, if JOB = 'B' or M = 0,
f = M + max( 2*r, M ), if JOB = 'D' and M > 0;
q = b + r*L;
and, if JOBX0 = 'X' and COMUSE <> 'C', then
LDW1 = 2,
LDW2 = t*L*(N + 1) + 2*N + max( 2*N*N, 4*N ),
LDW3 = N*(N + 1) + 2*N + max( q*(N + 1) + 2*N*N + L*N,
4*N ),
q = N*L.
For good performance, LDWORK should be larger.
If LDWORK >= LDW2, or if COMUSE = 'C' and
LDWORK >= t*L*(r + 1) + (b + r)*(r + 1) + N*N*M + c +
max( d, f ),
then standard QR factorizations of the matrices U and/or
W2, if COMUSE = 'C', or of the matrix Gamma, if
JOBX0 = 'X' and COMUSE <> 'C' (see METHOD), are used.
Otherwise, the QR factorizations are computed sequentially
by performing NCYCLE cycles, each cycle (except possibly
the last one) processing s < t samples, where s is
chosen by equating LDWORK to the first term of LDWb,
if COMUSE = 'C', or of LDW3, if COMUSE <> 'C', for
q replaced by s*L. (s is larger than or equal to the
minimum value of NSMP.) The computational effort may
increase and the accuracy may slightly decrease with the
decrease of s. Recommended value is LDWORK = LDW2,
assuming a large enough cache size, to also accommodate
A, (B,) C, (D,) U, and Y.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 4: the least squares problem to be solved has a
rank-deficient coefficient matrix;
= 6: the matrix A is unstable; the estimated x(0)
and/or B and D could be inaccurate.
NOTE: the value 4 of IWARN has no significance for the
identification problem.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the QR algorithm failed to compute all the
eigenvalues of the matrix A (see LAPACK Library
routine DGEES); the locations DWORK(i), for
i = g+1:g+N*N, contain the partially converged
Schur form;
= 2: the singular value decomposition (SVD) algorithm did
not converge.
Method
Matrix A is initially reduced to a real Schur form, A = V*At*V',
and the given system matrices are transformed accordingly. For the
reduced system, an extension and refinement of the method in [1,2]
is used. Specifically, for JOBX0 = 'X', COMUSE = 'C', and
JOB = 'D', denoting
X = [ vec(D')' vec(B)' x0' ]',
where vec(M) is the vector obtained by stacking the columns of
the matrix M, then X is the least squares solution of the
system S*X = vec(Y), with the matrix S = [ diag(U) W ],
defined by
( U | | ... | | | ... | | )
( U | 11 | ... | n1 | 12 | ... | nm | )
S = ( : | y | ... | y | y | ... | y | P*Gamma ),
( : | | ... | | | ... | | )
( U | | ... | | | ... | | )
ij
diag(U) having L block rows and columns. In this formula, y
are the outputs of the system for zero initial state computed
using the following model, for j = 1:m, and for i = 1:n,
ij ij ij
x (k+1) = Ax (k) + e_i u_j(k), x (0) = 0,
ij ij
y (k) = Cx (k),
where e_i is the i-th n-dimensional unit vector, Gamma is
given by
( C )
( C*A )
Gamma = ( C*A^2 ),
( : )
( C*A^(t-1) )
and P is a permutation matrix that groups together the rows of
Gamma depending on the same row of C, namely
[ c_j; c_j*A; c_j*A^2; ... c_j*A^(t-1) ], for j = 1:L.
The first block column, diag(U), is not explicitly constructed,
but its structure is exploited. The last block column is evaluated
using powers of A with exponents 2^k. No interchanges are applied.
A special QR decomposition of the matrix S is computed. Let
U = q*[ r' 0 ]' be the QR decomposition of U, if M > 0, where
r is M-by-M. Then, diag(q') is applied to W and vec(Y).
The block-rows of S and vec(Y) are implicitly permuted so that
matrix S becomes
( diag(r) W1 )
( 0 W2 ),
where W1 has L*M rows. Then, the QR decomposition of W2 is
computed (sequentially, if M > 0) and used to obtain B and x0.
The intermediate results and the QR decomposition of U are
needed to find D. If a triangular factor is too ill conditioned,
then singular value decomposition (SVD) is employed. SVD is not
generally needed if the input sequence is sufficiently
persistently exciting and NSMP is large enough.
If the matrix W cannot be stored in the workspace (i.e.,
LDWORK < LDW2), the QR decompositions of W2 and U are
computed sequentially.
For JOBX0 = 'N' and COMUSE = 'C', or JOB = 'B', a simpler
problem is solved efficiently.
For JOBX0 = 'X' and COMUSE <> 'C', a simpler method is used.
Specifically, the output y0(k) of the system for zero initial
state is computed for k = 0, 1, ..., t-1 using the given model.
Then the following least squares problem is solved for x(0)
( y(0) - y0(0) )
( y(1) - y0(1) )
Gamma * x(0) = ( : ).
( : )
( y(t-1) - y0(t-1) )
The coefficient matrix Gamma is evaluated using powers of A with
exponents 2^k. The QR decomposition of this matrix is computed.
If its triangular factor R is too ill conditioned, then singular
value decomposition of R is used.
If the coefficient matrix cannot be stored in the workspace (i.e.,
LDWORK < LDW2), the QR decomposition is computed sequentially.
References
[1] Verhaegen M., and Varga, A.
Some Experience with the MOESP Class of Subspace Model
Identification Methods in Identifying the BO105 Helicopter.
Report TR R165-94, DLR Oberpfaffenhofen, 1994.
[2] Sima, V., and Varga, A.
RASP-IDENT : Subspace Model Identification Programs.
Deutsche Forschungsanstalt fur Luft- und Raumfahrt e. V.,
Report TR R888-94, DLR Oberpfaffenhofen, Oct. 1994.
Numerical Aspects
The implemented method is numerically stable.Further Comments
The algorithm for computing the system matrices B and D is less efficient than the MOESP or N4SID algorithms implemented in SLICOT Library routines IB01BD/IB01PD, because a large least squares problem has to be solved, but the accuracy is better, as the computed matrices B and D are fitted to the input and output trajectories. However, if matrix A is unstable, the computed matrices B and D could be inaccurate.Example
Program Text
* IB01CD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER LDA, LDB, LDC, LDD, LDR, LDU, LDV, LDW1, LDW2,
$ LDW4, LDW5, LDWORK, LDY, LIWORK, LMAX, MMAX,
$ NMAX, NOBRMX, NSMPMX
PARAMETER ( LMAX = 5, MMAX = 5, NOBRMX = 20, NSMPMX = 2000,
$ NMAX = NOBRMX - 1, LDA = NMAX, LDB = NMAX,
$ LDC = LMAX, LDD = LMAX, LDV = NMAX,
$ LDR = MAX( 2*( MMAX + LMAX )*NOBRMX,
$ 3*MMAX*NOBRMX ), LDU = NSMPMX,
$ LDW1 = MAX( LMAX*( NOBRMX - 1 )*NMAX + NMAX +
$ MAX( 6*MMAX, 4*LMAX )*NOBRMX,
$ LMAX*NOBRMX*NMAX +
$ MAX( LMAX*( NOBRMX - 1 )*NMAX +
$ 3*NMAX + LMAX +
$ ( 2*MMAX + LMAX )*NOBRMX,
$ 2*LMAX*( NOBRMX - 1 )*NMAX +
$ NMAX*NMAX + 8*NMAX,
$ NMAX +
$ 4*( MMAX*NOBRMX + NMAX ) ) ),
$ LDW2 = LMAX*NOBRMX*NMAX +
$ MMAX*NOBRMX*( NMAX + LMAX )*
$ ( MMAX*( NMAX + LMAX ) + 1 ) +
$ MAX( ( NMAX + LMAX )**2,
$ 4*MMAX*( NMAX + LMAX ) + 1 ),
$ LDW4 = NSMPMX*LMAX*NMAX*( MMAX + 1 ) +
$ MAX( NMAX +
$ MAX( 2*NMAX*NMAX + NMAX,
$ MMAX +
$ MAX( 2*NMAX*( MMAX + 1 ),
$ MMAX ),
$ 6*NMAX*( MMAX + 1 ) ),
$ 2*MMAX*MMAX*NMAX + 6*MMAX ),
$ LDW5 = ( LMAX*MMAX + NMAX*( MMAX + 1 ) )*
$ NMAX*( MMAX + 1 ) +
$ MAX( ( LMAX*MMAX +
$ LMAX*NMAX*( MMAX + 1 ) )*
$ NMAX*( MMAX + 1 ) +
$ NMAX*NMAX*MMAX + LMAX*NMAX +
$ MAX( 2*NMAX*NMAX + NMAX,
$ MMAX +
$ MAX( 2*NMAX*( MMAX + 1 ),
$ MMAX ),
$ 6*NMAX*( MMAX + 1 ) ),
$ 2*MMAX*MMAX*NMAX + 6*MMAX ),
$ LDWORK = MAX( 6*( MMAX + LMAX )*NOBRMX,
$ ( MMAX + LMAX )*( 4*NOBRMX*
$ ( MMAX + LMAX + 2 ) - 2 ),
$ ( MMAX + LMAX )*4*NOBRMX*
$ ( NOBRMX + 1 ), LDW1, LDW2,
$ 3 + ( NMAX + MMAX + LMAX )*NMAX +
$ MAX( 5*NMAX, 3,
$ MIN( LDW4, LDW5 ) ) ),
$ LDY = NSMPMX,
$ LIWORK = MAX( ( MMAX + LMAX )*NOBRMX,
$ MMAX*NOBRMX + NMAX,
$ MMAX*( NMAX + LMAX ),
$ NMAX*MMAX + NMAX, MMAX )
$ )
* .. Local Scalars ..
LOGICAL NGIVEN
CHARACTER ALG, BATCH, COMUSE, CONCT, CTRL, JOB, JOBBD,
$ JOBCK, JOBD, JOBDA, JOBX0, METH, METHA
INTEGER I, ICYCLE, II, INFO, IWARN, J, L, M, N, NCYCLE,
$ NGIV, NOBR, NSAMPL, NSMP
DOUBLE PRECISION RCOND, TOL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA, NMAX), B(LDB, MMAX), C(LDC, NMAX),
$ D(LDD, MMAX), DUM(1), DWORK(LDWORK),
$ R(LDR, 2*(MMAX+LMAX)*NOBRMX),
$ SV(LMAX*NOBRMX), U(LDU, MMAX), V(LDV, NMAX),
$ X0(NMAX), Y(LDY, LMAX)
INTEGER IWORK(LIWORK)
LOGICAL BWORK(1)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL IB01AD, IB01BD, IB01CD
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
* If the value of N is positive, it will be taken as system order.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) NOBR, N, M, L, NSMP, RCOND, TOL
READ ( NIN, FMT = * ) METH, ALG, JOBD, BATCH, CONCT, CTRL, JOB,
$ COMUSE, JOBX0
IF ( LSAME( BATCH, 'F' ) ) THEN
READ ( NIN, FMT = * ) NCYCLE
ELSE
NCYCLE = 1
END IF
NSAMPL = NCYCLE*NSMP
*
NGIVEN = N.GT.0
IF( NGIVEN )
$ NGIV = N
IF ( NOBR.LE.0 .OR. NOBR.GT.NOBRMX ) THEN
WRITE ( NOUT, FMT = 99997 ) NOBR
ELSE IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99996 ) M
ELSE IF ( L.LE.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) L
ELSE IF ( NSMP.LT.0 .OR. NSMP.GT.NSMPMX .OR.
$ ( NSMP.LT.2*( M + L + 1 )*NOBR - 1 .AND.
$ LSAME( BATCH, 'O' ) ) .OR.
$ ( NSAMPL.LT.2*( M + L + 1 )*NOBR - 1 .AND.
$ LSAME( BATCH, 'L' ) ) .OR.
$ NSMP.LT.2*NOBR .AND. ( LSAME( BATCH, 'F' ) .OR.
$ LSAME( BATCH, 'I' ) ) ) THEN
WRITE ( NOUT, FMT = 99994 ) NSMP
ELSE IF ( NCYCLE.LE.0 .OR. NSAMPL.GT.NSMPMX ) THEN
WRITE ( NOUT, FMT = 99993 ) NCYCLE
ELSE IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99983 ) N
ELSE
* Read the matrices U and Y from the input file.
IF ( M.GT.0 )
$ READ ( NIN, FMT = * )
$ ( ( U(I,J), J = 1, M ), I = 1, NSAMPL )
READ ( NIN, FMT = * ) ( ( Y(I,J), J = 1, L ), I = 1, NSAMPL )
* Force some options, depending on the specifications.
IF ( LSAME( METH, 'C' ) ) THEN
METHA = 'M'
JOBDA = 'N'
ELSE
METHA = METH
JOBDA = JOBD
END IF
* The covariances and Kalman gain matrix are not computed.
JOBCK = 'N'
IF ( LSAME( JOB, 'A' ) .OR. LSAME( JOB, 'C' ) ) THEN
JOBBD = 'D'
ELSE
JOBBD = JOB
END IF
IF ( LSAME( COMUSE, 'C' ) ) THEN
JOB = 'C'
ELSE IF ( LSAME( COMUSE, 'U' ) ) THEN
JOB = 'A'
END IF
* Compute the R factor from a QR (or Cholesky) factorization
* of the Hankel-like matrix (or correlation matrix).
DO 10 ICYCLE = 1, NCYCLE
II = ( ICYCLE - 1 )*NSMP + 1
IF ( NCYCLE.GT.1 ) THEN
IF ( ICYCLE.GT.1 ) BATCH = 'I'
IF ( ICYCLE.EQ.NCYCLE ) BATCH = 'L'
END IF
CALL IB01AD( METHA, ALG, JOBDA, BATCH, CONCT, CTRL, NOBR, M,
$ L, NSMP, U(II,1), LDU, Y(II,1), LDY, N, R, LDR,
$ SV, RCOND, TOL, IWORK, DWORK, LDWORK, IWARN,
$ INFO )
10 CONTINUE
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( IWARN.NE.0 )
$ WRITE ( NOUT, FMT = 99990 ) IWARN
IF( NGIVEN )
$ N = NGIV
* Compute the system matrices and x0.
CALL IB01BD( METH, JOB, JOBCK, NOBR, N, M, L, NSMP, R,
$ LDR, A, LDA, C, LDC, B, LDB, D, LDD, DUM, 1,
$ DUM, 1, DUM, 1, DUM, 1, RCOND, IWORK, DWORK,
$ LDWORK, BWORK, IWARN, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99982 ) INFO
ELSE
IF ( IWARN.NE.0 )
$ WRITE ( NOUT, FMT = 99981 ) IWARN
CALL IB01CD( JOBX0, COMUSE, JOBBD, N, M, L, NSMP, A, LDA,
$ B, LDB, C, LDC, D, LDD, U, LDU, Y, LDY, X0,
$ V, LDV, RCOND, IWORK, DWORK, LDWORK, IWARN,
$ INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99992 ) INFO
ELSE
IF ( IWARN.NE.0 )
$ WRITE ( NOUT, FMT = 99991 ) IWARN
IF ( LSAME( JOB, 'A' ) .OR. LSAME( JOB, 'C' ) ) THEN
WRITE ( NOUT, FMT = 99989 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99988 ) ( A(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99987 )
DO 30 I = 1, L
WRITE ( NOUT, FMT = 99988 ) ( C(I,J), J = 1,N )
30 CONTINUE
END IF
IF ( LSAME( COMUSE, 'C' ) ) THEN
WRITE ( NOUT, FMT = 99986 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99988 ) ( B(I,J), J = 1,M )
40 CONTINUE
IF ( LSAME( JOBBD, 'D' ) ) THEN
WRITE ( NOUT, FMT = 99985 )
DO 50 I = 1, L
WRITE ( NOUT, FMT = 99988 )
$ ( D(I,J), J = 1,M )
50 CONTINUE
END IF
END IF
IF ( LSAME( JOBX0, 'X' ) ) THEN
WRITE ( NOUT, FMT = 99984 )
WRITE ( NOUT, FMT = 99988 ) ( X0(I), I = 1,N )
END IF
END IF
END IF
END IF
END IF
STOP
99999 FORMAT ( ' IB01CD EXAMPLE PROGRAM RESULTS', /1X)
99998 FORMAT ( ' INFO on exit from IB01AD = ',I2)
99997 FORMAT (/' NOBR is out of range.',/' NOBR = ', I5)
99996 FORMAT (/' M is out of range.',/' M = ', I5)
99995 FORMAT (/' L is out of range.',/' L = ', I5)
99994 FORMAT (/' NSMP is out of range.',/' NSMP = ', I5)
99993 FORMAT (/' NCYCLE is out of range.',/' NCYCLE = ', I5)
99992 FORMAT ( ' INFO on exit from IB01CD = ',I2)
99991 FORMAT ( ' IWARN on exit from IB01CD = ',I2)
99990 FORMAT ( ' IWARN on exit from IB01AD = ',I2)
99989 FORMAT (/' The system state matrix A is ')
99988 FORMAT (20(1X,F8.4))
99987 FORMAT (/' The system output matrix C is ')
99986 FORMAT (/' The system input matrix B is ')
99985 FORMAT (/' The system input-output matrix D is ')
99984 FORMAT (/' The initial state vector x0 is ')
99983 FORMAT (/' N is out of range.',/' N = ', I5)
99982 FORMAT ( ' INFO on exit from IB01BD = ',I2)
99981 FORMAT ( ' IWARN on exit from IB01BD = ',I2)
END
Program Data
IB01CD EXAMPLE PROGRAM DATA 15 0 1 1 1000 0.0 -1.0 C C N O N N A C X 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 6.41 3.41 3.41 3.41 6.41 3.41 3.41 3.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 3.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 6.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 6.41 6.41 4.766099 4.763659 4.839359 5.002979 5.017629 5.056699 5.154379 5.361949 5.425439 5.569519 5.681849 5.742899 5.803949 5.918729 5.821049 5.447419 5.061589 4.629349 4.267939 4.011519 3.850349 3.711159 3.569519 3.518239 3.652549 3.818609 3.862559 4.011519 4.353409 4.705049 5.083559 5.344859 5.274039 5.127519 4.761219 4.451089 4.221539 4.045709 3.874769 3.730689 3.662319 3.576849 3.542659 3.479169 3.454749 3.359509 3.298459 3.225199 3.200779 3.225199 3.227639 3.274039 3.457189 3.867449 4.321659 4.492599 4.431549 4.243519 4.050599 3.857679 3.730689 3.791739 3.921169 3.955359 3.847909 3.725809 3.611039 3.716039 4.092109 4.480389 4.814939 5.054259 5.303339 5.486489 5.672089 5.779529 5.799069 5.664759 5.291129 4.880879 4.558529 4.184909 3.889419 3.708719 3.623249 3.569519 3.718479 4.033499 4.412009 4.629349 4.558529 4.394919 4.180019 4.197119 4.431549 4.714819 4.961459 5.300899 5.567079 5.681849 5.545099 5.188569 4.883319 4.600049 4.270379 4.038389 3.838139 3.711159 3.591499 3.535329 3.486489 3.476729 3.425439 3.381489 3.369279 3.364389 3.347299 3.381489 3.420559 3.413229 3.452309 3.635459 4.038389 4.375379 4.727029 5.056699 5.298459 5.532889 5.466959 5.195899 4.885759 4.763659 4.875989 5.042049 5.283809 5.491379 5.596379 5.672089 5.772209 5.830819 5.933379 5.899189 5.935819 5.894309 5.918729 5.994429 5.957799 6.031059 6.062809 6.040829 6.096999 6.123859 6.162929 6.040829 5.845469 5.772209 5.799069 5.923609 5.928499 6.001759 6.001759 6.060369 5.882099 5.510909 5.322879 5.371719 5.454749 5.437649 5.159269 4.902859 4.587839 4.502369 4.595159 4.824709 5.064029 5.271599 5.466959 5.615919 5.528009 5.254499 4.883319 4.517019 4.197119 4.001759 3.806399 3.904079 3.923609 3.869889 3.806399 3.720929 3.818609 4.140949 4.529229 4.805179 5.086009 5.339969 5.532889 5.576849 5.667199 5.791739 5.850349 5.923609 5.921169 5.977339 5.740459 5.388809 5.000539 4.849129 4.944369 5.173919 5.369279 5.447419 5.603709 5.730689 5.850349 5.979779 5.991989 6.084789 5.940709 5.803949 5.791739 5.603709 5.264269 4.946809 4.619579 4.514579 4.433989 4.285029 4.121419 3.945589 3.984659 4.219099 4.546319 4.873549 5.154379 5.388809 5.613479 5.835699 5.884539 5.955359 5.762439 5.459629 5.061589 4.707499 4.458409 4.267939 4.053039 3.943149 3.825929 3.967569 4.280149 4.480389 4.492599 4.390039 4.197119 4.111649 3.982219 3.867449 3.767319 3.872329 4.236189 4.663539 4.971229 5.066469 4.902859 4.675749 4.392479 4.099439 4.114089 4.326539 4.643999 4.971229 5.159269 5.388809 5.576849 5.652549 5.803949 5.913839 5.886979 5.799069 5.730689 5.762439 5.813719 5.821049 5.928499 6.013969 5.764879 5.413229 5.098219 4.678189 4.372939 4.392479 4.590279 4.919949 5.017629 4.858899 4.675749 4.619579 4.834479 5.090889 5.376599 5.681849 5.823489 5.952919 6.062809 6.089669 6.075019 6.026179 5.994429 6.077459 5.857679 5.701389 5.730689 5.784419 5.823489 5.894309 5.762439 5.415679 4.961459 4.595159 4.331429 4.297239 4.582949 4.861339 5.173919 5.166589 4.919949 4.607369 4.370499 4.182469 4.038389 4.145839 4.431549 4.556089 4.480389 4.375379 4.370499 4.558529 4.858899 4.895529 4.741679 4.744129 4.875989 5.105539 5.239849 5.518239 5.652549 5.723369 5.855239 5.962679 5.984659 5.984659 6.055479 6.062809 6.055479 6.070129 5.784419 5.440099 5.056699 4.941929 5.010299 5.134849 5.313109 5.479169 5.623249 5.562199 5.330209 5.010299 4.665979 4.414459 4.201999 4.048159 4.079899 4.189789 4.131179 4.004199 3.916289 3.960239 4.199559 4.624469 4.883319 5.137289 5.379049 5.623249 5.762439 5.833259 5.686739 5.366839 5.225199 5.239849 5.354629 5.508469 5.596379 5.752669 5.874769 5.906519 5.894309 5.742899 5.447419 5.024959 4.883319 4.885759 4.893089 4.714819 4.451089 4.233749 4.043269 3.864999 3.757559 3.669639 3.593939 3.547539 3.506029 3.454749 3.398579 3.361949 3.339969 3.374159 3.520679 3.713599 3.757559 3.779529 3.696509 3.777089 3.886979 3.904079 3.850349 3.965129 4.282589 4.521899 4.714819 4.971229 5.220319 5.532889 5.652549 5.781979 5.955359 6.035939 6.118969 6.133629 6.153159 6.192229 6.143389 6.167809 5.991989 5.652549 5.459629 5.437649 5.339969 5.098219 4.785639 4.492599 4.236189 4.067689 3.933379 3.823489 3.730689 3.611039 3.564639 3.549989 3.557309 3.513359 3.515799 3.694059 4.072579 4.480389 4.705049 4.612259 4.385149 4.201999 4.026179 3.904079 3.774649 3.691619 3.845469 4.201999 4.585399 4.902859 5.256949 5.510909 5.640339 5.843029 5.974889 5.935819 5.821049 5.528009 5.171479 4.810059 4.453529 4.380269 4.565859 4.805179 5.125079 5.354629 5.589059 5.764879 5.923609 5.940709 5.857679 5.694059 5.486489 5.149499 4.844249 4.541439 4.267939 4.060369 3.960239 3.789299 3.642779 3.525569 3.498699 3.454749 3.408349 3.379049 3.376599 3.361949 3.359509 3.369279 3.398579 3.579289 3.948029 4.412009 4.585399 4.514579 4.343639 4.155599 3.984659 4.043269 4.307009 4.421779 4.353409 4.223979 4.053039 3.940709 3.838139 3.730689 3.652549 3.611039 3.564639 3.496259 3.462069 3.454749 3.425439 3.379049 3.432769 3.623249 3.974889 4.380269 4.714819 5.073799 5.369279 5.603709 5.745349 5.652549 5.401019 5.015189 4.709939 4.416899 4.236189 4.236189 4.248399 4.221539 4.297239 4.590279 4.893089 5.134849 5.427889 5.379049 5.364389 5.452309 5.567079 5.672089 5.769769 5.830819 5.923609 5.965129 6.057919 6.050599 6.072579 6.111649 6.070129 5.896749 5.755109 5.718479 5.821049 6.001759 6.001759 5.901629 5.557309 5.173919 4.800289 4.431549 4.194679 4.006639 3.850349 3.747789 3.642779 3.591499 3.569519 3.528009 3.537779 3.554869 3.493819 3.447419 3.440099 3.408349 3.410789 3.452309 3.681849 4.060369 4.441319 4.854019 5.154379 5.425439 5.596379 5.586619 5.354629 5.027399 4.863779 4.761219 4.570739 4.368059 4.397359 4.573189 4.841809 5.203219 5.452309 5.652549 5.855239 5.906519 5.952919 5.828369 5.791739 5.799069 5.813719 5.877209 5.955359 5.781979 5.518239 5.127519 4.763659 4.492599 4.233749 4.011519 3.855239 3.691619 3.635459 3.818609 4.155599 4.590279 4.988329 5.076239 4.907739 4.648889 4.377829 4.216649 4.287469 4.590279 4.846689 5.139729 5.388809 5.689179 5.884539 6.043269 6.170259 6.211769 6.250839 6.209329 6.013969 5.701389 5.469399 5.479169 5.557309 5.728249 5.882099 5.984659 5.901629 5.581729 5.371719 5.418119 5.510909 5.667199 5.791739 5.698949 5.484049 5.154379 4.980999 5.061589 5.195899 5.359509 5.615919 5.762439 5.857679 5.948029 5.835699 5.706269 5.498699 5.188569 5.117749 5.191009 5.315549 5.532889 5.444979 5.396139 5.274039 5.027399 4.744129 4.668419 4.651329 4.514579 4.267939 4.260609 4.263049 4.189789 4.277699 4.600049 4.932159 5.283809 5.528009 5.740459 5.874769 5.955359 5.991989 5.845469 5.528009 5.061589 4.734359 4.534109 4.534109 4.697729 4.744129 4.619579 4.643999 4.832039 5.132399 5.410789 5.625689 5.603709 5.315549 4.961459 4.619579 4.358289 4.155599 4.033499 3.886979 3.772209 3.640339 3.532889 3.435209 3.427889 3.422999 3.398579 3.603709 4.023729 4.451089 4.792969 4.902859 4.780759 4.590279 4.336309 4.145839 4.216649 4.433989 4.714819 5.098219 5.359509 5.569519 5.772209 5.921169 6.055479 5.962679 5.642779 5.435209 5.388809 5.537779 5.681849 5.701389 5.615919 5.667199 5.740459 5.803949 5.882099 5.950469 6.072579 6.148279 6.116529 6.177579 6.201999 6.206889 5.991989 5.564639 5.178799 4.998089 5.051819 5.232529 5.484049 5.686739 5.899189 5.869889 5.977339 6.053039 6.079899 6.128739 6.079899 6.167809 6.194679 6.236189 6.053039 5.652549 5.274039 4.858899 4.534109 4.455969 4.619579 4.866229 5.117749 5.166589 5.056699 5.002979 5.098219 5.325319 5.567079 5.466959 5.252059 4.946809 4.880879 4.980999 5.225199 5.459629 5.723369 5.791739 5.906519 5.991989 5.835699 5.528009 5.142169 4.775869 4.490159 4.236189 4.023729 3.886979 3.752669 3.681849 3.806399 4.145839 4.600049 5.002979 5.303339 5.552429 5.615919 5.523119 5.611039 5.713599 5.845469 5.899189 5.994429 6.092109 6.092109 6.143389 6.153159 6.233749 6.187349 6.013969 5.835699 5.774649 5.686739 5.537779 5.327759 5.054259 4.700169 4.394919 4.180019 4.043269 3.877209 3.752669 3.728249 3.869889 4.206889 4.355849 4.426669 4.453529 4.521899 4.392479 4.155599 3.965129 3.877209 3.970009 4.258169 4.421779 4.336309 4.299679 4.392479 4.675749 4.761219 4.658659 4.490159 4.307009 4.126299 3.972449 4.077459 4.372939 4.741679 5.088449 5.186129 5.037169 4.785639 4.563419 4.534109 4.705049 4.741679 4.648889 4.431549 4.238629 4.065249 3.943149 3.811279 3.691619 3.652549 3.825929 4.223979 4.424219 4.429109 4.319219 4.138509 3.965129 3.886979 3.801509 3.701389 3.640339 3.767319 4.150719 4.648889 4.990769 5.088449 5.022509 4.783199 4.685519 4.665979 4.707499 4.912619 5.195899 5.415679 5.623249 5.740459 5.899189 5.928499 6.050599 6.153159 5.965129 5.586619 5.381489 5.371719 5.486489 5.567079 5.821049 5.913839 5.994429 6.011519 5.999309 6.018849 5.821049 5.728249 5.740459 5.764879 5.882099 5.926049 5.750229 5.415679 4.995649 4.861339 4.902859 5.103099 5.364389 5.596379 5.752669 5.845469 5.928499 6.006639 5.840579 5.518239 5.173919 4.739239 4.458409 4.426669 4.602489 4.822269 5.183689 5.430329 5.652549 5.821049 5.706269 5.369279 5.027399 4.705049 4.414459 4.145839 3.965129 4.033499 4.372939 4.683079Program Results
IB01CD EXAMPLE PROGRAM RESULTS The system state matrix A is 0.8924 0.3887 0.1285 0.1716 -0.0837 0.6186 -0.6273 -0.4582 0.0052 0.1307 0.6685 -0.6755 0.0055 0.0734 -0.2148 0.4788 The system output matrix C is -0.4442 0.6663 0.3961 0.4102 The system input matrix B is -0.2150 -0.1962 0.0511 0.0373 The system input-output matrix D is -0.0018 The initial state vector x0 is -11.4329 -0.6767 0.0472 0.3600
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/IB01MD.html 0000664 0000000 0000000 00000036260 14560147231 0020335 0 ustar 00root root 0000000 0000000
Purpose
To construct an upper triangular factor R of the concatenated block Hankel matrices using input-output data. The input-output data can, optionally, be processed sequentially.Specification
SUBROUTINE IB01MD( METH, ALG, BATCH, CONCT, NOBR, M, L, NSMP, U,
$ LDU, Y, LDY, R, LDR, IWORK, DWORK, LDWORK,
$ IWARN, INFO )
C .. Scalar Arguments ..
INTEGER INFO, IWARN, L, LDR, LDU, LDWORK, LDY, M, NOBR,
$ NSMP
CHARACTER ALG, BATCH, CONCT, METH
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION DWORK(*), R(LDR, *), U(LDU, *), Y(LDY, *)
Arguments
Mode Parameters
METH CHARACTER*1
Specifies the subspace identification method to be used,
as follows:
= 'M': MOESP algorithm with past inputs and outputs;
= 'N': N4SID algorithm.
ALG CHARACTER*1
Specifies the algorithm for computing the triangular
factor R, as follows:
= 'C': Cholesky algorithm applied to the correlation
matrix of the input-output data;
= 'F': Fast QR algorithm;
= 'Q': QR algorithm applied to the concatenated block
Hankel matrices.
BATCH CHARACTER*1
Specifies whether or not sequential data processing is to
be used, and, for sequential processing, whether or not
the current data block is the first block, an intermediate
block, or the last block, as follows:
= 'F': the first block in sequential data processing;
= 'I': an intermediate block in sequential data
processing;
= 'L': the last block in sequential data processing;
= 'O': one block only (non-sequential data processing).
NOTE that when 100 cycles of sequential data processing
are completed for BATCH = 'I', a warning is
issued, to prevent for an infinite loop.
CONCT CHARACTER*1
Specifies whether or not the successive data blocks in
sequential data processing belong to a single experiment,
as follows:
= 'C': the current data block is a continuation of the
previous data block and/or it will be continued
by the next data block;
= 'N': there is no connection between the current data
block and the previous and/or the next ones.
This parameter is not used if BATCH = 'O'.
Input/Output Parameters
NOBR (input) INTEGER
The number of block rows, s, in the input and output
block Hankel matrices to be processed. NOBR > 0.
(In the MOESP theory, NOBR should be larger than n,
the estimated dimension of state vector.)
M (input) INTEGER
The number of system inputs. M >= 0.
When M = 0, no system inputs are processed.
L (input) INTEGER
The number of system outputs. L > 0.
NSMP (input) INTEGER
The number of rows of matrices U and Y (number of
samples, t). (When sequential data processing is used,
NSMP is the number of samples of the current data
block.)
NSMP >= 2*(M+L+1)*NOBR - 1, for non-sequential
processing;
NSMP >= 2*NOBR, for sequential processing.
The total number of samples when calling the routine with
BATCH = 'L' should be at least 2*(M+L+1)*NOBR - 1.
The NSMP argument may vary from a cycle to another in
sequential data processing, but NOBR, M, and L should
be kept constant. For efficiency, it is advisable to use
NSMP as large as possible.
U (input) DOUBLE PRECISION array, dimension (LDU,M)
The leading NSMP-by-M part of this array must contain the
t-by-m input-data sequence matrix U,
U = [u_1 u_2 ... u_m]. Column j of U contains the
NSMP values of the j-th input component for consecutive
time increments.
If M = 0, this array is not referenced.
LDU INTEGER
The leading dimension of the array U.
LDU >= NSMP, if M > 0;
LDU >= 1, if M = 0.
Y (input) DOUBLE PRECISION array, dimension (LDY,L)
The leading NSMP-by-L part of this array must contain the
t-by-l output-data sequence matrix Y,
Y = [y_1 y_2 ... y_l]. Column j of Y contains the
NSMP values of the j-th output component for consecutive
time increments.
LDY INTEGER
The leading dimension of the array Y. LDY >= NSMP.
R (output or input/output) DOUBLE PRECISION array, dimension
( LDR,2*(M+L)*NOBR )
On exit, if INFO = 0 and ALG = 'Q', or (ALG = 'C' or 'F',
and BATCH = 'L' or 'O'), the leading
2*(M+L)*NOBR-by-2*(M+L)*NOBR upper triangular part of
this array contains the (current) upper triangular factor
R from the QR factorization of the concatenated block
Hankel matrices. The diagonal elements of R are positive
when the Cholesky algorithm was successfully used.
On exit, if ALG = 'C' and BATCH = 'F' or 'I', the leading
2*(M+L)*NOBR-by-2*(M+L)*NOBR upper triangular part of this
array contains the current upper triangular part of the
correlation matrix in sequential data processing.
If ALG = 'F' and BATCH = 'F' or 'I', the array R is not
referenced.
On entry, if ALG = 'C', or ALG = 'Q', and BATCH = 'I' or
'L', the leading 2*(M+L)*NOBR-by-2*(M+L)*NOBR upper
triangular part of this array must contain the upper
triangular matrix R computed at the previous call of this
routine in sequential data processing. The array R need
not be set on entry if ALG = 'F' or if BATCH = 'F' or 'O'.
LDR INTEGER
The leading dimension of the array R.
LDR >= 2*(M+L)*NOBR.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK >= MAX(3,M+L), if ALG = 'F';
LIWORK >= 3, if ALG = 'C' or 'Q'.
On entry with BATCH = 'I' or BATCH = 'L', IWORK(1:3)
must contain the values of ICYCLE, MAXWRK, and NSMPSM
set by the previous call of this routine.
On exit with BATCH = 'F' or BATCH = 'I', IWORK(1:3)
contains the values of ICYCLE, MAXWRK, and NSMPSM to be
used by the next call of the routine.
ICYCLE counts the cycles for BATCH = 'I'.
MAXWRK stores the current optimal workspace.
NSMPSM sums up the NSMP values for BATCH <> 'O'.
The first three elements of IWORK should be preserved
during successive calls of the routine with BATCH = 'F'
or BATCH = 'I', till the final call with BATCH = 'L'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -17, DWORK(1) returns the minimum
value of LDWORK.
Let
k = 0, if CONCT = 'N' and ALG = 'C' or 'Q';
k = 2*NOBR-1, if CONCT = 'C' and ALG = 'C' or 'Q';
k = 2*NOBR*(M+L+1), if CONCT = 'N' and ALG = 'F';
k = 2*NOBR*(M+L+2), if CONCT = 'C' and ALG = 'F'.
The first (M+L)*k elements of DWORK should be preserved
during successive calls of the routine with BATCH = 'F'
or 'I', till the final call with BATCH = 'L'.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= (4*NOBR-2)*(M+L), if ALG = 'C', BATCH <> 'O' and
CONCT = 'C';
LDWORK >= 1, if ALG = 'C', BATCH = 'O' or
CONCT = 'N';
LDWORK >= (M+L)*2*NOBR*(M+L+3), if ALG = 'F',
BATCH <> 'O' and CONCT = 'C';
LDWORK >= (M+L)*2*NOBR*(M+L+1), if ALG = 'F',
BATCH = 'F', 'I' and CONCT = 'N';
LDWORK >= (M+L)*4*NOBR*(M+L+1)+(M+L)*2*NOBR, if ALG = 'F',
BATCH = 'L' and CONCT = 'N', or
BATCH = 'O';
LDWORK >= 4*(M+L)*NOBR, if ALG = 'Q', BATCH = 'F' or 'O',
and LDR >= NS = NSMP - 2*NOBR + 1;
LDWORK >= 6*(M+L)*NOBR, if ALG = 'Q', BATCH = 'F' or 'O',
and LDR < NS, or BATCH = 'I' or
'L' and CONCT = 'N';
LDWORK >= 4*(NOBR+1)*(M+L)*NOBR, if ALG = 'Q', BATCH = 'I'
or 'L' and CONCT = 'C'.
The workspace used for ALG = 'Q' is
LDRWRK*2*(M+L)*NOBR + 4*(M+L)*NOBR,
where LDRWRK = LDWORK/(2*(M+L)*NOBR) - 2; recommended
value LDRWRK = NS, assuming a large enough cache size.
For good performance, LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: the number of 100 cycles in sequential data
processing has been exhausted without signaling
that the last block of data was get; the cycle
counter was reinitialized;
= 2: a fast algorithm was requested (ALG = 'C' or 'F'),
but it failed, and the QR algorithm was then used
(non-sequential data processing).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: a fast algorithm was requested (ALG = 'C', or 'F')
in sequential data processing, but it failed. The
routine can be repeatedly called again using the
standard QR algorithm.
Method
1) For non-sequential data processing using QR algorithm, a
t x 2(m+l)s matrix H is constructed, where
H = [ Uf' Up' Y' ], for METH = 'M',
s+1,2s,t 1,s,t 1,2s,t
H = [ U' Y' ], for METH = 'N',
1,2s,t 1,2s,t
and Up , Uf , U , and Y are block Hankel
1,s,t s+1,2s,t 1,2s,t 1,2s,t
matrices defined in terms of the input and output data [3].
A QR factorization is used to compress the data.
The fast QR algorithm uses a QR factorization which exploits
the block-Hankel structure. Actually, the Cholesky factor of H'*H
is computed.
2) For sequential data processing using QR algorithm, the QR
decomposition is done sequentially, by updating the upper
triangular factor R. This is also performed internally if the
workspace is not large enough to accommodate an entire batch.
3) For non-sequential or sequential data processing using
Cholesky algorithm, the correlation matrix of input-output data is
computed (sequentially, if requested), taking advantage of the
block Hankel structure [7]. Then, the Cholesky factor of the
correlation matrix is found, if possible.
References
[1] Verhaegen M., and Dewilde, P.
Subspace Model Identification. Part 1: The output-error
state-space model identification class of algorithms.
Int. J. Control, 56, pp. 1187-1210, 1992.
[2] Verhaegen M.
Subspace Model Identification. Part 3: Analysis of the
ordinary output-error state-space model identification
algorithm.
Int. J. Control, 58, pp. 555-586, 1993.
[3] Verhaegen M.
Identification of the deterministic part of MIMO state space
models given in innovations form from input-output data.
Automatica, Vol.30, No.1, pp.61-74, 1994.
[4] Van Overschee, P., and De Moor, B.
N4SID: Subspace Algorithms for the Identification of
Combined Deterministic-Stochastic Systems.
Automatica, Vol.30, No.1, pp. 75-93, 1994.
[5] Peternell, K., Scherrer, W. and Deistler, M.
Statistical Analysis of Novel Subspace Identification Methods.
Signal Processing, 52, pp. 161-177, 1996.
[6] Sima, V.
Subspace-based Algorithms for Multivariable System
Identification.
Studies in Informatics and Control, 5, pp. 335-344, 1996.
[7] Sima, V.
Cholesky or QR Factorization for Data Compression in
Subspace-based Identification ?
Proceedings of the Second NICONET Workshop on ``Numerical
Control Software: SLICOT, a Useful Tool in Industry'',
December 3, 1999, INRIA Rocquencourt, France, pp. 75-80, 1999.
Numerical Aspects
The implemented method is numerically stable (when QR algorithm is
used), reliable and efficient. The fast Cholesky or QR algorithms
are more efficient, but the accuracy could diminish by forming the
correlation matrix.
2
The QR algorithm needs 0(t(2(m+l)s) ) floating point operations.
2 3
The Cholesky algorithm needs 0(2t(m+l) s)+0((2(m+l)s) ) floating
point operations.
2 3 2
The fast QR algorithm needs 0(2t(m+l) s)+0(4(m+l) s ) floating
point operations.
Further Comments
For ALG = 'Q', BATCH = 'O' and LDR < NS, or BATCH <> 'O', the calculations could be rather inefficient if only minimal workspace (see argument LDWORK) is provided. It is advisable to provide as much workspace as possible. Almost optimal efficiency can be obtained for LDWORK = (NS+2)*(2*(M+L)*NOBR), assuming that the cache size is large enough to accommodate R, U, Y, and DWORK.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To construct an upper triangular factor R of the concatenated block Hankel matrices using input-output data, via a fast QR algorithm based on displacement rank. The input-output data can, optionally, be processed sequentially.Specification
SUBROUTINE IB01MY( METH, BATCH, CONCT, NOBR, M, L, NSMP, U, LDU,
$ Y, LDY, R, LDR, IWORK, DWORK, LDWORK, IWARN,
$ INFO )
C .. Scalar Arguments ..
INTEGER INFO, IWARN, L, LDR, LDU, LDWORK, LDY, M, NOBR,
$ NSMP
CHARACTER BATCH, CONCT, METH
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION DWORK(*), R(LDR, *), U(LDU, *), Y(LDY, *)
Arguments
Mode Parameters
METH CHARACTER*1
Specifies the subspace identification method to be used,
as follows:
= 'M': MOESP algorithm with past inputs and outputs;
= 'N': N4SID algorithm.
BATCH CHARACTER*1
Specifies whether or not sequential data processing is to
be used, and, for sequential processing, whether or not
the current data block is the first block, an intermediate
block, or the last block, as follows:
= 'F': the first block in sequential data processing;
= 'I': an intermediate block in sequential data
processing;
= 'L': the last block in sequential data processing;
= 'O': one block only (non-sequential data processing).
NOTE that when 100 cycles of sequential data processing
are completed for BATCH = 'I', a warning is
issued, to prevent for an infinite loop.
CONCT CHARACTER*1
Specifies whether or not the successive data blocks in
sequential data processing belong to a single experiment,
as follows:
= 'C': the current data block is a continuation of the
previous data block and/or it will be continued
by the next data block;
= 'N': there is no connection between the current data
block and the previous and/or the next ones.
This parameter is not used if BATCH = 'O'.
Input/Output Parameters
NOBR (input) INTEGER
The number of block rows, s, in the input and output
block Hankel matrices to be processed. NOBR > 0.
(In the MOESP theory, NOBR should be larger than n, the
estimated dimension of state vector.)
M (input) INTEGER
The number of system inputs. M >= 0.
When M = 0, no system inputs are processed.
L (input) INTEGER
The number of system outputs. L > 0.
NSMP (input) INTEGER
The number of rows of matrices U and Y (number of
samples, t). (When sequential data processing is used,
NSMP is the number of samples of the current data
block.)
NSMP >= 2*(M+L+1)*NOBR - 1, for non-sequential
processing;
NSMP >= 2*NOBR, for sequential processing.
The total number of samples when calling the routine with
BATCH = 'L' should be at least 2*(M+L+1)*NOBR - 1.
The NSMP argument may vary from a cycle to another in
sequential data processing, but NOBR, M, and L should
be kept constant. For efficiency, it is advisable to use
NSMP as large as possible.
U (input) DOUBLE PRECISION array, dimension (LDU,M)
The leading NSMP-by-M part of this array must contain the
t-by-m input-data sequence matrix U,
U = [u_1 u_2 ... u_m]. Column j of U contains the
NSMP values of the j-th input component for consecutive
time increments.
If M = 0, this array is not referenced.
LDU INTEGER
The leading dimension of the array U.
LDU >= NSMP, if M > 0;
LDU >= 1, if M = 0.
Y (input) DOUBLE PRECISION array, dimension (LDY,L)
The leading NSMP-by-L part of this array must contain the
t-by-l output-data sequence matrix Y,
Y = [y_1 y_2 ... y_l]. Column j of Y contains the
NSMP values of the j-th output component for consecutive
time increments.
LDY INTEGER
The leading dimension of the array Y. LDY >= NSMP.
R (output) DOUBLE PRECISION array, dimension
( LDR,2*(M+L)*NOBR )
If INFO = 0 and BATCH = 'L' or 'O', the leading
2*(M+L)*NOBR-by-2*(M+L)*NOBR upper triangular part of this
array contains the upper triangular factor R from the
QR factorization of the concatenated block Hankel
matrices.
LDR INTEGER
The leading dimension of the array R.
LDR >= 2*(M+L)*NOBR.
Workspace
IWORK INTEGER array, dimension MAX(3,M+L)
On entry with BATCH = 'I' or BATCH = 'L', IWORK(1:3)
must contain the values of ICYCLE, MAXWRK, and NSMPSM
set by the previous call of this routine.
On exit with BATCH = 'F' or BATCH = 'I', IWORK(1:3)
contains the values of ICYCLE, MAXWRK, and NSMPSM to be
used by the next call of the routine.
ICYCLE counts the cycles for BATCH = 'I'.
MAXWRK stores the current optimal workspace.
NSMPSM sums up the NSMP values for BATCH <> 'O'.
The first three elements of IWORK should be preserved
during successive calls of the routine with BATCH = 'F'
or BATCH = 'I', till the final call with BATCH = 'L'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -16, DWORK(1) returns the minimum
value of LDWORK.
The first (M+L)*2*NOBR*(M+L+c) elements of DWORK should
be preserved during successive calls of the routine
with BATCH = 'F' or 'I', till the final call with
BATCH = 'L', where
c = 1, if the successive data blocks do not belong to a
single experiment (CONCT = 'N');
c = 2, if the successive data blocks belong to a single
experiment (CONCT = 'C').
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= (M+L)*2*NOBR*(M+L+3),
if BATCH <> 'O' and CONCT = 'C';
LDWORK >= (M+L)*2*NOBR*(M+L+1),
if BATCH = 'F' or 'I' and CONCT = 'N';
LDWORK >= (M+L)*4*NOBR*(M+L+1)+(M+L)*2*NOBR,
if BATCH = 'L' and CONCT = 'N',
or BATCH = 'O'.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA. The workspace query should be done
for BATCH = 'L' or BATCH = 'O'. To get it in advance, use
BATCH = 'O'.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: the number of 100 cycles in sequential data
processing has been exhausted without signaling
that the last block of data was get.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the fast QR factorization algorithm failed. The
matrix H'*H is not (numerically) positive definite.
Method
Consider the t x 2(m+l)s matrix H of concatenated block Hankel
matrices
H = [ Uf' Up' Y' ], for METH = 'M',
s+1,2s,t 1,s,t 1,2s,t
H = [ U' Y' ], for METH = 'N',
1,2s,t 1,2s,t
where Up , Uf , U , and Y are block
1,s,t s+1,2s,t 1,2s,t 1,2s,t
Hankel matrices defined in terms of the input and output data [3].
The fast QR algorithm uses a factorization of H'*H which exploits
the block-Hankel structure, via a displacement rank technique [5].
References
[1] Verhaegen M., and Dewilde, P.
Subspace Model Identification. Part 1: The output-error
state-space model identification class of algorithms.
Int. J. Control, 56, pp. 1187-1210, 1992.
[2] Verhaegen M.
Subspace Model Identification. Part 3: Analysis of the
ordinary output-error state-space model identification
algorithm.
Int. J. Control, 58, pp. 555-586, 1993.
[3] Verhaegen M.
Identification of the deterministic part of MIMO state space
models given in innovations form from input-output data.
Automatica, Vol.30, No.1, pp.61-74, 1994.
[4] Van Overschee, P., and De Moor, B.
N4SID: Subspace Algorithms for the Identification of
Combined Deterministic-Stochastic Systems.
Automatica, Vol.30, No.1, pp. 75-93, 1994.
[5] Kressner, D., Mastronardi, N., Sima, V., Van Dooren, P., and
Van Huffel, S.
A Fast Algorithm for Subspace State-space System
Identification via Exploitation of the Displacement Structure.
J. Comput. Appl. Math., Vol.132, No.1, pp. 71-81, 2001.
Numerical Aspects
The implemented method is reliable and efficient. Numerical
difficulties are possible when the matrix H'*H is nearly rank
defficient. The method cannot be used if the matrix H'*H is not
numerically positive definite.
2 3 2
The algorithm requires 0(2t(m+l) s)+0(4(m+l) s ) floating point
operations.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To find the singular value decomposition (SVD) giving the system order, using the triangular factor of the concatenated block Hankel matrices. Related preliminary calculations needed for computing the system matrices are also performed.Specification
SUBROUTINE IB01ND( METH, JOBD, NOBR, M, L, R, LDR, SV, TOL, IWORK,
$ DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
DOUBLE PRECISION TOL
INTEGER INFO, IWARN, L, LDR, LDWORK, M, NOBR
CHARACTER JOBD, METH
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), R(LDR, *), SV(*)
INTEGER IWORK(*)
Arguments
Mode Parameters
METH CHARACTER*1
Specifies the subspace identification method to be used,
as follows:
= 'M': MOESP algorithm with past inputs and outputs;
= 'N': N4SID algorithm.
JOBD CHARACTER*1
Specifies whether or not the matrices B and D should later
be computed using the MOESP approach, as follows:
= 'M': the matrices B and D should later be computed
using the MOESP approach;
= 'N': the matrices B and D should not be computed using
the MOESP approach.
This parameter is not relevant for METH = 'N'.
Input/Output Parameters
NOBR (input) INTEGER
The number of block rows, s, in the input and output
block Hankel matrices. NOBR > 0.
M (input) INTEGER
The number of system inputs. M >= 0.
L (input) INTEGER
The number of system outputs. L > 0.
R (input/output) DOUBLE PRECISION array, dimension
( LDR,2*(M+L)*NOBR )
On entry, the leading 2*(M+L)*NOBR-by-2*(M+L)*NOBR upper
triangular part of this array must contain the upper
triangular factor R from the QR factorization of the
concatenated block Hankel matrices. Denote R_ij,
i,j = 1:4, the ij submatrix of R, partitioned by
M*NOBR, M*NOBR, L*NOBR, and L*NOBR rows and columns.
On exit, if INFO = 0, the leading
2*(M+L)*NOBR-by-2*(M+L)*NOBR upper triangular part of this
array contains the matrix S, the processed upper
triangular factor R, as required by other subroutines.
Specifically, let S_ij, i,j = 1:4, be the ij submatrix
of S, partitioned by M*NOBR, L*NOBR, M*NOBR, and
L*NOBR rows and columns. The submatrix S_22 contains
the matrix of left singular vectors needed subsequently.
Useful information is stored in S_11 and in the
block-column S_14 : S_44. For METH = 'M' and JOBD = 'M',
the upper triangular part of S_31 contains the upper
triangular factor in the QR factorization of the matrix
R_1c = [ R_12' R_22' R_11' ]', and S_12 contains the
corresponding leading part of the transformed matrix
R_2c = [ R_13' R_23' R_14' ]'. For METH = 'N', the
subarray S_41 : S_43 contains the transpose of the
matrix contained in S_14 : S_34.
LDR INTEGER
The leading dimension of the array R.
LDR >= MAX( 2*(M+L)*NOBR, 3*M*NOBR ),
for METH = 'M' and JOBD = 'M';
LDR >= 2*(M+L)*NOBR, for METH = 'M' and JOBD = 'N' or
for METH = 'N'.
SV (output) DOUBLE PRECISION array, dimension ( L*NOBR )
The singular values of the relevant part of the triangular
factor from the QR factorization of the concatenated block
Hankel matrices.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used for estimating the rank of
matrices. If the user sets TOL > 0, then the given value
of TOL is used as a lower bound for the reciprocal
condition number; an m-by-n matrix whose estimated
condition number is less than 1/TOL is considered to
be of full rank. If the user sets TOL <= 0, then an
implicitly computed, default tolerance, defined by
TOLDEF = m*n*EPS, is used instead, where EPS is the
relative machine precision (see LAPACK Library routine
DLAMCH).
This parameter is not used for METH = 'M'.
Workspace
IWORK INTEGER array, dimension ((M+L)*NOBR)
This parameter is not referenced for METH = 'M'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK, and, for METH = 'N', DWORK(2) and DWORK(3)
contain the reciprocal condition numbers of the
triangular factors of the matrices U_f and r_1 [6].
On exit, if INFO = -12, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= max( (2*M-1)*NOBR, (M+L)*NOBR, 5*L*NOBR ),
if METH = 'M' and JOBD = 'M';
LDWORK >= 5*L*NOBR, if METH = 'M' and JOBD = 'N';
LDWORK >= 5*(M+L)*NOBR+1, if METH = 'N'.
For good performance, LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 4: the least squares problems with coefficient matrix
U_f, used for computing the weighted oblique
projection (for METH = 'N'), have a rank-deficient
coefficient matrix;
= 5: the least squares problem with coefficient matrix
r_1 [6], used for computing the weighted oblique
projection (for METH = 'N'), has a rank-deficient
coefficient matrix.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 2: the singular value decomposition (SVD) algorithm did
not converge.
Method
A singular value decomposition (SVD) of a certain matrix is computed, which reveals the order n of the system as the number of "non-zero" singular values. For the MOESP approach, this matrix is [ R_24' R_34' ]' := R(ms+1:(2m+l)s,(2m+l)s+1:2(m+l)s), where R is the upper triangular factor R constructed by SLICOT Library routine IB01MD. For the N4SID approach, a weighted oblique projection is computed from the upper triangular factor R and its SVD is then found.References
[1] Verhaegen M., and Dewilde, P.
Subspace Model Identification. Part 1: The output-error
state-space model identification class of algorithms.
Int. J. Control, 56, pp. 1187-1210, 1992.
[2] Verhaegen M.
Subspace Model Identification. Part 3: Analysis of the
ordinary output-error state-space model identification
algorithm.
Int. J. Control, 58, pp. 555-586, 1993.
[3] Verhaegen M.
Identification of the deterministic part of MIMO state space
models given in innovations form from input-output data.
Automatica, Vol.30, No.1, pp.61-74, 1994.
[4] Van Overschee, P., and De Moor, B.
N4SID: Subspace Algorithms for the Identification of
Combined Deterministic-Stochastic Systems.
Automatica, Vol.30, No.1, pp. 75-93, 1994.
[5] Van Overschee, P., and De Moor, B.
Subspace Identification for Linear Systems: Theory -
Implementation - Applications.
Kluwer Academic Publishers, Boston/London/Dordrecht, 1996.
[6] Sima, V.
Subspace-based Algorithms for Multivariable System
Identification.
Studies in Informatics and Control, 5, pp. 335-344, 1996.
Numerical Aspects
The implemented method is numerically stable.
3
The algorithm requires 0(((m+l)s) ) floating point operations.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To estimate the system order, based on the singular values of the relevant part of the triangular factor of the concatenated block Hankel matrices.Specification
SUBROUTINE IB01OD( CTRL, NOBR, L, SV, N, TOL, IWARN, INFO )
C .. Scalar Arguments ..
DOUBLE PRECISION TOL
INTEGER INFO, IWARN, L, N, NOBR
CHARACTER CTRL
C .. Array Arguments ..
DOUBLE PRECISION SV(*)
Arguments
Mode Parameters
CTRL CHARACTER*1
Specifies whether or not the user's confirmation of the
system order estimate is desired, as follows:
= 'C': user's confirmation;
= 'N': no confirmation.
If CTRL = 'C', a reverse communication routine, IB01OY,
is called, and, after inspecting the singular values and
system order estimate, n, the user may accept n or set
a new value.
IB01OY is not called by the routine if CTRL = 'N'.
Input/Output Parameters
NOBR (input) INTEGER
The number of block rows, s, in the processed input and
output block Hankel matrices. NOBR > 0.
L (input) INTEGER
The number of system outputs. L > 0.
SV (input) DOUBLE PRECISION array, dimension ( L*NOBR )
The singular values of the relevant part of the triangular
factor from the QR factorization of the concatenated block
Hankel matrices.
N (output) INTEGER
The estimated order of the system.
Tolerances
TOL DOUBLE PRECISION
Absolute tolerance used for determining an estimate of
the system order. If TOL >= 0, the estimate is
indicated by the index of the last singular value greater
than or equal to TOL. (Singular values less than TOL
are considered as zero.) When TOL = 0, an internally
computed default value, TOL = NOBR*EPS*SV(1), is used,
where SV(1) is the maximal singular value, and EPS is
the relative machine precision (see LAPACK Library routine
DLAMCH). When TOL < 0, the estimate is indicated by the
index of the singular value that has the largest
logarithmic gap to its successor.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 3: all singular values were exactly zero, hence N = 0.
(Both input and output were identically zero.)
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The singular values are compared to the given, or default TOL, and the estimated order n is returned, possibly after user's confirmation.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To ask for user's confirmation of the system order found by SLICOT Library routine IB01OD. This routine may be modified, but its interface must be preserved.Specification
SUBROUTINE IB01OY( NS, NMAX, N, SV, INFO )
C .. Scalar Arguments ..
INTEGER INFO, N, NMAX, NS
C .. Array Arguments ..
DOUBLE PRECISION SV( * )
Arguments
Input/Output Parameters
NS (input) INTEGER
The number of singular values. NS > 0.
NMAX (input) INTEGER
The maximum value of the system order. 0 <= NMAX <= NS.
N (input/output) INTEGER
On entry, the estimate of the system order computed by
IB01OD routine. 0 <= N <= NS.
On exit, the user's estimate of the system order, which
could be identical with the input value of N.
Note that the output value of N should be less than
or equal to NMAX.
SV (input) DOUBLE PRECISION array, dimension ( NS )
The singular values, in descending order, used for
determining the system order.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To estimate the matrices A, C, B, and D of a linear time-invariant (LTI) state space model, using the singular value decomposition information provided by other routines. Optionally, the system and noise covariance matrices, needed for the Kalman gain, are also determined.Specification
SUBROUTINE IB01PD( METH, JOB, JOBCV, NOBR, N, M, L, NSMPL, R,
$ LDR, A, LDA, C, LDC, B, LDB, D, LDD, Q, LDQ,
$ RY, LDRY, S, LDS, O, LDO, TOL, IWORK, DWORK,
$ LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
DOUBLE PRECISION TOL
INTEGER INFO, IWARN, L, LDA, LDB, LDC, LDD, LDO, LDQ,
$ LDR, LDRY, LDS, LDWORK, M, N, NOBR, NSMPL
CHARACTER JOB, JOBCV, METH
C .. Array Arguments ..
DOUBLE PRECISION A(LDA, *), B(LDB, *), C(LDC, *), D(LDD, *),
$ DWORK(*), O(LDO, *), Q(LDQ, *), R(LDR, *),
$ RY(LDRY, *), S(LDS, *)
INTEGER IWORK( * )
Arguments
Mode Parameters
METH CHARACTER*1
Specifies the subspace identification method to be used,
as follows:
= 'M': MOESP algorithm with past inputs and outputs;
= 'N': N4SID algorithm.
JOB CHARACTER*1
Specifies which matrices should be computed, as follows:
= 'A': compute all system matrices, A, B, C, and D;
= 'C': compute the matrices A and C only;
= 'B': compute the matrix B only;
= 'D': compute the matrices B and D only.
JOBCV CHARACTER*1
Specifies whether or not the covariance matrices are to
be computed, as follows:
= 'C': the covariance matrices should be computed;
= 'N': the covariance matrices should not be computed.
Input/Output Parameters
NOBR (input) INTEGER
The number of block rows, s, in the input and output
Hankel matrices processed by other routines. NOBR > 1.
N (input) INTEGER
The order of the system. NOBR > N > 0.
M (input) INTEGER
The number of system inputs. M >= 0.
L (input) INTEGER
The number of system outputs. L > 0.
NSMPL (input) INTEGER
If JOBCV = 'C', the total number of samples used for
calculating the covariance matrices.
NSMPL >= 2*(M+L)*NOBR.
This parameter is not meaningful if JOBCV = 'N'.
R (input/workspace) DOUBLE PRECISION array, dimension
( LDR,2*(M+L)*NOBR )
On entry, the leading 2*(M+L)*NOBR-by-2*(M+L)*NOBR part
of this array must contain the relevant data for the MOESP
or N4SID algorithms, as constructed by SLICOT Library
routines IB01AD or IB01ND. Let R_ij, i,j = 1:4, be the
ij submatrix of R (denoted S in IB01AD and IB01ND),
partitioned by M*NOBR, L*NOBR, M*NOBR, and L*NOBR
rows and columns. The submatrix R_22 contains the matrix
of left singular vectors used. Also needed, for
METH = 'N' or JOBCV = 'C', are the submatrices R_11,
R_14 : R_44, and, for METH = 'M' and JOB <> 'C', the
submatrices R_31 and R_12, containing the processed
matrices R_1c and R_2c, respectively, as returned by
SLICOT Library routines IB01AD or IB01ND.
Moreover, if METH = 'N' and JOB = 'A' or 'C', the
block-row R_41 : R_43 must contain the transpose of the
block-column R_14 : R_34 as returned by SLICOT Library
routines IB01AD or IB01ND.
The remaining part of R is used as workspace.
On exit, part of this array is overwritten. Specifically,
if METH = 'M', R_22 and R_31 are overwritten if
JOB = 'B' or 'D', and R_12, R_22, R_14 : R_34,
and possibly R_11 are overwritten if JOBCV = 'C';
if METH = 'N', all needed submatrices are overwritten.
LDR INTEGER
The leading dimension of the array R.
LDR >= 2*(M+L)*NOBR.
A (input or output) DOUBLE PRECISION array, dimension
(LDA,N)
On entry, if METH = 'N' and JOB = 'B' or 'D', the
leading N-by-N part of this array must contain the system
state matrix.
If METH = 'M' or (METH = 'N' and JOB = 'A' or 'C'),
this array need not be set on input.
On exit, if JOB = 'A' or 'C' and INFO = 0, the
leading N-by-N part of this array contains the system
state matrix.
LDA INTEGER
The leading dimension of the array A.
LDA >= N, if JOB = 'A' or 'C', or METH = 'N' and
JOB = 'B' or 'D';
LDA >= 1, otherwise.
C (input or output) DOUBLE PRECISION array, dimension
(LDC,N)
On entry, if METH = 'N' and JOB = 'B' or 'D', the
leading L-by-N part of this array must contain the system
output matrix.
If METH = 'M' or (METH = 'N' and JOB = 'A' or 'C'),
this array need not be set on input.
On exit, if JOB = 'A' or 'C' and INFO = 0, or
INFO = 3 (or INFO >= 0, for METH = 'M'), the leading
L-by-N part of this array contains the system output
matrix.
LDC INTEGER
The leading dimension of the array C.
LDC >= L, if JOB = 'A' or 'C', or METH = 'N' and
JOB = 'B' or 'D';
LDC >= 1, otherwise.
B (output) DOUBLE PRECISION array, dimension (LDB,M)
If M > 0, JOB = 'A', 'B', or 'D' and INFO = 0, the
leading N-by-M part of this array contains the system
input matrix. If M = 0 or JOB = 'C', this array is
not referenced.
LDB INTEGER
The leading dimension of the array B.
LDB >= N, if M > 0 and JOB = 'A', 'B', or 'D';
LDB >= 1, if M = 0 or JOB = 'C'.
D (output) DOUBLE PRECISION array, dimension (LDD,M)
If M > 0, JOB = 'A' or 'D' and INFO = 0, the leading
L-by-M part of this array contains the system input-output
matrix. If M = 0 or JOB = 'C' or 'B', this array is
not referenced.
LDD INTEGER
The leading dimension of the array D.
LDD >= L, if M > 0 and JOB = 'A' or 'D';
LDD >= 1, if M = 0 or JOB = 'C' or 'B'.
Q (output) DOUBLE PRECISION array, dimension (LDQ,N)
If JOBCV = 'C', the leading N-by-N part of this array
contains the positive semidefinite state covariance matrix
to be used as state weighting matrix when computing the
Kalman gain.
This parameter is not referenced if JOBCV = 'N'.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= N, if JOBCV = 'C';
LDQ >= 1, if JOBCV = 'N'.
RY (output) DOUBLE PRECISION array, dimension (LDRY,L)
If JOBCV = 'C', the leading L-by-L part of this array
contains the positive (semi)definite output covariance
matrix to be used as output weighting matrix when
computing the Kalman gain.
This parameter is not referenced if JOBCV = 'N'.
LDRY INTEGER
The leading dimension of the array RY.
LDRY >= L, if JOBCV = 'C';
LDRY >= 1, if JOBCV = 'N'.
S (output) DOUBLE PRECISION array, dimension (LDS,L)
If JOBCV = 'C', the leading N-by-L part of this array
contains the state-output cross-covariance matrix to be
used as cross-weighting matrix when computing the Kalman
gain.
This parameter is not referenced if JOBCV = 'N'.
LDS INTEGER
The leading dimension of the array S.
LDS >= N, if JOBCV = 'C';
LDS >= 1, if JOBCV = 'N'.
O (output) DOUBLE PRECISION array, dimension ( LDO,N )
If METH = 'M' and JOBCV = 'C', or METH = 'N',
the leading L*NOBR-by-N part of this array contains
the estimated extended observability matrix, i.e., the
first N columns of the relevant singular vectors.
If METH = 'M' and JOBCV = 'N', this array is not
referenced.
LDO INTEGER
The leading dimension of the array O.
LDO >= L*NOBR, if JOBCV = 'C' or METH = 'N';
LDO >= 1, otherwise.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used for estimating the rank of
matrices. If the user sets TOL > 0, then the given value
of TOL is used as a lower bound for the reciprocal
condition number; an m-by-n matrix whose estimated
condition number is less than 1/TOL is considered to
be of full rank. If the user sets TOL <= 0, then an
implicitly computed, default tolerance, defined by
TOLDEF = m*n*EPS, is used instead, where EPS is the
relative machine precision (see LAPACK Library routine
DLAMCH).
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK = N, if METH = 'M' and M = 0
or JOB = 'C' and JOBCV = 'N';
LIWORK = M*NOBR+N, if METH = 'M', JOB = 'C',
and JOBCV = 'C';
LIWORK = max(L*NOBR,M*NOBR), if METH = 'M', JOB <> 'C',
and JOBCV = 'N';
LIWORK = max(L*NOBR,M*NOBR+N), if METH = 'M', JOB <> 'C',
and JOBCV = 'C';
LIWORK = max(M*NOBR+N,M*(N+L)), if METH = 'N'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK, and DWORK(2), DWORK(3), DWORK(4), and
DWORK(5) contain the reciprocal condition numbers of the
triangular factors of the matrices, defined in the code,
GaL (GaL = Un(1:(s-1)*L,1:n)), R_1c (if METH = 'M'),
M (if JOBCV = 'C' or METH = 'N'), and Q or T (see
SLICOT Library routines IB01PY or IB01PX), respectively.
If METH = 'N', DWORK(3) is set to one without any
calculations. Similarly, if METH = 'M' and JOBCV = 'N',
DWORK(4) is set to one. If M = 0 or JOB = 'C',
DWORK(3) and DWORK(5) are set to one.
On exit, if INFO = -30, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= max( LDW1,LDW2 ), where, if METH = 'M',
LDW1 >= max( 2*(L*NOBR-L)*N+2*N, (L*NOBR-L)*N+N*N+7*N ),
if JOB = 'C' or JOB = 'A' and M = 0;
LDW1 >= max( 2*(L*NOBR-L)*N+N*N+7*N,
(L*NOBR-L)*N+N+6*M*NOBR, (L*NOBR-L)*N+N+
max( L+M*NOBR, L*NOBR +
max( 3*L*NOBR+1, M ) ) )
if M > 0 and JOB = 'A', 'B', or 'D';
LDW2 >= 0, if JOBCV = 'N';
LDW2 >= max( (L*NOBR-L)*N+Aw+2*N+max(5*N,(2*M+L)*NOBR+L),
4*(M*NOBR+N)+1, M*NOBR+2*N+L ),
if JOBCV = 'C',
where Aw = N+N*N, if M = 0 or JOB = 'C';
Aw = 0, otherwise;
and, if METH = 'N',
LDW1 >= max( (L*NOBR-L)*N+2*N+(2*M+L)*NOBR+L,
2*(L*NOBR-L)*N+N*N+8*N, N+4*(M*NOBR+N)+1,
M*NOBR+3*N+L );
LDW2 >= 0, if M = 0 or JOB = 'C';
LDW2 >= M*NOBR*(N+L)*(M*(N+L)+1)+
max( (N+L)**2, 4*M*(N+L)+1 ),
if M > 0 and JOB = 'A', 'B', or 'D'.
For good performance, LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 4: a least squares problem to be solved has a
rank-deficient coefficient matrix;
= 5: the computed covariance matrices are too small.
The problem seems to be a deterministic one.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 2: the singular value decomposition (SVD) algorithm did
not converge;
= 3: a singular upper triangular matrix was found.
Method
In the MOESP approach, the matrices A and C are first computed from an estimated extended observability matrix [1], and then, the matrices B and D are obtained by solving an extended linear system in a least squares sense. In the N4SID approach, besides the estimated extended observability matrix, the solutions of two least squares problems are used to build another least squares problem, whose solution is needed to compute the system matrices A, C, B, and D. The solutions of the two least squares problems are also optionally used by both approaches to find the covariance matrices.References
[1] Verhaegen M., and Dewilde, P.
Subspace Model Identification. Part 1: The output-error state-
space model identification class of algorithms.
Int. J. Control, 56, pp. 1187-1210, 1992.
[2] Van Overschee, P., and De Moor, B.
N4SID: Two Subspace Algorithms for the Identification
of Combined Deterministic-Stochastic Systems.
Automatica, Vol.30, No.1, pp. 75-93, 1994.
[3] Van Overschee, P.
Subspace Identification : Theory - Implementation -
Applications.
Ph. D. Thesis, Department of Electrical Engineering,
Katholieke Universiteit Leuven, Belgium, Feb. 1995.
[4] Sima, V.
Subspace-based Algorithms for Multivariable System
Identification.
Studies in Informatics and Control, 5, pp. 335-344, 1996.
Numerical Aspects
The implemented method is numerically stable.Further Comments
In some applications, it is useful to compute the system matrices
using two calls to this routine, the first one with JOB = 'C',
and the second one with JOB = 'B' or 'D'. This is slightly less
efficient than using a single call with JOB = 'A', because some
calculations are repeated. If METH = 'N', all the calculations
at the first call are performed again at the second call;
moreover, it is required to save the needed submatrices of R
before the first call and restore them before the second call.
If the covariance matrices are desired, JOBCV should be set
to 'C' at the second call. If B and D are both needed, they
should be computed at once.
It is possible to compute the matrices A and C using the MOESP
algorithm (METH = 'M'), and the matrices B and D using the N4SID
algorithm (METH = 'N'). This combination could be slightly more
efficient than N4SID algorithm alone and it could be more accurate
than MOESP algorithm. No saving/restoring is needed in such a
combination, provided JOBCV is set to 'N' at the first call.
Recommended usage: either one call with JOB = 'A', or
first call with METH = 'M', JOB = 'C', JOBCV = 'N',
second call with METH = 'M', JOB = 'D', JOBCV = 'C', or
first call with METH = 'M', JOB = 'C', JOBCV = 'N',
second call with METH = 'N', JOB = 'D', JOBCV = 'C'.
Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To build and solve the least squares problem T*X = Kv, and
estimate the matrices B and D of a linear time-invariant (LTI)
state space model, using the solution X, and the singular
value decomposition information and other intermediate results,
provided by other routines.
The matrix T is computed as a sum of Kronecker products,
T = T + kron(Uf(:,(i-1)*m+1:i*m),N_i), for i = 1 : s,
(with T initialized by zero), where Uf is the triangular
factor of the QR factorization of the future input part (see
SLICOT Library routine IB01ND), N_i is given by the i-th block
row of the matrix
[ Q_11 Q_12 ... Q_1,s-2 Q_1,s-1 Q_1s ] [ I_L 0 ]
[ Q_12 Q_13 ... Q_1,s-1 Q_1s 0 ] [ ]
N = [ Q_13 Q_14 ... Q_1s 0 0 ] * [ ],
[ : : : : : ] [ ]
[ Q_1s 0 ... 0 0 0 ] [ 0 GaL ]
and where
[ -L_1|1 ] [ M_i-1 - L_1|i ]
Q_11 = [ ], Q_1i = [ ], i = 2:s,
[ I_L - L_2|1 ] [ -L_2|i ]
are (n+L)-by-L matrices, and GaL is built from the first n
relevant singular vectors, GaL = Un(1:L(s-1),1:n), computed
by IB01ND.
The vector Kv is vec(K), with the matrix K defined by
K = [ K_1 K_2 K_3 ... K_s ],
where K_i = K(:,(i-1)*m+1:i*m), i = 1:s, is (n+L)-by-m.
The given matrices are Uf, GaL, and
[ L_1|1 ... L_1|s ]
L = [ ], (n+L)-by-L*s,
[ L_2|1 ... L_2|s ]
M = [ M_1 ... M_s-1 ], n-by-L*(s-1), and
K, (n+L)-by-m*s.
Matrix M is the pseudoinverse of the matrix GaL, computed by
SLICOT Library routine IB01PD.
Specification
SUBROUTINE IB01PX( JOB, NOBR, N, M, L, UF, LDUF, UN, LDUN, UL,
$ LDUL, PGAL, LDPGAL, K, LDK, R, LDR, X, B, LDB,
$ D, LDD, TOL, IWORK, DWORK, LDWORK, IWARN,
$ INFO )
C .. Scalar Arguments ..
DOUBLE PRECISION TOL
INTEGER INFO, IWARN, L, LDB, LDD, LDK, LDPGAL, LDR,
$ LDUF, LDUL, LDUN, LDWORK, M, N, NOBR
CHARACTER JOB
C .. Array Arguments ..
DOUBLE PRECISION B(LDB, *), D(LDD, *), DWORK(*), K(LDK, *),
$ PGAL(LDPGAL, *), R(LDR, *), UF(LDUF, *),
$ UL(LDUL, *), UN(LDUN, *), X(*)
INTEGER IWORK( * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies which of the matrices B and D should be
computed, as follows:
= 'B': compute the matrix B, but not the matrix D;
= 'D': compute both matrices B and D.
Input/Output Parameters
NOBR (input) INTEGER
The number of block rows, s, in the input and output
Hankel matrices processed by other routines. NOBR > 1.
N (input) INTEGER
The order of the system. NOBR > N > 0.
M (input) INTEGER
The number of system inputs. M >= 0.
L (input) INTEGER
The number of system outputs. L > 0.
UF (input/output) DOUBLE PRECISION array, dimension
( LDUF,M*NOBR )
On entry, the leading M*NOBR-by-M*NOBR upper triangular
part of this array must contain the upper triangular
factor of the QR factorization of the future input part,
as computed by SLICOT Library routine IB01ND.
The strict lower triangle need not be set to zero.
On exit, the leading M*NOBR-by-M*NOBR upper triangular
part of this array is unchanged, and the strict lower
triangle is set to zero.
LDUF INTEGER
The leading dimension of the array UF.
LDUF >= MAX( 1, M*NOBR ).
UN (input) DOUBLE PRECISION array, dimension ( LDUN,N )
The leading L*(NOBR-1)-by-N part of this array must
contain the matrix GaL, i.e., the leading part of the
first N columns of the matrix Un of relevant singular
vectors.
LDUN INTEGER
The leading dimension of the array UN.
LDUN >= L*(NOBR-1).
UL (input/output) DOUBLE PRECISION array, dimension
( LDUL,L*NOBR )
On entry, the leading (N+L)-by-L*NOBR part of this array
must contain the given matrix L.
On exit, if M > 0, the leading (N+L)-by-L*NOBR part of
this array is overwritten by the matrix
[ Q_11 Q_12 ... Q_1,s-2 Q_1,s-1 Q_1s ].
LDUL INTEGER
The leading dimension of the array UL. LDUL >= N+L.
PGAL (input) DOUBLE PRECISION array, dimension
( LDPGAL,L*(NOBR-1) )
The leading N-by-L*(NOBR-1) part of this array must
contain the pseudoinverse of the matrix GaL, computed by
SLICOT Library routine IB01PD.
LDPGAL INTEGER
The leading dimension of the array PGAL. LDPGAL >= N.
K (input) DOUBLE PRECISION array, dimension ( LDK,M*NOBR )
The leading (N+L)-by-M*NOBR part of this array must
contain the given matrix K.
LDK INTEGER
The leading dimension of the array K. LDK >= N+L.
R (output) DOUBLE PRECISION array, dimension ( LDR,M*(N+L) )
The leading (N+L)*M*NOBR-by-M*(N+L) part of this array
contains details of the complete orthogonal factorization
of the coefficient matrix T of the least squares problem
which is solved for getting the system matrices B and D.
LDR INTEGER
The leading dimension of the array R.
LDR >= MAX( 1, (N+L)*M*NOBR ).
X (output) DOUBLE PRECISION array, dimension
( (N+L)*M*NOBR )
The leading M*(N+L) elements of this array contain the
least squares solution of the system T*X = Kv.
The remaining elements are used as workspace (to store the
corresponding part of the vector Kv = vec(K)).
B (output) DOUBLE PRECISION array, dimension ( LDB,M )
The leading N-by-M part of this array contains the system
input matrix.
LDB INTEGER
The leading dimension of the array B. LDB >= N.
D (output) DOUBLE PRECISION array, dimension ( LDD,M )
If JOB = 'D', the leading L-by-M part of this array
contains the system input-output matrix.
If JOB = 'B', this array is not referenced.
LDD INTEGER
The leading dimension of the array D.
LDD >= L, if JOB = 'D';
LDD >= 1, if JOB = 'B'.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used for estimating the rank of
matrices. If the user sets TOL > 0, then the given value
of TOL is used as a lower bound for the reciprocal
condition number; an m-by-n matrix whose estimated
condition number is less than 1/TOL is considered to
be of full rank. If the user sets TOL <= 0, then an
implicitly computed, default tolerance, defined by
TOLDEF = m*n*EPS, is used instead, where EPS is the
relative machine precision (see LAPACK Library routine
DLAMCH).
Workspace
IWORK INTEGER array, dimension ( M*(N+L) )
DWORK DOUBLE PRECISION array, dimension ( LDWORK )
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK, and, if M > 0, DWORK(2) contains the
reciprocal condition number of the triangular factor of
the matrix T.
On exit, if INFO = -26, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( (N+L)*(N+L), 4*M*(N+L)+1 ).
For good performance, LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 4: the least squares problem to be solved has a
rank-deficient coefficient matrix.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The matrix T is computed, evaluating the sum of Kronecker products, and then the linear system T*X = Kv is solved in a least squares sense. The matrices B and D are then directly obtained from the least squares solution.References
[1] Verhaegen M., and Dewilde, P.
Subspace Model Identification. Part 1: The output-error
state-space model identification class of algorithms.
Int. J. Control, 56, pp. 1187-1210, 1992.
[2] Van Overschee, P., and De Moor, B.
N4SID: Two Subspace Algorithms for the Identification
of Combined Deterministic-Stochastic Systems.
Automatica, Vol.30, No.1, pp. 75-93, 1994.
[3] Van Overschee, P.
Subspace Identification : Theory - Implementation -
Applications.
Ph. D. Thesis, Department of Electrical Engineering,
Katholieke Universiteit Leuven, Belgium, Feb. 1995.
Numerical Aspects
The implemented method is numerically stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
1. To compute the triangular (QR) factor of the p-by-L*s
structured matrix Q,
[ Q_1s Q_1,s-1 Q_1,s-2 ... Q_12 Q_11 ]
[ 0 Q_1s Q_1,s-1 ... Q_13 Q_12 ]
Q = [ 0 0 Q_1s ... Q_14 Q_13 ],
[ : : : : : ]
[ 0 0 0 ... 0 Q_1s ]
and apply the transformations to the p-by-m matrix Kexpand,
[ K_1 ]
[ K_2 ]
Kexpand = [ K_3 ],
[ : ]
[ K_s ]
where, for MOESP approach (METH = 'M'), p = s*(L*s-n), and
Q_1i = u2(L*(i-1)+1:L*i,:)' is (Ls-n)-by-L, for i = 1:s,
u2 = Un(1:L*s,n+1:L*s), K_i = K(:,(i-1)*m+1:i*m) (i = 1:s)
is (Ls-n)-by-m, and for N4SID approach (METH = 'N'), p = s*(n+L),
and
[ -L_1|1 ] [ M_i-1 - L_1|i ]
Q_11 = [ ], Q_1i = [ ], i = 2:s,
[ I_L - L_2|1 ] [ -L_2|i ]
are (n+L)-by-L matrices, and
K_i = K(:,(i-1)*m+1:i*m), i = 1:s, is (n+L)-by-m.
The given matrices are:
For METH = 'M', u2 = Un(1:L*s,n+1:L*s),
K(1:Ls-n,1:m*s);
[ L_1|1 ... L_1|s ]
For METH = 'N', L = [ ], (n+L)-by-L*s,
[ L_2|1 ... L_2|s ]
M = [ M_1 ... M_s-1 ], n-by-L*(s-1), and
K, (n+L)-by-m*s.
Matrix M is the pseudoinverse of the matrix GaL,
built from the first n relevant singular
vectors, GaL = Un(1:L(s-1),1:n), and computed
by SLICOT Library routine IB01PD for METH = 'N'.
Matrix Q is triangularized (in R), exploiting its structure,
and the transformations are applied from the left to Kexpand.
2. To estimate the matrices B and D of a linear time-invariant
(LTI) state space model, using the factor R, transformed matrix
Kexpand, and the singular value decomposition information provided
by other routines.
IB01PY routine is intended for speed and efficient use of the
memory space. It is generally not recommended for METH = 'N', as
IB01PX routine can produce more accurate results.
Specification
SUBROUTINE IB01PY( METH, JOB, NOBR, N, M, L, RANKR1, UL, LDUL,
$ R1, LDR1, TAU1, PGAL, LDPGAL, K, LDK, R, LDR,
$ H, LDH, B, LDB, D, LDD, TOL, IWORK, DWORK,
$ LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
DOUBLE PRECISION TOL
INTEGER INFO, IWARN, L, LDB, LDD, LDH, LDK, LDPGAL,
$ LDR, LDR1, LDUL, LDWORK, M, N, NOBR, RANKR1
CHARACTER JOB, METH
C .. Array Arguments ..
DOUBLE PRECISION B(LDB, *), D(LDD, *), DWORK(*), H(LDH, *),
$ K(LDK, *), PGAL(LDPGAL, *), R(LDR, *),
$ R1(LDR1, *), TAU1(*), UL(LDUL, *)
INTEGER IWORK( * )
Arguments
Mode Parameters
METH CHARACTER*1
Specifies the subspace identification method to be used,
as follows:
= 'M': MOESP algorithm with past inputs and outputs;
= 'N': N4SID algorithm.
JOB CHARACTER*1
Specifies whether or not the matrices B and D should be
computed, as follows:
= 'B': compute the matrix B, but not the matrix D;
= 'D': compute both matrices B and D;
= 'N': do not compute the matrices B and D, but only the
R factor of Q and the transformed Kexpand.
Input/Output Parameters
NOBR (input) INTEGER
The number of block rows, s, in the input and output
Hankel matrices processed by other routines. NOBR > 1.
N (input) INTEGER
The order of the system. NOBR > N > 0.
M (input) INTEGER
The number of system inputs. M >= 0.
L (input) INTEGER
The number of system outputs. L > 0.
RANKR1 (input) INTEGER
The effective rank of the upper triangular matrix r1,
i.e., the triangular QR factor of the matrix GaL,
computed by SLICOT Library routine IB01PD. It is also
the effective rank of the matrix GaL. 0 <= RANKR1 <= N.
If JOB = 'N', or M = 0, or METH = 'N', this
parameter is not used.
UL (input/workspace) DOUBLE PRECISION array, dimension
( LDUL,L*NOBR )
On entry, if METH = 'M', the leading L*NOBR-by-L*NOBR
part of this array must contain the matrix Un of
relevant singular vectors. The first N columns of UN
need not be specified for this routine.
On entry, if METH = 'N', the leading (N+L)-by-L*NOBR
part of this array must contain the given matrix L.
On exit, the leading LDF-by-L*(NOBR-1) part of this array
is overwritten by the matrix F of the algorithm in [4],
where LDF = MAX( 1, L*NOBR-N-L ), if METH = 'M';
LDF = N, if METH = 'N'.
LDUL INTEGER
The leading dimension of the array UL.
LDUL >= L*NOBR, if METH = 'M';
LDUL >= N+L, if METH = 'N'.
R1 (input) DOUBLE PRECISION array, dimension ( LDR1,N )
If JOB <> 'N', M > 0, METH = 'M', and RANKR1 = N,
the leading L*(NOBR-1)-by-N part of this array must
contain details of the QR factorization of the matrix
GaL, as computed by SLICOT Library routine IB01PD.
Specifically, the leading N-by-N upper triangular part
must contain the upper triangular factor r1 of GaL,
and the lower L*(NOBR-1)-by-N trapezoidal part, together
with array TAU1, must contain the factored form of the
orthogonal matrix Q1 in the QR factorization of GaL.
If JOB = 'N', or M = 0, or METH = 'N', or METH = 'M'
and RANKR1 < N, this array is not referenced.
LDR1 INTEGER
The leading dimension of the array R1.
LDR1 >= L*(NOBR-1), if JOB <> 'N', M > 0, METH = 'M',
and RANKR1 = N;
LDR1 >= 1, otherwise.
TAU1 (input) DOUBLE PRECISION array, dimension ( N )
If JOB <> 'N', M > 0, METH = 'M', and RANKR1 = N,
this array must contain the scalar factors of the
elementary reflectors used in the QR factorization of the
matrix GaL, computed by SLICOT Library routine IB01PD.
If JOB = 'N', or M = 0, or METH = 'N', or METH = 'M'
and RANKR1 < N, this array is not referenced.
PGAL (input) DOUBLE PRECISION array, dimension
( LDPGAL,L*(NOBR-1) )
If METH = 'N', or JOB <> 'N', M > 0, METH = 'M' and
RANKR1 < N, the leading N-by-L*(NOBR-1) part of this
array must contain the pseudoinverse of the matrix GaL,
as computed by SLICOT Library routine IB01PD.
If METH = 'M' and JOB = 'N', or M = 0, or
RANKR1 = N, this array is not referenced.
LDPGAL INTEGER
The leading dimension of the array PGAL.
LDPGAL >= N, if METH = 'N', or JOB <> 'N', M > 0,
and METH = 'M' and RANKR1 < N;
LDPGAL >= 1, otherwise.
K (input/output) DOUBLE PRECISION array, dimension
( LDK,M*NOBR )
On entry, the leading (p/s)-by-M*NOBR part of this array
must contain the given matrix K defined above.
On exit, the leading (p/s)-by-M*NOBR part of this array
contains the transformed matrix K.
LDK INTEGER
The leading dimension of the array K. LDK >= p/s.
R (output) DOUBLE PRECISION array, dimension ( LDR,L*NOBR )
If JOB = 'N', or M = 0, or Q has full rank, the
leading L*NOBR-by-L*NOBR upper triangular part of this
array contains the R factor of the QR factorization of
the matrix Q.
If JOB <> 'N', M > 0, and Q has not a full rank, the
leading L*NOBR-by-L*NOBR upper trapezoidal part of this
array contains details of the complete orhogonal
factorization of the matrix Q, as constructed by SLICOT
Library routines MB03OD and MB02QY.
LDR INTEGER
The leading dimension of the array R. LDR >= L*NOBR.
H (output) DOUBLE PRECISION array, dimension ( LDH,M )
If JOB = 'N' or M = 0, the leading L*NOBR-by-M part
of this array contains the updated part of the matrix
Kexpand corresponding to the upper triangular factor R
in the QR factorization of the matrix Q.
If JOB <> 'N', M > 0, and METH = 'N' or METH = 'M'
and RANKR1 < N, the leading L*NOBR-by-M part of this
array contains the minimum norm least squares solution of
the linear system Q*X = Kexpand, from which the matrices
B and D are found. The first NOBR-1 row blocks of X
appear in the reverse order in H.
If JOB <> 'N', M > 0, METH = 'M' and RANKR1 = N, the
leading L*(NOBR-1)-by-M part of this array contains the
matrix product Q1'*X, and the subarray
L*(NOBR-1)+1:L*NOBR-by-M contains the corresponding
submatrix of X, with X defined in the phrase above.
LDH INTEGER
The leading dimension of the array H. LDH >= L*NOBR.
B (output) DOUBLE PRECISION array, dimension ( LDB,M )
If M > 0, JOB = 'B' or 'D' and INFO = 0, the leading
N-by-M part of this array contains the system input
matrix.
If M = 0 or JOB = 'N', this array is not referenced.
LDB INTEGER
The leading dimension of the array B.
LDB >= N, if M > 0 and JOB = 'B' or 'D';
LDB >= 1, if M = 0 or JOB = 'N'.
D (output) DOUBLE PRECISION array, dimension ( LDD,M )
If M > 0, JOB = 'D' and INFO = 0, the leading
L-by-M part of this array contains the system input-output
matrix.
If M = 0 or JOB = 'B' or 'N', this array is not
referenced.
LDD INTEGER
The leading dimension of the array D.
LDD >= L, if M > 0 and JOB = 'D';
LDD >= 1, if M = 0 or JOB = 'B' or 'N'.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used for estimating the rank of
matrices. If the user sets TOL > 0, then the given value
of TOL is used as a lower bound for the reciprocal
condition number; an m-by-n matrix whose estimated
condition number is less than 1/TOL is considered to
be of full rank. If the user sets TOL <= 0, then an
implicitly computed, default tolerance, defined by
TOLDEF = m*n*EPS, is used instead, where EPS is the
relative machine precision (see LAPACK Library routine
DLAMCH).
This parameter is not used if M = 0 or JOB = 'N'.
Workspace
IWORK INTEGER array, dimension ( LIWORK )
where LIWORK >= 0, if JOB = 'N', or M = 0;
LIWORK >= L*NOBR, if JOB <> 'N', and M > 0.
DWORK DOUBLE PRECISION array, dimension ( LDWORK )
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK, and, if JOB <> 'N', and M > 0, DWORK(2)
contains the reciprocal condition number of the triangular
factor of the matrix R.
On exit, if INFO = -28, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( 2*L, L*NOBR, L+M*NOBR ),
if JOB = 'N', or M = 0;
LDWORK >= MAX( L+M*NOBR, L*NOBR + MAX( 3*L*NOBR+1, M ) ),
if JOB <> 'N', and M > 0.
For good performance, LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 4: the least squares problem to be solved has a
rank-deficient coefficient matrix.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 3: a singular upper triangular matrix was found.
Method
The QR factorization is computed exploiting the structure, as described in [4]. The matrices B and D are then obtained by solving certain linear systems in a least squares sense.References
[1] Verhaegen M., and Dewilde, P.
Subspace Model Identification. Part 1: The output-error
state-space model identification class of algorithms.
Int. J. Control, 56, pp. 1187-1210, 1992.
[2] Van Overschee, P., and De Moor, B.
N4SID: Two Subspace Algorithms for the Identification
of Combined Deterministic-Stochastic Systems.
Automatica, Vol.30, No.1, pp. 75-93, 1994.
[3] Van Overschee, P.
Subspace Identification : Theory - Implementation -
Applications.
Ph. D. Thesis, Department of Electrical Engineering,
Katholieke Universiteit Leuven, Belgium, Feb. 1995.
[4] Sima, V.
Subspace-based Algorithms for Multivariable System
Identification.
Studies in Informatics and Control, 5, pp. 335-344, 1996.
Numerical Aspects
The implemented method for computing the triangular factor and updating Kexpand is numerically stable.Further Comments
The computed matrices B and D are not the least squares solutions delivered by either MOESP or N4SID algorithms, except for the special case n = s - 1, L = 1. However, the computed B and D are frequently good enough estimates, especially for METH = 'M'. Better estimates could be obtained by calling SLICOT Library routine IB01PX, but it is less efficient, and requires much more workspace.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To estimate the initial state and the system matrices B and D
of a linear time-invariant (LTI) discrete-time system, given the
matrix pair (A,C) and the input and output trajectories of the
system. The model structure is :
x(k+1) = Ax(k) + Bu(k), k >= 0,
y(k) = Cx(k) + Du(k),
where x(k) is the n-dimensional state vector (at time k),
u(k) is the m-dimensional input vector,
y(k) is the l-dimensional output vector,
and A, B, C, and D are real matrices of appropriate dimensions.
Matrix A is assumed to be in a real Schur form.
Specification
SUBROUTINE IB01QD( JOBX0, JOB, N, M, L, NSMP, A, LDA, C, LDC, U,
$ LDU, Y, LDY, X0, B, LDB, D, LDD, TOL, IWORK,
$ DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
DOUBLE PRECISION TOL
INTEGER INFO, IWARN, L, LDA, LDB, LDC, LDD, LDU,
$ LDWORK, LDY, M, N, NSMP
CHARACTER JOB, JOBX0
C .. Array Arguments ..
DOUBLE PRECISION A(LDA, *), B(LDB, *), C(LDC, *), D(LDD, *),
$ DWORK(*), U(LDU, *), X0(*), Y(LDY, *)
INTEGER IWORK(*)
Arguments
Mode Parameters
JOBX0 CHARACTER*1
Specifies whether or not the initial state should be
computed, as follows:
= 'X': compute the initial state x(0);
= 'N': do not compute the initial state (x(0) is known
to be zero).
JOB CHARACTER*1
Specifies which matrices should be computed, as follows:
= 'B': compute the matrix B only (D is known to be zero);
= 'D': compute the matrices B and D.
Input/Output Parameters
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
L (input) INTEGER
The number of system outputs. L > 0.
NSMP (input) INTEGER
The number of rows of matrices U and Y (number of
samples, t).
NSMP >= N*M + a + e, where
a = 0, if JOBX0 = 'N';
a = N, if JOBX0 = 'X';
e = 0, if JOBX0 = 'X' and JOB = 'B';
e = 1, if JOBX0 = 'N' and JOB = 'B';
e = M, if JOB = 'D'.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A in a real Schur form.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading L-by-N part of this array must contain the
system output matrix C (corresponding to the real Schur
form of A).
LDC INTEGER
The leading dimension of the array C. LDC >= L.
U (input/output) DOUBLE PRECISION array, dimension (LDU,M)
On entry, the leading NSMP-by-M part of this array must
contain the t-by-m input-data sequence matrix U,
U = [u_1 u_2 ... u_m]. Column j of U contains the
NSMP values of the j-th input component for consecutive
time increments.
On exit, if JOB = 'D', the leading NSMP-by-M part of
this array contains details of the QR factorization of
the t-by-m matrix U, possibly computed sequentially
(see METHOD).
If JOB = 'B', this array is unchanged on exit.
If M = 0, this array is not referenced.
LDU INTEGER
The leading dimension of the array U.
LDU >= MAX(1,NSMP), if M > 0;
LDU >= 1, if M = 0.
Y (input) DOUBLE PRECISION array, dimension (LDY,L)
The leading NSMP-by-L part of this array must contain the
t-by-l output-data sequence matrix Y,
Y = [y_1 y_2 ... y_l]. Column j of Y contains the
NSMP values of the j-th output component for consecutive
time increments.
LDY INTEGER
The leading dimension of the array Y. LDY >= MAX(1,NSMP).
X0 (output) DOUBLE PRECISION array, dimension (N)
If JOBX0 = 'X', the estimated initial state of the
system, x(0).
If JOBX0 = 'N', x(0) is set to zero without any
calculations.
B (output) DOUBLE PRECISION array, dimension (LDB,M)
If N > 0, M > 0, and INFO = 0, the leading N-by-M
part of this array contains the system input matrix B
in the coordinates corresponding to the real Schur form
of A.
If N = 0 or M = 0, this array is not referenced.
LDB INTEGER
The leading dimension of the array B.
LDB >= N, if N > 0 and M > 0;
LDB >= 1, if N = 0 or M = 0.
D (output) DOUBLE PRECISION array, dimension (LDD,M)
If M > 0, JOB = 'D', and INFO = 0, the leading
L-by-M part of this array contains the system input-output
matrix D.
If M = 0 or JOB = 'B', this array is not referenced.
LDD INTEGER
The leading dimension of the array D.
LDD >= L, if M > 0 and JOB = 'D';
LDD >= 1, if M = 0 or JOB = 'B'.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used for estimating the rank of
matrices. If the user sets TOL > 0, then the given value
of TOL is used as a lower bound for the reciprocal
condition number; a matrix whose estimated condition
number is less than 1/TOL is considered to be of full
rank. If the user sets TOL <= 0, then EPS is used
instead, where EPS is the relative machine precision
(see LAPACK Library routine DLAMCH). TOL <= 1.
Workspace
IWORK INTEGER array, dimension (LIWORK), where
LIWORK >= N*M + a, if JOB = 'B',
LIWORK >= max( N*M + a, M ), if JOB = 'D',
with a = 0, if JOBX0 = 'N';
a = N, if JOBX0 = 'X'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK; DWORK(2) contains the reciprocal condition
number of the triangular factor of the QR factorization of
the matrix W2 (see METHOD); if M > 0 and JOB = 'D',
DWORK(3) contains the reciprocal condition number of the
triangular factor of the QR factorization of U.
On exit, if INFO = -23, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= max( LDW1, min( LDW2, LDW3 ) ), where
LDW1 = 2, if M = 0 or JOB = 'B',
LDW1 = 3, if M > 0 and JOB = 'D',
LDWa = t*L*(r + 1) + max( N + max( d, f ), 6*r ),
LDW2 = LDWa, if M = 0 or JOB = 'B',
LDW2 = max( LDWa, t*L*(r + 1) + 2*M*M + 6*M ),
if M > 0 and JOB = 'D',
LDWb = (b + r)*(r + 1) +
max( q*(r + 1) + N*N*M + c + max( d, f ), 6*r ),
LDW3 = LDWb, if M = 0 or JOB = 'B',
LDW3 = max( LDWb, (b + r)*(r + 1) + 2*M*M + 6*M ),
if M > 0 and JOB = 'D',
r = N*M + a,
a = 0, if JOBX0 = 'N',
a = N, if JOBX0 = 'X';
b = 0, if JOB = 'B',
b = L*M, if JOB = 'D';
c = 0, if JOBX0 = 'N',
c = L*N, if JOBX0 = 'X';
d = 0, if JOBX0 = 'N',
d = 2*N*N + N, if JOBX0 = 'X';
f = 2*r, if JOB = 'B' or M = 0,
f = M + max( 2*r, M ), if JOB = 'D' and M > 0;
q = b + r*L.
For good performance, LDWORK should be larger.
If LDWORK >= LDW2 or
LDWORK >= t*L*(r + 1) + (b + r)*(r + 1) + N*N*M + c +
max( d, f ),
then standard QR factorizations of the matrices U and/or
W2 (see METHOD) are used.
Otherwise, the QR factorizations are computed sequentially
by performing NCYCLE cycles, each cycle (except possibly
the last one) processing s < t samples, where s is
chosen from the equation
LDWORK = s*L*(r + 1) + (b + r)*(r + 1) + N*N*M + c +
max( d, f ).
(s is at least N*M+a+e, the minimum value of NSMP.)
The computational effort may increase and the accuracy may
decrease with the decrease of s. Recommended value is
LDWORK = LDW2, assuming a large enough cache size, to
also accommodate A, C, U, and Y.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 4: the least squares problem to be solved has a
rank-deficient coefficient matrix.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 2: the singular value decomposition (SVD) algorithm did
not converge.
Method
An extension and refinement of the method in [1,2] is used.
Specifically, denoting
X = [ vec(D')' vec(B)' x0' ]',
where vec(M) is the vector obtained by stacking the columns of
the matrix M, then X is the least squares solution of the
system S*X = vec(Y), with the matrix S = [ diag(U) W ],
defined by
( U | | ... | | | ... | | )
( U | 11 | ... | n1 | 12 | ... | nm | )
S = ( : | y | ... | y | y | ... | y | P*Gamma ),
( : | | ... | | | ... | | )
( U | | ... | | | ... | | )
ij
diag(U) having L block rows and columns. In this formula, y
are the outputs of the system for zero initial state computed
using the following model, for j = 1:m, and for i = 1:n,
ij ij ij
x (k+1) = Ax (k) + e_i u_j(k), x (0) = 0,
ij ij
y (k) = Cx (k),
where e_i is the i-th n-dimensional unit vector, Gamma is
given by
( C )
( C*A )
Gamma = ( C*A^2 ),
( : )
( C*A^(t-1) )
and P is a permutation matrix that groups together the rows of
Gamma depending on the same row of C, namely
[ c_j; c_j*A; c_j*A^2; ... c_j*A^(t-1) ], for j = 1:L.
The first block column, diag(U), is not explicitly constructed,
but its structure is exploited. The last block column is evaluated
using powers of A with exponents 2^k. No interchanges are applied.
A special QR decomposition of the matrix S is computed. Let
U = q*[ r' 0 ]' be the QR decomposition of U, if M > 0, where
r is M-by-M. Then, diag(q') is applied to W and vec(Y).
The block-rows of S and vec(Y) are implicitly permuted so that
matrix S becomes
( diag(r) W1 )
( 0 W2 ),
where W1 has L*M rows. Then, the QR decomposition of W2 is
computed (sequentially, if M > 0) and used to obtain B and x0.
The intermediate results and the QR decomposition of U are
needed to find D. If a triangular factor is too ill conditioned,
then singular value decomposition (SVD) is employed. SVD is not
generally needed if the input sequence is sufficiently
persistently exciting and NSMP is large enough.
If the matrix W cannot be stored in the workspace (i.e.,
LDWORK < LDW2), the QR decompositions of W2 and U are
computed sequentially.
References
[1] Verhaegen M., and Varga, A.
Some Experience with the MOESP Class of Subspace Model
Identification Methods in Identifying the BO105 Helicopter.
Report TR R165-94, DLR Oberpfaffenhofen, 1994.
[2] Sima, V., and Varga, A.
RASP-IDENT : Subspace Model Identification Programs.
Deutsche Forschungsanstalt fur Luft- und Raumfahrt e. V.,
Report TR R888-94, DLR Oberpfaffenhofen, Oct. 1994.
Numerical Aspects
The implemented method is numerically stable.Further Comments
The algorithm for computing the system matrices B and D is less efficient than the MOESP or N4SID algorithms implemented in SLICOT Library routine IB01PD, because a large least squares problem has to be solved, but the accuracy is better, as the computed matrices B and D are fitted to the input and output trajectories. However, if matrix A is unstable, the computed matrices B and D could be inaccurate.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To estimate the initial state of a linear time-invariant (LTI)
discrete-time system, given the system matrices (A,B,C,D) and
the input and output trajectories of the system. The model
structure is :
x(k+1) = Ax(k) + Bu(k), k >= 0,
y(k) = Cx(k) + Du(k),
where x(k) is the n-dimensional state vector (at time k),
u(k) is the m-dimensional input vector,
y(k) is the l-dimensional output vector,
and A, B, C, and D are real matrices of appropriate dimensions.
Matrix A is assumed to be in a real Schur form.
Specification
SUBROUTINE IB01RD( JOB, N, M, L, NSMP, A, LDA, B, LDB, C, LDC, D,
$ LDD, U, LDU, Y, LDY, X0, TOL, IWORK, DWORK,
$ LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
DOUBLE PRECISION TOL
INTEGER INFO, IWARN, L, LDA, LDB, LDC, LDD, LDU,
$ LDWORK, LDY, M, N, NSMP
CHARACTER JOB
C .. Array Arguments ..
DOUBLE PRECISION A(LDA, *), B(LDB, *), C(LDC, *), D(LDD, *),
$ DWORK(*), U(LDU, *), X0(*), Y(LDY, *)
INTEGER IWORK(*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies whether or not the matrix D is zero, as follows:
= 'Z': the matrix D is zero;
= 'N': the matrix D is not zero.
Input/Output Parameters
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
L (input) INTEGER
The number of system outputs. L > 0.
NSMP (input) INTEGER
The number of rows of matrices U and Y (number of
samples used, t). NSMP >= N.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A in a real Schur form.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
system input matrix B (corresponding to the real Schur
form of A).
If N = 0 or M = 0, this array is not referenced.
LDB INTEGER
The leading dimension of the array B.
LDB >= N, if N > 0 and M > 0;
LDB >= 1, if N = 0 or M = 0.
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading L-by-N part of this array must contain the
system output matrix C (corresponding to the real Schur
form of A).
LDC INTEGER
The leading dimension of the array C. LDC >= L.
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading L-by-M part of this array must contain the
system input-output matrix.
If M = 0 or JOB = 'Z', this array is not referenced.
LDD INTEGER
The leading dimension of the array D.
LDD >= L, if M > 0 and JOB = 'N';
LDD >= 1, if M = 0 or JOB = 'Z'.
U (input) DOUBLE PRECISION array, dimension (LDU,M)
If M > 0, the leading NSMP-by-M part of this array must
contain the t-by-m input-data sequence matrix U,
U = [u_1 u_2 ... u_m]. Column j of U contains the
NSMP values of the j-th input component for consecutive
time increments.
If M = 0, this array is not referenced.
LDU INTEGER
The leading dimension of the array U.
LDU >= MAX(1,NSMP), if M > 0;
LDU >= 1, if M = 0.
Y (input) DOUBLE PRECISION array, dimension (LDY,L)
The leading NSMP-by-L part of this array must contain the
t-by-l output-data sequence matrix Y,
Y = [y_1 y_2 ... y_l]. Column j of Y contains the
NSMP values of the j-th output component for consecutive
time increments.
LDY INTEGER
The leading dimension of the array Y. LDY >= MAX(1,NSMP).
X0 (output) DOUBLE PRECISION array, dimension (N)
The estimated initial state of the system, x(0).
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used for estimating the rank of
matrices. If the user sets TOL > 0, then the given value
of TOL is used as a lower bound for the reciprocal
condition number; a matrix whose estimated condition
number is less than 1/TOL is considered to be of full
rank. If the user sets TOL <= 0, then EPS is used
instead, where EPS is the relative machine precision
(see LAPACK Library routine DLAMCH). TOL <= 1.
Workspace
IWORK INTEGER array, dimension (N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK and DWORK(2) contains the reciprocal condition
number of the triangular factor of the QR factorization of
the matrix Gamma (see METHOD).
On exit, if INFO = -22, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= max( 2, min( LDW1, LDW2 ) ), where
LDW1 = t*L*(N + 1) + 2*N + max( 2*N*N, 4*N ),
LDW2 = N*(N + 1) + 2*N +
max( q*(N + 1) + 2*N*N + L*N, 4*N ),
q = N*L.
For good performance, LDWORK should be larger.
If LDWORK >= LDW1, then standard QR factorization of
the matrix Gamma (see METHOD) is used. Otherwise, the
QR factorization is computed sequentially by performing
NCYCLE cycles, each cycle (except possibly the last one)
processing s samples, where s is chosen by equating
LDWORK to LDW2, for q replaced by s*L.
The computational effort may increase and the accuracy may
decrease with the decrease of s. Recommended value is
LDRWRK = LDW1, assuming a large enough cache size, to
also accommodate A, B, C, D, U, and Y.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 4: the least squares problem to be solved has a
rank-deficient coefficient matrix.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 2: the singular value decomposition (SVD) algorithm did
not converge.
Method
An extension and refinement of the method in [1] is used.
Specifically, the output y0(k) of the system for zero initial
state is computed for k = 0, 1, ..., t-1 using the given model.
Then the following least squares problem is solved for x(0)
( C ) ( y(0) - y0(0) )
( C*A ) ( y(1) - y0(1) )
Gamma * x(0) = ( : ) * x(0) = ( : ).
( : ) ( : )
( C*A^(t-1) ) ( y(t-1) - y0(t-1) )
The coefficient matrix Gamma is evaluated using powers of A with
exponents 2^k. The QR decomposition of this matrix is computed.
If its triangular factor R is too ill conditioned, then singular
value decomposition of R is used.
If the coefficient matrix cannot be stored in the workspace (i.e.,
LDWORK < LDW1), the QR decomposition is computed sequentially.
References
[1] Verhaegen M., and Varga, A.
Some Experience with the MOESP Class of Subspace Model
Identification Methods in Identifying the BO105 Helicopter.
Report TR R165-94, DLR Oberpfaffenhofen, 1994.
Numerical Aspects
The implemented method is numerically stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute a set of parameters for approximating a Wiener system
in a least-squares sense, using a neural network approach and a
Levenberg-Marquardt algorithm. Conjugate gradients (CG) or
Cholesky algorithms are used to solve linear systems of equations.
The Wiener system is represented as
x(t+1) = A*x(t) + B*u(t)
z(t) = C*x(t) + D*u(t),
y(t) = f(z(t),wb(1:L)),
where t = 1, 2, ..., NSMP, and f is a nonlinear function,
evaluated by the SLICOT Library routine NF01AY. The parameter
vector X is partitioned as X = ( wb(1), ..., wb(L), theta ),
where wb(i), i = 1 : L, correspond to the nonlinear part, and
theta corresponds to the linear part. See SLICOT Library routine
NF01AD for further details.
The sum of squares of the error functions, defined by
e(t) = y(t) - Y(t), t = 1, 2, ..., NSMP,
is minimized, where Y(t) is the measured output vector. The
functions and their Jacobian matrices are evaluated by SLICOT
Library routine NF01BB (the FCN routine in the call of MD03AD).
Specification
SUBROUTINE IB03AD( INIT, ALG, STOR, NOBR, M, L, NSMP, N, NN,
$ ITMAX1, ITMAX2, NPRINT, U, LDU, Y, LDY, X, LX,
$ TOL1, TOL2, IWORK, DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER ALG, INIT, STOR
INTEGER INFO, ITMAX1, ITMAX2, IWARN, L, LDU, LDWORK,
$ LDY, LX, M, N, NN, NOBR, NPRINT, NSMP
DOUBLE PRECISION TOL1, TOL2
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), U(LDU, *), X(*), Y(LDY, *)
INTEGER IWORK(*)
Arguments
Mode Parameters
INIT CHARACTER*1
Specifies which parts have to be initialized, as follows:
= 'L' : initialize the linear part only, X already
contains an initial approximation of the
nonlinearity;
= 'S' : initialize the static nonlinearity only, X
already contains an initial approximation of the
linear part;
= 'B' : initialize both linear and nonlinear parts;
= 'N' : do not initialize anything, X already contains
an initial approximation.
If INIT = 'S' or 'B', the error functions for the
nonlinear part, and their Jacobian matrices, are evaluated
by SLICOT Library routine NF01BA (used as a second FCN
routine in the MD03AD call for the initialization step,
see METHOD).
ALG CHARACTER*1
Specifies the algorithm used for solving the linear
systems involving a Jacobian matrix J, as follows:
= 'D' : a direct algorithm, which computes the Cholesky
factor of the matrix J'*J + par*I is used, where
par is the Levenberg factor;
= 'I' : an iterative Conjugate Gradients algorithm, which
only needs the matrix J, is used.
In both cases, matrix J is stored in a compressed form.
STOR CHARACTER*1
If ALG = 'D', specifies the storage scheme for the
symmetric matrix J'*J, as follows:
= 'F' : full storage is used;
= 'P' : packed storage is used.
The option STOR = 'F' usually ensures a faster execution.
This parameter is not relevant if ALG = 'I'.
Input/Output Parameters
NOBR (input) INTEGER
If INIT = 'L' or 'B', NOBR is the number of block rows, s,
in the input and output block Hankel matrices to be
processed for estimating the linear part. NOBR > 0.
(In the MOESP theory, NOBR should be larger than n,
the estimated dimension of state vector.)
This parameter is ignored if INIT is 'S' or 'N'.
M (input) INTEGER
The number of system inputs. M >= 0.
L (input) INTEGER
The number of system outputs. L >= 0, and L > 0, if
INIT = 'L' or 'B'.
NSMP (input) INTEGER
The number of input and output samples, t. NSMP >= 0, and
NSMP >= 2*(M+L+1)*NOBR - 1, if INIT = 'L' or 'B'.
N (input/output) INTEGER
The order of the linear part.
If INIT = 'L' or 'B', and N < 0 on entry, the order is
assumed unknown and it will be found by the routine.
Otherwise, the input value will be used. If INIT = 'S'
or 'N', N must be non-negative. The values N >= NOBR,
or N = 0, are not acceptable if INIT = 'L' or 'B'.
NN (input) INTEGER
The number of neurons which shall be used to approximate
the nonlinear part. NN >= 0.
ITMAX1 (input) INTEGER
The maximum number of iterations for the initialization of
the static nonlinearity.
This parameter is ignored if INIT is 'N' or 'L'.
Otherwise, ITMAX1 >= 0.
ITMAX2 (input) INTEGER
The maximum number of iterations. ITMAX2 >= 0.
NPRINT (input) INTEGER
This parameter enables controlled printing of iterates if
it is positive. In this case, FCN is called with IFLAG = 0
at the beginning of the first iteration and every NPRINT
iterations thereafter and immediately prior to return,
and the current error norm is printed. Other intermediate
results could be printed by modifying the corresponding
FCN routine (NF01BA and/or NF01BB). If NPRINT <= 0, no
special calls of FCN with IFLAG = 0 are made.
U (input) DOUBLE PRECISION array, dimension (LDU, M)
The leading NSMP-by-M part of this array must contain the
set of input samples,
U = ( U(1,1),...,U(1,M); ...; U(NSMP,1),...,U(NSMP,M) ).
LDU INTEGER
The leading dimension of array U. LDU >= MAX(1,NSMP).
Y (input) DOUBLE PRECISION array, dimension (LDY, L)
The leading NSMP-by-L part of this array must contain the
set of output samples,
Y = ( Y(1,1),...,Y(1,L); ...; Y(NSMP,1),...,Y(NSMP,L) ).
LDY INTEGER
The leading dimension of array Y. LDY >= MAX(1,NSMP).
X (input/output) DOUBLE PRECISION array dimension (LX)
On entry, if INIT = 'L', the leading (NN*(L+2) + 1)*L part
of this array must contain the initial parameters for
the nonlinear part of the system.
On entry, if INIT = 'S', the elements lin1 : lin2 of this
array must contain the initial parameters for the linear
part of the system, corresponding to the output normal
form, computed by SLICOT Library routine TB01VD, where
lin1 = (NN*(L+2) + 1)*L + 1;
lin2 = (NN*(L+2) + 1)*L + N*(L+M+1) + L*M.
On entry, if INIT = 'N', the elements 1 : lin2 of this
array must contain the initial parameters for the
nonlinear part followed by the initial parameters for the
linear part of the system, as specified above.
This array need not be set on entry if INIT = 'B'.
On exit, the elements 1 : lin2 of this array contain the
optimal parameters for the nonlinear part followed by the
optimal parameters for the linear part of the system, as
specified above.
LX (input/output) INTEGER
On entry, this parameter must contain the intended length
of X. If N >= 0, then LX >= NX := lin2 (see parameter X).
If N is unknown (N < 0 on entry), a large enough estimate
of N should be used in the formula of lin2.
On exit, if N < 0 on entry, but LX is not large enough,
then this parameter contains the actual length of X,
corresponding to the computed N. Otherwise, its value
is unchanged.
Tolerances
TOL1 DOUBLE PRECISION
If INIT = 'S' or 'B' and TOL1 >= 0, TOL1 is the tolerance
which measures the relative error desired in the sum of
squares, for the initialization step of nonlinear part.
Termination occurs when the actual relative reduction in
the sum of squares is at most TOL1. In addition, if
ALG = 'I', TOL1 also measures the relative residual of
the solutions computed by the CG algorithm (for the
initialization step). Termination of a CG process occurs
when the relative residual is at most TOL1.
If the user sets TOL1 < 0, then SQRT(EPS) is used
instead TOL1, where EPS is the machine precision
(see LAPACK Library routine DLAMCH).
This parameter is ignored if INIT is 'N' or 'L'.
TOL2 DOUBLE PRECISION
If TOL2 >= 0, TOL2 is the tolerance which measures the
relative error desired in the sum of squares, for the
whole optimization process. Termination occurs when the
actual relative reduction in the sum of squares is at
most TOL2.
If ALG = 'I', TOL2 also measures the relative residual of
the solutions computed by the CG algorithm (for the whole
optimization). Termination of a CG process occurs when the
relative residual is at most TOL2.
If the user sets TOL2 < 0, then SQRT(EPS) is used
instead TOL2. This default value could require many
iterations, especially if TOL1 is larger. If INIT = 'S'
or 'B', it is advisable that TOL2 be larger than TOL1,
and spend more time with cheaper iterations.
Workspace
IWORK INTEGER array, dimension (MAX( 3, LIW1, LIW2 )), where
LIW1 = LIW2 = 0, if INIT = 'S' or 'N'; otherwise,
LIW1 = M+L;
LIW2 = MAX(M*NOBR+N,M*(N+L)).
On output, if INFO = 0, IWORK(1) and IWORK(2) return the
(total) number of function and Jacobian evaluations,
respectively (including the initialization step, if it was
performed), and if INIT = 'L' or INIT = 'B', IWORK(3)
specifies how many locations of DWORK contain reciprocal
condition number estimates (see below); otherwise,
IWORK(3) = 0.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On entry, if desired, and if INIT = 'S' or 'B', the
entries DWORK(1:4) are set to initialize the random
numbers generator for the nonlinear part parameters (see
the description of the argument XINIT of SLICOT Library
routine MD03AD); this enables to obtain reproducible
results. The same seed is used for all outputs.
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK, DWORK(2) returns the residual error norm (the
sum of squares), DWORK(3) returns the number of iterations
performed, DWORK(4) returns the number of conjugate
gradients iterations performed, and DWORK(5) returns the
final Levenberg factor, for optimizing the parameters of
both the linear part and the static nonlinearity part.
If INIT = 'S' or INIT = 'B' and INFO = 0, then the
elements DWORK(6) to DWORK(10) contain the corresponding
five values for the initialization step (see METHOD).
(If L > 1, DWORK(10) contains the maximum of the Levenberg
factors for all outputs.) If INIT = 'L' or INIT = 'B', and
INFO = 0, DWORK(11) to DWORK(10+IWORK(3)) contain
reciprocal condition number estimates set by SLICOT
Library routines IB01AD, IB01BD, and IB01CD.
On exit, if INFO = -23, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
In the formulas below, N should be taken not larger than
NOBR - 1, if N < 0 on entry.
LDWORK = MAX( LW1, LW2, LW3, LW4 ), where
LW1 = 0, if INIT = 'S' or 'N'; otherwise,
LW1 = MAX( 2*(M+L)*NOBR*(2*(M+L)*(NOBR+1)+3) + L*NOBR,
4*(M+L)*NOBR*(M+L)*NOBR + (N+L)*(N+M) +
MAX( LDW1, LDW2 ),
(N+L)*(N+M) + N + N*N + 2 + N*(N+M+L) +
MAX( 5*N, 2, MIN( LDW3, LDW4 ), LDW5, LDW6 ),
where,
LDW1 >= MAX( 2*(L*NOBR-L)*N+2*N, (L*NOBR-L)*N+N*N+7*N,
L*NOBR*N +
MAX( (L*NOBR-L)*N+2*N + (2*M+L)*NOBR+L,
2*(L*NOBR-L)*N+N*N+8*N,
N+4*(M*NOBR+N)+1, M*NOBR+3*N+L ) )
LDW2 >= 0, if M = 0;
LDW2 >= L*NOBR*N + M*NOBR*(N+L)*(M*(N+L)+1) +
MAX( (N+L)**2, 4*M*(N+L)+1 ), if M > 0;
LDW3 = NSMP*L*(N+1) + 2*N + MAX( 2*N*N, 4*N ),
LDW4 = N*(N+1) + 2*N +
MAX( N*L*(N+1) + 2*N*N + L*N, 4*N );
LDW5 = NSMP*L + (N+L)*(N+M) + 3*N+M+L;
LDW6 = NSMP*L + (N+L)*(N+M) + N +
MAX(1, N*N*L + N*L + N, N*N +
MAX(N*N + N*MAX(N,L) + 6*N + MIN(N,L),
N*M));
LW2 = LW3 = 0, if INIT = 'L' or 'N'; otherwise,
LW2 = NSMP*L +
MAX( 5, NSMP + 2*BSN + NSMP*BSN +
MAX( 2*NN + BSN, LDW7 ) );
LDW7 = BSN*BSN, if ALG = 'D' and STOR = 'F';
LDW7 = BSN*(BSN+1)/2, if ALG = 'D' and STOR = 'P';
LDW7 = 3*BSN + NSMP, if ALG = 'I';
LW3 = MAX( LDW8, NSMP*L + (N+L)*(2*N+M) + 2*N );
LDW8 = NSMP*L + (N+L)*(N+M) + 3*N+M+L, if M > 0;
LDW8 = NSMP*L + (N+L)*N + 2*N+L, if M = 0;
LW4 = MAX( 5, NSMP*L + 2*NX + NSMP*L*( BSN + LTHS ) +
MAX( L1 + NX, NSMP*L + L1, L2 ) ),
L0 = MAX( N*(N+L), N+M+L ), if M > 0;
L0 = MAX( N*(N+L), L ), if M = 0;
L1 = NSMP*L + MAX( 2*NN, (N+L)*(N+M) + 2*N + L0);
L2 = NX*NX, if ALG = 'D' and STOR = 'F';
L2 = NX*(NX+1)/2, if ALG = 'D' and STOR = 'P';
L2 = 3*NX + NSMP*L, if ALG = 'I',
with BSN = NN*( L + 2 ) + 1,
LTHS = N*( L + M + 1 ) + L*M.
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
< 0: the user set IFLAG = IWARN in (one of) the
subroutine(s) FCN, i.e., NF01BA, if INIT = 'S'
or 'B', and/or NF01BB; this value cannot be returned
without changing the FCN routine(s);
otherwise, IWARN has the value k*100 + j*10 + i,
where k is defined below, i refers to the whole
optimization process, and j refers to the
initialization step (j = 0, if INIT = 'L' or 'N'),
and the possible values for i and j have the
following meaning (where TOL* denotes TOL1 or TOL2,
and similarly for ITMAX*):
= 1: the number of iterations has reached ITMAX* without
satisfying the convergence condition;
= 2: if alg = 'I' and in an iteration of the Levenberg-
Marquardt algorithm, the CG algorithm finished
after 3*NX iterations (or 3*(lin1-1) iterations, for
the initialization phase), without achieving the
precision required in the call;
= 3: the cosine of the angle between the vector of error
function values and any column of the Jacobian is at
most FACTOR*EPS in absolute value (FACTOR = 100);
= 4: TOL* is too small: no further reduction in the sum
of squares is possible.
The digit k is normally 0, but if INIT = 'L' or 'B', it
can have a value in the range 1 to 6 (see IB01AD, IB01BD
and IB01CD). In all these cases, the entries DWORK(1:5),
DWORK(6:10) (if INIT = 'S' or 'B'), and
DWORK(11:10+IWORK(3)) (if INIT = 'L' or 'B'), are set as
described above.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
otherwise, INFO has the value k*100 + j*10 + i,
where k is defined below, i refers to the whole
optimization process, and j refers to the
initialization step (j = 0, if INIT = 'L' or 'N'),
and the possible values for i and j have the
following meaning:
= 1: the routine FCN returned with INFO <> 0 for
IFLAG = 1;
= 2: the routine FCN returned with INFO <> 0 for
IFLAG = 2;
= 3: ALG = 'D' and SLICOT Library routines MB02XD or
NF01BU (or NF01BV, if INIT = 'S' or 'B') or
ALG = 'I' and SLICOT Library routines MB02WD or
NF01BW (or NF01BX, if INIT = 'S' or 'B') returned
with INFO <> 0.
In addition, if INIT = 'L' or 'B', i could also be
= 4: if a Lyapunov equation could not be solved;
= 5: if the identified linear system is unstable;
= 6: if the QR algorithm failed on the state matrix
of the identified linear system.
The digit k is normally 0, but if INIT = 'L' or 'B', it
can have a value in the range 1 to 10 (see IB01AD/IB01BD).
Method
If INIT = 'L' or 'B', the linear part of the system is approximated using the combined MOESP and N4SID algorithm. If necessary, this algorithm can also choose the order, but it is advantageous if the order is already known. If INIT = 'S' or 'B', the output of the approximated linear part is computed and used to calculate an approximation of the static nonlinearity using the Levenberg-Marquardt algorithm [1]. This step is referred to as the (nonlinear) initialization step. As last step, the Levenberg-Marquardt algorithm is used again to optimize the parameters of the linear part and the static nonlinearity as a whole. Therefore, it is necessary to parametrise the matrices of the linear part. The output normal form [2] parameterisation is used. The Jacobian is computed analytically, for the nonlinear part, and numerically, for the linear part.References
[1] Kelley, C.T.
Iterative Methods for Optimization.
Society for Industrial and Applied Mathematics (SIAM),
Philadelphia (Pa.), 1999.
[2] Peeters, R.L.M., Hanzon, B., and Olivi, M.
Balanced realizations of discrete-time stable all-pass
systems and the tangential Schur algorithm.
Proceedings of the European Control Conference,
31 August - 3 September 1999, Karlsruhe, Germany.
Session CP-6, Discrete-time Systems, 1999.
Further Comments
NoneExample
Program Text
* IB03AD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER LDU, LDY, LIWORK, LMAX, MMAX, NMAX, NNMAX,
$ NOBRMX, NSMPMX
PARAMETER ( LMAX = 2, MMAX = 3, NOBRMX = 10, NNMAX = 12,
$ NMAX = 4, NSMPMX = 1024,
$ LDU = NSMPMX, LDY = NSMPMX,
$ LIWORK = MAX( MMAX + LMAX, MMAX*NOBRMX + NMAX,
$ MMAX*( NMAX + LMAX ) ) )
INTEGER BSNM, L0, L1M, L2M, LDW1, LDW2, LDW3, LDW4,
$ LDW5, LDW6, LDW7, LDW8, LDWORK, LTHS, LW1, LW2,
$ LW3, LW4, LXM
PARAMETER ( BSNM = NNMAX*( LMAX + 2 ) + 1,
$ LTHS = NMAX*( LMAX + MMAX + 1 ) + LMAX*MMAX,
$ L0 = MAX( NMAX*( NMAX + LMAX ),
$ NMAX + MMAX + LMAX ),
$ L1M = NSMPMX*LMAX +
$ MAX( 2*NNMAX,
$ ( NMAX + LMAX )*( NMAX + MMAX ) +
$ 2*NMAX + L0 ),
$ LXM = BSNM*LMAX + LTHS,
$ L2M = MAX( LXM*LXM, 3*LXM + NSMPMX*LMAX ),
$ LDW1 = MAX( 2*( LMAX*NOBRMX - LMAX )*NMAX +
$ 2*NMAX,
$ ( LMAX*NOBRMX - LMAX )*NMAX +
$ NMAX*NMAX + 7*NMAX,
$ LMAX*NOBRMX*NMAX +
$ MAX( ( LMAX*NOBRMX - LMAX )*NMAX +
$ 2*NMAX + LMAX +
$ ( 2*MMAX + LMAX )*NOBRMX,
$ 2*( LMAX*NOBRMX - LMAX )*NMAX
$ + NMAX*NMAX + 8*NMAX,
$ NMAX + 4*( MMAX*NOBRMX +
$ NMAX ) + 1,
$ MMAX*NOBRMX + 3*NMAX + LMAX )
$ ),
$ LDW2 = LMAX*NOBRMX*NMAX +
$ MMAX*NOBRMX*( NMAX + LMAX )*
$ ( MMAX*( NMAX + LMAX ) + 1 ) +
$ MAX( ( NMAX + LMAX )**2,
$ 4*MMAX*( NMAX + LMAX ) + 1 ),
$ LDW3 = NSMPMX*LMAX*( NMAX + 1 ) + 2*NMAX +
$ MAX( 2*NMAX*NMAX, 4*NMAX ),
$ LDW4 = NMAX*( NMAX + 1 ) + 2*NMAX +
$ MAX( NMAX*LMAX*( NMAX + 1 ) +
$ 2*NMAX*NMAX + LMAX*NMAX, 4*NMAX ),
$ LDW5 = NSMPMX*LMAX + ( NMAX + LMAX )*
$ ( NMAX + MMAX ) + 3*NMAX + MMAX + LMAX,
$ LDW6 = NSMPMX*LMAX + ( NMAX + LMAX )*
$ ( NMAX + MMAX ) + NMAX +
$ MAX( 1, NMAX*NMAX*LMAX + NMAX*LMAX +
$ NMAX, NMAX*NMAX +
$ MAX( NMAX*NMAX +
$ NMAX*MAX( NMAX, LMAX ) +
$ 6*NMAX + MIN( NMAX, LMAX ),
$ NMAX*MMAX ) ),
$ LDW7 = MAX( BSNM*BSNM, 3*BSNM + NSMPMX ),
$ LDW8 = NSMPMX*LMAX + ( NMAX + LMAX )*
$ ( NMAX + MMAX ) + 3*NMAX + MMAX + LMAX,
$ LW1 = MAX( 2*( MMAX + LMAX )*NOBRMX*
$ ( 2*( MMAX + LMAX )*( NOBRMX + 1 )
$ + 3 ) + LMAX*NOBRMX,
$ 4*( MMAX + LMAX )*NOBRMX*
$ ( MMAX + LMAX )*NOBRMX +
$ ( NMAX + LMAX )*( NMAX + MMAX ) +
$ MAX( LDW1, LDW2 ),
$ ( NMAX + LMAX )*( NMAX + MMAX ) +
$ NMAX + NMAX*NMAX + 2 +
$ NMAX*( NMAX + MMAX + LMAX ) +
$ MAX( 5*NMAX, 2, MIN( LDW3, LDW4 ),
$ LDW5, LDW6 ) ),
$ LW2 = NSMPMX*LMAX +
$ MAX( 5, NSMPMX + 2*BSNM + NSMPMX*BSNM +
$ MAX( 2*NNMAX + BSNM, LDW7 ) ),
$ LW3 = MAX( LDW8, NSMPMX*LMAX +
$ ( NMAX + LMAX )*( 2*NMAX + MMAX )+
$ 2*NMAX ),
$ LW4 = MAX( 5, NSMPMX*LMAX + 2*LXM +
$ NSMPMX*LMAX*( BSNM + LTHS ) +
$ MAX( L1M + LXM, NSMPMX*LMAX + L1M,
$ L2M ) ),
$ LDWORK = MAX( LW1, LW2, LW3, LW4 ) )
* .. Local Scalars ..
LOGICAL INIT1, INITB, INITL, INITN, INITS
CHARACTER*1 ALG, INIT, STOR
INTEGER BSN, I, INFO, INI, ITER, ITERCG, ITMAX1, ITMAX2,
$ IWARN, J, L, L1, L2, LPAR, LX, M, N, NN, NOBR,
$ NPRINT, NS, NSMP
DOUBLE PRECISION TOL1, TOL2
* .. Array Arguments ..
INTEGER IWORK(LIWORK)
DOUBLE PRECISION DWORK(LDWORK), U(LDU,MMAX), X(LXM), Y(LDY,LMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL IB03AD
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) NOBR, M, L, NSMP, N, NN, ITMAX1, ITMAX2,
$ NPRINT, TOL1, TOL2, INIT, ALG, STOR
INITL = LSAME( INIT, 'L' )
INITS = LSAME( INIT, 'S' )
INITB = LSAME( INIT, 'B' )
INITN = LSAME( INIT, 'N' )
INIT1 = INITL .OR. INITB
IF( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) M
ELSE
IF( L.LE.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) L
ELSE
NS = N
IF( INIT1 ) THEN
IF( NOBR.LE.0 .OR. NOBR.GT.NOBRMX ) THEN
WRITE ( NOUT, FMT = 99991 ) NOBR
STOP
ELSEIF( NSMP.LT.2*( M + L + 1 )*NOBR - 1 ) THEN
WRITE ( NOUT, FMT = 99990 ) NSMP
STOP
ELSEIF( N.EQ.0 .OR. N.GE.NOBR ) THEN
WRITE ( NOUT, FMT = 99989 ) N
STOP
END IF
IF ( N.LT.0 )
$ N = NOBR - 1
ELSE
IF( NSMP.LT.0 ) THEN
WRITE ( NOUT, FMT = 99990 ) NSMP
STOP
ELSEIF( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) N
STOP
END IF
END IF
IF( NN.LT.0 .OR. NN.GT.NNMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) NN
ELSE
BSN = NN*( L + 2 ) + 1
L1 = BSN*L
L2 = N*( L + M + 1 ) + L*M
LX = L1 + L2
INI = 1
IF ( INITL ) THEN
LPAR = L1
ELSEIF ( INITS ) THEN
INI = L1 + 1
LPAR = L2
ELSEIF ( INITN ) THEN
LPAR = LX
END IF
IF( INIT1 )
$ N = NS
* Read the input-output data, initial parameters, and seed.
READ ( NIN, FMT = * ) ( ( U(I,J), J = 1,M ), I = 1,NSMP )
READ ( NIN, FMT = * ) ( ( Y(I,J), J = 1,L ), I = 1,NSMP )
IF ( .NOT.INITB )
$ READ ( NIN, FMT = * ) ( X(I), I = INI,INI+LPAR-1 )
IF ( INITS .OR. INITB )
$ READ ( NIN, FMT = * ) ( DWORK(I), I = 1,4 )
* Solve a Wiener system identification problem.
CALL IB03AD( INIT, ALG, STOR, NOBR, M, L, NSMP, N, NN,
$ ITMAX1, ITMAX2, NPRINT, U, LDU, Y, LDY,
$ X, LX, TOL1, TOL2, IWORK, DWORK, LDWORK,
$ IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF( IWARN.NE.0 ) WRITE ( NOUT, FMT = 99987 ) IWARN
ITER = DWORK(3)
ITERCG = DWORK(4)
WRITE ( NOUT, FMT = 99997 ) DWORK(2)
WRITE ( NOUT, FMT = 99996 ) ITER, ITERCG,
$ IWORK(1), IWORK(2)
* Recompute LX is necessary.
IF ( INIT1 .AND. NS.LT.0 )
$ LX = L1 + N*( L + M + 1 ) + L*M
WRITE ( NOUT, FMT = 99994 )
WRITE ( NOUT, FMT = 99995 ) ( X(I), I = 1, LX )
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' IB03AD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from IB03AD = ',I4)
99997 FORMAT (/' Final 2-norm of the residuals = ',D15.7)
99996 FORMAT (/' Number of iterations = ', I7,
$ /' Number of conjugate gradients iterations = ', I7,
$ /' Number of function evaluations = ', I7,
$ /' Number of Jacobian evaluations = ', I7)
99995 FORMAT (10(1X,F8.4))
99994 FORMAT (/' Final approximate solution is ' )
99993 FORMAT (/' M is out of range.',/' M = ',I5)
99992 FORMAT (/' L is out of range.',/' L = ',I5)
99991 FORMAT (/' NOBR is out of range.',/' NOBR = ',I5)
99990 FORMAT (/' NSMP is out of range.',/' NSMP = ',I5)
99989 FORMAT (/' N is out of range.',/' N = ',I5)
99988 FORMAT (/' NN is out of range.',/' NN = ',I5)
99987 FORMAT (' IWARN on exit from IB03AD = ',I4)
END
Program Data
IB03AD EXAMPLE PROGRAM DATA 10 1 1 1024 4 12 500 1000 0 .00001 .00001 B D F 2.2183165e-01 3.9027807e-02 -5.0295887e-02 8.5386224e-03 7.2431159e-02 -1.7082198e-03 -1.7176287e-01 -2.6198104e-01 -1.7194108e-01 1.8566868e-02 1.5625362e-01 1.7463811e-01 1.1564450e-01 2.8779248e-02 -8.4265993e-02 -2.0978501e-01 -2.6591828e-01 -1.7268680e-01 2.1525013e-02 1.4363602e-01 7.3101431e-02 -1.0259212e-01 -1.6380473e-01 -1.0021167e-02 2.0263451e-01 2.1983417e-01 -2.1636523e-02 -3.0986057e-01 -3.8521982e-01 -2.1785179e-01 -1.4761096e-02 3.7005180e-02 -2.8119028e-02 -4.2167901e-02 5.2117694e-02 1.2023747e-01 1.8863385e-02 -1.9506434e-01 -3.0192175e-01 -1.7000747e-01 8.0740471e-02 2.0188076e-01 8.5108288e-02 -1.3270970e-01 -2.3646822e-01 -1.6505385e-01 -4.7448014e-02 -2.7886815e-02 -1.0152026e-01 -1.4155374e-01 -6.1650823e-02 8.3519614e-02 1.5926650e-01 8.6142760e-02 -9.4385381e-02 -2.6609066e-01 -3.2883874e-01 -2.5908050e-01 -1.1648940e-01 -3.0653766e-03 1.0326675e-02 -5.3445909e-02 -9.2412724e-02 -3.0279541e-02 8.4846832e-02 1.1133075e-01 -3.2135250e-02 -2.5308181e-01 -3.5670882e-01 -2.4458860e-01 -2.5254261e-02 9.3714332e-02 1.8643667e-02 -1.4592119e-01 -2.2730880e-01 -1.7140060e-01 -7.4131665e-02 -3.9669515e-02 -5.1266129e-02 -1.1752833e-02 1.0785565e-01 2.0665525e-01 1.6117322e-01 -2.6938653e-02 -2.1941152e-01 -2.7753567e-01 -1.8805912e-01 -4.6845025e-02 5.8585698e-02 1.2218407e-01 1.7838638e-01 2.2169815e-01 1.9825589e-01 8.0215288e-02 -7.2135308e-02 -1.4381520e-01 -6.8724371e-02 1.0191205e-01 2.3766633e-01 2.3876101e-01 1.1678077e-01 -2.0428168e-02 -5.8973233e-02 3.1326900e-02 1.7391495e-01 2.4558570e-01 1.7650262e-01 1.2444292e-02 -1.1538234e-01 -9.5917970e-02 6.4762165e-02 2.4258524e-01 3.0102251e-01 2.1222960e-01 7.8706189e-02 3.1500466e-02 1.0297577e-01 1.9875173e-01 1.9434906e-01 5.8146667e-02 -1.1941921e-01 -2.1038478e-01 -1.5594967e-01 1.8552198e-03 1.6878529e-01 2.5937416e-01 2.2516346e-01 6.6144472e-02 -1.5623019e-01 -3.3161105e-01 -3.6695732e-01 -2.6565333e-01 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-6.1784801e-01 -1.8264666e-01 3.2014735e-01 6.1135123e-01 5.4895999e-01 1.9768580e-01 -2.2062099e-01 -4.6220719e-01 -4.0211731e-01 -9.9950534e-02 2.4465654e-01 4.1872319e-01 3.2500596e-01 3.2810917e-02 -2.7440750e-01 -4.1536442e-01 -3.1832701e-01 -5.5989066e-02 2.0726049e-01 3.1798239e-01 2.2484797e-01 5.1703651e-03 -1.8889751e-01 -2.2927380e-01 -9.1914974e-02 1.3314428e-01 3.0513495e-01 3.2224987e-01 1.7778028e-01 -4.7100451e-02 -2.4007922e-01 -3.2145867e-01 -2.7615883e-01 -1.4545755e-01 4.2822900e-03 1.1399372e-01 1.5138712e-01 1.1530153e-01 3.0234280e-02 -6.4234624e-02 -1.2615802e-01 -1.2407054e-01 -4.9317670e-02 7.5619816e-02 2.0015044e-01 2.6472178e-01 2.3118708e-01 1.0699863e-01 -5.5412012e-02 -1.8550876e-01 -2.3096135e-01 -1.8218227e-01 -7.2615500e-02 4.0881922e-02 1.0372451e-01 8.6362391e-02 -1.1351454e-03 -1.0889033e-01 -1.6548976e-01 -1.1405709e-01 4.6560657e-02 2.4386985e-01 3.6111476e-01 3.0662373e-01 8.1468123e-02 -2.0497551e-01 -3.9165036e-01 -3.6309524e-01 -1.2535574e-01 1.8954273e-01 3.9793935e-01 3.7486538e-01 1.3124068e-01 -1.9174474e-01 -4.0848802e-01 -4.0149539e-01 -1.8960477e-01 9.0301438e-02 2.7507284e-01 2.7972729e-01 1.4341274e-01 -1.2566755e-02 -7.8032703e-02 -2.7425697e-02 7.5351759e-02 1.3487633e-01 9.5488652e-02 -2.4590018e-02 -1.5233210e-01 -2.1189289e-01 -1.7248897e-01 -6.2455423e-02 5.4933614e-02 1.2398028e-01 1.2778044e-01 8.7386392e-02 3.4966577e-02 -1.0850501e-02 -4.6716543e-02 -6.9020828e-02 -6.3681635e-02 -1.6203206e-02 6.7394491e-02 1.5127737e-01 1.8399090e-01 1.2920707e-01 -7.0434827e-03 -1.7216342e-01 -2.8937677e-01 -2.9509198e-01 -1.7314710e-01 3.2745183e-02 2.3542177e-01 3.4097958e-01 2.9247721e-01 1.0411948e-01 -1.3495077e-01 -2.9868629e-01 -2.9240849e-01 -1.1517683e-01 1.2871323e-01 2.8803761e-01 2.6146766e-01 6.7234759e-02 -1.6729947e-01 -2.9180077e-01 -2.3297675e-01 -3.8493954e-02 1.6188055e-01 2.4607750e-01 1.7580193e-01 1.0770499e-02 -1.3917580e-01 -1.8630712e-01 -1.1496682e-01 1.8120146e-02 1.2605380e-01 1.4532251e-01 6.9056099e-02 -5.5814690e-02 -1.6001831e-01 -1.8912751e-01 -1.2778372e-01 -4.4698128e-03 1.2208903e-01 1.8963074e-01 1.6384408e-01 6.0799128e-02 -5.7339158e-02 -1.1860919e-01 -9.0086196e-02 -4.5798607e-03 6.0280807e-02 4.1676388e-02 -5.5180320e-02 -1.5518201e-01 -1.6828578e-01 -6.2049884e-02 1.0561621e-01 2.2337555e-01 2.0643187e-01 5.9839911e-02 -1.2043322e-01 -2.1083864e-01 -1.4415945e-01 4.3538937e-02 2.3203364e-01 2.9044234e-01 1.6171416e-01 -9.5674666e-02 -3.3749265e-01 -4.1795872e-01 -2.7746809e-01 2.0648626e-02 3.2603206e-01 4.8410918e-01 4.1672303e-01 1.5905611e-01 -1.6318595e-01 -3.9931562e-01 -4.4568803e-01 -2.9169291e-01 -2.0960934e-02 2.3175866e-01 3.4693819e-01 2.7877641e-01 7.7125945e-02 -1.4069530e-01 -2.5367798e-01 -2.0150506e-01 -1.6778161e-02 1.9116819e-01 2.9409556e-01 2.1593628e-01 -1.9610708e-02 -2.9401135e-01 -4.5512990e-01 -4.0311941e-01 -1.5075705e-01 1.7921653e-01 4.2153577e-01 4.6143206e-01 2.9688389e-01 3.5275834e-02 -1.7206796e-01 -2.2040717e-01 -1.1280250e-01 4.6014479e-02 1.2005000e-01 3.5297082e-02 -1.6459920e-01 -3.4121448e-01 -3.5130088e-01 -1.4787707e-01 1.7615712e-01 4.3972643e-01 4.8949447e-01 2.9899548e-01 -1.6059656e-02 -2.7414987e-01 -3.4124596e-01 -2.0476598e-01 3.1287353e-02 2.1535118e-01 2.3693813e-01 8.7039128e-02 -1.3914592e-01 -2.9731202e-01 -2.8057123e-01 -8.9244625e-02 1.6445576e-01 3.2621002e-01 2.9949560e-01 1.0678193e-01 -1.3016725e-01 -2.7225661e-01 -2.4687907e-01 -8.3173776e-02 1.1381888e-01 2.2819642e-01 1.9830143e-01 4.8505476e-02 -1.2763594e-01 -2.2560309e-01 -1.9560311e-01 -7.1212054e-02 6.0380807e-02 1.2445307e-01 1.0835168e-01 5.5609724e-02 1.7269294e-02 9.3997346e-03 1.1223045e-02 -4.3543819e-03 -4.2668837e-02 -8.5657964e-02 -1.0909342e-01 -9.7154374e-02 -4.6781850e-02 3.1101930e-02 1.0973840e-01 1.5122945e-01 1.2531404e-01 3.3620966e-02 -8.3194568e-02 -1.6716420e-01 1998. 1999. 2000. 2001.Program Results
IB03AD EXAMPLE PROGRAM RESULTS Final 2-norm of the residuals = 0.2970365D+00 Number of iterations = 87 Number of conjugate gradients iterations = 0 Number of function evaluations = 1322 Number of Jacobian evaluations = 105 Final approximate solution is -0.9728 0.6465 -1.2888 -0.4296 -0.8530 0.3181 0.9778 0.4570 -0.1420 0.8984 -0.6031 0.0697 -1.0822 0.4465 0.6036 0.3792 0.2532 -0.0285 0.4129 0.4833 0.1746 0.5626 0.2150 -0.3343 0.4013 -0.3679 0.5653 0.8092 -0.2363 -0.6361 -0.6818 0.6110 -0.5506 0.9914 0.0352 0.1968 -0.2502 7.0067 -10.7378 2.6900 -59.8756 -0.9898 -0.8296 2.3429 1.3456 -0.2531 -1.1265 0.0326 0.5617 0.1045
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/IB03BD.html 0000664 0000000 0000000 00000204106 14560147231 0020320 0 ustar 00root root 0000000 0000000
Purpose
To compute a set of parameters for approximating a Wiener system
in a least-squares sense, using a neural network approach and a
MINPACK-like Levenberg-Marquardt algorithm. The Wiener system
consists of a linear part and a static nonlinearity, and it is
represented as
x(t+1) = A*x(t) + B*u(t)
z(t) = C*x(t) + D*u(t),
y(t) = f(z(t),wb(1:L)),
where t = 1, 2, ..., NSMP, and f is a nonlinear function,
evaluated by the SLICOT Library routine NF01AY. The parameter
vector X is partitioned as X = ( wb(1), ..., wb(L), theta ),
where theta corresponds to the linear part, and wb(i), i = 1 : L,
correspond to the nonlinear part. See SLICOT Library routine
NF01AD for further details.
The sum of squares of the error functions, defined by
e(t) = y(t) - Y(t), t = 1, 2, ..., NSMP,
is minimized, where Y(t) is the measured output vector. The
functions and their Jacobian matrices are evaluated by SLICOT
Library routine NF01BF (the FCN routine in the call of MD03BD).
Specification
SUBROUTINE IB03BD( INIT, NOBR, M, L, NSMP, N, NN, ITMAX1, ITMAX2,
$ NPRINT, U, LDU, Y, LDY, X, LX, TOL1, TOL2,
$ IWORK, DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER INIT
INTEGER INFO, ITMAX1, ITMAX2, IWARN, L, LDU, LDWORK,
$ LDY, LX, M, N, NN, NOBR, NPRINT, NSMP
DOUBLE PRECISION TOL1, TOL2
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), U(LDU, *), X(*), Y(LDY, *)
INTEGER IWORK(*)
Arguments
Mode Parameters
INIT CHARACTER*1
Specifies which parts have to be initialized, as follows:
= 'L' : initialize the linear part only, X already
contains an initial approximation of the
nonlinearity;
= 'S' : initialize the static nonlinearity only, X
already contains an initial approximation of the
linear part;
= 'B' : initialize both linear and nonlinear parts;
= 'N' : do not initialize anything, X already contains
an initial approximation.
If INIT = 'S' or 'B', the error functions for the
nonlinear part, and their Jacobian matrices, are evaluated
by SLICOT Library routine NF01BE (used as a second FCN
routine in the MD03BD call for the initialization step,
see METHOD).
Input/Output Parameters
NOBR (input) INTEGER
If INIT = 'L' or 'B', NOBR is the number of block rows, s,
in the input and output block Hankel matrices to be
processed for estimating the linear part. NOBR > 0.
(In the MOESP theory, NOBR should be larger than n,
the estimated dimension of state vector.)
This parameter is ignored if INIT is 'S' or 'N'.
M (input) INTEGER
The number of system inputs. M >= 0.
L (input) INTEGER
The number of system outputs. L >= 0, and L > 0, if
INIT = 'L' or 'B'.
NSMP (input) INTEGER
The number of input and output samples, t. NSMP >= 0, and
NSMP >= 2*(M+L+1)*NOBR - 1, if INIT = 'L' or 'B'.
N (input/output) INTEGER
The order of the linear part.
If INIT = 'L' or 'B', and N < 0 on entry, the order is
assumed unknown and it will be found by the routine.
Otherwise, the input value will be used. If INIT = 'S'
or 'N', N must be non-negative. The values N >= NOBR,
or N = 0, are not acceptable if INIT = 'L' or 'B'.
NN (input) INTEGER
The number of neurons which shall be used to approximate
the nonlinear part. NN >= 0.
ITMAX1 (input) INTEGER
The maximum number of iterations for the initialization of
the static nonlinearity.
This parameter is ignored if INIT is 'N' or 'L'.
Otherwise, ITMAX1 >= 0.
ITMAX2 (input) INTEGER
The maximum number of iterations. ITMAX2 >= 0.
NPRINT (input) INTEGER
This parameter enables controlled printing of iterates if
it is positive. In this case, FCN is called with IFLAG = 0
at the beginning of the first iteration and every NPRINT
iterations thereafter and immediately prior to return,
and the current error norm is printed. Other intermediate
results could be printed by modifying the corresponding
FCN routine (NF01BE and/or NF01BF). If NPRINT <= 0, no
special calls of FCN with IFLAG = 0 are made.
U (input) DOUBLE PRECISION array, dimension (LDU, M)
The leading NSMP-by-M part of this array must contain the
set of input samples,
U = ( U(1,1),...,U(1,M); ...; U(NSMP,1),...,U(NSMP,M) ).
LDU INTEGER
The leading dimension of array U. LDU >= MAX(1,NSMP).
Y (input) DOUBLE PRECISION array, dimension (LDY, L)
The leading NSMP-by-L part of this array must contain the
set of output samples,
Y = ( Y(1,1),...,Y(1,L); ...; Y(NSMP,1),...,Y(NSMP,L) ).
LDY INTEGER
The leading dimension of array Y. LDY >= MAX(1,NSMP).
X (input/output) DOUBLE PRECISION array dimension (LX)
On entry, if INIT = 'L', the leading (NN*(L+2) + 1)*L part
of this array must contain the initial parameters for
the nonlinear part of the system.
On entry, if INIT = 'S', the elements lin1 : lin2 of this
array must contain the initial parameters for the linear
part of the system, corresponding to the output normal
form, computed by SLICOT Library routine TB01VD, where
lin1 = (NN*(L+2) + 1)*L + 1;
lin2 = (NN*(L+2) + 1)*L + N*(L+M+1) + L*M.
On entry, if INIT = 'N', the elements 1 : lin2 of this
array must contain the initial parameters for the
nonlinear part followed by the initial parameters for the
linear part of the system, as specified above.
This array need not be set on entry if INIT = 'B'.
On exit, the elements 1 : lin2 of this array contain the
optimal parameters for the nonlinear part followed by the
optimal parameters for the linear part of the system, as
specified above.
LX (input/output) INTEGER
On entry, this parameter must contain the intended length
of X. If N >= 0, then LX >= NX := lin2 (see parameter X).
If N is unknown (N < 0 on entry), a large enough estimate
of N should be used in the formula of lin2.
On exit, if N < 0 on entry, but LX is not large enough,
then this parameter contains the actual length of X,
corresponding to the computed N. Otherwise, its value
is unchanged.
Tolerances
TOL1 DOUBLE PRECISION
If INIT = 'S' or 'B' and TOL1 >= 0, TOL1 is the tolerance
which measures the relative error desired in the sum of
squares, as well as the relative error desired in the
approximate solution, for the initialization step of
nonlinear part. Termination occurs when either both the
actual and predicted relative reductions in the sum of
squares, or the relative error between two consecutive
iterates are at most TOL1. If the user sets TOL1 < 0,
then SQRT(EPS) is used instead TOL1, where EPS is the
machine precision (see LAPACK Library routine DLAMCH).
This parameter is ignored if INIT is 'N' or 'L'.
TOL2 DOUBLE PRECISION
If TOL2 >= 0, TOL2 is the tolerance which measures the
relative error desired in the sum of squares, as well as
the relative error desired in the approximate solution,
for the whole optimization process. Termination occurs
when either both the actual and predicted relative
reductions in the sum of squares, or the relative error
between two consecutive iterates are at most TOL2. If the
user sets TOL2 < 0, then SQRT(EPS) is used instead TOL2.
This default value could require many iterations,
especially if TOL1 is larger. If INIT = 'S' or 'B', it is
advisable that TOL2 be larger than TOL1, and spend more
time with cheaper iterations.
Workspace
IWORK INTEGER array, dimension (MAX( LIW1, LIW2, LIW3 )), where
LIW1 = LIW2 = 0, if INIT = 'S' or 'N'; otherwise,
LIW1 = M+L;
LIW2 = MAX(M*NOBR+N,M*(N+L));
LIW3 = 3+MAX(NN*(L+2)+2,NX+L), if INIT = 'S' or 'B';
LIW3 = 3+NX+L, if INIT = 'L' or 'N'.
On output, if INFO = 0, IWORK(1) and IWORK(2) return the
(total) number of function and Jacobian evaluations,
respectively (including the initialization step, if it was
performed), and if INIT = 'L' or INIT = 'B', IWORK(3)
specifies how many locations of DWORK contain reciprocal
condition number estimates (see below); otherwise,
IWORK(3) = 0. If INFO = 0, the entries 4 to 3+NX of IWORK
define a permutation matrix P such that J*P = Q*R, where
J is the final calculated Jacobian, Q is an orthogonal
matrix (not stored), and R is upper triangular with
diagonal elements of nonincreasing magnitude (possibly
for each block column of J). Column j of P is column
IWORK(3+j) of the identity matrix. Moreover, the entries
4+NX:3+NX+L of this array contain the ranks of the final
submatrices S_k (see description of LMPARM in MD03BD).
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On entry, if desired, and if INIT = 'S' or 'B', the
entries DWORK(1:4) are set to initialize the random
numbers generator for the nonlinear part parameters (see
the description of the argument XINIT of SLICOT Library
routine MD03BD); this enables to obtain reproducible
results. The same seed is used for all outputs.
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK, DWORK(2) returns the residual error norm (the
sum of squares), DWORK(3) returns the number of iterations
performed, and DWORK(4) returns the final Levenberg
factor, for optimizing the parameters of both the linear
part and the static nonlinearity part. If INIT = 'S' or
INIT = 'B' and INFO = 0, then the elements DWORK(5) to
DWORK(8) contain the corresponding four values for the
initialization step (see METHOD). (If L > 1, DWORK(8)
contains the maximum of the Levenberg factors for all
outputs.) If INIT = 'L' or INIT = 'B', and INFO = 0,
DWORK(9) to DWORK(8+IWORK(3)) contain reciprocal condition
number estimates set by SLICOT Library routines IB01AD,
IB01BD, and IB01CD.
On exit, if INFO = -21, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
In the formulas below, N should be taken not larger than
NOBR - 1, if N < 0 on entry.
LDWORK = MAX( LW1, LW2, LW3, LW4 ), where
LW1 = 0, if INIT = 'S' or 'N'; otherwise,
LW1 = MAX( 2*(M+L)*NOBR*(2*(M+L)*(NOBR+1)+3) + L*NOBR,
4*(M+L)*NOBR*(M+L)*NOBR + (N+L)*(N+M) +
MAX( LDW1, LDW2 ),
(N+L)*(N+M) + N + N*N + 2 + N*(N+M+L) +
MAX( 5*N, 2, MIN( LDW3, LDW4 ), LDW5, LDW6 ),
where,
LDW1 >= MAX( 2*(L*NOBR-L)*N+2*N, (L*NOBR-L)*N+N*N+7*N,
L*NOBR*N +
MAX( (L*NOBR-L)*N+2*N + (2*M+L)*NOBR+L,
2*(L*NOBR-L)*N+N*N+8*N,
N+4*(M*NOBR+N)+1, M*NOBR+3*N+L ) )
LDW2 >= 0, if M = 0;
LDW2 >= L*NOBR*N + M*NOBR*(N+L)*(M*(N+L)+1) +
MAX( (N+L)**2, 4*M*(N+L)+1 ), if M > 0;
LDW3 = NSMP*L*(N+1) + 2*N + MAX( 2*N*N, 4*N ),
LDW4 = N*(N+1) + 2*N +
MAX( N*L*(N+1) + 2*N*N + L*N, 4*N );
LDW5 = NSMP*L + (N+L)*(N+M) + 3*N+M+L;
LDW6 = NSMP*L + (N+L)*(N+M) + N +
MAX(1, N*N*L + N*L + N, N*N +
MAX(N*N + N*MAX(N,L) + 6*N + MIN(N,L),
N*M));
LW2 = LW3 = 0, if INIT = 'L' or 'N'; otherwise,
LW2 = NSMP*L + BSN +
MAX( 4, NSMP +
MAX( NSMP*BSN + MAX( 2*NN, 5*BSN + 1 ),
BSN**2 + BSN +
MAX( NSMP + 2*NN, 5*BSN ) ) );
LW3 = MAX( LDW7, NSMP*L + (N+L)*(2*N+M) + 2*N );
LDW7 = NSMP*L + (N+L)*(N+M) + 3*N+M+L, if M > 0;
LDW7 = NSMP*L + (N+L)*N + 2*N+L, if M = 0;
LW4 = NSMP*L + NX +
MAX( 4, NSMP*L +
MAX( NSMP*L*( BSN + LTHS ) +
MAX( NSMP*L + L1, L2 + NX ),
NX*( BSN + LTHS ) + NX +
MAX( NSMP*L + L1, NX + L3 ) ) ),
L0 = MAX( N*(N+L), N+M+L ), if M > 0;
L0 = MAX( N*(N+L), L ), if M = 0;
L1 = NSMP*L + MAX( 2*NN, (N+L)*(N+M) + 2*N + L0);
L2 = 4*NX + 1, if L <= 1 or BSN = 0; otherwise,
L2 = BSN + MAX(3*BSN+1,LTHS);
L2 = MAX(L2,4*LTHS+1), if NSMP > BSN;
L2 = MAX(L2,(NSMP-BSN)*(L-1)), if BSN < NSMP < 2*BSN;
L3 = 4*NX, if L <= 1 or BSN = 0;
L3 = LTHS*BSN + 2*NX + 2*MAX(BSN,LTHS),
if L > 1 and BSN > 0,
with BSN = NN*( L + 2 ) + 1,
LTHS = N*( L + M + 1 ) + L*M.
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
< 0: the user set IFLAG = IWARN in (one of) the
subroutine(s) FCN, i.e., NF01BE, if INIT = 'S'
or 'B', and/or NF01BF; this value cannot be returned
without changing the FCN routine(s);
otherwise, IWARN has the value k*100 + j*10 + i,
where k is defined below, i refers to the whole
optimization process, and j refers to the
initialization step (j = 0, if INIT = 'L' or 'N'),
and the possible values for i and j have the
following meaning (where TOL* denotes TOL1 or TOL2,
and similarly for ITMAX*):
= 1: both actual and predicted relative reductions in
the sum of squares are at most TOL*;
= 2: relative error between two consecutive iterates is
at most TOL*;
= 3: conditions for i or j = 1 and i or j = 2 both hold;
= 4: the cosine of the angle between the vector of error
function values and any column of the Jacobian is at
most EPS in absolute value;
= 5: the number of iterations has reached ITMAX* without
satisfying any convergence condition;
= 6: TOL* is too small: no further reduction in the sum
of squares is possible;
= 7: TOL* is too small: no further improvement in the
approximate solution X is possible;
= 8: the vector of function values e is orthogonal to the
columns of the Jacobian to machine precision.
The digit k is normally 0, but if INIT = 'L' or 'B', it
can have a value in the range 1 to 6 (see IB01AD, IB01BD
and IB01CD). In all these cases, the entries DWORK(1:4),
DWORK(5:8) (if INIT = 'S' or 'B'), and DWORK(9:8+IWORK(3))
(if INIT = 'L' or 'B'), are set as described above.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
otherwise, INFO has the value k*100 + j*10 + i,
where k is defined below, i refers to the whole
optimization process, and j refers to the
initialization step (j = 0, if INIT = 'L' or 'N'),
and the possible values for i and j have the
following meaning:
= 1: the routine FCN returned with INFO <> 0 for
IFLAG = 1;
= 2: the routine FCN returned with INFO <> 0 for
IFLAG = 2;
= 3: the routine QRFACT returned with INFO <> 0;
= 4: the routine LMPARM returned with INFO <> 0.
In addition, if INIT = 'L' or 'B', i could also be
= 5: if a Lyapunov equation could not be solved;
= 6: if the identified linear system is unstable;
= 7: if the QR algorithm failed on the state matrix
of the identified linear system.
QRFACT and LMPARM are generic names for SLICOT Library
routines NF01BS and NF01BP, respectively, for the whole
optimization process, and MD03BA and MD03BB, respectively,
for the initialization step (if INIT = 'S' or 'B').
The digit k is normally 0, but if INIT = 'L' or 'B', it
can have a value in the range 1 to 10 (see IB01AD/IB01BD).
Method
If INIT = 'L' or 'B', the linear part of the system is approximated using the combined MOESP and N4SID algorithm. If necessary, this algorithm can also choose the order, but it is advantageous if the order is already known. If INIT = 'S' or 'B', the output of the approximated linear part is computed and used to calculate an approximation of the static nonlinearity using the Levenberg-Marquardt algorithm [1,3]. This step is referred to as the (nonlinear) initialization step. As last step, the Levenberg-Marquardt algorithm is used again to optimize the parameters of the linear part and the static nonlinearity as a whole. Therefore, it is necessary to parametrise the matrices of the linear part. The output normal form [2] parameterisation is used. The Jacobian is computed analytically, for the nonlinear part, and numerically, for the linear part.References
[1] More, J.J., Garbow, B.S, and Hillstrom, K.E.
User's Guide for MINPACK-1.
Applied Math. Division, Argonne National Laboratory, Argonne,
Illinois, Report ANL-80-74, 1980.
[2] Peeters, R.L.M., Hanzon, B., and Olivi, M.
Balanced realizations of discrete-time stable all-pass
systems and the tangential Schur algorithm.
Proceedings of the European Control Conference,
31 August - 3 September 1999, Karlsruhe, Germany.
Session CP-6, Discrete-time Systems, 1999.
[3] More, J.J.
The Levenberg-Marquardt algorithm: implementation and theory.
In Watson, G.A. (Ed.), Numerical Analysis, Lecture Notes in
Mathematics, vol. 630, Springer-Verlag, Berlin, Heidelberg
and New York, pp. 105-116, 1978.
Numerical Aspects
The Levenberg-Marquardt algorithm described in [3] is scaling invariant and globally convergent to (maybe local) minima. The convergence rate near a local minimum is quadratic, if the Jacobian is computed analytically, and linear, if the Jacobian is computed numerically.Further Comments
NoneExample
Program Text
* IB03BD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER BSNM, LDU, LDY, LIWORK, LMAX, LTHS, LXM, MMAX,
$ NMAX, NNMAX, NOBRMX, NSMPMX
PARAMETER ( LMAX = 2, MMAX = 3, NOBRMX = 10, NNMAX = 12,
$ NMAX = 4, NSMPMX = 1024,
$ BSNM = NNMAX*( LMAX + 2 ) + 1,
$ LTHS = NMAX*( LMAX + MMAX + 1 ) + LMAX*MMAX,
$ LDU = NSMPMX, LDY = NSMPMX,
$ LXM = BSNM*LMAX + LTHS,
$ LIWORK = MAX( MMAX + LMAX, MMAX*NOBRMX + NMAX,
$ MMAX*( NMAX + LMAX ), 3 +
$ MAX( BSNM + 1, LXM + LMAX ) ) )
INTEGER L0, L1M, L2M, L3M, LDW1, LDW2, LDW3, LDW4, LDW5,
$ LDW6, LDW7, LDWORK, LW1, LW2, LW3, LW4
PARAMETER ( L0 = MAX( NMAX*( NMAX + LMAX ),
$ NMAX + MMAX + LMAX ),
$ L1M = NSMPMX*LMAX +
$ MAX( 2*NNMAX,
$ ( NMAX + LMAX )*( NMAX + MMAX ) +
$ 2*NMAX + L0 ),
$ L2M = MAX( 4*LXM + 1, BSNM +
$ MAX( 3*BSNM + 1, LTHS ),
$ NSMPMX*( LMAX - 1 ) ),
$ L3M = MAX( 4*LXM, LTHS*BSNM + 2*LXM +
$ 2*MAX( BSNM, LTHS ) ),
$ LDW1 = MAX( 2*( LMAX*NOBRMX - LMAX )*NMAX +
$ 2*NMAX,
$ ( LMAX*NOBRMX - LMAX )*NMAX +
$ NMAX*NMAX + 7*NMAX,
$ LMAX*NOBRMX*NMAX +
$ MAX( ( LMAX*NOBRMX - LMAX )*NMAX +
$ 2*NMAX + LMAX +
$ ( 2*MMAX + LMAX )*NOBRMX,
$ 2*( LMAX*NOBRMX - LMAX )*NMAX
$ + NMAX*NMAX + 8*NMAX,
$ NMAX + 4*( MMAX*NOBRMX +
$ NMAX ) + 1,
$ MMAX*NOBRMX + 3*NMAX + LMAX )
$ ),
$ LDW2 = LMAX*NOBRMX*NMAX +
$ MMAX*NOBRMX*( NMAX + LMAX )*
$ ( MMAX*( NMAX + LMAX ) + 1 ) +
$ MAX( ( NMAX + LMAX )**2,
$ 4*MMAX*( NMAX + LMAX ) + 1 ),
$ LDW3 = NSMPMX*LMAX*( NMAX + 1 ) + 2*NMAX +
$ MAX( 2*NMAX*NMAX, 4*NMAX ),
$ LDW4 = NMAX*( NMAX + 1 ) + 2*NMAX +
$ MAX( NMAX*LMAX*( NMAX + 1 ) +
$ 2*NMAX*NMAX + LMAX*NMAX, 4*NMAX ),
$ LDW5 = NSMPMX*LMAX + ( NMAX + LMAX )*
$ ( NMAX + MMAX ) + 3*NMAX + MMAX + LMAX,
$ LDW6 = NSMPMX*LMAX + ( NMAX + LMAX )*
$ ( NMAX + MMAX ) + NMAX +
$ MAX( 1, NMAX*NMAX*LMAX + NMAX*LMAX +
$ NMAX, NMAX*NMAX +
$ MAX( NMAX*NMAX +
$ NMAX*MAX( NMAX, LMAX ) +
$ 6*NMAX + MIN( NMAX, LMAX ),
$ NMAX*MMAX ) ),
$ LDW7 = NSMPMX*LMAX + ( NMAX + LMAX )*
$ ( NMAX + MMAX ) + 3*NMAX + MMAX + LMAX,
$ LW1 = MAX( 2*( MMAX + LMAX )*NOBRMX*
$ ( 2*( MMAX + LMAX )*( NOBRMX + 1 )
$ + 3 ) + LMAX*NOBRMX,
$ 4*( MMAX + LMAX )*NOBRMX*
$ ( MMAX + LMAX )*NOBRMX +
$ ( NMAX + LMAX )*( NMAX + MMAX ) +
$ MAX( LDW1, LDW2 ),
$ ( NMAX + LMAX )*( NMAX + MMAX ) +
$ NMAX + NMAX*NMAX + 2 +
$ NMAX*( NMAX + MMAX + LMAX ) +
$ MAX( 5*NMAX, 2, MIN( LDW3, LDW4 ),
$ LDW5, LDW6 ) ),
$ LW2 = NSMPMX*LMAX + BSNM +
$ MAX( 4, NSMPMX +
$ MAX( NSMPMX*BSNM +
$ MAX( 2*NNMAX, 5*BSNM + 1 ),
$ BSNM**2 + BSNM +
$ MAX( NSMPMX + 2*NNMAX,
$ 5*BSNM ) ) ),
$ LW3 = MAX( LDW7, NSMPMX*LMAX +
$ ( NMAX + LMAX )*( 2*NMAX + MMAX )+
$ 2*NMAX ),
$ LW4 = NSMPMX*LMAX + LXM +
$ MAX( 4, NSMPMX*LMAX +
$ MAX( NSMPMX*LMAX*( BSNM + LTHS ) +
$ MAX( NSMPMX*LMAX + L1M,
$ L2M + LXM ),
$ LXM*( BSNM + LTHS ) +
$ LXM +
$ MAX( NSMPMX*LMAX + L1M,
$ LXM + L3M ) ) ),
$ LDWORK = MAX( LW1, LW2, LW3, LW4 ) )
* .. Local Scalars ..
LOGICAL INIT1, INITB, INITL, INITN, INITS
CHARACTER*1 INIT
INTEGER BSN, I, INFO, INI, ITER, ITMAX1, ITMAX2, IWARN,
$ J, L, L1, L2, LPAR, LX, M, N, NN, NOBR, NPRINT,
$ NS, NSMP
DOUBLE PRECISION TOL1, TOL2
* .. Array Arguments ..
INTEGER IWORK(LIWORK)
DOUBLE PRECISION DWORK(LDWORK), U(LDU,MMAX), X(LXM), Y(LDY,LMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL IB03BD
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) NOBR, M, L, NSMP, N, NN, ITMAX1, ITMAX2,
$ NPRINT, TOL1, TOL2, INIT
INITL = LSAME( INIT, 'L' )
INITS = LSAME( INIT, 'S' )
INITB = LSAME( INIT, 'B' )
INITN = LSAME( INIT, 'N' )
INIT1 = INITL .OR. INITB
IF( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) M
ELSE
IF( L.LE.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) L
ELSE
NS = N
IF( INIT1 ) THEN
IF( NOBR.LE.0 .OR. NOBR.GT.NOBRMX ) THEN
WRITE ( NOUT, FMT = 99991 ) NOBR
STOP
ELSEIF( NSMP.LT.2*( M + L + 1 )*NOBR - 1 ) THEN
WRITE ( NOUT, FMT = 99990 ) NSMP
STOP
ELSEIF( N.EQ.0 .OR. N.GE.NOBR ) THEN
WRITE ( NOUT, FMT = 99989 ) N
STOP
END IF
IF ( N.LT.0 )
$ N = NOBR - 1
ELSE
IF( NSMP.LT.0 ) THEN
WRITE ( NOUT, FMT = 99990 ) NSMP
STOP
ELSEIF( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) N
STOP
END IF
END IF
IF( NN.LT.0 .OR. NN.GT.NNMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) NN
ELSE
BSN = NN*( L + 2 ) + 1
L1 = BSN*L
L2 = N*( L + M + 1 ) + L*M
LX = L1 + L2
INI = 1
IF ( INITL ) THEN
LPAR = L1
ELSEIF ( INITS ) THEN
INI = L1 + 1
LPAR = L2
ELSEIF ( INITN ) THEN
LPAR = LX
END IF
IF( INIT1 )
$ N = NS
* Read the input-output data, initial parameters, and seed.
READ ( NIN, FMT = * ) ( ( U(I,J), J = 1,M ), I = 1,NSMP )
READ ( NIN, FMT = * ) ( ( Y(I,J), J = 1,L ), I = 1,NSMP )
IF ( .NOT.INITB )
$ READ ( NIN, FMT = * ) ( X(I), I = INI,INI+LPAR-1 )
IF ( INITS .OR. INITB )
$ READ ( NIN, FMT = * ) ( DWORK(I), I = 1,4 )
* Solve a Wiener system identification problem.
CALL IB03BD( INIT, NOBR, M, L, NSMP, N, NN, ITMAX1,
$ ITMAX2, NPRINT, U, LDU, Y, LDY, X, LX, TOL1,
$ TOL2, IWORK, DWORK, LDWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF( IWARN.NE.0 ) WRITE ( NOUT, FMT = 99987 ) IWARN
ITER = DWORK(3)
WRITE ( NOUT, FMT = 99997 ) DWORK(2)
WRITE ( NOUT, FMT = 99996 ) ITER, IWORK(1), IWORK(2)
* Recompute LX is necessary.
IF ( INIT1 .AND. NS.LT.0 )
$ LX = L1 + N*( L + M + 1 ) + L*M
WRITE ( NOUT, FMT = 99994 )
WRITE ( NOUT, FMT = 99995 ) ( X(I), I = 1, LX )
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' IB03BD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from IB03BD = ',I4)
99997 FORMAT (/' Final 2-norm of the residuals = ',D15.7)
99996 FORMAT (/' Number of iterations = ', I7,
$ /' Number of function evaluations = ', I7,
$ /' Number of Jacobian evaluations = ', I7)
99995 FORMAT (10(1X,F9.4))
99994 FORMAT (/' Final approximate solution is ' )
99993 FORMAT (/' M is out of range.',/' M = ',I5)
99992 FORMAT (/' L is out of range.',/' L = ',I5)
99991 FORMAT (/' NOBR is out of range.',/' NOBR = ',I5)
99990 FORMAT (/' NSMP is out of range.',/' NSMP = ',I5)
99989 FORMAT (/' N is out of range.',/' N = ',I5)
99988 FORMAT (/' NN is out of range.',/' NN = ',I5)
99987 FORMAT (' IWARN on exit from IB03BD = ',I4)
END
Program Data
IB03BD EXAMPLE PROGRAM DATA 10 1 1 1024 4 12 500 1000 0 .00001 .00001 B 2.2183165e-01 3.9027807e-02 -5.0295887e-02 8.5386224e-03 7.2431159e-02 -1.7082198e-03 -1.7176287e-01 -2.6198104e-01 -1.7194108e-01 1.8566868e-02 1.5625362e-01 1.7463811e-01 1.1564450e-01 2.8779248e-02 -8.4265993e-02 -2.0978501e-01 -2.6591828e-01 -1.7268680e-01 2.1525013e-02 1.4363602e-01 7.3101431e-02 -1.0259212e-01 -1.6380473e-01 -1.0021167e-02 2.0263451e-01 2.1983417e-01 -2.1636523e-02 -3.0986057e-01 -3.8521982e-01 -2.1785179e-01 -1.4761096e-02 3.7005180e-02 -2.8119028e-02 -4.2167901e-02 5.2117694e-02 1.2023747e-01 1.8863385e-02 -1.9506434e-01 -3.0192175e-01 -1.7000747e-01 8.0740471e-02 2.0188076e-01 8.5108288e-02 -1.3270970e-01 -2.3646822e-01 -1.6505385e-01 -4.7448014e-02 -2.7886815e-02 -1.0152026e-01 -1.4155374e-01 -6.1650823e-02 8.3519614e-02 1.5926650e-01 8.6142760e-02 -9.4385381e-02 -2.6609066e-01 -3.2883874e-01 -2.5908050e-01 -1.1648940e-01 -3.0653766e-03 1.0326675e-02 -5.3445909e-02 -9.2412724e-02 -3.0279541e-02 8.4846832e-02 1.1133075e-01 -3.2135250e-02 -2.5308181e-01 -3.5670882e-01 -2.4458860e-01 -2.5254261e-02 9.3714332e-02 1.8643667e-02 -1.4592119e-01 -2.2730880e-01 -1.7140060e-01 -7.4131665e-02 -3.9669515e-02 -5.1266129e-02 -1.1752833e-02 1.0785565e-01 2.0665525e-01 1.6117322e-01 -2.6938653e-02 -2.1941152e-01 -2.7753567e-01 -1.8805912e-01 -4.6845025e-02 5.8585698e-02 1.2218407e-01 1.7838638e-01 2.2169815e-01 1.9825589e-01 8.0215288e-02 -7.2135308e-02 -1.4381520e-01 -6.8724371e-02 1.0191205e-01 2.3766633e-01 2.3876101e-01 1.1678077e-01 -2.0428168e-02 -5.8973233e-02 3.1326900e-02 1.7391495e-01 2.4558570e-01 1.7650262e-01 1.2444292e-02 -1.1538234e-01 -9.5917970e-02 6.4762165e-02 2.4258524e-01 3.0102251e-01 2.1222960e-01 7.8706189e-02 3.1500466e-02 1.0297577e-01 1.9875173e-01 1.9434906e-01 5.8146667e-02 -1.1941921e-01 -2.1038478e-01 -1.5594967e-01 1.8552198e-03 1.6878529e-01 2.5937416e-01 2.2516346e-01 6.6144472e-02 -1.5623019e-01 -3.3161105e-01 -3.6695732e-01 -2.6565333e-01 -1.3254832e-01 -8.0101064e-02 -1.2531889e-01 -1.8843171e-01 -1.9038956e-01 -1.3230055e-01 -7.0889306e-02 -3.9679280e-02 -2.6286077e-02 -2.3630770e-02 -6.0652834e-02 -1.4929250e-01 -2.2155095e-01 -1.7331044e-01 5.2693564e-03 1.7683919e-01 1.8244690e-01 2.5118458e-02 -1.1051051e-01 -5.1764984e-02 1.6342054e-01 3.1563281e-01 2.3808751e-01 -4.4871135e-03 -1.8778679e-01 -1.6017584e-01 2.3481991e-02 1.9209185e-01 2.4281065e-01 2.1224192e-01 1.8825017e-01 1.9811718e-01 2.0202486e-01 1.6812825e-01 1.1444796e-01 7.2452475e-02 4.0090973e-02 -6.7139529e-03 -6.8721730e-02 -1.1460099e-01 -1.1914168e-01 -8.9852521e-02 -4.5942222e-02 1.0932686e-02 8.1900393e-02 1.3092374e-01 9.0790221e-02 -6.3538148e-02 -2.5119963e-01 -3.2585173e-01 -2.0850925e-01 1.7922009e-02 1.6783753e-01 1.2518317e-01 -4.3517162e-02 -1.5783138e-01 -1.0686847e-01 4.4782565e-02 1.3893172e-01 9.8691579e-02 2.6311282e-03 -1.6073049e-02 7.8512306e-02 1.9453537e-01 2.2504627e-01 1.6121235e-01 7.8124056e-02 2.9774586e-02 -5.3899280e-03 -6.5745322e-02 -1.2329059e-01 -9.5096521e-02 5.5471394e-02 2.5017082e-01 3.4773286e-01 2.6656242e-01 5.3705965e-02 -1.6135006e-01 -2.7310977e-01 -2.6814818e-01 -2.1074926e-01 -1.7743213e-01 -1.9796482e-01 -2.4059041e-01 -2.4663820e-01 -1.8780129e-01 -9.8317382e-02 -4.7848155e-02 -7.3425069e-02 -1.3529842e-01 -1.4739094e-01 -6.2482366e-02 6.8729554e-02 1.3251322e-01 6.1482940e-02 -8.5065014e-02 -1.6074078e-01 -6.7974104e-02 1.3976672e-01 2.9838081e-01 2.8233998e-01 1.1391411e-01 -7.1966946e-02 -1.5876983e-01 -1.3805556e-01 -8.2998592e-02 -5.7864811e-02 -6.5300733e-02 -7.0590592e-02 -5.5847027e-02 -4.1219301e-02 -6.1578267e-02 -1.3176243e-01 -2.2968907e-01 -3.0193311e-01 -2.8770451e-01 -1.5729276e-01 5.4414593e-02 2.5362617e-01 3.4482230e-01 3.0119122e-01 1.8534835e-01 9.6712488e-02 9.3385279e-02 1.6057572e-01 2.4424680e-01 3.0164891e-01 3.1693510e-01 2.8441517e-01 1.9948758e-01 7.3600888e-02 -5.4291337e-02 -1.3721320e-01 -1.5626045e-01 -1.3464149e-01 -1.1510541e-01 -1.2587072e-01 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6.9056099e-02 -5.5814690e-02 -1.6001831e-01 -1.8912751e-01 -1.2778372e-01 -4.4698128e-03 1.2208903e-01 1.8963074e-01 1.6384408e-01 6.0799128e-02 -5.7339158e-02 -1.1860919e-01 -9.0086196e-02 -4.5798607e-03 6.0280807e-02 4.1676388e-02 -5.5180320e-02 -1.5518201e-01 -1.6828578e-01 -6.2049884e-02 1.0561621e-01 2.2337555e-01 2.0643187e-01 5.9839911e-02 -1.2043322e-01 -2.1083864e-01 -1.4415945e-01 4.3538937e-02 2.3203364e-01 2.9044234e-01 1.6171416e-01 -9.5674666e-02 -3.3749265e-01 -4.1795872e-01 -2.7746809e-01 2.0648626e-02 3.2603206e-01 4.8410918e-01 4.1672303e-01 1.5905611e-01 -1.6318595e-01 -3.9931562e-01 -4.4568803e-01 -2.9169291e-01 -2.0960934e-02 2.3175866e-01 3.4693819e-01 2.7877641e-01 7.7125945e-02 -1.4069530e-01 -2.5367798e-01 -2.0150506e-01 -1.6778161e-02 1.9116819e-01 2.9409556e-01 2.1593628e-01 -1.9610708e-02 -2.9401135e-01 -4.5512990e-01 -4.0311941e-01 -1.5075705e-01 1.7921653e-01 4.2153577e-01 4.6143206e-01 2.9688389e-01 3.5275834e-02 -1.7206796e-01 -2.2040717e-01 -1.1280250e-01 4.6014479e-02 1.2005000e-01 3.5297082e-02 -1.6459920e-01 -3.4121448e-01 -3.5130088e-01 -1.4787707e-01 1.7615712e-01 4.3972643e-01 4.8949447e-01 2.9899548e-01 -1.6059656e-02 -2.7414987e-01 -3.4124596e-01 -2.0476598e-01 3.1287353e-02 2.1535118e-01 2.3693813e-01 8.7039128e-02 -1.3914592e-01 -2.9731202e-01 -2.8057123e-01 -8.9244625e-02 1.6445576e-01 3.2621002e-01 2.9949560e-01 1.0678193e-01 -1.3016725e-01 -2.7225661e-01 -2.4687907e-01 -8.3173776e-02 1.1381888e-01 2.2819642e-01 1.9830143e-01 4.8505476e-02 -1.2763594e-01 -2.2560309e-01 -1.9560311e-01 -7.1212054e-02 6.0380807e-02 1.2445307e-01 1.0835168e-01 5.5609724e-02 1.7269294e-02 9.3997346e-03 1.1223045e-02 -4.3543819e-03 -4.2668837e-02 -8.5657964e-02 -1.0909342e-01 -9.7154374e-02 -4.6781850e-02 3.1101930e-02 1.0973840e-01 1.5122945e-01 1.2531404e-01 3.3620966e-02 -8.3194568e-02 -1.6716420e-01 1998. 1999. 2000. 2001.Program Results
IB03BD EXAMPLE PROGRAM RESULTS
IWARN on exit from IB03BD = 12
Final 2-norm of the residuals = 0.2995840D+00
Number of iterations = 42
Number of function evaluations = 898
Number of Jacobian evaluations = 295
Final approximate solution is
14.1294 1.1232 6.4322 -11.2418 7.6380 -33.4730 -64.7203 747.1515 -0.4623 -92.6092
6.1682 -0.7672 0.1194 0.3558 0.9091 0.2948 1.3465 0.0093 0.0560 -0.0035
-0.4179 -0.0455 -2.0871 -0.9196 1.0777 0.9213 0.5373 1.0412 -0.3978 7.6832
-6.8614 -31.6119 -0.1092 -9.8984 0.1257 0.4056 0.0472 7.5819 -13.3969 2.4869
-66.0727 -0.8411 -0.7040 1.9641 1.3059 -0.2046 -0.9326 0.0040 0.4032 0.1479
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/KINSOL.html 0000664 0000000 0000000 00000071276 14560147231 0020466 0 ustar 00root root 0000000 0000000
Purpose
To solve a nonlinear system of equations F(u)=0, where F(u) is
n n
a nonlinear function from R to R , using Krylov Inexact Newton
techniques.
Specification
SUBROUTINE KINSOL (GSTRAT, LINSU, NEQ, OPTIN, MAXL, MAXLRST,
& MSBPRE, UU, USCALE, FSCALE, CONSTR,
& IOPT, ROPT, TOL1, TOL2, INFO)
C .. Scalar Arguments ..
LOGICAL OPTIN
INTEGER INFO, GSTRAT, MAXL, MAXLRST, MSBPRE, NEQ
DOUBLE PRECISION TOL1, TOL2
C .. Array Arguments ..
INTEGER IOPT(40)
DOUBLE PRECISION CONSTR(NEQ), FSCALE(NEQ), ROPT(40), USCALE(NEQ),
$ UU(NEQ)
Arguments
Input/Output Parameters
GSTRAT (input) INTEGER
Indicates the global strategy to apply the computed
increment delta in the solution UU. Choices are:
0 - Inexact Newton.
1 - Linesearch.
LINSU SUBROUTINE
Linear Solver Set-up Routine. This is the KINSOL routine
to be called to set-up the linear solver. The user should
specify here one of the 6 different Fortran-callable
routines provided by KINSOL for this purpose. The choice
to be used depends on which of the optional user-defined
routines are provided by the user (see User-defined
routines below).
LINSU can be one of the following routines: FKINSPGMR00,
FKINSPGMR01, FKINSPGMR10, FKINSPGMR11, FKINSPGMR20, and
FKINSPGMR21, where the first digit in the name of the
function is: 0 if neither KPSOL nor KPRECO routines are
provided; 1 if only the preconditioner solve routine
(KPSOL) is provided; and 2 if both the preconditioner
solve (KPSOL) and setup (KPRECO) routines are provided.
The second digit is: 0 if a function FATIMES is not
provided; and 1 if a function FATIMES is provided.
NEQ (input) INTEGER
Number of equations (and unknowns) in the algebraic
system.
OPTIN (input) LOGICAL
Flag indicating whether optional inputs from the user in
the arrays IOPT and ROPT are to be used.
Pass FALSE to ignore all optional inputs and TRUE to use
all optional inputs that are present. Either choice does
NOT affect outputs in other positions of IOPT or ROPT.
MAXL (input) INTEGER
Maximum Krylov dimension for the Linear Solver. Pass 0
to use the default value MIN(Neq, 10).
MAXLRST (input) INTEGER
Maximum number of linear solver restarts allowed. Values
outside the range 0 to 2*NEQ/MAXL will be restricted to
that range. 0, meaning no restarts, is a safe starting
value.
MSBPRE (input) INTEGER
Maximum number of steps calling the solver KPSOL
without calling the preconditioner KPRECO. (The default is
10).
UU (input/output) DOUBLE PRECISION array, dimension (NEQ)
On entry, UU is the initial guess.
On exit, if no errors ocurr, UU is the solution of
the system KFUN(UU) = 0.
USCALE (input) DOUBLE PRECISION array, dimension (NEQ)
Array of diagonal elements of the scaling matrix for UU.
The elements of USCALE must be positive values. The
scaling matrix USCALE should be chosen so that
USCALE * UU (as a matrix multiplication) should have all
its components with roughly the same magnitude when UU is
close to a root of KFUN.
FSCALE (input) DOUBLE PRECISION array, dimension (NEQ)
Array of diagonal elements of the scaling matrix for
KFUN. The elements of FSCALE must be positive values.
The scaling matrix FSCALE should be chosen so that
FSCALE * KFUN(UU) (as a matrix multiplication) should
have all its components with roughly the same magnitude
when UU is NOT too near a root of KFUN.
CONSTR (input) DOUBLE PRECISION array, dimension (NEQ)
Constraints on UU.
A positive value in CONSTR(I) implies that the Ith
component of UU is to be constrained > 0.
A negative value in CONSTR(I) implies that the Ith
component of UU is to be constrained < 0.
A zero value in CONSTR(I) implies there is no constraint
on UU(I).
IOPT (input/output) INTEGER array, dimension (40)
Array of optional integer inputs and outputs.
If OPTIN is TRUE, the user should preset to 0 those
locations for which default values are to be used.
See Optional Inputs and Outputs, below.
ROPT (input/output) DOUBLE PRECISION array, dimension (40)
Array of optional double precision inputs and outputs.
If OPTIN is TRUE, the user should preset to 0 those
locations for which default values are to be used.
See Optional Inputs and Outputs, below.
Tolerances
TOL1 DOUBLE PRECISION
Stopping tolerance on maxnorm( FSCALE * KFUN(UU) ).
If TOL1 is input as 0., then a default value of
(uround) to the 1/3 power will be used. uround is the
unit roundoff for the machine in use for the calculation.
TOL2 DOUBLE PRECISION
Stopping tolerance on the maximum scaled step
UU(K) - UU(K-1).
If TOL2 is input as 0., then a default value of (uround)
to the 2/3 power will be used. uround is the unit
roundoff for the machine in use for the calculation.
Error Indicator
INFO (output) INTEGER
See Termination Codes below.
---------------------------------------------------------------
Termination Codes
(Note: in this documentation we use named constants for
certain integer constant values. To see the values of these
symbols see Named constants below.)
The termination values KINS_***** are now given. These are the
values of the INFO argument.
SUCCESS : means maxnorm(FSCALE*KFUN(UU) <= TOL1, where
maxnorm() is the maximum norm function N_VMaxNorm.
Therefore, UU is probably an approximate root of
KFUN.
INITIAL_GUESS_OK: means the initial guess UU has been found
to already satisfy the system to the desired
accuracy. No calculation was performed other
than testing UU.
STEP_LT_STPTOL: means the scaled distance between the last
two steps is less than TOL2. UU may be an
approximate root of KFUN, but it is also possible
that the algorithm is making very slow progress
and is not near a root or that TOL2 is too
large
LNSRCH_NONCONV: means the LineSearch module failed to reduce
norm(KFUN) sufficiently on the last global step.
Either UU is close to a root of F and no more
accuracy is possible, or the finite-difference
approximation to J*v is inaccurate, or TOL2
is too large. Check the outputs NCFL and NNI: if
NCFL is close to NNI, it may be the case that the
Krylov iteration is converging very slowly. In
this case, the user may want to use precondition-
ing and/or increase the MAXL argument (that is,
increase the max dimension of the Krylov subspace)
by setting MAXL to nonzero (thus not using the
default value of KINSPGMR_MAXL) or if MAXL is being
set, increase its value.
MAXITER_REACHED: means that the maximum allowable number of
nonlinear iterations has been reached. This is by
default 200, but may be changed through optional
input IOPT(MXITER).
MXNEWT_5X_EXCEEDED: means 5 consecutive steps of length mxnewt
(maximum Newton stepsize limit) have been taken.
Either norm(F) asymptotes from above to a finite
value in some direction, or mxnewt is too small.
Mxnewt is computed internally (by default) as
mxnewt = 1000*max(norm(USCALE*UU0),1), where
UU0 is the initial guess for UU, and norm() is
the Euclidean norm. Mxnewt can be set by the
user through optional input ROPT(MXNEWTSTEP).
LINESEARCH_BCFAIL: means that more than the allowed maximum
number of failures (MXNBCF) occurred when trying
to satisfy the beta condition in the linesearch
algorithm. It is likely that the iteration is
making poor progress.
KRYLOV_FAILURE: means there was a failure of the Krylov
iteration process to converge.
PRECONDSET_FAILURE: means there was a nonrecoverable
error in PrecondSet causing the iteration to halt.
PRECONDSOLVE_FAILURE: means there was a nonrecoverable
error in PrecondSolve causing the iteration to halt.
NO_MEM: the KINSol memory pointer received was NULL.
INPUT_ERROR: one or more input parameters or arrays was in
error. See the program output for further info.
LSOLV_NO_MEM: The linear solver memory pointer (lmem) was
received as NULL. The return value from the linear
solver needs to be checked and the cause found.
---------------------------------------------------------------
Optional inputs and outputs
(Note: in this documentation we use named constants for
certain integer constant values. To see the values of these
symbols see Named constants below.)
The user should declare two arrays for optional input and
output, an IOPT array for optional integer input and output
and an ROPT array for optional real input and output. These
arrays should both be of size OPT_SIZE.
So the user's declaration should look like:
INTEGER IOPT(OPT_SIZE)
DOUBLE PRECISION ROPT(OPT_SIZE)
The following definitions are indices into the IOPT and ROPT
arrays. A brief description of the contents of these positions
follows.
IOPT(PRINTFL) (input) Allows user to select from 4 levels
of output.
=0 no statistics printed (DEFAULT)
=1 output the nonlinear iteration count, the
scaled norm of KFUN(UU), and number of
KFUN calls.
=2 same as 1 with the addition of global
strategy statistics:
f1 = 0.5*norm(FSCALE*KFUN(UU))**2 and
f1new = 0.5*norm(FSCALE*KFUN(unew))**2 .
=3 same as 2 with the addition of further
Krylov iteration statistics.
IOPT(MXITER) (input) Maximum allowable number of nonlinear
iterations. The default is MXITER_DEFAULT.
IOPT(PRECOND_NO_INIT) (input) Set to 1 to prevent the initial
call to the routine KPRECO upon a given
call to KINSol. Set to 0 or leave unset to
force the initial call to KPRECO.
Use the choice of 1 only after beginning the
first of a series of calls with a 0 value.
If a value other than 0 or 1 is encountered,
the default, 0, is set in this element of
IOPT and thus the routine KPRECO will
be called upon every call to KINSol, unless
IOPT(PRECOND_NO_INIT) is changed by the user.
IOPT(ETACHOICE) (input) A flag indicating which of three
methods to use for computing eta, the
coefficient in the linear solver
convergence tolerance eps, given by
eps = (eta+u_round)*norm(KFUN(UU)).
Here, all norms are the scaled L2 norm.
The linear solver attempts to produce a step
p such that norm(KFUN(UU)+J(UU)*p) <= eps.
Two of the methods for computing eta
calculate a value based on the convergence
process in the routine KINForcingTerm.
The third method does not require
calculation; a constant eta is selected.
The default if IOPT(ETACHOICE) is not
specified is ETACHOICE1, (see below).
The allowed values (methods) are:
ETACONSTANT constant eta, default of 0.1 or user
supplied choice, for which see ROPT(ETACONST),
ETACHOICE1 (default) which uses choice 1 of
Eisenstat and Walker's paper of SIAM J. Sci.
Comput.,17 (1996), pp 16-32 wherein eta is:
eta(k) =
ABS( norm(KFUN(UU(k))) - norm(KFUN(UU(k-1))+J(UU(k-1))*p) )
/ norm(KFUN(UU(k-1))),
ETACHOICE2 which uses choice 2 of
Eisenstat and Walker wherein eta is:
eta(k) = egamma *
( norm(KFUN(UU(k))) / norm(KFUN(u(k-1))) )^ealpha
egamma and ealpha for choice 2, both required,
are from either defaults (egamma = 0.9 ,
ealpha = 2) or from user input,
see ROPT(ETAALPHA) and ROPT(ETAGAMMA), below.
For eta(k) determined by either Choice 1 or
Choice 2, a value eta_safe is determined, and
the safeguard eta(k) <- max(eta_safe,eta(k))
is applied to prevent eta(k) from becoming too
small too quickly.
For Choice 1,
eta_safe = eta(k-1)^((1.+sqrt(5.))/2.)
and for Choice 2,
eta_safe = egamma*eta(k-1)^ealpha.
(These safeguards are turned off if they drop
below 0.1 . Also, eta is never allowed to be
less than eta_min = 1.e-4).
IOPT(NO_MIN_EPS) (input) Set to 1 or greater to remove
protection agains eps becoming too small.
This option is useful for debugging linear
and nonlinear solver interactions. Set to 0
for standard eps minimum value testing.
IOPT(NNI) (output) Total number of nonlinear iterations.
IOPT(NFE) (output) Total number of calls to the user-
supplied system function KFUN.
IOPT(NBCF) (output) Total number of times the beta
condition could not be met in the linesearch
algorithm. The nonlinear iteration is halted
if this value ever exceeds MXNBCF (10).
IOPT(NBKTRK) (output) Total number of backtracks in the
linesearch algorithm.
IOPT(SPGMR_NLI) (output) Number of linear iterations.
IOPT(SPGMR_NPE) (output) Number of preconditioner evaluations.
IOPT(SPGMR_NPS) (output) Number of calls made to user's psolve
function.
IOPT(SPGMR_NCFL) (output) Number of linear convergence failures.
ROPT(MXNEWTSTEP) (input) Maximum allowable length of a Newton
step. The default value is calculated from
1000*max(norm(USCALE*UU(0),norm(USCALE)).
ROPT(RELFUNC) (input) Relative error in computing KFUN(UU)
if known. Default is the machine epsilon.
ROPT(RELU) (input) A scalar constraint which restricts
the update of UU to del(UU)/UU < ROPT(RELU)
The default is no constraint on the relative
step in UU.
ROPT(ETAGAMMA) (input) The coefficient egamma in the eta
computation. See routine KINForcingTerm
(SEE IOPT(ETACHOICE) above for additional info).
ROPT(ETAALPHA) (input) The coefficient ealpha in the eta
computation. See routine KINForcingTerm
(SEE IOPT(ETACHOICE) above for additional info).
ROPT(ETACONST) (input) A user specified constant value for
eta, used in lieu of that computed by
routine KINForcingTerm
(SEE IOPT(ETACHOICE) above for additional info).
ROPT(FNORM) (output) The scaled norm at a given iteration:
norm(FSCALE(KFUN(UU)).
ROPT(STEPL) (output) Last step length in the global
strategy routine:
KINLineSearch or KINInexactNewton.
---------------------------------------------------------------
User-defined routines
In order to use this routine, some user-defined routines have to
be provided. One of them is required, while the others are
optional. These routines are described next.
KFUN Required
SUBROUTINE KFUN (NEQ, UU, FVAL)
INTEGER NEQ
DOUBLE PRECISION UU(NEQ), FVAL(NEQ)
PURPOSE
Evaluates the KFUN function which defines the system
to be solved:
KFUN(UU)=0
ARGUMENTS
NEQ
(input) INTEGER
Number of equations (and unknowns) in the algebraic
system
UU
(input) DOUBLE PRECISION array, dimension (NEQ)
independent variable vector
FVAL
(output) DOUBLE PRECISION array, dimension (NEQ)
Result of KFUN(UU)
KPRECO Optional
SUBROUTINE KPRECO (NEQ, UU, USCALE, FVAL, FSCALE,
VTEMP1, VTEMP2, UROUND, NFE, IER)
INTEGER NEQ, NFE, IER
DOUBLE PRECISION UROUND
DOUBLE PRECISION UU(NEQ), USCALE(NEQ), FVAL(NEQ),
FSCALE(NEQ), VTEMP1(NEQ), VTEMP2(NEQ)
PURPOSE
The user-supplied preconditioner setup function KPRECO and
the user-supplied preconditioner solve function KPSOL
together must define the right preconditoner matrix P
chosen so as to provide an easier system for the Krylov
solver to solve. KPRECO is called to provide any matrix
data required by the subsequent call(s) to KPSOL. The
data is expected to be stored in variables within a
COMMON block and the definition of those variables is up
to the user. More specifically, the user-supplied
preconditioner setup function KPRECO is to evaluate and
preprocess any Jacobian-related data needed by the
preconditioner solve function KPSOL. This might include
forming a crude approximate Jacobian, and performing an
LU factorization on the resulting approximation to J.
This function will not be called in advance of every call
to KPSOL, but instead will be called only as often as
necessary to achieve convergence within the Newton
iteration in KINSol. If the KPSOL function needs no
preparation, the KPRECO function need not be provided.
KPRECO should not modify the contents of the arrays
UU or FVAL as those arrays are used elsewhere in the
iteration process.
Each call to the KPRECO function is preceded by a call to
the system function KFUN. Thus the KPRECO function can use
any auxiliary data that is computed by the KFUN function
and saved in a way accessible to KPRECO.
The two scaling arrays, FSCALE and USCALE, and unit
roundoff UROUND are provided to the KPRECO function for
possible use in approximating Jacobian data, e.g. by
difference quotients. These arrays should also not be
altered
ARGUMENTS
NEQ
(input) INTEGER
Number of equations (and unknowns) in the algebraic
system.
UU
(input) DOUBLE PRECISION array, dimension (NEQ)
Independent variable vector.
USCALE
(input) DOUBLE PRECISION array, dimension (NEQ)
See USCALE above.
FVAL
(input) DOUBLE PRECISION array, dimension (NEQ)
Current value of KFUN(UU).
FSCALE
(input) DOUBLE PRECISION array, dimension (NEQ)
See FSCALE above.
VTEMP1
DOUBLE PRECISION array, dimension (NEQ)
Temporary work array.
VTEMP2
DOUBLE PRECISION array, dimension (NEQ)
Temporary work array.
UROUND
(input) DOUBLE PRECISION
Machine unit roundoff.
NFE
(input/output) INTEGER
Number of calls to KFUN made by the package. The KPRECO
routine should update this counter by adding on the
number of KFUN calls made in order to approximate the
Jacobian, if any. For example, if the routine calls
KFUN a total of W times, then the update is
NFE = NFE + W.
IER
(output) INTEGER
Error indicator.
0 if successful,
1 if failure, in which case KINSOL stops.
KPSOL Optional
SUBROUTINE KPSOL (NEQ, UU, USCALE, FVAL, FSCALE, VTEM,
FTEM, UROUND, NFE, IER)
INTEGER NEQ, NFE, IER
DOUBLE PRECISION UU(NEQ), USCALE(NEQ), FVAL(NEQ),
FSCALE(NEQ), VTEM(NEQ), FTEM(NEQ)
PURPOSE
The user-supplied preconditioner solve function KPSOL
is to solve a linear system P x = r in which the matrix
P is the (right) preconditioner matrix P.
KPSOL should not modify the contents of the iterate
array UU or the current function value array FVAL as
those are used elsewhere in the iteration process.
ARGUMENTS
NEQ
(input) INTEGER
Number of equations (and unknowns) in the algebraic
system.
UU
(input) DOUBLE PRECISION array, dimension (NEQ)
Independent variable vector.
USCALE
(input) DOUBLE PRECISION array, dimension (NEQ)
See USCALE above.
FVAL
(input) DOUBLE PRECISION array, dimension (NEQ)
Current value of KFUN(UU).
FSCALE
(input) DOUBLE PRECISION array, dimension (NEQ)
See FSCALE above.
VTEM
(input/output) DOUBLE PRECISION array, dimension (NEQ)
On entry, holds the RHS vector r.
On exit, holds the result x.
FTEM
DOUBLE PRECISION array, dimension (NEQ)
Temporary work array.
UROUND
(input) DOUBLE PRECISION
Machine unit roundoff.
NFE
(input/output) INTEGER
Number of calls to KFUN made by the package. The KPRECO
routine should update this counter by adding on the
number of KFUN calls made in order to carry out the
solution, if any. For example, if the routine calls
KFUN a total of W times, then the update is
NFE = NFE + W.
IER
(output) INTEGER
Error indicator.
0 if successful,
1 if failure, in which case KINSOL stops.
FATIMES Optional
SUBROUTINE FATIMES(V, Z, NEWU, UU, IER)
INTEGER NEWU, IER
DOUBLE PRECISION V(:), Z(:), UU(:)
PURPOSE
The user-supplied A times V routine (optional) where
A is the Jacobian matrix dF/du, or an approximation to
it, and V is a given vector. This routine computes the
product Z = J V.
ARGUMENTS
V
(input) DOUBLE PRECISION array, dimension (NEQ)
Vector to be multiplied by J
(preconditioned and unscaled as received).
Z
(output) DOUBLE PRECISION array, dimension (NEQ)
Vector resulting from the application of J to V.
NEW_UU
(input) INTEGER
Flag indicating whether or not the UU vector has been
changed since the last call to this function (0 means
FALSE, 1 TRUE).
If this function computes and saves Jacobian data, then
this computation can be skipped if NEW_UU = FALSE.
UU
(input) DOUBLE PRECISION array, dimension (NEQ)
Current iterate u.
IER
(output) INTEGER
Error indicator.
0 if successful,
1 if failure, in which case KINSOL stops.
---------------------------------------------------------------
Named constants
Here we specify the value of the named integer constants used in this documentation. We use Fortran code for the specification, so that the user can copy and paste these lines in order to use the named constants in his/her programs. KINSOL return values Note that the value of these constants differ from those of the KINSOL package. This is due to the adaptation to the SLICOT standards. INTEGER KINS_NO_MEM, KINS_INPUT_ERROR, KINS_LSOLV_NO_MEM, & KINS_SUCCESS, KINS_INITIAL_GUESS_OK,KINS_STEP_LT_STPTOL, & KINS_LNSRCH_NONCONV, KINS_MAXITER_REACHED, & KINS_MXNEWT_5X_EXCEEDED, KINS_LINESEARCH_BCFAIL, & KINS_KRYLOV_FAILURE, KINS_PRECONDSET_FAILURE, & KINS_PRECONDSOLVE_FAILURE} PARAMETER(KINS_NO_MEM=101) PARAMETER(KINS_INPUT_ERROR=102) PARAMETER(KINS_LSOLV_NO_MEM=103) PARAMETER(KINS_SUCCESS=0) PARAMETER(KINS_INITIAL_GUESS_OK=2) PARAMETER(KINS_STEP_LT_STPTOL=3) PARAMETER(KINS_LNSRCH_NONCONV=4) PARAMETER(KINS_MAXITER_REACHED=5) PARAMETER(KINS_MXNEWT_5X_EXCEEDED=6) PARAMETER(KINS_LINESEARCH_BCFAIL=7) PARAMETER(KINS_KRYLOV_FAILURE = 8) PARAMETER(KINS_PRECONDSET_FAILURE=9) PARAMETER(KINS_PRECONDSOLVE_FAILURE=10) Size of IOPT, ROPT INTEGER OPT_SIZE PARAMETER(OPT_SIZE=40) IOPT indices INTEGER PRINTFL, MXITER, PRECOND_NO_INIT, NNI ,NFE ,NBCF, NBKTRK, & ETACHOICE, NO_MIN_EPS INTEGER SPGMR_NLI, SPGMR_NPE, SPGMR_NPS, SPGMR_NCFL PARAMETER(PRINTFL=1) PARAMETER(MXITER=2) PARAMETER(PRECOND_NO_INIT=3) PARAMETER(NNI=4) PARAMETER(NFE=5) PARAMETER(NBCF=6) PARAMETER(NBKTRK=7) PARAMETER(ETACHOICE=8) PARAMETER(NO_MIN_EPS=9) PARAMETER(SPGMR_NLI=11) PARAMETER(SPGMR_NPE=12) PARAMETER(SPGMR_NPS=13) PARAMETER(SPGMR_NCFL=14) ROPT indices INTEGER MXNEWTSTEP , RELFUNC , RELU , FNORM , STEPL, & ETACONST, ETAGAMMA, ETAALPHA PARAMETER(MXNEWTSTEP=1) PARAMETER(RELFUNC=2) PARAMETER(RELU=3) PARAMETER(FNORM=4) PARAMETER(STEPL=5) PARAMETER(ETACONST=6) PARAMETER(ETAGAMMA=7) PARAMETER(ETAALPHA=8) Values for IOPT(ETACHOICE) INTEGER ETACHOICE1, ETACHOICE2, ETACONSTANT PARAMETER(ETACHOICE1=0) PARAMETER(ETACHOICE2=1) PARAMETER(ETACONSTANT=2) ---------------------------------------------------------------Method
KINSOL (Krylov Inexact Newton SOLver) is a general purpose
solver for nonlinear systems of equations. Its most notable
feature is that it uses Krylov Inexact Newton techniques in the
system's approximate solution.
The Newton method used results in the solution of linear systems
of the form
J(u)*x = b
where J(u) is the Jacobian of F at u. The solution of these
systems by a Krylov method requires products of the form J(u)*v,
which are approximated by a difference quotient of the form
F(u+sigma*v)-F(u)
-----------------
sigma
Thus, the Jacobian need not be formed explicitly.
References
[1] Allan G. Taylor and Alan C. Hindmarsh, "User Documentation
for KINSOL, a Nonlinear Solver for Sequential and Parallel
Computers", Center for Applied Scientific Computing, L-561,
LLNL, Livermore, CA 94551.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MA01AD.html 0000664 0000000 0000000 00000004033 14560147231 0020315 0 ustar 00root root 0000000 0000000
Purpose
To compute the complex square root YR + i*YI of a complex number XR + i*XI in real arithmetic. The returned result is so that YR >= 0.0 and SIGN(YI) = SIGN(XI).Specification
SUBROUTINE MA01AD( XR, XI, YR, YI )
C .. Scalar Arguments ..
DOUBLE PRECISION XR, XI, YR, YI
Arguments
Input/Output Parameters
XR (input) DOUBLE PRECISION
XI (input) DOUBLE PRECISION
These scalars define the real and imaginary part of the
complex number of which the square root is sought.
YR (output) DOUBLE PRECISION
YI (output) DOUBLE PRECISION
These scalars define the real and imaginary part of the
complex square root.
Method
The complex square root YR + i*YI of the complex number XR + i*XI is computed in real arithmetic, taking care to avoid overflow.References
Adapted from EISPACK subroutine CSROOT.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the general product of K real scalars without over- or underflow.Specification
SUBROUTINE MA01BD( BASE, LGBAS, K, S, A, INCA, ALPHA, BETA, SCAL )
C .. Scalar Arguments ..
INTEGER INCA, K, SCAL
DOUBLE PRECISION ALPHA, BASE, BETA, LGBAS
C .. Array Arguments ..
INTEGER S(*)
DOUBLE PRECISION A(*)
Arguments
Input/Output Parameters
BASE (input) DOUBLE PRECISION
Machine base.
LGBAS (input) DOUBLE PRECISION
Logarithm of BASE.
K (input) INTEGER
The number of scalars. K >= 1.
S (input) INTEGER array, dimension (K)
The signature array. Each entry of S must be 1 or -1.
A (input) DOUBLE PRECISION array, dimension (K)
Vector of real scalars.
INCA (input) INTEGER
Increment for the array A. INCA <> 0.
ALPHA (output) DOUBLE PRECISION
ALPHA is a real scalar such that
ALPHA / BETA * BASE**(SCAL)
is the general product of the scalars in the array A.
BETA (output) DOUBLE PRECISION
BETA is either 0.0 or 1.0.
See also the description of ALPHA.
SCAL (output) INTEGER
Scaling factor exponent, see ALPHA.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the general product of K complex scalars trying to avoid over- and underflow.Specification
SUBROUTINE MA01BZ( BASE, K, S, A, INCA, ALPHA, BETA, SCAL )
C .. Scalar Arguments ..
INTEGER INCA, K, SCAL
DOUBLE PRECISION BASE
COMPLEX*16 ALPHA, BETA
C .. Array Arguments ..
INTEGER S(*)
COMPLEX*16 A(*)
Arguments
Input/Output Parameters
BASE (input) DOUBLE PRECISION
Machine base.
K (input) INTEGER
The number of scalars. K >= 1.
S (input) INTEGER array, dimension (K)
The signature array. Each entry of S must be 1 or -1.
A (input) COMPLEX*16 array, dimension (K)
Vector of complex scalars.
INCA (input) INTEGER
Increment for the array A. INCA <> 0.
ALPHA (output) COMPLEX*16
ALPHA is a complex scalar with ABS(ALPHA) = 0, or
1.0 <= ABS(ALPHA) < BASE, such that
ALPHA / BETA * BASE**(SCAL)
is the general product of the scalars in the array A.
BETA (output) COMPLEX*16
BETA is either 0.0 or 1.0.
See also the description of ALPHA.
SCAL (output) INTEGER
Scaling factor exponent, see ALPHA.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute, without over- or underflow, the sign of the sum of two real numbers represented using integer powers of a base (usually, the machine base). Any base can be used, but it should the same for both numbers. The result is an integer with value 1, 0, or -1, depending on the sum being found as positive, zero, or negative, respectively.Specification
INTEGER FUNCTION MA01CD( A, IA, B, IB )
C .. Scalar Arguments ..
INTEGER IA, IB
DOUBLE PRECISION A, B
Function Value
MA01CD INTEGER
The sign of the sum of the two numbers, which is usually
either 1, or -1. If both numbers are 0, or if they have
the same exponent and their sum is 0, the returned value
is 0.
Arguments
Input/Output Parameters
A (input) DOUBLE PRECISION
The first real scalar.
IA (input) INTEGER
Exponent of the base for the first real scalar. The scalar
is represented as A * BASE**(IA).
B (input) DOUBLE PRECISION
The first real scalar.
IB (input) INTEGER
Exponent of the base for the first real scalar. The scalar
is represented as B * BASE**(IB).
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute an approximate symmetric chordal metric for two complex numbers A1 and A2, with Aj = ARj + i*AIj, j = 1, 2.Specification
SUBROUTINE MA01DD( AR1, AI1, AR2, AI2, EPS, SAFEMN, D )
C .. Scalar Arguments ..
DOUBLE PRECISION AI1, AI2, AR1, AR2, D, EPS, SAFEMN
Arguments
Input/Output Parameters
AR1 (input) DOUBLE PRECISION
AI1 (input) DOUBLE PRECISION
These scalars define the real and imaginary parts of the
number A1.
AR2 (input) DOUBLE PRECISION
AI2 (input) DOUBLE PRECISION
These scalars define the real and imaginary parts of the
number A2.
EPS (input) DOUBLE PRECISION
The relative machine precision. See the LAPACK Library
routine DLAMCH.
SAFEMN (input) DOUBLE PRECISION
The "safe minimum", such that its reciprocal does not
overflow. See the LAPACK Library routine DLAMCH.
D (output) DOUBLE PRECISION
The approximate symmetric chordal metric D. D >= 0.
Method
The approximate symmetric chordal metric is evaluated using the
formula
D = MIN( | A1 - A2 |, |1/A1 - 1/A2| ).
The chordal metric is finite even if A1 and A2 are both infinite,
or if one of them is infinite and the other is finite, nonzero.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute an approximate symmetric chordal metric for two complex numbers A1 and A2 in real arithmetic. Each number Aj, j = 1, 2, is represented as a rational number with numerator ARj + i*AIj, and denominator Bj, with Bj >= 0. A value Bj = 0, when ARj or AIj is nonzero, means that the number Aj is infinite. The case when ARj = AIj = Bj = 0 means that the pair (Aj,Bj) is not a number.Specification
SUBROUTINE MA01DZ( AR1, AI1, B1, AR2, AI2, B2, EPS, SAFEMN, D1,
$ D2, IWARN )
C .. Scalar Arguments ..
DOUBLE PRECISION AI1, AI2, AR1, AR2, B1, B2, D1, D2, EPS, SAFEMN
INTEGER IWARN
Arguments
Input/Output Parameters
AR1 (input) DOUBLE PRECISION
AI1 (input) DOUBLE PRECISION
These scalars define the real and imaginary parts of the
numerator of A1.
B1 (input) DOUBLE PRECISION
The denominator of A1. B1 >= 0.
AR2 (input) DOUBLE PRECISION
AI2 (input) DOUBLE PRECISION
These scalars define the real and imaginary parts of the
numerator of A2.
B2 (input) DOUBLE PRECISION
The denominator of A2. B2 >= 0.
EPS (input) DOUBLE PRECISION
The relative machine precision. See the LAPACK Library
routine DLAMCH.
SAFEMN (input) DOUBLE PRECISION
The "safe minimum", such that its reciprocal does not
overflow. See the LAPACK Library routine DLAMCH.
D1 (output) DOUBLE PRECISION
The numerator of the chordal metric D. D1 >= 0.
D2 (output) DOUBLE PRECISION
The denominator of the chordal metric D. D2 is 0 or 1.
If D2 = 0, and D1 = 0, the chordal metric is undefined, so
either A1 and/or A2 are undefined.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: A1 or A2 is not a number (NaN); D1 and D2 are both
set to 0.
Method
The approximate symmetric chordal metric is evaluated using the
formula
D = MIN( | A1 - A2 |, |1/A1 - 1/A2| ),
taking into account the special cases of infinite or NaN values.
The chordal metric is finite even if A1 and A2 are both infinite,
or if one of them is infinite and the other is finite, nonzero.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To transpose all or part of a two-dimensional matrix A into another matrix B.Specification
SUBROUTINE MA02AD( JOB, M, N, A, LDA, B, LDB )
C .. Scalar Arguments ..
CHARACTER JOB
INTEGER LDA, LDB, M, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the part of the matrix A to be transposed into B
as follows:
= 'U': Upper triangular part;
= 'L': Lower triangular part;
Otherwise: All of the matrix A.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The m-by-n matrix A. If JOB = 'U', only the upper
triangle or trapezoid is accessed; if JOB = 'L', only the
lower triangle or trapezoid is accessed.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B (output) DOUBLE PRECISION array, dimension (LDB,M)
B = A' in the locations specified by JOB.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To (conjugate) transpose all or part of a two-dimensional complex matrix A into another matrix B.Specification
SUBROUTINE MA02AZ( TRANS, JOB, M, N, A, LDA, B, LDB )
C .. Scalar Arguments ..
CHARACTER JOB, TRANS
INTEGER LDA, LDB, M, N
C .. Array Arguments ..
COMPLEX*16 A(LDA,*), B(LDB,*)
Arguments
Mode Parameters
TRANS CHARACTER*1
Specifies if a transpose or conjugate transpose operation
should be performed as follows:
= 'T': transpose operation;
= 'C': conjugate transpose operation.
JOB CHARACTER*1
Specifies the part of the matrix A to be transposed into B
as follows:
= 'U': Upper triangular part;
= 'L': Lower triangular part;
Otherwise: All of the matrix A.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
A (input) COMPLEX*16 array, dimension (LDA,N)
The m-by-n matrix A. If JOB = 'U', only the upper
triangle or trapezoid is accessed; if JOB = 'L', only the
lower triangle or trapezoid is accessed.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B (output) COMPLEX*16 array, dimension (LDB,M)
B = A' in the locations specified by JOB, where ' denotes
the transpose or conjugate transpose operation, as
as specified by TRANS.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To reverse the order of rows and/or columns of a given matrix A by pre-multiplying and/or post-multiplying it, respectively, with a permutation matrix P, where P is a square matrix of appropriate order, with ones down the secondary diagonal.Specification
SUBROUTINE MA02BD( SIDE, M, N, A, LDA )
C .. Scalar Arguments ..
CHARACTER SIDE
INTEGER LDA, M, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*)
Arguments
Mode Parameters
SIDE CHARACTER*1
Specifies the operation to be performed, as follows:
= 'L': the order of rows of A is to be reversed by
pre-multiplying A with P;
= 'R': the order of columns of A is to be reversed by
post-multiplying A with P;
= 'B': both the order of rows and the order of columns
of A is to be reversed by pre-multiplying and
post-multiplying A with P.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading M-by-N part of this array must
contain the given matrix whose rows and/or columns are to
be permuted.
On exit, the leading M-by-N part of this array contains
the matrix P*A if SIDE = 'L', or A*P if SIDE = 'R', or
P*A*P if SIDE = 'B'.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To reverse the order of rows and/or columns of a given matrix A by pre-multiplying and/or post-multiplying it, respectively, with a permutation matrix P, where P is a square matrix of appropriate order, with ones down the secondary diagonal.Specification
SUBROUTINE MA02BZ( SIDE, M, N, A, LDA )
C .. Scalar Arguments ..
CHARACTER SIDE
INTEGER LDA, M, N
C .. Array Arguments ..
COMPLEX*16 A(LDA,*)
Arguments
Mode Parameters
SIDE CHARACTER*1
Specifies the operation to be performed, as follows:
= 'L': the order of rows of A is to be reversed by
pre-multiplying A with P;
= 'R': the order of columns of A is to be reversed by
post-multiplying A with P;
= 'B': both the order of rows and the order of columns
of A is to be reversed by pre-multiplying and
post-multiplying A with P.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the leading M-by-N part of this array must
contain the given matrix whose rows and/or columns are to
be permuted.
On exit, the leading M-by-N part of this array contains
the matrix P*A if SIDE = 'L', or A*P if SIDE = 'R', or
P*A*P if SIDE = 'B'.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the pertranspose of a central band of a square matrix.Specification
SUBROUTINE MA02CD( N, KL, KU, A, LDA )
C .. Scalar Arguments ..
INTEGER KL, KU, LDA, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the square matrix A. N >= 0.
KL (input) INTEGER
The number of subdiagonals of A to be pertransposed.
0 <= KL <= N-1.
KU (input) INTEGER
The number of superdiagonals of A to be pertransposed.
0 <= KU <= N-1.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain a square matrix whose central band formed from
the KL subdiagonals, the main diagonal and the KU
superdiagonals will be pertransposed.
On exit, the leading N-by-N part of this array contains
the matrix A with its central band (the KL subdiagonals,
the main diagonal and the KU superdiagonals) pertransposed
(that is the elements of each antidiagonal appear in
reversed order). This is equivalent to forming P*B'*P,
where B is the matrix formed from the central band of A
and P is a permutation matrix with ones down the secondary
diagonal.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the pertranspose of a central band of a square matrix.Specification
SUBROUTINE MA02CZ( N, KL, KU, A, LDA )
C .. Scalar Arguments ..
INTEGER KL, KU, LDA, N
C .. Array Arguments ..
COMPLEX*16 A(LDA,*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the square matrix A. N >= 0.
KL (input) INTEGER
The number of subdiagonals of A to be pertransposed.
0 <= KL <= N-1.
KU (input) INTEGER
The number of superdiagonals of A to be pertransposed.
0 <= KU <= N-1.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain a square matrix whose central band formed from
the KL subdiagonals, the main diagonal and the KU
superdiagonals will be pertransposed.
On exit, the leading N-by-N part of this array contains
the matrix A with its central band (the KL subdiagonals,
the main diagonal and the KU superdiagonals) pertransposed
(that is the elements of each antidiagonal appear in
reversed order). This is equivalent to forming P*B'*P,
where B is the matrix formed from the central band of A
and P is a permutation matrix with ones down the secondary
diagonal.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To pack/unpack the upper or lower triangle of a symmetric matrix. The packed matrix is stored column-wise in the one-dimensional array AP.Specification
SUBROUTINE MA02DD( JOB, UPLO, N, A, LDA, AP )
C .. Scalar Arguments ..
CHARACTER JOB, UPLO
INTEGER LDA, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), AP(*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies whether the matrix should be packed or unpacked,
as follows:
= 'P': The matrix should be packed;
= 'U': The matrix should be unpacked.
UPLO CHARACTER*1
Specifies the part of the matrix to be packed/unpacked,
as follows:
= 'U': Upper triangular part;
= 'L': Lower triangular part.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input or output) DOUBLE PRECISION array, dimension
(LDA,N)
This array is an input parameter if JOB = 'P', and an
output parameter if JOB = 'U'.
On entry, if JOB = 'P', the leading N-by-N upper
triangular part (if UPLO = 'U'), or lower triangular part
(if UPLO = 'L'), of this array must contain the
corresponding upper or lower triangle of the symmetric
matrix A, and the other strictly triangular part is not
referenced.
On exit, if JOB = 'U', the leading N-by-N upper triangular
part (if UPLO = 'U'), or lower triangular part (if
UPLO = 'L'), of this array contains the corresponding
upper or lower triangle of the symmetric matrix A; the
other strictly triangular part is not referenced.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AP (output or input) DOUBLE PRECISION array, dimension
(N*(N+1)/2)
This array is an output parameter if JOB = 'P', and an
input parameter if JOB = 'U'.
On entry, if JOB = 'U', the leading N*(N+1)/2 elements of
this array must contain the upper (if UPLO = 'U') or lower
(if UPLO = 'L') triangle of the symmetric matrix A, packed
column-wise. That is, the elements are stored in the order
11, 12, 22, ..., 1n, 2n, 3n, ..., nn, if UPLO = 'U';
11, 21, 31, ..., n1, 22, 32, ..., n2, ..., if UPLO = 'L'.
On exit, if JOB = 'P', the leading N*(N+1)/2 elements of
this array contain the upper (if UPLO = 'U') or lower
(if UPLO = 'L') triangle of the symmetric matrix A, packed
column-wise, as described above.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To store by symmetry the upper or lower triangle of a symmetric matrix, given the other triangle.Specification
SUBROUTINE MA02ED( UPLO, N, A, LDA )
C .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies which part of the matrix is given as follows:
= 'U': Upper triangular part;
= 'L': Lower triangular part.
For all other values, the array A is not referenced.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N upper triangular part
(if UPLO = 'U'), or lower triangular part (if UPLO = 'L'),
of this array must contain the corresponding upper or
lower triangle of the symmetric matrix A.
On exit, the leading N-by-N part of this array contains
the symmetric matrix A with all elements stored.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To store by skew-symmetry the upper or lower triangle of a skew-symmetric matrix, given the other triangle. The diagonal entries are set to zero.Specification
SUBROUTINE MA02ES( UPLO, N, A, LDA )
C .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies which part of the matrix is given as follows:
= 'U': Upper triangular part;
= 'L': Lower triangular part.
For all other values, the array A is not referenced.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N upper triangular part
(if UPLO = 'U'), or lower triangular part (if UPLO = 'L'),
of this array must contain the corresponding upper or
lower triangle of the skew-symmetric matrix A.
On exit, the leading N-by-N part of this array contains
the skew-symmetric matrix A with all elements stored.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To store by (skew-)symmetry the upper or lower triangle of a (skew-)symmetric/Hermitian complex matrix, given the other triangle. The option SKEW = 'G' allows to suitably deal with the diagonal of a general square triangular matrix.Specification
SUBROUTINE MA02EZ( UPLO, TRANS, SKEW, N, A, LDA )
C .. Scalar Arguments ..
CHARACTER SKEW, TRANS, UPLO
INTEGER LDA, N
C .. Array Arguments ..
COMPLEX*16 A(LDA,*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies which part of the matrix is given as follows:
= 'U': Upper triangular part;
= 'L': Lower triangular part.
For all other values, the array A is not referenced.
TRANS CHARACTER*1
Specifies whether to use transposition or conjugate
transposition as follows:
= 'T': Use transposition;
= 'C': Use conjugate transposition.
SKEW CHARACTER*1
Specifies whether the matrix is symmetric/Hermitian or
skew-symmetric/Hermitian as follows:
= 'G': The matrix is not symmetric/Hermitian (general);
= 'N': The matrix is symmetric/Hermitian;
= 'S': The matrix is skew-symmetric/Hermitian.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the leading N-by-N upper triangular part
(if UPLO = 'U'), or lower triangular part (if UPLO = 'L'),
of this array must contain the corresponding upper or
lower triangle of the (skew-)symmetric/Hermitian matrix A.
On exit, the leading N-by-N part of this array contains
the (skew-)symmetric/Hermitian matrix A with all elements
stored. If the resulted matrix should be Hermitian, the
imaginary parts of the diagonal entries are set to zero.
If the resulted matrix should be skew-Hermitian, the real
parts of the diagonal entries are set to zero.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the coefficients c and s (c^2 + s^2 = 1) for a modified
hyperbolic plane rotation, such that,
y1 := 1/c * x1 - s/c * x2 = sqrt(x1^2 - x2^2),
y2 := -s * y1 + c * x2 = 0,
given two real numbers x1 and x2, satisfying either x1 = x2 = 0,
or abs(x2) < abs(x1).
Specification
SUBROUTINE MA02FD( X1, X2, C, S, INFO )
C .. Scalar Arguments ..
DOUBLE PRECISION X1, X2, C, S
INTEGER INFO
Arguments
Input/Output Parameters
X1 (input/output) DOUBLE PRECISION
On entry, the real number x1.
On exit, the real number y1.
X2 (input) DOUBLE PRECISION
The real number x2.
The values x1 and x2 should satisfy either x1 = x2 = 0, or
abs(x2) < abs(x1).
C (output) DOUBLE PRECISION
The cosines c of the modified hyperbolic plane rotation.
S (output) DOUBLE PRECISION
The sines s of the modified hyperbolic plane rotation.
Error Indicator
INFO INTEGER
= 0: succesful exit;
= 1: if abs(x2) >= abs(x1) and either x1 <> 0 or x2 <> 0.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To perform a series of column interchanges on the matrix A. One column interchange is initiated for each of columns K1 through K2 of A. This is useful for solving linear systems X*A = B, when the matrix A has already been factored by LAPACK Library routine DGETRF.Specification
SUBROUTINE MA02GD( N, A, LDA, K1, K2, IPIV, INCX )
C .. Scalar Arguments ..
INTEGER INCX, K1, K2, LDA, N
C .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * )
Arguments
Input/Output Parameters
N (input) INTEGER
The number of rows of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,*)
On entry, the leading N-by-M part of this array must
contain the matrix A to which the column interchanges will
be applied, where M is the largest element of IPIV(K), for
K = K1, ..., K2.
On exit, the leading N-by-M part of this array contains
the permuted matrix.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
K1 (input) INTEGER
The first element of IPIV for which a column interchange
will be done.
K2 (input) INTEGER
The last element of IPIV for which a column interchange
will be done.
IPIV (input) INTEGER array, dimension (K1+(K2-K1)*abs(INCX))
The vector of interchanging (pivot) indices. Only the
elements in positions K1 through K2 of IPIV are accessed.
IPIV(K) = L implies columns K and L are to be
interchanged.
INCX (input) INTEGER
The increment between successive values of IPIV.
If INCX is negative, the interchanges are applied in
reverse order.
Method
The columns IPIV(K) and K are swapped for K = K1, ..., K2, for INCX = 1 (and similarly, for INCX <> 1).Further Comments
This routine is the column-oriented counterpart of the LAPACK
Library routine DLASWP. The LAPACK Library routine DLAPMT cannot
be used in this context. To solve the system X*A = B, where A and
B are N-by-N and M-by-N, respectively, the following statements
can be used:
CALL DGETRF( N, N, A, LDA, IPIV, INFO )
CALL DTRSM( 'R', 'U', 'N', 'N', M, N, ONE, A, LDA, B, LDB )
CALL DTRSM( 'R', 'L', 'N', 'U', M, N, ONE, A, LDA, B, LDB )
CALL MA02GD( M, B, LDB, 1, N, IPIV, -1 )
Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To perform a series of column interchanges on the matrix A. One column interchange is initiated for each of columns K1 through K2 of A. This is useful for solving linear systems X*A = B, when the matrix A has already been factored by LAPACK Library routine DGETRF.Specification
SUBROUTINE MA02GZ( N, A, LDA, K1, K2, IPIV, INCX )
C .. Scalar Arguments ..
INTEGER INCX, K1, K2, LDA, N
C .. Array Arguments ..
INTEGER IPIV( * )
COMPLEX*16 A( LDA, * )
Arguments
Input/Output Parameters
N (input) INTEGER
The number of rows of the matrix A. N >= 0.
A (input/output) COMPLEX*16 array, dimension (LDA,*)
On entry, the leading N-by-M part of this array must
contain the matrix A to which the column interchanges will
be applied, where M is the largest element of IPIV(K), for
K = K1, ..., K2.
On exit, the leading N-by-M part of this array contains
the permuted matrix.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
K1 (input) INTEGER
The first element of IPIV for which a column interchange
will be done.
K2 (input) INTEGER
The last element of IPIV for which a column interchange
will be done.
IPIV (input) INTEGER array, dimension (K1+(K2-K1)*abs(INCX))
The vector of interchanging (pivot) indices. Only the
elements in positions K1 through K2 of IPIV are accessed.
IPIV(K) = L implies columns K and L are to be
interchanged.
INCX (input) INTEGER
The increment between successive values of IPIV.
If INCX is negative, the interchanges are applied in
reverse order.
Method
The columns IPIV(K) and K are swapped for K = K1, ..., K2, for INCX = 1 (and similarly, for INCX <> 1).Further Comments
This routine is the column-oriented counterpart of the LAPACK
Library routine DLASWP. The LAPACK Library routine DLAPMT cannot
be used in this context. To solve the system X*A = B, where A and
B are N-by-N and M-by-N, respectively, the following statements
can be used:
CALL DGETRF( N, N, A, LDA, IPIV, INFO )
CALL DTRSM( 'R', 'U', 'N', 'N', M, N, ONE, A, LDA, B, LDB )
CALL DTRSM( 'R', 'L', 'N', 'U', M, N, ONE, A, LDA, B, LDB )
CALL MA02GZ( M, B, LDB, 1, N, IPIV, -1 )
Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To check if A = DIAG*I, where I is an M-by-N matrix with ones on the diagonal and zeros elsewhere.Specification
LOGICAL FUNCTION MA02HD( JOB, M, N, DIAG, A, LDA )
C .. Scalar Arguments ..
CHARACTER JOB
INTEGER LDA, M, N
DOUBLE PRECISION DIAG
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*)
Function Value
MA02HD LOGICAL
The function value is set to .TRUE. if A = DIAG*I, and to
.FALSE., otherwise. If min(M,N) = 0, the value is .FALSE.
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the part of the matrix A to be checked out,
as follows:
= 'U': Upper triangular/trapezoidal part;
= 'L': Lower triangular/trapezoidal part.
Otherwise: All of the matrix A.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
DIAG (input) DOUBLE PRECISION
The scalar DIAG.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading M-by-N part of this array must contain the
matrix A. If JOB = 'U', only the upper triangle or
trapezoid is accessed; if JOB = 'L', only the lower
triangle or trapezoid is accessed.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
Method
The routine returns immediately after detecting a diagonal element which differs from DIAG, or a nonzero off-diagonal element in the searched part of A.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To check if A = DIAG*I, where I is an M-by-N matrix with ones on the diagonal and zeros elsewhere, A is a complex matrix and DIAG is a complex scalar.Specification
LOGICAL FUNCTION MA02HZ( JOB, M, N, DIAG, A, LDA )
C .. Scalar Arguments ..
CHARACTER JOB
INTEGER LDA, M, N
COMPLEX*16 DIAG
C .. Array Arguments ..
COMPLEX*16 A(LDA,*)
Function Value
MA02HZ LOGICAL
The function value is set to .TRUE. if A = DIAG*I, and to
.FALSE., otherwise. If min(M,N) = 0, the value is .FALSE.
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the part of the matrix A to be checked out,
as follows:
= 'U': Upper triangular/trapezoidal part;
= 'L': Lower triangular/trapezoidal part.
Otherwise: All of the matrix A.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
DIAG (input) COMPLEX*16
The scalar DIAG.
A (input) COMPLEX*16 array, dimension (LDA,N)
The leading M-by-N part of this array must contain the
matrix A. If JOB = 'U', only the upper triangle or
trapezoid is accessed; if JOB = 'L', only the lower
triangle or trapezoid is accessed.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
Method
The routine returns immediately after detecting a diagonal element which differs from DIAG, or a nonzero off-diagonal element in the searched part of A.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the value of the one norm, or the Frobenius norm, or
the infinity norm, or the element of largest absolute value
of a real skew-Hamiltonian matrix
[ A G ] T T
X = [ T ], G = -G, Q = -Q,
[ Q A ]
or of a real Hamiltonian matrix
[ A G ] T T
X = [ T ], G = G, Q = Q,
[ Q -A ]
where A, G and Q are real n-by-n matrices.
Note that for this kind of matrices the infinity norm is equal
to the one norm.
Specification
DOUBLE PRECISION FUNCTION MA02ID( TYP, NORM, N, A, LDA, QG,
$ LDQG, DWORK )
C .. Scalar Arguments ..
CHARACTER NORM, TYP
INTEGER LDA, LDQG, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), QG(LDQG,*)
Function Value
MA02ID DOUBLE PRECISION
The computed norm.
Arguments
Mode Parameters
TYP CHARACTER*1
Specifies the type of the input matrix X:
= 'S': X is skew-Hamiltonian;
= 'H': X is Hamiltonian.
NORM CHARACTER*1
Specifies the value to be returned in MA02ID:
= '1' or 'O': one norm of X;
= 'F' or 'E': Frobenius norm of X;
= 'I': infinity norm of X;
= 'M': max(abs(X(i,j)).
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
QG (input) DOUBLE PRECISION array, dimension (LDQG,N+1)
On entry, the leading N-by-N+1 part of this array must
contain in columns 1:N the lower triangular part of the
matrix Q and in columns 2:N+1 the upper triangular part
of the matrix G. If TYP = 'S', the parts containing the
diagonal and the first supdiagonal of this array are not
referenced.
LDQG INTEGER
The leading dimension of the array QG. LDQG >= MAX(1,N).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
where LDWORK >= 2*N when NORM = '1', NORM = 'I' or
NORM = 'O'; otherwise, DWORK is not referenced.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the value of the one norm, or the Frobenius norm, or
the infinity norm, or the element of largest absolute value
of a complex skew-Hamiltonian matrix
[ A G ] H H
X = [ H ], G = -G, Q = -Q,
[ Q A ]
or of a complex Hamiltonian matrix
[ A G ] H H
X = [ H ], G = G, Q = Q,
[ Q -A ]
where A, G and Q are complex n-by-n matrices.
Note that for this kind of matrices the infinity norm is equal
to the one norm.
Specification
DOUBLE PRECISION FUNCTION MA02IZ( TYP, NORM, N, A, LDA, QG,
$ LDQG, DWORK )
C .. Scalar Arguments ..
CHARACTER NORM, TYP
INTEGER LDA, LDQG, N
C .. Array Arguments ..
COMPLEX*16 A(LDA,*), QG(LDQG,*)
DOUBLE PRECISION DWORK(*)
Function Value
MA02IZ DOUBLE PRECISION
The computed norm.
Arguments
Mode Parameters
TYP CHARACTER*1
Specifies the type of the input matrix X:
= 'S': X is skew-Hamiltonian;
= 'H': X is Hamiltonian.
NORM CHARACTER*1
Specifies the value to be returned in MA02IZ:
= '1' or 'O': one norm of X;
= 'F' or 'E': Frobenius norm of X;
= 'I': infinity norm of X;
= 'M': max(abs(X(i,j)).
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input) COMPLEX*16 array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
QG (input) COMPLEX*16 array, dimension (LDQG,N+1)
On entry, the leading N-by-N+1 part of this array must
contain in columns 1:N the lower triangular part of the
matrix Q and in columns 2:N+1 the upper triangular part
of the matrix G. If TYP = 'S', the real parts of the
entries on the diagonal and the first superdiagonal of
this array, which should be zero, need not be set, since
they are not used. Similarly, if TYP = 'H', the imaginary
parts of the entries on the diagonal and the first
superdiagonal of this array, which should be zero, need
not be set.
LDQG INTEGER
The leading dimension of the array QG. LDQG >= MAX(1,N).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
where LDWORK >= 2*N when NORM = '1', NORM = 'I' or
NORM = 'O'; otherwise, DWORK is not referenced.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute || Q^T Q - I ||_F for a matrix of the form
[ op( Q1 ) op( Q2 ) ]
Q = [ ],
[ -op( Q2 ) op( Q1 ) ]
where Q1 and Q2 are N-by-N matrices. This residual can be used to
test wether Q is numerically an orthogonal symplectic matrix.
Specification
DOUBLE PRECISION FUNCTION MA02JD( LTRAN1, LTRAN2, N, Q1, LDQ1, Q2,
$ LDQ2, RES, LDRES )
C .. Scalar Arguments ..
LOGICAL LTRAN1, LTRAN2
INTEGER LDQ1, LDQ2, LDRES, N
C .. Array Arguments ..
DOUBLE PRECISION Q1(LDQ1,*), Q2(LDQ2,*), RES(LDRES,*)
Function Value
MA02JD DOUBLE PRECISION
The computed residual.
Arguments
Mode Parameters
LTRAN1 LOGICAL
Specifies the form of op( Q1 ) as follows:
= .FALSE.: op( Q1 ) = Q1;
= .TRUE. : op( Q1 ) = Q1'.
LTRAN2 LOGICAL
Specifies the form of op( Q2 ) as follows:
= .FALSE.: op( Q2 ) = Q2;
= .TRUE. : op( Q2 ) = Q2'.
Input/Output Parameters
N (input) INTEGER
The order of the matrices Q1 and Q2. N >= 0.
Q1 (input) DOUBLE PRECISION array, dimension (LDQ1,N)
On entry, the leading N-by-N part of this array must
contain the matrix op( Q1 ).
LDQ1 INTEGER
The leading dimension of the array Q1. LDQ1 >= MAX(1,N).
Q2 (input) DOUBLE PRECISION array, dimension (LDQ2,N)
On entry, the leading N-by-N part of this array must
contain the matrix op( Q2 ).
LDQ2 INTEGER
The leading dimension of the array Q2. LDQ2 >= MAX(1,N).
Workspace
RES DOUBLE PRECISION array, dimension (LDRES,N)
LDRES INTEGER
The leading dimension of the array RES. LDRES >= MAX(1,N).
Method
The routine computes the residual by simple elementary operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute || Q^H Q - I ||_F for a complex matrix of the form
[ op( Q1 ) op( Q2 ) ]
Q = [ ],
[ -op( Q2 ) op( Q1 ) ]
where Q1 and Q2 are N-by-N matrices. This residual can be used to
test wether Q is numerically a unitary symplectic matrix.
Specification
DOUBLE PRECISION FUNCTION MA02JZ( LTRAN1, LTRAN2, N, Q1, LDQ1, Q2,
$ LDQ2, RES, LDRES )
C .. Scalar Arguments ..
LOGICAL LTRAN1, LTRAN2
INTEGER LDQ1, LDQ2, LDRES, N
C .. Array Arguments ..
COMPLEX*16 Q1(LDQ1,*), Q2(LDQ2,*), RES(LDRES,*)
Function Value
MA02JZ DOUBLE PRECISION
The computed residual.
Arguments
Mode Parameters
LTRAN1 LOGICAL
Specifies the form of op( Q1 ) as follows:
= .FALSE.: op( Q1 ) = Q1;
= .TRUE. : op( Q1 ) = Q1'.
LTRAN2 LOGICAL
Specifies the form of op( Q2 ) as follows:
= .FALSE.: op( Q2 ) = Q2;
= .TRUE. : op( Q2 ) = Q2'.
Input/Output Parameters
N (input) INTEGER
The order of the matrices Q1 and Q2. N >= 0.
Q1 (input) COMPLEX*16 array, dimension (LDQ1,N)
On entry, the leading N-by-N part of this array must
contain the matrix op( Q1 ).
LDQ1 INTEGER
The leading dimension of the array Q1. LDQ1 >= MAX(1,N).
Q2 (input) COMPLEX*16 array, dimension (LDQ2,N)
On entry, the leading N-by-N part of this array must
contain the matrix op( Q2 ).
LDQ2 INTEGER
The leading dimension of the array Q2. LDQ2 >= MAX(1,N).
Workspace
RES DOUBLE PRECISION array, dimension (LDRES,N)
LDRES INTEGER
The leading dimension of the array RES.
LDRES >= MAX(1,N).
Method
The routine computes the residual by simple elementary operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real skew-symmetric matrix. Note that for this kind of matrices the infinity norm is equal to the one norm.Specification
DOUBLE PRECISION FUNCTION MA02MD( NORM, UPLO, N, A, LDA, DWORK )
C .. Scalar Arguments ..
CHARACTER NORM, UPLO
INTEGER LDA, N
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), DWORK( * )
Function Value
MA02MD DOUBLE PRECISION
The computed norm.
Arguments
Mode Parameters
NORM CHARACTER*1
Specifies the value to be returned in MA02MD:
= '1' or 'O': one norm of A;
= 'F' or 'E': Frobenius norm of A;
= 'I': infinity norm of A;
= 'M': max(abs(A(i,j)).
UPLO CHARACTER*1
Specifies whether the upper or lower triangular part of
the skew-symmetric matrix A is to be referenced.
= 'U': Upper triangular part of A is referenced;
= 'L': Lower triangular part of A is referenced.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0. When N = 0, MA02MD is
set to zero.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The skew-symmetric matrix A. If UPLO = 'U', the leading
N-by-N strictly upper triangular part of A contains the
strictly upper triangular part of the matrix A, and the
lower triangular part of A is not referenced.
If UPLO = 'L', the leading N-by-N strictly lower
triangular part of A contains the strictly lower
triangular part of the matrix A, and the upper triangular
part of A is not referenced.
The diagonal of A need not be set to zero.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
Workspace
DWORK DOUBLE PRECISION array, dimension (MAX(1,LDWORK)),
where LDWORK >= N when NORM = 'I' or '1' or 'O';
otherwise, DWORK is not referenced.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex skew-Hermitian matrix. Note that for this kind of matrices the infinity norm is equal to the one norm.Specification
DOUBLE PRECISION FUNCTION MA02MZ( NORM, UPLO, N, A, LDA, DWORK )
C .. Scalar Arguments ..
CHARACTER NORM, UPLO
INTEGER LDA, N
C .. Array Arguments ..
DOUBLE PRECISION DWORK( * )
COMPLEX*16 A( LDA, * )
Function Value
MA02MZ DOUBLE PRECISION
The computed norm.
Arguments
Mode Parameters
NORM CHARACTER*1
Specifies the value to be returned in MA02MZ:
= '1' or 'O': one norm of A;
= 'F' or 'E': Frobenius norm of A;
= 'I': infinity norm of A;
= 'M': max(abs(A(i,j)).
UPLO CHARACTER*1
Specifies whether the upper or lower triangular part of
the skew-Hermitian matrix A is to be referenced.
= 'U': Upper triangular part of A is referenced;
= 'L': Lower triangular part of A is referenced.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0. When N = 0, MA02MZ is
set to zero.
A (input) COMPLEX*16 array, dimension (LDA,N)
The skew-Hermitian matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of A contains the
lower triangular part of the matrix A, and the strictly
upper triangular part of A is not referenced.
The real parts of the diagonal elements of A need not be
set and are assumed to be zero.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
Workspace
DWORK DOUBLE PRECISION array, dimension (MAX(1,LDWORK)),
where LDWORK >= N when NORM = 'I' or '1' or 'O';
otherwise, DWORK is not referenced.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To permute two specified rows and corresponding columns of a (skew-)symmetric/Hermitian complex matrix.Specification
SUBROUTINE MA02NZ( UPLO, TRANS, SKEW, N, K, L, A, LDA )
C .. Scalar Arguments ..
CHARACTER SKEW, TRANS, UPLO
INTEGER K, L, LDA, N
C .. Array Arguments ..
COMPLEX*16 A(LDA,*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies whether the upper or lower triangular part of
the (skew-)symmetric/Hermitian matrix A is to be
referenced, as follows:
= 'U': Upper triangular part of A is referenced;
= 'L': Lower triangular part of A is referenced.
TRANS CHARACTER*1
Specifies whether to use transposition or conjugate
transposition as follows:
= 'T': Use transposition;
= 'C': Use conjugate transposition.
SKEW CHARACTER*1
Specifies whether the matrix is symmetric/Hermitian or
skew-symmetric/Hermitian as follows:
= 'N': The matrix is symmetric/Hermitian;
= 'S': The matrix is skew-symmetric/skew-Hermitian.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
K (input) INTEGER
The smaller index of the pair of rows and columns to be
permuted. 0 <= K <= L. If K = 0, the routine returns.
L (input) INTEGER
The larger index of the pair of rows and columns to be
permuted. K <= L <= N.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the leading N-by-N upper triangular part
(if UPLO = 'U'), or lower triangular part (if UPLO = 'L'),
of this array must contain the corresponding upper or
lower triangle of the (skew-)symmetric/Hermitian matrix A.
On exit, the leading N-by-N upper or lower triangular part
of this array (depending on UPLO) contains the
corresponding part of the permuted matrix A.
Note that a Hermitian matrix has the imaginary parts of
the diagonal entries zero. Similarly, a skew-Hermitian
matrix has the real parts of the diagonal entries zero.
The routine does not check out this conditions.
LDA INTEGER
The leading dimension of the array A. LDA >= max(N,1).
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the number of zero rows (and zero columns) of a real
(skew-)Hamiltonian matrix,
( A D )
H = ( ).
( E +/-A' )
Specification
INTEGER FUNCTION MA02OD( SKEW, M, A, LDA, DE, LDDE )
C .. Scalar Arguments ..
CHARACTER SKEW
INTEGER LDA, LDDE, M
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), DE( LDDE, * )
Function Value
MA02OD INTEGER
The number of zero rows.
Arguments
Mode Parameters
SKEW CHARACTER*1
Specifies whether the matrix is Hamiltonian or skew-
Hamiltonian as follows:
= 'H': The matrix is Hamiltonian;
= 'S': The matrix is skew-Hamiltonian.
Input/Output Parameters
M (input) INTEGER
The order of the matrices A, D, and E. M >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,M)
The leading M-by-M part of this array must contain the
matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
DE (input) DOUBLE PRECISION array, dimension (LDDE,M+1)
The leading M-by-M lower triangular part of this array
must contain the lower triangular part of the (skew-)
symmetric matrix E, and the M-by-M upper triangular
part of the submatrix in the columns 2 to M+1 of this
array must contain the upper triangular part of the
(skew-)symmetric matrix D. If S is skew-Hamiltonian, the
parts containing the diagonal and the first superdiagonal
of this array, which should be zero, are not referenced.
LDDE INTEGER
The leading dimension of the array DE. LDDE >= MAX(1,M).
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the number of zero rows (and zero columns) of a complex
(skew-)Hamiltonian matrix,
( A D )
H = ( ).
( E +/-A' )
Specification
INTEGER FUNCTION MA02OZ( SKEW, M, A, LDA, DE, LDDE )
C .. Scalar Arguments ..
CHARACTER SKEW
INTEGER LDA, LDDE, M
C .. Array Arguments ..
COMPLEX*16 A( LDA, * ), DE( LDDE, * )
Function Value
MA02OZ INTEGER
The number of zero rows.
Arguments
Mode Parameters
SKEW CHARACTER*1
Specifies whether the matrix is Hamiltonian or skew-
Hamiltonian as follows:
= 'H': The matrix is Hamiltonian;
= 'S': The matrix is skew-Hamiltonian.
Input/Output Parameters
M (input) INTEGER
The order of the matrices A, D, and E. M >= 0.
A (input) COMPLEX*16 array, dimension (LDA,M)
The leading M-by-M part of this array must contain the
matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
DE (input) COMPLEX*16 array, dimension (LDDE,M+1)
The leading M-by-M lower triangular part of this array
must contain the lower triangular part of the (skew-)
Hermitian matrix E, and the M-by-M upper triangular
part of the submatrix in the columns 2 to M+1 of this
array must contain the upper triangular part of the
(skew-)Hermitian matrix D. If S is skew-Hamiltonian, the
real parts of the entries on the diagonal and the first
superdiagonal of this array, which should be zero, are
not used. If S is Hamiltonian, the imaginary parts of the
entries on the diagonal and the first superdiagonal of
this array, which should be zero, are not used.
LDDE INTEGER
The leading dimension of the array DE. LDDE >= MAX(1,M).
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the number of zero rows and zero columns of a real matrix.Specification
SUBROUTINE MA02PD( M, N, A, LDA, NZR, NZC )
C .. Scalar Arguments ..
INTEGER LDA, M, N, NZC, NZR
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
Arguments
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading M-by-N part of this array must contain the
matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
NZR (output) INTEGER
The number of zero rows of the matrix A.
NZC (output) INTEGER
The number of zero columns of the matrix A.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the number of zero rows and zero columns of a complex matrix.Specification
SUBROUTINE MA02PZ( M, N, A, LDA, NZR, NZC )
C .. Scalar Arguments ..
INTEGER LDA, M, N, NZC, NZR
C .. Array Arguments ..
COMPLEX*16 A( LDA, * )
Arguments
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
A (input) COMPLEX*16 array, dimension (LDA,N)
The leading M-by-N part of this array must contain the
matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
NZR (output) INTEGER
The number of zero rows of the matrix A.
NZC (output) INTEGER
The number of zero columns of the matrix A.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To perform one of the skew-symmetric rank 2k operations
C := alpha*A*B' - alpha*B*A' + beta*C,
or
C := alpha*A'*B - alpha*B'*A + beta*C,
where alpha and beta are scalars, C is a real N-by-N skew-
symmetric matrix and A, B are N-by-K matrices in the first case
and K-by-N matrices in the second case.
This is a modified version of the vanilla implemented BLAS
routine DSYR2K written by Jack Dongarra, Iain Duff,
Jeremy Du Croz and Sven Hammarling.
Specification
SUBROUTINE MB01KD( UPLO, TRANS, N, K, ALPHA, A, LDA, B, LDB, BETA,
$ C, LDC, INFO )
C .. Scalar Arguments ..
CHARACTER UPLO, TRANS
INTEGER INFO, K, LDA, LDB, LDC, N
DOUBLE PRECISION ALPHA, BETA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies whether the upper or lower triangular part of
the array C is to be referenced, as follows:
= 'U': only the strictly upper triangular part of C is to
be referenced;
= 'L': only the striclty lower triangular part of C is to
be referenced.
TRANS CHARACTER*1
Specifies the operation to be performed, as follows:
= 'N': C := alpha*A*B' - alpha*B*A' + beta*C;
= 'T' or 'C': C := alpha*A'*B - alpha*B'*A + beta*C.
Input/Output Parameters
N (input) INTEGER
The order of the matrix C. N >= 0.
K (input) INTEGER
If TRANS = 'N' the number of columns of A and B; and if
TRANS = 'T' or TRANS = 'C' the number of rows of A and B.
K >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. If alpha is zero, or N <= 1, or K = 0,
A and B are not referenced.
A (input) DOUBLE PRECISION array, dimension (LDA,KA),
where KA is K when TRANS = 'N', and is N otherwise.
On entry with TRANS = 'N', the leading N-by-K part of
of this array must contain the matrix A.
On entry with TRANS = 'T' or TRANS = 'C', the leading
K-by-N part of this array must contain the matrix A.
LDA INTEGER
The leading dimension of the array A.
LDA >= MAX(1,N), if TRANS = 'N';
LDA >= MAX(1,K), if TRANS = 'T' or TRANS = 'C'.
B (input) DOUBLE PRECISION array, dimension (LDB,KB),
where KB is K when TRANS = 'N', and is N otherwise.
On entry with TRANS = 'N', the leading N-by-K part of
of this array must contain the matrix B.
On entry with TRANS = 'T' or TRANS = 'C', the leading
K-by-N part of this array must contain the matrix B.
LDB INTEGER
The leading dimension of the array B.
LDB >= MAX(1,N), if TRANS = 'N';
LDB >= MAX(1,K), if TRANS = 'T' or TRANS = 'C'.
BETA (input) DOUBLE PRECISION
The scalar beta. If beta is zero C need not be set before
entry.
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry with UPLO = 'U', the leading N-by-N part of this
array must contain the strictly upper triangular part of
the matrix C. The lower triangular part of this array is
not referenced.
On entry with UPLO = 'L', the leading N-by-N part of this
array must contain the strictly lower triangular part of
the matrix C. The upper triangular part of this array is
not referenced.
On exit with UPLO = 'U', the leading N-by-N part of this
array contains the strictly upper triangular part of the
updated matrix C.
On exit with UPLO = 'L', the leading N-by-N part of this
array contains the strictly lower triangular part of the
updated matrix C.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,N)
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Numerical Aspects
Though being almost identical with the vanilla implementation of the BLAS routine DSYR2K the performance of this routine could be significantly lower in the case of vendor supplied, highly optimized BLAS.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the matrix formula
_
R = alpha*R + beta*op( A )*X*op( A )',
_
where alpha and beta are scalars, R, X, and R are skew-symmetric
matrices, A is a general matrix, and op( A ) is one of
op( A ) = A or op( A ) = A'.
The result is overwritten on R.
Specification
SUBROUTINE MB01LD( UPLO, TRANS, M, N, ALPHA, BETA, R, LDR, A, LDA,
$ X, LDX, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TRANS, UPLO
INTEGER INFO, LDA, LDR, LDWORK, LDX, M, N
DOUBLE PRECISION ALPHA, BETA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), R(LDR,*), X(LDX,*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies which triangles of the skew-symmetric matrices R
and X are given, as follows:
= 'U': the strictly upper triangular part is given;
= 'L': the strictly lower triangular part is given.
TRANS CHARACTER*1
Specifies the form of op( A ) to be used in the matrix
multiplication, as follows:
= 'N': op( A ) = A;
= 'T': op( A ) = A';
= 'C': op( A ) = A'.
Input/Output Parameters
M (input) INTEGER _
The order of the matrices R and R and the number of rows
of the matrix op( A ). M >= 0.
N (input) INTEGER
The order of the matrix X and the number of columns of the
matrix op( A ). N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then R need not be
set before entry, except when R is identified with X in
the call.
BETA (input) DOUBLE PRECISION
The scalar beta. When beta is zero or N <= 1, or M <= 1,
then A and X are not referenced.
R (input/output) DOUBLE PRECISION array, dimension (LDR,M)
On entry with UPLO = 'U', the leading M-by-M strictly
upper triangular part of this array must contain the
strictly upper triangular part of the skew-symmetric
matrix R. The lower triangle is not referenced.
On entry with UPLO = 'L', the leading M-by-M strictly
lower triangular part of this array must contain the
strictly lower triangular part of the skew-symmetric
matrix R. The upper triangle is not referenced.
On exit, the leading M-by-M strictly upper triangular part
(if UPLO = 'U'), or strictly lower triangular part
(if UPLO = 'L'), of this array contains the corresponding
_
strictly triangular part of the computed matrix R.
LDR INTEGER
The leading dimension of the array R. LDR >= MAX(1,M).
A (input) DOUBLE PRECISION array, dimension (LDA,k)
where k is N when TRANS = 'N' and is M when TRANS = 'T' or
TRANS = 'C'.
On entry with TRANS = 'N', the leading M-by-N part of this
array must contain the matrix A.
On entry with TRANS = 'T' or TRANS = 'C', the leading
N-by-M part of this array must contain the matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,k),
where k is M when TRANS = 'N' and is N when TRANS = 'T' or
TRANS = 'C'.
X (input or input/output) DOUBLE PRECISION array, dimension
(LDX,K), where K = N, if UPLO = 'U' or LDWORK >= M*(N-1),
or K = MAX(N,M), if UPLO = 'L' and LDWORK < M*(N-1).
On entry, if UPLO = 'U', the leading N-by-N strictly upper
triangular part of this array must contain the strictly
upper triangular part of the skew-symmetric matrix X and
the lower triangular part of the array is not referenced.
On entry, if UPLO = 'L', the leading N-by-N strictly lower
triangular part of this array must contain the strictly
lower triangular part of the skew-symmetric matrix X and
the upper triangular part of the array is not referenced.
If LDWORK < M*(N-1), this array is overwritten with the
matrix op(A)*X, if UPLO = 'U', or X*op(A)', if UPLO = 'L'.
LDX INTEGER
The leading dimension of the array X.
LDX >= MAX(1,N), if UPLO = 'L' or LDWORK >= M*(N-1);
LDX >= MAX(1,N,M), if UPLO = 'U' and LDWORK < M*(N-1).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
This array is not referenced when beta = 0, or M <= 1, or
N <= 1.
LDWORK The length of the array DWORK.
LDWORK >= N, if beta <> 0, and M > 0, and N > 1;
LDWORK >= 0, if beta = 0, or M = 0, or N <= 1.
For optimum performance, LDWORK >= M*(N-1), if beta <> 0,
M > 1, and N > 1.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -k, the k-th argument had an illegal
value.
Method
The matrix expression is efficiently evaluated taking the skew-
symmetry into account. If LDWORK >= M*(N-1), a BLAS 3 like
implementation is used. Specifically, let X = T - T', with T a
strictly upper or strictly lower triangular matrix, defined by
T = striu( X ), if UPLO = 'U',
T = stril( X ), if UPLO = 'L',
where striu and stril denote the strictly upper triangular part
and strictly lower triangular part of X, respectively. Then,
A*X*A' = ( A*T )*A' - A*( A*T )', for TRANS = 'N',
A'*X*A = A'*( T*A ) - ( T*A )'*A, for TRANS = 'T', or 'C',
which involve BLAS 3 operations DTRMM and the skew-symmetric
correspondent of DSYR2K (with a Fortran implementation available
in the SLICOT Library routine MB01KD).
If LDWORK < M*(N-1), a BLAS 2 implementation is used.
Numerical Aspects
The algorithm requires approximately
2 2
3/2 x M x N + 1/2 x M
operations.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To perform the matrix-vector operation
y := alpha*A*x + beta*y,
where alpha and beta are scalars, x and y are vectors of length
n and A is an n-by-n skew-symmetric matrix.
This is a modified version of the vanilla implemented BLAS
routine DSYMV written by Jack Dongarra, Jeremy Du Croz,
Sven Hammarling, and Richard Hanson.
Specification
SUBROUTINE MB01MD( UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y,
$ INCY )
C .. Scalar Arguments ..
DOUBLE PRECISION ALPHA, BETA
INTEGER INCX, INCY, LDA, N
CHARACTER UPLO
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), X(*), Y(*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies whether the upper or lower triangular part of
the array A is to be referenced as follows:
= 'U': only the strictly upper triangular part of A is to
be referenced;
= 'L': only the strictly lower triangular part of A is to
be referenced.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. If alpha is zero the array A is not
referenced.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
On entry with UPLO = 'U', the leading N-by-N part of this
array must contain the strictly upper triangular part of
the matrix A. The lower triangular part of this array is
not referenced.
On entry with UPLO = 'L', the leading N-by-N part of this
array must contain the strictly lower triangular part of
the matrix A. The upper triangular part of this array is
not referenced.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N)
X (input) DOUBLE PRECISION array, dimension
( 1 + ( N - 1 )*abs( INCX ) ).
On entry, elements 1, INCX+1, .., ( N - 1 )*INCX + 1 of
this array must contain the elements of the vector X.
INCX (input) INTEGER
The increment for the elements of X. IF INCX < 0 then the
elements of X are accessed in reversed order. INCX <> 0.
BETA (input) DOUBLE PRECISION
The scalar beta. If beta is zero then Y need not be set on
input.
Y (input/output) DOUBLE PRECISION array, dimension
( 1 + ( N - 1 )*abs( INCY ) ).
On entry, elements 1, INCY+1, .., ( N - 1 )*INCY + 1 of
this array must contain the elements of the vector Y.
On exit, elements 1, INCY+1, .., ( N - 1 )*INCY + 1 of
this array contain the updated elements of the vector Y.
INCY (input) INTEGER
The increment for the elements of Y. IF INCY < 0 then the
elements of Y are accessed in reversed order. INCY <> 0.
Numerical Aspects
Though being almost identical with the vanilla implementation of the BLAS routine DSYMV the performance of this routine could be significantly lower in the case of vendor supplied, highly optimized BLAS.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To perform the skew-symmetric rank 2 operation
A := alpha*x*y' - alpha*y*x' + A,
where alpha is a scalar, x and y are vectors of length n and A is
an n-by-n skew-symmetric matrix.
This is a modified version of the vanilla implemented BLAS
routine DSYR2 written by Jack Dongarra, Jeremy Du Croz,
Sven Hammarling, and Richard Hanson.
Specification
SUBROUTINE MB01ND( UPLO, N, ALPHA, X, INCX, Y, INCY, A, LDA )
C .. Scalar Arguments ..
DOUBLE PRECISION ALPHA
INTEGER INCX, INCY, LDA, N
CHARACTER UPLO
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), X( * ), Y( * )
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies whether the upper or lower triangular part of
the array A is to be referenced as follows:
= 'U': only the strictly upper triangular part of A is to
be referenced;
= 'L': only the strictly lower triangular part of A is to
be referenced.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. If alpha is zero X and Y are not
referenced.
X (input) DOUBLE PRECISION array, dimension
( 1 + ( N - 1 )*abs( INCX ) ).
On entry, elements 1, INCX+1, .., ( N - 1 )*INCX + 1 of
this array must contain the elements of the vector X.
INCX (input) INTEGER
The increment for the elements of X. IF INCX < 0 then the
elements of X are accessed in reversed order. INCX <> 0.
Y (input) DOUBLE PRECISION array, dimension
( 1 + ( N - 1 )*abs( INCY ) ).
On entry, elements 1, INCY+1, .., ( N - 1 )*INCY + 1 of
this array must contain the elements of the vector Y.
INCY (input) INTEGER
The increment for the elements of Y. IF INCY < 0 then the
elements of Y are accessed in reversed order. INCY <> 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry with UPLO = 'U', the leading N-by-N part of this
array must contain the strictly upper triangular part of
the matrix A. The lower triangular part of this array is
not referenced.
On entry with UPLO = 'L', the leading N-by-N part of this
array must contain the strictly lower triangular part of
the matrix A. The upper triangular part of this array is
not referenced.
On exit with UPLO = 'U', the leading N-by-N part of this
array contains the strictly upper triangular part of the
updated matrix A.
On exit with UPLO = 'L', the leading N-by-N part of this
array contains the strictly lower triangular part of the
updated matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N)
Numerical Aspects
Though being almost identical with the vanilla implementation of the BLAS routine DSYR2 the performance of this routine could be significantly lower in the case of vendor supplied, highly optimized BLAS.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To perform one of the special symmetric rank 2k operations
R := alpha*R + beta*H*X + beta*X*H',
or
R := alpha*R + beta*H'*X + beta*X*H,
where alpha and beta are scalars, R and X are N-by-N symmetric
matrices, and H is an N-by-N upper Hessenberg matrix.
Specification
SUBROUTINE MB01OC( UPLO, TRANS, N, ALPHA, BETA, R, LDR, H, LDH, X,
$ LDX, INFO )
C .. Scalar Arguments ..
CHARACTER TRANS, UPLO
INTEGER INFO, LDH, LDR, LDX, N
DOUBLE PRECISION ALPHA, BETA
C .. Array Arguments ..
DOUBLE PRECISION H(LDH,*), R(LDR,*), X(LDX,*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies which triangles of the symmetric matrices R
and X are given as follows:
= 'U': the upper triangular part is given;
= 'L': the lower triangular part is given.
TRANS CHARACTER*1
Specifies the operation to be performed as follows:
= 'N': R := alpha*R + beta*H*X + beta*X*H';
= 'T' or 'C': R := alpha*R + beta*H'*X + beta*X*H.
Input/Output Parameters
N (input) INTEGER
The order of the matrices R, H, and X. N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then R need not be
set before entry, except when R is identified with X in
the call.
BETA (input) DOUBLE PRECISION
The scalar beta. When beta is zero then H and X are not
referenced.
R (input/output) DOUBLE PRECISION array, dimension (LDR,N)
On entry with UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix R.
On entry with UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix R.
In both cases, the other strictly triangular part is not
referenced.
On exit, the leading N-by-N upper triangular part (if
UPLO = 'U'), or lower triangular part (if UPLO = 'L'), of
this array contains the corresponding triangular part of
the computed matrix R.
LDR INTEGER
The leading dimension of the array R. LDR >= MAX(1,N).
H (input) DOUBLE PRECISION array, dimension (LDH,N)
On entry, the leading N-by-N upper Hessenberg part of this
array must contain the upper Hessenberg matrix H.
The remaining part of this array is not referenced.
LDH INTEGER
The leading dimension of the array H. LDH >= MAX(1,N).
X (input) DOUBLE PRECISION array, dimension (LDX,N)
On entry, if UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix X and the strictly
lower triangular part of the array is not referenced.
On entry, if UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix X and the strictly
upper triangular part of the array is not referenced.
LDX INTEGER
The leading dimension of the array X. LDX >= MAX(1,N).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -k, the k-th argument had an illegal
value.
Method
The matrix expression is efficiently evaluated taking the structure into account, and using inline code and BLAS1 routines.Numerical Aspects
The algorithm requires approximately N**3/2 operations.Further Comments
This routine acts as a specialization of BLAS Library routine DSYR2K.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the matrix formula
R := alpha*R + beta*( op( H )*X*op( E )' + op( E )*X*op( H )' ),
where alpha and beta are scalars, R and X are symmetric matrices,
H is an upper Hessenberg matrix, E is an upper triangular matrix,
and op( M ) is one of
op( M ) = M or op( M ) = M'.
The result is overwritten on R.
Specification
SUBROUTINE MB01OD( UPLO, TRANS, N, ALPHA, BETA, R, LDR, H, LDH,
$ X, LDX, E, LDE, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TRANS, UPLO
INTEGER INFO, LDE, LDH, LDR, LDWORK, LDX, N
DOUBLE PRECISION ALPHA, BETA
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), E(LDE,*), H(LDH,*), R(LDR,*), X(LDX,*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies which triangles of the symmetric matrices R
and X are given as follows:
= 'U': the upper triangular part is given;
= 'L': the lower triangular part is given.
TRANS CHARACTER*1
Specifies the form of op( M ) to be used in the matrix
multiplication as follows:
= 'N': op( M ) = M;
= 'T': op( M ) = M';
= 'C': op( M ) = M'.
Input/Output Parameters
N (input) INTEGER
The order of the matrices R, H, E, and X. N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then R need not be
set before entry, except when R is identified with X in
the call.
BETA (input) DOUBLE PRECISION
The scalar beta. When beta is zero then H and X are not
referenced.
R (input/output) DOUBLE PRECISION array, dimension (LDR,N)
On entry with UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix R.
On entry with UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix R.
In both cases, the other strictly triangular part is not
referenced.
On exit, the leading N-by-N upper triangular part (if
UPLO = 'U'), or lower triangular part (if UPLO = 'L'), of
this array contains the corresponding triangular part of
the computed matrix R.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
H (input) DOUBLE PRECISION array, dimension (LDH,N)
On entry, the leading N-by-N upper Hessenberg part of this
array must contain the upper Hessenberg matrix H.
If TRANS = 'N', the entries 3, 4,..., N of the first
column are modified internally, but are restored on exit.
The remaining part of this array is not referenced.
LDH INTEGER
The leading dimension of array H. LDH >= MAX(1,N).
X (input) DOUBLE PRECISION array, dimension (LDX,N)
On entry, if UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix X and the strictly
lower triangular part of the array is not referenced.
On entry, if UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix X and the strictly
upper triangular part of the array is not referenced.
The diagonal elements of this array are modified
internally, but are restored on exit.
LDX INTEGER
The leading dimension of array X. LDX >= MAX(1,N).
E (input) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix E.
The remaining part of this array is not referenced.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,N).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
This array is not referenced when beta = 0, or N = 0.
LDWORK The length of the array DWORK.
LDWORK >= N*N, if beta <> 0;
LDWORK >= 0, if beta = 0.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -k, the k-th argument had an illegal
value.
Method
The matrix expression is efficiently evaluated taking the symmetry
into account. Specifically, let X = U + L, with U and L upper and
lower triangular matrices, defined by
U = triu( X ) - (1/2)*diag( X ),
L = tril( X ) - (1/2)*diag( X ),
where triu, tril, and diag denote the upper triangular part, lower
triangular part, and diagonal part of X, respectively. Then,
if UPLO = 'U',
H*X*E' + E*X*H' = (H*U)*E' + E*(H*U)' + H*(E*U)' + (E*U)*H',
for TRANS = 'N',
H'*X*E + E'*X*H = H'*(U*E) + (U*E)'*H + (U*H)'*E + E'*(U*H),
for TRANS = 'T', or 'C',
and if UPLO = 'L',
H*X*E' + E*X*H' = (H*L')*E' + E*(H*L')' + H*(E*L')' + (E*L')*H',
for TRANS = 'N',
H'*X*E + E'*X*H = H'*(L'*E) + (L'*E)'*H + (L'*H)'*E + E'*(L'*H),
for TRANS = 'T', or 'C',
which involve operations like in BLAS 2 and 3 (DTRMV and DSYR2K).
This approach ensures that the matrices H*U, U*H, H*L', or L'*H
are upper Hessenberg, and E*U, U*E, E*L', or L'*E are upper
triangular.
Numerical Aspects
The algorithm requires approximately N**3/2 operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute one of the symmetric rank 2k operations
R := alpha*R + beta*H*E' + beta*E*H',
or
R := alpha*R + beta*H'*E + beta*E'*H,
where alpha and beta are scalars, R, E, and H are N-by-N matrices,
with H upper Hessenberg and E upper triangular.
Specification
SUBROUTINE MB01OE( UPLO, TRANS, N, ALPHA, BETA, R, LDR, H, LDH, E,
$ LDE )
C .. Scalar Arguments ..
DOUBLE PRECISION ALPHA, BETA
INTEGER LDE, LDH, LDR, N
CHARACTER TRANS, UPLO
C .. Array Arguments ..
DOUBLE PRECISION E(LDE,*), H(LDH,*), R(LDR,*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies which triangle of the symmetric matrix R is
given as follows:
= 'U': the upper triangular part is given;
= 'L': the lower triangular part is given.
TRANS CHARACTER*1
Specifies the form of H to be used in the matrix
multiplication as follows:
= 'N': R := alpha*R + beta*H*E' + beta*E*H';
= 'T': R := alpha*R + beta*H'*E + beta*E'*H;
= 'C': R := alpha*R + beta*H'*E + beta*E'*H.
Input/Output Parameters
N (input) INTEGER
The order of the matrices R, E, and H. N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then R need not be
set before entry.
BETA (input) DOUBLE PRECISION
The scalar beta. When beta is zero then E and H are not
referenced.
R (input/output) DOUBLE PRECISION array, dimension (LDR,N)
On entry with UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix R.
On entry with UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix R.
In both cases, the other strictly triangular part is not
referenced.
On exit, the leading N-by-N upper triangular part (if
UPLO = 'U'), or lower triangular part (if UPLO = 'L'), of
this array contains the corresponding triangular part of
the computed matrix R.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
H (input) DOUBLE PRECISION array, dimension (LDH,N)
On entry, the leading N-by-N upper Hessenberg part of this
array must contain the upper Hessenberg matrix H.
The remaining part of this array is not referenced.
LDH INTEGER
The leading dimension of array H. LDH >= MAX(1,N).
E (input) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix E.
The remaining part of this array is not referenced.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,N).
Method
E particularization of the algorithm used in the BLAS 3 routine DSYR2K is used.Numerical Aspects
The algorithm requires approximately N**3/3 operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute one of the symmetric rank 2k operations
R := alpha*R + beta*H*A' + beta*A*H',
or
R := alpha*R + beta*H'*A + beta*A'*H,
where alpha and beta are scalars, R, A, and H are N-by-N matrices,
with A and H upper Hessenberg.
Specification
SUBROUTINE MB01OH( UPLO, TRANS, N, ALPHA, BETA, R, LDR, H, LDH, A,
$ LDA )
C .. Scalar Arguments ..
DOUBLE PRECISION ALPHA, BETA
INTEGER LDA, LDH, LDR, N
CHARACTER TRANS, UPLO
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), H(LDH,*), R(LDR,*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies which triangle of the symmetric matrix R is
given as follows:
= 'U': the upper triangular part is given;
= 'L': the lower triangular part is given.
TRANS CHARACTER*1
Specifies the form of H to be used in the matrix
multiplication as follows:
= 'N': R := alpha*R + beta*H*A' + beta*A*H';
= 'T': R := alpha*R + beta*H'*A + beta*A'*H;
= 'C': R := alpha*R + beta*H'*A + beta*A'*H.
Input/Output Parameters
N (input) INTEGER
The order of the matrices R, A, and H. N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then R need not be
set before entry.
BETA (input) DOUBLE PRECISION
The scalar beta. When beta is zero then A and H are not
referenced.
R (input/output) DOUBLE PRECISION array, dimension (LDR,N)
On entry with UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix R.
On entry with UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix R.
In both cases, the other strictly triangular part is not
referenced.
On exit, the leading N-by-N upper triangular part (if
UPLO = 'U'), or lower triangular part (if UPLO = 'L'), of
this array contains the corresponding triangular part of
the computed matrix R.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
H (input) DOUBLE PRECISION array, dimension (LDH,N)
On entry, the leading N-by-N upper Hessenberg part of this
array must contain the upper Hessenberg matrix H.
The remaining part of this array is not referenced.
LDH INTEGER
The leading dimension of array H. LDH >= MAX(1,N)
A (input) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N upper Hessenberg part of this
array must contain the upper Hessenberg matrix A.
The remaining part of this array is not referenced.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N)
Method
A particularization of the algorithm used in the BLAS 3 routine DSYR2K is used.Numerical Aspects
The algorithm requires approximately N**3/3 operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute either P or P', with P defined by the matrix formula
P = op( H )*X*op( E )',
where H is an upper Hessenberg matrix, X is a symmetric matrix,
E is an upper triangular matrix, and op( M ) is one of
op( M ) = M or op( M ) = M'.
Specification
SUBROUTINE MB01OO( UPLO, TRANS, N, H, LDH, X, LDX, E, LDE, P, LDP,
$ INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDE, LDH, LDP, LDX, N
CHARACTER TRANS, UPLO
C .. Array Arguments ..
DOUBLE PRECISION E(LDE,*), H(LDH,*), P(LDP,*), X(LDX,*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies which triangle of the symmetric matrix X is
given as follows:
= 'U': the upper triangular part is given;
= 'L': the lower triangular part is given.
TRANS CHARACTER*1
Specifies the operation to be performed as follows:
= 'N': compute P = H*X*E';
= 'T' or 'C': compute P' = E'*X*H.
Input/Output Parameters
N (input) INTEGER
The order of the matrices H, X, E, and P. N >= 0.
H (input) DOUBLE PRECISION array, dimension (LDH,N)
On entry, the leading N-by-N upper Hessenberg part of this
array must contain the upper Hessenberg matrix H.
The remaining part of this array is not referenced.
LDH INTEGER
The leading dimension of the array H. LDH >= MAX(1,N).
X (input) DOUBLE PRECISION array, dimension (LDX,N)
On entry, if UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix X and the strictly
lower triangular part of the array is not referenced.
On entry, if UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix X and the strictly
upper triangular part of the array is not referenced.
LDX INTEGER
The leading dimension of the array X. LDX >= MAX(1,N).
E (input) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix E.
The remaining part of this array is not referenced.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,N).
P (output) DOUBLE PRECISION array, dimension (LDP,N)
On exit, the leading N-by-N part of this array contains
the computed matrix P = H*X*E', if TRANS = 'N', or
the computed matrix P' = E'*X*H, if TRANS = 'T'.
LDP INTEGER
The leading dimension of the array P. LDP >= MAX(1,N).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -k, the k-th argument had an illegal
value.
Method
The matrix expression is efficiently evaluated taking the structure into account, and using BLAS and SLICOT routines. Let W = H*X, or W = X*H, computed using SLICOT Library routine MB01OS. The result is then obtained calling BLAS 3 routine DTRMM.Numerical Aspects
The algorithm requires approximately N**3 operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute P = H*X or P = X*H, where H is an upper Hessenberg matrix and X is a symmetric matrix.Specification
SUBROUTINE MB01OS( UPLO, TRANS, N, H, LDH, X, LDX, P, LDP, INFO )
C .. Scalar Arguments ..
CHARACTER TRANS, UPLO
INTEGER INFO, LDH, LDP, LDX, N
C .. Array Arguments ..
DOUBLE PRECISION H(LDH,*), P(LDP,*), X(LDX,*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies which triangle of the symmetric matrix X is
given as follows:
= 'U': the upper triangular part is given;
= 'L': the lower triangular part is given.
TRANS CHARACTER*1
Specifies the operation to be performed as follows:
= 'N': compute P = H*X;
= 'T' or 'C': compute P = X*H.
Input/Output Parameters
N (input) INTEGER
The order of the matrices H, X, and P. N >= 0.
H (input) DOUBLE PRECISION array, dimension (LDH,N)
On entry, the leading N-by-N upper Hessenberg part of this
array must contain the upper Hessenberg matrix H.
The remaining part of this array is not referenced.
LDH INTEGER
The leading dimension of the array H. LDH >= MAX(1,N).
X (input) DOUBLE PRECISION array, dimension (LDX,N)
On entry, if UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix X and the strictly
lower triangular part of the array is not referenced.
On entry, if UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix X and the strictly
upper triangular part of the array is not referenced.
LDX INTEGER
The leading dimension of the array X. LDX >= MAX(1,N).
P (output) DOUBLE PRECISION array, dimension (LDP,N)
On exit, the leading N-by-N part of this array contains
the computed matrix P.
LDP INTEGER
The leading dimension of the array P. LDP >= MAX(1,N).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -k, the k-th argument had an illegal
value.
Method
The matrix expression is efficiently evaluated taking the structure into account, and using inline code and BLAS routines. Let X = U + sL, where U is upper triangular and sL is strictly lower triangular. Then, P = H*X = H*U + H*sL = H*U + H*sU', where sU is the strictly upper triangular part of X. Similarly, P = X*H = L'*H + sL*H, where L is lower triangular, and X = L + sL'. Note that H*U and L'*H are both upper Hessenberg. However, when UPLO = 'L' and TRANS = 'N', or when UPLO = 'U' and TRANS = 'T', then the matrix P is full. The computations are done similarly.Numerical Aspects
The algorithm requires approximately N**3/2 operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute one of the symmetric rank 2k operations
R := alpha*R + beta*E*T' + beta*T*E',
or
R := alpha*R + beta*E'*T + beta*T'*E,
where alpha and beta are scalars, R, T, and E are N-by-N matrices,
with T and E upper triangular.
Specification
SUBROUTINE MB01OT( UPLO, TRANS, N, ALPHA, BETA, R, LDR, E, LDE, T,
$ LDT )
C .. Scalar Arguments ..
DOUBLE PRECISION ALPHA, BETA
INTEGER LDE, LDR, LDT, N
CHARACTER TRANS, UPLO
C .. Array Arguments ..
DOUBLE PRECISION E(LDE,*), R(LDR,*), T(LDT,*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies which triangle of the symmetric matrix R is
given as follows:
= 'U': the upper triangular part is given;
= 'L': the lower triangular part is given.
TRANS CHARACTER*1
Specifies the form of E to be used in the matrix
multiplication as follows:
= 'N': R := alpha*R + beta*E*T' + beta*T*E';
= 'T': R := alpha*R + beta*E'*T + beta*T'*E;
= 'C': R := alpha*R + beta*E'*T + beta*T'*E.
Input/Output Parameters
N (input) INTEGER
The order of the matrices R, T, and E. N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then R need not be
set before entry.
BETA (input) DOUBLE PRECISION
The scalar beta. When beta is zero then T and E are not
referenced.
R (input/output) DOUBLE PRECISION array, dimension (LDR,N)
On entry with UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix R.
On entry with UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix R.
In both cases, the other strictly triangular part is not
referenced.
On exit, the leading N-by-N upper triangular part (if
UPLO = 'U'), or lower triangular part (if UPLO = 'L'), of
this array contains the corresponding triangular part of
the computed matrix R.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
E (input) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix E.
The remaining part of this array is not referenced.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,N).
T (input) DOUBLE PRECISION array, dimension (LDT,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix T.
The remaining part of this array is not referenced.
LDT INTEGER
The leading dimension of array T. LDT >= MAX(1,N).
Method
A particularization of the algorithm used in the BLAS 3 routine DSYR2K is used.Numerical Aspects
The algorithm requires approximately N**3/3 operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To scale a matrix or undo scaling. Scaling is performed, if necessary, so that the matrix norm will be in a safe range of representable numbers.Specification
SUBROUTINE MB01PD( SCUN, TYPE, M, N, KL, KU, ANRM, NBL, NROWS, A,
$ LDA, INFO )
C .. Scalar Arguments ..
CHARACTER SCUN, TYPE
INTEGER INFO, KL, KU, LDA, M, MN, N, NBL
DOUBLE PRECISION ANRM
C .. Array Arguments ..
INTEGER NROWS ( * )
DOUBLE PRECISION A( LDA, * )
Arguments
Mode Parameters
SCUN CHARACTER*1
SCUN indicates the operation to be performed.
= 'S': scale the matrix.
= 'U': undo scaling of the matrix.
TYPE CHARACTER*1
TYPE indicates the storage type of the input matrix.
= 'G': A is a full matrix.
= 'L': A is a (block) lower triangular matrix.
= 'U': A is an (block) upper triangular matrix.
= 'H': A is an (block) upper Hessenberg matrix.
= 'B': A is a symmetric band matrix with lower bandwidth
KL and upper bandwidth KU and with the only the
lower half stored.
= 'Q': A is a symmetric band matrix with lower bandwidth
KL and upper bandwidth KU and with the only the
upper half stored.
= 'Z': A is a band matrix with lower bandwidth KL and
upper bandwidth KU.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
KL (input) INTEGER
The lower bandwidth of A. Referenced only if TYPE = 'B',
'Q' or 'Z'.
KU (input) INTEGER
The upper bandwidth of A. Referenced only if TYPE = 'B',
'Q' or 'Z'.
ANRM (input) DOUBLE PRECISION
The norm of the initial matrix A. ANRM >= 0.
When ANRM = 0 then an immediate return is effected.
ANRM should be preserved between the call of the routine
with SCUN = 'S' and the corresponding one with SCUN = 'U'.
NBL (input) INTEGER
The number of diagonal blocks of the matrix A, if it has a
block structure. To specify that matrix A has no block
structure, set NBL = 0. NBL >= 0.
NROWS (input) INTEGER array, dimension max(1,NBL)
NROWS(i) contains the number of rows and columns of the
i-th diagonal block of matrix A. The sum of the values
NROWS(i), for i = 1: NBL, should be equal to min(M,N).
The elements of the array NROWS are not referenced if
NBL = 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading M by N part of this array must
contain the matrix to be scaled/unscaled.
On exit, the leading M by N part of A will contain
the modified matrix.
The storage mode of A is specified by TYPE.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
Error Indicator
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Denote by ANRM the norm of the matrix, and by SMLNUM and BIGNUM, two positive numbers near the smallest and largest safely representable numbers, respectively. The matrix is scaled, if needed, such that the norm of the result is in the range [SMLNUM, BIGNUM]. The scaling factor is represented as a ratio of two numbers, one of them being ANRM, and the other one either SMLNUM or BIGNUM, depending on ANRM being less than SMLNUM or larger than BIGNUM, respectively. For undoing the scaling, the norm is again compared with SMLNUM or BIGNUM, and the reciprocal of the previous scaling factor is used.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB01QD.html 0000664 0000000 0000000 00000010576 14560147231 0020347 0 ustar 00root root 0000000 0000000
Purpose
To multiply the M by N real matrix A by the real scalar CTO/CFROM. This is done without over/underflow as long as the final result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that A may be full, (block) upper triangular, (block) lower triangular, (block) upper Hessenberg, or banded.Specification
SUBROUTINE MB01QD( TYPE, M, N, KL, KU, CFROM, CTO, NBL, NROWS, A,
$ LDA, INFO )
C .. Scalar Arguments ..
CHARACTER TYPE
INTEGER INFO, KL, KU, LDA, M, N, NBL
DOUBLE PRECISION CFROM, CTO
C .. Array Arguments ..
INTEGER NROWS ( * )
DOUBLE PRECISION A( LDA, * )
Arguments
Mode Parameters
TYPE CHARACTER*1
TYPE indices the storage type of the input matrix.
= 'G': A is a full matrix.
= 'L': A is a (block) lower triangular matrix.
= 'U': A is a (block) upper triangular matrix.
= 'H': A is a (block) upper Hessenberg matrix.
= 'B': A is a symmetric band matrix with lower bandwidth
KL and upper bandwidth KU and with the only the
lower half stored.
= 'Q': A is a symmetric band matrix with lower bandwidth
KL and upper bandwidth KU and with the only the
upper half stored.
= 'Z': A is a band matrix with lower bandwidth KL and
upper bandwidth KU.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
KL (input) INTEGER
The lower bandwidth of A. Referenced only if TYPE = 'B',
'Q' or 'Z'.
KU (input) INTEGER
The upper bandwidth of A. Referenced only if TYPE = 'B',
'Q' or 'Z'.
CFROM (input) DOUBLE PRECISION
CTO (input) DOUBLE PRECISION
The matrix A is multiplied by CTO/CFROM. A(I,J) is
computed without over/underflow if the final result
CTO*A(I,J)/CFROM can be represented without over/
underflow. CFROM must be nonzero.
NBL (input) INTEGER
The number of diagonal blocks of the matrix A, if it has a
block structure. To specify that matrix A has no block
structure, set NBL = 0. NBL >= 0.
NROWS (input) INTEGER array, dimension max(1,NBL)
NROWS(i) contains the number of rows and columns of the
i-th diagonal block of matrix A. The sum of the values
NROWS(i), for i = 1: NBL, should be equal to min(M,N).
The array NROWS is not referenced if NBL = 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
The matrix to be multiplied by CTO/CFROM. See TYPE for
the storage type.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
Error Indicator
INFO INTEGER
Not used in this implementation.
Method
Matrix A is multiplied by the real scalar CTO/CFROM, taking into account the specified storage mode of the matrix. MB01QD is a version of the LAPACK routine DLASCL, modified for dealing with block triangular, or block Hessenberg matrices. For efficiency, no tests of the input scalar parameters are performed.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB01RB.html 0000664 0000000 0000000 00000015223 14560147231 0020340 0 ustar 00root root 0000000 0000000
Purpose
To compute either the upper or lower triangular part of one of the
matrix formulas
_
R = alpha*R + beta*op( A )*B, (1)
_
R = alpha*R + beta*B*op( A ), (2)
_
where alpha and beta are scalars, R and R are m-by-m matrices,
op( A ) and B are m-by-n and n-by-m matrices for (1), or n-by-m
and m-by-n matrices for (2), respectively, and op( A ) is one of
op( A ) = A or op( A ) = A', the transpose of A.
The result is overwritten on R.
Specification
SUBROUTINE MB01RB( SIDE, UPLO, TRANS, M, N, ALPHA, BETA, R, LDR,
$ A, LDA, B, LDB, INFO )
C .. Scalar Arguments ..
CHARACTER SIDE, TRANS, UPLO
INTEGER INFO, LDA, LDB, LDR, M, N
DOUBLE PRECISION ALPHA, BETA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), R(LDR,*)
Arguments
Mode Parameters
SIDE CHARACTER*1
Specifies whether the matrix A appears on the left or
right in the matrix product as follows:
_
= 'L': R = alpha*R + beta*op( A )*B;
_
= 'R': R = alpha*R + beta*B*op( A ).
UPLO CHARACTER*1 _
Specifies which triangles of the matrices R and R are
computed and given, respectively, as follows:
= 'U': the upper triangular part;
= 'L': the lower triangular part.
TRANS CHARACTER*1
Specifies the form of op( A ) to be used in the matrix
multiplication as follows:
= 'N': op( A ) = A;
= 'T': op( A ) = A';
= 'C': op( A ) = A'.
Input/Output Parameters
M (input) INTEGER _
The order of the matrices R and R, the number of rows of
the matrix op( A ) and the number of columns of the
matrix B, for SIDE = 'L', or the number of rows of the
matrix B and the number of columns of the matrix op( A ),
for SIDE = 'R'. M >= 0.
N (input) INTEGER
The number of rows of the matrix B and the number of
columns of the matrix op( A ), for SIDE = 'L', or the
number of rows of the matrix op( A ) and the number of
columns of the matrix B, for SIDE = 'R'. N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then R need not be
set before entry.
BETA (input) DOUBLE PRECISION
The scalar beta. When beta is zero then A and B are not
referenced.
R (input/output) DOUBLE PRECISION array, dimension (LDR,M)
On entry with UPLO = 'U', the leading M-by-M upper
triangular part of this array must contain the upper
triangular part of the matrix R; the strictly lower
triangular part of the array is not referenced.
On entry with UPLO = 'L', the leading M-by-M lower
triangular part of this array must contain the lower
triangular part of the matrix R; the strictly upper
triangular part of the array is not referenced.
On exit, the leading M-by-M upper triangular part (if
UPLO = 'U'), or lower triangular part (if UPLO = 'L') of
this array contains the corresponding triangular part of
_
the computed matrix R.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,M).
A (input) DOUBLE PRECISION array, dimension (LDA,k), where
k = N when SIDE = 'L', and TRANS = 'N', or
SIDE = 'R', and TRANS <> 'T';
k = M when SIDE = 'R', and TRANS = 'N', or
SIDE = 'L', and TRANS <> 'T'.
On entry, if SIDE = 'L', and TRANS = 'N', or
SIDE = 'R', and TRANS <> 'T',
the leading M-by-N part of this array must contain the
matrix A.
On entry, if SIDE = 'R', and TRANS = 'N', or
SIDE = 'L', and TRANS <> 'T',
the leading N-by-M part of this array must contain the
matrix A.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,l), where
l = M when SIDE = 'L', and TRANS = 'N', or
SIDE = 'R', and TRANS <> 'T';
l = N when SIDE = 'R', and TRANS = 'N', or
SIDE = 'L', and TRANS <> 'T'.
B (input) DOUBLE PRECISION array, dimension (LDB,p), where
p = M when SIDE = 'L';
p = N when SIDE = 'R'.
On entry, the leading N-by-M part, if SIDE = 'L', or
M-by-N part, if SIDE = 'R', of this array must contain the
matrix B.
LDB INTEGER
The leading dimension of array B.
LDB >= MAX(1,N), if SIDE = 'L';
LDB >= MAX(1,M), if SIDE = 'R'.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The matrix expression is evaluated taking the triangular structure into account. A block algorithm is used.Further Comments
The main application of this routine is when the result should be a symmetric matrix, e.g., when B = X*op( A )', for (1), or B = op( A )'*X, for (2), where B is already available and X = X'. The required triangle only is computed and overwritten, contrary to a general matrix multiplication operation. This is a BLAS 3 version of the SLICOT Library routine MB01RX.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB01RD.html 0000664 0000000 0000000 00000016477 14560147231 0020356 0 ustar 00root root 0000000 0000000
Purpose
To compute the matrix formula
_
R = alpha*R + beta*op( A )*X*op( A )',
_
where alpha and beta are scalars, R, X, and R are symmetric
matrices, A is a general matrix, and op( A ) is one of
op( A ) = A or op( A ) = A'.
The result is overwritten on R.
Specification
SUBROUTINE MB01RD( UPLO, TRANS, M, N, ALPHA, BETA, R, LDR, A, LDA,
$ X, LDX, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TRANS, UPLO
INTEGER INFO, LDA, LDR, LDWORK, LDX, M, N
DOUBLE PRECISION ALPHA, BETA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), R(LDR,*), X(LDX,*)
Arguments
Mode Parameters
UPLO CHARACTER*1 _
Specifies which triangles of the symmetric matrices R, R,
and X are given as follows:
= 'U': the upper triangular part is given;
= 'L': the lower triangular part is given.
TRANS CHARACTER*1
Specifies the form of op( A ) to be used in the matrix
multiplication as follows:
= 'N': op( A ) = A;
= 'T': op( A ) = A';
= 'C': op( A ) = A'.
Input/Output Parameters
M (input) INTEGER _
The order of the matrices R and R and the number of rows
of the matrix op( A ). M >= 0.
N (input) INTEGER
The order of the matrix X and the number of columns of the
the matrix op( A ). N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then R need not be
set before entry, except when R is identified with X in
the call (which is possible only in this case).
BETA (input) DOUBLE PRECISION
The scalar beta. When beta is zero then A and X are not
referenced.
R (input/output) DOUBLE PRECISION array, dimension (LDR,M)
On entry with UPLO = 'U', the leading M-by-M upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix R; the strictly
lower triangular part of the array is used as workspace.
On entry with UPLO = 'L', the leading M-by-M lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix R; the strictly
upper triangular part of the array is used as workspace.
On exit, the leading M-by-M upper triangular part (if
UPLO = 'U'), or lower triangular part (if UPLO = 'L'), of
this array contains the corresponding triangular part of
_
the computed matrix R. If beta <> 0, the remaining
strictly triangular part of this array contains the
corresponding part of the matrix expression
beta*op( A )*T*op( A )', where T is the triangular matrix
defined in the Method section.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,M).
A (input) DOUBLE PRECISION array, dimension (LDA,k)
where k is N when TRANS = 'N' and is M when TRANS = 'T' or
TRANS = 'C'.
On entry with TRANS = 'N', the leading M-by-N part of this
array must contain the matrix A.
On entry with TRANS = 'T' or TRANS = 'C', the leading
N-by-M part of this array must contain the matrix A.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,l),
where l is M when TRANS = 'N' and is N when TRANS = 'T' or
TRANS = 'C'.
X (input/output) DOUBLE PRECISION array, dimension (LDX,N)
On entry, if UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix X and the strictly
lower triangular part of the array is not referenced.
On entry, if UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix X and the strictly
upper triangular part of the array is not referenced.
On exit, each diagonal element of this array has half its
input value, but the other elements are not modified.
LDX INTEGER
The leading dimension of array X. LDX >= MAX(1,N).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, the leading M-by-N part of this
array (with the leading dimension MAX(1,M)) returns the
matrix product beta*op( A )*T, where T is the triangular
matrix defined in the Method section.
This array is not referenced when beta = 0.
LDWORK The length of the array DWORK.
LDWORK >= MAX(1,M*N), if beta <> 0;
LDWORK >= 1, if beta = 0.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -k, the k-th argument had an illegal
value.
Method
The matrix expression is efficiently evaluated taking the symmetry
into account. Specifically, let X = T + T', with T an upper or
lower triangular matrix, defined by
T = triu( X ) - (1/2)*diag( X ), if UPLO = 'U',
T = tril( X ) - (1/2)*diag( X ), if UPLO = 'L',
where triu, tril, and diag denote the upper triangular part, lower
triangular part, and diagonal part of X, respectively. Then,
op( A )*X*op( A )' = B + B',
where B := op( A )*T*op( A )'. Matrix B is not symmetric, but it
can be written as tri( B ) + stri( B ), where tri denotes the
triangular part specified by UPLO, and stri denotes the remaining
strictly triangular part. Let R = V + V', with V defined as T
above. Then, the required triangular part of the result can be
written as
alpha*V + beta*tri( B ) + beta*(stri( B ))' +
alpha*diag( V ) + beta*diag( tri( B ) ).
References
None.Numerical Aspects
The algorithm requires approximately
2 2
M x N + 1/2 x N x M
operations.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB01RH.html 0000664 0000000 0000000 00000014154 14560147231 0020350 0 ustar 00root root 0000000 0000000
Purpose
To compute the matrix formula
R := alpha*R + beta*op( H )*X*op( H )',
where alpha and beta are scalars, R and X are symmetric matrices,
H is an upper Hessenberg matrix, and op( H ) is one of
op( H ) = H or op( H ) = H'.
The result is overwritten on R.
Specification
SUBROUTINE MB01RH( UPLO, TRANS, N, ALPHA, BETA, R, LDR, H, LDH,
$ X, LDX, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TRANS, UPLO
INTEGER INFO, LDH, LDR, LDWORK, LDX, N
DOUBLE PRECISION ALPHA, BETA
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), H(LDH,*), R(LDR,*), X(LDX,*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies which triangles of the symmetric matrices R
and X are given as follows:
= 'U': the upper triangular part is given;
= 'L': the lower triangular part is given.
TRANS CHARACTER*1
Specifies the form of op( H ) to be used in the matrix
multiplication as follows:
= 'N': op( H ) = H;
= 'T': op( H ) = H';
= 'C': op( H ) = H'.
Input/Output Parameters
N (input) INTEGER
The order of the matrices R, H, and X. N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then R need not be
set before entry, except when R is identified with X in
the call.
BETA (input) DOUBLE PRECISION
The scalar beta. When beta is zero then H and X are not
referenced.
R (input/output) DOUBLE PRECISION array, dimension (LDR,N)
On entry with UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix R.
On entry with UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix R.
In both cases, the other strictly triangular part is not
referenced.
On exit, the leading N-by-N upper triangular part (if
UPLO = 'U'), or lower triangular part (if UPLO = 'L'), of
this array contains the corresponding triangular part of
the computed matrix R.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
H (input) DOUBLE PRECISION array, dimension (LDH,N)
On entry, the leading N-by-N upper Hessenberg part of this
array must contain the upper Hessenberg matrix H.
If TRANS = 'N', the entries 3, 4,..., N of the first
column are modified internally, but are restored on exit.
The remaining part of this array is not referenced.
LDH INTEGER
The leading dimension of array H. LDH >= MAX(1,N).
X (input) DOUBLE PRECISION array, dimension (LDX,N)
On entry, if UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix X and the strictly
lower triangular part of the array is not referenced.
On entry, if UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix X and the strictly
upper triangular part of the array is not referenced.
The diagonal elements of this array are modified
internally, but are restored on exit.
LDX INTEGER
The leading dimension of array X. LDX >= MAX(1,N).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
This array is not referenced when beta = 0, or N = 0.
LDWORK The length of the array DWORK.
LDWORK >= N*N, if beta <> 0;
LDWORK >= 0, if beta = 0.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -k, the k-th argument had an illegal
value.
Method
The matrix expression is efficiently evaluated taking the symmetry
into account. Specifically, let X = U + L, with U and L upper and
lower triangular matrices, defined by
U = triu( X ) - (1/2)*diag( X ),
L = tril( X ) - (1/2)*diag( X ),
where triu, tril, and diag denote the upper triangular part, lower
triangular part, and diagonal part of X, respectively. Then,
if UPLO = 'U',
H*X*H' = ( H*U )*H' + H*( H*U )', for TRANS = 'N',
H'*X*H = H'*( U*H ) + ( U*H )'*H, for TRANS = 'T', or 'C',
and if UPLO = 'L',
H*X*H' = ( H*L' )*H' + H*( H*L' )', for TRANS = 'N',
H'*X*H = H'*( L'*H ) + ( L'*H )'*H, for TRANS = 'T', or 'C',
which involve operations like in BLAS 2 and 3 (DTRMV and DSYR2K).
This approach ensures that the matrices H*U, U*H, H*L', or L'*H
are upper Hessenberg.
Numerical Aspects
The algorithm requires approximately N**3/2 operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the matrix formula
R := alpha*R + beta*op( E )*X*op( E )',
where alpha and beta are scalars, R and X are symmetric matrices,
E is an upper triangular matrix, and op( E ) is one of
op( E ) = E or op( E ) = E'.
The result is overwritten on R.
Specification
SUBROUTINE MB01RT( UPLO, TRANS, N, ALPHA, BETA, R, LDR, E, LDE,
$ X, LDX, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TRANS, UPLO
INTEGER INFO, LDE, LDR, LDWORK, LDX, N
DOUBLE PRECISION ALPHA, BETA
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), E(LDE,*), R(LDR,*), X(LDX,*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies which triangles of the symmetric matrices R
and X are given as follows:
= 'U': the upper triangular part is given;
= 'L': the lower triangular part is given.
TRANS CHARACTER*1
Specifies the form of op( E ) to be used in the matrix
multiplication as follows:
= 'N': op( E ) = E;
= 'T': op( E ) = E';
= 'C': op( E ) = E'.
Input/Output Parameters
N (input) INTEGER
The order of the matrices R, E, and X. N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then R need not be
set before entry, except when R is identified with X in
the call.
BETA (input) DOUBLE PRECISION
The scalar beta. When beta is zero then E and X are not
referenced.
R (input/output) DOUBLE PRECISION array, dimension (LDR,N)
On entry with UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix R.
On entry with UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix R.
In both cases, the other strictly triangular part is not
referenced.
On exit, the leading N-by-N upper triangular part (if
UPLO = 'U'), or lower triangular part (if UPLO = 'L'), of
this array contains the corresponding triangular part of
the computed matrix R.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
E (input) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix E.
The remaining part of this array is not referenced.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,N).
X (input) DOUBLE PRECISION array, dimension (LDX,N)
On entry, if UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix X and the strictly
lower triangular part of the array is not referenced.
On entry, if UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix X and the strictly
upper triangular part of the array is not referenced.
The diagonal elements of this array are modified
internally, but are restored on exit.
LDX INTEGER
The leading dimension of array X. LDX >= MAX(1,N).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
This array is not referenced when beta = 0, or N = 0.
LDWORK The length of the array DWORK.
LDWORK >= N*N, if beta <> 0;
LDWORK >= 0, if beta = 0.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -k, the k-th argument had an illegal
value.
Method
The matrix expression is efficiently evaluated taking the symmetry
into account. Specifically, let X = U + L, with U and L upper and
lower triangular matrices, defined by
U = triu( X ) - (1/2)*diag( X ),
L = tril( X ) - (1/2)*diag( X ),
where triu, tril, and diag denote the upper triangular part, lower
triangular part, and diagonal part of X, respectively. Then,
if UPLO = 'U',
E*X*E' = ( E*U )*E' + E*( E*U )', for TRANS = 'N',
E'*X*E = E'*( U*E ) + ( U*E )'*E, for TRANS = 'T', or 'C',
and if UPLO = 'L',
E*X*E' = ( E*L' )*E' + E*( E*L' )', for TRANS = 'N',
E'*X*E = E'*( L'*E ) + ( L'*E )'*E, for TRANS = 'T', or 'C',
which involve operations like in BLAS 2 and 3 (DTRMV and DSYR2K).
This approach ensures that the matrices E*U, U*E, E*L', or L'*E
are upper triangular.
Numerical Aspects
The algorithm requires approximately N**3/2 operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the matrix formula
_
R = alpha*R + beta*op( A )*X*op( A )',
_
where alpha and beta are scalars, R, X, and R are symmetric
matrices, A is a general matrix, and op( A ) is one of
op( A ) = A or op( A ) = A'.
The result is overwritten on R.
Specification
SUBROUTINE MB01RU( UPLO, TRANS, M, N, ALPHA, BETA, R, LDR, A, LDA,
$ X, LDX, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TRANS, UPLO
INTEGER INFO, LDA, LDR, LDWORK, LDX, M, N
DOUBLE PRECISION ALPHA, BETA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), R(LDR,*), X(LDX,*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies which triangles of the symmetric matrices R
and X are given as follows:
= 'U': the upper triangular part is given;
= 'L': the lower triangular part is given.
TRANS CHARACTER*1
Specifies the form of op( A ) to be used in the matrix
multiplication as follows:
= 'N': op( A ) = A;
= 'T': op( A ) = A';
= 'C': op( A ) = A'.
Input/Output Parameters
M (input) INTEGER _
The order of the matrices R and R and the number of rows
of the matrix op( A ). M >= 0.
N (input) INTEGER
The order of the matrix X and the number of columns of the
the matrix op( A ). N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then R need not be
set before entry, except when R is identified with X in
the call.
BETA (input) DOUBLE PRECISION
The scalar beta. When beta is zero then A and X are not
referenced.
R (input/output) DOUBLE PRECISION array, dimension (LDR,M)
On entry with UPLO = 'U', the leading M-by-M upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix R.
On entry with UPLO = 'L', the leading M-by-M lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix R.
On exit, the leading M-by-M upper triangular part (if
UPLO = 'U'), or lower triangular part (if UPLO = 'L'), of
this array contains the corresponding triangular part of
_
the computed matrix R. When R is identified with X in
the call, after exit, the diagonal entries of R must be
divided by 2.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,M).
A (input) DOUBLE PRECISION array, dimension (LDA,k)
where k is N when TRANS = 'N' and is M when TRANS = 'T' or
TRANS = 'C'.
On entry with TRANS = 'N', the leading M-by-N part of this
array must contain the matrix A.
On entry with TRANS = 'T' or TRANS = 'C', the leading
N-by-M part of this array must contain the matrix A.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,k),
where k is M when TRANS = 'N' and is N when TRANS = 'T' or
TRANS = 'C'.
X (input) DOUBLE PRECISION array, dimension (LDX,N)
On entry, if UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix X and the strictly
lower triangular part of the array is not referenced.
On entry, if UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix X and the strictly
upper triangular part of the array is not referenced.
The diagonal elements of this array are modified
internally, but are restored on exit.
LDX INTEGER
The leading dimension of array X. LDX >= MAX(1,N).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
This array is not referenced when beta = 0, or M*N = 0.
LDWORK The length of the array DWORK.
LDWORK >= M*N, if beta <> 0;
LDWORK >= 0, if beta = 0.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -k, the k-th argument had an illegal
value.
Method
The matrix expression is efficiently evaluated taking the symmetry
into account. Specifically, let X = T + T', with T an upper or
lower triangular matrix, defined by
T = triu( X ) - (1/2)*diag( X ), if UPLO = 'U',
T = tril( X ) - (1/2)*diag( X ), if UPLO = 'L',
where triu, tril, and diag denote the upper triangular part, lower
triangular part, and diagonal part of X, respectively. Then,
A*X*A' = ( A*T )*A' + A*( A*T )', for TRANS = 'N',
A'*X*A = A'*( T*A ) + ( T*A )'*A, for TRANS = 'T', or 'C',
which involve BLAS 3 operations (DTRMM and DSYR2K).
Numerical Aspects
The algorithm requires approximately
2 2
M x N + 1/2 x N x M
operations.
Further Comments
This is a simpler version for MB01RD.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the transformation of the symmetric matrix A by the
matrix Z in the form
A := op(Z)*A*op(Z)',
where op(Z) is either Z or its transpose, Z'.
Specification
SUBROUTINE MB01RW( UPLO, TRANS, M, N, A, LDA, Z, LDZ, DWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER TRANS, UPLO
INTEGER INFO, LDA, LDZ, M, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), Z(LDZ,*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies whether the upper or lower triangle of A
is stored:
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
TRANS CHARACTER*1
Specifies whether op(Z) is Z or its transpose Z':
= 'N': op(Z) = Z;
= 'T': op(Z) = Z'.
Input/Output Parameters
M (input) INTEGER
The order of the resulting symmetric matrix op(Z)*A*op(Z)'
and the number of rows of the matrix Z, if TRANS = 'N',
or the number of columns of the matrix Z, if TRANS = 'T'.
M >= 0.
N (input) INTEGER
The order of the symmetric matrix A and the number of
columns of the matrix Z, if TRANS = 'N', or the number of
rows of the matrix Z, if TRANS = 'T'. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension
(LDA,MAX(M,N))
On entry, the leading N-by-N upper or lower triangular
part of this array must contain the upper (UPLO = 'U')
or lower (UPLO = 'L') triangular part of the symmetric
matrix A.
On exit, the leading M-by-M upper or lower triangular
part of this array contains the upper (UPLO = 'U') or
lower (UPLO = 'L') triangular part of the symmetric
matrix op(Z)*A*op(Z)'.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,M,N).
Z (input) DOUBLE PRECISION array, dimension (LDQ,K)
where K = N if TRANS = 'N' and K = M if TRANS = 'T'.
The leading M-by-N part, if TRANS = 'N', or N-by-M part,
if TRANS = 'T', of this array contains the matrix Z.
LDZ INTEGER
The leading dimension of the array Z.
LDZ >= MAX(1,M) if TRANS = 'N' and
LDZ >= MAX(1,N) if TRANS = 'T'.
Workspace
DWORK DOUBLE PRECISION array, dimension (N)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Further Comments
This is a simpler, BLAS 2 version for MB01RD.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute either the upper or lower triangular part of one of the
matrix formulas
_
R = alpha*R + beta*op( A )*B, (1)
_
R = alpha*R + beta*B*op( A ), (2)
_
where alpha and beta are scalars, R and R are m-by-m matrices,
op( A ) and B are m-by-n and n-by-m matrices for (1), or n-by-m
and m-by-n matrices for (2), respectively, and op( A ) is one of
op( A ) = A or op( A ) = A', the transpose of A.
The result is overwritten on R.
Specification
SUBROUTINE MB01RX( SIDE, UPLO, TRANS, M, N, ALPHA, BETA, R, LDR,
$ A, LDA, B, LDB, INFO )
C .. Scalar Arguments ..
CHARACTER SIDE, TRANS, UPLO
INTEGER INFO, LDA, LDB, LDR, M, N
DOUBLE PRECISION ALPHA, BETA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), R(LDR,*)
Arguments
Mode Parameters
SIDE CHARACTER*1
Specifies whether the matrix A appears on the left or
right in the matrix product as follows:
_
= 'L': R = alpha*R + beta*op( A )*B;
_
= 'R': R = alpha*R + beta*B*op( A ).
UPLO CHARACTER*1 _
Specifies which triangles of the matrices R and R are
computed and given, respectively, as follows:
= 'U': the upper triangular part;
= 'L': the lower triangular part.
TRANS CHARACTER*1
Specifies the form of op( A ) to be used in the matrix
multiplication as follows:
= 'N': op( A ) = A;
= 'T': op( A ) = A';
= 'C': op( A ) = A'.
Input/Output Parameters
M (input) INTEGER _
The order of the matrices R and R, the number of rows of
the matrix op( A ) and the number of columns of the
matrix B, for SIDE = 'L', or the number of rows of the
matrix B and the number of columns of the matrix op( A ),
for SIDE = 'R'. M >= 0.
N (input) INTEGER
The number of rows of the matrix B and the number of
columns of the matrix op( A ), for SIDE = 'L', or the
number of rows of the matrix op( A ) and the number of
columns of the matrix B, for SIDE = 'R'. N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then R need not be
set before entry.
BETA (input) DOUBLE PRECISION
The scalar beta. When beta is zero then A and B are not
referenced.
R (input/output) DOUBLE PRECISION array, dimension (LDR,M)
On entry with UPLO = 'U', the leading M-by-M upper
triangular part of this array must contain the upper
triangular part of the matrix R; the strictly lower
triangular part of the array is not referenced.
On entry with UPLO = 'L', the leading M-by-M lower
triangular part of this array must contain the lower
triangular part of the matrix R; the strictly upper
triangular part of the array is not referenced.
On exit, the leading M-by-M upper triangular part (if
UPLO = 'U'), or lower triangular part (if UPLO = 'L') of
this array contains the corresponding triangular part of
_
the computed matrix R.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,M).
A (input) DOUBLE PRECISION array, dimension (LDA,k), where
k = N when SIDE = 'L', and TRANS = 'N', or
SIDE = 'R', and TRANS = 'T';
k = M when SIDE = 'R', and TRANS = 'N', or
SIDE = 'L', and TRANS = 'T'.
On entry, if SIDE = 'L', and TRANS = 'N', or
SIDE = 'R', and TRANS = 'T',
the leading M-by-N part of this array must contain the
matrix A.
On entry, if SIDE = 'R', and TRANS = 'N', or
SIDE = 'L', and TRANS = 'T',
the leading N-by-M part of this array must contain the
matrix A.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,l), where
l = M when SIDE = 'L', and TRANS = 'N', or
SIDE = 'R', and TRANS = 'T';
l = N when SIDE = 'R', and TRANS = 'N', or
SIDE = 'L', and TRANS = 'T'.
B (input) DOUBLE PRECISION array, dimension (LDB,p), where
p = M when SIDE = 'L';
p = N when SIDE = 'R'.
On entry, the leading N-by-M part, if SIDE = 'L', or
M-by-N part, if SIDE = 'R', of this array must contain the
matrix B.
LDB INTEGER
The leading dimension of array B.
LDB >= MAX(1,N), if SIDE = 'L';
LDB >= MAX(1,M), if SIDE = 'R'.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The matrix expression is evaluated taking the triangular structure into account. BLAS 2 operations are used. A block algorithm can be easily constructed; it can use BLAS 3 GEMM operations for most computations, and calls of this BLAS 2 algorithm for computing the triangles.Further Comments
The main application of this routine is when the result should be a symmetric matrix, e.g., when B = X*op( A )', for (1), or B = op( A )'*X, for (2), where B is already available and X = X'.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute either the upper or lower triangular part of one of the
matrix formulas
_
R = alpha*R + beta*op( H )*B, (1)
_
R = alpha*R + beta*B*op( H ), (2)
_
where alpha and beta are scalars, H, B, R, and R are m-by-m
matrices, H is an upper Hessenberg matrix, and op( H ) is one of
op( H ) = H or op( H ) = H', the transpose of H.
The result is overwritten on R.
Specification
SUBROUTINE MB01RY( SIDE, UPLO, TRANS, M, ALPHA, BETA, R, LDR, H,
$ LDH, B, LDB, DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER SIDE, TRANS, UPLO
INTEGER INFO, LDB, LDH, LDR, M
DOUBLE PRECISION ALPHA, BETA
C .. Array Arguments ..
DOUBLE PRECISION B(LDB,*), DWORK(*), H(LDH,*), R(LDR,*)
Arguments
Mode Parameters
SIDE CHARACTER*1
Specifies whether the Hessenberg matrix H appears on the
left or right in the matrix product as follows:
_
= 'L': R = alpha*R + beta*op( H )*B;
_
= 'R': R = alpha*R + beta*B*op( H ).
UPLO CHARACTER*1 _
Specifies which triangles of the matrices R and R are
computed and given, respectively, as follows:
= 'U': the upper triangular part;
= 'L': the lower triangular part.
TRANS CHARACTER*1
Specifies the form of op( H ) to be used in the matrix
multiplication as follows:
= 'N': op( H ) = H;
= 'T': op( H ) = H';
= 'C': op( H ) = H'.
Input/Output Parameters
M (input) INTEGER _
The order of the matrices R, R, H and B. M >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then R need not be
set before entry.
BETA (input) DOUBLE PRECISION
The scalar beta. When beta is zero then H and B are not
referenced.
R (input/output) DOUBLE PRECISION array, dimension (LDR,M)
On entry with UPLO = 'U', the leading M-by-M upper
triangular part of this array must contain the upper
triangular part of the matrix R; the strictly lower
triangular part of the array is not referenced.
On entry with UPLO = 'L', the leading M-by-M lower
triangular part of this array must contain the lower
triangular part of the matrix R; the strictly upper
triangular part of the array is not referenced.
On exit, the leading M-by-M upper triangular part (if
UPLO = 'U'), or lower triangular part (if UPLO = 'L') of
this array contains the corresponding triangular part of
_
the computed matrix R.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,M).
H (input) DOUBLE PRECISION array, dimension (LDH,M)
On entry, the leading M-by-M upper Hessenberg part of
this array must contain the upper Hessenberg part of the
matrix H.
The elements below the subdiagonal are not referenced,
except possibly for those in the first column, which
could be overwritten, but are restored on exit.
LDH INTEGER
The leading dimension of array H. LDH >= MAX(1,M).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading M-by-M part of this array must
contain the matrix B.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,M).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK >= M, if beta <> 0 and SIDE = 'L';
LDWORK >= 0, if beta = 0 or SIDE = 'R'.
This array is not referenced when beta = 0 or SIDE = 'R'.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The matrix expression is efficiently evaluated taking the Hessenberg/triangular structure into account. BLAS 2 operations are used. A block algorithm can be constructed; it can use BLAS 3 GEMM operations for most computations, and calls of this BLAS 2 algorithm for computing the triangles.Further Comments
The main application of this routine is when the result should be a symmetric matrix, e.g., when B = X*op( H )', for (1), or B = op( H )'*X, for (2), where B is already available and X = X'.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To scale a general M-by-N matrix A using the row and column scaling factors in the vectors R and C.Specification
SUBROUTINE MB01SD( JOBS, M, N, A, LDA, R, C )
C .. Scalar Arguments ..
CHARACTER JOBS
INTEGER LDA, M, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), C(*), R(*)
Arguments
Mode Parameters
JOBS CHARACTER*1
Specifies the scaling operation to be done, as follows:
= 'R': row scaling, i.e., A will be premultiplied
by diag(R);
= 'C': column scaling, i.e., A will be postmultiplied
by diag(C);
= 'B': both row and column scaling, i.e., A will be
replaced by diag(R) * A * diag(C).
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the scaled matrix. See JOBS for the form of the
scaled matrix.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
R (input) DOUBLE PRECISION array, dimension (M)
The row scale factors for A.
R is not referenced if JOBS = 'C'.
C (input) DOUBLE PRECISION array, dimension (N)
The column scale factors for A.
C is not referenced if JOBS = 'R'.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To scale a symmetric N-by-N matrix A using the row and column scaling factors stored in the vector D.Specification
SUBROUTINE MB01SS( JOBS, UPLO, N, A, LDA, D )
C .. Scalar Arguments ..
CHARACTER JOBS, UPLO
INTEGER LDA, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), D(*)
Arguments
Mode Parameters
JOBS CHARACTER*1
Specifies the scaling operation to be done, as follows:
= 'D': row and column scaling with D, i.e., A will be
transformed to diag(D)*A*diag(D);
= 'I': row and column scaling with inv(D), i.e., A will
be transformed to inv(diag(D))*A*inv(diag(D)).
UPLO CHARACTER*1
Specifies which triangle of the matrix A is stored, as
follows:
= 'U': Upper triangle is stored;
= 'L': Lower triangle is stored.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N upper triangular part
(if UPLO = 'U') or lower triangular part (if UPLO = 'L')
of this array must contain the upper triangular part or
lower triangular part, respectively, of the symmetric
matrix A.
On exit, the leading N-by-N upper triangular part
(if UPLO = 'U') or lower triangular part (if UPLO = 'L')
of this array contains the corresponding triangular part
of the matrix diag(D)*A*diag(D), if JOBS = 'D', or of the
matrix inv(diag(D))*A*inv(diag(D)), JOBS = 'I'.
The stricly lower triangular part (if UPLO = 'U') or
stricly upper triangular part (if UPLO = 'L') is not
referenced.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
D (input) DOUBLE PRECISION array, dimension (N)
The diagonal elements of the diagonal matrix D.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the matrix product A * B, where A and B are upper quasi-triangular matrices (that is, block upper triangular with 1-by-1 or 2-by-2 diagonal blocks) with the same structure. The result is returned in the array B.Specification
SUBROUTINE MB01TD( N, A, LDA, B, LDB, DWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrices A and B. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
upper quasi-triangular matrix A. The elements below the
subdiagonal are not referenced.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the leading N-by-N part of this array must
contain the upper quasi-triangular matrix B, with the same
structure as matrix A.
On exit, the leading N-by-N part of this array contains
the computed product A * B, with the same structure as
on entry.
The elements below the subdiagonal are not referenced.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Workspace
DWORK DOUBLE PRECISION array, dimension (N-1)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the matrices A and B have not the same structure,
and/or A and B are not upper quasi-triangular.
Method
The matrix product A * B is computed column by column, using BLAS 2 and BLAS 1 operations.Further Comments
This routine can be used, for instance, for computing powers of a real Schur form matrix.Example
Program Text
* MB01TD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA, LDB
PARAMETER ( LDA = NMAX, LDB = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX-1 )
* .. Local Scalars ..
INTEGER I, INFO, J, N
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,NMAX), DWORK(LDWORK)
* .. External Subroutines ..
EXTERNAL MB01TD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read in the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,N ), I = 1,N )
* Compute the matrix product A*B.
CALL MB01TD( N, A, LDA, B, LDB, DWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( B(I,J), J = 1,N )
20 CONTINUE
END IF
END IF
STOP
*
99999 FORMAT (' MB01TD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB01TD = ',I2)
99997 FORMAT (' The matrix product A*B is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
MB01TD EXAMPLE PROGRAM DATA 5 1. 2. 6. 3. 5. -2. -1. -1. 0. -2. 0. 0. 1. 5. 1. 0. 0. 0. 0. -4. 0. 0. 0. 20. 4. 5. 5. 1. 5. 1. -2. 1. 3. 0. -4. 0. 0. 4. 20. 4. 0. 0. 0. 3. 5. 0. 0. 0. 1. -2.Program Results
MB01TD EXAMPLE PROGRAM RESULTS The matrix product A*B is 1.0000 7.0000 31.0000 139.0000 22.0000 -8.0000 -11.0000 -9.0000 -32.0000 2.0000 0.0000 0.0000 4.0000 36.0000 27.0000 0.0000 0.0000 0.0000 -4.0000 8.0000 0.0000 0.0000 0.0000 64.0000 92.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB01UD.html 0000664 0000000 0000000 00000010245 14560147231 0020344 0 ustar 00root root 0000000 0000000
Purpose
To compute one of the matrix products
B = alpha*op( H ) * A, or B = alpha*A * op( H ),
where alpha is a scalar, A and B are m-by-n matrices, H is an
upper Hessenberg matrix, and op( H ) is one of
op( H ) = H or op( H ) = H', the transpose of H.
Specification
SUBROUTINE MB01UD( SIDE, TRANS, M, N, ALPHA, H, LDH, A, LDA, B,
$ LDB, INFO )
C .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER INFO, LDA, LDB, LDH, M, N
DOUBLE PRECISION ALPHA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), H(LDH,*)
Arguments
Mode Parameters
SIDE CHARACTER*1
Specifies whether the Hessenberg matrix H appears on the
left or right in the matrix product as follows:
= 'L': B = alpha*op( H ) * A;
= 'R': B = alpha*A * op( H ).
TRANS CHARACTER*1
Specifies the form of op( H ) to be used in the matrix
multiplication as follows:
= 'N': op( H ) = H;
= 'T': op( H ) = H';
= 'C': op( H ) = H'.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrices A and B. M >= 0.
N (input) INTEGER
The number of columns of the matrices A and B. N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then H is not
referenced and A need not be set before entry.
H (input) DOUBLE PRECISION array, dimension (LDH,k)
where k is M when SIDE = 'L' and is N when SIDE = 'R'.
On entry with SIDE = 'L', the leading M-by-M upper
Hessenberg part of this array must contain the upper
Hessenberg matrix H.
On entry with SIDE = 'R', the leading N-by-N upper
Hessenberg part of this array must contain the upper
Hessenberg matrix H.
The elements below the subdiagonal are not referenced,
except possibly for those in the first column, which
could be overwritten, but are restored on exit.
LDH INTEGER
The leading dimension of the array H. LDH >= max(1,k),
where k is M when SIDE = 'L' and is N when SIDE = 'R'.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading M-by-N part of this array must contain the
matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B (output) DOUBLE PRECISION array, dimension (LDB,N)
The leading M-by-N part of this array contains the
computed product.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,M).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The required matrix product is computed in two steps. In the first step, the upper triangle of H is used; in the second step, the contribution of the subdiagonal is added. A fast BLAS 3 DTRMM operation is used in the first step.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB01UW.html 0000664 0000000 0000000 00000011233 14560147231 0020365 0 ustar 00root root 0000000 0000000
Purpose
To compute one of the matrix products
A : = alpha*op( H ) * A, or A : = alpha*A * op( H ),
where alpha is a scalar, A is an m-by-n matrix, H is an upper
Hessenberg matrix, and op( H ) is one of
op( H ) = H or op( H ) = H', the transpose of H.
Specification
SUBROUTINE MB01UW( SIDE, TRANS, M, N, ALPHA, H, LDH, A, LDA,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER INFO, LDA, LDH, LDWORK, M, N
DOUBLE PRECISION ALPHA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), H(LDH,*)
Arguments
Mode Parameters
SIDE CHARACTER*1
Specifies whether the Hessenberg matrix H appears on the
left or right in the matrix product as follows:
= 'L': A := alpha*op( H ) * A;
= 'R': A := alpha*A * op( H ).
TRANS CHARACTER*1
Specifies the form of op( H ) to be used in the matrix
multiplication as follows:
= 'N': op( H ) = H;
= 'T': op( H ) = H';
= 'C': op( H ) = H'.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then H is not
referenced and A need not be set before entry.
H (input) DOUBLE PRECISION array, dimension (LDH,k)
where k is M when SIDE = 'L' and is N when SIDE = 'R'.
On entry with SIDE = 'L', the leading M-by-M upper
Hessenberg part of this array must contain the upper
Hessenberg matrix H.
On entry with SIDE = 'R', the leading N-by-N upper
Hessenberg part of this array must contain the upper
Hessenberg matrix H.
The elements below the subdiagonal are not referenced,
except possibly for those in the first column, which
could be overwritten, but are restored on exit.
LDH INTEGER
The leading dimension of the array H. LDH >= max(1,k),
where k is M when SIDE = 'L' and is N when SIDE = 'R'.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading M-by-N part of this array must
contain the matrix A.
On exit, the leading M-by-N part of this array contains
the computed product.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, alpha <> 0, and LDWORK >= M*N > 0,
DWORK contains a copy of the matrix A, having the leading
dimension M.
This array is not referenced when alpha = 0.
LDWORK The length of the array DWORK.
LDWORK >= 0, if alpha = 0 or MIN(M,N) = 0;
LDWORK >= M-1, if SIDE = 'L';
LDWORK >= N-1, if SIDE = 'R'.
For maximal efficiency LDWORK should be at least M*N.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The required matrix product is computed in two steps. In the first step, the upper triangle of H is used; in the second step, the contribution of the subdiagonal is added. If the workspace can accomodate a copy of A, a fast BLAS 3 DTRMM operation is used in the first step.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute one of the matrix products
A : = alpha*op( T ) * A, or A : = alpha*A * op( T ),
where alpha is a scalar, A is an m-by-n matrix, T is a quasi-
triangular matrix, and op( T ) is one of
op( T ) = T or op( T ) = T', the transpose of T.
Specification
SUBROUTINE MB01UX( SIDE, UPLO, TRANS, M, N, ALPHA, T, LDT, A, LDA,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER SIDE, TRANS, UPLO
INTEGER INFO, LDA, LDT, LDWORK, M, N
DOUBLE PRECISION ALPHA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), T(LDT,*)
Arguments
Mode Parameters
SIDE CHARACTER*1
Specifies whether the upper quasi-triangular matrix H
appears on the left or right in the matrix product as
follows:
= 'L': A := alpha*op( T ) * A;
= 'R': A := alpha*A * op( T ).
UPLO CHARACTER*1.
Specifies whether the matrix T is an upper or lower
quasi-triangular matrix as follows:
= 'U': T is an upper quasi-triangular matrix;
= 'L': T is a lower quasi-triangular matrix.
TRANS CHARACTER*1
Specifies the form of op( T ) to be used in the matrix
multiplication as follows:
= 'N': op( T ) = T;
= 'T': op( T ) = T';
= 'C': op( T ) = T'.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then T is not
referenced and A need not be set before entry.
T (input) DOUBLE PRECISION array, dimension (LDT,k)
where k is M when SIDE = 'L' and is N when SIDE = 'R'.
On entry with UPLO = 'U', the leading k-by-k upper
Hessenberg part of this array must contain the upper
quasi-triangular matrix T. The elements below the
subdiagonal are not referenced.
On entry with UPLO = 'L', the leading k-by-k lower
Hessenberg part of this array must contain the lower
quasi-triangular matrix T. The elements above the
supdiagonal are not referenced.
LDT INTEGER
The leading dimension of the array T. LDT >= max(1,k),
where k is M when SIDE = 'L' and is N when SIDE = 'R'.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading M-by-N part of this array must
contain the matrix A.
On exit, the leading M-by-N part of this array contains
the computed product.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 and ALPHA<>0, DWORK(1) returns the
optimal value of LDWORK.
On exit, if INFO = -12, DWORK(1) returns the minimum
value of LDWORK.
This array is not referenced when alpha = 0.
LDWORK The length of the array DWORK.
LDWORK >= 1, if alpha = 0 or MIN(M,N) = 0;
LDWORK >= 2*(M-1), if SIDE = 'L';
LDWORK >= 2*(N-1), if SIDE = 'R'.
For maximal efficiency LDWORK should be at least
NOFF*N + M - 1, if SIDE = 'L';
NOFF*M + N - 1, if SIDE = 'R';
where NOFF is the number of nonzero elements on the
subdiagonal (if UPLO = 'U') or supdiagonal (if UPLO = 'L')
of T.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The technique used in this routine is similiar to the technique used in the SLICOT [1] subroutine MB01UW developed by Vasile Sima. The required matrix product is computed in two steps. In the first step, the triangle of T specified by UPLO is used; in the second step, the contribution of the sub-/supdiagonal is added. If the workspace can accommodate parts of A, a fast BLAS 3 DTRMM operation is used in the first step.References
[1] Benner, P., Mehrmann, V., Sima, V., Van Huffel, S., and
Varga, A.
SLICOT - A subroutine library in systems and control theory.
In: Applied and computational control, signals, and circuits,
Vol. 1, pp. 499-539, Birkhauser, Boston, 1999.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB01UY.html 0000664 0000000 0000000 00000012077 14560147231 0020376 0 ustar 00root root 0000000 0000000
Purpose
To compute one of the matrix products
T : = alpha*op( T ) * A, or T : = alpha*A * op( T ),
where alpha is a scalar, A is an M-by-N matrix, T is a triangular
matrix, and op( T ) is one of
op( T ) = T or op( T ) = T', the transpose of T.
A block-row/column algorithm is used, if possible. The result
overwrites the array T.
Specification
SUBROUTINE MB01UY( SIDE, UPLO, TRANS, M, N, ALPHA, T, LDT, A, LDA,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER SIDE, TRANS, UPLO
INTEGER INFO, LDA, LDT, LDWORK, M, N
DOUBLE PRECISION ALPHA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), T(LDT,*)
Arguments
Mode Parameters
SIDE CHARACTER*1
Specifies whether the triangular matrix T appears on the
left or right in the matrix product, as follows:
= 'L': T := alpha*op( T ) * A;
= 'R': T := alpha*A * op( T ).
UPLO CHARACTER*1.
Specifies whether the matrix T is an upper or lower
triangular matrix, as follows:
= 'U': T is an upper triangular matrix;
= 'L': T is a lower triangular matrix.
TRANS CHARACTER*1
Specifies the form of op( T ) to be used in the matrix
multiplication as follows:
= 'N': op( T ) = T;
= 'T': op( T ) = T';
= 'C': op( T ) = T'.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then T and A need not
be set before entry.
T (input/output) DOUBLE PRECISION array, dimension
(LDT,max(K,N)), when SIDE = 'L', and
(LDT,K), when SIDE = 'R',
where K is M if SIDE = 'L' and is N if SIDE = 'R'.
On entry with UPLO = 'U', the leading K-by-K upper
triangular part of this array must contain the upper
triangular matrix T. The elements below the diagonal
do not need to be zero.
On entry with UPLO = 'L', the leading K-by-K lower
triangular part of this array must contain the lower
triangular matrix T. The elements above the diagonal
do not need to be zero.
On exit, the leading M-by-N part of this array contains
the corresponding product defined by SIDE, UPLO, and
TRANS.
LDT INTEGER
The leading dimension of the array T.
LDT >= max(1,M), if SIDE = 'L';
LDT >= max(1,M,N), if SIDE = 'R'.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading M-by-N part of this array must contain the
matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
On exit, if INFO = -12, DWORK(1) returns the minimum
value of LDWORK.
LDWORK The length of the array DWORK.
LDWORK >= 1, if alpha = 0 or MIN(M,N) = 0;
LDWORK >= M, if SIDE = 'L';
LDWORK >= N, if SIDE = 'R'.
For good performance, LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
A block-row/column size is found based on the available workspace. BLAS 3 gemm and trmm are used if possible.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB01UZ.html 0000664 0000000 0000000 00000012213 14560147231 0020367 0 ustar 00root root 0000000 0000000
Purpose
To compute one of the matrix products
T : = alpha*op( T ) * A, or T : = alpha*A * op( T ),
where alpha is a scalar, A is an M-by-N matrix, T is a triangular
matrix, and op( T ) is one of
op( T ) = T, or op( T ) = T', the transpose of T, or
op( T ) = conj(T'), the conjugate transpose of T.
A block-row/column algorithm is used, if possible. The result
overwrites the array T.
Specification
SUBROUTINE MB01UZ( SIDE, UPLO, TRANS, M, N, ALPHA, T, LDT, A, LDA,
$ ZWORK, LZWORK, INFO )
C .. Scalar Arguments ..
CHARACTER SIDE, TRANS, UPLO
INTEGER INFO, LDA, LDT, LZWORK, M, N
COMPLEX*16 ALPHA
C .. Array Arguments ..
COMPLEX*16 A(LDA,*), T(LDT,*), ZWORK(*)
Arguments
Mode Parameters
SIDE CHARACTER*1
Specifies whether the triangular matrix T appears on the
left or right in the matrix product, as follows:
= 'L': T := alpha * op( T ) * A;
= 'R': T := alpha * A * op( T ).
UPLO CHARACTER*1.
Specifies whether the matrix T is an upper or lower
triangular matrix, as follows:
= 'U': T is an upper triangular matrix;
= 'L': T is a lower triangular matrix.
TRANS CHARACTER*1
Specifies the form of op( T ) to be used in the matrix
multiplication as follows:
= 'N': op( T ) = T;
= 'T': op( T ) = T';
= 'C': op( T ) = conj(T').
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
ALPHA (input) COMPLEX*16
The scalar alpha. When alpha is zero then T and A need not
be set before entry.
T (input/output) COMPLEX*16 array, dimension
(LDT,max(K,N)), when SIDE = 'L', and
(LDT,K), when SIDE = 'R',
where K is M if SIDE = 'L' and is N if SIDE = 'R'.
On entry with UPLO = 'U', the leading K-by-K upper
triangular part of this array must contain the upper
triangular matrix T. The elements below the diagonal
do not need to be zero.
On entry with UPLO = 'L', the leading K-by-K lower
triangular part of this array must contain the lower
triangular matrix T. The elements above the diagonal
do not need to be zero.
On exit, the leading M-by-N part of this array contains
the corresponding product defined by SIDE, UPLO, and
TRANS.
LDT INTEGER
The leading dimension of the array T.
LDT >= max(1,M), if SIDE = 'L';
LDT >= max(1,M,N), if SIDE = 'R'.
A (input) COMPLEX*16 array, dimension (LDA,N)
The leading M-by-N part of this array must contain the
matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
Workspace
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) returns the optimal value
of LZWORK.
On exit, if INFO = -12, ZWORK(1) returns the minimum
value of LZWORK.
LZWORK The length of the array ZWORK.
LZWORK >= 1, if alpha = 0 or MIN(M,N) = 0;
LZWORK >= M, if SIDE = 'L';
LZWORK >= N, if SIDE = 'R'.
For good performance, LZWORK should be larger.
If LZWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
ZWORK array, returns this value as the first entry of
the ZWORK array, and no error message related to LZWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
A block-row/column size is found based on the available workspace. BLAS 3 gemm and trmm are used if possible.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB01VD.html 0000664 0000000 0000000 00000012030 14560147231 0020337 0 ustar 00root root 0000000 0000000
Purpose
To perform the following matrix operation
C = alpha*kron( op(A), op(B) ) + beta*C,
where alpha and beta are real scalars, op(M) is either matrix M or
its transpose, M', and kron( X, Y ) denotes the Kronecker product
of the matrices X and Y.
Specification
SUBROUTINE MB01VD( TRANA, TRANB, MA, NA, MB, NB, ALPHA, BETA,
$ A, LDA, B, LDB, C, LDC, MC, NC, INFO )
C .. Scalar Arguments ..
CHARACTER TRANA, TRANB
INTEGER INFO, LDA, LDB, LDC, MA, MB, MC, NA, NB, NC
DOUBLE PRECISION ALPHA, BETA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*)
Arguments
Mode Parameters
TRANA CHARACTER*1
Specifies the form of op(A) to be used as follows:
= 'N': op(A) = A;
= 'T': op(A) = A';
= 'C': op(A) = A'.
TRANB CHARACTER*1
Specifies the form of op(B) to be used as follows:
= 'N': op(B) = B;
= 'T': op(B) = B';
= 'C': op(B) = B'.
Input/Output Parameters
MA (input) INTEGER
The number of rows of the matrix op(A). MA >= 0.
NA (input) INTEGER
The number of columns of the matrix op(A). NA >= 0.
MB (input) INTEGER
The number of rows of the matrix op(B). MB >= 0.
NB (input) INTEGER
The number of columns of the matrix op(B). NB >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then A and B need not
be set before entry.
BETA (input) DOUBLE PRECISION
The scalar beta. When beta is zero then C need not be
set before entry.
A (input) DOUBLE PRECISION array, dimension (LDA,ka),
where ka is NA when TRANA = 'N', and is MA otherwise.
If TRANA = 'N', the leading MA-by-NA part of this array
must contain the matrix A; otherwise, the leading NA-by-MA
part of this array must contain the matrix A.
LDA INTEGER
The leading dimension of the array A.
LDA >= max(1,MA), if TRANA = 'N';
LDA >= max(1,NA), if TRANA = 'T' or 'C'.
B (input) DOUBLE PRECISION array, dimension (LDB,kb)
where kb is NB when TRANB = 'N', and is MB otherwise.
If TRANB = 'N', the leading MB-by-NB part of this array
must contain the matrix B; otherwise, the leading NB-by-MB
part of this array must contain the matrix B.
LDB INTEGER
The leading dimension of the array B.
LDB >= max(1,MB), if TRANB = 'N';
LDB >= max(1,NB), if TRANB = 'T' or 'C'.
C (input/output) DOUBLE PRECISION array, dimension (LDC,NC)
On entry, if beta is nonzero, the leading MC-by-NC part of
this array must contain the given matric C, where
MC = MA*MB and NC = NA*NB.
On exit, the leading MC-by-NC part of this array contains
the computed matrix expression
C = alpha*kron( op(A), op(B) ) + beta*C.
LDC INTEGER
The leading dimension of the array C.
LDC >= max(1,MC).
MC (output) INTEGER
The number of rows of the matrix C. MC = MA*MB.
NC (output) INTEGER
The number of columns of the matrix C. NC = NA*NB.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The Kronecker product of the matrices op(A) and op(B) is computed column by column.Further Comments
The multiplications by zero elements in A are avoided, if the matrix A is considered to be sparse, i.e., if (number of zeros in A)/(MA*NA) >= SPARST = 0.8. The code makes NB+1 passes through the matrix A, and MA*NA passes through the matrix B. If LDA and/or LDB are very large, and op(A) = A' and/or op(B) = B', it could be more efficient to transpose A and/or B before calling this routine, and use the 'N' values for TRANA and/or TRANB.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the matrix formula
_
R = alpha*( op( A )'*op( T )'*op( T ) + op( T )'*op( T )*op( A ) )
+ beta*R, (1)
if DICO = 'C', or
_
R = alpha*( op( A )'*op( T )'*op( T )*op( A ) - op( T )'*op( T ))
+ beta*R, (2)
_
if DICO = 'D', where alpha and beta are scalars, R, and R are
symmetric matrices, T is a triangular matrix, A is a general or
Hessenberg matrix, and op( M ) is one of
op( M ) = M or op( M ) = M'.
The result is overwritten on R.
Specification
SUBROUTINE MB01WD( DICO, UPLO, TRANS, HESS, N, ALPHA, BETA, R,
$ LDR, A, LDA, T, LDT, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, HESS, TRANS, UPLO
INTEGER INFO, LDA, LDR, LDT, N
DOUBLE PRECISION ALPHA, BETA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), R(LDR,*), T(LDT,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the formula to be evaluated, as follows:
= 'C': formula (1), "continuous-time" case;
= 'D': formula (2), "discrete-time" case.
UPLO CHARACTER*1
Specifies which triangles of the symmetric matrix R and
triangular matrix T are given, as follows:
= 'U': the upper triangular parts of R and T are given;
= 'L': the lower triangular parts of R and T are given;
TRANS CHARACTER*1
Specifies the form of op( M ) to be used, as follows:
= 'N': op( M ) = M;
= 'T': op( M ) = M';
= 'C': op( M ) = M'.
HESS CHARACTER*1
Specifies the form of the matrix A, as follows:
= 'F': matrix A is full;
= 'H': matrix A is Hessenberg (or Schur), either upper
(if UPLO = 'U'), or lower (if UPLO = 'L').
Input/Output Parameters
N (input) INTEGER
The order of the matrices R, A, and T. N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then the arrays A
and T are not referenced.
BETA (input) DOUBLE PRECISION
The scalar beta. When beta is zero then the array R need
not be set before entry.
R (input/output) DOUBLE PRECISION array, dimension (LDR,N)
On entry with UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix R.
On entry with UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix R.
On exit, the leading N-by-N upper triangular part (if
UPLO = 'U'), or lower triangular part (if UPLO = 'L'), of
this array contains the corresponding triangular part of
_
the computed matrix R.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A. If HESS = 'H' the elements below the
first subdiagonal, if UPLO = 'U', or above the first
superdiagonal, if UPLO = 'L', need not be set to zero,
and are not referenced if DICO = 'D'.
On exit, the leading N-by-N part of this array contains
the following matrix product
alpha*T'*T*A, if TRANS = 'N', or
alpha*A*T*T', otherwise,
if DICO = 'C', or
T*A, if TRANS = 'N', or
A*T, otherwise,
if DICO = 'D' (and in this case, these products have a
Hessenberg form, if HESS = 'H').
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
T (input) DOUBLE PRECISION array, dimension (LDT,N)
If UPLO = 'U', the leading N-by-N upper triangular part of
this array must contain the upper triangular matrix T and
the strictly lower triangular part need not be set to zero
(and it is not referenced).
If UPLO = 'L', the leading N-by-N lower triangular part of
this array must contain the lower triangular matrix T and
the strictly upper triangular part need not be set to zero
(and it is not referenced).
LDT INTEGER
The leading dimension of array T. LDT >= MAX(1,N).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -k, the k-th argument had an illegal
value.
Method
The matrix expression (1) or (2) is efficiently evaluated taking the structure into account. BLAS 3 operations (DTRMM, DSYRK and their specializations) are used throughout.Numerical Aspects
If A is a full matrix, the algorithm requires approximately
3
N operations, if DICO = 'C';
3
7/6 x N operations, if DICO = 'D'.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB01XD.html 0000664 0000000 0000000 00000006620 14560147231 0020351 0 ustar 00root root 0000000 0000000
Purpose
To compute the matrix product U' * U or L * L', where U and L are upper and lower triangular matrices, respectively, stored in the corresponding upper or lower triangular part of the array A. If UPLO = 'U' then the upper triangle of the result is stored, overwriting the matrix U in A. If UPLO = 'L' then the lower triangle of the result is stored, overwriting the matrix L in A.Specification
SUBROUTINE MB01XD( UPLO, N, A, LDA, INFO )
C .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies which triangle (U or L) is given in the array A,
as follows:
= 'U': the upper triangular part U is given;
= 'L': the lower triangular part L is given.
Input/Output Parameters
N (input) INTEGER
The order of the triangular matrices U or L. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, if UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular matrix U.
On entry, if UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular matrix L.
On exit, if UPLO = 'U', the leading N-by-N upper
triangular part of this array contains the upper
triangular part of the product U' * U. The strictly lower
triangular part is not referenced.
On exit, if UPLO = 'L', the leading N-by-N lower
triangular part of this array contains the lower
triangular part of the product L * L'. The strictly upper
triangular part is not referenced.
LDA INTEGER
The leading dimension of array A. LDA >= max(1,N).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The matrix product U' * U or L * L' is computed using BLAS 3 operations as much as possible (a block algorithm).Further Comments
This routine is a counterpart of LAPACK Library routine DLAUUM, which computes the matrix product U * U' or L' * L.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB01XY.html 0000664 0000000 0000000 00000006632 14560147231 0020401 0 ustar 00root root 0000000 0000000
Purpose
To compute the matrix product U' * U or L * L', where U and L are upper and lower triangular matrices, respectively, stored in the corresponding upper or lower triangular part of the array A. If UPLO = 'U' then the upper triangle of the result is stored, overwriting the matrix U in A. If UPLO = 'L' then the lower triangle of the result is stored, overwriting the matrix L in A.Specification
SUBROUTINE MB01XY( UPLO, N, A, LDA, INFO )
C .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies which triangle (U or L) is given in the array A,
as follows:
= 'U': the upper triangular part U is given;
= 'L': the lower triangular part L is given.
Input/Output Parameters
N (input) INTEGER
The order of the triangular matrices U or L. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, if UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular matrix U.
On entry, if UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular matrix L.
On exit, if UPLO = 'U', the leading N-by-N upper
triangular part of this array contains the upper
triangular part of the product U' * U. The strictly lower
triangular part is not referenced.
On exit, if UPLO = 'L', the leading N-by-N lower
triangular part of this array contains the lower
triangular part of the product L * L'. The strictly upper
triangular part is not referenced.
LDA INTEGER
The leading dimension of array A. LDA >= max(1,N).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The matrix product U' * U or L * L' is computed using BLAS 2 and BLAS 1 operations (an unblocked algorithm).Further Comments
This routine is a counterpart of LAPACK Library routine DLAUU2, which computes the matrix product U * U' or L' * L.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To perform the symmetric rank k operations
C := alpha*op( A )*op( A )' + beta*C,
where alpha and beta are scalars, C is an n-by-n symmetric matrix,
op( A ) is an n-by-k matrix, and op( A ) is one of
op( A ) = A or op( A ) = A'.
The matrix A has l nonzero codiagonals, either upper or lower.
Specification
SUBROUTINE MB01YD( UPLO, TRANS, N, K, L, ALPHA, BETA, A, LDA, C,
$ LDC, INFO )
C .. Scalar Arguments ..
CHARACTER TRANS, UPLO
INTEGER INFO, LDA, LDC, K, L, N
DOUBLE PRECISION ALPHA, BETA
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * )
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies which triangle of the symmetric matrix C
is given and computed, as follows:
= 'U': the upper triangular part is given/computed;
= 'L': the lower triangular part is given/computed.
UPLO also defines the pattern of the matrix A (see below).
TRANS CHARACTER*1
Specifies the form of op( A ) to be used, as follows:
= 'N': op( A ) = A;
= 'T': op( A ) = A';
= 'C': op( A ) = A'.
Input/Output Parameters
N (input) INTEGER
The order of the matrix C. N >= 0.
K (input) INTEGER
The number of columns of the matrix op( A ). K >= 0.
L (input) INTEGER
If UPLO = 'U', matrix A has L nonzero subdiagonals.
If UPLO = 'L', matrix A has L nonzero superdiagonals.
MAX(0,NR-1) >= L >= 0, if UPLO = 'U',
MAX(0,NC-1) >= L >= 0, if UPLO = 'L',
where NR and NC are the numbers of rows and columns of the
matrix A, respectively.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then the array A is
not referenced.
BETA (input) DOUBLE PRECISION
The scalar beta. When beta is zero then the array C need
not be set before entry.
A (input) DOUBLE PRECISION array, dimension (LDA,NC), where
NC is K when TRANS = 'N', and is N otherwise.
If TRANS = 'N', the leading N-by-K part of this array must
contain the matrix A, otherwise the leading K-by-N part of
this array must contain the matrix A.
If UPLO = 'U', only the upper triangular part and the
first L subdiagonals are referenced, and the remaining
subdiagonals are assumed to be zero.
If UPLO = 'L', only the lower triangular part and the
first L superdiagonals are referenced, and the remaining
superdiagonals are assumed to be zero.
LDA INTEGER
The leading dimension of array A. LDA >= max(1,NR),
where NR = N, if TRANS = 'N', and NR = K, otherwise.
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry with UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix C.
On entry with UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix C.
On exit, the leading N-by-N upper triangular part (if
UPLO = 'U'), or lower triangular part (if UPLO = 'L'), of
this array contains the corresponding triangular part of
the updated matrix C.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,N).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The calculations are efficiently performed taking the symmetry and structure into account.Further Comments
The matrix A may have the following patterns, when n = 7, k = 5,
and l = 2 are used for illustration:
UPLO = 'U', TRANS = 'N' UPLO = 'L', TRANS = 'N'
[ x x x x x ] [ x x x 0 0 ]
[ x x x x x ] [ x x x x 0 ]
[ x x x x x ] [ x x x x x ]
A = [ 0 x x x x ], A = [ x x x x x ],
[ 0 0 x x x ] [ x x x x x ]
[ 0 0 0 x x ] [ x x x x x ]
[ 0 0 0 0 x ] [ x x x x x ]
UPLO = 'U', TRANS = 'T' UPLO = 'L', TRANS = 'T'
[ x x x x x x x ] [ x x x 0 0 0 0 ]
[ x x x x x x x ] [ x x x x 0 0 0 ]
A = [ x x x x x x x ], A = [ x x x x x 0 0 ].
[ 0 x x x x x x ] [ x x x x x x 0 ]
[ 0 0 x x x x x ] [ x x x x x x x ]
If N = K, the matrix A is upper or lower triangular, for L = 0,
and upper or lower Hessenberg, for L = 1.
This routine is a specialization of the BLAS 3 routine DSYRK.
BLAS 1 calls are used when appropriate, instead of in-line code,
in order to increase the efficiency. If the matrix A is full, or
its zero triangle has small order, an optimized DSYRK code could
be faster than MB01YD.
Example
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB01ZD.html 0000664 0000000 0000000 00000015160 14560147231 0020352 0 ustar 00root root 0000000 0000000
Purpose
To compute the matrix product
H := alpha*op( T )*H, or H := alpha*H*op( T ),
where alpha is a scalar, H is an m-by-n upper or lower
Hessenberg-like matrix (with l nonzero subdiagonals or
superdiagonals, respectively), T is a unit, or non-unit,
upper or lower triangular matrix, and op( T ) is one of
op( T ) = T or op( T ) = T'.
Specification
SUBROUTINE MB01ZD( SIDE, UPLO, TRANST, DIAG, M, N, L, ALPHA, T,
$ LDT, H, LDH, INFO )
C .. Scalar Arguments ..
CHARACTER DIAG, SIDE, TRANST, UPLO
INTEGER INFO, L, LDH, LDT, M, N
DOUBLE PRECISION ALPHA
C .. Array Arguments ..
DOUBLE PRECISION H( LDH, * ), T( LDT, * )
Arguments
Mode Parameters
SIDE CHARACTER*1
Specifies whether the triangular matrix T appears on the
left or right in the matrix product, as follows:
= 'L': the product alpha*op( T )*H is computed;
= 'R': the product alpha*H*op( T ) is computed.
UPLO CHARACTER*1
Specifies the form of the matrices T and H, as follows:
= 'U': the matrix T is upper triangular and the matrix H
is upper Hessenberg-like;
= 'L': the matrix T is lower triangular and the matrix H
is lower Hessenberg-like.
TRANST CHARACTER*1
Specifies the form of op( T ) to be used, as follows:
= 'N': op( T ) = T;
= 'T': op( T ) = T';
= 'C': op( T ) = T'.
DIAG CHARACTER*1.
Specifies whether or not T is unit triangular, as follows:
= 'U': the matrix T is assumed to be unit triangular;
= 'N': the matrix T is not assumed to be unit triangular.
Input/Output Parameters
M (input) INTEGER
The number of rows of H. M >= 0.
N (input) INTEGER
The number of columns of H. N >= 0.
L (input) INTEGER
If UPLO = 'U', matrix H has L nonzero subdiagonals.
If UPLO = 'L', matrix H has L nonzero superdiagonals.
MAX(0,M-1) >= L >= 0, if UPLO = 'U';
MAX(0,N-1) >= L >= 0, if UPLO = 'L'.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then T is not
referenced and H need not be set before entry.
T (input) DOUBLE PRECISION array, dimension (LDT,k), where
k is m when SIDE = 'L' and is n when SIDE = 'R'.
If UPLO = 'U', the leading k-by-k upper triangular part
of this array must contain the upper triangular matrix T
and the strictly lower triangular part is not referenced.
If UPLO = 'L', the leading k-by-k lower triangular part
of this array must contain the lower triangular matrix T
and the strictly upper triangular part is not referenced.
Note that when DIAG = 'U', the diagonal elements of T are
not referenced either, but are assumed to be unity.
LDT INTEGER
The leading dimension of array T.
LDT >= MAX(1,M), if SIDE = 'L';
LDT >= MAX(1,N), if SIDE = 'R'.
H (input/output) DOUBLE PRECISION array, dimension (LDH,N)
On entry, if UPLO = 'U', the leading M-by-N upper
Hessenberg part of this array must contain the upper
Hessenberg-like matrix H.
On entry, if UPLO = 'L', the leading M-by-N lower
Hessenberg part of this array must contain the lower
Hessenberg-like matrix H.
On exit, the leading M-by-N part of this array contains
the matrix product alpha*op( T )*H, if SIDE = 'L',
or alpha*H*op( T ), if SIDE = 'R'. If TRANST = 'N', this
product has the same pattern as the given matrix H;
the elements below the L-th subdiagonal (if UPLO = 'U'),
or above the L-th superdiagonal (if UPLO = 'L'), are not
referenced in this case. If TRANST = 'T', the elements
below the (N+L)-th row (if UPLO = 'U', SIDE = 'R', and
M > N+L), or at the right of the (M+L)-th column
(if UPLO = 'L', SIDE = 'L', and N > M+L), are not set to
zero nor referenced.
LDH INTEGER
The leading dimension of array H. LDH >= max(1,M).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The calculations are efficiently performed taking the problem structure into account.Further Comments
The matrix H may have the following patterns, when m = 7, n = 6,
and l = 2 are used for illustration:
UPLO = 'U' UPLO = 'L'
[ x x x x x x ] [ x x x 0 0 0 ]
[ x x x x x x ] [ x x x x 0 0 ]
[ x x x x x x ] [ x x x x x 0 ]
H = [ 0 x x x x x ], H = [ x x x x x x ].
[ 0 0 x x x x ] [ x x x x x x ]
[ 0 0 0 x x x ] [ x x x x x x ]
[ 0 0 0 0 x x ] [ x x x x x x ]
The products T*H or H*T have the same pattern as H, but the
products T'*H or H*T' may be full matrices.
If m = n, the matrix H is upper or lower triangular, for l = 0,
and upper or lower Hessenberg, for l = 1.
This routine is a specialization of the BLAS 3 routine DTRMM.
BLAS 1 calls are used when appropriate, instead of in-line code,
in order to increase the efficiency. If the matrix H is full, or
its zero triangle has small order, an optimized DTRMM code could
be faster than MB01ZD.
Example
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02CD.html 0000664 0000000 0000000 00000031354 14560147231 0020327 0 ustar 00root root 0000000 0000000
Purpose
To compute the Cholesky factor and the generator and/or the Cholesky factor of the inverse of a symmetric positive definite (s.p.d.) block Toeplitz matrix T, defined by either its first block row, or its first block column, depending on the routine parameter TYPET. Transformation information is stored.Specification
SUBROUTINE MB02CD( JOB, TYPET, K, N, T, LDT, G, LDG, R, LDR, L,
$ LDL, CS, LCS, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOB, TYPET
INTEGER INFO, K, LCS, LDG, LDL, LDR, LDT, LDWORK, N
C .. Array Arguments ..
DOUBLE PRECISION CS(*), DWORK(*), G(LDG, *), L(LDL,*), R(LDR,*),
$ T(LDT,*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the output of the routine, as follows:
= 'G': only computes the generator G of the inverse;
= 'R': computes the generator G of the inverse and the
Cholesky factor R of T, i.e., if TYPET = 'R',
then R'*R = T, and if TYPET = 'C', then R*R' = T;
= 'L': computes the generator G and the Cholesky factor L
of the inverse, i.e., if TYPET = 'R', then
L'*L = inv(T), and if TYPET = 'C', then
L*L' = inv(T);
= 'A': computes the generator G, the Cholesky factor L
of the inverse and the Cholesky factor R of T;
= 'O': only computes the Cholesky factor R of T.
TYPET CHARACTER*1
Specifies the type of T, as follows:
= 'R': T contains the first block row of an s.p.d. block
Toeplitz matrix; if demanded, the Cholesky factors
R and L are upper and lower triangular,
respectively, and G contains the transposed
generator of the inverse;
= 'C': T contains the first block column of an s.p.d.
block Toeplitz matrix; if demanded, the Cholesky
factors R and L are lower and upper triangular,
respectively, and G contains the generator of the
inverse. This choice results in a column oriented
algorithm which is usually faster.
Note: in the sequel, the notation x / y means that
x corresponds to TYPET = 'R' and y corresponds to
TYPET = 'C'.
Input/Output Parameters
K (input) INTEGER
The number of rows / columns in T, which should be equal
to the blocksize. K >= 0.
N (input) INTEGER
The number of blocks in T. N >= 0.
T (input/output) DOUBLE PRECISION array, dimension
(LDT,N*K) / (LDT,K)
On entry, the leading K-by-N*K / N*K-by-K part of this
array must contain the first block row / column of an
s.p.d. block Toeplitz matrix.
On exit, if INFO = 0, then the leading K-by-N*K / N*K-by-K
part of this array contains, in the first K-by-K block,
the upper / lower Cholesky factor of T(1:K,1:K), and in
the remaining part, the Householder transformations
applied during the process.
LDT INTEGER
The leading dimension of the array T.
LDT >= MAX(1,K), if TYPET = 'R';
LDT >= MAX(1,N*K), if TYPET = 'C'.
G (output) DOUBLE PRECISION array, dimension
(LDG,N*K) / (LDG,2*K)
If INFO = 0 and JOB = 'G', 'R', 'L', or 'A', the leading
2*K-by-N*K / N*K-by-2*K part of this array contains, in
the first K-by-K block of the second block row / column,
the lower right block of L (necessary for updating
factorizations in SLICOT Library routine MB02DD), and
in the remaining part, the generator of the inverse of T.
Actually, to obtain a generator one has to set
G(K+1:2*K, 1:K) = 0, if TYPET = 'R';
G(1:K, K+1:2*K) = 0, if TYPET = 'C'.
LDG INTEGER
The leading dimension of the array G.
LDG >= MAX(1,2*K), if TYPET = 'R' and
JOB = 'G', 'R', 'L', or 'A';
LDG >= MAX(1,N*K), if TYPET = 'C' and
JOB = 'G', 'R', 'L', or 'A';
LDG >= 1, if JOB = 'O'.
R (output) DOUBLE PRECISION array, dimension (LDR,N*K)
If INFO = 0 and JOB = 'R', 'A', or 'O', then the leading
N*K-by-N*K part of this array contains the upper / lower
Cholesky factor of T.
The elements in the strictly lower / upper triangular part
are not referenced.
LDR INTEGER
The leading dimension of the array R.
LDR >= MAX(1,N*K), if JOB = 'R', 'A', or 'O';
LDR >= 1, if JOB = 'G', or 'L'.
L (output) DOUBLE PRECISION array, dimension (LDL,N*K)
If INFO = 0 and JOB = 'L', or 'A', then the leading
N*K-by-N*K part of this array contains the lower / upper
Cholesky factor of the inverse of T.
The elements in the strictly upper / lower triangular part
are not referenced.
LDL INTEGER
The leading dimension of the array L.
LDL >= MAX(1,N*K), if JOB = 'L', or 'A';
LDL >= 1, if JOB = 'G', 'R', or 'O'.
CS (output) DOUBLE PRECISION array, dimension (LCS)
If INFO = 0, then the leading 3*(N-1)*K part of this
array contains information about the hyperbolic rotations
and Householder transformations applied during the
process. This information is needed for updating the
factorizations in SLICOT Library routine MB02DD.
LCS INTEGER
The length of the array CS. LCS >= 3*(N-1)*K.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -16, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,(N-1)*K).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction algorithm failed. The Toeplitz matrix
associated with T is not (numerically) positive
definite.
Method
Householder transformations and modified hyperbolic rotations are used in the Schur algorithm [1], [2].References
[1] Kailath, T. and Sayed, A.
Fast Reliable Algorithms for Matrices with Structure.
SIAM Publications, Philadelphia, 1999.
[2] Kressner, D. and Van Dooren, P.
Factorizations and linear system solvers for matrices with
Toeplitz structure.
SLICOT Working Note 2000-2, 2000.
Numerical Aspects
The implemented method is numerically stable.
3 2
The algorithm requires 0(K N ) floating point operations.
Further Comments
NoneExample
Program Text
* MB02CD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER KMAX, NMAX
PARAMETER ( KMAX = 20, NMAX = 20 )
INTEGER LCS, LDG, LDL, LDR, LDT, LDWORK
PARAMETER ( LDG = 2*KMAX, LDL = NMAX*KMAX, LDR = NMAX*KMAX,
$ LDT = KMAX, LDWORK = ( NMAX - 1 )*KMAX )
PARAMETER ( LCS = 3*LDWORK )
* .. Local Scalars ..
INTEGER I, INFO, J, K, M, N
CHARACTER JOB, TYPET
* .. Local Arrays .. (Dimensioned for TYPET = 'R'.)
DOUBLE PRECISION CS(LCS), DWORK(LDWORK), G(LDG, NMAX*KMAX),
$ L(LDL, NMAX*KMAX), R(LDR, NMAX*KMAX),
$ T(LDT, NMAX*KMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL DLASET, MB02CD
*
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, K, JOB
TYPET = 'R'
M = N*K
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
IF( K.LE.0 .OR. K.GT.KMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) K
ELSE
READ ( NIN, FMT = * ) ( ( T(I,J), J = 1,M ), I = 1,K )
* Compute the Cholesky factor(s) and/or the generator.
CALL MB02CD( JOB, TYPET, K, N, T, LDT, G, LDG, R, LDR, L,
$ LDL, CS, LCS, DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( LSAME( JOB, 'G' ) .OR. LSAME( JOB, 'A' ) .OR.
$ LSAME( JOB, 'L' ) .OR. LSAME( JOB, 'R' ) ) THEN
WRITE ( NOUT, FMT = 99997 )
CALL DLASET( 'Full', K, K, ZERO, ZERO, G(K+1,1), LDG )
DO 10 I = 1, 2*K
WRITE ( NOUT, FMT = 99994 ) ( G(I,J), J = 1, M )
10 CONTINUE
END IF
IF ( LSAME( JOB, 'L' ) .OR. LSAME( JOB, 'A' ) ) THEN
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, M
WRITE ( NOUT, FMT = 99994 ) ( L(I,J), J = 1, M )
20 CONTINUE
END IF
IF ( LSAME( JOB, 'R' ) .OR. LSAME( JOB, 'A' )
$ .OR. LSAME( JOB, 'O' ) ) THEN
WRITE ( NOUT, FMT = 99995 )
DO 30 I = 1, M
WRITE ( NOUT, FMT = 99994 ) ( R(I,J), J = 1, M )
30 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' MB02CD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB02CD = ',I2)
99997 FORMAT (' The generator of the inverse of block Toeplitz matrix',
$ ' is ')
99996 FORMAT (/' The lower Cholesky factor of the inverse is ')
99995 FORMAT (/' The upper Cholesky factor of block Toeplitz matrix is '
$ )
99994 FORMAT (20(1X,F8.4))
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' K is out of range.',/' K = ',I5)
END
Program Data
MB02CD EXAMPLE PROGRAM DATA 3 2 A 3.0000 1.0000 0.1000 0.1000 0.2000 0.0500 1.0000 4.0000 0.4000 0.1000 0.0400 0.2000Program Results
MB02CD EXAMPLE PROGRAM RESULTS The generator of the inverse of block Toeplitz matrix is -0.2355 0.5231 -0.0642 0.0077 0.0187 -0.0265 -0.5568 -0.0568 0.0229 0.0060 0.0363 0.0000 0.0000 0.0000 -0.0387 0.0052 0.0003 -0.0575 0.0000 0.0000 0.0119 -0.0265 -0.0110 0.0076 The lower Cholesky factor of the inverse is 0.5774 0.0000 0.0000 0.0000 0.0000 0.0000 -0.1741 0.5222 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0581 0.5812 0.0000 0.0000 0.0000 -0.0142 0.0080 -0.1747 0.5224 0.0000 0.0000 -0.0387 0.0052 0.0003 -0.0575 0.5825 0.0000 0.0119 -0.0265 -0.0110 0.0076 -0.1754 0.5231 The upper Cholesky factor of block Toeplitz matrix is 1.7321 0.5774 0.0577 0.0577 0.1155 0.0289 0.0000 1.9149 0.1915 0.0348 -0.0139 0.0957 0.0000 0.0000 1.7205 0.5754 0.0558 0.0465 0.0000 0.0000 0.0000 1.9142 0.1890 0.0357 0.0000 0.0000 0.0000 0.0000 1.7169 0.5759 0.0000 0.0000 0.0000 0.0000 0.0000 1.9118
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02CU.html 0000664 0000000 0000000 00000023002 14560147231 0020337 0 ustar 00root root 0000000 0000000
Purpose
To bring the first blocks of a generator to proper form. The positive part of the generator is contained in the arrays A1 and A2. The negative part of the generator is contained in B. Transformation information will be stored and can be applied via SLICOT Library routine MB02CV.Specification
SUBROUTINE MB02CU( TYPEG, K, P, Q, NB, A1, LDA1, A2, LDA2, B, LDB,
$ RNK, IPVT, CS, TOL, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TYPEG
INTEGER INFO, K, LDA1, LDA2, LDB, LDWORK, NB, P, Q, RNK
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IPVT(*)
DOUBLE PRECISION A1(LDA1,*), A2(LDA2,*), B(LDB,*), CS(*),
$ DWORK(*)
Arguments
Mode Parameters
TYPEG CHARACTER*1
Specifies the type of the generator, as follows:
= 'D': generator is column oriented and rank
deficiencies are expected;
= 'C': generator is column oriented and rank
deficiencies are not expected;
= 'R': generator is row oriented and rank
deficiencies are not expected.
Input/Output Parameters
K (input) INTEGER
The number of rows in A1 to be processed. K >= 0.
P (input) INTEGER
The number of columns of the positive generator. P >= K.
Q (input) INTEGER
The number of columns in B containing the negative
generators.
If TYPEG = 'D', Q >= K;
If TYPEG = 'C' or 'R', Q >= 0.
NB (input) INTEGER
On entry, if TYPEG = 'C' or TYPEG = 'R', NB specifies
the block size to be used in the blocked parts of the
algorithm. If NB <= 0, an unblocked algorithm is used.
A1 (input/output) DOUBLE PRECISION array, dimension
(LDA1, K)
On entry, the leading K-by-K part of this array must
contain the leading submatrix of the positive part of the
generator. If TYPEG = 'C', A1 is assumed to be lower
triangular and the strictly upper triangular part is not
referenced. If TYPEG = 'R', A1 is assumed to be upper
triangular and the strictly lower triangular part is not
referenced.
On exit, if TYPEG = 'D', the leading K-by-RNK part of this
array contains the lower trapezoidal part of the proper
generator and information for the Householder
transformations applied during the reduction process.
On exit, if TYPEG = 'C', the leading K-by-K part of this
array contains the leading lower triangular part of the
proper generator.
On exit, if TYPEG = 'R', the leading K-by-K part of this
array contains the leading upper triangular part of the
proper generator.
LDA1 INTEGER
The leading dimension of the array A1. LDA1 >= MAX(1,K).
A2 (input/output) DOUBLE PRECISION array,
if TYPEG = 'D' or TYPEG = 'C', dimension (LDA2, P-K);
if TYPEG = 'R', dimension (LDA2, K).
On entry, if TYPEG = 'D' or TYPEG = 'C', the leading
K-by-(P-K) part of this array must contain the (K+1)-st
to P-th columns of the positive part of the generator.
On entry, if TYPEG = 'R', the leading (P-K)-by-K part of
this array must contain the (K+1)-st to P-th rows of the
positive part of the generator.
On exit, if TYPEG = 'D' or TYPEG = 'C', the leading
K-by-(P-K) part of this array contains information for
Householder transformations.
On exit, if TYPEG = 'R', the leading (P-K)-by-K part of
this array contains information for Householder
transformations.
LDA2 INTEGER
The leading dimension of the array A2.
If P = K, LDA2 >= 1;
If P > K and (TYPEG = 'D' or TYPEG = 'C'),
LDA2 >= MAX(1,K);
if P > K and TYPEG = 'R', LDA2 >= P-K.
B (input/output) DOUBLE PRECISION array,
if TYPEG = 'D' or TYPEG = 'C', dimension (LDB, Q);
if TYPEG = 'R', dimension (LDB, K).
On entry, if TYPEG = 'D' or TYPEG = 'C', the leading
K-by-Q part of this array must contain the negative part
of the generator.
On entry, if TYPEG = 'R', the leading Q-by-K part of this
array must contain the negative part of the generator.
On exit, if TYPEG = 'D' or TYPEG = 'C', the leading
K-by-Q part of this array contains information for
Householder transformations.
On exit, if TYPEG = 'R', the leading Q-by-K part of this
array contains information for Householder transformations.
LDB INTEGER
The leading dimension of the array B.
If Q = 0, LDB >= 1;
if Q > 0 and (TYPEG = 'D' or TYPEG = 'C'),
LDB >= MAX(1,K);
if Q > 0 and TYPEG = 'R', LDB >= Q.
RNK (output) INTEGER
If TYPEG = 'D', the number of columns in the reduced
generator which are found to be linearly independent.
If TYPEG = 'C' or TYPEG = 'R', then RNK is not set.
IPVT (output) INTEGER array, dimension (K)
If TYPEG = 'D', then if IPVT(i) = k, the k-th row of the
proper generator is the reduced i-th row of the input
generator.
If TYPEG = 'C' or TYPEG = 'R', this array is not
referenced.
CS (output) DOUBLE PRECISION array, dimension (x)
If TYPEG = 'D' and P = K, x = 3*K;
if TYPEG = 'D' and P > K, x = 5*K;
if (TYPEG = 'C' or TYPEG = 'R') and P = K, x = 4*K;
if (TYPEG = 'C' or TYPEG = 'R') and P > K, x = 6*K.
On exit, the first x elements of this array contain
necessary information for the SLICOT library routine
MB02CV (Givens and modified hyperbolic rotation
parameters, scalar factors of the Householder
transformations).
Tolerances
TOL DOUBLE PRECISION
If TYPEG = 'D', this number specifies the used tolerance
for handling deficiencies. If the hyperbolic norm
of two diagonal elements in the positive and negative
generators appears to be less than or equal to TOL, then
the corresponding columns are not reduced.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = -17, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,4*K), if TYPEG = 'D';
LDWORK >= MAX(1,MAX(NB,1)*K), if TYPEG = 'C' or 'R'.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if TYPEG = 'D', the generator represents a
(numerically) indefinite matrix; and if TYPEG = 'C'
or TYPEG = 'R', the generator represents a
(numerically) semidefinite matrix.
Method
If TYPEG = 'C' or TYPEG = 'R', blocked Householder transformations and modified hyperbolic rotations are used to downdate the matrix [ A1 A2 sqrt(-1)*B ], cf. [1], [2]. If TYPEG = 'D', then an algorithm with row pivoting is used. In the first stage it maximizes the hyperbolic norm of the active row. As soon as the hyperbolic norm is below the threshold TOL, the strategy is changed. Now, in the second stage, the algorithm applies an LQ decomposition with row pivoting on B such that the Euclidean norm of the active row is maximized.References
[1] Kailath, T. and Sayed, A.
Fast Reliable Algorithms for Matrices with Structure.
SIAM Publications, Philadelphia, 1999.
[2] Kressner, D. and Van Dooren, P.
Factorizations and linear system solvers for matrices with
Toeplitz structure.
SLICOT Working Note 2000-2, 2000.
Numerical Aspects
2 The algorithm requires 0(K *( P + Q )) floating point operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To apply the transformations created by the SLICOT Library routine MB02CU on other columns / rows of the generator, contained in the arrays F1, F2 and G.Specification
SUBROUTINE MB02CV( TYPEG, STRUCG, K, N, P, Q, NB, RNK, A1, LDA1,
$ A2, LDA2, B, LDB, F1, LDF1, F2, LDF2, G, LDG,
$ CS, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER STRUCG, TYPEG
INTEGER INFO, K, LDA1, LDA2, LDB, LDF1, LDF2, LDG,
$ LDWORK, N, NB, P, Q, RNK
C .. Array Arguments ..
DOUBLE PRECISION A1(LDA1,*), A2(LDA2,*), B(LDB,*), CS(*),
$ DWORK(*), F1(LDF1,*), F2(LDF2,*), G(LDG,*)
Arguments
Mode Parameters
TYPEG CHARACTER*1
Specifies the type of the generator, as follows:
= 'D': generator is column oriented and rank
deficient;
= 'C': generator is column oriented and not rank
deficient;
= 'R': generator is row oriented and not rank
deficient.
Note that this parameter must be equivalent with the
used TYPEG in the call of MB02CU.
STRUCG CHARACTER*1
Information about the structure of the generators,
as follows:
= 'T': the trailing block of the positive generator
is upper / lower triangular, and the trailing
block of the negative generator is zero;
= 'N': no special structure to mention.
Input/Output Parameters
K (input) INTEGER
The number of rows in A1 to be processed. K >= 0.
N (input) INTEGER
If TYPEG = 'D' or TYPEG = 'C', the number of rows in F1;
if TYPEG = 'R', the number of columns in F1. N >= 0.
P (input) INTEGER
The number of columns of the positive generator. P >= K.
Q (input) INTEGER
The number of columns in B.
If TYPEG = 'D', Q >= K;
If TYPEG = 'C' or 'R', Q >= 0.
NB (input) INTEGER
On entry, if TYPEG = 'C' or TYPEG = 'R', NB specifies
the block size to be used in the blocked parts of the
algorithm. NB must be equivalent with the used block size
in the routine MB02CU.
RNK (input) INTEGER
If TYPEG = 'D', the number of linearly independent columns
in the generator as returned by MB02CU. 0 <= RNK <= K.
If TYPEG = 'C' or 'R', the value of this parameter is
irrelevant.
A1 (input) DOUBLE PRECISION array, dimension
(LDA1, K)
On entry, if TYPEG = 'D', the leading K-by-K part of this
array must contain the matrix A1 as returned by MB02CU.
If TYPEG = 'C' or 'R', this array is not referenced.
LDA1 INTEGER
The leading dimension of the array A1.
If TYPEG = 'D', LDA1 >= MAX(1,K);
if TYPEG = 'C' or TYPEG = 'R', LDA1 >= 1.
A2 (input) DOUBLE PRECISION array,
if TYPEG = 'D' or TYPEG = 'C', dimension (LDA2, P-K);
if TYPEG = 'R', dimension (LDA2, K).
On entry, if TYPEG = 'D' or TYPEG = 'C', the leading
K-by-(P-K) part of this array must contain the matrix
A2 as returned by MB02CU.
On entry, if TYPEG = 'R', the leading (P-K)-by-K part of
this array must contain the matrix A2 as returned by
MB02CU.
LDA2 INTEGER
The leading dimension of the array A2.
If P = K, LDA2 >= 1;
If P > K and (TYPEG = 'D' or TYPEG = 'C'),
LDA2 >= MAX(1,K);
if P > K and TYPEG = 'R', LDA2 >= P-K.
B (input) DOUBLE PRECISION array,
if TYPEG = 'D' or TYPEG = 'C', dimension (LDB, Q);
if TYPEG = 'R', dimension (LDB, K).
On entry, if TYPEG = 'D' or TYPEG = 'C', the leading
K-by-Q part of this array must contain the matrix B as
returned by MB02CU.
On entry, if TYPEG = 'R', the leading Q-by-K part of this
array must contain the matrix B as returned by MB02CU.
LDB INTEGER
The leading dimension of the array B.
If Q = 0, LDB >= 1;
If Q > 0 and (TYPEG = 'D' or TYPEG = 'C'),
LDB >= MAX(1,K);
if Q > 0 and TYPEG = 'R', LDB >= Q.
F1 (input/output) DOUBLE PRECISION array,
if TYPEG = 'D' or TYPEG = 'C', dimension (LDF1, K);
if TYPEG = 'R', dimension (LDF1, N).
On entry, if TYPEG = 'D' or TYPEG = 'C', the leading
N-by-K part of this array must contain the first part
of the positive generator to be processed.
On entry, if TYPEG = 'R', the leading K-by-N part of this
array must contain the first part of the positive
generator to be processed.
On exit, if TYPEG = 'D' or TYPEG = 'C', the leading
N-by-K part of this array contains the first part of the
transformed positive generator.
On exit, if TYPEG = 'R', the leading K-by-N part of this
array contains the first part of the transformed positive
generator.
LDF1 INTEGER
The leading dimension of the array F1.
If TYPEG = 'D' or TYPEG = 'C', LDF1 >= MAX(1,N);
if TYPEG = 'R', LDF1 >= MAX(1,K).
F2 (input/output) DOUBLE PRECISION array,
if TYPEG = 'D' or TYPEG = 'C', dimension (LDF2, P-K);
if TYPEG = 'R', dimension (LDF2, N).
On entry, if TYPEG = 'D' or TYPEG = 'C', the leading
N-by-(P-K) part of this array must contain the second part
of the positive generator to be processed.
On entry, if TYPEG = 'R', the leading (P-K)-by-N part of
this array must contain the second part of the positive
generator to be processed.
On exit, if TYPEG = 'D' or TYPEG = 'C', the leading
N-by-(P-K) part of this array contains the second part of
the transformed positive generator.
On exit, if TYPEG = 'R', the leading (P-K)-by-N part of
this array contains the second part of the transformed
positive generator.
LDF2 INTEGER
The leading dimension of the array F2.
If P = K, LDF2 >= 1;
If P > K and (TYPEG = 'D' or TYPEG = 'C'),
LDF2 >= MAX(1,N);
if P > K and TYPEG = 'R', LDF2 >= P-K.
G (input/output) DOUBLE PRECISION array,
if TYPEG = 'D' or TYPEG = 'C', dimension (LDG, Q);
if TYPEG = 'R', dimension (LDG, N).
On entry, if TYPEG = 'D' or TYPEG = 'C', the leading
N-by-Q part of this array must contain the negative part
of the generator to be processed.
On entry, if TYPEG = 'R', the leading Q-by-N part of this
array must contain the negative part of the generator to
be processed.
On exit, if TYPEG = 'D' or TYPEG = 'C', the leading
N-by-Q part of this array contains the transformed
negative generator.
On exit, if TYPEG = 'R', the leading Q-by-N part of this
array contains the transformed negative generator.
LDG INTEGER
The leading dimension of the array G.
If Q = 0, LDG >= 1;
If Q > 0 and (TYPEG = 'D' or TYPEG = 'C'),
LDG >= MAX(1,N);
if Q > 0 and TYPEG = 'R', LDG >= Q.
CS (input) DOUBLE PRECISION array, dimension (x)
If TYPEG = 'D' and P = K, x = 3*K;
If TYPEG = 'D' and P > K, x = 5*K;
If (TYPEG = 'C' or TYPEG = 'R') and P = K, x = 4*K;
If (TYPEG = 'C' or TYPEG = 'R') and P > K, x = 6*K.
On entry, the first x elements of this array must contain
Givens and modified hyperbolic rotation parameters, and
scalar factors of the Householder transformations as
returned by MB02CU.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = -23, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
TYPEG = 'D': LDWORK >= MAX(1,N);
(TYPEG = 'C' or TYPEG = 'R') and NB <= 0:
LDWORK >= MAX(1,N);
(TYPEG = 'C' or TYPEG = 'R') and NB >= 1:
LDWORK >= MAX(1,( N + K )*NB).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Numerical Aspects
The algorithm requires 0(N*K*( P + Q )) floating point operations.Method
The Householder transformations and modified hyperbolic rotations computed by SLICOT Library routine MB02CU are applied to the corresponding parts of the matrices F1, F2 and G.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To bring the first blocks of a generator in proper form. The columns / rows of the positive and negative generators are contained in the arrays A and B, respectively. Transformation information will be stored and can be applied via SLICOT Library routine MB02CY.Specification
SUBROUTINE MB02CX( TYPET, P, Q, K, A, LDA, B, LDB, CS, LCS,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TYPET
INTEGER INFO, K, LDA, LDB, LCS, LDWORK, P, Q
C .. Array Arguments ..
DOUBLE PRECISION A(LDA, *), B(LDB, *), CS(*), DWORK(*)
Arguments
Mode Parameters
TYPET CHARACTER*1
Specifies the type of the generator, as follows:
= 'R': A and B are the first blocks of the rows of the
positive and negative generators;
= 'C': A and B are the first blocks of the columns of the
positive and negative generators.
Note: in the sequel, the notation x / y means that
x corresponds to TYPET = 'R' and y corresponds to
TYPET = 'C'.
Input/Output Parameters
P (input) INTEGER
The number of rows / columns in A containing the positive
generators. P >= 0.
Q (input) INTEGER
The number of rows / columns in B containing the negative
generators. Q >= 0.
K (input) INTEGER
The number of columns / rows in A and B to be processed.
Normally, the size of the first block. P >= K >= 0.
A (input/output) DOUBLE PRECISION array, dimension
(LDA, K) / (LDA, P)
On entry, the leading P-by-K upper / K-by-P lower
triangular part of this array must contain the rows /
columns of the positive part in the first block of the
generator.
On exit, the leading P-by-K upper / K-by-P lower
triangular part of this array contains the rows / columns
of the positive part in the first block of the proper
generator.
The lower / upper trapezoidal part is not referenced.
LDA INTEGER
The leading dimension of the array A.
LDA >= MAX(1,P), if TYPET = 'R';
LDA >= MAX(1,K), if TYPET = 'C'.
B (input/output) DOUBLE PRECISION array, dimension
(LDB, K) / (LDB, Q)
On entry, the leading Q-by-K / K-by-Q part of this array
must contain the rows / columns of the negative part in
the first block of the generator.
On exit, the leading Q-by-K / K-by-Q part of this array
contains part of the necessary information for the
Householder transformations.
LDB INTEGER
The leading dimension of the array B.
LDB >= MAX(1,Q), if TYPET = 'R';
LDB >= MAX(1,K), if TYPET = 'C'.
CS (output) DOUBLE PRECISION array, dimension (LCS)
On exit, the leading 2*K + MIN(K,Q) part of this array
contains necessary information for the SLICOT Library
routine MB02CY (modified hyperbolic rotation parameters
and scalar factors of the Householder transformations).
LCS INTEGER
The length of the array CS. LCS >= 2*K + MIN(K,Q).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -12, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,K).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction algorithm failed. The matrix
associated with the generator is not (numerically)
positive definite.
Method
If TYPET = 'R', a QR decomposition of B is first computed. Then, the elements below the first row of each column i of B are annihilated by a Householder transformation modifying the first element in that column. This first element, in turn, is then annihilated by a modified hyperbolic rotation, acting also on the i-th row of A. If TYPET = 'C', an LQ decomposition of B is first computed. Then, the elements on the right of the first column of each row i of B are annihilated by a Householder transformation modifying the first element in that row. This first element, in turn, is then annihilated by a modified hyperbolic rotation, acting also on the i-th column of A.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To apply the transformations created by the SLICOT Library routine MB02CX on other columns / rows of the generator, contained in the arrays A and B of positive and negative generators, respectively.Specification
SUBROUTINE MB02CY( TYPET, STRUCG, P, Q, N, K, A, LDA, B, LDB, H,
$ LDH, CS, LCS, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, K, LDA, LDB, LCS, LDH, LDWORK, N, P, Q
CHARACTER STRUCG, TYPET
C .. Array Arguments ..
DOUBLE PRECISION A(LDA, *), B(LDB, *), CS(*), DWORK(*), H(LDH,*)
Arguments
Mode Parameters
TYPET CHARACTER*1
Specifies the type of the generator, as follows:
= 'R': A and B are additional columns of the generator;
= 'C': A and B are additional rows of the generator.
Note: in the sequel, the notation x / y means that
x corresponds to TYPET = 'R' and y corresponds to
TYPET = 'C'.
STRUCG CHARACTER*1
Information about the structure of the two generators,
as follows:
= 'T': the trailing block of the positive generator
is lower / upper triangular, and the trailing
block of the negative generator is zero;
= 'N': no special structure to mention.
Input/Output Parameters
P (input) INTEGER
The number of rows / columns in A containing the positive
generators. P >= 0.
Q (input) INTEGER
The number of rows / columns in B containing the negative
generators. Q >= 0.
N (input) INTEGER
The number of columns / rows in A and B to be processed.
N >= 0.
K (input) INTEGER
The number of columns / rows in H. P >= K >= 0.
A (input/output) DOUBLE PRECISION array, dimension
(LDA, N) / (LDA, P)
On entry, the leading P-by-N / N-by-P part of this array
must contain the positive part of the generator.
On exit, the leading P-by-N / N-by-P part of this array
contains the transformed positive part of the generator.
LDA INTEGER
The leading dimension of the array A.
LDA >= MAX(1,P), if TYPET = 'R';
LDA >= MAX(1,N), if TYPET = 'C'.
B (input/output) DOUBLE PRECISION array, dimension
(LDB, N) / (LDB, Q)
On entry, the leading Q-by-N / N-by-Q part of this array
must contain the negative part of the generator.
On exit, the leading Q-by-N / N-by-Q part of this array
contains the transformed negative part of the generator.
LDB INTEGER
The leading dimension of the array B.
LDB >= MAX(1,Q), if TYPET = 'R';
LDB >= MAX(1,N), if TYPET = 'C'.
H (input) DOUBLE PRECISION array, dimension
(LDH, K) / (LDH, Q)
The leading Q-by-K / K-by-Q part of this array must
contain part of the necessary information for the
Householder transformations computed by SLICOT Library
routine MB02CX.
LDH INTEGER
The leading dimension of the array H.
LDH >= MAX(1,Q), if TYPET = 'R';
LDH >= MAX(1,K), if TYPET = 'C'.
CS (input) DOUBLE PRECISION array, dimension (LCS)
The leading 2*K + MIN(K,Q) part of this array must
contain the necessary information for modified hyperbolic
rotations and the scalar factors of the Householder
transformations computed by SLICOT Library routine MB02CX.
LCS INTEGER
The length of the array CS. LCS >= 2*K + MIN(K,Q).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -16, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,N).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The Householder transformations and modified hyperbolic rotations computed by SLICOT Library routine MB02CX are applied to the corresponding parts of the matrices A and B.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To update the Cholesky factor and the generator and/or the Cholesky factor of the inverse of a symmetric positive definite (s.p.d.) block Toeplitz matrix T, given the information from a previous factorization and additional blocks in TA of its first block row, or its first block column, depending on the routine parameter TYPET. Transformation information is stored.Specification
SUBROUTINE MB02DD( JOB, TYPET, K, M, N, TA, LDTA, T, LDT, G,
$ LDG, R, LDR, L, LDL, CS, LCS, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER JOB, TYPET
INTEGER INFO, K, LCS, LDG, LDL, LDR, LDT, LDTA, LDWORK,
$ M, N
C .. Array Arguments ..
DOUBLE PRECISION CS(*), DWORK(*), G(LDG, *), L(LDL,*), R(LDR,*),
$ T(LDT,*), TA(LDTA,*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the output of the routine, as follows:
= 'R': updates the generator G of the inverse and
computes the new columns / rows for the Cholesky
factor R of T;
= 'A': updates the generator G, computes the new
columns / rows for the Cholesky factor R of T and
the new rows / columns for the Cholesky factor L
of the inverse;
= 'O': only computes the new columns / rows for the
Cholesky factor R of T.
TYPET CHARACTER*1
Specifies the type of T, as follows:
= 'R': the first block row of an s.p.d. block Toeplitz
matrix was/is defined; if demanded, the Cholesky
factors R and L are upper and lower triangular,
respectively, and G contains the transposed
generator of the inverse;
= 'C': the first block column of an s.p.d. block Toeplitz
matrix was/is defined; if demanded, the Cholesky
factors R and L are lower and upper triangular,
respectively, and G contains the generator of the
inverse. This choice results in a column oriented
algorithm which is usually faster.
Note: in this routine, the notation x / y means that
x corresponds to TYPET = 'R' and y corresponds to
TYPET = 'C'.
Input/Output Parameters
K (input) INTEGER
The number of rows / columns in T, which should be equal
to the blocksize. K >= 0.
M (input) INTEGER
The number of blocks in TA. M >= 0.
N (input) INTEGER
The number of blocks in T. N >= 0.
TA (input/output) DOUBLE PRECISION array, dimension
(LDTA,M*K) / (LDTA,K)
On entry, the leading K-by-M*K / M*K-by-K part of this
array must contain the (N+1)-th to (N+M)-th blocks in the
first block row / column of an s.p.d. block Toeplitz
matrix.
On exit, if INFO = 0, the leading K-by-M*K / M*K-by-K part
of this array contains information on the Householder
transformations used, such that the array
[ T TA ] / [ T ]
[ TA ]
serves as the new transformation matrix T for further
applications of this routine.
LDTA INTEGER
The leading dimension of the array TA.
LDTA >= MAX(1,K), if TYPET = 'R';
LDTA >= MAX(1,M*K), if TYPET = 'C'.
T (input) DOUBLE PRECISION array, dimension (LDT,N*K) /
(LDT,K)
The leading K-by-N*K / N*K-by-K part of this array must
contain transformation information generated by the SLICOT
Library routine MB02CD, i.e., in the first K-by-K block,
the upper / lower Cholesky factor of T(1:K,1:K), and in
the remaining part, the Householder transformations
applied during the initial factorization process.
LDT INTEGER
The leading dimension of the array T.
LDT >= MAX(1,K), if TYPET = 'R';
LDT >= MAX(1,N*K), if TYPET = 'C'.
G (input/output) DOUBLE PRECISION array, dimension
(LDG,( N + M )*K) / (LDG,2*K)
On entry, if JOB = 'R', or 'A', then the leading
2*K-by-N*K / N*K-by-2*K part of this array must contain,
in the first K-by-K block of the second block row /
column, the lower right block of the Cholesky factor of
the inverse of T, and in the remaining part, the generator
of the inverse of T.
On exit, if INFO = 0 and JOB = 'R', or 'A', then the
leading 2*K-by-( N + M )*K / ( N + M )*K-by-2*K part of
this array contains the same information as on entry, now
for the updated Toeplitz matrix. Actually, to obtain a
generator of the inverse one has to set
G(K+1:2*K, 1:K) = 0, if TYPET = 'R';
G(1:K, K+1:2*K) = 0, if TYPET = 'C'.
LDG INTEGER
The leading dimension of the array G.
LDG >= MAX(1,2*K), if TYPET = 'R' and JOB = 'R', or 'A';
LDG >= MAX(1,( N + M )*K),
if TYPET = 'C' and JOB = 'R', or 'A';
LDG >= 1, if JOB = 'O'.
R (input/output) DOUBLE PRECISION array, dimension
(LDR,M*K) / (LDR,( N + M )*K)
On input, the leading N*K-by-K part of R(K+1,1) /
K-by-N*K part of R(1,K+1) contains the last block column /
row of the previous Cholesky factor R.
On exit, if INFO = 0, then the leading
( N + M )*K-by-M*K / M*K-by-( N + M )*K part of this
array contains the last M*K columns / rows of the upper /
lower Cholesky factor of T. The elements in the strictly
lower / upper triangular part are not referenced.
LDR INTEGER
The leading dimension of the array R.
LDR >= MAX(1, ( N + M )*K), if TYPET = 'R';
LDR >= MAX(1, M*K), if TYPET = 'C'.
L (output) DOUBLE PRECISION array, dimension
(LDL,( N + M )*K) / (LDL,M*K)
If INFO = 0 and JOB = 'A', then the leading
M*K-by-( N + M )*K / ( N + M )*K-by-M*K part of this
array contains the last M*K rows / columns of the lower /
upper Cholesky factor of the inverse of T. The elements
in the strictly upper / lower triangular part are not
referenced.
LDL INTEGER
The leading dimension of the array L.
LDL >= MAX(1, M*K), if TYPET = 'R' and JOB = 'A';
LDL >= MAX(1, ( N + M )*K), if TYPET = 'C' and JOB = 'A';
LDL >= 1, if JOB = 'R', or 'O'.
CS (input/output) DOUBLE PRECISION array, dimension (LCS)
On input, the leading 3*(N-1)*K part of this array must
contain the necessary information about the hyperbolic
rotations and Householder transformations applied
previously by SLICOT Library routine MB02CD.
On exit, if INFO = 0, then the leading 3*(N+M-1)*K part of
this array contains information about all the hyperbolic
rotations and Householder transformations applied during
the whole process.
LCS INTEGER
The length of the array CS. LCS >= 3*(N+M-1)*K.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -19, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,(N+M-1)*K).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction algorithm failed. The block Toeplitz
matrix associated with [ T TA ] / [ T' TA' ]' is
not (numerically) positive definite.
Method
Householder transformations and modified hyperbolic rotations are used in the Schur algorithm [1], [2].References
[1] Kailath, T. and Sayed, A.
Fast Reliable Algorithms for Matrices with Structure.
SIAM Publications, Philadelphia, 1999.
[2] Kressner, D. and Van Dooren, P.
Factorizations and linear system solvers for matrices with
Toeplitz structure.
SLICOT Working Note 2000-2, 2000.
Numerical Aspects
The implemented method is numerically stable.
3 2
The algorithm requires 0(K ( N M + M ) ) floating point
operations.
Further Comments
For min(K,N,M) = 0, the routine sets DWORK(1) = 1 and returns. Although the calculations could still be performed when N = 0, but min(K,M) > 0, this case is not considered as an "update". SLICOT Library routine MB02CD should be called with the argument M instead of N.Example
Program Text
* MB02DD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER KMAX, MMAX, NMAX
PARAMETER ( KMAX = 20, MMAX = 20, NMAX = 20 )
INTEGER LCS, LDG, LDL, LDR, LDT, LDWORK
PARAMETER ( LDG = KMAX*( MMAX + NMAX ),
$ LDL = KMAX*( MMAX + NMAX ),
$ LDR = KMAX*( MMAX + NMAX ),
$ LDT = KMAX*( MMAX + NMAX ),
$ LDWORK = ( MMAX + NMAX - 1 )*KMAX )
PARAMETER ( LCS = 3*LDWORK )
* .. Local Scalars ..
INTEGER I, INFO, J, K, M, N, S
CHARACTER JOB, TYPET
* .. Local Arrays ..
* The arrays are dimensioned for both TYPET = 'R' and TYPET = 'C'.
* Arrays G and T could be smaller.
* For array G, it is assumed that MMAX + NMAX >= 2.
* The matrix TA is also stored in the array T.
DOUBLE PRECISION CS(LCS), DWORK(LDWORK),
$ G(LDG, KMAX*( MMAX + NMAX )),
$ L(LDL, KMAX*( MMAX + NMAX )),
$ R(LDR, KMAX*( MMAX + NMAX )),
$ T(LDT, KMAX*( MMAX + NMAX ))
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL DLACPY, MB02CD, MB02DD
*
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, K, M, JOB, TYPET
S = ( N + M )*K
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) N
ELSE
IF ( K.LE.0 .OR. K.GT.KMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) K
ELSE
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) M
ELSE
IF ( LSAME( TYPET, 'R' ) ) THEN
READ ( NIN, FMT = * ) ( ( T(I,J), J = 1,S ), I = 1,K )
ELSE
READ ( NIN, FMT = * ) ( ( T(I,J), J = 1,K ), I = 1,S )
END IF
* Compute the Cholesky factors.
CALL MB02CD( JOB, TYPET, K, N, T, LDT, G, LDG, R, LDR, L,
$ LDL, CS, LCS, DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99996 )
DO 10 I = 1, N*K
WRITE ( NOUT, FMT = 99990 ) ( R(I,J), J = 1, N*K )
10 CONTINUE
IF ( LSAME( JOB, 'R' ) .OR. LSAME( JOB, 'A' ) ) THEN
WRITE ( NOUT, FMT = 99995 )
IF ( LSAME( TYPET, 'R' ) ) THEN
DO 20 I = 1, 2*K
WRITE ( NOUT, FMT = 99990 )
$ ( G(I,J), J = 1, N*K )
20 CONTINUE
ELSE
DO 30 I = 1, N*K
WRITE ( NOUT, FMT = 99990 )
$ ( G(I,J), J = 1, 2*K )
30 CONTINUE
END IF
END IF
IF ( LSAME( JOB, 'A' ) ) THEN
WRITE ( NOUT, FMT = 99994 )
DO 40 I = 1, N*K
WRITE ( NOUT, FMT = 99990 )
$ ( L(I,J), J = 1, N*K )
40 CONTINUE
END IF
* Update the Cholesky factors.
IF ( LSAME( TYPET, 'R' ) ) THEN
* Copy the last block column of R.
CALL DLACPY( 'All', N*K, K, R(1,(N-1)*K+1), LDR,
$ R(K+1,N*K+1), LDR )
CALL MB02DD( JOB, TYPET, K, M, N, T(1,N*K+1), LDT,
$ T, LDT, G, LDG, R(1,N*K+1), LDR,
$ L(N*K+1,1), LDL, CS, LCS, DWORK,
$ LDWORK, INFO )
ELSE
* Copy the last block row of R.
CALL DLACPY( 'All', K, N*K, R((N-1)*K+1,1), LDR,
$ R(N*K+1,K+1), LDR )
CALL MB02DD( JOB, TYPET, K, M, N, T(N*K+1,1), LDT,
$ T, LDT, G, LDG, R(N*K+1,1), LDR,
$ L(1,N*K+1), LDL, CS, LCS, DWORK,
$ LDWORK, INFO )
END IF
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99993 )
DO 50 I = 1, S
WRITE ( NOUT, FMT = 99990 ) ( R(I,J), J = 1, S )
50 CONTINUE
IF ( LSAME( JOB, 'R' ) .OR. LSAME( JOB, 'A' ) )
$ THEN
WRITE ( NOUT, FMT = 99992 )
IF ( LSAME( TYPET, 'R' ) ) THEN
DO 60 I = 1, 2*K
WRITE ( NOUT, FMT = 99990 )
$ ( G(I,J), J = 1, S )
60 CONTINUE
ELSE
DO 70 I = 1, S
WRITE ( NOUT, FMT = 99990 )
$ ( G(I,J), J = 1, 2*K )
70 CONTINUE
END IF
END IF
IF ( LSAME( JOB, 'A' ) ) THEN
WRITE ( NOUT, FMT = 99991 )
DO 80 I = 1, S
WRITE ( NOUT, FMT = 99990 )
$ ( L(I,J), J = 1, S )
80 CONTINUE
END IF
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT ( ' MB02DD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT ( ' INFO on exit from MB02CD = ',I2)
99997 FORMAT ( ' INFO on exit from MB02DD = ',I2)
99996 FORMAT ( ' The Cholesky factor is ')
99995 FORMAT (/' The inverse generator is ')
99994 FORMAT (/' The inverse Cholesky factor is ')
99993 FORMAT (/' The updated Cholesky factor is ')
99992 FORMAT (/' The updated inverse generator is ')
99991 FORMAT (/' The updated inverse Cholesky factor is ')
99990 FORMAT (20(1X,F8.4))
99989 FORMAT (/' N is out of range.',/' N = ',I5)
99988 FORMAT (/' K is out of range.',/' K = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
END
Program Data
MB02DD EXAMPLE PROGRAM DATA
3 2 2 A R
3.0000 1.0000 0.1000 0.1000 0.2000 0.0500 0.1000 0.0400 0.01 0.02
1.0000 4.0000 0.4000 0.1000 0.0400 0.2000 0.0300 0.0200 0.03 0.01
Program Results
MB02DD EXAMPLE PROGRAM RESULTS The Cholesky factor is 1.7321 0.5774 0.0577 0.0577 0.1155 0.0289 0.0000 1.9149 0.1915 0.0348 -0.0139 0.0957 0.0000 0.0000 1.7205 0.5754 0.0558 0.0465 0.0000 0.0000 0.0000 1.9142 0.1890 0.0357 0.0000 0.0000 0.0000 0.0000 1.7169 0.5759 0.0000 0.0000 0.0000 0.0000 0.0000 1.9118 The inverse generator is -0.2355 0.5231 -0.0642 0.0077 0.0187 -0.0265 -0.5568 -0.0568 0.0229 0.0060 0.0363 0.0000 0.5825 0.0000 -0.0387 0.0052 0.0003 -0.0575 -0.1754 0.5231 0.0119 -0.0265 -0.0110 0.0076 The inverse Cholesky factor is 0.5774 0.0000 0.0000 0.0000 0.0000 0.0000 -0.1741 0.5222 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0581 0.5812 0.0000 0.0000 0.0000 -0.0142 0.0080 -0.1747 0.5224 0.0000 0.0000 -0.0387 0.0052 0.0003 -0.0575 0.5825 0.0000 0.0119 -0.0265 -0.0110 0.0076 -0.1754 0.5231 The updated Cholesky factor is 1.7321 0.5774 0.0577 0.0577 0.1155 0.0289 0.0577 0.0231 0.0058 0.0115 0.0000 1.9149 0.1915 0.0348 -0.0139 0.0957 -0.0017 0.0035 0.0139 0.0017 0.0000 0.0000 1.7205 0.5754 0.0558 0.0465 0.1145 0.0279 0.0564 0.0227 0.0000 0.0000 0.0000 1.9142 0.1890 0.0357 -0.0152 0.0953 -0.0017 0.0033 0.0000 0.0000 0.0000 0.0000 1.7169 0.5759 0.0523 0.0453 0.1146 0.0273 0.0000 0.0000 0.0000 0.0000 0.0000 1.9118 0.1902 0.0357 -0.0157 0.0955 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.7159 0.5757 0.0526 0.0450 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.9118 0.1901 0.0357 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.7159 0.5757 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.9117 The updated inverse generator is -0.5599 0.3310 -0.0305 0.0098 0.0392 -0.0209 0.0191 -0.0010 -0.0045 0.0035 -0.2289 -0.4091 0.0612 -0.0012 0.0125 0.0182 0.0042 0.0017 0.0014 0.0000 0.5828 0.0000 0.0027 -0.0029 -0.0195 0.0072 -0.0393 0.0057 0.0016 -0.0580 -0.1755 0.5231 -0.0037 0.0022 0.0005 -0.0022 0.0125 -0.0266 -0.0109 0.0077 The updated inverse Cholesky factor is 0.5774 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.1741 0.5222 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0581 0.5812 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0142 0.0080 -0.1747 0.5224 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0387 0.0052 0.0003 -0.0575 0.5825 0.0000 0.0000 0.0000 0.0000 0.0000 0.0119 -0.0265 -0.0110 0.0076 -0.1754 0.5231 0.0000 0.0000 0.0000 0.0000 -0.0199 0.0073 -0.0391 0.0056 0.0017 -0.0580 0.5828 0.0000 0.0000 0.0000 0.0007 -0.0023 0.0122 -0.0265 -0.0110 0.0077 -0.1755 0.5231 0.0000 0.0000 0.0027 -0.0029 -0.0195 0.0072 -0.0393 0.0057 0.0016 -0.0580 0.5828 0.0000 -0.0037 0.0022 0.0005 -0.0022 0.0125 -0.0266 -0.0109 0.0077 -0.1755 0.5231
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02ED.html 0000664 0000000 0000000 00000022423 14560147231 0020326 0 ustar 00root root 0000000 0000000
Purpose
To solve a system of linear equations T*X = B or X*T = B with a symmetric positive definite (s.p.d.) block Toeplitz matrix T. T is defined either by its first block row or its first block column, depending on the parameter TYPET.Specification
SUBROUTINE MB02ED( TYPET, K, N, NRHS, T, LDT, B, LDB, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TYPET
INTEGER INFO, K, LDB, LDT, LDWORK, N, NRHS
C .. Array Arguments ..
DOUBLE PRECISION B(LDB,*), DWORK(*), T(LDT,*)
Arguments
Mode Parameters
TYPET CHARACTER*1
Specifies the type of T, as follows:
= 'R': T contains the first block row of an s.p.d. block
Toeplitz matrix, and the system X*T = B is solved;
= 'C': T contains the first block column of an s.p.d.
block Toeplitz matrix, and the system T*X = B is
solved.
Note: in the sequel, the notation x / y means that
x corresponds to TYPET = 'R' and y corresponds to
TYPET = 'C'.
Input/Output Parameters
K (input) INTEGER
The number of rows / columns in T, which should be equal
to the blocksize. K >= 0.
N (input) INTEGER
The number of blocks in T. N >= 0.
NRHS (input) INTEGER
The number of right hand sides. NRHS >= 0.
T (input/output) DOUBLE PRECISION array, dimension
(LDT,N*K) / (LDT,K)
On entry, the leading K-by-N*K / N*K-by-K part of this
array must contain the first block row / column of an
s.p.d. block Toeplitz matrix.
On exit, if INFO = 0 and NRHS > 0, then the leading
K-by-N*K / N*K-by-K part of this array contains the last
row / column of the Cholesky factor of inv(T).
LDT INTEGER
The leading dimension of the array T.
LDT >= MAX(1,K), if TYPET = 'R';
LDT >= MAX(1,N*K), if TYPET = 'C'.
B (input/output) DOUBLE PRECISION array, dimension
(LDB,N*K) / (LDB,NRHS)
On entry, the leading NRHS-by-N*K / N*K-by-NRHS part of
this array must contain the right hand side matrix B.
On exit, the leading NRHS-by-N*K / N*K-by-NRHS part of
this array contains the solution matrix X.
LDB INTEGER
The leading dimension of the array B.
LDB >= MAX(1,NRHS), if TYPET = 'R';
LDB >= MAX(1,N*K), if TYPET = 'C'.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -10, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,N*K*K+(N+2)*K).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction algorithm failed. The Toeplitz matrix
associated with T is not (numerically) positive
definite.
Method
Householder transformations, modified hyperbolic rotations and block Gaussian eliminations are used in the Schur algorithm [1], [2].References
[1] Kailath, T. and Sayed, A.
Fast Reliable Algorithms for Matrices with Structure.
SIAM Publications, Philadelphia, 1999.
[2] Kressner, D. and Van Dooren, P.
Factorizations and linear system solvers for matrices with
Toeplitz structure.
SLICOT Working Note 2000-2, 2000.
Numerical Aspects
The implemented method is numerically equivalent with forming
the Cholesky factor R and the inverse Cholesky factor of T, using
the generalized Schur algorithm, and solving the systems of
equations R*X = L*B or X*R = B*L by a blocked backward
substitution algorithm.
3 2 2 2
The algorithm requires 0(K N + K N NRHS) floating point
operations.
Further Comments
NoneExample
Program Text
* MB02ED EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER KMAX, NMAX
PARAMETER ( KMAX = 20, NMAX = 20 )
INTEGER LDB, LDT, LDWORK
PARAMETER ( LDB = KMAX*NMAX, LDT = KMAX*NMAX,
$ LDWORK = NMAX*KMAX*KMAX + ( NMAX+2 )*KMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, K, M, N, NRHS
CHARACTER TYPET
* .. Local Arrays ..
* The arrays B and T are dimensioned for both TYPET = 'R' and
* TYPET = 'C'.
* NRHS is assumed to be not larger than KMAX*NMAX.
DOUBLE PRECISION B(LDB, KMAX*NMAX), DWORK(LDWORK),
$ T(LDT, KMAX*NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL MB02ED
*
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, K, NRHS, TYPET
M = N*K
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
IF ( K.LE.0 .OR. K.GT.KMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) K
ELSE
IF ( NRHS.LE.0 .OR. NRHS.GT.KMAX*NMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) NRHS
ELSE
IF ( LSAME( TYPET, 'R' ) ) THEN
READ ( NIN, FMT = * ) ( ( T(I,J), J = 1,M ), I = 1,K )
ELSE
READ ( NIN, FMT = * ) ( ( T(I,J), J = 1,K ), I = 1,M )
END IF
IF ( LSAME( TYPET, 'R' ) ) THEN
READ (NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,
$ NRHS )
ELSE
READ (NIN, FMT = * ) ( ( B(I,J), J = 1,NRHS ), I = 1,
$ M )
END IF
* Compute the solution of X T = B or T X = B.
CALL MB02ED( TYPET, K, N, NRHS, T, LDT, B, LDB, DWORK,
$ LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( LSAME( TYPET, 'R' ) ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, NRHS
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1, M )
10 CONTINUE
ELSE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, M
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,
$ NRHS )
20 CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' MB02ED EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB02ED = ',I2)
99997 FORMAT (' The solution of X*T = B is ')
99996 FORMAT (' The solution of T*X = B is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' K is out of range.',/' K = ',I5)
99992 FORMAT (/' NRHS is out of range.',/' NRHS = ',I5)
END
Program Data
MB02ED EXAMPLE PROGRAM DATA
3 3 2 C
3.0000 1.0000 0.2000
1.0000 4.0000 0.4000
0.2000 0.4000 5.0000
0.1000 0.1000 0.2000
0.2000 0.0400 0.0300
0.0500 0.2000 0.1000
0.1000 0.0300 0.1000
0.0400 0.0200 0.2000
0.0100 0.0300 0.0200
1.0000 2.0000
1.0000 2.0000
1.0000 2.0000
1.0000 2.0000
1.0000 2.0000
1.0000 2.0000
1.0000 2.0000
1.0000 2.0000
1.0000 2.0000
Program Results
MB02ED EXAMPLE PROGRAM RESULTS The solution of T*X = B is 0.2408 0.4816 0.1558 0.3116 0.1534 0.3068 0.2302 0.4603 0.1467 0.2934 0.1537 0.3075 0.2349 0.4698 0.1498 0.2995 0.1653 0.3307
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02FD.html 0000664 0000000 0000000 00000033212 14560147231 0020325 0 ustar 00root root 0000000 0000000
Purpose
To compute the incomplete Cholesky (ICC) factor of a symmetric positive definite (s.p.d.) block Toeplitz matrix T, defined by either its first block row, or its first block column, depending on the routine parameter TYPET. By subsequent calls of this routine, further rows / columns of the Cholesky factor can be added. Furthermore, the generator of the Schur complement of the leading (P+S)*K-by-(P+S)*K block in T is available, which can be used, e.g., for measuring the quality of the ICC factorization.Specification
SUBROUTINE MB02FD( TYPET, K, N, P, S, T, LDT, R, LDR, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TYPET
INTEGER INFO, K, LDR, LDT, LDWORK, N, P, S
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), R(LDR,*), T(LDT,*)
Arguments
Mode Parameters
TYPET CHARACTER*1
Specifies the type of T, as follows:
= 'R': T contains the first block row of an s.p.d. block
Toeplitz matrix; the ICC factor R is upper
trapezoidal;
= 'C': T contains the first block column of an s.p.d.
block Toeplitz matrix; the ICC factor R is lower
trapezoidal; this choice leads to better
localized memory references and hence a faster
algorithm.
Note: in the sequel, the notation x / y means that
x corresponds to TYPET = 'R' and y corresponds to
TYPET = 'C'.
Input/Output Parameters
K (input) INTEGER
The number of rows / columns in T, which should be equal
to the blocksize. K >= 0.
N (input) INTEGER
The number of blocks in T. N >= 0.
P (input) INTEGER
The number of previously computed block rows / columns
of R. 0 <= P <= N.
S (input) INTEGER
The number of block rows / columns of R to compute.
0 <= S <= N-P.
T (input/output) DOUBLE PRECISION array, dimension
(LDT,(N-P)*K) / (LDT,K)
On entry, if P = 0, then the leading K-by-N*K / N*K-by-K
part of this array must contain the first block row /
column of an s.p.d. block Toeplitz matrix.
If P > 0, the leading K-by-(N-P)*K / (N-P)*K-by-K must
contain the negative generator of the Schur complement of
the leading P*K-by-P*K part in T, computed from previous
calls of this routine.
On exit, if INFO = 0, then the leading K-by-(N-P)*K /
(N-P)*K-by-K part of this array contains, in the first
K-by-K block, the upper / lower Cholesky factor of
T(1:K,1:K), in the following S-1 K-by-K blocks, the
Householder transformations applied during the process,
and in the remaining part, the negative generator of the
Schur complement of the leading (P+S)*K-by(P+S)*K part
in T.
LDT INTEGER
The leading dimension of the array T.
LDT >= MAX(1,K), if TYPET = 'R';
LDT >= MAX(1,(N-P)*K), if TYPET = 'C'.
R (input/output) DOUBLE PRECISION array, dimension
(LDR, N*K) / (LDR, S*K ) if P = 0;
(LDR, (N-P+1)*K) / (LDR, (S+1)*K ) if P > 0.
On entry, if P > 0, then the leading K-by-(N-P+1)*K /
(N-P+1)*K-by-K part of this array must contain the
nonzero blocks of the last block row / column in the
ICC factor from a previous call of this routine. Note that
this part is identical with the positive generator of
the Schur complement of the leading P*K-by-P*K part in T.
If P = 0, then R is only an output parameter.
On exit, if INFO = 0 and P = 0, then the leading
S*K-by-N*K / N*K-by-S*K part of this array contains the
upper / lower trapezoidal ICC factor.
On exit, if INFO = 0 and P > 0, then the leading
(S+1)*K-by-(N-P+1)*K / (N-P+1)*K-by-(S+1)*K part of this
array contains the upper / lower trapezoidal part of the
P-th to (P+S)-th block rows / columns of the ICC factor.
The elements in the strictly lower / upper trapezoidal
part are not referenced.
LDR INTEGER
The leading dimension of the array R.
LDR >= MAX(1, S*K ), if TYPET = 'R' and P = 0;
LDR >= MAX(1, (S+1)*K ), if TYPET = 'R' and P > 0;
LDR >= MAX(1, N*K ), if TYPET = 'C' and P = 0;
LDR >= MAX(1, (N-P+1)*K ), if TYPET = 'C' and P > 0.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -11, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,(N+1)*K,4*K), if P = 0;
LDWORK >= MAX(1,(N-P+2)*K,4*K), if P > 0.
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction algorithm failed; the Toeplitz matrix
associated with T is not (numerically) positive
definite in its leading (P+S)*K-by-(P+S)*K part.
Method
Householder transformations and modified hyperbolic rotations are used in the Schur algorithm [1], [2].References
[1] Kailath, T. and Sayed, A.
Fast Reliable Algorithms for Matrices with Structure.
SIAM Publications, Philadelphia, 1999.
[2] Kressner, D. and Van Dooren, P.
Factorizations and linear system solvers for matrices with
Toeplitz structure.
SLICOT Working Note 2000-2, 2000.
Numerical Aspects
The implemented method is numerically stable.
3
The algorithm requires 0(K S (N-P)) floating point operations.
Further Comments
NoneExample
Program Text
* MB02FD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER ITMAX, KMAX, NMAX
PARAMETER ( ITMAX = 10, KMAX = 20, NMAX = 20 )
INTEGER LDR, LDT, LDWORK
PARAMETER ( LDR = NMAX*KMAX, LDT = KMAX,
$ LDWORK = ( NMAX + 1 )*KMAX )
* .. Local Scalars ..
INTEGER I, INFO, IT, J, K, LEN, M, N, P, PIT, POS, POSR,
$ S1, SCIT
CHARACTER TYPET
DOUBLE PRECISION NNRM
* .. Local Arrays .. (Dimensioned for TYPET = 'R'.)
INTEGER S(ITMAX)
DOUBLE PRECISION DWORK(LDWORK), R(LDR, NMAX*KMAX),
$ T(LDT, NMAX*KMAX), V(NMAX*KMAX), W(NMAX*KMAX),
$ Z(NMAX*KMAX)
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DNRM2
EXTERNAL DNRM2, LSAME
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGEMV, DLASET, DSCAL, DTRMV, MB02FD
*
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, K, IT
TYPET = 'R'
M = N*K
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE IF( K.LE.0 .OR. K.GT.KMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) K
ELSE IF( IT.LE.0 .OR. IT.GT.ITMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) IT
ELSE
READ ( NIN, FMT = * ) ( S(I), I = 1, IT )
READ ( NIN, FMT = * ) ( ( T(I,J), J = 1,M ), I = 1,K )
P = 0
POS = 1
WRITE ( NOUT, FMT = 99997 )
DO 90 SCIT = 1, IT
CALL MB02FD( TYPET, K, N, P, S(SCIT), T(1,POS), LDT,
$ R(POS,POS), LDR, DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
STOP
END IF
S1 = S(SCIT) + P
IF ( S1.EQ.0 ) THEN
* Estimate the 2-norm of the Toeplitz matrix with 5 power
* iterations.
LEN = N*K
CALL DLASET( 'All', LEN, 1, ONE, ONE, V, 1 )
DO 30 PIT = 1, 5
DO 10 I = 1, N
CALL DGEMV( 'NoTranspose', K, LEN-(I-1)*K, ONE, T,
$ LDT, V((I-1)*K+1), 1, ZERO,
$ W((I-1)*K+1), 1 )
10 CONTINUE
DO 20 I = 1, N-1
CALL DGEMV( 'Transpose', K, (N-I)*K, ONE,
$ T(1,K+1), LDT, V((I-1)*K+1), 1,
$ ONE, W(I*K+1), 1 )
20 CONTINUE
CALL DCOPY( LEN, W, 1, V, 1 )
NNRM = DNRM2( LEN, V, 1 )
CALL DSCAL( LEN, ONE/NNRM, V, 1 )
30 CONTINUE
ELSE
* Estimate the 2-norm of the Schur complement with 5 power
* iterations.
LEN = ( N - S1 )*K
CALL DLASET( 'All', LEN, 1, ONE, ONE, V, 1 )
DO 80 PIT = 1, 5
POSR = ( S1 - 1 )*K + 1
DO 40 I = 1, N - S1
CALL DGEMV( 'NoTranspose', K, LEN-(I-1)*K, ONE,
$ T(1,POSR+K), LDT, V((I-1)*K+1), 1,
$ ZERO, W((I-1)*K+1), 1 )
40 CONTINUE
DO 50 I = 1, N - S1
CALL DTRMV( 'Upper', 'NoTranspose', 'NonUnit', K,
$ R(POSR,POSR), LDR, V((I-1)*K+1), 1 )
CALL DGEMV( 'NoTranspose', K, LEN-I*K, ONE,
$ R(POSR,POSR+K), LDR, V(I*K+1), 1, ONE,
$ V((I-1)*K+1), 1 )
50 CONTINUE
CALL DLASET( 'All', LEN, 1, ZERO, ZERO, Z, 1 )
DO 60 I = 1, N - S1
CALL DGEMV( 'Transpose', K, LEN-I*K, ONE,
$ R(POSR,POSR+K), LDR, V((I-1)*K+1), 1,
$ ONE, Z(I*K+1), 1 )
CALL DTRMV( 'Upper', 'Transpose', 'NonUnit', K,
$ R(POSR,POSR), LDR, V((I-1)*K+1), 1 )
CALL DAXPY( K, ONE, V((I-1)*K+1), 1, Z((I-1)*K+1),
$ 1 )
60 CONTINUE
CALL DLASET( 'All', LEN, 1, ZERO, ZERO, V, 1 )
DO 70 I = 1, N - S1
CALL DGEMV( 'Transpose', K, LEN-(I-1)*K, ONE,
$ T(1,POSR+K), LDT, W((I-1)*K+1), 1,
$ ONE, V((I-1)*K+1), 1 )
70 CONTINUE
CALL DAXPY( LEN, -ONE, Z, 1, V, 1 )
NNRM = DNRM2( LEN, V, 1 )
CALL DSCAL( LEN, -ONE/NNRM, V, 1 )
80 CONTINUE
POS = ( S1 - 1 )*K + 1
P = S1
END IF
WRITE ( NOUT, FMT = 99995 ) P*K, NNRM
90 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 100 I = 1, P*K
WRITE ( NOUT, FMT = 99994 ) ( R(I,J), J = 1, M )
100 CONTINUE
END IF
STOP
*
99999 FORMAT (' MB02FD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB02FD = ',I2)
99997 FORMAT (' Incomplete Cholesky factorization ',
$ //' rows norm(Schur complement)',/)
99996 FORMAT (/' The upper ICC factor of the block Toeplitz matrix is '
$ )
99995 FORMAT (I4,5X,F8.4)
99994 FORMAT (20(1X,F8.4))
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' K is out of range.',/' K = ',I5)
99991 FORMAT (/' IT is out of range.',/' IT = ',I5)
END
Program Data
MB02FD EXAMPLE
4 2 3
0 1 1
3.0000 1.0000 0.1000 0.1000 0.2000 0.0500 0.2000 0.3000
1.0000 4.0000 0.4000 0.1000 0.0400 0.2000 0.1000 0.2000
Program Results
MB02FD EXAMPLE PROGRAM RESULTS Incomplete Cholesky factorization rows norm(Schur complement) 0 5.5509 2 5.1590 4 4.8766 The upper ICC factor of the block Toeplitz matrix is 1.7321 0.5774 0.0577 0.0577 0.1155 0.0289 0.1155 0.1732 0.0000 1.9149 0.1915 0.0348 -0.0139 0.0957 0.0174 0.0522 0.0000 0.0000 1.7205 0.5754 0.0558 0.0465 0.1104 0.0174 0.0000 0.0000 0.0000 1.9142 0.1890 0.0357 -0.0161 0.0931
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02GD.html 0000664 0000000 0000000 00000026010 14560147231 0020324 0 ustar 00root root 0000000 0000000
Purpose
To compute the Cholesky factor of a banded symmetric positive definite (s.p.d.) block Toeplitz matrix, defined by either its first block row, or its first block column, depending on the routine parameter TYPET. By subsequent calls of this routine the Cholesky factor can be computed block column by block column.Specification
SUBROUTINE MB02GD( TYPET, TRIU, K, N, NL, P, S, T, LDT, RB, LDRB,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TRIU, TYPET
INTEGER INFO, K, LDRB, LDT, LDWORK, N, NL, P, S
C .. Array Arguments ..
DOUBLE PRECISION DWORK(LDWORK), RB(LDRB,*), T(LDT,*)
Arguments
Mode Parameters
TYPET CHARACTER*1
Specifies the type of T, as follows:
= 'R': T contains the first block row of an s.p.d. block
Toeplitz matrix; the Cholesky factor is upper
triangular;
= 'C': T contains the first block column of an s.p.d.
block Toeplitz matrix; the Cholesky factor is
lower triangular. This choice results in a column
oriented algorithm which is usually faster.
Note: in the sequel, the notation x / y means that
x corresponds to TYPET = 'R' and y corresponds to
TYPET = 'C'.
TRIU CHARACTER*1
Specifies the structure of the last block in T, as
follows:
= 'N': the last block has no special structure;
= 'T': the last block is lower / upper triangular.
Input/Output Parameters
K (input) INTEGER
The number of rows / columns in T, which should be equal
to the blocksize. K >= 0.
N (input) INTEGER
The number of blocks in T. N >= 1.
If TRIU = 'N', N >= 1;
if TRIU = 'T', N >= 2.
NL (input) INTEGER
The lower block bandwidth, i.e., NL + 1 is the number of
nonzero blocks in the first block column of the block
Toeplitz matrix.
If TRIU = 'N', 0 <= NL < N;
if TRIU = 'T', 1 <= NL < N.
P (input) INTEGER
The number of previously computed block rows / columns of
the Cholesky factor. 0 <= P <= N.
S (input) INTEGER
The number of block rows / columns of the Cholesky factor
to compute. 0 <= S <= N - P.
T (input/output) DOUBLE PRECISION array, dimension
(LDT,(NL+1)*K) / (LDT,K)
On entry, if P = 0, the leading K-by-(NL+1)*K /
(NL+1)*K-by-K part of this array must contain the first
block row / column of an s.p.d. block Toeplitz matrix.
On entry, if P > 0, the leading K-by-(NL+1)*K /
(NL+1)*K-by-K part of this array must contain the P-th
block row / column of the Cholesky factor.
On exit, if INFO = 0, then the leading K-by-(NL+1)*K /
(NL+1)*K-by-K part of this array contains the (P+S)-th
block row / column of the Cholesky factor.
LDT INTEGER
The leading dimension of the array T.
LDT >= MAX(1,K) / MAX(1,(NL+1)*K).
RB (input/output) DOUBLE PRECISION array, dimension
(LDRB,MIN(P+NL+S,N)*K) / (LDRB,MIN(P+S,N)*K)
On entry, if TYPET = 'R' and TRIU = 'N' and P > 0,
the leading (NL+1)*K-by-MIN(NL,N-P)*K part of this array
must contain the (P*K+1)-st to ((P+NL)*K)-th columns
of the upper Cholesky factor in banded format from a
previous call of this routine.
On entry, if TYPET = 'R' and TRIU = 'T' and P > 0,
the leading (NL*K+1)-by-MIN(NL,N-P)*K part of this array
must contain the (P*K+1)-st to (MIN(P+NL,N)*K)-th columns
of the upper Cholesky factor in banded format from a
previous call of this routine.
On exit, if TYPET = 'R' and TRIU = 'N', the leading
(NL+1)*K-by-MIN(NL+S,N-P)*K part of this array contains
the (P*K+1)-st to (MIN(P+NL+S,N)*K)-th columns of the
upper Cholesky factor in banded format.
On exit, if TYPET = 'R' and TRIU = 'T', the leading
(NL*K+1)-by-MIN(NL+S,N-P)*K part of this array contains
the (P*K+1)-st to (MIN(P+NL+S,N)*K)-th columns of the
upper Cholesky factor in banded format.
On exit, if TYPET = 'C' and TRIU = 'N', the leading
(NL+1)*K-by-MIN(S,N-P)*K part of this array contains
the (P*K+1)-st to (MIN(P+S,N)*K)-th columns of the lower
Cholesky factor in banded format.
On exit, if TYPET = 'C' and TRIU = 'T', the leading
(NL*K+1)-by-MIN(S,N-P)*K part of this array contains
the (P*K+1)-st to (MIN(P+S,N)*K)-th columns of the lower
Cholesky factor in banded format.
For further details regarding the band storage scheme see
the documentation of the LAPACK routine DPBTF2.
LDRB INTEGER
The leading dimension of the array RB.
If TRIU = 'N', LDRB >= MAX( (NL+1)*K,1 );
if TRIU = 'T', LDRB >= NL*K+1.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -13, DWORK(1) returns the minimum
value of LDWORK.
The first 1 + ( NL + 1 )*K*K elements of DWORK should be
preserved during successive calls of the routine.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 1 + ( NL + 1 )*K*K + NL*K.
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction algorithm failed. The Toeplitz matrix
associated with T is not (numerically) positive
definite.
Method
Householder transformations and modified hyperbolic rotations are used in the Schur algorithm [1], [2].References
[1] Kailath, T. and Sayed, A.
Fast Reliable Algorithms for Matrices with Structure.
SIAM Publications, Philadelphia, 1999.
[2] Kressner, D. and Van Dooren, P.
Factorizations and linear system solvers for matrices with
Toeplitz structure.
SLICOT Working Note 2000-2, 2000.
Numerical Aspects
The implemented method is numerically stable.
3
The algorithm requires O( K *N*NL ) floating point operations.
Further Comments
NoneExample
Program Text
* MB02GD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER KMAX, NMAX, NLMAX
PARAMETER ( KMAX = 20, NMAX = 20, NLMAX = 20 )
INTEGER LDRB, LDT, LDWORK
PARAMETER ( LDRB = ( NLMAX + 1 )*KMAX, LDT = KMAX*NMAX,
$ LDWORK = ( NLMAX + 1 )*KMAX*KMAX +
$ ( 3 + NLMAX )*KMAX )
* .. Local Scalars ..
INTEGER I, J, INFO, K, M, N, NL, SIZR
CHARACTER TRIU, TYPET
* .. Local Arrays dimensioned for TYPET = 'R' ..
DOUBLE PRECISION DWORK(LDWORK), RB(LDRB, NMAX*KMAX),
$ T(LDT, NMAX*KMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL MB02GD
*
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) K, N, NL, TRIU
TYPET = 'R'
M = ( NL + 1 )*K
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) N
ELSE IF( NL.LE.0 .OR. NL.GT.NLMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) NL
ELSE IF( K.LE.0 .OR. K.GT.KMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) K
ELSE
READ ( NIN, FMT = * ) ( ( T(I,J), J = 1,M ), I = 1,K )
* Compute the banded Cholesky factor.
CALL MB02GD( TYPET, TRIU, K, N, NL, 0, N, T, LDT, RB, LDRB,
$ DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
IF ( LSAME( TRIU, 'T' ) ) THEN
SIZR = NL*K + 1
ELSE
SIZR = ( NL + 1 )*K
END IF
DO 10 I = 1, SIZR
WRITE ( NOUT, FMT = 99996 ) ( RB(I,J), J = 1, N*K )
10 CONTINUE
END IF
END IF
STOP
*
99999 FORMAT (' MB02GD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB02GD = ',I2)
99997 FORMAT (/' The upper Cholesky factor in banded storage format ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' N is out of range.',/' N = ',I5)
99994 FORMAT (/' NL is out of range.',/' NL = ',I5)
99993 FORMAT (/' K is out of range.',/' K = ',I5)
END
Program Data
MB02GD EXAMPLE PROGRAM DATA 2 4 2 T 3.0000 1.0000 0.1000 0.4000 0.2000 0.0000 0.0000 4.0000 0.1000 0.1000 0.0500 0.2000Program Results
MB02GD EXAMPLE PROGRAM RESULTS The upper Cholesky factor in banded storage format 0.0000 0.0000 0.0000 0.0000 0.1155 0.1044 0.1156 0.1051 0.0000 0.0000 0.0000 0.2309 -0.0087 0.2290 -0.0084 0.2302 0.0000 0.0000 0.0577 -0.0174 0.0541 -0.0151 0.0544 -0.0159 0.0000 0.5774 0.0348 0.5704 0.0222 0.5725 0.0223 0.5724 1.7321 1.9149 1.7307 1.9029 1.7272 1.8996 1.7272 1.8995
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02HD.html 0000664 0000000 0000000 00000030206 14560147231 0020327 0 ustar 00root root 0000000 0000000
Purpose
To compute, for a banded K*M-by-L*N block Toeplitz matrix T with
block size (K,L), specified by the nonzero blocks of its first
block column TC and row TR, a LOWER triangular matrix R (in band
storage scheme) such that
T T
T T = R R . (1)
It is assumed that the first MIN(M*K, N*L) columns of T are
linearly independent.
By subsequent calls of this routine, the matrix R can be computed
block column by block column.
Specification
SUBROUTINE MB02HD( TRIU, K, L, M, ML, N, NU, P, S, TC, LDTC, TR,
$ LDTR, RB, LDRB, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TRIU
INTEGER INFO, K, L, LDRB, LDTC, LDTR, LDWORK, M, ML, N,
$ NU, P, S
C .. Array Arguments ..
DOUBLE PRECISION DWORK(LDWORK), RB(LDRB,*), TC(LDTC,*),
$ TR(LDTR,*)
Arguments
Mode Parameters
TRIU CHARACTER*1
Specifies the structure, if any, of the last blocks in TC
and TR, as follows:
= 'N': TC and TR have no special structure;
= 'T': TC and TR are upper and lower triangular,
respectively. Depending on the block sizes, two
different shapes of the last blocks in TC and TR
are possible, as illustrated below:
1) TC TR 2) TC TR
x x x x 0 0 x x x x x 0 0 0
0 x x x x 0 0 x x x x x 0 0
0 0 x x x x 0 0 x x x x x 0
0 0 0 x x x
Input/Output Parameters
K (input) INTEGER
The number of rows in the blocks of T. K >= 0.
L (input) INTEGER
The number of columns in the blocks of T. L >= 0.
M (input) INTEGER
The number of blocks in the first block column of T.
M >= 1.
ML (input) INTEGER
The lower block bandwidth, i.e., ML + 1 is the number of
nonzero blocks in the first block column of T.
0 <= ML < M and (ML + 1)*K >= L and
if ( M*K <= N*L ), ML >= M - INT( ( M*K - 1 )/L ) - 1;
ML >= M - INT( M*K/L ) or
MOD( M*K, L ) >= K;
if ( M*K >= N*L ), ML*K >= N*( L - K ).
N (input) INTEGER
The number of blocks in the first block row of T.
N >= 1.
NU (input) INTEGER
The upper block bandwidth, i.e., NU + 1 is the number of
nonzero blocks in the first block row of T.
If TRIU = 'N', 0 <= NU < N and
(M + NU)*L >= MIN( M*K, N*L );
if TRIU = 'T', MAX(1-ML,0) <= NU < N and
(M + NU)*L >= MIN( M*K, N*L ).
P (input) INTEGER
The number of previously computed block columns of R.
P*L < MIN( M*K,N*L ) + L and P >= 0.
S (input) INTEGER
The number of block columns of R to compute.
(P+S)*L < MIN( M*K,N*L ) + L and S >= 0.
TC (input) DOUBLE PRECISION array, dimension (LDTC,L)
On entry, if P = 0, the leading (ML+1)*K-by-L part of this
array must contain the nonzero blocks in the first block
column of T.
LDTC INTEGER
The leading dimension of the array TC.
LDTC >= MAX(1,(ML+1)*K), if P = 0.
TR (input) DOUBLE PRECISION array, dimension (LDTR,NU*L)
On entry, if P = 0, the leading K-by-NU*L part of this
array must contain the 2nd to the (NU+1)-st blocks of
the first block row of T.
LDTR INTEGER
The leading dimension of the array TR.
LDTR >= MAX(1,K), if P = 0.
RB (output) DOUBLE PRECISION array, dimension
(LDRB,MIN( S*L,MIN( M*K,N*L )-P*L ))
On exit, if INFO = 0 and TRIU = 'N', the leading
MIN( ML+NU+1,N )*L-by-MIN( S*L,MIN( M*K,N*L )-P*L ) part
of this array contains the (P+1)-th to (P+S)-th block
column of the lower R factor (1) in band storage format.
On exit, if INFO = 0 and TRIU = 'T', the leading
MIN( (ML+NU)*L+1,N*L )-by-MIN( S*L,MIN( M*K,N*L )-P*L )
part of this array contains the (P+1)-th to (P+S)-th block
column of the lower R factor (1) in band storage format.
For further details regarding the band storage scheme see
the documentation of the LAPACK routine DPBTF2.
LDRB INTEGER
The leading dimension of the array RB.
LDRB >= MAX( MIN( ML+NU+1,N )*L,1 ), if TRIU = 'N';
LDRB >= MAX( MIN( (ML+NU)*L+1,N*L ),1 ), if TRIU = 'T'.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -17, DWORK(1) returns the minimum
value of LDWORK.
The first 1 + 2*MIN( ML+NU+1,N )*L*(K+L) elements of DWORK
should be preserved during successive calls of the routine.
LDWORK INTEGER
The length of the array DWORK.
Let x = MIN( ML+NU+1,N ), then
LDWORK >= 1 + MAX( x*L*L + (2*NU+1)*L*K,
2*x*L*(K+L) + (6+x)*L ), if P = 0;
LDWORK >= 1 + 2*x*L*(K+L) + (6+x)*L, if P > 0.
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the full rank condition for the first MIN(M*K, N*L)
columns of T is (numerically) violated.
Method
Householder transformations and modified hyperbolic rotations are used in the Schur algorithm [1], [2].References
[1] Kailath, T. and Sayed, A.
Fast Reliable Algorithms for Matrices with Structure.
SIAM Publications, Philadelphia, 1999.
[2] Kressner, D. and Van Dooren, P.
Factorizations and linear system solvers for matrices with
Toeplitz structure.
SLICOT Working Note 2000-2, 2000.
Numerical Aspects
The implemented method yields a factor R which has comparable
accuracy with the Cholesky factor of T^T * T.
The algorithm requires
2 2
O( L *K*N*( ML + NU ) + N*( ML + NU )*L *( L + K ) )
floating point operations.
Further Comments
NoneExample
Program Text
* MB02HD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER KMAX, LMAX, MMAX, MLMAX, NMAX, NUMAX
PARAMETER ( KMAX = 20, LMAX = 20, MMAX = 20, MLMAX = 10,
$ NMAX = 20, NUMAX = 10 )
INTEGER LDRB, LDTC, LDTR, LDWORK
PARAMETER ( LDRB = ( MLMAX + NUMAX + 1 )*LMAX,
$ LDTC = ( MLMAX + 1 )*KMAX, LDTR = KMAX )
PARAMETER ( LDWORK = LDRB*LMAX + ( 2*NUMAX + 1 )*LMAX*KMAX
$ + 2*LDRB*( KMAX + LMAX ) + LDRB
$ + 6*LMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, K, L, LENR, M, ML, N, NU, S
CHARACTER TRIU
* .. Local Arrays ..
DOUBLE PRECISION DWORK(LDWORK), RB(LDRB,NMAX*LMAX),
$ TC(LDTC,LMAX), TR(LDTR,NMAX*LMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL MB02HD
* .. Intrinsic Functions ..
INTRINSIC MIN
*
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) K, L, M, ML, N, NU, TRIU
IF( K.LT.0 .OR. K.GT.KMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) K
ELSE IF( L.LT.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) L
ELSE IF( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) M
ELSE IF( ML.LT.0 .OR. ML.GT.MLMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) ML
ELSE IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE IF( NU.LT.0 .OR. NU.GT.NUMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) NU
ELSE
READ ( NIN, FMT = * ) ( ( TC(I,J), J = 1,L ), I = 1,(ML+1)*K )
READ ( NIN, FMT = * ) ( ( TR(I,J), J = 1,NU*L ), I = 1,K )
S = ( MIN( M*K, N*L ) + L - 1 ) / L
* Compute the banded R factor.
CALL MB02HD( TRIU, K, L, M, ML, N, NU, 0, S, TC, LDTC, TR,
$ LDTR, RB, LDRB, DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
LENR = ( ML + NU + 1 )*L
IF ( LSAME( TRIU, 'T' ) ) LENR = ( ML + NU )*L + 1
LENR = MIN( LENR, N*L )
DO 10 I = 1, LENR
WRITE ( NOUT, FMT = 99996 ) ( RB(I,J), J = 1,
$ MIN( N*L, M*K ) )
10 CONTINUE
END IF
END IF
STOP
*
99999 FORMAT (' MB02HD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB02HD = ',I2)
99997 FORMAT (/' The lower triangular factor R in banded storage ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' NU is out of range.',/' NU = ',I5)
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' ML is out of range.',/' ML = ',I5)
99992 FORMAT (/' M is out of range.',/' M = ',I5)
99991 FORMAT (/' L is out of range.',/' L = ',I5)
99990 FORMAT (/' K is out of range.',/' K = ',I5)
END
Program Data
MB02HD EXAMPLE PROGRAM DATA
2 2 6 2 5 1 N
4.0 4.0
1.0 3.0
2.0 1.0
2.0 2.0
4.0 4.0
3.0 4.0
1.0 3.0
2.0 1.0
Program Results
MB02HD EXAMPLE PROGRAM RESULTS The lower triangular factor R in banded storage -7.0711 -2.4125 6.0822 2.9967 5.9732 2.8593 5.8497 2.7914 2.7298 1.9557 -7.4953 -0.0829 5.8986 -0.5571 5.5329 0.2059 5.6797 0.3414 0.9565 0.0000 -4.2426 0.9202 2.4747 -1.6425 2.9472 -1.0052 2.4396 -0.7785 0.0000 0.0000 -5.2326 0.6218 2.8391 -0.0820 3.2670 0.6327 2.7067 0.0000 0.0000 0.0000 -3.5355 0.8207 3.1160 -0.4451 3.5758 0.5701 0.0000 0.0000 0.0000 0.0000 -4.6669 -0.5803 3.9454 0.7682 4.5481 0.0000 0.0000 0.0000 0.0000 0.0000 -1.4142 -0.0415 1.6441 0.4848 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -2.1213 0.0000 2.4662 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02ID.html 0000664 0000000 0000000 00000027607 14560147231 0020343 0 ustar 00root root 0000000 0000000
Purpose
To solve the overdetermined or underdetermined real linear systems
involving an M*K-by-N*L block Toeplitz matrix T that is specified
by its first block column and row. It is assumed that T has full
rank.
The following options are provided:
1. If JOB = 'O' or JOB = 'A' : find the least squares solution of
an overdetermined system, i.e., solve the least squares problem
minimize || B - T*X ||. (1)
2. If JOB = 'U' or JOB = 'A' : find the minimum norm solution of
the undetermined system
T
T * X = C. (2)
Specification
SUBROUTINE MB02ID( JOB, K, L, M, N, RB, RC, TC, LDTC, TR, LDTR, B,
$ LDB, C, LDC, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOB
INTEGER INFO, K, L, LDB, LDC, LDTC, LDTR, LDWORK, M, N,
$ RB, RC
C .. Array Arguments ..
DOUBLE PRECISION B(LDB,*), C(LDC,*), DWORK(LDWORK), TC(LDTC,*),
$ TR(LDTR,*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the problem to be solved as follows
= 'O': solve the overdetermined system (1);
= 'U': solve the underdetermined system (2);
= 'A': solve (1) and (2).
Input/Output Parameters
K (input) INTEGER
The number of rows in the blocks of T. K >= 0.
L (input) INTEGER
The number of columns in the blocks of T. L >= 0.
M (input) INTEGER
The number of blocks in the first block column of T.
M >= 0.
N (input) INTEGER
The number of blocks in the first block row of T.
0 <= N <= M*K / L.
RB (input) INTEGER
If JOB = 'O' or 'A', the number of columns in B. RB >= 0.
RC (input) INTEGER
If JOB = 'U' or 'A', the number of columns in C. RC >= 0.
TC (input) DOUBLE PRECISION array, dimension (LDTC,L)
On entry, the leading M*K-by-L part of this array must
contain the first block column of T.
LDTC INTEGER
The leading dimension of the array TC. LDTC >= MAX(1,M*K)
TR (input) DOUBLE PRECISION array, dimension (LDTR,(N-1)*L)
On entry, the leading K-by-(N-1)*L part of this array must
contain the 2nd to the N-th blocks of the first block row
of T.
LDTR INTEGER
The leading dimension of the array TR. LDTR >= MAX(1,K).
B (input/output) DOUBLE PRECISION array, dimension (LDB,RB)
On entry, if JOB = 'O' or JOB = 'A', the leading M*K-by-RB
part of this array must contain the right hand side
matrix B of the overdetermined system (1).
On exit, if JOB = 'O' or JOB = 'A', the leading N*L-by-RB
part of this array contains the solution of the
overdetermined system (1).
This array is not referenced if JOB = 'U'.
LDB INTEGER
The leading dimension of the array B.
LDB >= MAX(1,M*K), if JOB = 'O' or JOB = 'A';
LDB >= 1, if JOB = 'U'.
C (input) DOUBLE PRECISION array, dimension (LDC,RC)
On entry, if JOB = 'U' or JOB = 'A', the leading N*L-by-RC
part of this array must contain the right hand side
matrix C of the underdetermined system (2).
On exit, if JOB = 'U' or JOB = 'A', the leading M*K-by-RC
part of this array contains the solution of the
underdetermined system (2).
This array is not referenced if JOB = 'O'.
LDC INTEGER
The leading dimension of the array C.
LDB >= 1, if JOB = 'O';
LDB >= MAX(1,M*K), if JOB = 'U' or JOB = 'A'.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -17, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
Let x = MAX( 2*N*L*(L+K) + (6+N)*L,(N*L+M*K+1)*L + M*K )
and y = N*M*K*L + N*L, then
if MIN( M,N ) = 1 and JOB = 'O',
LDWORK >= MAX( y + MAX( M*K,RB ),1 );
if MIN( M,N ) = 1 and JOB = 'U',
LDWORK >= MAX( y + MAX( M*K,RC ),1 );
if MIN( M,N ) = 1 and JOB = 'A',
LDWORK >= MAX( y +MAX( M*K,MAX( RB,RC ),1 );
if MIN( M,N ) > 1 and JOB = 'O',
LDWORK >= MAX( x,N*L*RB + 1 );
if MIN( M,N ) > 1 and JOB = 'U',
LDWORK >= MAX( x,N*L*RC + 1 );
if MIN( M,N ) > 1 and JOB = 'A',
LDWORK >= MAX( x,N*L*MAX( RB,RC ) + 1 ).
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction algorithm failed. The Toeplitz matrix
associated with T is (numerically) not of full rank.
Method
Householder transformations and modified hyperbolic rotations are used in the Schur algorithm [1], [2].References
[1] Kailath, T. and Sayed, A.
Fast Reliable Algorithms for Matrices with Structure.
SIAM Publications, Philadelphia, 1999.
[2] Kressner, D. and Van Dooren, P.
Factorizations and linear system solvers for matrices with
Toeplitz structure.
SLICOT Working Note 2000-2, 2000.
Numerical Aspects
The algorithm requires O( L*L*K*(N+M)*log(N+M) + N*N*L*L*(L+K) )
and additionally
if JOB = 'O' or JOB = 'A',
O( (K*L+RB*L+K*RB)*(N+M)*log(N+M) + N*N*L*L*RB );
if JOB = 'U' or JOB = 'A',
O( (K*L+RC*L+K*RC)*(N+M)*log(N+M) + N*N*L*L*RC );
floating point operations.
Further Comments
NoneExample
Program Text
* MB02ID EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER KMAX, LMAX, MMAX, NMAX, RBMAX, RCMAX
PARAMETER ( KMAX = 20, LMAX = 20, MMAX = 20, NMAX = 20,
$ RBMAX = 20, RCMAX = 20 )
INTEGER LDB, LDC, LDTC, LDTR, LDWORK
PARAMETER ( LDB = KMAX*MMAX, LDC = KMAX*MMAX,
$ LDTC = MMAX*KMAX, LDTR = KMAX,
$ LDWORK = 2*NMAX*LMAX*( LMAX + KMAX ) +
$ ( 6 + NMAX )*LMAX +
$ MMAX*KMAX*( LMAX + 1 ) +
$ RBMAX + RCMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, K, L, M, N, RB, RC
CHARACTER JOB
DOUBLE PRECISION B(LDB,RBMAX), C(LDC,RCMAX), DWORK(LDWORK),
$ TC(LDTC,LMAX), TR(LDTR,NMAX*LMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL MB02ID
*
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) K, L, M, N, RB, RC, JOB
IF( K.LE.0 .OR. K.GT.KMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) K
ELSE IF( L.LE.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) L
ELSE IF( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) M
ELSE IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) N
ELSE IF ( ( LSAME( JOB, 'O' ) .OR. LSAME( JOB, 'A' ) )
$ .AND. ( ( RB.LE.0 ) .OR. ( RB.GT.RBMAX ) ) ) THEN
WRITE ( NOUT, FMT = 99990 ) RB
ELSE IF ( ( LSAME( JOB, 'U' ) .OR. LSAME( JOB, 'A' ) )
$ .AND. ( ( RC.LE.0 ) .OR. ( RC.GT.RCMAX ) ) ) THEN
WRITE ( NOUT, FMT = 99989 ) RC
ELSE
READ ( NIN, FMT = * ) ( ( TC(I,J), J = 1,L ), I = 1,M*K )
READ ( NIN, FMT = * ) ( ( TR(I,J), J = 1,(N-1)*L ), I = 1,K )
IF ( LSAME( JOB, 'O' ) .OR. LSAME( JOB, 'A' ) ) THEN
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,RB ), I = 1,M*K )
END IF
IF ( LSAME( JOB, 'U' ) .OR. LSAME( JOB, 'A' ) ) THEN
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,RC ), I = 1,N*L )
END IF
CALL MB02ID( JOB, K, L, M, N, RB, RC, TC, LDTC, TR, LDTR, B,
$ LDB, C, LDC, DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( LSAME( JOB, 'O' ) .OR. LSAME( JOB, 'A' ) ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N*L
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1, RB )
10 CONTINUE
END IF
IF ( LSAME( JOB, 'U' ) .OR. LSAME( JOB, 'A' ) ) THEN
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, M*K
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1, RC )
20 CONTINUE
END IF
END IF
END IF
STOP
*
99999 FORMAT (' MB02ID EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB02ID = ',I2)
99997 FORMAT (' The least squares solution of T * X = B is ')
99996 FORMAT (' The minimum norm solution of T^T * X = C is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' K is out of range.',/' K = ',I5)
99993 FORMAT (/' L is out of range.',/' L = ',I5)
99992 FORMAT (/' M is out of range.',/' M = ',I5)
99991 FORMAT (/' N is out of range.',/' N = ',I5)
99990 FORMAT (/' RB is out of range.',/' RB = ',I5)
99989 FORMAT (/' RC is out of range.',/' RC = ',I5)
END
Program Data
MB02ID EXAMPLE PROGRAM DATA
3 2 4 3 1 1 A
5.0 2.0
1.0 2.0
4.0 3.0
4.0 0.0
2.0 2.0
3.0 3.0
5.0 1.0
3.0 3.0
1.0 1.0
2.0 3.0
1.0 3.0
2.0 2.0
1.0 4.0 2.0 3.0
2.0 2.0 2.0 4.0
3.0 1.0 0.0 1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
Program Results
MB02ID EXAMPLE PROGRAM RESULTS The least squares solution of T * X = B is 0.0379 0.1677 0.0485 -0.0038 0.0429 0.1365 The minimum norm solution of T^T * X = C is 0.0509 0.0547 0.0218 0.0008 0.0436 0.0404 0.0031 0.0451 0.0421 0.0243 0.0556 0.0472
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02JD.html 0000664 0000000 0000000 00000030075 14560147231 0020335 0 ustar 00root root 0000000 0000000
Purpose
To compute a lower triangular matrix R and a matrix Q with
Q^T Q = I such that
T
T = Q R ,
where T is a K*M-by-L*N block Toeplitz matrix with blocks of size
(K,L). The first column of T will be denoted by TC and the first
row by TR. It is assumed that the first MIN(M*K, N*L) columns of T
have full rank.
By subsequent calls of this routine the factors Q and R can be
computed block column by block column.
Specification
SUBROUTINE MB02JD( JOB, K, L, M, N, P, S, TC, LDTC, TR, LDTR, Q,
$ LDQ, R, LDR, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOB
INTEGER INFO, K, L, LDQ, LDR, LDTC, LDTR, LDWORK,
$ M, N, P, S
C .. Array Arguments ..
DOUBLE PRECISION DWORK(LDWORK), Q(LDQ,*), R(LDR,*), TC(LDTC,*),
$ TR(LDTR,*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the output of the routine as follows:
= 'Q': computes Q and R;
= 'R': only computes R.
Input/Output Parameters
K (input) INTEGER
The number of rows in one block of T. K >= 0.
L (input) INTEGER
The number of columns in one block of T. L >= 0.
M (input) INTEGER
The number of blocks in one block column of T. M >= 0.
N (input) INTEGER
The number of blocks in one block row of T. N >= 0.
P (input) INTEGER
The number of previously computed block columns of R.
P*L < MIN( M*K,N*L ) + L and P >= 0.
S (input) INTEGER
The number of block columns of R to compute.
(P+S)*L < MIN( M*K,N*L ) + L and S >= 0.
TC (input) DOUBLE PRECISION array, dimension (LDTC, L)
On entry, if P = 0, the leading M*K-by-L part of this
array must contain the first block column of T.
LDTC INTEGER
The leading dimension of the array TC.
LDTC >= MAX(1,M*K).
TR (input) DOUBLE PRECISION array, dimension (LDTR,(N-1)*L)
On entry, if P = 0, the leading K-by-(N-1)*L part of this
array must contain the first block row of T without the
leading K-by-L block.
LDTR INTEGER
The leading dimension of the array TR.
LDTR >= MAX(1,K).
Q (input/output) DOUBLE PRECISION array, dimension
(LDQ,MIN( S*L, MIN( M*K,N*L )-P*L ))
On entry, if JOB = 'Q' and P > 0, the leading M*K-by-L
part of this array must contain the last block column of Q
from a previous call of this routine.
On exit, if JOB = 'Q' and INFO = 0, the leading
M*K-by-MIN( S*L, MIN( M*K,N*L )-P*L ) part of this array
contains the P-th to (P+S)-th block columns of the factor
Q.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= MAX(1,M*K), if JOB = 'Q';
LDQ >= 1, if JOB = 'R'.
R (input/output) DOUBLE PRECISION array, dimension
(LDR,MIN( S*L, MIN( M*K,N*L )-P*L ))
On entry, if P > 0, the leading (N-P+1)*L-by-L
part of this array must contain the nozero part of the
last block column of R from a previous call of this
routine.
One exit, if INFO = 0, the leading
MIN( N, N-P+1 )*L-by-MIN( S*L, MIN( M*K,N*L )-P*L )
part of this array contains the nonzero parts of the P-th
to (P+S)-th block columns of the lower triangular
factor R.
Note that elements in the strictly upper triangular part
will not be referenced.
LDR INTEGER
The leading dimension of the array R.
LDR >= MAX( 1, MIN( N, N-P+1 )*L )
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
On exit, if INFO = -17, DWORK(1) returns the minimum
value of LDWORK.
If JOB = 'Q', the first 1 + ( (N-1)*L + M*K )*( 2*K + L )
elements of DWORK should be preserved during successive
calls of the routine.
If JOB = 'R', the first 1 + (N-1)*L*( 2*K + L ) elements
of DWORK should be preserved during successive calls of
the routine.
LDWORK INTEGER
The length of the array DWORK.
JOB = 'Q':
LDWORK >= 1 + ( M*K + ( N - 1 )*L )*( L + 2*K ) + 6*L
+ MAX( M*K,( N - MAX( 1,P )*L ) );
JOB = 'R':
If P = 0,
LDWORK >= MAX( 1 + ( N - 1 )*L*( L + 2*K ) + 6*L
+ (N-1)*L, M*K*( L + 1 ) + L );
If P > 0,
LDWORK >= 1 + (N-1)*L*( L + 2*K ) + 6*L + (N-P)*L.
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the full rank condition for the first MIN(M*K, N*L)
columns of T is (numerically) violated.
Method
Block Householder transformations and modified hyperbolic rotations are used in the Schur algorithm [1], [2].References
[1] Kailath, T. and Sayed, A.
Fast Reliable Algorithms for Matrices with Structure.
SIAM Publications, Philadelphia, 1999.
[2] Kressner, D. and Van Dooren, P.
Factorizations and linear system solvers for matrices with
Toeplitz structure.
SLICOT Working Note 2000-2, 2000.
Numerical Aspects
The implemented method yields a factor R which has comparable
accuracy with the Cholesky factor of T^T * T. Q is implicitly
computed from the formula Q = T * inv(R^T R) * R, i.e., for ill
conditioned problems this factor is of very limited value.
2
The algorithm requires 0(K*L *M*N) floating point operations.
Further Comments
NoneExample
Program Text
* MB02JD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER KMAX, LMAX, MMAX, NMAX
PARAMETER ( KMAX = 10, LMAX = 10, MMAX = 20, NMAX = 20 )
INTEGER LDR, LDQ, LDTC, LDTR, LDWORK
PARAMETER ( LDR = NMAX*LMAX, LDQ = MMAX*KMAX,
$ LDTC = MMAX*KMAX, LDTR = KMAX,
$ LDWORK = ( MMAX*KMAX + NMAX*LMAX )
$ *( LMAX + 2*KMAX ) + 6*LMAX
$ + MMAX*KMAX + NMAX*LMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, K, L, M, N, S
CHARACTER JOB
* .. Local Arrays ..
DOUBLE PRECISION DWORK(LDWORK), Q(LDQ,NMAX*LMAX),
$ R(LDR,NMAX*LMAX), TC(LDTC,LMAX),
$ TR(LDTR,NMAX*LMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL MB02JD
* .. Intrinsic Functions ..
INTRINSIC MIN
*
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) K, L, M, N, JOB
IF( K.LE.0 .OR. K.GT.KMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) K
ELSE IF( L.LE.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) L
ELSE IF( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) M
ELSE IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) N
ELSE
READ ( NIN, FMT = * ) ( ( TC(I,J), J = 1,L ), I = 1,M*K )
READ ( NIN, FMT = * ) ( ( TR(I,J), J = 1,( N - 1 )*L ),
$ I = 1,K )
S = ( MIN( M*K, N*L ) + L - 1 ) / L
* Compute the required part of the QR factorization.
CALL MB02JD( JOB, K, L, M, N, 0, S, TC, LDTC, TR, LDTR, Q, LDQ,
$ R, LDR, DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( LSAME( JOB, 'Q' ) ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, M*K
WRITE ( NOUT, FMT = 99995 )
$ ( Q(I,J), J = 1, MIN( N*L, M*K ) )
10 CONTINUE
END IF
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N*L
WRITE ( NOUT, FMT = 99995 )
$ ( R(I,J), J = 1, MIN( N*L, M*K ) )
20 CONTINUE
END IF
END IF
*
STOP
*
99999 FORMAT (' MB02JD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB02JD = ',I2)
99997 FORMAT (/' The factor Q is ')
99996 FORMAT (/' The factor R is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' K is out of range.',/' K = ',I5)
99993 FORMAT (/' L is out of range.',/' L = ',I5)
99992 FORMAT (/' M is out of range.',/' M = ',I5)
99991 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
MB02JD EXAMPLE PROGRAM DATA
2 3 4 3 Q
1.0 4.0 0.0
4.0 1.0 2.0
4.0 2.0 2.0
5.0 3.0 2.0
2.0 4.0 4.0
5.0 3.0 4.0
2.0 2.0 5.0
4.0 2.0 3.0
3.0 4.0 2.0 5.0 0.0 4.0
5.0 1.0 1.0 2.0 4.0 1.0
Program Results
MB02JD EXAMPLE PROGRAM RESULTS The factor Q is -0.0967 0.7166 -0.4651 0.1272 0.4357 0.0435 0.2201 0.0673 -0.3867 -0.3108 -0.0534 0.5251 0.0963 -0.3894 0.1466 0.5412 -0.3867 -0.0990 -0.1443 -0.7021 0.3056 -0.3367 -0.3233 0.1249 -0.4834 -0.0178 -0.3368 -0.1763 -0.5446 0.5100 0.1503 0.2054 -0.1933 0.5859 0.3214 0.1156 -0.4670 -0.3199 -0.4185 0.0842 -0.4834 -0.0178 0.1072 0.0357 -0.0575 -0.2859 0.4339 -0.6928 -0.1933 0.1623 0.7251 -0.1966 0.2736 0.3058 0.3398 0.2968 -0.3867 -0.0990 0.0777 0.3615 0.3386 0.4421 -0.5693 -0.2641 The factor R is -10.3441 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -6.3805 4.7212 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -7.3472 1.9320 4.5040 0.0000 0.0000 0.0000 0.0000 0.0000 -10.0541 2.5101 0.5065 3.6550 0.0000 0.0000 0.0000 0.0000 -6.5738 3.6127 1.2702 -1.3146 3.5202 0.0000 0.0000 0.0000 -5.2204 2.4764 2.4113 1.3890 1.2780 2.4976 0.0000 0.0000 -9.6674 3.2445 -0.5099 -0.0224 2.6548 2.9491 1.0049 0.0000 -6.3805 0.6968 1.9483 0.3050 0.7002 -2.0220 -2.8246 2.3147 -4.1570 2.4309 -0.7190 -0.1455 3.0149 0.5454 0.9394 -0.0548
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02JX.html 0000664 0000000 0000000 00000032270 14560147231 0020360 0 ustar 00root root 0000000 0000000
Purpose
To compute a low rank QR factorization with column pivoting of a
K*M-by-L*N block Toeplitz matrix T with blocks of size (K,L);
specifically,
T
T P = Q R ,
where R is lower trapezoidal, P is a block permutation matrix
and Q^T Q = I. The number of columns in R is equivalent to the
numerical rank of T with respect to the given tolerance TOL1.
Note that the pivoting scheme is local, i.e., only columns
belonging to the same block in T are permuted.
Specification
SUBROUTINE MB02JX( JOB, K, L, M, N, TC, LDTC, TR, LDTR, RNK, Q,
$ LDQ, R, LDR, JPVT, TOL1, TOL2, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER JOB
INTEGER INFO, K, L, LDQ, LDR, LDTC, LDTR, LDWORK, M, N,
$ RNK
DOUBLE PRECISION TOL1, TOL2
C .. Array Arguments ..
DOUBLE PRECISION DWORK(LDWORK), Q(LDQ,*), R(LDR,*), TC(LDTC,*),
$ TR(LDTR,*)
INTEGER JPVT(*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the output of the routine as follows:
= 'Q': computes Q and R;
= 'R': only computes R.
Input/Output Parameters
K (input) INTEGER
The number of rows in one block of T. K >= 0.
L (input) INTEGER
The number of columns in one block of T. L >= 0.
M (input) INTEGER
The number of blocks in one block column of T. M >= 0.
N (input) INTEGER
The number of blocks in one block row of T. N >= 0.
TC (input) DOUBLE PRECISION array, dimension (LDTC, L)
The leading M*K-by-L part of this array must contain
the first block column of T.
LDTC INTEGER
The leading dimension of the array TC.
LDTC >= MAX(1,M*K).
TR (input) DOUBLE PRECISION array, dimension (LDTR,(N-1)*L)
The leading K-by-(N-1)*L part of this array must contain
the first block row of T without the leading K-by-L
block.
LDTR INTEGER
The leading dimension of the array TR. LDTR >= MAX(1,K).
RNK (output) INTEGER
The number of columns in R, which is equivalent to the
numerical rank of T.
Q (output) DOUBLE PRECISION array, dimension (LDQ,RNK)
If JOB = 'Q', then the leading M*K-by-RNK part of this
array contains the factor Q.
If JOB = 'R', then this array is not referenced.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= MAX(1,M*K), if JOB = 'Q';
LDQ >= 1, if JOB = 'R'.
R (output) DOUBLE PRECISION array, dimension (LDR,RNK)
The leading N*L-by-RNK part of this array contains the
lower trapezoidal factor R.
LDR INTEGER
The leading dimension of the array R.
LDR >= MAX(1,N*L)
JPVT (output) INTEGER array, dimension (MIN(M*K,N*L))
This array records the column pivoting performed.
If JPVT(j) = k, then the j-th column of T*P was
the k-th column of T.
Tolerances
TOL1 DOUBLE PRECISION
If TOL1 >= 0.0, the user supplied diagonal tolerance;
if TOL1 < 0.0, a default diagonal tolerance is used.
TOL2 DOUBLE PRECISION
If TOL2 >= 0.0, the user supplied offdiagonal tolerance;
if TOL2 < 0.0, a default offdiagonal tolerance is used.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK; DWORK(2) and DWORK(3) return the used values
for TOL1 and TOL2, respectively.
On exit, if INFO = -19, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( 3, ( M*K + ( N - 1 )*L )*( L + 2*K ) + 9*L
+ MAX(M*K,(N-1)*L) ), if JOB = 'Q';
LDWORK >= MAX( 3, ( N - 1 )*L*( L + 2*K + 1 ) + 9*L,
M*K*( L + 1 ) + L ), if JOB = 'R'.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: due to perturbations induced by roundoff errors, or
removal of nearly linearly dependent columns of the
generator, the Schur algorithm encountered a
situation where a diagonal element in the negative
generator is larger in magnitude than the
corresponding diagonal element in the positive
generator (modulo TOL1);
= 2: due to perturbations induced by roundoff errors, or
removal of nearly linearly dependent columns of the
generator, the Schur algorithm encountered a
situation where diagonal elements in the positive
and negative generator are equal in magnitude
(modulo TOL1), but the offdiagonal elements suggest
that these columns are not linearly dependent
(modulo TOL2*ABS(diagonal element)).
Method
Householder transformations and modified hyperbolic rotations are used in the Schur algorithm [1], [2]. If, during the process, the hyperbolic norm of a row in the leading part of the generator is found to be less than or equal to TOL1, then this row is not reduced. If the difference of the corresponding columns has a norm less than or equal to TOL2 times the magnitude of the leading element, then this column is removed from the generator, as well as from R. Otherwise, the algorithm breaks down. TOL1 is set to norm(TC)*sqrt(eps) and TOL2 is set to N*L*sqrt(eps) by default. If M*K > L, the columns of T are permuted so that the diagonal elements in one block column of R have decreasing magnitudes.References
[1] Kailath, T. and Sayed, A.
Fast Reliable Algorithms for Matrices with Structure.
SIAM Publications, Philadelphia, 1999.
[2] Kressner, D. and Van Dooren, P.
Factorizations and linear system solvers for matrices with
Toeplitz structure.
SLICOT Working Note 2000-2, 2000.
Numerical Aspects
The algorithm requires 0(K*RNK*L*M*N) floating point operations.Further Comments
NoneExample
Program Text
* MB02JX EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER KMAX, LMAX, MMAX, NMAX
PARAMETER ( KMAX = 20, LMAX = 20, MMAX = 20, NMAX = 20 )
INTEGER LDR, LDQ, LDTC, LDTR, LDWORK
PARAMETER ( LDR = NMAX*LMAX, LDQ = MMAX*KMAX,
$ LDTC = MMAX*KMAX, LDTR = KMAX,
$ LDWORK = ( MMAX*KMAX + NMAX*LMAX )
$ *( LMAX + 2*KMAX ) + 5*LMAX
$ + MMAX*KMAX + NMAX*LMAX )
* .. Local Scalars ..
CHARACTER JOB
INTEGER I, INFO, J, K, L, M, N, RNK
DOUBLE PRECISION TOL1, TOL2
* .. Local Arrays ..
INTEGER JPVT(NMAX*LMAX)
DOUBLE PRECISION DWORK(LDWORK), Q(LDQ,NMAX*LMAX),
$ R(LDR,NMAX*LMAX), TC(LDTC,LMAX),
$ TR(LDTR,NMAX*LMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL MB02JX
* .. Intrinsic Functions ..
INTRINSIC MIN
*
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) K, L, M, N, TOL1, TOL2, JOB
IF( K.LE.0 .OR. K.GT.KMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) K
ELSE IF( L.LE.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) L
ELSE IF( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
READ ( NIN, FMT = * ) ( ( TC(I,J), J = 1,L ), I = 1,M*K )
READ ( NIN, FMT = * ) ( ( TR(I,J), J = 1,( N - 1 )*L ),
$ I = 1,K )
* Compute the required part of the QR factorization.
CALL MB02JX( JOB, K, L, M, N, TC, LDTC, TR, LDTR, RNK, Q, LDQ,
$ R, LDR, JPVT, TOL1, TOL2, DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) RNK
IF ( LSAME( JOB, 'Q' ) ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, M*K
WRITE ( NOUT, FMT = 99993 ) ( Q(I,J), J = 1, RNK )
10 CONTINUE
END IF
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N*L
WRITE ( NOUT, FMT = 99993 ) ( R(I,J), J = 1, RNK )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
WRITE ( NOUT, FMT = 99992 ) ( JPVT(I),
$ I = 1, MIN( M*K, N*L ) )
END IF
END IF
STOP
*
99999 FORMAT (' MB02JX EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB02JX = ',I2)
99997 FORMAT (/' The factor Q is ')
99996 FORMAT (/' The factor R is ')
99995 FORMAT (/' The column permutation is ')
99994 FORMAT (/' Numerical rank ',/' RNK = ',I5)
99993 FORMAT (20(1X,F8.4))
99992 FORMAT (20(1X,I4))
99991 FORMAT (/' K is out of range.',/' K = ',I5)
99990 FORMAT (/' L is out of range.',/' L = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
MB02JX EXAMPLE PROGRAM DATA
3 3 4 4 -1.0D0 -1.0D0 Q
1.0 2.0 3.0
1.0 2.0 3.0
1.0 2.0 3.0
1.0 2.0 3.0
1.0 2.0 3.0
1.0 2.0 3.0
1.0 2.0 3.0
1.0 2.0 3.0
1.0 2.0 3.0
1.0 0.0 1.0
1.0 1.0 0.0
2.0 2.0 0.0
1.0 2.0 3.0 1.0 2.0 3.0 0.0 1.0 1.0
1.0 2.0 3.0 1.0 2.0 3.0 1.0 2.0 1.0
1.0 2.0 3.0 1.0 2.0 3.0 1.0 1.0 1.0
1.0 2.0 3.0 1.0 2.0 3.0 0.0 1.0 0.0
Program Results
MB02JX EXAMPLE PROGRAM RESULTS
Numerical rank
RNK = 7
The factor Q is
-0.3313 -0.0105 -0.0353 0.0000 -0.4714 -0.8165 0.0000
-0.3313 -0.0105 -0.0353 0.0000 -0.4714 0.4082 0.7071
-0.3313 -0.0105 -0.0353 0.0000 -0.4714 0.4082 -0.7071
-0.3313 -0.0105 -0.0353 0.0000 0.2357 0.0000 0.0000
-0.3313 -0.0105 -0.0353 0.0000 0.2357 0.0000 0.0000
-0.3313 -0.0105 -0.0353 0.0000 0.2357 0.0000 0.0000
-0.3313 -0.0105 -0.0353 0.0000 0.2357 0.0000 0.0000
-0.3313 -0.0105 -0.0353 0.0000 0.2357 0.0000 0.0000
-0.3313 -0.0105 -0.0353 0.0000 0.2357 0.0000 0.0000
-0.1104 0.2824 0.9529 0.0000 0.0000 0.0000 0.0000
0.0000 0.4288 -0.1271 0.8944 0.0000 0.0000 0.0000
0.0000 0.8576 -0.2541 -0.4472 0.0000 0.0000 0.0000
The factor R is
-9.0554 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
-3.0921 2.3322 0.0000 0.0000 0.0000 0.0000 0.0000
-5.9633 1.9557 -1.2706 0.0000 0.0000 0.0000 0.0000
-9.2762 4.4238 0.7623 1.3416 0.0000 0.0000 0.0000
-6.1842 2.9492 0.5082 0.8944 0.0000 0.0000 0.0000
-3.0921 1.4746 0.2541 0.4472 0.0000 0.0000 0.0000
-9.2762 4.4238 0.7623 1.3416 0.0000 0.0000 0.0000
-6.1842 2.9492 0.5082 0.8944 0.0000 0.0000 0.0000
-3.0921 1.4746 0.2541 0.4472 0.0000 0.0000 0.0000
-7.2885 4.4866 0.9741 1.3416 2.8284 0.0000 0.0000
-2.7608 1.4851 0.2894 0.4472 0.4714 0.8165 0.0000
-5.5216 2.9701 0.5788 0.8944 0.9428 0.4082 0.7071
The column permutation is
3 1 2 6 5 4 9 8 7 12 10 11
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02KD.html 0000664 0000000 0000000 00000026224 14560147231 0020337 0 ustar 00root root 0000000 0000000
Purpose
To compute the matrix product
C = alpha*op( T )*B + beta*C,
where alpha and beta are scalars and T is a block Toeplitz matrix
specified by its first block column TC and first block row TR;
B and C are general matrices of appropriate dimensions.
Specification
SUBROUTINE MB02KD( LDBLK, TRANS, K, L, M, N, R, ALPHA, BETA,
$ TC, LDTC, TR, LDTR, B, LDB, C, LDC, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER LDBLK, TRANS
INTEGER INFO, K, L, LDB, LDC, LDTC, LDTR, LDWORK, M, N,
$ R
DOUBLE PRECISION ALPHA, BETA
C .. Array Arguments ..
DOUBLE PRECISION B(LDB,*), C(LDC,*), DWORK(*), TC(LDTC,*),
$ TR(LDTR,*)
Arguments
Mode Parameters
LDBLK CHARACTER*1
Specifies where the (1,1)-block of T is stored, as
follows:
= 'C': in the first block of TC;
= 'R': in the first block of TR.
TRANS CHARACTER*1
Specifies the form of op( T ) to be used in the matrix
multiplication as follows:
= 'N': op( T ) = T;
= 'T': op( T ) = T';
= 'C': op( T ) = T'.
Input/Output Parameters
K (input) INTEGER
The number of rows in the blocks of T. K >= 0.
L (input) INTEGER
The number of columns in the blocks of T. L >= 0.
M (input) INTEGER
The number of blocks in the first block column of T.
M >= 0.
N (input) INTEGER
The number of blocks in the first block row of T. N >= 0.
R (input) INTEGER
The number of columns in B and C. R >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then TC, TR and B
are not referenced.
BETA (input) DOUBLE PRECISION
The scalar beta. When beta is zero then C need not be set
before entry.
TC (input) DOUBLE PRECISION array, dimension (LDTC,L)
On entry with LDBLK = 'C', the leading M*K-by-L part of
this array must contain the first block column of T.
On entry with LDBLK = 'R', the leading (M-1)*K-by-L part
of this array must contain the 2nd to the M-th blocks of
the first block column of T.
LDTC INTEGER
The leading dimension of the array TC.
LDTC >= MAX(1,M*K), if LDBLK = 'C';
LDTC >= MAX(1,(M-1)*K), if LDBLK = 'R'.
TR (input) DOUBLE PRECISION array, dimension (LDTR,k)
where k is (N-1)*L when LDBLK = 'C' and is N*L when
LDBLK = 'R'.
On entry with LDBLK = 'C', the leading K-by-(N-1)*L part
of this array must contain the 2nd to the N-th blocks of
the first block row of T.
On entry with LDBLK = 'R', the leading K-by-N*L part of
this array must contain the first block row of T.
LDTR INTEGER
The leading dimension of the array TR. LDTR >= MAX(1,K).
B (input) DOUBLE PRECISION array, dimension (LDB,R)
On entry with TRANS = 'N', the leading N*L-by-R part of
this array must contain the matrix B.
On entry with TRANS = 'T' or TRANS = 'C', the leading
M*K-by-R part of this array must contain the matrix B.
LDB INTEGER
The leading dimension of the array B.
LDB >= MAX(1,N*L), if TRANS = 'N';
LDB >= MAX(1,M*K), if TRANS = 'T' or TRANS = 'C'.
C (input/output) DOUBLE PRECISION array, dimension (LDC,R)
On entry with TRANS = 'N', the leading M*K-by-R part of
this array must contain the matrix C.
On entry with TRANS = 'T' or TRANS = 'C', the leading
N*L-by-R part of this array must contain the matrix C.
On exit with TRANS = 'N', the leading M*K-by-R part of
this array contains the updated matrix C.
On exit with TRANS = 'T' or TRANS = 'C', the leading
N*L-by-R part of this array contains the updated matrix C.
LDC INTEGER
The leading dimension of the array C.
LDC >= MAX(1,M*K), if TRANS = 'N';
LDC >= MAX(1,N*L), if TRANS = 'T' or TRANS = 'C'.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -19, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= 1.
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
For point Toeplitz matrices or sufficiently large block Toeplitz matrices, this algorithm uses convolution algorithms based on the fast Hartley transforms [1]. Otherwise, TC is copied in reversed order into the workspace such that C can be computed from barely M matrix-by-matrix multiplications.References
[1] Van Loan, Charles.
Computational frameworks for the fast Fourier transform.
SIAM, 1992.
Numerical Aspects
The algorithm requires O( (K*L+R*L+K*R)*(N+M)*log(N+M) + K*L*R ) floating point operations.Further Comments
NoneExample
Program Text
* MB02KD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER KMAX, LMAX, MMAX, NMAX, RMAX
PARAMETER ( KMAX = 20, LMAX = 20, MMAX = 20, NMAX = 20,
$ RMAX = 20 )
INTEGER LDB, LDC, LDTC, LDTR, LDWORK
PARAMETER ( LDB = LMAX*NMAX, LDC = KMAX*MMAX,
$ LDTC = MMAX*KMAX, LDTR = KMAX,
$ LDWORK = 2*( KMAX*LMAX + KMAX*RMAX
$ + LMAX*RMAX + 1 )*( MMAX + NMAX ) )
* .. Local Scalars ..
INTEGER I, INFO, J, K, L, M, N, R
CHARACTER LDBLK, TRANS
DOUBLE PRECISION ALPHA, BETA
* .. Local Arrays .. (Dimensioned for TRANS = 'N'.)
DOUBLE PRECISION B(LDB,RMAX), C(LDC,RMAX), DWORK(LDWORK),
$ TC(LDTC,LMAX), TR(LDTR,NMAX*LMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL MB02KD
* .. Intrinsic Functions ..
INTRINSIC MAX
*
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) K, L, M, N, R, LDBLK, TRANS
IF( K.LE.0 .OR. K.GT.KMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) K
ELSE IF( L.LE.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) L
ELSE IF( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) M
ELSE IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) N
ELSE IF( R.LE.0 .OR. R.GT.RMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
IF ( LSAME( LDBLK, 'R' ) ) THEN
READ ( NIN, FMT = * ) ( ( TC(I,J), J = 1,L ),
$ I = 1,(M-1)*K )
READ ( NIN, FMT = * ) ( ( TR(I,J), J = 1,N*L ), I = 1,K )
ELSE
READ ( NIN, FMT = * ) ( ( TC(I,J), J = 1,L ), I = 1,M*K )
READ ( NIN, FMT = * ) ( ( TR(I,J), J = 1,(N-1)*L ),
$ I = 1,K )
END IF
IF ( LSAME( TRANS, 'N' ) ) THEN
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,R ), I = 1,N*L )
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,R ), I = 1,M*K )
END IF
ALPHA = ONE
BETA = ZERO
CALL MB02KD( LDBLK, TRANS, K, L, M, N, R, ALPHA, BETA, TC,
$ LDTC, TR, LDTR, B, LDB, C, LDC, DWORK, LDWORK,
$ INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( LSAME( TRANS, 'N' ) ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, M*K
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,R )
10 CONTINUE
ELSE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N*L
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,R )
20 CONTINUE
END IF
END IF
END IF
STOP
*
99999 FORMAT (' MB02KD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB02KD = ',I2)
99997 FORMAT (' The product C = T * B is ')
99996 FORMAT (' The product C = T^T * B is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' K is out of range.',/' K = ',I5)
99993 FORMAT (/' L is out of range.',/' L = ',I5)
99992 FORMAT (/' M is out of range.',/' M = ',I5)
99991 FORMAT (/' N is out of range.',/' N = ',I5)
99990 FORMAT (/' R is out of range.',/' R = ',I5)
END
Program Data
MB02KD EXAMPLE PROGRAM DATA
3 2 4 5 1 C N
4.0 1.0
3.0 5.0
2.0 1.0
4.0 1.0
3.0 4.0
2.0 4.0
3.0 1.0
3.0 0.0
4.0 4.0
5.0 1.0
3.0 1.0
4.0 3.0
5.0 2.0 2.0 2.0 2.0 1.0 1.0 3.0
4.0 1.0 5.0 4.0 5.0 4.0 1.0 2.0
2.0 3.0 4.0 1.0 3.0 3.0 3.0 3.0
0.0
2.0
2.0
2.0
1.0
3.0
3.0
4.0
2.0
3.0
Program Results
MB02KD EXAMPLE PROGRAM RESULTS The product C = T * B is 45.0000 76.0000 55.0000 44.0000 84.0000 56.0000 52.0000 70.0000 54.0000 49.0000 63.0000 59.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02MD.html 0000664 0000000 0000000 00000041311 14560147231 0020333 0 ustar 00root root 0000000 0000000
Purpose
To solve the Total Least Squares (TLS) problem using a Singular Value Decomposition (SVD) approach. The TLS problem assumes an overdetermined set of linear equations AX = B, where both the data matrix A as well as the observation matrix B are inaccurate. The routine also solves determined and underdetermined sets of equations by computing the minimum norm solution. It is assumed that all preprocessing measures (scaling, coordinate transformations, whitening, ... ) of the data have been performed in advance.Specification
SUBROUTINE MB02MD( JOB, M, N, L, RANK, C, LDC, S, X, LDX, TOL,
$ IWORK, DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER JOB
INTEGER INFO, IWARN, L, LDC, LDWORK, LDX, M, N, RANK
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION C(LDC,*), DWORK(*), S(*), X(LDX,*)
Arguments
Mode Parameters
JOB CHARACTER*1
Determines whether the values of the parameters RANK and
TOL are to be specified by the user or computed by the
routine as follows:
= 'R': Compute RANK only;
= 'T': Compute TOL only;
= 'B': Compute both RANK and TOL;
= 'N': Compute neither RANK nor TOL.
Input/Output Parameters
M (input) INTEGER
The number of rows in the data matrix A and the
observation matrix B. M >= 0.
N (input) INTEGER
The number of columns in the data matrix A. N >= 0.
L (input) INTEGER
The number of columns in the observation matrix B.
L >= 0.
RANK (input/output) INTEGER
On entry, if JOB = 'T' or JOB = 'N', then RANK must
specify r, the rank of the TLS approximation [A+DA|B+DB].
RANK <= min(M,N).
Otherwise, r is computed by the routine.
On exit, if JOB = 'R' or JOB = 'B', and INFO = 0, then
RANK contains the computed (effective) rank of the TLS
approximation [A+DA|B+DB].
Otherwise, the user-supplied value of RANK may be
changed by the routine on exit if the RANK-th and the
(RANK+1)-th singular values of C = [A|B] are considered
to be equal, or if the upper triangular matrix F (as
defined in METHOD) is (numerically) singular.
C (input/output) DOUBLE PRECISION array, dimension (LDC,N+L)
On entry, the leading M-by-(N+L) part of this array must
contain the matrices A and B. Specifically, the first N
columns must contain the data matrix A and the last L
columns the observation matrix B (right-hand sides).
On exit, the leading (N+L)-by-(N+L) part of this array
contains the (transformed) right singular vectors,
including null space vectors, if any, of C = [A|B].
Specifically, the leading (N+L)-by-RANK part of this array
always contains the first RANK right singular vectors,
corresponding to the largest singular values of C. If
L = 0, or if RANK = 0 and IWARN <> 2, the remaining
(N+L)-by-(N+L-RANK) top-right part of this array contains
the remaining N+L-RANK right singular vectors. Otherwise,
this part contains the matrix V2 transformed as described
in Step 3 of the TLS algorithm (see METHOD).
LDC INTEGER
The leading dimension of array C. LDC >= max(1,M,N+L).
S (output) DOUBLE PRECISION array, dimension (min(M,N+L))
If INFO = 0, the singular values of matrix C, ordered
such that S(1) >= S(2) >= ... >= S(p-1) >= S(p) >= 0,
where p = min(M,N+L).
X (output) DOUBLE PRECISION array, dimension (LDX,L)
If INFO = 0, the leading N-by-L part of this array
contains the solution X to the TLS problem specified
by A and B.
LDX INTEGER
The leading dimension of array X. LDX >= max(1,N).
Tolerances
TOL DOUBLE PRECISION
A tolerance used to determine the rank of the TLS
approximation [A+DA|B+DB] and to check the multiplicity
of the singular values of matrix C. Specifically, S(i)
and S(j) (i < j) are considered to be equal if
SQRT(S(i)**2 - S(j)**2) <= TOL, and the TLS approximation
[A+DA|B+DB] has rank r if S(i) > TOL*S(1) (or S(i) > TOL,
if TOL specifies sdev (see below)), for i = 1,2,...,r.
TOL is also used to check the singularity of the upper
triangular matrix F (as defined in METHOD).
If JOB = 'R' or JOB = 'N', then TOL must specify the
desired tolerance. If the user sets TOL to be less than or
equal to 0, the tolerance is taken as EPS, where EPS is
the machine precision (see LAPACK Library routine DLAMCH).
Otherwise, the tolerance is computed by the routine and
the user must supply the non-negative value sdev, i.e. the
estimated standard deviation of the error on each element
of the matrix C, as input value of TOL.
Workspace
IWORK INTEGER array, dimension (L)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK, and DWORK(2) returns the reciprocal of the
condition number of the matrix F.
If INFO > 0, DWORK(1:min(M,N+L)-1) contain the unconverged
non-diagonal elements of the bidiagonal matrix whose
diagonal is in S (see LAPACK Library routine DGESVD).
LDWORK INTEGER
The length of the array DWORK.
LDWORK = max(2, 3*(N+L) + M, 5*(N+L)), if M >= N+L;
LDWORK = max(2, M*(N+L) + max( 3M+N+L, 5*M), 3*L),
if M < N+L.
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Warning Indicator
IWARN INTEGER
= 0: no warnings;
= 1: if the rank of matrix C has been lowered because a
singular value of multiplicity greater than 1 was
found;
= 2: if the rank of matrix C has been lowered because the
upper triangular matrix F is (numerically) singular.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if the SVD algorithm (in LAPACK Library routine
DBDSQR) has failed to converge. In this case, S(1),
S(2), ..., S(INFO) may not have been found
correctly and the remaining singular values may
not be the smallest. This failure is not likely
to occur.
Method
The method used is an extension (see [3,4,5]) of the classical
TLS algorithm proposed by Golub and Van Loan [1].
Let [A|B] denote the matrix formed by adjoining the columns of B
to the columns of A on the right.
Total Least Squares (TLS) definition:
-------------------------------------
Given matrices A and B, find a matrix X satisfying
(A + DA) X = B + DB,
where A and DA are M-by-N matrices, B and DB are M-by-L matrices
and X is an N-by-L matrix.
The solution X must be such that the Frobenius norm of [DA|DB]
is a minimum and each column of B + DB is in the range of
A + DA. Whenever the solution is not unique, the routine singles
out the minimum norm solution X.
Define matrix C = [A|B] and s(i) as its i-th singular value for
i = 1,2,...,min(M,NL), where NL = N + L. If M < NL, then s(j) = 0
for j = M+1,...,NL.
The Classical TLS algorithm proceeds as follows (see [3,4,5]):
Step 1: Compute part of the singular value decomposition (SVD)
USV' of C = [A|B], namely compute S and V'. (An initial
QR factorization of C is used when M is larger enough
than NL.)
Step 2: If not fixed by the user, compute the rank r0 of the data
[A|B] based on TOL as follows: if JOB = 'R' or JOB = 'N',
s(1) >= ... >= s(r0) > TOL*s(1) >= ... >= s(NL).
Otherwise, using [2], TOL can be computed from the
standard deviation sdev of the errors on [A|B]:
TOL = SQRT(2 * max(M,NL)) * sdev,
and the rank r0 is determined (if JOB = 'R' or 'B') using
s(1) >= ... >= s(r0) > TOL >= ... >= s(NL).
The rank r of the approximation [A+DA|B+DB] is then equal
to the minimum of N and r0.
Step 3: Let V2 be the matrix of the columns of V corresponding to
the (NL - r) smallest singular values of C, i.e. the last
(NL - r) columns of V.
Compute with Householder transformations the orthogonal
matrix Q such that:
|VH Y|
V2 x Q = | |
|0 F|
where VH is an N-by-(N - r) matrix, Y is an N-by-L matrix
and F is an L-by-L upper triangular matrix.
If F is singular, then lower the rank r with the
multiplicity of s(r) and repeat this step.
Step 4: If F is nonsingular then the solution X is obtained by
solving the following equations by forward elimination:
X F = -Y.
Notes :
The TLS solution is unique if r = N, F is nonsingular and
s(N) > s(N+1).
If F is singular, however, then the computed solution is infinite
and hence does not satisfy the second TLS criterion (see TLS
definition). For these cases, Golub and Van Loan [1] claim that
the TLS problem has no solution. The properties of these so-called
nongeneric problems are described in [4] and the TLS computations
are generalized in order to solve them. As proven in [4], the
proposed generalization satisfies the TLS criteria for any
number L of observation vectors in B provided that, in addition,
the solution | X| is constrained to be orthogonal to all vectors
|-I|
of the form |w| which belong to the space generated by the columns
|0|
of the submatrix |Y|.
|F|
References
[1] Golub, G.H. and Van Loan, C.F.
An Analysis of the Total Least-Squares Problem.
SIAM J. Numer. Anal., 17, pp. 883-893, 1980.
[2] Staar, J., Vandewalle, J. and Wemans, M.
Realization of Truncated Impulse Response Sequences with
Prescribed Uncertainty.
Proc. 8th IFAC World Congress, Kyoto, I, pp. 7-12, 1981.
[3] Van Huffel, S.
Analysis of the Total Least Squares Problem and its Use in
Parameter Estimation.
Doctoral dissertation, Dept. of Electr. Eng., Katholieke
Universiteit Leuven, Belgium, June 1987.
[4] Van Huffel, S. and Vandewalle, J.
Analysis and Solution of the Nongeneric Total Least Squares
Problem.
SIAM J. Matr. Anal. and Appl., 9, pp. 360-372, 1988.
[5] Van Huffel, S. and Vandewalle, J.
The Total Least Squares Problem: Computational Aspects and
Analysis.
Series "Frontiers in Applied Mathematics", Vol. 9,
SIAM, Philadelphia, 1991.
Numerical Aspects
The algorithm consists in (backward) stable steps.Further Comments
NoneExample
Program Text
* MB02MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, NMAX, LMAX
PARAMETER ( MMAX = 20, NMAX = 20, LMAX = 20 )
INTEGER LDC, LDX
PARAMETER ( LDC = MAX( MMAX,NMAX+LMAX ), LDX = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MMAX*(NMAX+LMAX) +
$ MAX( 3*MIN(MMAX,NMAX+LMAX) +
$ MAX(MMAX,NMAX+LMAX),
$ 5*MIN(MMAX,NMAX+LMAX),
$ 3*LMAX ) )
INTEGER LIWORK
PARAMETER ( LIWORK = LMAX )
INTEGER LENGS
PARAMETER ( LENGS = MIN( MMAX, NMAX+LMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION SDEV, TOL
INTEGER I, INFO, IWARN, J, L, M, N, RANK
CHARACTER*1 JOB
* .. Local Arrays ..
DOUBLE PRECISION C(LDC,NMAX+LMAX), DWORK(LDWORK), S(LENGS),
$ X(LDX,LMAX)
INTEGER IWORK(LIWORK)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL MB02MD
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, N, L, JOB
*
IF ( LSAME( JOB, 'R' ) ) THEN
READ ( NIN, FMT = * ) TOL
ELSE IF ( LSAME( JOB, 'T' ) ) THEN
READ ( NIN, FMT = * ) RANK, SDEV
TOL = SDEV
ELSE IF ( LSAME( JOB, 'N' ) ) THEN
READ ( NIN, FMT = * ) RANK, TOL
ELSE
READ ( NIN, FMT = * ) SDEV
TOL = SDEV
END IF
*
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) M
ELSE IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) N
ELSE IF ( L.LT.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) L
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N+L ), I = 1,M )
* Compute the solution to the TLS problem Ax = b.
CALL MB02MD( JOB, M, N, L, RANK, C, LDC, S, X, LDX, TOL, IWORK,
$ DWORK, LDWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( IWARN.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) IWARN
WRITE ( NOUT, FMT = 99996 ) RANK
ELSE
IF ( ( LSAME( JOB, 'R' ) ) .OR. ( LSAME( JOB, 'B' ) ) )
$ WRITE ( NOUT, FMT = 99996 ) RANK
END IF
WRITE ( NOUT, FMT = 99995 )
DO 40 J = 1, L
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99994 ) X(I,J)
20 CONTINUE
IF ( J.LT.L ) WRITE ( NOUT, FMT = 99993 )
40 CONTINUE
WRITE ( NOUT, FMT = 99992 ) ( S(J),J = 1, MIN( M, N+L ) )
END IF
END IF
STOP
*
99999 FORMAT (' MB02MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB02MD = ',I2)
99997 FORMAT (' IWARN on exit from MB02MD = ',I2,/)
99996 FORMAT (' The computed rank of the TLS approximation = ',I3,/)
99995 FORMAT (' The solution X to the TLS problem is ',/)
99994 FORMAT (1X,F8.4)
99993 FORMAT (' ')
99992 FORMAT (/' The singular values of C are ',//(1X,F8.4))
99991 FORMAT (/' N is out of range.',/' N = ',I5)
99990 FORMAT (/' M is out of range.',/' M = ',I5)
99989 FORMAT (/' L is out of range.',/' L = ',I5)
END
Program Data
MB02MD EXAMPLE PROGRAM DATA 6 3 1 B 0.0 0.80010 0.39985 0.60005 0.89999 0.29996 0.69990 0.39997 0.82997 0.49994 0.60003 0.20012 0.79011 0.90013 0.20016 0.79995 0.85002 0.39998 0.80006 0.49985 0.99016 0.20002 0.90007 0.70009 1.02994Program Results
MB02MD EXAMPLE PROGRAM RESULTS The computed rank of the TLS approximation = 3 The solution X to the TLS problem is 0.5003 0.8003 0.2995 The singular values of C are 3.2281 0.8716 0.3697 0.0001
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02ND.html 0000664 0000000 0000000 00000055234 14560147231 0020345 0 ustar 00root root 0000000 0000000
Purpose
To solve the Total Least Squares (TLS) problem using a Partial Singular Value Decomposition (PSVD) approach. The TLS problem assumes an overdetermined set of linear equations AX = B, where both the data matrix A as well as the observation matrix B are inaccurate. The routine also solves determined and underdetermined sets of equations by computing the minimum norm solution. It is assumed that all preprocessing measures (scaling, coordinate transformations, whitening, ... ) of the data have been performed in advance.Specification
SUBROUTINE MB02ND( M, N, L, RANK, THETA, C, LDC, X, LDX, Q, INUL,
$ TOL, RELTOL, IWORK, DWORK, LDWORK, BWORK,
$ IWARN, INFO )
C .. Scalar Arguments ..
INTEGER INFO, IWARN, L, LDC, LDWORK, LDX, M, N, RANK
DOUBLE PRECISION RELTOL, THETA, TOL
C .. Array Arguments ..
LOGICAL BWORK(*), INUL(*)
INTEGER IWORK(*)
DOUBLE PRECISION C(LDC,*), DWORK(*), Q(*), X(LDX,*)
Arguments
Input/Output Parameters
M (input) INTEGER
The number of rows in the data matrix A and the
observation matrix B. M >= 0.
N (input) INTEGER
The number of columns in the data matrix A. N >= 0.
L (input) INTEGER
The number of columns in the observation matrix B.
L >= 0.
RANK (input/output) INTEGER
On entry, if RANK < 0, then the rank of the TLS
approximation [A+DA|B+DB] (r say) is computed by the
routine.
Otherwise, RANK must specify the value of r.
RANK <= min(M,N).
On exit, if RANK < 0 on entry and INFO = 0, then RANK
contains the computed rank of the TLS approximation
[A+DA|B+DB].
Otherwise, the user-supplied value of RANK may be
changed by the routine on exit if the RANK-th and the
(RANK+1)-th singular values of C = [A|B] are considered
to be equal, or if the upper triangular matrix F (as
defined in METHOD) is (numerically) singular.
THETA (input/output) DOUBLE PRECISION
On entry, if RANK < 0, then the rank of the TLS
approximation [A+DA|B+DB] is computed using THETA as
(min(M,N+L) - d), where d is the number of singular
values of [A|B] <= THETA. THETA >= 0.0.
Otherwise, THETA is an initial estimate (t say) for
computing a lower bound on the RANK largest singular
values of [A|B]. If THETA < 0.0 on entry however, then
t is computed by the routine.
On exit, if RANK >= 0 on entry, then THETA contains the
computed bound such that precisely RANK singular values
of C = [A|B] are greater than THETA + TOL.
Otherwise, THETA is unchanged.
C (input/output) DOUBLE PRECISION array, dimension (LDC,N+L)
On entry, the leading M-by-(N+L) part of this array must
contain the matrices A and B. Specifically, the first N
columns must contain the data matrix A and the last L
columns the observation matrix B (right-hand sides).
On exit, if INFO = 0, the first N+L components of the
columns of this array whose index i corresponds with
INUL(i) = .TRUE., are the possibly transformed (N+L-RANK)
base vectors of the right singular subspace corresponding
to the singular values of C = [A|B] which are less than or
equal to THETA. Specifically, if L = 0, or if RANK = 0 and
IWARN <> 2, these vectors are indeed the base vectors
above. Otherwise, these vectors form the matrix V2,
transformed as described in Step 4 of the PTLS algorithm
(see METHOD). The TLS solution is computed from these
vectors. The other columns of array C contain no useful
information.
LDC INTEGER
The leading dimension of array C. LDC >= max(1,M,N+L).
X (output) DOUBLE PRECISION array, dimension (LDX,L)
If INFO = 0, the leading N-by-L part of this array
contains the solution X to the TLS problem specified by
A and B.
LDX INTEGER
The leading dimension of array X. LDX >= max(1,N).
Q (output) DOUBLE PRECISION array, dimension
(max(1,2*min(M,N+L)-1))
This array contains the partially diagonalized bidiagonal
matrix J computed from C, at the moment that the desired
singular subspace has been found. Specifically, the
leading p = min(M,N+L) entries of Q contain the diagonal
elements q(1),q(2),...,q(p) and the entries Q(p+1),Q(p+2),
...,Q(2*p-1) contain the superdiagonal elements e(1),e(2),
...,e(p-1) of J.
INUL (output) LOGICAL array, dimension (N+L)
The indices of the elements of this array with value
.TRUE. indicate the columns in C containing the base
vectors of the right singular subspace of C from which
the TLS solution has been computed.
Tolerances
TOL DOUBLE PRECISION
This parameter defines the multiplicity of singular values
by considering all singular values within an interval of
length TOL as coinciding. TOL is used in checking how many
singular values are less than or equal to THETA. Also in
computing an appropriate upper bound THETA by a bisection
method, TOL is used as a stopping criterion defining the
minimum (absolute) subinterval width. TOL is also taken
as an absolute tolerance for negligible elements in the
QR/QL iterations. If the user sets TOL to be less than or
equal to 0, then the tolerance is taken as specified in
SLICOT Library routine MB04YD document.
RELTOL DOUBLE PRECISION
This parameter specifies the minimum relative width of an
interval. When an interval is narrower than TOL, or than
RELTOL times the larger (in magnitude) endpoint, then it
is considered to be sufficiently small and bisection has
converged. If the user sets RELTOL to be less than
BASE * EPS, where BASE is machine radix and EPS is machine
precision (see LAPACK Library routine DLAMCH), then the
tolerance is taken as BASE * EPS.
Workspace
IWORK INTEGER array, dimension (N+2*L)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK, and DWORK(2) returns the reciprocal of the
condition number of the matrix F.
LDWORK INTEGER
The length of the array DWORK.
LDWORK = max(2, max(M,N+L) + 2*min(M,N+L),
min(M,N+L) + LW + max(6*(N+L)-5,
L*L+max(N+L,3*L)),
where
LW = (N+L)*(N+L-1)/2, if M >= N+L,
LW = M*(N+L-(M-1)/2), if M < N+L.
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
BWORK LOGICAL array, dimension (N+L)
Warning Indicator
IWARN INTEGER
= 0: no warnings;
= 1: if the rank of matrix C has been lowered because a
singular value of multiplicity greater than 1 was
found;
= 2: if the rank of matrix C has been lowered because the
upper triangular matrix F is (numerically) singular.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the maximum number of QR/QL iteration steps
(30*MIN(M,N)) has been exceeded;
= 2: if the computed rank of the TLS approximation
[A+DA|B+DB] exceeds MIN(M,N). Try increasing the
value of THETA or set the value of RANK to min(M,N).
Method
The method used is the Partial Total Least Squares (PTLS) approach
proposed by Van Huffel and Vandewalle [5].
Let C = [A|B] denote the matrix formed by adjoining the columns of
B to the columns of A on the right.
Total Least Squares (TLS) definition:
-------------------------------------
Given matrices A and B, find a matrix X satisfying
(A + DA) X = B + DB,
where A and DA are M-by-N matrices, B and DB are M-by-L matrices
and X is an N-by-L matrix.
The solution X must be such that the Frobenius norm of [DA|DB]
is a minimum and each column of B + DB is in the range of
A + DA. Whenever the solution is not unique, the routine singles
out the minimum norm solution X.
Let V denote the right singular subspace of C. Since the TLS
solution can be computed from any orthogonal basis of the subspace
of V corresponding to the smallest singular values of C, the
Partial Singular Value Decomposition (PSVD) can be used instead of
the classical SVD. The dimension of this subspace of V may be
determined by the rank of C or by an upper bound for those
smallest singular values.
The PTLS algorithm proceeds as follows (see [2 - 5]):
Step 1: Bidiagonalization phase
-----------------------
(a) If M is large enough than N + L, transform C into upper
triangular form R by Householder transformations.
(b) Transform C (or R) into upper bidiagonal form
(p = min(M,N+L)):
|q(1) e(1) 0 ... 0 |
(0) | 0 q(2) e(2) . |
J = | . . |
| . e(p-1)|
| 0 ... q(p) |
if M >= N + L, or lower bidiagonal form:
|q(1) 0 0 ... 0 0 |
(0) |e(1) q(2) 0 . . |
J = | . . . |
| . q(p) . |
| 0 ... e(p-1) q(p)|
if M < N + L, using Householder transformations.
In the second case, transform the matrix to the upper
bidiagonal form by applying Givens rotations.
(c) Initialize the right singular base matrix with the identity
matrix.
Step 2: Partial diagonalization phase
-----------------------------
If the upper bound THETA is not given, then compute THETA such
that precisely p - RANK singular values (p=min(M,N+L)) of the
bidiagonal matrix are less than or equal to THETA, using a
bisection method [5]. Diagonalize the given bidiagonal matrix J
partially, using either QL iterations (if the upper left diagonal
element of the considered bidiagonal submatrix is smaller than the
lower right diagonal element) or QR iterations, such that J is
split into unreduced bidiagonal submatrices whose singular values
are either all larger than THETA or are all less than or equal
to THETA. Accumulate the Givens rotations in V.
Step 3: Back transformation phase
-------------------------
Apply the Householder transformations of Step 1(b) onto the base
vectors of V associated with the bidiagonal submatrices with all
singular values less than or equal to THETA.
Step 4: Computation of F and Y
----------------------
Let V2 be the matrix of the columns of V corresponding to the
(N + L - RANK) smallest singular values of C.
Compute with Householder transformations the matrices F and Y
such that:
|VH Y|
V2 x Q = | |
|0 F|
where Q is an orthogonal matrix, VH is an N-by-(N-RANK) matrix,
Y is an N-by-L matrix and F is an L-by-L upper triangular matrix.
If F is singular, then reduce the value of RANK by one and repeat
Steps 2, 3 and 4.
Step 5: Computation of the TLS solution
-------------------------------
If F is non-singular then the solution X is obtained by solving
the following equations by forward elimination:
X F = -Y.
Notes:
If RANK is lowered in Step 4, some additional base vectors must
be computed in Step 2. The additional computations are kept to
a minimum.
If RANK is lowered in Step 4 but the multiplicity of the RANK-th
singular value is larger than 1, then the value of RANK is further
lowered with its multiplicity defined by the parameter TOL. This
is done at the beginning of Step 2 by calling SLICOT Library
routine MB03MD (from MB04YD), which estimates THETA using a
bisection method. If F in Step 4 is singular, then the computed
solution is infinite and hence does not satisfy the second TLS
criterion (see TLS definition). For these cases, Golub and
Van Loan [1] claim that the TLS problem has no solution. The
properties of these so-called nongeneric problems are described
in [6] and the TLS computations are generalized in order to solve
them. As proven in [6], the proposed generalization satisfies the
TLS criteria for any number L of observation vectors in B provided
that, in addition, the solution | X| is constrained to be
|-I|
orthogonal to all vectors of the form |w| which belong to the
|0|
space generated by the columns of the submatrix |Y|.
|F|
References
[1] Golub, G.H. and Van Loan, C.F.
An Analysis of the Total Least-Squares Problem.
SIAM J. Numer. Anal., 17, pp. 883-893, 1980.
[2] Van Huffel, S., Vandewalle, J. and Haegemans, A.
An Efficient and Reliable Algorithm for Computing the
Singular Subspace of a Matrix Associated with its Smallest
Singular Values.
J. Comput. and Appl. Math., 19, pp. 313-330, 1987.
[3] Van Huffel, S.
Analysis of the Total Least Squares Problem and its Use in
Parameter Estimation.
Doctoral dissertation, Dept. of Electr. Eng., Katholieke
Universiteit Leuven, Belgium, June 1987.
[4] Chan, T.F.
An Improved Algorithm for Computing the Singular Value
Decomposition.
ACM TOMS, 8, pp. 72-83, 1982.
[5] Van Huffel, S. and Vandewalle, J.
The Partial Total Least Squares Algorithm.
J. Comput. Appl. Math., 21, pp. 333-341, 1988.
[6] Van Huffel, S. and Vandewalle, J.
Analysis and Solution of the Nongeneric Total Least Squares
Problem.
SIAM J. Matr. Anal. and Appl., 9, pp. 360-372, 1988.
Numerical Aspects
The computational efficiency of the PTLS algorithm compared with
the classical TLS algorithm (see [2 - 5]) is obtained by making
use of PSVD (see [1]) instead of performing the entire SVD.
Depending on the gap between the RANK-th and the (RANK+1)-th
singular values of C, the number (N + L - RANK) of base vectors to
be computed with respect to the column dimension (N + L) of C and
the desired accuracy RELTOL, the algorithm used by this routine is
approximately twice as fast as the classical TLS algorithm at the
expense of extra storage requirements, namely:
(N + L) x (N + L - 1)/2 if M >= N + L or
M x (N + L - (M - 1)/2) if M < N + L.
This is because the Householder transformations performed on the
rows of C in the bidiagonalization phase (see Step 1) must be kept
until the end (Step 5).
Further Comments
NoneExample
Program Text
* MB02ND EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, NMAX, LMAX
PARAMETER ( MMAX = 20, NMAX = 20, LMAX = 20 )
INTEGER LDC, LDX
PARAMETER ( LDC = MAX( MMAX, NMAX+LMAX ), LDX = NMAX )
INTEGER LENGQ
PARAMETER ( LENGQ = 2*MIN(MMAX,NMAX+LMAX)-1 )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX+2*LMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX(2, MAX( MMAX, NMAX+LMAX ) +
$ 2*MIN( MMAX, NMAX+LMAX ),
$ MIN( MMAX, NMAX+LMAX ) +
$ MAX( ( NMAX+LMAX )*( NMAX+LMAX-1 )/2,
$ MMAX*( NMAX+LMAX-( MMAX-1 )/2 ) ) +
$ MAX( 6*(NMAX+LMAX)-5, LMAX*LMAX +
$ MAX( NMAX+LMAX, 3*LMAX ) ) ) )
INTEGER LBWORK
PARAMETER ( LBWORK = NMAX+LMAX )
* .. Local Scalars ..
DOUBLE PRECISION RELTOL, THETA, THETA1, TOL
INTEGER I, INFO, IWARN, J, K, L, LOOP, M, MINMNL, N,
$ RANK, RANK1
* .. Local Arrays ..
DOUBLE PRECISION C(LDC,NMAX+LMAX), DWORK(LDWORK),
$ Q(LENGQ), X(LDX,LMAX)
INTEGER IWORK(LIWORK)
LOGICAL BWORK(LBWORK), INUL(NMAX+LMAX)
* .. External Subroutines ..
EXTERNAL MB02ND
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, N, L, RANK, THETA, TOL, RELTOL
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99982 ) M
ELSE IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99983 ) N
ELSE IF ( L.LT.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99981 ) L
ELSE IF ( RANK.GT.MIN( MMAX, NMAX ) ) THEN
WRITE ( NOUT, FMT = 99980 ) RANK
ELSE IF ( RANK.LT.0 .AND. THETA.LT.ZERO ) THEN
WRITE ( NOUT, FMT = 99979 ) THETA
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N+L ), I = 1,M )
RANK1 = RANK
THETA1 = THETA
* Compute the solution to the TLS problem Ax = b.
CALL MB02ND( M, N, L, RANK, THETA, C, LDC, X, LDX, Q, INUL,
$ TOL, RELTOL, IWORK, DWORK, LDWORK, BWORK, IWARN,
$ INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( IWARN.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) IWARN
WRITE ( NOUT, FMT = 99996 ) RANK
ELSE
IF ( RANK1.LT.0 ) WRITE ( NOUT, FMT = 99996 ) RANK
END IF
IF ( THETA1.LT.ZERO ) WRITE ( NOUT, FMT = 99995 ) THETA
WRITE ( NOUT, FMT = 99994 )
MINMNL = MIN( M, N+L )
LOOP = MINMNL - 1
DO 20 I = 1, LOOP
K = I + MINMNL
WRITE ( NOUT, FMT = 99993 ) I, I, Q(I), I, I + 1, Q(K)
20 CONTINUE
WRITE ( NOUT, FMT = 99992 ) MINMNL, MINMNL, Q(MINMNL)
WRITE ( NOUT, FMT = 99991 )
DO 60 J = 1, L
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99990 ) X(I,J)
40 CONTINUE
IF ( J.LT.L ) WRITE ( NOUT, FMT = 99989 )
60 CONTINUE
WRITE ( NOUT, FMT = 99987 ) N + L, N + L
WRITE ( NOUT, FMT = 99985 )
DO 80 I = 1, MAX( M, N + L )
WRITE ( NOUT, FMT = 99984 ) ( C(I,J), J = 1,N+L )
80 CONTINUE
WRITE ( NOUT, FMT = 99986 )
DO 100 J = 1, N + L
WRITE ( NOUT, FMT = 99988 ) J, INUL(J)
100 CONTINUE
END IF
END IF
STOP
*
99999 FORMAT (' MB02ND EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB02ND = ',I2)
99997 FORMAT (' IWARN on exit from MB02ND = ',I2,/)
99996 FORMAT (' The computed rank of the TLS approximation = ',I3,/)
99995 FORMAT (' The computed value of THETA = ',F7.4,/)
99994 FORMAT (' The elements of the partially diagonalized bidiagonal ',
$ 'matrix are',/)
99993 FORMAT (2(' (',I1,',',I1,') = ',F7.4,2X))
99992 FORMAT (' (',I1,',',I1,') = ',F7.4,/)
99991 FORMAT (' The solution X to the TLS problem is ',/)
99990 FORMAT (1X,F8.4)
99989 FORMAT (' ')
99988 FORMAT (I3,L8)
99987 FORMAT (/' Right singular subspace corresponds to the first ',I2,
$ ' components of the j-th ',/' column of C for which INUL(',
$ 'j) = .TRUE., j = 1,...,',I2,/)
99986 FORMAT (/' j INUL(j)',/)
99985 FORMAT (' Matrix C',/)
99984 FORMAT (20(1X,F8.4))
99983 FORMAT (/' N is out of range.',/' N = ',I5)
99982 FORMAT (/' M is out of range.',/' M = ',I5)
99981 FORMAT (/' L is out of range.',/' L = ',I5)
99980 FORMAT (/' RANK is out of range.',/' RANK = ',I5)
99979 FORMAT (/' THETA must be at least zero.',/' THETA = ',F8.4)
END
Program Data
MB02ND EXAMPLE PROGRAM DATA 6 3 1 -1 0.001 0.0 0.0 0.80010 0.39985 0.60005 0.89999 0.29996 0.69990 0.39997 0.82997 0.49994 0.60003 0.20012 0.79011 0.90013 0.20016 0.79995 0.85002 0.39998 0.80006 0.49985 0.99016 0.20002 0.90007 0.70009 1.02994Program Results
MB02ND EXAMPLE PROGRAM RESULTS The computed rank of the TLS approximation = 3 The elements of the partially diagonalized bidiagonal matrix are (1,1) = 3.2280 (1,2) = -0.0287 (2,2) = 0.8714 (2,3) = 0.0168 (3,3) = 0.3698 (3,4) = 0.0000 (4,4) = 0.0001 The solution X to the TLS problem is 0.5003 0.8003 0.2995 Right singular subspace corresponds to the first 4 components of the j-th column of C for which INUL(j) = .TRUE., j = 1,..., 4 Matrix C -0.3967 -0.7096 0.4612 -0.3555 0.9150 -0.2557 0.2414 -0.5687 -0.0728 0.6526 0.5215 -0.2128 0.0000 0.0720 0.6761 0.7106 0.1809 0.3209 0.0247 -0.4139 0.0905 0.4609 -0.3528 0.5128 j INUL(j) 1 F 2 F 3 F 4 T
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02NY.html 0000664 0000000 0000000 00000015470 14560147231 0020370 0 ustar 00root root 0000000 0000000
Purpose
To separate a zero singular value of a bidiagonal submatrix of
order k, k <= p, of the bidiagonal matrix
|Q(1) E(1) 0 ... 0 |
| 0 Q(2) E(2) . |
J = | . . |
| . E(p-1)|
| 0 ... ... ... Q(p) |
with p = MIN(M,N), by annihilating one or two superdiagonal
elements E(i-1) (if i > 1) and/or E(i) (if i < k).
Specification
SUBROUTINE MB02NY( UPDATU, UPDATV, M, N, I, K, Q, E, U, LDU, V,
$ LDV, DWORK )
C .. Scalar Arguments ..
LOGICAL UPDATU, UPDATV
INTEGER I, K, LDU, LDV, M, N
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), E(*), Q(*), U(LDU,*), V(LDV,*)
Arguments
Mode Parameters
UPDATU LOGICAL
Indicates whether the user wishes to accumulate in a
matrix U the left-hand Givens rotations S, as follows:
= .FALSE.: Do not form U;
= .TRUE. : The given matrix U is updated (postmultiplied)
by the left-hand Givens rotations S.
UPDATV LOGICAL
Indicates whether the user wishes to accumulate in a
matrix V the right-hand Givens rotations T, as follows:
= .FALSE.: Do not form V;
= .TRUE. : The given matrix V is updated (postmultiplied)
by the right-hand Givens rotations T.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix U. M >= 0.
N (input) INTEGER
The number of rows of the matrix V. N >= 0.
I (input) INTEGER
The index of the negligible diagonal entry Q(I) of the
bidiagonal matrix J, I <= p.
K (input) INTEGER
The index of the last diagonal entry of the considered
bidiagonal submatrix of J, i.e., E(K-1) is considered
negligible, K <= p.
Q (input/output) DOUBLE PRECISION array, dimension (p)
where p = MIN(M,N).
On entry, Q must contain the diagonal entries of the
bidiagonal matrix J.
On exit, Q contains the diagonal entries of the
transformed bidiagonal matrix S' J T.
E (input/output) DOUBLE PRECISION array, dimension (p-1)
On entry, E must contain the superdiagonal entries of J.
On exit, E contains the superdiagonal entries of the
transformed bidiagonal matrix S' J T.
U (input/output) DOUBLE PRECISION array, dimension (LDU,p)
On entry, if UPDATU = .TRUE., U must contain the M-by-p
left transformation matrix.
On exit, if UPDATU = .TRUE., the Givens rotations S on the
left, annihilating E(i) if i < k, have been postmultiplied
into U.
U is not referenced if UPDATU = .FALSE..
LDU INTEGER
The leading dimension of the array U.
LDU >= max(1,M) if UPDATU = .TRUE.;
LDU >= 1 if UPDATU = .FALSE..
V (input/output) DOUBLE PRECISION array, dimension (LDV,p)
On entry, if UPDATV = .TRUE., V must contain the N-by-p
right transformation matrix.
On exit, if UPDATV = .TRUE., the Givens rotations T on the
right, annihilating E(i-1) if i > 1, have been
postmultiplied into V.
V is not referenced if UPDATV = .FALSE..
LDV INTEGER
The leading dimension of the array V.
LDV >= max(1,N) if UPDATV = .TRUE.;
LDV >= 1 if UPDATV = .FALSE..
Workspace
DWORK DOUBLE PRECISION array, dimension (MAX(1,LDWORK))
LDWORK >= 2*MAX(K-I,I-1), if UPDATV = UPDATU = .TRUE.;
LDWORK >= 2*(K-I), if UPDATU = .TRUE., UPDATV = .FALSE.;
LDWORK >= 2*(I-1), if UPDATV = .TRUE., UPDATU = .FALSE.;
LDWORK >= 1, if UPDATU = UPDATV = .FALSE..
Method
Let the considered bidiagonal submatrix be
|Q(1) E(1) 0 ... 0 |
| 0 Q(2) E(2) . |
| . . |
| . Q(i-1) E(i-1) . |
Jk = | . Q(i) E(i) . |.
| . Q(i+1) . . |
| . .. . |
| . E(k-1)|
| 0 ... ... Q(k) |
A zero singular value of Jk manifests itself by a zero diagonal
entry Q(i) or in practice, a negligible value of Q(i).
When a negligible diagonal element Q(i) in Jk is present, the
bidiagonal submatrix Jk is split by the routine into 2 or 3
unreduced bidiagonal submatrices by annihilating E(i) (if i < k)
using Givens rotations S on the left and by annihilating E(i-1)
(if i > 1) using Givens rotations T on the right until Jk is
reduced to the form:
|Q(1) E(1) 0 ... 0 |
| 0 . ... . |
| . ... . |
| . Q(i-1) 0 . |
S' Jk T = | . 0 0 . |.
| . Q(i+1) . . |
| . .. . |
| . E(k-1)|
| 0 ... ... Q(k) |
For more details, see [1, pp.11.12-11.14].
References
[1] Dongarra, J.J., Bunch, J.R., Moler C.B. and Stewart, G.W.
LINPACK User's Guide.
SIAM, Philadelphia, 1979.
Numerical Aspects
The algorithm is backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve (if well-conditioned) one of the matrix equations
op( A )*X = alpha*B, or X*op( A ) = alpha*B,
where alpha is a scalar, X and B are m-by-n matrices, A is a unit,
or non-unit, upper or lower triangular matrix and op( A ) is one
of
op( A ) = A or op( A ) = A'.
An estimate of the reciprocal of the condition number of the
triangular matrix A, in either the 1-norm or the infinity-norm, is
also computed as
RCOND = 1 / ( norm(A) * norm(inv(A)) ).
and the specified matrix equation is solved only if RCOND is
larger than a given tolerance TOL. In that case, the matrix X is
overwritten on B.
Specification
SUBROUTINE MB02OD( SIDE, UPLO, TRANS, DIAG, NORM, M, N, ALPHA, A,
$ LDA, B, LDB, RCOND, TOL, IWORK, DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DIAG, NORM, SIDE, TRANS, UPLO
INTEGER INFO, LDA, LDB, M, N
DOUBLE PRECISION ALPHA, RCOND, TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*)
Arguments
Mode Parameters
SIDE CHARACTER*1
Specifies whether op( A ) appears on the left or right
of X as follows:
= 'L': op( A )*X = alpha*B;
= 'R': X*op( A ) = alpha*B.
UPLO CHARACTER*1
Specifies whether the matrix A is an upper or lower
triangular matrix as follows:
= 'U': A is an upper triangular matrix;
= 'L': A is a lower triangular matrix.
TRANS CHARACTER*1
Specifies the form of op( A ) to be used in the matrix
multiplication as follows:
= 'N': op( A ) = A;
= 'T': op( A ) = A';
= 'C': op( A ) = A'.
DIAG CHARACTER*1
Specifies whether or not A is unit triangular as follows:
= 'U': A is assumed to be unit triangular;
= 'N': A is not assumed to be unit triangular.
NORM CHARACTER*1
Specifies whether the 1-norm condition number or the
infinity-norm condition number is required:
= '1' or 'O': 1-norm;
= 'I': Infinity-norm.
Input/Output Parameters
M (input) INTEGER
The number of rows of B. M >= 0.
N (input) INTEGER
The number of columns of B. N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then A is not
referenced and B need not be set before entry.
A (input) DOUBLE PRECISION array, dimension (LDA,k),
where k is M when SIDE = 'L' and is N when SIDE = 'R'.
On entry with UPLO = 'U', the leading k-by-k upper
triangular part of this array must contain the upper
triangular matrix and the strictly lower triangular part
of A is not referenced.
On entry with UPLO = 'L', the leading k-by-k lower
triangular part of this array must contain the lower
triangular matrix and the strictly upper triangular part
of A is not referenced.
Note that when DIAG = 'U', the diagonal elements of A are
not referenced either, but are assumed to be unity.
LDA INTEGER
The leading dimension of array A.
LDA >= max(1,M) when SIDE = 'L';
LDA >= max(1,N) when SIDE = 'R'.
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the leading M-by-N part of this array must
contain the right-hand side matrix B.
On exit, if INFO = 0, the leading M-by-N part of this
array contains the solution matrix X.
Otherwise, this array is not modified by the routine.
LDB INTEGER
The leading dimension of array B. LDB >= max(1,M).
RCOND (output) DOUBLE PRECISION
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(norm(A) * norm(inv(A))).
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used to test for near singularity of
the matrix A. If the user sets TOL > 0, then the given
value of TOL is used as a lower bound for the reciprocal
condition number of that matrix; a matrix whose estimated
condition number is less than 1/TOL is considered to be
nonsingular. If the user sets TOL <= 0, then an implicitly
computed, default tolerance, defined by TOLDEF = k*k*EPS,
is used instead, where EPS is the machine precision (see
LAPACK Library routine DLAMCH).
Workspace
IWORK INTEGER array, dimension (k) DWORK DOUBLE PRECISION array, dimension (3*k)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the matrix A is numerically singular, i.e. the
condition number estimate of A (in the specified
norm) exceeds 1/TOL.
Method
An estimate of the reciprocal of the condition number of the triangular matrix A (in the specified norm) is computed, and if this estimate is larger then the given (or default) tolerance, the specified matrix equation is solved using Level 3 BLAS routine DTRSM.References
None.Numerical Aspects
2 The algorithm requires k N/2 operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02PD.html 0000664 0000000 0000000 00000033441 14560147231 0020343 0 ustar 00root root 0000000 0000000
Purpose
To solve (if well-conditioned) the matrix equations
op( A )*X = B,
where X and B are N-by-NRHS matrices, A is an N-by-N matrix and
op( A ) is one of
op( A ) = A or op( A ) = A'.
Error bounds on the solution and a condition estimate are also
provided.
Specification
SUBROUTINE MB02PD( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
$ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
$ IWORK, DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER EQUED, FACT, TRANS
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
C .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
$ BERR( * ), C( * ), DWORK( * ), FERR( * ),
$ R( * ), X( LDX, * )
Arguments
Mode Parameters
FACT CHARACTER*1
Specifies whether or not the factored form of the matrix A
is supplied on entry, and if not, whether the matrix A
should be equilibrated before it is factored.
= 'F': On entry, AF and IPIV contain the factored form
of A. If EQUED is not 'N', the matrix A has been
equilibrated with scaling factors given by R
and C. A, AF, and IPIV are not modified.
= 'N': The matrix A will be copied to AF and factored.
= 'E': The matrix A will be equilibrated if necessary,
then copied to AF and factored.
TRANS CHARACTER*1
Specifies the form of the system of equations as follows:
= 'N': A * X = B (No transpose);
= 'T': A**T * X = B (Transpose);
= 'C': A**H * X = B (Transpose).
Input/Output Parameters
N (input) INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of
columns of the matrices B and X. NRHS >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A. If FACT = 'F' and EQUED is not 'N',
then A must have been equilibrated by the scaling factors
in R and/or C. A is not modified if FACT = 'F' or 'N',
or if FACT = 'E' and EQUED = 'N' on exit.
On exit, if EQUED .NE. 'N', the leading N-by-N part of
this array contains the matrix A scaled as follows:
EQUED = 'R': A := diag(R) * A;
EQUED = 'C': A := A * diag(C);
EQUED = 'B': A := diag(R) * A * diag(C).
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF (input or output) DOUBLE PRECISION array, dimension
(LDAF,N)
If FACT = 'F', then AF is an input argument and on entry
the leading N-by-N part of this array must contain the
factors L and U from the factorization A = P*L*U as
computed by DGETRF. If EQUED .NE. 'N', then AF is the
factored form of the equilibrated matrix A.
If FACT = 'N', then AF is an output argument and on exit
the leading N-by-N part of this array contains the factors
L and U from the factorization A = P*L*U of the original
matrix A.
If FACT = 'E', then AF is an output argument and on exit
the leading N-by-N part of this array contains the factors
L and U from the factorization A = P*L*U of the
equilibrated matrix A (see the description of A for the
form of the equilibrated matrix).
LDAF (input) INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV (input or output) INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and on entry
it must contain the pivot indices from the factorization
A = P*L*U as computed by DGETRF; row i of the matrix was
interchanged with row IPIV(i).
If FACT = 'N', then IPIV is an output argument and on exit
it contains the pivot indices from the factorization
A = P*L*U of the original matrix A.
If FACT = 'E', then IPIV is an output argument and on exit
it contains the pivot indices from the factorization
A = P*L*U of the equilibrated matrix A.
EQUED (input or output) CHARACTER*1
Specifies the form of equilibration that was done as
follows:
= 'N': No equilibration (always true if FACT = 'N');
= 'R': Row equilibration, i.e., A has been premultiplied
by diag(R);
= 'C': Column equilibration, i.e., A has been
postmultiplied by diag(C);
= 'B': Both row and column equilibration, i.e., A has
been replaced by diag(R) * A * diag(C).
EQUED is an input argument if FACT = 'F'; otherwise, it is
an output argument.
R (input or output) DOUBLE PRECISION array, dimension (N)
The row scale factors for A. If EQUED = 'R' or 'B', A is
multiplied on the left by diag(R); if EQUED = 'N' or 'C',
R is not accessed. R is an input argument if FACT = 'F';
otherwise, R is an output argument. If FACT = 'F' and
EQUED = 'R' or 'B', each element of R must be positive.
C (input or output) DOUBLE PRECISION array, dimension (N)
The column scale factors for A. If EQUED = 'C' or 'B',
A is multiplied on the right by diag(C); if EQUED = 'N'
or 'R', C is not accessed. C is an input argument if
FACT = 'F'; otherwise, C is an output argument. If
FACT = 'F' and EQUED = 'C' or 'B', each element of C must
be positive.
B (input/output) DOUBLE PRECISION array, dimension
(LDB,NRHS)
On entry, the leading N-by-NRHS part of this array must
contain the right-hand side matrix B.
On exit,
if EQUED = 'N', B is not modified;
if TRANS = 'N' and EQUED = 'R' or 'B', the leading
N-by-NRHS part of this array contains diag(R)*B;
if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', the leading
N-by-NRHS part of this array contains diag(C)*B.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, the leading N-by-NRHS part of
this array contains the solution matrix X to the original
system of equations. Note that A and B are modified on
exit if EQUED .NE. 'N', and the solution to the
equilibrated system is inv(diag(C))*X if TRANS = 'N' and
EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or
'C' and EQUED = 'R' or 'B'.
LDX (input) INTEGER
The leading dimension of the array X. LDX >= max(1,N).
RCOND (output) DOUBLE PRECISION
The estimate of the reciprocal condition number of the
matrix A after equilibration (if done). If RCOND is less
than the machine precision (in particular, if RCOND = 0),
the matrix is singular to working precision. This
condition is indicated by a return code of INFO > 0.
For efficiency reasons, RCOND is computed only when the
matrix A is factored, i.e., for FACT = 'N' or 'E'. For
FACT = 'F', RCOND is not used, but it is assumed that it
has been computed and checked before the routine call.
FERR (output) DOUBLE PRECISION array, dimension (NRHS)
The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X).
If XTRUE is the true solution corresponding to X(j),
FERR(j) is an estimated upper bound for the magnitude of
the largest element in (X(j) - XTRUE) divided by the
magnitude of the largest element in X(j). The estimate
is as reliable as the estimate for RCOND, and is almost
always a slight overestimate of the true error.
BERR (output) DOUBLE PRECISION array, dimension (NRHS)
The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in
any element of A or B that makes X(j) an exact solution).
Workspace
IWORK INTEGER array, dimension (N)
DWORK DOUBLE PRECISION array, dimension (4*N)
On entry, if FACT = 'F', DWORK(1) contains the reciprocal
pivot growth factor norm(A)/norm(U), computed previously
by this routine, with FACT <> 'N', for the same matrix A.
On exit, DWORK(1) contains the reciprocal pivot growth
factor norm(A)/norm(U). The "max absolute element" norm is
used. If DWORK(1) is much less than 1, then the stability
of the LU factorization of the (equilibrated) matrix A
could be poor. This also means that the solution X,
condition estimator RCOND, and forward error bound FERR
could be unreliable. If factorization fails with
0 < INFO <= N, then DWORK(1) contains the reciprocal pivot
growth factor for the leading INFO columns of A.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, and i is
<= N: U(i,i) is exactly zero. The factorization
has been completed, but the factor U is
exactly singular, so the solution and error
bounds could not be computed. RCOND = 0 is
returned.
= N+1: U is nonsingular, but RCOND is less than
machine precision, meaning that the matrix is
singular to working precision. Nevertheless,
the solution and error bounds are computed
because there are a number of situations
where the computed solution can be more
accurate than the value of RCOND would
suggest.
The positive values for INFO are set only when the
matrix A is factored, i.e., for FACT = 'N' or 'E'.
Method
The following steps are performed:
1. If FACT = 'E', real scaling factors are computed to equilibrate
the system:
TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
Whether or not the system will be equilibrated depends on the
scaling of the matrix A, but if equilibration is used, A is
overwritten by diag(R)*A*diag(C) and B by diag(R)*B
(if TRANS='N') or diag(C)*B (if TRANS = 'T' or 'C').
2. If FACT = 'N' or 'E', the LU decomposition is used to factor
the matrix A (after equilibration if FACT = 'E') as
A = P * L * U,
where P is a permutation matrix, L is a unit lower triangular
matrix, and U is upper triangular.
3. If some U(i,i)=0, so that U is exactly singular, then the
routine returns with INFO = i. Otherwise, the factored form
of A is used to estimate the condition number of the matrix A.
If the reciprocal of the condition number is less than machine
precision, INFO = N+1 is returned as a warning, but the routine
still goes on to solve for X and compute error bounds as
described below.
4. The system of equations is solved for X using the factored form
of A.
5. Iterative refinement is applied to improve the computed
solution matrix and calculate error bounds and backward error
estimates for it.
6. If equilibration was used, the matrix X is premultiplied by
diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
that it solves the original system before equilibration.
References
[1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., Sorensen, D.
LAPACK Users' Guide: Second Edition, SIAM, Philadelphia, 1995.
Further Comments
This is a simplified version of the LAPACK Library routine DGESVX, useful when several sets of matrix equations with the same coefficient matrix A and/or A' should be solved.Numerical Aspects
3 The algorithm requires 0(N ) operations.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02QD.html 0000664 0000000 0000000 00000031611 14560147231 0020341 0 ustar 00root root 0000000 0000000
Purpose
To compute a solution, optionally corresponding to specified free
elements, to a real linear least squares problem:
minimize || A * X - B ||
using a complete orthogonal factorization of the M-by-N matrix A,
which may be rank-deficient.
Several right hand side vectors b and solution vectors x can be
handled in a single call; they are stored as the columns of the
M-by-NRHS right hand side matrix B and the N-by-NRHS solution
matrix X.
Specification
SUBROUTINE MB02QD( JOB, INIPER, M, N, NRHS, RCOND, SVLMAX, A, LDA,
$ B, LDB, Y, JPVT, RANK, SVAL, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER INIPER, JOB
INTEGER INFO, LDA, LDB, LDWORK, M, N, NRHS, RANK
DOUBLE PRECISION RCOND, SVLMAX
C .. Array Arguments ..
INTEGER JPVT( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DWORK( * ),
$ SVAL( 3 ), Y ( * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies whether or not a standard least squares solution
must be computed, as follows:
= 'L': Compute a standard least squares solution (Y = 0);
= 'F': Compute a solution with specified free elements
(given in Y).
INIPER CHARACTER*1
Specifies whether an initial column permutation, defined
by JPVT, must be performed, as follows:
= 'P': Perform an initial column permutation;
= 'N': Do not perform an initial column permutation.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of
columns of the matrices B and X. NRHS >= 0.
RCOND (input) DOUBLE PRECISION
RCOND is used to determine the effective rank of A, which
is defined as the order of the largest leading triangular
submatrix R11 in the QR factorization with pivoting of A,
whose estimated condition number is less than 1/RCOND.
0 <= RCOND <= 1.
SVLMAX (input) DOUBLE PRECISION
If A is a submatrix of another matrix C, and the rank
decision should be related to that matrix, then SVLMAX
should be an estimate of the largest singular value of C
(for instance, the Frobenius norm of C). If this is not
the case, the input value SVLMAX = 0 should work.
SVLMAX >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading M-by-N part of this array must
contain the given matrix A.
On exit, the leading M-by-N part of this array contains
details of its complete orthogonal factorization:
the leading RANK-by-RANK upper triangular part contains
the upper triangular factor T11 (see METHOD);
the elements below the diagonal, with the entries 2 to
min(M,N)+1 of the array DWORK, represent the orthogonal
matrix Q as a product of min(M,N) elementary reflectors
(see METHOD);
the elements of the subarray A(1:RANK,RANK+1:N), with the
next RANK entries of the array DWORK, represent the
orthogonal matrix Z as a product of RANK elementary
reflectors (see METHOD).
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B (input/output) DOUBLE PRECISION array, dimension
(LDB,NRHS)
On entry, the leading M-by-NRHS part of this array must
contain the right hand side matrix B.
On exit, the leading N-by-NRHS part of this array contains
the solution matrix X.
If M >= N and RANK = N, the residual sum-of-squares for
the solution in the i-th column is given by the sum of
squares of elements N+1:M in that column.
If NRHS = 0, this array is not referenced, and the routine
returns the effective rank of A, and its QR factorization.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,M,N).
Y (input) DOUBLE PRECISION array, dimension ( N*NRHS )
If JOB = 'F', the elements Y(1:(N-RANK)*NRHS) are used as
free elements in computing the solution (see METHOD).
The remaining elements are not referenced.
If JOB = 'L', or NRHS = 0, this array is not referenced.
JPVT (input/output) INTEGER array, dimension (N)
On entry with INIPER = 'P', if JPVT(i) <> 0, the i-th
column of A is an initial column, otherwise it is a free
column. Before the QR factorization of A, all initial
columns are permuted to the leading positions; only the
remaining free columns are moved as a result of column
pivoting during the factorization.
If INIPER = 'N', JPVT need not be set on entry.
On exit, if JPVT(i) = k, then the i-th column of A*P
was the k-th column of A.
RANK (output) INTEGER
The effective rank of A, i.e., the order of the submatrix
R11. This is the same as the order of the submatrix T11
in the complete orthogonal factorization of A.
SVAL (output) DOUBLE PRECISION array, dimension ( 3 )
The estimates of some of the singular values of the
triangular factor R11:
SVAL(1): largest singular value of R(1:RANK,1:RANK);
SVAL(2): smallest singular value of R(1:RANK,1:RANK);
SVAL(3): smallest singular value of R(1:RANK+1,1:RANK+1),
if RANK < MIN( M, N ), or of R(1:RANK,1:RANK),
otherwise.
If the triangular factorization is a rank-revealing one
(which will be the case if the leading columns were well-
conditioned), then SVAL(1) will also be an estimate for
the largest singular value of A, and SVAL(2) and SVAL(3)
will be estimates for the RANK-th and (RANK+1)-st singular
values of A, respectively.
By examining these values, one can confirm that the rank
is well defined with respect to the chosen value of RCOND.
The ratio SVAL(1)/SVAL(2) is an estimate of the condition
number of R(1:RANK,1:RANK).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK, and the entries 2 to min(M,N) + RANK + 1
contain the scalar factors of the elementary reflectors
used in the complete orthogonal factorization of A.
Among the entries 2 to min(M,N) + 1, only the first RANK
elements are useful, if INIPER = 'N'.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= max( min(M,N)+3*N+1, 2*min(M,N)+NRHS )
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
If INIPER = 'P', the routine first computes a QR factorization
with column pivoting:
A * P = Q * [ R11 R12 ]
[ 0 R22 ]
with R11 defined as the largest leading submatrix whose estimated
condition number is less than 1/RCOND. The order of R11, RANK,
is the effective rank of A.
If INIPER = 'N', the effective rank is estimated during a
truncated QR factorization (with column pivoting) process, and
the submatrix R22 is not upper triangular, but full and of small
norm. (See SLICOT Library routines MB03OD or MB03OY, respectively,
for further details.)
Then, R22 is considered to be negligible, and R12 is annihilated
by orthogonal transformations from the right, arriving at the
complete orthogonal factorization:
A * P = Q * [ T11 0 ] * Z
[ 0 0 ]
The solution is then
X = P * Z' [ inv(T11)*Q1'*B ]
[ Y ]
where Q1 consists of the first RANK columns of Q, and Y contains
free elements (if JOB = 'F'), or is zero (if JOB = 'L').
Numerical Aspects
The algorithm is backward stable.Further Comments
Significant gain in efficiency is possible for small-rank problems using truncated QR factorization (option INIPER = 'N').Example
Program Text
* MB02QD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, NRHSMX
PARAMETER ( NMAX = 20, MMAX = 20, NRHSMX = 20 )
INTEGER LDA, LDB
PARAMETER ( LDA = MMAX, LDB = MAX( MMAX, NMAX ) )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( MIN( MMAX, NMAX) + 3*NMAX + 1,
$ 2*MIN( MMAX, NMAX) + NRHSMX ) )
* .. Local Scalars ..
DOUBLE PRECISION RCOND, SVLMAX
INTEGER I, INFO, J, M, N, NRHS, RANK
CHARACTER*1 INIPER, JOB
* .. Local Arrays ..
INTEGER JPVT(NMAX)
DOUBLE PRECISION A(LDA,NMAX), B(LDB,NRHSMX), DWORK(LDWORK),
$ SVAL(3), Y(NMAX*NRHSMX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL MB02QD
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, N, NRHS, RCOND, SVLMAX, JOB, INIPER
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) M
ELSE
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
IF ( NRHS.LT.0 .OR. NRHS.GT.NRHSMX ) THEN
WRITE ( NOUT, FMT = 99992 ) NRHS
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,M )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,NRHS ), I = 1,M )
IF ( LSAME( JOB, 'F' ) )
$ READ ( NIN, FMT = * ) ( Y(I), I = 1,N*NRHS )
IF ( LSAME( INIPER, 'P' ) )
$ READ ( NIN, FMT = * ) ( JPVT(I), I = 1,N )
* Find the least squares solution.
CALL MB02QD( JOB, INIPER, M, N, NRHS, RCOND, SVLMAX, A,
$ LDA, B, LDB, Y, JPVT, RANK, SVAL, DWORK,
$ LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) RANK, SVAL
WRITE ( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,NRHS )
10 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' MB02QD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB02QD =',I2)
99997 FORMAT (' The effective rank of A =',I2,/
$ ' Estimates of the singular values SVAL = '/3(1X,F8.4))
99996 FORMAT (' The least squares solution is')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' M is out of range.',/' M = ',I5)
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' NRHS is out of range.',/' NRHS = ',I5)
END
Program Data
MB02QD EXAMPLE PROGRAM DATA 4 3 2 2.3D-16 0.0 L N 2.0 2.0 -3.0 3.0 3.0 -1.0 4.0 4.0 -5.0 -1.0 -1.0 -2.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0Program Results
MB02QD EXAMPLE PROGRAM RESULTS The effective rank of A = 2 Estimates of the singular values SVAL = 7.8659 2.6698 0.0000 The least squares solution is -0.0034 -0.1054 -0.0034 -0.1054 -0.0816 -0.1973
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02QY.html 0000664 0000000 0000000 00000014120 14560147231 0020362 0 ustar 00root root 0000000 0000000
Purpose
To determine the minimum-norm solution to a real linear least
squares problem:
minimize || A * X - B ||,
using the rank-revealing QR factorization of a real general
M-by-N matrix A, computed by SLICOT Library routine MB03OD.
Specification
SUBROUTINE MB02QY( M, N, NRHS, RANK, A, LDA, JPVT, B, LDB, TAU,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDWORK, M, N, NRHS, RANK
C .. Array Arguments ..
INTEGER JPVT( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DWORK( * ), TAU( * )
Arguments
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrices A and B. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
NRHS (input) INTEGER
The number of columns of the matrix B. NRHS >= 0.
RANK (input) INTEGER
The effective rank of A, as returned by SLICOT Library
routine MB03OD. min(M,N) >= RANK >= 0.
A (input/output) DOUBLE PRECISION array, dimension
( LDA, N )
On entry, the leading min(M,N)-by-N upper trapezoidal
part of this array contains the triangular factor R, as
returned by SLICOT Library routine MB03OD. The strict
lower trapezoidal part of A is not referenced.
On exit, if RANK < N, the leading RANK-by-RANK upper
triangular part of this array contains the upper
triangular matrix R of the complete orthogonal
factorization of A, and the submatrix (1:RANK,RANK+1:N)
of this array, with the array TAU, represent the
orthogonal matrix Z (of the complete orthogonal
factorization of A), as a product of RANK elementary
reflectors.
On exit, if RANK = N, this array is unchanged.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
JPVT (input) INTEGER array, dimension ( N )
The recorded permutations performed by SLICOT Library
routine MB03OD; if JPVT(i) = k, then the i-th column
of A*P was the k-th column of the original matrix A.
B (input/output) DOUBLE PRECISION array, dimension
( LDB, NRHS )
On entry, if NRHS > 0, the leading M-by-NRHS part of
this array must contain the matrix B (corresponding to
the transformed matrix A, returned by SLICOT Library
routine MB03OD).
On exit, if NRHS > 0, the leading N-by-NRHS part of this
array contains the solution matrix X.
If M >= N and RANK = N, the residual sum-of-squares
for the solution in the i-th column is given by the sum
of squares of elements N+1:M in that column.
If NRHS = 0, the array B is not referenced.
LDB INTEGER
The leading dimension of the array B.
LDB >= max(1,M,N), if NRHS > 0.
LDB >= 1, if NRHS = 0.
TAU (output) DOUBLE PRECISION array, dimension ( min(M,N) )
The scalar factors of the elementary reflectors.
If RANK = N, the array TAU is not referenced.
Workspace
DWORK DOUBLE PRECISION array, dimension ( LDWORK )
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= max( 1, N, NRHS ).
For good performance, LDWORK should sometimes be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine uses a QR factorization with column pivoting:
A * P = Q * R = Q * [ R11 R12 ],
[ 0 R22 ]
where R11 is an upper triangular submatrix of estimated rank
RANK, the effective rank of A. The submatrix R22 can be
considered as negligible.
If RANK < N, then R12 is annihilated by orthogonal
transformations from the right, arriving at the complete
orthogonal factorization:
A * P = Q * [ T11 0 ] * Z.
[ 0 0 ]
The minimum-norm solution is then
X = P * Z' [ inv(T11)*Q1'*B ],
[ 0 ]
where Q1 consists of the first RANK columns of Q.
The input data for MB02QY are the transformed matrices Q' * A
(returned by SLICOT Library routine MB03OD) and Q' * B.
Matrix Q is not needed.
Numerical Aspects
The implemented method is numerically stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve a system of linear equations
H * X = B or H' * X = B
with an upper Hessenberg N-by-N matrix H using the LU
factorization computed by MB02SD.
Specification
SUBROUTINE MB02RD( TRANS, N, NRHS, H, LDH, IPIV, B, LDB, INFO )
C .. Scalar Arguments ..
CHARACTER TRANS
INTEGER INFO, LDB, LDH, N, NRHS
C .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION B( LDB, * ), H( LDH, * )
Arguments
Mode Parameters
TRANS CHARACTER*1
Specifies the form of the system of equations:
= 'N': H * X = B (No transpose)
= 'T': H'* X = B (Transpose)
= 'C': H'* X = B (Conjugate transpose = Transpose)
Input/Output Parameters
N (input) INTEGER
The order of the matrix H. N >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of
columns of the matrix B. NRHS >= 0.
H (input) DOUBLE PRECISION array, dimension (LDH,N)
The factors L and U from the factorization H = P*L*U
as computed by MB02SD.
LDH INTEGER
The leading dimension of the array H. LDH >= max(1,N).
IPIV (input) INTEGER array, dimension (N)
The pivot indices from MB02SD; for 1<=i<=N, row i of the
matrix was interchanged with row IPIV(i).
B (input/output) DOUBLE PRECISION array, dimension
(LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Error Indicator
INFO (output) INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine uses the factorization
H = P * L * U
where P is a permutation matrix, L is lower triangular with unit
diagonal elements (and one nonzero subdiagonal), and U is upper
triangular.
References
-Numerical Aspects
2 The algorithm requires 0( N x NRHS ) operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02RZ.html 0000664 0000000 0000000 00000006500 14560147231 0020367 0 ustar 00root root 0000000 0000000
Purpose
To solve a system of linear equations
H * X = B, H' * X = B or H**H * X = B
with a complex upper Hessenberg N-by-N matrix H using the LU
factorization computed by MB02SZ.
Specification
SUBROUTINE MB02RZ( TRANS, N, NRHS, H, LDH, IPIV, B, LDB, INFO )
C .. Scalar Arguments ..
CHARACTER TRANS
INTEGER INFO, LDB, LDH, N, NRHS
C .. Array Arguments ..
INTEGER IPIV( * )
COMPLEX*16 B( LDB, * ), H( LDH, * )
Arguments
Mode Parameters
TRANS CHARACTER*1
Specifies the form of the system of equations:
= 'N': H * X = B (No transpose)
= 'T': H'* X = B (Transpose)
= 'C': H**H * X = B (Conjugate transpose)
Input/Output Parameters
N (input) INTEGER
The order of the matrix H. N >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of
columns of the matrix B. NRHS >= 0.
H (input) COMPLEX*16 array, dimension (LDH,N)
The factors L and U from the factorization H = P*L*U
as computed by MB02SZ.
LDH INTEGER
The leading dimension of the array H. LDH >= max(1,N).
IPIV (input) INTEGER array, dimension (N)
The pivot indices from MB02SZ; for 1<=i<=N, row i of the
matrix was interchanged with row IPIV(i).
B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO (output) INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine uses the factorization
H = P * L * U
where P is a permutation matrix, L is lower triangular with unit
diagonal elements (and one nonzero subdiagonal), and U is upper
triangular.
References
-Numerical Aspects
2 The algorithm requires 0( N x NRHS ) complex operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02SD.html 0000664 0000000 0000000 00000014623 14560147231 0020347 0 ustar 00root root 0000000 0000000
Purpose
To compute an LU factorization of an n-by-n upper Hessenberg matrix H using partial pivoting with row interchanges.Specification
SUBROUTINE MB02SD( N, H, LDH, IPIV, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDH, N
C .. Array Arguments ..
INTEGER IPIV(*)
DOUBLE PRECISION H(LDH,*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix H. N >= 0.
H (input/output) DOUBLE PRECISION array, dimension (LDH,N)
On entry, the n-by-n upper Hessenberg matrix to be
factored.
On exit, the factors L and U from the factorization
H = P*L*U; the unit diagonal elements of L are not stored,
and L is lower bidiagonal.
LDH INTEGER
The leading dimension of the array H. LDH >= max(1,N).
IPIV (output) INTEGER array, dimension (N)
The pivot indices; for 1 <= i <= N, row i of the matrix
was interchanged with row IPIV(i).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, U(i,i) is exactly zero. The
factorization has been completed, but the factor U
is exactly singular, and division by zero will occur
if it is used to solve a system of equations.
Method
The factorization has the form
H = P * L * U
where P is a permutation matrix, L is lower triangular with unit
diagonal elements (and one nonzero subdiagonal), and U is upper
triangular.
This is the right-looking Level 1 BLAS version of the algorithm
(adapted after DGETF2).
References
-Numerical Aspects
2 The algorithm requires 0( N ) operations.Further Comments
NoneExample
Program Text
* MB02SD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, NRHMAX
PARAMETER ( NMAX = 20, NRHMAX = 20 )
INTEGER LDB, LDH
PARAMETER ( LDB = NMAX, LDH = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 3*NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX )
* .. Local Scalars ..
DOUBLE PRECISION HNORM, RCOND
INTEGER I, INFO, INFO1, J, N, NRHS
CHARACTER*1 NORM, TRANS
* .. Local Arrays ..
DOUBLE PRECISION H(LDH,NMAX), B(LDB,NRHMAX), DWORK(LDWORK)
INTEGER IPIV(NMAX), IWORK(LIWORK)
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DLANHS
EXTERNAL DLAMCH, DLANHS
* .. External Subroutines ..
EXTERNAL DLASET, MB02RD, MB02SD, MB02TD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read in the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, NRHS, NORM, TRANS
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( ( H(I,J), J = 1,N ), I = 1,N )
IF ( NRHS.LT.0 .OR. NRHS.GT.NRHMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) NRHS
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,NRHS ), I = 1,N )
IF ( N.GT.2 )
$ CALL DLASET( 'Lower', N-2, N-2, ZERO, ZERO, H(3,1), LDH )
* Compute the LU factorization of the upper Hessenberg matrix.
CALL MB02SD( N, H, LDH, IPIV, INFO )
* Estimate the reciprocal condition number of the matrix.
HNORM = DLANHS( NORM, N, H, LDH, DWORK )
CALL MB02TD( NORM, N, HNORM, H, LDH, IPIV, RCOND, IWORK,
$ DWORK, INFO1 )
IF ( INFO.EQ.0 .AND. RCOND.GT.DLAMCH( 'Epsilon' ) ) THEN
* Solve the linear system.
CALL MB02RD( TRANS, N, NRHS, H, LDH, IPIV, B, LDB, INFO )
*
WRITE ( NOUT, FMT = 99997 )
ELSE
WRITE ( NOUT, FMT = 99998 ) INFO
END IF
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( B(I,J), J = 1,NRHS )
10 CONTINUE
WRITE ( NOUT, FMT = 99995 ) RCOND
END IF
END IF
STOP
*
99999 FORMAT (' MB02SD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB02SD = ',I2)
99997 FORMAT (' The solution matrix is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' Reciprocal condition number = ',D12.4)
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' NRHS is out of range.',/' NRHS = ',I5)
END
Program Data
MB02SD EXAMPLE PROGRAM DATA 5 4 O N 1. 2. 6. 3. 5. -2. -1. -1. 0. -2. 0. 3. 1. 5. 1. 0. 0. 2. 0. -4. 0. 0. 0. 1. 4. 5. 5. 1. 5. -2. 1. 3. 1. 0. 0. 4. 5. 2. 1. 1. 3. -1. 3. 3. 1.Program Results
MB02SD EXAMPLE PROGRAM RESULTS The solution matrix is 0.0435 1.2029 1.6377 1.1014 1.0870 -4.4275 -5.5580 -2.9638 0.9130 0.7609 -0.1087 0.6304 -0.8261 2.4783 4.2174 2.7391 -0.0435 0.1304 -0.3043 -0.4348 Reciprocal condition number = 0.1554D-01
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02SZ.html 0000664 0000000 0000000 00000006054 14560147231 0020374 0 ustar 00root root 0000000 0000000
Purpose
To compute an LU factorization of a complex n-by-n upper Hessenberg matrix H using partial pivoting with row interchanges.Specification
SUBROUTINE MB02SZ( N, H, LDH, IPIV, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDH, N
C .. Array Arguments ..
INTEGER IPIV(*)
COMPLEX*16 H(LDH,*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix H. N >= 0.
H (input/output) COMPLEX*16 array, dimension (LDH,N)
On entry, the n-by-n upper Hessenberg matrix to be
factored.
On exit, the factors L and U from the factorization
H = P*L*U; the unit diagonal elements of L are not stored,
and L is lower bidiagonal.
LDH INTEGER
The leading dimension of the array H. LDH >= max(1,N).
IPIV (output) INTEGER array, dimension (N)
The pivot indices; for 1 <= i <= N, row i of the matrix
was interchanged with row IPIV(i).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, U(i,i) is exactly zero. The
factorization has been completed, but the factor U
is exactly singular, and division by zero will occur
if it is used to solve a system of equations.
Method
The factorization has the form
H = P * L * U
where P is a permutation matrix, L is lower triangular with unit
diagonal elements (and one nonzero subdiagonal), and U is upper
triangular.
This is the right-looking Level 2 BLAS version of the algorithm
(adapted after ZGETF2).
References
-Numerical Aspects
2 The algorithm requires 0( N ) complex operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02TD.html 0000664 0000000 0000000 00000006632 14560147231 0020351 0 ustar 00root root 0000000 0000000
Purpose
To estimate the reciprocal of the condition number of an upper Hessenberg matrix H, in either the 1-norm or the infinity-norm, using the LU factorization computed by MB02SD.Specification
SUBROUTINE MB02TD( NORM, N, HNORM, H, LDH, IPIV, RCOND, IWORK,
$ DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER NORM
INTEGER INFO, LDH, N
DOUBLE PRECISION HNORM, RCOND
C .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION DWORK( * ), H( LDH, * )
Arguments
Mode Parameters
NORM CHARACTER*1
Specifies whether the 1-norm condition number or the
infinity-norm condition number is required:
= '1' or 'O': 1-norm;
= 'I': Infinity-norm.
Input/Output Parameters
N (input) INTEGER
The order of the matrix H. N >= 0.
HNORM (input) DOUBLE PRECISION
If NORM = '1' or 'O', the 1-norm of the original matrix H.
If NORM = 'I', the infinity-norm of the original matrix H.
H (input) DOUBLE PRECISION array, dimension (LDH,N)
The factors L and U from the factorization H = P*L*U
as computed by MB02SD.
LDH INTEGER
The leading dimension of the array H. LDH >= max(1,N).
IPIV (input) INTEGER array, dimension (N)
The pivot indices; for 1 <= i <= N, row i of the matrix
was interchanged with row IPIV(i).
RCOND (output) DOUBLE PRECISION
The reciprocal of the condition number of the matrix H,
computed as RCOND = 1/(norm(H) * norm(inv(H))).
Workspace
IWORK INTEGER array, dimension (N) DWORK DOUBLE PRECISION array, dimension (3*N)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
An estimate is obtained for norm(inv(H)), and the reciprocal of
the condition number is computed as
RCOND = 1 / ( norm(H) * norm(inv(H)) ).
References
-Numerical Aspects
2 The algorithm requires 0( N ) operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02TZ.html 0000664 0000000 0000000 00000006703 14560147231 0020376 0 ustar 00root root 0000000 0000000
Purpose
To estimate the reciprocal of the condition number of a complex upper Hessenberg matrix H, in either the 1-norm or the infinity-norm, using the LU factorization computed by MB02SZ.Specification
SUBROUTINE MB02TZ( NORM, N, HNORM, H, LDH, IPIV, RCOND, DWORK,
$ ZWORK, INFO )
C .. Scalar Arguments ..
CHARACTER NORM
INTEGER INFO, LDH, N
DOUBLE PRECISION HNORM, RCOND
C .. Array Arguments ..
INTEGER IPIV(*)
DOUBLE PRECISION DWORK( * )
COMPLEX*16 H( LDH, * ), ZWORK( * )
Arguments
Mode Parameters
NORM CHARACTER*1
Specifies whether the 1-norm condition number or the
infinity-norm condition number is required:
= '1' or 'O': 1-norm;
= 'I': Infinity-norm.
Input/Output Parameters
N (input) INTEGER
The order of the matrix H. N >= 0.
HNORM (input) DOUBLE PRECISION
If NORM = '1' or 'O', the 1-norm of the original matrix H.
If NORM = 'I', the infinity-norm of the original matrix H.
H (input) COMPLEX*16 array, dimension (LDH,N)
The factors L and U from the factorization H = P*L*U
as computed by MB02SZ.
LDH INTEGER
The leading dimension of the array H. LDH >= max(1,N).
IPIV (input) INTEGER array, dimension (N)
The pivot indices; for 1 <= i <= N, row i of the matrix
was interchanged with row IPIV(i).
RCOND (output) DOUBLE PRECISION
The reciprocal of the condition number of the matrix H,
computed as RCOND = 1/(norm(H) * norm(inv(H))).
Workspace
DWORK DOUBLE PRECISION array, dimension (N) ZWORK COMPLEX*16 array, dimension (2*N)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
An estimate is obtained for norm(inv(H)), and the reciprocal of
the condition number is computed as
RCOND = 1 / ( norm(H) * norm(inv(H)) ).
References
-Numerical Aspects
2 The algorithm requires 0( N ) complex operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02UD.html 0000664 0000000 0000000 00000023340 14560147231 0020345 0 ustar 00root root 0000000 0000000
Purpose
To compute the minimum norm least squares solution of one of the
following linear systems
op(R)*X = alpha*B, (1)
X*op(R) = alpha*B, (2)
where alpha is a real scalar, op(R) is either R or its transpose,
R', R is an L-by-L real upper triangular matrix, B is an M-by-N
real matrix, and L = M for (1), or L = N for (2). Singular value
decomposition, R = Q*S*P', is used, assuming that R is rank
deficient.
Specification
SUBROUTINE MB02UD( FACT, SIDE, TRANS, JOBP, M, N, ALPHA, RCOND,
$ RANK, R, LDR, Q, LDQ, SV, B, LDB, RP, LDRP,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER FACT, JOBP, SIDE, TRANS
INTEGER INFO, LDB, LDQ, LDR, LDRP, LDWORK, M, N, RANK
DOUBLE PRECISION ALPHA, RCOND
C .. Array Arguments ..
DOUBLE PRECISION B(LDB,*), DWORK(*), Q(LDQ,*), R(LDR,*),
$ RP(LDRP,*), SV(*)
Arguments
Mode Parameters
FACT CHARACTER*1
Specifies whether R has been previously factored or not,
as follows:
= 'F': R has been factored and its rank and singular
value decomposition, R = Q*S*P', are available;
= 'N': R has not been factored and its singular value
decomposition, R = Q*S*P', should be computed.
SIDE CHARACTER*1
Specifies whether op(R) appears on the left or right
of X as follows:
= 'L': Solve op(R)*X = alpha*B (op(R) is on the left);
= 'R': Solve X*op(R) = alpha*B (op(R) is on the right).
TRANS CHARACTER*1
Specifies the form of op(R) to be used as follows:
= 'N': op(R) = R;
= 'T': op(R) = R';
= 'C': op(R) = R'.
JOBP CHARACTER*1
Specifies whether or not the pseudoinverse of R is to be
computed or it is available as follows:
= 'P': Compute pinv(R), if FACT = 'N', or
use pinv(R), if FACT = 'F';
= 'N': Do not compute or use pinv(R).
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix B. M >= 0.
N (input) INTEGER
The number of columns of the matrix B. N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then B need not be
set before entry.
RCOND (input) DOUBLE PRECISION
RCOND is used to determine the effective rank of R.
Singular values of R satisfying Sv(i) <= RCOND*Sv(1) are
treated as zero. If RCOND <= 0, then EPS is used instead,
where EPS is the relative machine precision (see LAPACK
Library routine DLAMCH). RCOND <= 1.
RCOND is not used if FACT = 'F'.
RANK (input or output) INTEGER
The rank of matrix R.
RANK is an input parameter when FACT = 'F', and an output
parameter when FACT = 'N'. L >= RANK >= 0.
R (input/output) DOUBLE PRECISION array, dimension (LDR,L)
On entry, if FACT = 'F', the leading L-by-L part of this
array must contain the L-by-L orthogonal matrix P' from
singular value decomposition, R = Q*S*P', of the matrix R;
if JOBP = 'P', the first RANK rows of P' are assumed to be
scaled by inv(S(1:RANK,1:RANK)).
On entry, if FACT = 'N', the leading L-by-L upper
triangular part of this array must contain the upper
triangular matrix R.
On exit, if INFO = 0, the leading L-by-L part of this
array contains the L-by-L orthogonal matrix P', with its
first RANK rows scaled by inv(S(1:RANK,1:RANK)), when
JOBP = 'P'.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,L).
Q (input or output) DOUBLE PRECISION array, dimension
(LDQ,L)
On entry, if FACT = 'F', the leading L-by-L part of this
array must contain the L-by-L orthogonal matrix Q from
singular value decomposition, R = Q*S*P', of the matrix R.
If FACT = 'N', this array need not be set on entry, and
on exit, if INFO = 0, the leading L-by-L part of this
array contains the orthogonal matrix Q.
LDQ INTEGER
The leading dimension of array Q. LDQ >= MAX(1,L).
SV (input or output) DOUBLE PRECISION array, dimension (L)
On entry, if FACT = 'F', the first RANK entries of this
array must contain the reciprocal of the largest RANK
singular values of the matrix R, and the last L-RANK
entries of this array must contain the remaining singular
values of R sorted in descending order.
If FACT = 'N', this array need not be set on input, and
on exit, if INFO = 0, the first RANK entries of this array
contain the reciprocal of the largest RANK singular values
of the matrix R, and the last L-RANK entries of this array
contain the remaining singular values of R sorted in
descending order.
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, if ALPHA <> 0, the leading M-by-N part of this
array must contain the matrix B.
On exit, if INFO = 0 and RANK > 0, the leading M-by-N part
of this array contains the M-by-N solution matrix X.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,M).
RP (input or output) DOUBLE PRECISION array, dimension
(LDRP,L)
On entry, if FACT = 'F', JOBP = 'P', and RANK > 0, the
leading L-by-L part of this array must contain the L-by-L
matrix pinv(R), the Moore-Penrose pseudoinverse of R.
On exit, if FACT = 'N', JOBP = 'P', and RANK > 0, the
leading L-by-L part of this array contains the L-by-L
matrix pinv(R), the Moore-Penrose pseudoinverse of R.
If JOBP = 'N', this array is not referenced.
LDRP INTEGER
The leading dimension of array RP.
LDRP >= MAX(1,L), if JOBP = 'P'.
LDRP >= 1, if JOBP = 'N'.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK;
if INFO = i, 1 <= i <= L, then DWORK(2:L) contain the
unconverged superdiagonal elements of an upper bidiagonal
matrix D whose diagonal is in SV (not necessarily sorted).
D satisfies R = Q*D*P', so it has the same singular
values as R, and singular vectors related by Q and P'.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,L), if FACT = 'F';
LDWORK >= MAX(1,5*L), if FACT = 'N'.
For optimum performance LDWORK should be larger than
MAX(1,L,M*N), if FACT = 'F';
MAX(1,5*L,M*N), if FACT = 'N'.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, i = 1:L, the SVD algorithm has failed
to converge. In this case INFO specifies how many
superdiagonals did not converge (see the description
of DWORK); this failure is not likely to occur.
Method
The L-by-L upper triangular matrix R is factored as R = Q*S*P', if FACT = 'N', using SLICOT Library routine MB03UD, where Q and P are L-by-L orthogonal matrices and S is an L-by-L diagonal matrix with non-negative diagonal elements, SV(1), SV(2), ..., SV(L), ordered decreasingly. Then, the effective rank of R is estimated, and matrix (or matrix-vector) products and scalings are used to compute X. If FACT = 'F', only matrix (or matrix-vector) products and scalings are performed.Further Comments
Option JOBP = 'P' should be used only if the pseudoinverse is really needed. Usually, it is possible to avoid the use of pseudoinverse, by computing least squares solutions. The routine uses BLAS 3 calculations if LDWORK >= M*N, and BLAS 2 calculations, otherwise. No advantage of any additional workspace larger than L is taken for matrix products, but the routine can be called repeatedly for chunks of columns of B, if LDWORK < M*N.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02UU.html 0000664 0000000 0000000 00000005574 14560147231 0020377 0 ustar 00root root 0000000 0000000
Purpose
To solve for x in A * x = scale * RHS, using the LU factorization of the N-by-N matrix A computed by SLICOT Library routine MB02UV. The factorization has the form A = P * L * U * Q, where P and Q are permutation matrices, L is unit lower triangular and U is upper triangular.Specification
SUBROUTINE MB02UU( N, A, LDA, RHS, IPIV, JPIV, SCALE )
C .. Scalar Arguments ..
INTEGER LDA, N
DOUBLE PRECISION SCALE
C .. Array Arguments ..
INTEGER IPIV( * ), JPIV( * )
DOUBLE PRECISION A( LDA, * ), RHS( * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix A.
A (input) DOUBLE PRECISION array, dimension (LDA, N)
The leading N-by-N part of this array must contain
the LU part of the factorization of the matrix A computed
by SLICOT Library routine MB02UV: A = P * L * U * Q.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1, N).
RHS (input/output) DOUBLE PRECISION array, dimension (N)
On entry, this array must contain the right hand side
of the system.
On exit, this array contains the solution of the system.
IPIV (input) INTEGER array, dimension (N)
The pivot indices; for 1 <= i <= N, row i of the
matrix has been interchanged with row IPIV(i).
JPIV (input) INTEGER array, dimension (N)
The pivot indices; for 1 <= j <= N, column j of the
matrix has been interchanged with column JPIV(j).
SCALE (output) DOUBLE PRECISION
The scale factor, chosen 0 < SCALE <= 1 to prevent
overflow in the solution.
Further Comments
In the interest of speed, this routine does not check the input for errors. It should only be used if the order of the matrix A is very small.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute an LU factorization, using complete pivoting, of the N-by-N matrix A. The factorization has the form A = P * L * U * Q, where P and Q are permutation matrices, L is lower triangular with unit diagonal elements and U is upper triangular.Specification
SUBROUTINE MB02UV( N, A, LDA, IPIV, JPIV, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, N
C .. Array Arguments ..
INTEGER IPIV( * ), JPIV( * )
DOUBLE PRECISION A( LDA, * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix A.
A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
On entry, the leading N-by-N part of this array must
contain the matrix A to be factored.
On exit, the leading N-by-N part of this array contains
the factors L and U from the factorization A = P*L*U*Q;
the unit diagonal elements of L are not stored. If U(k, k)
appears to be less than SMIN, U(k, k) is given the value
of SMIN, giving a nonsingular perturbed system.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1, N).
IPIV (output) INTEGER array, dimension (N)
The pivot indices; for 1 <= i <= N, row i of the
matrix has been interchanged with row IPIV(i).
JPIV (output) INTEGER array, dimension (N)
The pivot indices; for 1 <= j <= N, column j of the
matrix has been interchanged with column JPIV(j).
Error Indicator
INFO INTEGER
= 0: successful exit;
= k: U(k, k) is likely to produce owerflow if one tries
to solve for x in Ax = b. So U is perturbed to get
a nonsingular system. This is a warning.
Further Comments
In the interests of speed, this routine does not check the input for errors. It should only be used to factorize matrices A of very small order.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve a system of the form A X = s B or A' X = s B with
possible scaling ("s") and perturbation of A. (A' means
A-transpose.) A is an N-by-N real matrix, and X and B are
N-by-M matrices. N may be 1 or 2. The scalar "s" is a scaling
factor (.LE. 1), computed by this subroutine, which is so chosen
that X can be computed without overflow. X is further scaled if
necessary to assure that norm(A)*norm(X) is less than overflow.
Specification
SUBROUTINE MB02UW( LTRANS, N, M, PAR, A, LDA, B, LDB, SCALE,
$ IWARN )
C .. Scalar Arguments ..
LOGICAL LTRANS
INTEGER IWARN, LDA, LDB, N, M
DOUBLE PRECISION SCALE, SMIN
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), PAR( * )
Arguments
Mode Parameters
LTRANS LOGICAL
Specifies if A or A-transpose is to be used, as follows:
=.TRUE. : A-transpose will be used;
=.FALSE.: A will be used (not transposed).
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. It may (only) be 1 or 2.
M (input) INTEGER
The number of right hand size vectors.
PAR (input) DOUBLE PRECISION array, dimension (3)
Machine related parameters:
PAR(1) =: PREC (machine precision)*base, DLAMCH( 'P' );
PAR(2) =: SFMIN safe minimum, DLAMCH( 'S' );
PAR(3) =: SMIN The desired lower bound on the singular
values of A. This should be a safe
distance away from underflow or overflow,
say, between (underflow/machine precision)
and (machine precision * overflow).
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= N.
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the matrix B (right-hand side).
On exit, the leading N-by-M part of this array contains
the N-by-M matrix X (unknowns).
LDB INTEGER
The leading dimension of the array B. LDB >= N.
SCALE (output) DOUBLE PRECISION
The scale factor that B must be multiplied by to insure
that overflow does not occur when computing X. Thus,
A X will be SCALE*B, not B (ignoring perturbations of A).
SCALE will be at most 1.
Warning Indicator
IWARN INTEGER
= 0: no warnings (A did not have to be perturbed);
= 1: A had to be perturbed to make its smallest (or only)
singular value greater than SMIN (see below).
Method
Gaussian elimination with complete pivoting is used. The matrix A is slightly perturbed if it is (close to being) singular.Further Comments
If both singular values of A are less than SMIN, SMIN*identity will be used instead of A. If only one singular value is less than SMIN, one element of A will be perturbed enough to make the smallest singular value roughly SMIN. If both singular values are at least SMIN, A will not be perturbed. In any case, the perturbation will be at most some small multiple of max( SMIN, EPS*norm(A) ), where EPS is the machine precision (see LAPACK Library routine DLAMCH). The singular values are computed by infinity-norm approximations, and thus will only be correct to a factor of 2 or so. Note: all input quantities are assumed to be smaller than overflow by a reasonable factor. (See BIGNUM.) In the interests of speed, this routine does not check the inputs for errors.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the solution to a real system of linear equations
X * op(A) = B,
where op(A) is either A or its transpose, A is an N-by-N matrix,
and X and B are M-by-N matrices.
The LU decomposition with partial pivoting and row interchanges,
A = P * L * U, is used, where P is a permutation matrix, L is unit
lower triangular, and U is upper triangular.
Specification
SUBROUTINE MB02VD( TRANS, M, N, A, LDA, IPIV, B, LDB, INFO )
C .. Scalar Arguments ..
CHARACTER TRANS
INTEGER INFO, LDA, LDB, M, N
C .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * )
Arguments
Mode Parameters
TRANS CHARACTER*1
Specifies the form of op(A) to be used as follows:
= 'N': op(A) = A;
= 'T': op(A) = A';
= 'C': op(A) = A'.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix B. M >= 0.
N (input) INTEGER
The number of columns of the matrix B, and the order of
the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the coefficient matrix A.
On exit, the leading N-by-N part of this array contains
the factors L and U from the factorization A = P*L*U;
the unit diagonal elements of L are not stored.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
IPIV (output) INTEGER array, dimension (N)
The pivot indices that define the permutation matrix P;
row i of the matrix was interchanged with row IPIV(i).
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the leading M-by-N part of this array must
contain the right hand side matrix B.
On exit, if INFO = 0, the leading M-by-N part of this
array contains the solution matrix X.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,M).
INFO (output) INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, U(i,i) is exactly zero. The
factorization has been completed, but the factor U
is exactly singular, so the solution could not be
computed.
Method
The LU decomposition with partial pivoting and row interchanges is
used to factor A as
A = P * L * U,
where P is a permutation matrix, L is unit lower triangular, and
U is upper triangular. The factored form of A is then used to
solve the system of equations X * A = B or X * A' = B.
Further Comments
This routine enables to solve the system X * A = B or X * A' = B as easily and efficiently as possible; it is similar to the LAPACK Library routine DGESV, which solves A * X = B.Example
Program Text
* MB02VD EXAMPLE PROGRAM TEXT
*
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, NMAX
PARAMETER ( MMAX = 20, NMAX = 20 )
INTEGER LDA, LDB
PARAMETER ( LDA = NMAX, LDB = MMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, M, N
CHARACTER*1 TRANS
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,NMAX)
INTEGER IPIV(NMAX)
* .. External Subroutines ..
EXTERNAL MB02VD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read in the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, N, TRANS
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,N ), I = 1,M )
* Solve the linear system using the LU factorization.
CALL MB02VD( TRANS, M, N, A, LDA, IPIV, B, LDB, INFO )
*
IF ( INFO.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, M
WRITE ( NOUT, FMT = 99996 ) ( B(I,J), J = 1,N )
10 CONTINUE
ELSE
WRITE ( NOUT, FMT = 99998 ) INFO
END IF
END IF
END IF
STOP
*
99999 FORMAT (' MB02VD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB02VD = ',I2)
99997 FORMAT (' The solution matrix is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' N is out of range.',/' N = ',I5)
99994 FORMAT (/' M is out of range.',/' M = ',I5)
END
Program Data
MB02VD EXAMPLE PROGRAM DATA 5 4 N 1. 2. 6. 3. -2. -1. -1. 0. 2. 3. 1. 5. 1. -1. 2. 0. 0. 0. 0. 1. 5. 5. 1. 5. -2. 1. 3. 1. 0. 0. 4. 5. 2. 1. 1. 3.Program Results
MB02VD EXAMPLE PROGRAM RESULTS The solution matrix is -0.0690 0.3333 0.2414 0.2529 -0.1724 -1.6667 1.1034 -0.3678 0.9655 0.6667 -0.3793 -0.8736 0.3448 1.6667 0.7931 1.4023 -0.2069 0.0000 0.7241 0.7586
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB02WD.html 0000664 0000000 0000000 00000025046 14560147231 0020354 0 ustar 00root root 0000000 0000000
Purpose
To solve the system of linear equations Ax = b, with A symmetric, positive definite, or, in the implicit form, f(A, x) = b, where y = f(A, x) is a symmetric positive definite linear mapping from x to y, using the conjugate gradient (CG) algorithm without preconditioning.Specification
SUBROUTINE MB02WD( FORM, F, N, IPAR, LIPAR, DPAR, LDPAR, ITMAX,
$ A, LDA, B, INCB, X, INCX, TOL, DWORK, LDWORK,
$ IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER FORM
INTEGER INCB, INCX, INFO, ITMAX, IWARN, LDA, LDPAR,
$ LDWORK, LIPAR, N
DOUBLE PRECISION TOL
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(*), DPAR(*), DWORK(*), X(*)
INTEGER IPAR(*)
Arguments
Mode Parameters
FORM CHARACTER*1
Specifies the form of the system of equations, as
follows:
= 'U' : Ax = b, the upper triagular part of A is used;
= 'L' : Ax = b, the lower triagular part of A is used;
= 'F' : the implicit, function form, f(A, x) = b.
Function Parameters
F EXTERNAL
If FORM = 'F', then F is a subroutine which calculates the
value of f(A, x), for given A and x.
If FORM <> 'F', then F is not called.
F must have the following interface:
SUBROUTINE F( N, IPAR, LIPAR, DPAR, LDPAR, A, LDA, X,
$ INCX, DWORK, LDWORK, INFO )
where
N (input) INTEGER
The dimension of the vector x. N >= 0.
IPAR (input) INTEGER array, dimension (LIPAR)
The integer parameters describing the structure of
the matrix A.
LIPAR (input) INTEGER
The length of the array IPAR. LIPAR >= 0.
DPAR (input) DOUBLE PRECISION array, dimension (LDPAR)
The real parameters needed for solving the
problem.
LDPAR (input) INTEGER
The length of the array DPAR. LDPAR >= 0.
A (input) DOUBLE PRECISION array, dimension
(LDA, NC), where NC is the number of columns.
The leading NR-by-NC part of this array must
contain the (compressed) representation of the
matrix A, where NR is the number of rows of A
(function of IPAR entries).
LDA (input) INTEGER
The leading dimension of the array A.
LDA >= MAX(1,NR).
X (input/output) DOUBLE PRECISION array, dimension
(1+(N-1)*INCX)
On entry, this incremented array must contain the
vector x.
On exit, this incremented array contains the value
of the function f, y = f(A, x).
INCX (input) INTEGER
The increment for the elements of X. INCX > 0.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
The workspace array for subroutine F.
LDWORK (input) INTEGER
The size of the array DWORK (as large as needed
in the subroutine F).
INFO INTEGER
Error indicator, set to a negative value if an
input scalar argument is erroneous, and to
positive values for other possible errors in the
subroutine F. The LAPACK Library routine XERBLA
should be used in conjunction with negative INFO.
INFO must be zero if the subroutine finished
successfully.
Parameters marked with "(input)" must not be changed.
Input/Output Parameters
N (input) INTEGER
The dimension of the vector x. N >= 0.
If FORM = 'U' or FORM = 'L', N is also the number of rows
and columns of the matrix A.
IPAR (input) INTEGER array, dimension (LIPAR)
If FORM = 'F', the integer parameters describing the
structure of the matrix A.
This parameter is ignored if FORM = 'U' or FORM = 'L'.
LIPAR (input) INTEGER
The length of the array IPAR. LIPAR >= 0.
DPAR (input) DOUBLE PRECISION array, dimension (LDPAR)
If FORM = 'F', the real parameters needed for solving
the problem.
This parameter is ignored if FORM = 'U' or FORM = 'L'.
LDPAR (input) INTEGER
The length of the array DPAR. LDPAR >= 0.
ITMAX (input) INTEGER
The maximal number of iterations to do. ITMAX >= 0.
A (input) DOUBLE PRECISION array,
dimension (LDA, NC), if FORM = 'F',
dimension (LDA, N), otherwise.
If FORM = 'F', the leading NR-by-NC part of this array
must contain the (compressed) representation of the
matrix A, where NR and NC are the number of rows and
columns, respectively, of the matrix A. The array A is
not referenced by this routine itself, except in the
calls to the routine F.
If FORM <> 'F', the leading N-by-N part of this array
must contain the matrix A, assumed to be symmetric;
only the triangular part specified by FORM is referenced.
LDA (input) INTEGER
The leading dimension of array A.
LDA >= MAX(1,NR), if FORM = 'F';
LDA >= MAX(1,N), if FORM = 'U' or FORM = 'L'.
B (input) DOUBLE PRECISION array, dimension (1+(N-1)*INCB)
The incremented vector b.
INCB (input) INTEGER
The increment for the elements of B. INCB > 0.
X (input/output) DOUBLE PRECISION array, dimension
(1+(N-1)*INCX)
On entry, this incremented array must contain an initial
approximation of the solution. If an approximation is not
known, setting all elements of x to zero is recommended.
On exit, this incremented array contains the computed
solution x of the system of linear equations.
INCX (input) INTEGER
The increment for the elements of X. INCX > 0.
Tolerances
TOL DOUBLE PRECISION
If TOL > 0, absolute tolerance for the iterative process.
The algorithm will stop if || Ax - b ||_2 <= TOL. Since
it is advisable to use a relative tolerance, say TOLER,
TOL should be chosen as TOLER*|| b ||_2.
If TOL <= 0, a default relative tolerance,
TOLDEF = N*EPS*|| b ||_2, is used, where EPS is the
machine precision (see LAPACK Library routine DLAMCH).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the number of
iterations performed and DWORK(2) returns the remaining
residual, || Ax - b ||_2.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(2,3*N + DWORK(F)), if FORM = 'F',
where DWORK(F) is the workspace needed by F;
LDWORK >= MAX(2,3*N), if FORM = 'U' or FORM = 'L'.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: the algorithm finished after ITMAX > 0 iterations,
without achieving the desired precision TOL;
= 2: ITMAX is zero; in this case, DWORK(2) is not set.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, then F returned with INFO = i.
Method
The following CG iteration is used for solving Ax = b:
Start: q(0) = r(0) = Ax - b
< q(k), r(k) >
ALPHA(k) = - ----------------
< q(k), Aq(k) >
x(k+1) = x(k) - ALPHA(k) * q(k)
r(k+1) = r(k) - ALPHA(k) * Aq(k)
< r(k+1), r(k+1) >
BETA(k) = --------------------
< r(k) , r(k) >
q(k+1) = r(k+1) + BETA(k) * q(k)
where <.,.> denotes the scalar product.
References
[1] Golub, G.H. and van Loan, C.F.
Matrix Computations. Third Edition.
M. D. Johns Hopkins University Press, Baltimore, pp. 520-528,
1996.
[2] Luenberger, G.
Introduction to Linear and Nonlinear Programming.
Addison-Wesley, Reading, MA, p.187, York, 1973.
Numerical Aspects
Since the residuals are orthogonal in the scalar product
<x, y> = y'Ax, the algorithm is theoretically finite. But rounding
errors cause a loss of orthogonality, so a finite termination
cannot be guaranteed. However, one can prove [2] that
|| x-x_k ||_A := sqrt( (x-x_k)' * A * (x-x_k) )
sqrt( kappa_2(A) ) - 1
<= 2 || x-x_0 ||_A * ------------------------ ,
sqrt( kappa_2(A) ) + 1
where kappa_2 is the condition number.
The approximate number of floating point operations is
(k*(N**2 + 15*N) + N**2 + 3*N)/2, if FORM <> 'F',
k*(f + 7*N) + f, if FORM = 'F',
where k is the number of CG iterations performed, and f is the
number of floating point operations required by the subroutine F.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve a set of systems of linear equations, A'*A*X = B, or, in the implicit form, f(A)*X = B, with A'*A or f(A) positive definite, using symmetric Gaussian elimination.Specification
SUBROUTINE MB02XD( FORM, STOR, UPLO, F, M, N, NRHS, IPAR, LIPAR,
$ DPAR, LDPAR, A, LDA, B, LDB, ATA, LDATA, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER FORM, STOR, UPLO
INTEGER INFO, LDA, LDATA, LDB, LDPAR, LDWORK, LIPAR, M,
$ N, NRHS
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), ATA(*), B(LDB,*), DPAR(*), DWORK(*)
INTEGER IPAR(*)
Arguments
Mode Parameters
FORM CHARACTER*1
Specifies the form in which the matrix A is provided, as
follows:
= 'S' : standard form, the matrix A is given;
= 'F' : the implicit, function form f(A) is provided.
If FORM = 'F', then the routine F is called to compute the
matrix A'*A.
STOR CHARACTER*1
Specifies the storage scheme for the symmetric
matrix A'*A, as follows:
= 'F' : full storage is used;
= 'P' : packed storage is used.
UPLO CHARACTER*1
Specifies which part of the matrix A'*A is stored, as
follows:
= 'U' : the upper triagular part is stored;
= 'L' : the lower triagular part is stored.
Function Parameters
F EXTERNAL
If FORM = 'F', then F is a subroutine which calculates the
value of f(A) = A'*A, for given A.
If FORM = 'S', then F is not called.
F must have the following interface:
SUBROUTINE F( STOR, UPLO, N, IPAR, LIPAR, DPAR, LDPAR, A,
$ LDA, ATA, LDATA, DWORK, LDWORK, INFO )
where
STOR (input) CHARACTER*1
Specifies the storage scheme for the symmetric
matrix A'*A, as follows:
= 'F' : full storage is used;
= 'P' : packed storage is used.
UPLO (input) CHARACTER*1
Specifies which part of the matrix A'*A is stored,
as follows:
= 'U' : the upper triagular part is stored;
= 'L' : the lower triagular part is stored.
N (input) INTEGER
The order of the matrix A'*A. N >= 0.
IPAR (input) INTEGER array, dimension (LIPAR)
The integer parameters describing the structure of
the matrix A.
LIPAR (input) INTEGER
The length of the array IPAR. LIPAR >= 0.
DPAR (input) DOUBLE PRECISION array, dimension (LDPAR)
The real parameters needed for solving the
problem.
LDPAR (input) INTEGER
The length of the array DPAR. LDPAR >= 0.
A (input) DOUBLE PRECISION array, dimension
(LDA, NC), where NC is the number of columns.
The leading NR-by-NC part of this array must
contain the (compressed) representation of the
matrix A, where NR is the number of rows of A
(function of IPAR entries).
LDA (input) INTEGER
The leading dimension of the array A.
LDA >= MAX(1,NR).
ATA (output) DOUBLE PRECISION array,
dimension (LDATA,N), if STOR = 'F',
dimension (N*(N+1)/2), if STOR = 'P'.
The leading N-by-N (if STOR = 'F'), or N*(N+1)/2
(if STOR = 'P') part of this array contains the
upper or lower triangle of the matrix A'*A,
depending on UPLO = 'U', or UPLO = 'L',
respectively, stored either as a two-dimensional,
or one-dimensional array, depending on STOR.
LDATA (input) INTEGER
The leading dimension of the array ATA.
LDATA >= MAX(1,N), if STOR = 'F'.
LDATA >= 1, if STOR = 'P'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
The workspace array for subroutine F.
LDWORK (input) INTEGER
The size of the array DWORK (as large as needed
in the subroutine F).
INFO INTEGER
Error indicator, set to a negative value if an
input scalar argument is erroneous, and to
positive values for other possible errors in the
subroutine F. The LAPACK Library routine XERBLA
should be used in conjunction with negative INFO.
INFO must be zero if the subroutine finished
successfully.
Parameters marked with "(input)" must not be changed.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The order of the matrix A'*A, the number of columns of the
matrix A, and the number of rows of the matrix X. N >= 0.
NRHS (input) INTEGER
The number of columns of the matrices B and X. NRHS >= 0.
IPAR (input) INTEGER array, dimension (LIPAR)
If FORM = 'F', the integer parameters describing the
structure of the matrix A.
This parameter is ignored if FORM = 'S'.
LIPAR (input) INTEGER
The length of the array IPAR. LIPAR >= 0.
DPAR (input) DOUBLE PRECISION array, dimension (LDPAR)
If FORM = 'F', the real parameters needed for solving
the problem.
This parameter is ignored if FORM = 'S'.
LDPAR (input) INTEGER
The length of the array DPAR. LDPAR >= 0.
A (input) DOUBLE PRECISION array,
dimension (LDA, N), if FORM = 'S',
dimension (LDA, NC), if FORM = 'F', where NC is
the number of columns.
If FORM = 'S', the leading M-by-N part of this array
must contain the matrix A.
If FORM = 'F', the leading NR-by-NC part of this array
must contain an appropriate representation of matrix A,
where NR is the number of rows.
If FORM = 'F', this array is not referenced by this
routine itself, except in the call to the routine F.
LDA INTEGER
The leading dimension of array A.
LDA >= MAX(1,M), if FORM = 'S';
LDA >= MAX(1,NR), if FORM = 'F'.
B (input/output) DOUBLE PRECISION array, dimension
(LDB, NRHS)
On entry, the leading N-by-NRHS part of this array must
contain the right hand side matrix B.
On exit, if INFO = 0 and M (or NR) is nonzero, the leading
N-by-NRHS part of this array contains the solution X of
the set of systems of linear equations A'*A*X = B or
f(A)*X = B. If M (or NR) is zero, then B is unchanged.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
ATA (output) DOUBLE PRECISION array,
dimension (LDATA,N), if STOR = 'F',
dimension (N*(N+1)/2), if STOR = 'P'.
The leading N-by-N (if STOR = 'F'), or N*(N+1)/2 (if
STOR = 'P') part of this array contains the upper or lower
triangular Cholesky factor of the matrix A'*A, depending
on UPLO = 'U', or UPLO = 'L', respectively, stored either
as a two-dimensional, or one-dimensional array, depending
on STOR.
LDATA INTEGER
The leading dimension of the array ATA.
LDATA >= MAX(1,N), if STOR = 'F'.
LDATA >= 1, if STOR = 'P'.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, then the (i,i) element of the
triangular factor of the matrix A'*A is exactly
zero (the matrix A'*A is exactly singular);
if INFO = j > n, then F returned with INFO = j-n.
Method
The matrix A'*A is built either directly (if FORM = 'S'), or implicitly, by calling the routine F. Then, A'*A is Cholesky factored and its factor is used to solve the set of systems of linear equations, A'*A*X = B.References
[1] Golub, G.H. and van Loan, C.F.
Matrix Computations. Third Edition.
M. D. Johns Hopkins University Press, Baltimore, 1996.
[2] Anderson, E., Bai, Z., Bischof, C., Blackford, Demmel, J.,
Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S.,
McKenney, A., Sorensen, D.
LAPACK Users' Guide: Third Edition, SIAM, Philadelphia, 1999.
Numerical Aspects
For speed, this routine does not check for near singularity of the
matrix A'*A. If the matrix A is nearly rank deficient, then the
computed X could be inaccurate. Estimates of the reciprocal
condition numbers of the matrices A and A'*A can be obtained
using LAPACK routines DGECON and DPOCON (DPPCON), respectively.
The approximate number of floating point operations is
(M+3)*N**2/2 + N**3/6 + NRHS*N**2, if FORM = 'S',
f + N**3/6 + NRHS*N**2, if FORM = 'F',
where M is the number of rows of A, and f is the number of
floating point operations required by the subroutine F.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To determine a vector x which solves the system of linear
equations
A*x = b , D*x = 0 ,
in the least squares sense, where A is an m-by-n matrix,
D is an n-by-n diagonal matrix, and b is an m-vector.
It is assumed that a QR factorization, with column pivoting, of A
is available, that is, A*P = Q*R, where P is a permutation matrix,
Q has orthogonal columns, and R is an upper triangular matrix
with diagonal elements of nonincreasing magnitude.
The routine needs the full upper triangle of R, the permutation
matrix P, and the first n components of Q'*b (' denotes the
transpose). The system A*x = b, D*x = 0, is then equivalent to
R*z = Q'*b , P'*D*P*z = 0 , (1)
where x = P*z. If this system does not have full rank, then a
least squares solution is obtained. On output, MB02YD also
provides an upper triangular matrix S such that
P'*(A'*A + D*D)*P = S'*S .
The system (1) is equivalent to S*z = c , where c contains the
first n components of the vector obtained by applying to
[ (Q'*b)' 0 ]' the transformations which triangularized
[ R' P'*D*P ]', getting S.
Specification
SUBROUTINE MB02YD( COND, N, R, LDR, IPVT, DIAG, QTB, RANK, X, TOL,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COND
INTEGER INFO, LDR, LDWORK, N, RANK
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IPVT(*)
DOUBLE PRECISION DIAG(*), DWORK(*), QTB(*), R(LDR,*), X(*)
Arguments
Mode Parameters
COND CHARACTER*1
Specifies whether the condition of the matrix S should be
estimated, as follows:
= 'E' : use incremental condition estimation and store
the numerical rank of S in RANK;
= 'N' : do not use condition estimation, but check the
diagonal entries of S for zero values;
= 'U' : use the rank already stored in RANK.
Input/Output Parameters
N (input) INTEGER
The order of the matrix R. N >= 0.
R (input/output) DOUBLE PRECISION array, dimension (LDR, N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix R.
On exit, the full upper triangle is unaltered, and the
strict lower triangle contains the strict upper triangle
(transposed) of the upper triangular matrix S.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
IPVT (input) INTEGER array, dimension (N)
This array must define the permutation matrix P such that
A*P = Q*R. Column j of P is column IPVT(j) of the identity
matrix.
DIAG (input) DOUBLE PRECISION array, dimension (N)
This array must contain the diagonal elements of the
matrix D.
QTB (input) DOUBLE PRECISION array, dimension (N)
This array must contain the first n elements of the
vector Q'*b.
RANK (input or output) INTEGER
On entry, if COND = 'U', this parameter must contain the
(numerical) rank of the matrix S.
On exit, if COND = 'E' or 'N', this parameter contains
the numerical rank of the matrix S, estimated according
to the value of COND.
X (output) DOUBLE PRECISION array, dimension (N)
This array contains the least squares solution of the
system A*x = b, D*x = 0.
Tolerances
TOL DOUBLE PRECISION
If COND = 'E', the tolerance to be used for finding the
rank of the matrix S. If the user sets TOL > 0, then the
given value of TOL is used as a lower bound for the
reciprocal condition number; a (sub)matrix whose
estimated condition number is less than 1/TOL is
considered to be of full rank. If the user sets TOL <= 0,
then an implicitly computed, default tolerance, defined by
TOLDEF = N*EPS, is used instead, where EPS is the machine
precision (see LAPACK Library routine DLAMCH).
This parameter is not relevant if COND = 'U' or 'N'.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, the first N elements of this array contain the
diagonal elements of the upper triangular matrix S, and
the next N elements contain the solution z.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 4*N, if COND = 'E';
LDWORK >= 2*N, if COND <> 'E'.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Standard plane rotations are used to annihilate the elements of the diagonal matrix D, updating the upper triangular matrix R and the first n elements of the vector Q'*b. A basic least squares solution is computed.References
[1] More, J.J., Garbow, B.S, and Hillstrom, K.E.
User's Guide for MINPACK-1.
Applied Math. Division, Argonne National Laboratory, Argonne,
Illinois, Report ANL-80-74, 1980.
Numerical Aspects
2 The algorithm requires 0(N ) operations and is backward stable.Further Comments
This routine is a LAPACK-based modification of QRSOLV from the MINPACK package [1], and with optional condition estimation. The option COND = 'U' is useful when dealing with several right-hand side vectors.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute two Givens rotations (C1,S1) and (C2,S2) such that the
orthogonal matrix
[ Q 0 ] [ C1 S1 0 ] [ 1 0 0 ]
Z = [ ], Q := [ -S1 C1 0 ] * [ 0 C2 S2 ],
[ 0 I ] [ 0 0 1 ] [ 0 -S2 C2 ]
makes the first column of the real Wilkinson double shift
polynomial of the product of matrices in periodic upper Hessenberg
form, stored in the array A, parallel to the first unit vector.
Only the rotation defined by C1 and S1 is needed for the real
Wilkinson single shift polynomial (see the SLICOT Library routines
MB03BE or MB03BF). The shifts are defined based on the eigenvalues
(computed externally by the SLICOT Library routine MB03BB) of the
trailing 2-by-2 submatrix of the matrix product. See the
definitions of the arguments W1 and W2.
Specification
SUBROUTINE MB03AB( SHFT, K, N, AMAP, S, SINV, A, LDA1, LDA2, W1,
$ W2, C1, S1, C2, S2 )
C .. Scalar Arguments ..
CHARACTER SHFT
INTEGER K, LDA1, LDA2, N, SINV
DOUBLE PRECISION C1, C2, S1, S2, W1, W2
C .. Array Arguments ..
INTEGER AMAP(*), S(*)
DOUBLE PRECISION A(LDA1,LDA2,*)
Arguments
Mode Parameters
SHFT CHARACTER*1
Specifies the number and type of shifts employed by the
shift polynomial, as follows:
= 'C': two complex conjugate shifts;
= 'D': two real identical shifts;
= 'R': two real shifts;
= 'S': one real shift.
When the eigenvalues are complex conjugate, this argument
must be set to 'C'.
Input/Output Parameters
K (input) INTEGER
The number of factors. K >= 1.
N (input) INTEGER
The order of the factors in the array A.
N >= 2, for a single shift polynomial;
N >= 3, for a double shift polynomial.
AMAP (input) INTEGER array, dimension (K)
The map for accessing the factors, i.e., if AMAP(I) = J,
then the factor A_I is stored at the J-th position in A.
AMAP(1) is the pointer to the Hessenberg matrix, defined
by A(:,:,AMAP(1)).
S (input) INTEGER array, dimension (K)
The signature array. Each entry of S must be 1 or -1.
SINV (input) INTEGER
Signature multiplier. Entries of S are virtually
multiplied by SINV.
A (input) DOUBLE PRECISION array, dimension (LDA1,LDA2,K)
The leading N-by-N-by-K part of this array must contain an
n-by-n product (implicitly represented by its K factors)
in periodic upper Hessenberg form.
LDA1 INTEGER
The first leading dimension of the array A. LDA1 >= N.
LDA2 INTEGER
The second leading dimension of the array A. LDA2 >= N.
W1 (input) DOUBLE PRECISION
The real part of the first eigenvalue.
If SHFT = 'S', this argument is not used.
W2 (input) DOUBLE PRECISION
The second eigenvalue, if both eigenvalues are real, else
the imaginary part of the complex conjugate pair.
C1 (output) DOUBLE PRECISION
S1 (output) DOUBLE PRECISION
On exit, C1 and S1 contain the parameters for the first
Givens rotation.
C2 (output) DOUBLE PRECISION
S2 (output) DOUBLE PRECISION
On exit, C2 and S2 contain the parameters for the second
Givens rotation. If SHFT = 'S', C2 = 1, S2 = 0.
Method
Givens rotations are properly computed and applied.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute two Givens rotations (C1,S1) and (C2,S2) such that the
orthogonal matrix
[ Q 0 ] [ C1 S1 0 ] [ 1 0 0 ]
Z = [ ], Q := [ -S1 C1 0 ] * [ 0 C2 S2 ],
[ 0 I ] [ 0 0 1 ] [ 0 -S2 C2 ]
makes the first column of the real Wilkinson double shift
polynomial of the product of matrices in periodic upper Hessenberg
form, stored in the array A, parallel to the first unit vector.
Only the rotation defined by C1 and S1 is used for the real
Wilkinson single shift polynomial (see SLICOT Library routine
MB03BE).
Specification
SUBROUTINE MB03AD( SHFT, K, N, AMAP, S, SINV, A, LDA1, LDA2, C1,
$ S1, C2, S2 )
C .. Scalar Arguments ..
CHARACTER SHFT
INTEGER K, LDA1, LDA2, N, SINV
DOUBLE PRECISION C1, C2, S1, S2
C .. Array Arguments ..
INTEGER AMAP(*), S(*)
DOUBLE PRECISION A(LDA1,LDA2,*)
Arguments
Mode Parameters
SHFT CHARACTER*1
Specifies the number of shifts employed by the shift
polynomial, as follows:
= 'D': two shifts (assumes N > 2);
= 'S': one real shift.
Input/Output Parameters
K (input) INTEGER
The number of factors. K >= 1.
N (input) INTEGER
The order of the factors in the array A.
N >= 2, for a single shift polynomial;
N >= 3, for a double shift polynomial.
AMAP (input) INTEGER array, dimension (K)
The map for accessing the factors, i.e., if AMAP(I) = J,
then the factor A_I is stored at the J-th position in A.
AMAP(1) is the pointer to the Hessenberg matrix.
S (input) INTEGER array, dimension (K)
The signature array. Each entry of S must be 1 or -1.
SINV (input) INTEGER
Signature multiplier. Entries of S are virtually
multiplied by SINV.
A (input) DOUBLE PRECISION array, dimension (LDA1,LDA2,K)
The leading N-by-N-by-K part of this array must contain
the product (implicitly represented by its K factors)
in periodic upper Hessenberg form.
LDA1 INTEGER
The first leading dimension of the array A. LDA1 >= N.
LDA2 INTEGER
The second leading dimension of the array A. LDA2 >= N.
C1 (output) DOUBLE PRECISION
S1 (output) DOUBLE PRECISION
On exit, C1 and S1 contain the parameters for the first
Givens rotation.
C2 (output) DOUBLE PRECISION
S2 (output) DOUBLE PRECISION
On exit, if SHFT = 'D' and N > 2, C2 and S2 contain the
parameters for the second Givens rotation. Otherwise,
C2 = 1, S2 = 0.
Method
Two Givens rotations are properly computed and applied.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute two Givens rotations (C1,S1) and (C2,S2) such that the
orthogonal matrix
[ Q 0 ] [ C1 S1 0 ] [ 1 0 0 ]
Z = [ ], Q := [ -S1 C1 0 ] * [ 0 C2 S2 ],
[ 0 I ] [ 0 0 1 ] [ 0 -S2 C2 ]
makes the first column of the real Wilkinson double shift
polynomial of the product of matrices in periodic upper Hessenberg
form, stored in the array A, parallel to the first unit vector.
Only the rotation defined by C1 and S1 is used for the real
Wilkinson single shift polynomial (see SLICOT Library routines
MB03BE or MB03BF). All factors whose exponents differ from that of
the Hessenberg factor are assumed nonsingular. The trailing 2-by-2
submatrix and the five nonzero elements in the first two columns
of the matrix product are evaluated when a double shift is used.
Specification
SUBROUTINE MB03AE( SHFT, K, N, AMAP, S, SINV, A, LDA1, LDA2, C1,
$ S1, C2, S2 )
C .. Scalar Arguments ..
CHARACTER SHFT
INTEGER K, LDA1, LDA2, N, SINV
DOUBLE PRECISION C1, C2, S1, S2
C .. Array Arguments ..
INTEGER AMAP(*), S(*)
DOUBLE PRECISION A(LDA1,LDA2,*)
Arguments
Mode Parameters
SHFT CHARACTER*1
Specifies the number of shifts employed by the shift
polynomial, as follows:
= 'D': two shifts (assumes N > 2);
= 'S': one real shift.
Input/Output Parameters
K (input) INTEGER
The number of factors. K >= 1.
N (input) INTEGER
The order of the factors. N >= 2.
AMAP (input) INTEGER array, dimension (K)
The map for accessing the factors, i.e., if AMAP(I) = J,
then the factor A_I is stored at the J-th position in A.
AMAP(1) is the pointer to the Hessenberg matrix.
S (input) INTEGER array, dimension (K)
The signature array. Each entry of S must be 1 or -1.
SINV (input) INTEGER
Signature multiplier. Entries of S are virtually
multiplied by SINV.
A (input) DOUBLE PRECISION array, dimension (LDA1,LDA2,K)
The leading N-by-N-by-K part of this array must contain
the product (implicitly represented by its K factors)
in periodic upper Hessenberg form.
LDA1 INTEGER
The first leading dimension of the array A. LDA1 >= N.
LDA2 INTEGER
The second leading dimension of the array A. LDA2 >= N.
C1 (output) DOUBLE PRECISION
S1 (output) DOUBLE PRECISION
On exit, C1 and S1 contain the parameters for the first
Givens rotation.
C2 (output) DOUBLE PRECISION
S2 (output) DOUBLE PRECISION
On exit, if SHFT = 'D' and N > 2, C2 and S2 contain the
parameters for the second Givens rotation. Otherwise,
C2 = 1, S2 = 0.
Method
The necessary elements of the real Wilkinson double/single shift polynomial are computed, and suitable Givens rotations are found. For numerical reasons, this routine should be called when convergence difficulties are encountered.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute two Givens rotations (C1,S1) and (C2,S2) such that the
orthogonal matrix
[ Q 0 ] [ C1 S1 0 ] [ 1 0 0 ]
Z = [ ], Q := [ -S1 C1 0 ] * [ 0 C2 S2 ],
[ 0 I ] [ 0 0 1 ] [ 0 -S2 C2 ]
makes the first column of the real Wilkinson double shift
polynomial of the product of matrices in periodic upper Hessenberg
form, stored in the array A, parallel to the first unit vector.
Only the rotation defined by C1 and S1 is used for the real
Wilkinson single shift polynomial (see SLICOT Library routines
MB03BE or MB03BF).
Specification
SUBROUTINE MB03AF( SHFT, K, N, AMAP, S, SINV, A, LDA1, LDA2, C1,
$ S1, C2, S2 )
C .. Scalar Arguments ..
CHARACTER SHFT
INTEGER K, LDA1, LDA2, N, SINV
DOUBLE PRECISION C1, C2, S1, S2
C .. Array Arguments ..
INTEGER AMAP(*), S(*)
DOUBLE PRECISION A(LDA1,LDA2,*)
Arguments
Mode Parameters
SHFT CHARACTER*1
Specifies the number of shifts employed by the shift
polynomial, as follows:
= 'D': two real shifts;
= 'S': one real shift.
Input/Output Parameters
K (input) INTEGER
The number of factors. K >= 1.
N (input) INTEGER
The order of the factors in the array A.
N >= 2, for a single shift polynomial;
N >= 3, for a double shift polynomial.
AMAP (input) INTEGER array, dimension (K)
The map for accessing the factors, i.e., if AMAP(I) = J,
then the factor A_I is stored at the J-th position in A.
AMAP(K) is the pointer to the Hessenberg matrix.
S (input) INTEGER array, dimension (K)
The signature array. Each entry of S must be 1 or -1.
S(K) is not used, but assumed to be 1.
SINV (input) INTEGER
Signature multiplier. Entries of S are virtually
multiplied by SINV.
A (input) DOUBLE PRECISION array, dimension (LDA1,LDA2,K)
On entry, the leading N-by-N-by-K part of this array must
contain a n-by-n product (implicitly represented by its K
factors) in periodic upper Hessenberg form. The Hessenberg
matrix is assumed to be the last one in the product.
LDA1 INTEGER
The first leading dimension of the array A. LDA1 >= N.
LDA2 INTEGER
The second leading dimension of the array A. LDA2 >= N.
C1 (output) DOUBLE PRECISION
S1 (output) DOUBLE PRECISION
On exit, C1 and S1 contain the parameters for the first
Givens rotation.
C2 (output) DOUBLE PRECISION
S2 (output) DOUBLE PRECISION
On exit, if SHFT = 'D', C2 and S2 contain the parameters
for the second Givens rotation. Otherwise, C2 = 1, S2 = 0.
Method
Givens rotations are properly computed and applied.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute two Givens rotations (C1,S1) and (C2,S2) such that the
orthogonal matrix
[ Q 0 ] [ C1 S1 0 ] [ 1 0 0 ]
Z = [ ], Q := [ -S1 C1 0 ] * [ 0 C2 S2 ],
[ 0 I ] [ 0 0 1 ] [ 0 -S2 C2 ]
makes the first column of the real Wilkinson double shift
polynomial of the product of matrices in periodic upper Hessenberg
form, stored in the array A, parallel to the first unit vector.
Only the rotation defined by C1 and S1 is used for the real
Wilkinson single shift polynomial (see SLICOT Library routines
MB03BE or MB03BF). All factors whose exponents differ from that of
the Hessenberg factor are assumed nonsingular. The matrix product
is evaluated.
Specification
SUBROUTINE MB03AG( SHFT, K, N, AMAP, S, SINV, A, LDA1, LDA2, C1,
$ S1, C2, S2, IWORK, DWORK )
C .. Scalar Arguments ..
CHARACTER SHFT
INTEGER K, LDA1, LDA2, N, SINV
DOUBLE PRECISION C1, C2, S1, S2
C .. Array Arguments ..
INTEGER AMAP(*), IWORK(*), S(*)
DOUBLE PRECISION A(LDA1,LDA2,*), DWORK(*)
Arguments
Mode Parameters
SHFT CHARACTER*1
Specifies the number of shifts employed by the shift
polynomial, as follows:
= 'D': two shifts (assumes N > 2);
= 'S': one real shift.
Input/Output Parameters
K (input) INTEGER
The number of factors. K >= 1.
N (input) INTEGER
The order of the factors. N >= 2.
AMAP (input) INTEGER array, dimension (K)
The map for accessing the factors, i.e., if AMAP(I) = J,
then the factor A_I is stored at the J-th position in A.
AMAP(1) is the pointer to the Hessenberg matrix.
S (input) INTEGER array, dimension (K)
The signature array. Each entry of S must be 1 or -1.
SINV (input) INTEGER
Signature multiplier. Entries of S are virtually
multiplied by SINV.
A (input) DOUBLE PRECISION array, dimension (LDA1,LDA2,K)
The leading N-by-N-by-K part of this array must contain
the product (implicitly represented by its K factors)
in periodic upper Hessenberg form.
LDA1 INTEGER
The first leading dimension of the array A. LDA1 >= N.
LDA2 INTEGER
The second leading dimension of the array A. LDA2 >= N.
C1 (output) DOUBLE PRECISION
S1 (output) DOUBLE PRECISION
On exit, C1 and S1 contain the parameters for the first
Givens rotation.
C2 (output) DOUBLE PRECISION
S2 (output) DOUBLE PRECISION
On exit, if SHFT = 'D', C2 and S2 contain the parameters
for the second Givens rotation. Otherwise, C2 = 1, S2 = 0.
Workspace
IWORK INTEGER array, dimension (2*N)
DWORK DOUBLE PRECISION array, dimension (2*N*N)
On exit, DWORK(N*N+1:N*N+N) and DWORK(N*N+N+1:N*N+2*N)
contain the real and imaginary parts, respectively, of the
eigenvalues of the matrix product.
Method
The necessary elements of the real Wilkinson double shift polynomial are computed, and suitable Givens rotations are found. For numerical reasons, this routine should be called when convergence difficulties are encountered for small order matrices and small K, e.g., N, K <= 6.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute two Givens rotations (C1,S1) and (C2,S2) such that the
orthogonal matrix
[ Q 0 ] [ C1 S1 0 ] [ 1 0 0 ]
Z = [ ], Q := [ -S1 C1 0 ] * [ 0 C2 S2 ],
[ 0 I ] [ 0 0 1 ] [ 0 -S2 C2 ]
makes the first column of the real Wilkinson double shift
polynomial of the product of matrices in periodic upper Hessenberg
form, stored in the array A, parallel to the first unit vector.
Only the rotation defined by C1 and S1 is used for the real
Wilkinson single shift polynomial (see SLICOT Library routines
MB03BE or MB03BF). All factors whose exponents differ from that of
the Hessenberg factor are assumed nonsingular. The trailing 2-by-2
submatrix and the five nonzero elements in the first two columns
of the matrix product are evaluated when a double shift is used.
Specification
SUBROUTINE MB03AH( SHFT, K, N, AMAP, S, SINV, A, LDA1, LDA2, C1,
$ S1, C2, S2 )
C .. Scalar Arguments ..
CHARACTER SHFT
INTEGER K, LDA1, LDA2, N, SINV
DOUBLE PRECISION C1, C2, S1, S2
C .. Array Arguments ..
INTEGER AMAP(*), S(*)
DOUBLE PRECISION A(LDA1,LDA2,*)
Arguments
Mode Parameters
SHFT CHARACTER*1
Specifies the number of shifts employed by the shift
polynomial, as follows:
= 'D': two shifts (assumes N > 2);
= 'S': one real shift.
Input/Output Parameters
K (input) INTEGER
The number of factors. K >= 1.
N (input) INTEGER
The order of the factors. N >= 2.
AMAP (input) INTEGER array, dimension (K)
The map for accessing the factors, i.e., if AMAP(I) = J,
then the factor A_I is stored at the J-th position in A.
AMAP(K) is the pointer to the Hessenberg matrix.
S (input) INTEGER array, dimension (K)
The signature array. Each entry of S must be 1 or -1.
SINV (input) INTEGER
Signature multiplier. Entries of S are virtually
multiplied by SINV.
A (input) DOUBLE PRECISION array, dimension (LDA1,LDA2,K)
The leading N-by-N-by-K part of this array must contain
the product (implicitly represented by its K factors)
in periodic upper Hessenberg form.
LDA1 INTEGER
The first leading dimension of the array A. LDA1 >= N.
LDA2 INTEGER
The second leading dimension of the array A. LDA2 >= N.
C1 (output) DOUBLE PRECISION
S1 (output) DOUBLE PRECISION
On exit, C1 and S1 contain the parameters for the first
Givens rotation.
C2 (output) DOUBLE PRECISION
S2 (output) DOUBLE PRECISION
On exit, if SHFT = 'D' and N > 2, C2 and S2 contain the
parameters for the second Givens rotation. Otherwise,
C2 = 1, S2 = 0.
Method
The necessary elements of the real Wilkinson double/single shift polynomial are computed, and suitable Givens rotations are found. For numerical reasons, this routine should be called when convergence difficulties are encountered. For a double shift, if there are two real eigenvalues of the trailing 2-by-2 part of the product, both shifts are chosen equal to the eigenvalue with minimum modulus. The trailing element of the product is used as a single shift. If SINV is negative, the shift(s) correspond to the reciprocals of the eigenvalues of the product, as required by the SLICOT Library routine MB03BD.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute two Givens rotations (C1,S1) and (C2,S2)
such that the orthogonal matrix
[ Q 0 ] [ C1 S1 0 ] [ 1 0 0 ]
Z = [ ], Q := [ -S1 C1 0 ] * [ 0 C2 S2 ],
[ 0 I ] [ 0 0 1 ] [ 0 -S2 C2 ]
makes the first column of the real Wilkinson double shift
polynomial of the product of matrices in periodic upper Hessenberg
form, stored in the array A, parallel to the first unit vector.
Only the rotation defined by C1 and S1 is used for the real
Wilkinson single shift polynomial (see SLICOT Library routines
MB03BE or MB03BF). All factors whose exponents differ from that of
the Hessenberg factor are assumed nonsingular. The matrix product
is evaluated.
Specification
SUBROUTINE MB03AI( SHFT, K, N, AMAP, S, SINV, A, LDA1, LDA2, C1,
$ S1, C2, S2, DWORK )
C .. Scalar Arguments ..
CHARACTER SHFT
INTEGER K, LDA1, LDA2, N, SINV
DOUBLE PRECISION C1, C2, S1, S2
C .. Array Arguments ..
INTEGER AMAP(*), S(*)
DOUBLE PRECISION A(LDA1,LDA2,*), DWORK(*)
Arguments
Mode Parameters
SHFT CHARACTER*1
Specifies the number of shifts employed by the shift
polynomial, as follows:
= 'D': two shifts (assumes N > 2);
= 'S': one real shift.
Input/Output Parameters
K (input) INTEGER
The number of factors. K >= 1.
N (input) INTEGER
The order of the factors. N >= 2.
AMAP (input) INTEGER array, dimension (K)
The map for accessing the factors, i.e., if AMAP(I) = J,
then the factor A_I is stored at the J-th position in A.
AMAP(K) is the pointer to the Hessenberg matrix.
S (input) INTEGER array, dimension (K)
The signature array. Each entry of S must be 1 or -1.
SINV (input) INTEGER
Signature multiplier. Entries of S are virtually
multiplied by SINV.
A (input) DOUBLE PRECISION array, dimension (LDA1,LDA2,K)
The leading N-by-N-by-K part of this array must contain
the product (implicitly represented by its K factors)
in periodic upper Hessenberg form.
LDA1 INTEGER
The first leading dimension of the array A. LDA1 >= N.
LDA2 INTEGER
The second leading dimension of the array A. LDA2 >= N.
C1 (output) DOUBLE PRECISION
S1 (output) DOUBLE PRECISION
On exit, C1 and S1 contain the parameters for the first
Givens rotation.
C2 (output) DOUBLE PRECISION
S2 (output) DOUBLE PRECISION
On exit, if SHFT = 'D', C2 and S2 contain the parameters
for the second Givens rotation. Otherwise, C2 = 1, S2 = 0.
Workspace
DWORK DOUBLE PRECISION array, dimension (N*(N+2))Method
The necessary elements of the real Wilkinson double shift polynomial are computed, and suitable Givens rotations are found. For numerical reasons, this routine should be called when convergence difficulties are encountered for small order matrices and small K, e.g., N, K <= 6.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the suitable maps for Hessenberg index H and signature array S. Auxiliary routine for the periodic QZ algorithms.Specification
SUBROUTINE MB03BA( K, H, S, SMULT, AMAP, QMAP )
C .. Scalar Arguments ..
INTEGER K, H, SMULT
C .. Array Arguments ..
INTEGER AMAP(*), QMAP(*), S(*)
Arguments
Input/Output Parameters
K (input) INTEGER
The number of factors. K >= 1.
H (input) INTEGER
Index which corresponds to A_1.
S (input) INTEGER array, dimension (K)
The signature array. Each entry of S must be 1 or -1.
SMULT (output) INTEGER
Signature multiplier. Entries of S are virtually
multiplied by SMULT.
AMAP (output) INTEGER array, dimension (K)
The map for accessing the factors, that is,
if AMAP(I) = J, then the factor A_I is stored at the J-th
position in A.
QMAP (output) INTEGER array, dimension (K)
The map for accessing the orthognal transformation
matrices, that is, if QMAP(I) = J, then the matrix Q_I is
stored at the J-th position in Q.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the eigenvalues of a general 2-by-2 matrix product via a complex single shifted periodic QZ algorithm.Specification
SUBROUTINE MB03BB( BASE, LGBAS, ULP, K, AMAP, S, SINV, A, LDA1,
$ LDA2, ALPHAR, ALPHAI, BETA, SCAL, DWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, K, LDA1, LDA2, SINV
DOUBLE PRECISION BASE, LGBAS, ULP
C .. Array Arguments ..
DOUBLE PRECISION A(LDA1,LDA2,*), ALPHAI(2), ALPHAR(2), BETA(2),
$ DWORK(*)
INTEGER AMAP(*), S(*), SCAL(2)
Arguments
Input/Output Parameters
BASE (input) DOUBLE PRECISION
Machine base.
LGBAS (input) DOUBLE PRECISION
Logarithm of BASE.
ULP (input) DOUBLE PRECISION
Machine precision.
K (input) INTEGER
The number of factors. K >= 1.
AMAP (input) INTEGER array, dimension (K)
The map for accessing the factors, i.e., if AMAP(I) = J,
then the factor A_I is stored at the J-th position in A.
S (input) INTEGER array, dimension (K)
The signature array. Each entry of S must be 1 or -1.
SINV (input) INTEGER
Signature multiplier. Entries of S are virtually
multiplied by SINV.
A (input) DOUBLE PRECISION array, dimension (LDA1,LDA2,K)
On entry, the leading 2-by-2-by-K part of this array must
contain a 2-by-2 product (implicitly represented by its K
factors) in upper Hessenberg-triangular form.
LDA1 INTEGER
The first leading dimension of the array A. LDA1 >= 2.
LDA2 INTEGER
The second leading dimension of the array A. LDA2 >= 2.
ALPHAR (output) DOUBLE PRECISION array, dimension (2)
On exit, this array contains the scaled real part of the
two eigenvalues. If BETA(I) <> 0, then the I-th eigenvalue
(I = 1 : 2) is given by
(ALPHAR(I) + ALPHAI(I)*SQRT(-1) ) * (BASE)**SCAL(I).
ALPHAI (output) DOUBLE PRECISION array, dimension (2)
On exit, this array contains the scaled imaginary part of
the two eigenvalues. ALPHAI(1) >= 0.
BETA (output) DOUBLE PRECISION array, dimension (2)
On exit, this array contains information about infinite
eigenvalues. If BETA(I) = 0, then the I-th eigenvalue is
infinite. Otherwise, BETA(I) = 1.0.
SCAL (output) INTEGER array, dimension (2)
On exit, this array contains the scaling exponents for the
two eigenvalues.
Workspace
DWORK DOUBLE PRECISION array, dimension (8*K)Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: the periodic QZ algorithm did not converge;
= 2: the computed eigenvalues might be inaccurate.
Both values might be taken as warnings, since
approximations of eigenvalues are returned.
Method
A complex single shifted periodic QZ iteration is applied.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the product singular value decomposition of the K-1
triangular factors corresponding to a 2-by-2 product of K
factors in upper Hessenberg-triangular form.
For a general product of 2-by-2 triangular matrices
S(2) S(3) S(K)
A = A(:,:,2) A(:,:,3) ... A(:,:,K),
Givens rotations are computed so that
S(i)
[ CV(i-1) SV(i-1) ] [ A(1,1,i)(in) A(1,2,i)(in) ]
[ -SV(i-1) CV(i-1) ] [ 0 A(2,2,i)(in) ]
S(i)
[ A(1,1,i)(out) A(1,2,i)(out) ] [ CV(i) SV(i) ]
= [ 0 A(2,2,i)(out) ] [ -SV(i) CV(i) ]
stays upper triangular and
[ CV(1) SV(1) ] [ CV(K) -SV(K) ]
[ -SV(1) CV(1) ] * A * [ SV(K) CV(K) ]
is diagonal.
Specification
SUBROUTINE MB03BC( K, AMAP, S, SINV, A, LDA1, LDA2, MACPAR, CV,
$ SV, DWORK )
C .. Scalar Arguments ..
INTEGER K, LDA1, LDA2, SINV
C .. Array Arguments ..
INTEGER AMAP(*), S(*)
DOUBLE PRECISION A(LDA1,LDA2,*), CV(*), DWORK(*), MACPAR(*),
$ SV(*)
Arguments
Input/Output Parameters
K (input) INTEGER
The number of factors. K >= 1.
AMAP (input) INTEGER array, dimension (K)
The map for accessing the factors, i.e., if AMAP(I) = J,
then the factor A_I is stored at the J-th position in A.
S (input) INTEGER array, dimension (K)
The signature array. Each entry of S must be 1 or -1.
SINV (input) INTEGER
Signature multiplier. Entries of S are virtually
multiplied by SINV.
A (input/output) DOUBLE PRECISION array, dimension
(LDA1,LDA2,K)
On entry, the leading 2-by-2-by-K part of this array must
contain a 2-by-2 product (implicitly represented by its K
factors) in upper Hessenberg-triangular form.
On exit, the leading 2-by-2-by-K part of this array
contains modified triangular factors such that their
product is diagonal.
LDA1 INTEGER
The first leading dimension of the array A. LDA1 >= 2.
LDA2 INTEGER
The second leading dimension of the array A. LDA2 >= 2.
MACPAR (input) DOUBLE PRECISION array, dimension (5)
Machine parameters:
MACPAR(1) overflow threshold, DLAMCH( 'O' );
MACPAR(2) underflow threshold, DLAMCH( 'U' );
MACPAR(3) safe minimum, DLAMCH( 'S' );
MACPAR(4) relative machine precision, DLAMCH( 'E' );
MACPAR(5) base of the machine, DLAMCH( 'B' ).
CV (output) DOUBLE PRECISION array, dimension (K)
On exit, the first K elements of this array contain the
cosines of the Givens rotations.
SV (output) DOUBLE PRECISION array, dimension (K)
On exit, the first K elements of this array contain the
sines of the Givens rotations.
Workspace
DWORK DOUBLE PRECISION array, dimension (3*(K-1))Method
The product singular value decomposition of the K-1 triangular factors are computed as described in [1].References
[1] Bojanczyk, A. and Van Dooren, P.
On propagating orthogonal transformations in a product of 2x2
triangular matrices.
In Reichel, Ruttan and Varga: 'Numerical Linear Algebra',
pp. 1-9, 1993.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To find the eigenvalues of the generalized matrix product
S(1) S(2) S(K)
A(:,:,1) * A(:,:,2) * ... * A(:,:,K)
where A(:,:,H) is upper Hessenberg and A(:,:,i), i <> H, is upper
triangular, using a double-shift version of the periodic
QZ method. In addition, A may be reduced to periodic Schur form:
A(:,:,H) is upper quasi-triangular and all the other factors
A(:,:,I) are upper triangular. Optionally, the 2-by-2 triangular
matrices corresponding to 2-by-2 diagonal blocks in A(:,:,H)
are so reduced that their product is a 2-by-2 diagonal matrix.
If COMPQ = 'U' or COMPQ = 'I', then the orthogonal factors are
computed and stored in the array Q so that for S(I) = 1,
T
Q(:,:,I)(in) A(:,:,I)(in) Q(:,:,MOD(I,K)+1)(in)
T (1)
= Q(:,:,I)(out) A(:,:,I)(out) Q(:,:,MOD(I,K)+1)(out),
and for S(I) = -1,
T
Q(:,:,MOD(I,K)+1)(in) A(:,:,I)(in) Q(:,:,I)(in)
T (2)
= Q(:,:,MOD(I,K)+1)(out) A(:,:,I)(out) Q(:,:,I)(out).
A partial generation of the orthogonal factors can be realized
via the array QIND.
Specification
SUBROUTINE MB03BD( JOB, DEFL, COMPQ, QIND, K, N, H, ILO, IHI, S,
$ A, LDA1, LDA2, Q, LDQ1, LDQ2, ALPHAR, ALPHAI,
$ BETA, SCAL, IWORK, LIWORK, DWORK, LDWORK,
$ IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, DEFL, JOB
INTEGER H, IHI, ILO, INFO, IWARN, K, LDA1, LDA2, LDQ1,
$ LDQ2, LDWORK, LIWORK, N
C .. Array Arguments ..
INTEGER IWORK(*), QIND(*), S(*), SCAL(*)
DOUBLE PRECISION A(LDA1,LDA2,*), ALPHAI(*), ALPHAR(*), BETA(*),
$ DWORK(*), Q(LDQ1,LDQ2,*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'E': compute the eigenvalues only; A will not
necessarily be put into periodic Schur form;
= 'S': put A into periodic Schur form, and return the
eigenvalues in ALPHAR, ALPHAI, BETA, and SCAL;
= 'T': as JOB = 'S', but A is put into standardized
periodic Schur form, that is, the general product
of the 2-by-2 triangular matrices corresponding to
a complex eigenvalue is diagonal.
DEFL CHARACTER*1
Specifies the deflation strategy to be used, as follows:
= 'C': apply a careful deflation strategy, that is,
the criteria are based on the magnitudes of
neighboring elements and infinite eigenvalues are
only deflated at the top; this is the recommended
option;
= 'A': apply a more aggressive strategy, that is,
elements on the subdiagonal or diagonal are set
to zero as soon as they become smaller in magnitude
than eps times the norm of the corresponding
factor; this option is only recommended if
balancing is applied beforehand and convergence
problems are observed.
COMPQ CHARACTER*1
Specifies whether or not the orthogonal transformations
should be accumulated in the array Q, as follows:
= 'N': do not modify Q;
= 'U': modify (update) the array Q by the orthogonal
transformations that are applied to the matrices in
the array A to reduce them to periodic Schur form;
= 'I': like COMPQ = 'U', except that each matrix in the
array Q will be first initialized to the identity
matrix;
= 'P': use the parameters as encoded in QIND.
QIND INTEGER array, dimension (K)
If COMPQ = 'P', then this array describes the generation
of the orthogonal factors as follows:
If QIND(I) > 0, then the array Q(:,:,QIND(I)) is
modified by the transformations corresponding to the
i-th orthogonal factor in (1) and (2).
If QIND(I) < 0, then the array Q(:,:,-QIND(I)) is
initialized to the identity and modified by the
transformations corresponding to the i-th orthogonal
factor in (1) and (2).
If QIND(I) = 0, then the transformations corresponding
to the i-th orthogonal factor in (1), (2) are not applied.
Input/Output Parameters
K (input) INTEGER
The number of factors. K >= 1.
N (input) INTEGER
The order of each factor in the array A. N >= 0.
H (input) INTEGER
Hessenberg index. The factor A(:,:,H) is on entry in upper
Hessenberg form. 1 <= H <= K.
ILO (input) INTEGER
IHI (input) INTEGER
It is assumed that each factor in A is already upper
triangular in rows and columns 1:ILO-1 and IHI+1:N.
1 <= ILO <= IHI <= N, if N > 0;
ILO = 1 and IHI = 0, if N = 0.
S (input) INTEGER array, dimension (K)
The leading K elements of this array must contain the
signatures of the factors. Each entry in S must be either
1 or -1.
A (input/output) DOUBLE PRECISION array, dimension
(LDA1,LDA2,K)
On entry, the leading N-by-N-by-K part of this array
must contain the factors in upper Hessenberg-triangular
form, that is, A(:,:,H) is upper Hessenberg and the other
factors are upper triangular.
On exit, if JOB = 'S' and INFO = 0, the leading
N-by-N-by-K part of this array contains the factors of
A in periodic Schur form, that is, A(:,:,H) is upper quasi
triangular and the other factors are upper triangular.
On exit, if JOB = 'T' and INFO = 0, the leading
N-by-N-by-K part of this array contains the factors of
A as for the option JOB = 'S', but the product of the
triangular factors corresponding to a 2-by-2 block in
A(:,:,H) is diagonal.
On exit, if JOB = 'E', then the leading N-by-N-by-K part
of this array contains meaningless elements in the off-
diagonal blocks. Consequently, the formulas (1) and (2)
do not hold for the returned A and Q (if COMPQ <> 'N')
in this case.
LDA1 INTEGER
The first leading dimension of the array A.
LDA1 >= MAX(1,N).
LDA2 INTEGER
The second leading dimension of the array A.
LDA2 >= MAX(1,N).
Q (input/output) DOUBLE PRECISION array, dimension
(LDQ1,LDQ2,K)
On entry, if COMPQ = 'U', the leading N-by-N-by-K part
of this array must contain the initial orthogonal factors
as described in (1) and (2).
On entry, if COMPQ = 'P', only parts of the leading
N-by-N-by-K part of this array must contain some
orthogonal factors as described by the parameters QIND.
If COMPQ = 'I', this array should not be set on entry.
On exit, if COMPQ = 'U' or COMPQ = 'I', the leading
N-by-N-by-K part of this array contains the modified
orthogonal factors as described in (1) and (2).
On exit, if COMPQ = 'P', only parts of the leading
N-by-N-by-K part contain some modified orthogonal factors
as described by the parameters QIND.
This array is not referenced if COMPQ = 'N'.
LDQ1 INTEGER
The first leading dimension of the array Q. LDQ1 >= 1,
and, if COMPQ <> 'N', LDQ1 >= MAX(1,N).
LDQ2 INTEGER
The second leading dimension of the array Q. LDQ2 >= 1,
and, if COMPQ <> 'N', LDQ2 >= MAX(1,N).
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
On exit, if INFO = 0, the leading N elements of this array
contain the scaled real parts of the eigenvalues of the
matrix product A. The i-th eigenvalue of A is given by
(ALPHAR(I) + ALPHAI(I)*SQRT(-1))/BETA(I) * BASE**SCAL(I),
where BASE is the machine base (often 2.0). Complex
conjugate eigenvalues appear in consecutive locations.
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
On exit, if INFO = 0, the leading N elements of this array
contain the scaled imaginary parts of the eigenvalues
of A.
BETA (output) DOUBLE PRECISION array, dimension (N)
On exit, if INFO = 0, the leading N elements of this array
contain indicators for infinite eigenvalues. That is, if
BETA(I) = 0.0, then the i-th eigenvalue is infinite.
Otherwise BETA(I) is set to 1.0.
SCAL (output) INTEGER array, dimension (N)
On exit, if INFO = 0, the leading N elements of this array
contain the scaling parameters for the eigenvalues of A.
Workspace
IWORK INTEGER array, dimension (LIWORK)
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK,
and if IWARN > N, the nonzero absolute values in IWORK(2),
..., IWORK(N+1) are indices of the possibly inaccurate
eigenvalues, as well as of the corresponding 1-by-1 or
2-by-2 diagonal blocks of the factors in the array A.
The 2-by-2 blocks correspond to negative values in IWORK.
One negative value is stored for each such eigenvalue
pair. Its modulus indicates the starting index of a
2-by-2 block. This is also done for any value of IWARN,
if a 2-by-2 block is found to have two real eigenvalues.
On exit, if INFO = -22, IWORK(1) returns the minimum value
of LIWORK.
LIWORK INTEGER
The length of the array IWORK. LIWORK >= 2*K+N.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK,
and DWORK(2), ..., DWORK(1+K) contain the Frobenius norms
of the factors of the formal matrix product used by the
algorithm.
On exit, if INFO = -24, DWORK(1) returns the minimum value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= K + MAX( 2*N, 8*K ).
Warning Indicator
IWARN INTEGER
= 0 : no warnings;
= 1,..,N-1 : A is in periodic Schur form, but the
algorithm was not able to reveal information
about the eigenvalues from the 2-by-2
blocks.
ALPHAR(i), ALPHAI(i), BETA(i) and SCAL(i),
can be incorrect for i = 1, ..., IWARN+1;
= N : some eigenvalues might be inaccurate;
= N+1 : some eigenvalues might be inaccurate, and
details can be found in IWORK.
Error Indicator
INFO INTEGER
= 0 : succesful exit;
< 0 : if INFO = -i, the i-th argument had an illegal
value;
= 1,..,N : the periodic QZ iteration did not converge.
A is not in periodic Schur form, but
ALPHAR(i), ALPHAI(i), BETA(i) and SCAL(i), for
i = INFO+1,...,N should be correct.
Method
A modified version of the periodic QZ algorithm is used [1], [2].References
[1] Bojanczyk, A., Golub, G. H. and Van Dooren, P.
The periodic Schur decomposition: algorithms and applications.
In F.T. Luk (editor), Advanced Signal Processing Algorithms,
Architectures, and Implementations III, Proc. SPIE Conference,
vol. 1770, pp. 31-42, 1992.
[2] Kressner, D.
An efficient and reliable implementation of the periodic QZ
algorithm. IFAC Workshop on Periodic Control Systems (PSYCO
2001), Como (Italy), August 27-28 2001. Periodic Control
Systems 2001 (IFAC Proceedings Volumes), Pergamon.
Numerical Aspects
The implemented method is numerically backward stable.
3
The algorithm requires 0(K N ) floating point operations.
Further Comments
NoneExample
Program Text
* MB03BD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER KMAX, NMAX
PARAMETER ( KMAX = 6, NMAX = 50 )
INTEGER LDA1, LDA2, LDQ1, LDQ2, LDWORK, LIWORK
PARAMETER ( LDA1 = NMAX, LDA2 = NMAX, LDQ1 = NMAX,
$ LDQ2 = NMAX,
$ LDWORK = KMAX + MAX( 2*NMAX, 8*KMAX ),
$ LIWORK = 2*KMAX + NMAX )
*
* .. Local Scalars ..
CHARACTER COMPQ, DEFL, JOB
INTEGER H, I, IHI, ILO, INFO, IWARN, J, K, L, N
*
* .. Local Arrays ..
INTEGER IWORK( LIWORK ), QIND( KMAX ), S( KMAX ),
$ SCAL( NMAX )
DOUBLE PRECISION A( LDA1, LDA2, KMAX ), ALPHAI( NMAX ),
$ ALPHAR( NMAX ), BETA( NMAX ), DWORK( LDWORK),
$ Q( LDQ1, LDQ2, KMAX )
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
*
* .. External Subroutines ..
EXTERNAL MB03BD
*
* .. Intrinsic Functions ..
INTRINSIC MAX
*
* .. Executable Statements ..
*
WRITE( NOUT, FMT = 99999 )
* Skip the heading in the data file and read in the data.
READ( NIN, FMT = * )
READ( NIN, FMT = * ) JOB, DEFL, COMPQ, K, N, H, ILO, IHI
IF( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE( NOUT, FMT = 99998 ) N
ELSE
READ( NIN, FMT = * ) ( S( I ), I = 1, K )
READ( NIN, FMT = * ) ( ( ( A( I, J, L ), J = 1, N ),
$ I = 1, N ), L = 1, K )
IF( LSAME( COMPQ, 'U' ) )
$ READ( NIN, FMT = * ) ( ( ( Q( I, J, L ), J = 1, N ),
$ I = 1, N ), L = 1, K )
IF( LSAME( COMPQ, 'P' ) ) THEN
READ( NIN, FMT = * ) ( QIND( I ), I = 1, K )
DO 10 L = 1, K
IF( QIND( L ).GT.0 )
$ READ( NIN, FMT = * ) ( ( Q( I, J, QIND( L ) ),
$ J = 1, N ), I = 1, N )
10 CONTINUE
END IF
* Compute the eigenvalues and the transformed matrices, if
* required.
CALL MB03BD( JOB, DEFL, COMPQ, QIND, K, N, H, ILO, IHI, S, A,
$ LDA1, LDA2, Q, LDQ1, LDQ2, ALPHAR, ALPHAI, BETA,
$ SCAL, IWORK, LIWORK, DWORK, LDWORK, IWARN, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99997 ) INFO
ELSE IF( IWARN.EQ.0 ) THEN
IF( LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'T' ) ) THEN
WRITE( NOUT, FMT = 99996 )
DO 30 L = 1, K
WRITE( NOUT, FMT = 99988 ) L
DO 20 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( A( I, J, L ), J = 1, N
$ )
20 CONTINUE
30 CONTINUE
END IF
IF( LSAME( COMPQ, 'U' ) .OR. LSAME( COMPQ, 'I' ) ) THEN
WRITE( NOUT, FMT = 99994 )
DO 50 L = 1, K
WRITE( NOUT, FMT = 99988 ) L
DO 40 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( Q( I, J, L ), J = 1, N
$ )
40 CONTINUE
50 CONTINUE
ELSE IF( LSAME( COMPQ, 'P' ) ) THEN
WRITE( NOUT, FMT = 99994 )
DO 70 L = 1, K
IF( QIND( L ).GT.0 ) THEN
WRITE( NOUT, FMT = 99988 ) QIND( L )
DO 60 I = 1, N
WRITE( NOUT, FMT = 99995 )
$ ( Q( I, J, QIND( L ) ), J = 1, N )
60 CONTINUE
END IF
70 CONTINUE
END IF
WRITE( NOUT, FMT = 99993 )
WRITE( NOUT, FMT = 99995 ) ( ALPHAR( I ), I = 1, N )
WRITE( NOUT, FMT = 99992 )
WRITE( NOUT, FMT = 99995 ) ( ALPHAI( I ), I = 1, N )
WRITE( NOUT, FMT = 99991 )
WRITE( NOUT, FMT = 99995 ) ( BETA( I ), I = 1, N )
WRITE( NOUT, FMT = 99990 )
WRITE( NOUT, FMT = 99989 ) ( SCAL( I ), I = 1, N )
ELSE
WRITE( NOUT, FMT = 99987 ) IWARN
END IF
END IF
STOP
*
99999 FORMAT( 'MB03BD EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT( 'INFO on exit from MB03BD = ', I2 )
99996 FORMAT( 'The matrix A on exit is ' )
99995 FORMAT( 50( 1X, F8.4 ) )
99994 FORMAT( 'The matrix Q on exit is ' )
99993 FORMAT( 'The vector ALPHAR is ' )
99992 FORMAT( 'The vector ALPHAI is ' )
99991 FORMAT( 'The vector BETA is ' )
99990 FORMAT( 'The vector SCAL is ' )
99989 FORMAT( 50( 1X, I8 ) )
99988 FORMAT( 'The factor ', I2, ' is ' )
99987 FORMAT( 'IWARN on exit from MB03BD = ', I2 )
END
Program Data
MB03BD EXAMPLE PROGRAM DATA S C I 3 3 2 1 3 -1 1 -1 2.0 0.0 1.0 0.0 -2.0 -1.0 0.0 0.0 3.0 1.0 2.0 0.0 4.0 -1.0 3.0 0.0 3.0 1.0 1.0 0.0 1.0 0.0 4.0 -1.0 0.0 0.0 -2.0Program Results
MB03BD EXAMPLE PROGRAM RESULTS
The matrix A on exit is
The factor 1 is
-2.1306 0.8205 0.7462
0.0000 2.8786 1.0564
0.0000 0.0000 1.9566
The factor 2 is
-4.0763 -1.0376 -2.6948
-1.9525 1.8283 2.2987
0.0000 0.0000 1.8990
The factor 3 is
3.3463 -2.3239 -0.5623
0.0000 1.0778 -0.0646
0.0000 0.0000 -2.2180
The matrix Q on exit is
The factor 1 is
0.2594 0.7715 -0.5809
-0.9552 0.1162 -0.2723
-0.1426 0.6255 0.7671
The factor 2 is
-0.1766 0.8037 -0.5683
-0.9636 -0.0234 0.2664
0.2008 0.5946 0.7785
The factor 3 is
0.6295 0.7315 0.2619
-0.7394 0.4605 0.4911
0.2386 -0.5028 0.8308
The vector ALPHAR is
0.3230 0.3230 -0.8752
The vector ALPHAI is
0.5694 -0.5694 0.0000
The vector BETA is
1.0000 1.0000 1.0000
The vector SCAL is
0 0 -1
Purpose
To apply at most 20 iterations of a real single shifted periodic QZ algorithm to the 2-by-2 product of matrices stored in the array A.Specification
SUBROUTINE MB03BE( K, AMAP, S, SINV, A, LDA1, LDA2 )
C .. Scalar Arguments ..
INTEGER K, LDA1, LDA2, SINV
C .. Array Arguments ..
INTEGER AMAP(*), S(*)
DOUBLE PRECISION A(LDA1,LDA2,*)
Arguments
Input/Output Parameters
K (input) INTEGER
The number of factors. K >= 1.
AMAP (input) INTEGER array, dimension (K)
The map for accessing the factors, i.e., if AMAP(I) = J,
then the factor A_I is stored at the J-th position in A.
S (input) INTEGER array, dimension (K)
The signature array. Each entry of S must be 1 or -1.
SINV (input) INTEGER
Signature multiplier. Entries of S are virtually
multiplied by SINV.
A (input/output) DOUBLE PRECISION array, dimension
(LDA1,LDA2,K)
On entry, the leading 2-by-2-by-K part of this array must
contain a 2-by-2 product (implicitly represented by its K
factors) in upper Hessenberg form.
On exit, the leading 2-by-2-by-K part of this array
contains the product after at most 20 iterations of a real
shifted periodic QZ algorithm.
LDA1 INTEGER
The first leading dimension of the array A. LDA1 >= 2.
LDA2 INTEGER
The second leading dimension of the array A. LDA2 >= 2.
Method
Ten iterations of a real single shifted periodic QZ algorithm are applied to the 2-by-2 matrix product A.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To apply at most 20 iterations of a real single shifted periodic QZ algorithm to the 2-by-2 product of matrices stored in the array A. The Hessenberg matrix is the last one of the formal matrix product.Specification
SUBROUTINE MB03BF( K, AMAP, S, SINV, A, LDA1, LDA2, ULP )
C .. Scalar Arguments ..
INTEGER K, LDA1, LDA2, SINV
DOUBLE PRECISION ULP
C .. Array Arguments ..
INTEGER AMAP(*), S(*)
DOUBLE PRECISION A(LDA1,LDA2,*)
Arguments
Input/Output Parameters
K (input) INTEGER
The number of factors. K >= 1.
AMAP (input) INTEGER array, dimension (K)
The map for accessing the factors, i.e., if AMAP(I) = J,
then the factor A_I is stored at the J-th position in A.
S (input) INTEGER array, dimension (K)
The signature array. Each entry of S must be 1 or -1.
SINV (input) INTEGER
Signature multiplier. Entries of S are virtually
multiplied by SINV.
A (input/output) DOUBLE PRECISION array, dimension
(LDA1,LDA2,K)
On entry, the leading 2-by-2-by-K part of this array must
contain a 2-by-2 product (implicitly represented by its K
factors) in upper Hessenberg form. The Hessenberg matrix
is the last one of the formal matrix product.
On exit, the leading 2-by-2-by-K part of this array
contains the product after at most 20 iterations of a real
shifted periodic QZ algorithm.
LDA1 INTEGER
The first leading dimension of the array A. LDA1 >= 2.
LDA2 INTEGER
The second leading dimension of the array A. LDA2 >= 2.
ULP INTEGER
The machine relation precision.
Method
Twenty iterations of a real single shifted periodic QZ algorithm are applied to the 2-by-2 matrix product A.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To find the eigenvalues of the complex generalized matrix product
S(1) S(2) S(K)
A(:,:,1) * A(:,:,2) * ... * A(:,:,K) , S(1) = 1,
where A(:,:,1) is upper Hessenberg and A(:,:,i) is upper
triangular, i = 2, ..., K, using a single-shift version of the
periodic QZ method. In addition, A may be reduced to periodic
Schur form by unitary transformations: all factors A(:,:,i) become
upper triangular.
If COMPQ = 'V' or COMPQ = 'I', then the unitary factors are
computed and stored in the array Q so that for S(I) = 1,
H
Q(:,:,I)(in) A(:,:,I)(in) Q(:,:,MOD(I,K)+1)(in)
H (1)
= Q(:,:,I)(out) A(:,:,I)(out) Q(:,:,MOD(I,K)+1)(out),
and for S(I) = -1,
H
Q(:,:,MOD(I,K)+1)(in) A(:,:,I)(in) Q(:,:,I)(in)
H (2)
= Q(:,:,MOD(I,K)+1)(out) A(:,:,I)(out) Q(:,:,I)(out).
Specification
SUBROUTINE MB03BZ( JOB, COMPQ, K, N, ILO, IHI, S, A, LDA1, LDA2,
$ Q, LDQ1, LDQ2, ALPHA, BETA, SCAL, DWORK,
$ LDWORK, ZWORK, LZWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, JOB
INTEGER IHI, ILO, INFO, K, LDA1, LDA2, LDQ1, LDQ2,
$ LDWORK, LZWORK, N
C .. Array Arguments ..
INTEGER S(*), SCAL(*)
DOUBLE PRECISION DWORK(*)
COMPLEX*16 A(LDA1, LDA2, *), ALPHA(*), BETA(*),
$ Q(LDQ1, LDQ2, *), ZWORK(*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'E': compute the eigenvalues only; A will not
necessarily be put into periodic Schur form;
= 'S': put A into periodic Schur form, and return the
eigenvalues in ALPHA, BETA, and SCAL.
COMPQ CHARACTER*1
Specifies whether or not the unitary transformations
should be accumulated in the array Q, as follows:
= 'N': do not modify Q;
= 'V': modify the array Q by the unitary transformations
that are applied to the matrices in the array A to
reduce them to periodic Schur form;
= 'I': like COMPQ = 'V', except that each matrix in the
array Q will be first initialized to the identity
matrix.
Input/Output Parameters
K (input) INTEGER
The number of factors. K >= 1.
N (input) INTEGER
The order of each factor in the array A. N >= 0.
ILO (input) INTEGER
IHI (input) INTEGER
It is assumed that each factor in A is already upper
triangular in rows and columns 1:ILO-1 and IHI+1:N.
1 <= ILO <= IHI <= N, if N > 0;
ILO = 1 and IHI = 0, if N = 0.
S (input) INTEGER array, dimension (K)
The leading K elements of this array must contain the
signatures of the factors. Each entry in S must be either
1 or -1. By definition, S(1) must be set to 1.
A (input/output) COMPLEX*16 array, dimension (LDA1,LDA2,K)
On entry, the leading N-by-N-by-K part of this array
must contain the factors in upper Hessenberg-triangular
form, that is, A(:,:,1) is upper Hessenberg and the other
factors are upper triangular.
On exit, if JOB = 'S' and INFO = 0, the leading
N-by-N-by-K part of this array contains the factors of
A in periodic Schur form. All factors are reduced to
upper triangular form and, moreover, A(:,:,2), ...,
A(:,:,K) are normalized so that their diagonals contain
nonnegative real numbers.
On exit, if JOB = 'E', then the leading N-by-N-by-K part
of this array contains meaningless elements.
LDA1 INTEGER
The first leading dimension of the array A.
LDA1 >= MAX(1,N).
LDA2 INTEGER
The second leading dimension of the array A.
LDA2 >= MAX(1,N).
Q (input/output) COMPLEX*16 array, dimension (LDQ1,LDQ2,K)
On entry, if COMPQ = 'V', the leading N-by-N-by-K part
of this array must contain the initial unitary factors
as described in (1) and (2).
On exit, if COMPQ = 'V' or COMPQ = 'I', the leading
N-by-N-by-K part of this array contains the modified
unitary factors as described in (1) and (2).
This array is not referenced if COMPQ = 'N'.
LDQ1 INTEGER
The first leading dimension of the array Q. LDQ1 >= 1,
and, if COMPQ <> 'N', LDQ1 >= MAX(1,N).
LDQ2 INTEGER
The second leading dimension of the array Q. LDQ2 >= 1,
and, if COMPQ <> 'N', LDQ2 >= MAX(1,N).
ALPHA (output) COMPLEX*16 array, dimension (N)
On exit, if INFO = 0, the leading N elements of this
array contain the scaled eigenvalues of the matrix
product A. The i-th eigenvalue of A is given by
ALPHA(I) / BETA(I) * BASE**(SCAL(I)),
where ABS(ALPHA(I)) = 0.0 or 1.0 <= ABS(ALPHA(I)) < BASE,
and BASE is the machine base (normally 2.0).
BETA (output) COMPLEX*16 array, dimension (N)
On exit, if INFO = 0, the leading N elements of this
array contain indicators for infinite eigenvalues. That
is, if BETA(I) = 0.0, then the i-th eigenvalue is
infinite. Otherwise BETA(I) is set to 1.0.
SCAL (output) INTEGER array, dimension (N)
On exit, if INFO = 0, the leading N elements of this
array contain the scaling parameters for the eigenvalues
of A.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the minimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,N).
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) returns the minimal value
of LZWORK.
LZWORK INTEGER
The length of the array ZWORK. LZWORK >= MAX(1,N).
Error Indicator
INFO INTEGER
= 0 : succesful exit;
< 0 : if INFO = -i, the i-th argument had an illegal
value;
= 1,..,N : the periodic QZ iteration did not converge.
A is not in periodic Schur form, but
ALPHA(I), BETA(I), and SCAL(I), for
I = INFO+1,...,N should be correct.
Method
A slightly modified version of the periodic QZ algorithm is used. For more details, see [2].References
[1] Bojanczyk, A., Golub, G. H. and Van Dooren, P.
The periodic Schur decomposition: algorithms and applications.
In F.T. Luk (editor), Advanced Signal Processing Algorithms,
Architectures, and Implementations III, Proc. SPIE Conference,
vol. 1770, pp. 31-42, 1992.
[2] Kressner, D.
An efficient and reliable implementation of the periodic QZ
algorithm. IFAC Workshop on Periodic Control Systems (PSYCO
2001), Como (Italy), August 27-28 2001. Periodic Control
Systems 2001 (IFAC Proceedings Volumes), Pergamon.
Numerical Aspects
The implemented method is numerically backward stable.
3
The algorithm requires 0(K N ) floating point operations.
Further Comments
NoneExample
Program Text
* MB03BZ EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER KMAX, NMAX
PARAMETER ( KMAX = 6, NMAX = 50 )
INTEGER LDA1, LDA2, LDQ1, LDQ2, LDWORK, LZWORK
PARAMETER ( LDA1 = NMAX, LDA2 = NMAX, LDQ1 = NMAX,
$ LDQ2 = NMAX, LDWORK = NMAX, LZWORK = NMAX )
*
* .. Local Scalars ..
CHARACTER COMPQ, JOB
INTEGER I, IHI, ILO, INFO, J, K, L, N
*
* .. Local Arrays ..
COMPLEX*16 A( LDA1, LDA2, KMAX ), ALPHA( NMAX ),
$ BETA( NMAX ), Q( LDQ1, LDQ2, KMAX ),
$ ZWORK( LZWORK )
DOUBLE PRECISION DWORK( LDWORK)
INTEGER S( KMAX ), SCAL( NMAX )
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
*
* .. External Subroutines ..
EXTERNAL MB03BZ
*
* .. Executable Statements ..
*
WRITE( NOUT, FMT = 99999 )
* Skip the heading in the data file and read in the data.
READ( NIN, FMT = * )
READ( NIN, FMT = * ) JOB, COMPQ, K, N, ILO, IHI
IF( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE( NOUT, FMT = 99998 ) N
ELSE
READ( NIN, FMT = * ) ( S( I ), I = 1, K )
READ( NIN, FMT = * ) ( ( ( A( I, J, L ), J = 1, N ),
$ I = 1, N ), L = 1, K )
IF( LSAME( COMPQ, 'V' ) )
$ READ( NIN, FMT = * ) ( ( ( Q( I, J, L ), J = 1, N ),
$ I = 1, N ), L = 1, K )
* Compute the eigenvalues and the transformed matrices, if
* required.
CALL MB03BZ( JOB, COMPQ, K, N, ILO, IHI, S, A, LDA1, LDA2,
$ Q, LDQ1, LDQ2, ALPHA, BETA, SCAL, DWORK, LDWORK,
$ ZWORK, LZWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99997 ) INFO
ELSE
IF( LSAME( JOB, 'S' ) ) THEN
WRITE( NOUT, FMT = 99996 )
DO 20 L = 1, K
WRITE( NOUT, FMT = 99995 ) L
DO 10 I = 1, N
WRITE( NOUT, FMT = 99994 ) ( A( I, J, L ), J = 1, N
$ )
10 CONTINUE
20 CONTINUE
END IF
IF( .NOT.LSAME( COMPQ, 'N' ) ) THEN
WRITE( NOUT, FMT = 99993 )
DO 40 L = 1, K
WRITE( NOUT, FMT = 99995 ) L
DO 30 I = 1, N
WRITE( NOUT, FMT = 99994 ) ( Q( I, J, L ), J = 1, N
$ )
30 CONTINUE
40 CONTINUE
END IF
WRITE( NOUT, FMT = 99992 )
WRITE( NOUT, FMT = 99994 ) ( ALPHA( I ), I = 1, N )
WRITE( NOUT, FMT = 99991 )
WRITE( NOUT, FMT = 99994 ) ( BETA( I ), I = 1, N )
WRITE( NOUT, FMT = 99990 )
WRITE( NOUT, FMT = 99989 ) ( SCAL( I ), I = 1, N )
END IF
END IF
STOP
*
99999 FORMAT( 'MB03BZ EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT( 'INFO on exit from MB03BZ = ', I2 )
99996 FORMAT(/'The matrix A on exit is ' )
99995 FORMAT( 'The factor ', I2, ' is ' )
99994 FORMAT( 50( 1X, F9.4, SP, F9.4, S, 'i ') )
99993 FORMAT(/'The matrix Q on exit is ' )
99992 FORMAT(/'The vector ALPHA is ' )
99991 FORMAT( 'The vector BETA is ' )
99990 FORMAT( 'The vector SCAL is ' )
99989 FORMAT( 50( 1X, I8 ) )
END
Program Data
MB03BZ EXAMPLE PROGRAM DATA
S I 3 4 1 4
1 -1 1
(0.8637,0.9326) (0.8819,0.4850) (0.5920,0.8826) (0.8991,0.9040)
(0.6994,0.8588) (0.9527,0.2672) (0.5087,0.0621) (0.9653,0.5715)
0 (0.1561,0.1898) (0.9514,0.9266) (0.6582,0.3102)
0 0 (0.8649,0.1265) (0.1701,0.0013)
(0.5113,0.7375) (0.6869,0.7692) (0.7812,0.1467) (0.7216,0.9498)
0 (0.1319,0.9137) (0.5879,0.0201) (0.9834,0.0549)
0 0 (0.7711,0.2422) (0.9468,0.3280)
0 0 0 (0.2219,0.3971)
(0.0158,0.4042) (0.0082,0.2033) (0.1028,0.9913) (0.6954,0.1987)
0 (0.5066,0.4587) (0.1060,0.6949) (0.5402,0.0970)
0 0 (0.4494,0.3700) (0.8492,0.4882)
0 0 0 (0.2110,0.5824)
Program Results
MB03BZ EXAMPLE PROGRAM RESULTS
The matrix A on exit is
The factor 1 is
0.6053 +1.0311i -1.7227 -0.5753i 1.2428 -1.2632i 0.9445 -0.4317i
0.0000 +0.0000i -0.2596 +1.0235i 0.4673 -0.2403i -0.5579 -1.1564i
0.0000 +0.0000i 0.0000 +0.0000i -0.3336 -0.3367i 0.0687 +0.0261i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i -0.2014 +0.0057i
The factor 2 is
1.1118 +0.0000i -1.4173 +1.1607i 0.3271 -0.5800i 0.5291 -0.6341i
0.0000 +0.0000i 0.9051 +0.0000i 0.1710 +0.1014i -0.2696 -0.3549i
0.0000 +0.0000i 0.0000 +0.0000i 0.3599 +0.0000i 0.0231 -0.5865i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.8410 +0.0000i
The factor 3 is
1.1554 +0.0000i -0.7577 +0.0825i 0.1284 -0.0063i 1.1175 -0.0778i
0.0000 +0.0000i 0.5216 +0.0000i -0.5761 +0.2972i -0.3534 -0.3595i
0.0000 +0.0000i 0.0000 +0.0000i 0.2750 +0.0000i 0.2587 -0.1664i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.6015 +0.0000i
The matrix Q on exit is
The factor 1 is
0.6033 -0.4021i 0.3478 -0.1834i -0.0029 -0.4458i -0.3157 +0.1462i
0.1519 -0.3002i 0.6010 +0.1411i 0.0134 +0.5692i 0.4223 -0.0574i
0.1002 -0.5387i -0.4199 +0.3933i 0.1148 +0.2694i -0.3713 -0.3810i
-0.0599 -0.2395i -0.0968 +0.3521i 0.2342 -0.5801i 0.6247 -0.1555i
The factor 2 is
0.3325 +0.6289i 0.0930 +0.3421i -0.1849 -0.1715i -0.5470 -0.0734i
0.3247 -0.4396i 0.1766 -0.1691i 0.2307 -0.3746i -0.2617 -0.6160i
0.2180 -0.2074i -0.3853 +0.1978i -0.6852 +0.2781i 0.1492 -0.3909i
-0.0268 -0.3223i -0.0623 +0.7893i 0.0957 -0.4323i 0.1716 +0.2072i
The factor 3 is
0.6791 +0.1138i 0.0183 +0.2703i 0.3289 -0.2770i -0.2355 -0.4605i
0.5111 -0.1775i 0.3990 +0.1941i -0.0851 +0.4779i 0.4563 +0.2580i
0.2687 -0.3788i -0.7974 -0.0691i -0.2836 +0.1192i 0.1765 -0.1340i
0.1098 -0.0739i -0.2845 -0.0903i 0.6651 +0.2062i -0.2810 +0.5741i
The vector ALPHA is
0.6290 +1.0715i -0.2992 +1.1797i -1.0195 -1.0290i -1.1523 +0.0326i
The vector BETA is
1.0000 +0.0000i 1.0000 +0.0000i 1.0000 +0.0000i 1.0000 +0.0000i
The vector SCAL is
0 -1 -2 -3
Purpose
To compute orthogonal matrices Q1, Q2, Q3 for a real 2-by-2,
3-by-3, or 4-by-4 regular block upper triangular pencil
( A11 A12 ) ( B11 B12 ) ( D11 D12 )
aAB - bD = a ( ) ( ) - b ( ), (1)
( 0 A22 ) ( 0 B22 ) ( 0 D22 )
such that the pencil a(Q3' A Q2 )(Q2' B Q1 ) - b(Q3' D Q1) is
still in block upper triangular form, but the eigenvalues in
Spec(A11 B11, D11), Spec(A22 B22, D22) are exchanged, where
Spec(X,Y) denotes the spectrum of the matrix pencil (X,Y), and M'
denotes the transpose of the matrix M.
Optionally, to upper triangularize the real regular pencil in
block lower triangular form
( A11 0 ) ( B11 0 ) ( D11 0 )
aAB - bD = a ( ) ( ) - b ( ), (2)
( A21 A22 ) ( B21 B22 ) ( D21 D22 )
while keeping the eigenvalues in the same diagonal position.
Specification
SUBROUTINE MB03CD( UPLO, N1, N2, PREC, A, LDA, B, LDB, D, LDD, Q1,
$ LDQ1, Q2, LDQ2, Q3, LDQ3, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LDB, LDD, LDQ1, LDQ2, LDQ3, LDWORK,
$ N1, N2
DOUBLE PRECISION PREC
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), D( LDD, * ),
$ DWORK( * ), Q1( LDQ1, * ), Q2( LDQ2, * ),
$ Q3( LDQ3, * )
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies if the pencil is in lower or upper block
triangular form on entry, as follows:
= 'U': Upper block triangular, eigenvalues are exchanged
on exit;
= 'L': Lower block triangular, eigenvalues are not
exchanged on exit.
Input/Output Parameters
N1 (input/output) INTEGER
Size of the upper left block, N1 <= 2.
If UPLO = 'U' and INFO = 0, or UPLO = 'L' and INFO <> 0,
N1 and N2 are exchanged on exit; otherwise, N1 is
unchanged on exit.
N2 (input/output) INTEGER
Size of the lower right block, N2 <= 2.
If UPLO = 'U' and INFO = 0, or UPLO = 'L' and INFO <> 0,
N1 and N2 are exchanged on exit; otherwise, N2 is
unchanged on exit.
PREC (input) DOUBLE PRECISION
The machine precision, (relative machine precision)*base.
See the LAPACK Library routine DLAMCH.
A (input or input/output) DOUBLE PRECISION array, dimension
(LDA, N1+N2)
On entry, the leading (N1+N2)-by-(N1+N2) part of this
array must contain the matrix A of the pencil aAB - bD.
The (2,1) block, if UPLO = 'U', or the (1,2) block, if
UPLO = 'L', need not be set to zero.
On exit, if N1 = N2 = 1, this array contains the matrix
[ 0 1 ]
J' A J, where J = [ -1 0 ]; otherwise, this array is
unchanged on exit.
LDA INTEGER
The leading dimension of the array A. LDA >= N1+N2.
B (input or input/output) DOUBLE PRECISION array, dimension
(LDB, N1+N2)
On entry, the leading (N1+N2)-by-(N1+N2) part of this
array must contain the matrix B of the pencil aAB - bD.
The (2,1) block, if UPLO = 'U', or the (1,2) block, if
UPLO = 'L', need not be set to zero.
On exit, if N1 = N2 = 1, this array contains the matrix
J' B J; otherwise, this array is unchanged on exit.
LDB INTEGER
The leading dimension of the array B. LDB >= N1+N2.
D (input/output) DOUBLE PRECISION array, dimension
(LDD, N1+N2)
On entry, the leading (N1+N2)-by-(N1+N2) part of this
array must contain the matrix D of the pencil aAB - bD.
On exit, if N1 = 2 or N2 = 2, the leading
(N1+N2)-by-(N1+N2) part of this array contains the
transformed matrix D in real Schur form. If N1 = 1 and
N2 = 1, this array contains the matrix J' D J.
LDD INTEGER
The leading dimension of the array D. LDD >= N1+N2.
Q1 (output) DOUBLE PRECISION array, dimension (LDQ1, N1+N2)
The leading (N1+N2)-by-(N1+N2) part of this array contains
the first orthogonal transformation matrix.
LDQ1 INTEGER
The leading dimension of the array Q1. LDQ1 >= N1+N2.
Q2 (output) DOUBLE PRECISION array, dimension (LDQ2, N1+N2)
The leading (N1+N2)-by-(N1+N2) part of this array contains
the second orthogonal transformation matrix.
LDQ2 INTEGER
The leading dimension of the array Q2. LDQ2 >= N1+N2.
Q3 (output) DOUBLE PRECISION array, dimension (LDQ3, N1+N2)
The leading (N1+N2)-by-(N1+N2) part of this array contains
the third orthogonal transformation matrix.
LDQ3 INTEGER
The leading dimension of the array Q3. LDQ3 >= N1+N2.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
If N1+N2 = 2 then DWORK is not referenced.
LDWORK INTEGER
The dimension of the array DWORK.
If N1+N2 = 2, then LDWORK = 0; otherwise,
LDWORK >= 16*N1 + 10*N2 + 23, UPLO = 'U';
LDWORK >= 10*N1 + 16*N2 + 23, UPLO = 'L'.
Error Indicator
INFO INTEGER
= 0: succesful exit;
= 1: the QZ iteration failed in the LAPACK routine DGGEV;
= 2: another error occured while executing a routine in
DGGEV;
= 3: the QZ iteration failed in the LAPACK routine DGGES;
= 4: another error occured during execution of DGGES;
= 5: reordering of aA*B - bD in the LAPACK routine DTGSEN
failed because the transformed matrix pencil
aA*B - bD would be too far from generalized Schur
form; the problem is very ill-conditioned.
Method
The algorithm uses orthogonal transformations as described in [2] (page 21). The QZ algorithm is used for N1 = 2 or N2 = 2, but it always acts on an upper block triangular pencil.References
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical computation of deflating subspaces of skew-
Hamiltonian/Hamiltonian pencils.
SIAM J. Matrix Anal. Appl., 24 (1), pp. 165-190, 2002.
[2] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
Eigenproblems.
Tech. Rep., Technical University Chemnitz, Germany,
Nov. 2007.
Numerical Aspects
The algorithm is numerically backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute unitary matrices Q1, Q2, and Q3 for a complex 2-by-2
regular pencil aAB - bD, with A, B, D upper triangular, such that
Q3' A Q2, Q2' B Q1, Q3' D Q1 are still upper triangular, but the
eigenvalues are in reversed order. The matrices Q1, Q2, and Q3 are
represented by
( CO1 SI1 ) ( CO2 SI2 ) ( CO3 SI3 )
Q1 = ( ), Q2 = ( ), Q3 = ( ).
( -SI1' CO1 ) ( -SI2' CO2 ) ( -SI3' CO3 )
The notation M' denotes the conjugate transpose of the matrix M.
Specification
SUBROUTINE MB03CZ( A, LDA, B, LDB, D, LDD, CO1, SI1, CO2, SI2,
$ CO3, SI3 )
C .. Scalar Arguments ..
INTEGER LDA, LDB, LDD
DOUBLE PRECISION CO1, CO2, CO3
COMPLEX*16 SI1, SI2, SI3
C .. Array Arguments ..
COMPLEX*16 A( LDA, * ), B( LDB, * ), D( LDD, * )
Arguments
Input/Output Parameters
A (input) COMPLEX*16 array, dimension (LDA, 2)
On entry, the leading 2-by-2 upper triangular part of
this array must contain the matrix A of the pencil.
The (2,1) entry is not referenced.
LDA INTEGER
The leading dimension of the array A. LDA >= 2.
B (input) COMPLEX*16 array, dimension (LDB, 2)
On entry, the leading 2-by-2 upper triangular part of
this array must contain the matrix B of the pencil.
The (2,1) entry is not referenced.
LDB INTEGER
The leading dimension of the array B. LDB >= 2.
D (input) COMPLEX*16 array, dimension (LDD, 2)
On entry, the leading 2-by-2 upper triangular part of
this array must contain the matrix D of the pencil.
The (2,1) entry is not referenced.
LDD INTEGER
The leading dimension of the array D. LDD >= 2.
CO1 (output) DOUBLE PRECISION
The upper left element of the unitary matrix Q1.
SI1 (output) COMPLEX*16
The upper right element of the unitary matrix Q1.
CO2 (output) DOUBLE PRECISION
The upper left element of the unitary matrix Q2.
SI2 (output) COMPLEX*16
The upper right element of the unitary matrix Q2.
CO3 (output) DOUBLE PRECISION
The upper left element of the unitary matrix Q3.
SI3 (output) COMPLEX*16
The upper right element of the unitary matrix Q3.
Method
The algorithm uses unitary transformations as described on page 37 in [1].References
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical Computation of Deflating Subspaces of Embedded
Hamiltonian Pencils.
Tech. Rep. SFB393/99-15, Technical University Chemnitz,
Germany, June 1999.
Numerical Aspects
The algorithm is numerically backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute orthogonal matrices Q1 and Q2 for a real 2-by-2,
3-by-3, or 4-by-4 regular block upper triangular pencil
( A11 A12 ) ( B11 B12 )
aA - bB = a ( ) - b ( ), (1)
( 0 A22 ) ( 0 B22 )
such that the pencil a(Q2' A Q1) - b(Q2' B Q1) is still in block
upper triangular form, but the eigenvalues in Spec(A11, B11),
Spec(A22, B22) are exchanged, where Spec(X,Y) denotes the spectrum
of the matrix pencil (X,Y) and the notation M' denotes the
transpose of the matrix M.
Optionally, to upper triangularize the real regular pencil in
block lower triangular form
( A11 0 ) ( B11 0 )
aA - bB = a ( ) - b ( ), (2)
( A21 A22 ) ( B21 B22 )
while keeping the eigenvalues in the same diagonal position.
Specification
SUBROUTINE MB03DD( UPLO, N1, N2, PREC, A, LDA, B, LDB, Q1, LDQ1,
$ Q2, LDQ2, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LDB, LDQ1, LDQ2, LDWORK, N1, N2
DOUBLE PRECISION PREC
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DWORK( * ),
$ Q1( LDQ1, * ), Q2( LDQ2, * )
Arguments
Mode Parameters
UPLO CHARACTER*1
Specifies if the pencil is in lower or upper block
triangular form on entry, as follows:
= 'U': Upper block triangular, eigenvalues are exchanged
on exit;
= 'T': Upper block triangular, B triangular, eigenvalues
are exchanged on exit;
= 'L': Lower block triangular, eigenvalues are not
exchanged on exit.
Input/Output Parameters
N1 (input/output) INTEGER
Size of the upper left block, N1 <= 2.
If UPLO = 'U' or UPLO = 'T' and INFO = 0, or UPLO = 'L'
and INFO <> 0, N1 and N2 are exchanged on exit; otherwise,
N1 is unchanged on exit.
N2 (input/output) INTEGER
Size of the lower right block, N2 <= 2.
If UPLO = 'U' or UPLO = 'T' and INFO = 0, or UPLO = 'L'
and INFO <> 0, N1 and N2 are exchanged on exit; otherwise,
N2 is unchanged on exit.
PREC (input) DOUBLE PRECISION
The machine precision, (relative machine precision)*base.
See the LAPACK Library routine DLAMCH.
A (input/output) DOUBLE PRECISION array, dimension
(LDA, N1+N2)
On entry, the leading (N1+N2)-by-(N1+N2) part of this
array must contain the matrix A of the pencil aA - bB.
On exit, if N1 = N2 = 1, this array is unchanged, if
UPLO = 'U' or UPLO = 'T', but, if UPLO = 'L', it contains
[ 0 1 ]
the matrix J' A J, where J = [ -1 0 ]; otherwise, this
array contains the transformed quasi-triangular matrix in
generalized real Schur form.
LDA INTEGER
The leading dimension of the array A. LDA >= N1+N2.
B (input/output) DOUBLE PRECISION array, dimension
(LDB, N1+N2)
On entry, the leading (N1+N2)-by-(N1+N2) part of this
array must contain the matrix B of the pencil aA - bB.
On exit, if N1 = N2 = 1, this array is unchanged, if
UPLO = 'U' or UPLO = 'T', but, if UPLO = 'L', it contains
the matrix J' B J; otherwise, this array contains the
transformed upper triangular matrix in generalized real
Schur form.
LDB INTEGER
The leading dimension of the array B. LDB >= N1+N2.
Q1 (output) DOUBLE PRECISION array, dimension (LDQ1, N1+N2)
The leading (N1+N2)-by-(N1+N2) part of this array contains
the first orthogonal transformation matrix.
LDQ1 INTEGER
The leading dimension of the array Q1. LDQ1 >= N1+N2.
Q2 (output) DOUBLE PRECISION array, dimension (LDQ2, N1+N2)
The leading (N1+N2)-by-(N1+N2) part of this array contains
the second orthogonal transformation matrix.
LDQ2 INTEGER
The leading dimension of the array Q2. LDQ2 >= N1+N2.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
If N1+N2 = 2 then DWORK is not referenced.
LDWORK INTEGER
The dimension of the array DWORK.
If N1+N2 = 2, then LDWORK = 0; otherwise,
LDWORK >= 16*N1 + 10*N2 + 23, if UPLO = 'U';
LDWORK >= 7*N1 + 7*N2 + 16, if UPLO = 'T';
LDWORK >= 10*N1 + 16*N2 + 23, if UPLO = 'L'.
For good performance LDWORK should be generally larger.
Error Indicator
INFO INTEGER
= 0: succesful exit;
= 3: the QZ iteration failed in the LAPACK routine DGGES
(if UPLO <> 'T') or DHGEQZ (if UPLO = 'T');
= 4: another error occured during execution of DHGEQZ;
= 5: reordering of aA - bB in the LAPACK routine DTGSEN
failed because the transformed matrix pencil aA - bB
would be too far from generalized Schur form;
the problem is very ill-conditioned.
Method
The algorithm uses orthogonal transformations as described in [2] (page 30). The QZ algorithm is used for N1 = 2 or N2 = 2, but it always acts on an upper block triangular pencil.References
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical computation of deflating subspaces of skew-
Hamiltonian/Hamiltonian pencils.
SIAM J. Matrix Anal. Appl., 24 (1), pp. 165-190, 2002.
[2] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
Eigenproblems.
Tech. Rep., Technical University Chemnitz, Germany,
Nov. 2007.
Numerical Aspects
The algorithm is numerically backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute unitary matrices Q1 and Q2 for a complex 2-by-2 regular
pencil aA - bB with A, B upper triangular, such that
Q2' (aA - bB) Q1 is still upper triangular but the eigenvalues are
in reversed order. The matrices Q1 and Q2 are represented by
( CO1 SI1 ) ( CO2 SI2 )
Q1 = ( ), Q2 = ( ).
( -SI1' CO1 ) ( -SI2' CO2 )
The notation M' denotes the conjugate transpose of the matrix M.
Specification
SUBROUTINE MB03DZ( A, LDA, B, LDB, CO1, SI1, CO2, SI2 )
C .. Scalar Arguments ..
INTEGER LDA, LDB
DOUBLE PRECISION CO1, CO2
COMPLEX*16 SI1, SI2
C .. Array Arguments ..
COMPLEX*16 A( LDA, * ), B( LDB, * )
Arguments
Input/Output Parameters
A (input) COMPLEX*16 array, dimension (LDA, 2)
On entry, the leading 2-by-2 upper triangular part of
this array must contain the matrix A of the pencil.
The (2,1) entry is not referenced.
LDA INTEGER
The leading dimension of the array A. LDA >= 2.
B (input) COMPLEX*16 array, dimension (LDB, 2)
On entry, the leading 2-by-2 upper triangular part of
this array must contain the matrix B of the pencil.
The (2,1) entry is not referenced.
LDB INTEGER
The leading dimension of the array B. LDB >= 2.
CO1 (output) DOUBLE PRECISION
The upper left element of the unitary matrix Q1.
SI1 (output) COMPLEX*16
The upper right element of the unitary matrix Q1.
CO2 (output) DOUBLE PRECISION
The upper left element of the unitary matrix Q2.
SI2 (output) COMPLEX*16
The upper right element of the unitary matrix Q2.
Method
The algorithm uses unitary transformations as described on page 42 in [1].References
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical Computation of Deflating Subspaces of Embedded
Hamiltonian Pencils.
Tech. Rep. SFB393/99-15, Technical University Chemnitz,
Germany, June 1999.
Numerical Aspects
The algorithm is numerically backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute orthogonal matrices Q1, Q2, Q3 for a real 2-by-2 or
4-by-4 regular pencil
( A11 0 ) ( B11 0 ) ( 0 D12 )
aAB - bD = a ( ) ( ) - b ( ), (1)
( 0 A22 ) ( 0 B22 ) ( D21 0 )
such that Q3' A Q2 and Q2' B Q1 are upper triangular, Q3' D Q1 is
upper quasi-triangular, and the eigenvalues with negative real
parts (if there are any) are allocated on the top. The notation M'
denotes the transpose of the matrix M. The submatrices A11, A22,
B11, B22 and D12 are upper triangular. If D21 is 2-by-2, then all
other blocks are nonsingular and the product
-1 -1 -1 -1
A22 D21 B11 A11 D12 B22 has a pair of complex conjugate
eigenvalues.
Specification
SUBROUTINE MB03ED( N, PREC, A, LDA, B, LDB, D, LDD, Q1, LDQ1, Q2,
$ LDQ2, Q3, LDQ3, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDD, LDQ1, LDQ2, LDQ3, LDWORK,
$ N
DOUBLE PRECISION PREC
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), D( LDD, * ),
$ DWORK( * ), Q1( LDQ1, * ), Q2( LDQ2, * ),
$ Q3( LDQ3, * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the input pencil, N = 2 or N = 4.
PREC (input) DOUBLE PRECISION
The machine precision, (relative machine precision)*base.
See the LAPACK Library routine DLAMCH.
A (input) DOUBLE PRECISION array, dimension (LDA, N)
The leading N-by-N upper triangular part of this array
must contain the upper triangular matrix A of the pencil
aAB - bD. The strictly lower triangular part and the
entries of the (1,2) block are not referenced.
LDA INTEGER
The leading dimension of the array A. LDA >= N.
B (input) DOUBLE PRECISION array, dimension (LDB, N)
The leading N-by-N upper triangular part of this array
must contain the upper triangular matrix B of the pencil
aAB - bD. The strictly lower triangular part and the
entries of the (1,2) block are not referenced.
LDB INTEGER
The leading dimension of the array B. LDB >= N.
D (input/output) DOUBLE PRECISION array, dimension (LDD, N)
On entry, the leading N-by-N part of this array must
contain the matrix D of the pencil aAB - bD.
On exit, if N = 4, the leading N-by-N part of this array
contains the transformed matrix D in real Schur form.
If N = 2, this array is unchanged on exit.
LDD INTEGER
The leading dimension of the array D. LDD >= N.
Q1 (output) DOUBLE PRECISION array, dimension (LDQ1, N)
The leading N-by-N part of this array contains the first
orthogonal transformation matrix.
LDQ1 INTEGER
The leading dimension of the array Q1. LDQ1 >= N.
Q2 (output) DOUBLE PRECISION array, dimension (LDQ2, N)
The leading N-by-N part of this array contains the second
orthogonal transformation matrix.
LDQ2 INTEGER
The leading dimension of the array Q2. LDQ2 >= N.
Q3 (output) DOUBLE PRECISION array, dimension (LDQ3, N)
The leading N-by-N part of this array contains the third
orthogonal transformation matrix.
LDQ3 INTEGER
The leading dimension of the array Q3. LDQ3 >= N.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
If N = 2, then DWORK is not referenced.
LDWORK INTEGER
The dimension of the array DWORK.
If N = 4, then LDWORK >= 79. For good performance LDWORK
should be generally larger.
If N = 2, then LDWORK >= 0.
Error Indicator
INFO INTEGER
= 0: succesful exit;
= 1: the QZ iteration failed in the LAPACK routine DGGES;
= 2: another error occured during execution of DGGES.
Method
The algorithm uses orthogonal transformations as described on page 20 in [2].References
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical computation of deflating subspaces of skew-
Hamiltonian/Hamiltonian pencils.
SIAM J. Matrix Anal. Appl., 24 (1), pp. 165-190, 2002.
[2] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
Eigenproblems.
Tech. Rep., Technical University Chemnitz, Germany,
Nov. 2007.
Numerical Aspects
The algorithm is numerically backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute orthogonal matrices Q1 and Q2 for a real 2-by-2 or
4-by-4 regular pencil
( A11 0 ) ( 0 B12 )
aA - bB = a ( ) - b ( ), (1)
( 0 A22 ) ( B21 0 )
such that Q2' A Q1 is upper triangular, Q2' B Q1 is upper quasi-
triangular, and the eigenvalues with negative real parts (if there
are any) are allocated on the top. The notation M' denotes the
transpose of the matrix M. The submatrices A11, A22, and B12 are
upper triangular. If B21 is 2-by-2, then all the other blocks are
-1 -1
nonsingular and the product A11 B12 A22 B21 has a pair of
complex conjugate eigenvalues.
Specification
SUBROUTINE MB03FD( N, PREC, A, LDA, B, LDB, Q1, LDQ1, Q2, LDQ2,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDQ1, LDQ2, LDWORK, N
DOUBLE PRECISION PREC
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DWORK( * ),
$ Q1( LDQ1, * ), Q2( LDQ2, * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the input pencil, N = 2 or N = 4.
PREC (input) DOUBLE PRECISION
The machine precision, (relative machine precision)*base.
See the LAPACK Library routine DLAMCH.
A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
On entry, the leading N-by-N part of this array must
contain the matrix A of the pencil aA - bB.
If N = 2, the diagonal elements only are referenced.
On exit, if N = 4, the leading N-by-N part of this array
contains the transformed upper triangular matrix of the
generalized real Schur form of the pencil aA - bB.
If N = 2, this array is unchanged on exit.
LDA INTEGER
The leading dimension of the array A. LDA >= N.
B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
On entry, the leading N-by-N part of this array must
contain the matrix B of the pencil aA - bB.
If N = 2, the anti-diagonal elements only are referenced.
On exit, if N = 4, the leading N-by-N part of this array
contains the transformed real Schur matrix of the
generalized real Schur form of the pencil aA - bB.
If N = 2, this array is unchanged on exit.
LDB INTEGER
The leading dimension of the array B. LDB >= N.
Q1 (output) DOUBLE PRECISION array, dimension (LDQ1, N)
The leading N-by-N part of this array contains the first
orthogonal transformation matrix.
LDQ1 INTEGER
The leading dimension of the array Q1. LDQ1 >= N.
Q2 (output) DOUBLE PRECISION array, dimension (LDQ2, N)
The leading N-by-N part of this array contains the second
orthogonal transformation matrix.
LDQ2 INTEGER
The leading dimension of the array Q2. LDQ2 >= N.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
If N = 2, then DWORK is not referenced.
LDWORK INTEGER
The dimension of the array DWORK.
If N = 4, then LDWORK >= 63. For good performance LDWORK
should be generally larger.
If N = 2, then LDWORK >= 0.
Error Indicator
INFO INTEGER
= 0: succesful exit;
= 1: the QZ iteration failed in the LAPACK routine DGGES;
= 2: another error occured during execution of DGGES.
Method
The algorithm uses orthogonal transformations as described on page 29 in [2].References
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical computation of deflating subspaces of skew-
Hamiltonian/Hamiltonian pencils.
SIAM J. Matrix Anal. Appl., 24 (1), pp. 165-190, 2002.
[2] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
Eigenproblems.
Tech. Rep., Technical University Chemnitz, Germany,
Nov. 2007.
Numerical Aspects
The algorithm is numerically backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the eigenvalues of a complex N-by-N skew-Hamiltonian/
Hamiltonian pencil aS - bH, with
( B F ) ( Z11 Z12 )
S = J Z' J' Z and H = ( ), Z = ( ),
( G -B' ) ( Z21 Z22 )
(1)
( 0 I )
J = ( ).
( -I 0 )
The structured Schur form of the embedded real skew-Hamiltonian/
skew-Hamiltonian pencil, aB_S - bB_T, with B_S = J B_Z' J' B_Z,
( Re(Z11) -Im(Z11) | Re(Z12) -Im(Z12) )
( | )
( Im(Z11) Re(Z11) | Im(Z12) Re(Z12) )
( | )
B_Z = (---------------------+---------------------) ,
( | )
( Re(Z21) -Im(Z21) | Re(Z22) -Im(Z22) )
( | )
( Im(Z21) Re(Z21) | Im(Z22) Re(Z22) )
(2)
( -Im(B) -Re(B) | -Im(F) -Re(F) )
( | )
( Re(B) -Im(B) | Re(F) -Im(F) )
( | )
B_T = (-----------------+-----------------) , T = i*H,
( | )
( -Im(G) -Re(G) | -Im(B') Re(B') )
( | )
( Re(G) -Im(G) | -Re(B') -Im(B') )
is determined and used to compute the eigenvalues. Optionally, if
COMPQ = 'C', an orthonormal basis of the right deflating subspace,
Def_-(S, H), of the pencil aS - bH in (1), corresponding to the
eigenvalues with strictly negative real part, is computed. Namely,
after transforming aB_S - bB_H, in the factored form, by unitary
matrices, we have B_Sout = J B_Zout' J' B_Zout,
( BA BD ) ( BB BF )
B_Zout = ( ) and B_Hout = ( ), (3)
( 0 BC ) ( 0 -BB' )
and the eigenvalues with strictly negative real part of the
complex pencil aB_Sout - bB_Hout are moved to the top. The
notation M' denotes the conjugate transpose of the matrix M.
Optionally, if COMPU = 'C', an orthonormal basis of the companion
subspace, range(P_U) [1], which corresponds to the eigenvalues
with negative real part, is computed. The embedding doubles the
multiplicities of the eigenvalues of the pencil aS - bH.
Specification
SUBROUTINE MB03FZ( COMPQ, COMPU, ORTH, N, Z, LDZ, B, LDB, FG,
$ LDFG, NEIG, D, LDD, C, LDC, Q, LDQ, U, LDU,
$ ALPHAR, ALPHAI, BETA, IWORK, LIWORK, DWORK,
$ LDWORK, ZWORK, LZWORK, BWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, COMPU, ORTH
INTEGER INFO, LDB, LDC, LDD, LDFG, LDQ, LDU, LDWORK,
$ LDZ, LIWORK, LZWORK, N, NEIG
C .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION ALPHAI( * ), ALPHAR( * ), BETA( * ), DWORK( * )
COMPLEX*16 B( LDB, * ), C( LDC, * ), D( LDD, * ),
$ FG( LDFG, * ), Q( LDQ, * ), U( LDU, * ),
$ Z( LDZ, * ), ZWORK( * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
Specifies whether to compute the right deflating subspace
corresponding to the eigenvalues of aS - bH with strictly
negative real part.
= 'N': do not compute the deflating subspace;
= 'C': compute the deflating subspace and store it in the
leading subarray of Q.
COMPU CHARACTER*1
Specifies whether to compute the companion subspace
corresponding to the eigenvalues of aS - bH with strictly
negative real part.
= 'N': do not compute the companion subspace;
= 'C': compute the companion subspace and store it in the
leading subarray of U.
ORTH CHARACTER*1
If COMPQ = 'C' or COMPU = 'C', specifies the technique for
computing the orthonormal bases of the deflating subspace
and companion subspace, as follows:
= 'P': QR factorization with column pivoting;
= 'S': singular value decomposition.
If COMPQ = 'N' and COMPU = 'N', the ORTH value is not
used.
Input/Output Parameters
N (input) INTEGER
Order of the pencil aS - bH. N >= 0, even.
Z (input/output) COMPLEX*16 array, dimension (LDZ, N)
On entry, the leading N-by-N part of this array must
contain the non-trivial factor Z in the factorization
S = J Z' J' Z of the skew-Hamiltonian matrix S.
On exit, if COMPQ = 'C' or COMPU = 'C', the leading
N-by-N part of this array contains the upper triangular
matrix BA in (3) (see also METHOD). The strictly lower
triangular part is not zeroed.
If COMPQ = 'N' and COMPU = 'N', this array is unchanged
on exit.
LDZ INTEGER
The leading dimension of the array Z. LDZ >= MAX(1, N).
B (input/output) COMPLEX*16 array, dimension (LDB, N)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix B.
On exit, if COMPQ = 'C' or COMPU = 'C', the leading
N-by-N part of this array contains the upper triangular
matrix BB in (3) (see also METHOD). The strictly lower
triangular part is not zeroed.
If COMPQ = 'N' and COMPU = 'N', this array is unchanged
on exit.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N).
FG (input/output) COMPLEX*16 array, dimension (LDFG, N)
On entry, the leading N/2-by-N/2 lower triangular part of
this array must contain the lower triangular part of the
Hermitian matrix G, and the N/2-by-N/2 upper triangular
part of the submatrix in the columns 2 to N/2+1 of this
array must contain the upper triangular part of the
Hermitian matrix F.
On exit, if COMPQ = 'C' or COMPU = 'C', the leading
N-by-N part of this array contains the Hermitian matrix
BF in (3) (see also METHOD). The strictly lower triangular
part of the input matrix is preserved. The diagonal
elements might have tiny imaginary parts.
If COMPQ = 'N' and COMPU = 'N', this array is unchanged
on exit.
LDFG INTEGER
The leading dimension of the array FG. LDFG >= MAX(1, N).
NEIG (output) INTEGER
If COMPQ = 'C' or COMPU = 'C', the number of eigenvalues
in aS - bH with strictly negative real part.
D (output) COMPLEX*16 array, dimension (LDD, N)
If COMPQ = 'C' or COMPU = 'C', the leading N-by-N part of
this array contains the matrix BD in (3) (see METHOD).
If COMPQ = 'N' and COMPU = 'N', this array is not
referenced.
LDD INTEGER
The leading dimension of the array D.
LDD >= 1, if COMPQ = 'N' and COMPU = 'N';
LDD >= MAX(1, N), if COMPQ = 'C' or COMPU = 'C'.
C (output) COMPLEX*16 array, dimension (LDC, N)
If COMPQ = 'C' or COMPU = 'C', the leading N-by-N part of
this array contains the lower triangular matrix BC in (3)
(see also METHOD). The strictly upper triangular part is
not zeroed.
If COMPQ = 'N' and COMPU = 'N', this array is not
referenced.
LDC INTEGER
The leading dimension of the array C.
LDC >= 1, if COMPQ = 'N' and COMPU = 'N';
LDC >= MAX(1, N), if COMPQ = 'C' or COMPU = 'C'.
Q (output) COMPLEX*16 array, dimension (LDQ, 2*N)
On exit, if COMPQ = 'C', the leading N-by-NEIG part of
this array contains an orthonormal basis of the right
deflating subspace corresponding to the eigenvalues of the
pencil aS - bH with strictly negative real part.
The remaining entries are meaningless.
If COMPQ = 'N', this array is not referenced.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1, 2*N), if COMPQ = 'C'.
U (output) COMPLEX*16 array, dimension (LDU, 2*N)
On exit, if COMPU = 'C', the leading N-by-NEIG part of
this array contains an orthonormal basis of the companion
subspace corresponding to the eigenvalues of the
pencil aS - bH with strictly negative real part. The
remaining entries are meaningless.
If COMPU = 'N', this array is not referenced.
LDU INTEGER
The leading dimension of the array U.
LDU >= 1, if COMPU = 'N';
LDU >= MAX(1, N), if COMPU = 'C'.
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
The real parts of each scalar alpha defining an eigenvalue
of the pencil aS - bH.
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
The imaginary parts of each scalar alpha defining an
eigenvalue of the pencil aS - bH.
If ALPHAI(j) is zero, then the j-th eigenvalue is real.
BETA (output) DOUBLE PRECISION array, dimension (N)
The scalars beta that define the eigenvalues of the pencil
aS - bH.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent the j-th eigenvalue of the pencil
aS - bH, in the form lambda = alpha/beta. Since lambda may
overflow, the ratios should not, in general, be computed.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK INTEGER
The dimension of the array IWORK. LIWORK >= 2*N+9.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
On exit, if INFO = -26, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= c*N**2 + N + MAX(6*N, 27), where
c = 18, if COMPU = 'C';
c = 16, if COMPQ = 'C' and COMPU = 'N';
c = 13, if COMPQ = 'N' and COMPU = 'N'.
For good performance LDWORK should be generally larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) returns the optimal LZWORK.
On exit, if INFO = -28, ZWORK(1) returns the minimum
value of LZWORK.
LZWORK INTEGER
The dimension of the array ZWORK.
LZWORK >= 8*N + 28, if COMPQ = 'C';
LZWORK >= 6*N + 28, if COMPQ = 'N' and COMPU = 'C';
LZWORK >= 1, if COMPQ = 'N' and COMPU = 'N'.
For good performance LZWORK should be generally larger.
If LZWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
ZWORK array, returns this value as the first entry of
the ZWORK array, and no error message related to LZWORK
is issued by XERBLA.
BWORK LOGICAL array, dimension (LBWORK)
LBWORK >= 0, if COMPQ = 'N' and COMPU = 'N';
LBWORK >= N, if COMPQ = 'C' or COMPU = 'C'.
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: the algorithm was not able to reveal information
about the eigenvalues from the 2-by-2 blocks in the
SLICOT Library routine MB03BD (called by MB04ED);
= 2: periodic QZ iteration failed in the SLICOT Library
routines MB03BD or MB03BZ when trying to
triangularize the 2-by-2 blocks;
= 3: the singular value decomposition failed in the LAPACK
routine ZGESVD (for ORTH = 'S').
Method
First T = i*H is set. Then, the embeddings, B_Z and B_T, of the
matrices S and T, are determined and, subsequently, the SLICOT
Library routine MB04ED is applied to compute the structured Schur
form, i.e., the factorizations
~ ( BZ11 BZ12 )
B_Z = U' B_Z Q = ( ) and
( 0 BZ22 )
~ ( T11 T12 )
B_T = J Q' J' B_T Q = ( ),
( 0 T11' )
where Q is real orthogonal, U is real orthogonal symplectic, BZ11,
BZ22' are upper triangular and T11 is upper quasi-triangular.
Second, the SLICOT Library routine MB03IZ is applied, to compute a
~ ~
unitary matrix Q and a unitary symplectic matrix U, such that
~ ~
~ ~ ~ ( Z11 Z12 )
U' B_Z Q = ( ~ ) =: B_Zout,
( 0 Z22 )
~ ~ ~ ( H11 H12 )
J Q' J'(-i*B_T) Q = ( ) =: B_Hout,
( 0 -H11' )
~ ~
with Z11, Z22', H11 upper triangular, and such that the spectrum
~ ~ ~
Spec_-(J B_Z' J' B_Z, -i*B_T) is contained in the spectrum of the
~ ~
2*NEIG-by-2*NEIG leading principal subpencil aZ22'*Z11 - bH11.
Finally, the right deflating subspace and the companion subspace
are computed. See also page 21 in [1] for more details.
References
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical Computation of Deflating Subspaces of Embedded
Hamiltonian Pencils.
Tech. Rep. SFB393/99-15, Technical University Chemnitz,
Germany, June 1999.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) complex floating point operations.Further Comments
This routine does not perform any scaling of the matrices. Scaling might sometimes be useful, and it should be done externally.Example
Program Text
* MB03FZ EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 50 )
INTEGER LDB, LDC, LDD, LDFG, LDQ, LDU, LDWORK, LDZ,
$ LIWORK, LZWORK
PARAMETER ( LDB = NMAX, LDC = NMAX, LDD = NMAX,
$ LDFG = NMAX, LDQ = 2*NMAX, LDU = NMAX,
$ LDWORK = 18*NMAX*NMAX + NMAX + 3 +
$ MAX( 2*NMAX, 24 ), LDZ = NMAX,
$ LIWORK = 2*NMAX + 9, LZWORK = 8*NMAX + 28 )
*
* .. Local Scalars ..
CHARACTER COMPQ, COMPU, ORTH
INTEGER I, INFO, J, M, N, NEIG
*
* .. Local Arrays ..
COMPLEX*16 B( LDB, NMAX ), C( LDC, NMAX ), D( LDD, NMAX ),
$ FG( LDFG, NMAX ), Q( LDQ, 2*NMAX ),
$ U( LDU, 2*NMAX ), Z( LDZ, NMAX ),
$ ZWORK( LZWORK )
DOUBLE PRECISION ALPHAI( NMAX ), ALPHAR( NMAX ), BETA( NMAX ),
$ DWORK( LDWORK )
INTEGER IWORK( LIWORK )
LOGICAL BWORK( NMAX )
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
*
* .. External Subroutines ..
EXTERNAL MB03FZ
*
* .. Intrinsic Functions ..
INTRINSIC MAX, MOD
*
* .. Executable Statements ..
*
WRITE( NOUT, FMT = 99999 )
* Skip the heading in the data file and read in the data.
READ( NIN, FMT = * )
READ( NIN, FMT = * ) COMPQ, COMPU, ORTH, N
IF( N.LT.0 .OR. N.GT.NMAX .OR. MOD( N, 2 ).NE.0 ) THEN
WRITE( NOUT, FMT = 99998 ) N
ELSE
M = N/2
READ( NIN, FMT = * ) ( ( Z( I, J ), J = 1, N ), I = 1, N )
READ( NIN, FMT = * ) ( ( B( I, J ), J = 1, M ), I = 1, M )
READ( NIN, FMT = * ) ( ( FG( I, J ), J = 1, M+1 ), I = 1, M )
* Compute the eigenvalues and orthogonal bases of the right
* deflating subspace and companion subspace of a complex
* skew-Hamiltonian/Hamiltonian pencil, corresponding to the
* eigenvalues with strictly negative real part.
CALL MB03FZ( COMPQ, COMPU, ORTH, N, Z, LDZ, B, LDB, FG, LDFG,
$ NEIG, D, LDD, C, LDC, Q, LDQ, U, LDU, ALPHAR,
$ ALPHAI, BETA, IWORK, LIWORK, DWORK, LDWORK, ZWORK,
$ LZWORK, BWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99997 ) INFO
ELSE
WRITE( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( Z( I, J ), J = 1, N )
10 CONTINUE
IF( LSAME( COMPQ, 'C' ) .OR. LSAME( COMPU, 'C' ) ) THEN
WRITE( NOUT, FMT = 99994 )
DO 20 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( D( I, J ), J = 1, N )
20 CONTINUE
WRITE( NOUT, FMT = 99993 )
DO 30 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( C( I, J ), J = 1, N )
30 CONTINUE
WRITE( NOUT, FMT = 99992 )
DO 40 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( B( I, J ), J = 1, N )
40 CONTINUE
WRITE( NOUT, FMT = 99991 )
DO 50 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( FG( I, J ), J = 1, N )
50 CONTINUE
END IF
WRITE( NOUT, FMT = 99990 )
WRITE( NOUT, FMT = 99989 ) ( ALPHAR( I ), I = 1, N )
WRITE( NOUT, FMT = 99988 )
WRITE( NOUT, FMT = 99989 ) ( ALPHAI( I ), I = 1, N )
WRITE( NOUT, FMT = 99987 )
WRITE( NOUT, FMT = 99989 ) ( BETA( I ), I = 1, N )
IF( LSAME( COMPQ, 'C' ) .AND. NEIG.GT.0 ) THEN
WRITE( NOUT, FMT = 99986 )
DO 60 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( Q( I, J ), J = 1, NEIG )
60 CONTINUE
END IF
IF( LSAME( COMPU, 'C' ) .AND. NEIG.GT.0 ) THEN
WRITE( NOUT, FMT = 99985 )
DO 70 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( U( I, J ), J = 1, NEIG )
70 CONTINUE
END IF
IF( LSAME( COMPQ, 'C' ) .OR. LSAME( COMPU, 'C' ) )
$ WRITE( NOUT, FMT = 99984 ) NEIG
END IF
END IF
STOP
*
99999 FORMAT ( 'MB03FZ EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT ( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT ( 'INFO on exit from MB03FZ = ', I2 )
99996 FORMAT (/'The matrix Z on exit is ' )
99995 FORMAT ( 20( 1X, F9.4, SP, F9.4, S, 'i ') )
99994 FORMAT (/'The matrix D is ' )
99993 FORMAT (/'The matrix C is ' )
99992 FORMAT (/'The matrix B on exit is ' )
99991 FORMAT (/'The matrix F on exit is ' )
99990 FORMAT (/'The vector ALPHAR is ' )
99989 FORMAT ( 50( 1X, F8.4 ) )
99988 FORMAT (/'The vector ALPHAI is ' )
99987 FORMAT (/'The vector BETA is ' )
99986 FORMAT (/'The deflating subspace corresponding to the ',
$ 'eigenvalues with negative real part is ' )
99985 FORMAT (/'The companion subspace corresponding to the ',
$ 'eigenvalues with negative real part is ' )
99984 FORMAT (/'The number of eigenvalues in the initial pencil with ',
$ 'negative real part is ', I2 )
END
Program Data
MB03FZ EXAMPLE PROGRAM DATA
C C P 4
(0.0328,0.9611) (0.6428,0.2585) (0.7033,0.4254) (0.2552,0.7053)
(0.0501,0.2510) (0.2827,0.8865) (0.4719,0.5387) (0.0389,0.5676)
(0.5551,0.4242) (0.0643,0.2716) (0.1165,0.7875) (0.9144,0.3891)
(0.0539,0.7931) (0.0408,0.2654) (0.9912,0.0989) (0.0991,0.6585)
(0.0547,0.8726) (0.4008,0.8722)
(0.7423,0.6166) (0.2631,0.5872)
0.8740 0.3697 (0.9178,0.6418)
(0.7748,0.5358) 0.1652 0.2441
Program Results
MB03FZ EXAMPLE PROGRAM RESULTS
The matrix Z on exit is
1.1347 -0.1694i 0.0920 -0.0894i 0.5253 +0.0280i -0.0597 +0.1098i
0.0000 +0.0000i -0.9874 -0.6015i 0.2523 -0.0600i 0.3178 -0.0902i
0.5551 +0.4242i 0.0643 +0.2716i 0.7553 -0.3356i 0.4772 -0.3177i
0.0539 +0.7931i 0.0408 +0.2654i 0.9912 +0.0989i 0.9064 -0.1055i
The matrix D is
-0.7634 -0.2773i -0.8466 -0.9586i -0.0308 -0.0175i -0.2754 -0.0715i
1.2612 -0.2643i -0.7291 -0.3165i 0.0282 -0.1748i 0.4091 +0.0233i
0.3773 -0.1536i -0.3937 -0.0480i -0.1635 +0.1617i -0.1775 +0.1277i
0.7540 -0.0280i -0.6860 -0.8306i -0.2446 +0.0943i -0.0722 +0.0517i
The matrix C is
0.5063 +0.1548i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i
-0.0046 +0.1049i 0.3884 +0.3420i 0.0000 +0.0000i 0.0000 +0.0000i
-1.1206 +0.1313i -0.2270 -0.1753i 0.4300 -0.6107i 0.0000 +0.0000i
-0.6127 -0.1939i -0.5713 -0.7913i 0.3739 -0.2943i -1.1501 -0.0850i
The matrix B on exit is
0.3322 +1.9093i -0.1216 -0.1193i -0.0030 +0.0330i 0.0405 +0.0592i
0.0000 +0.0000i 0.1863 -1.8998i 0.2983 +0.2974i 0.6636 +0.5916i
0.0000 +0.0000i 0.0000 +0.0000i 0.4459 -0.7452i -0.0625 +0.2197i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.1418 +0.7392i
The matrix F on exit is
0.0258 +0.0000i -0.0878 +0.1090i 0.3547 +0.5306i -0.0138 -0.8770i
0.7748 +0.5358i 0.0864 +0.0000i -0.3788 -0.2829i -0.3303 -0.0415i
0.0000 +0.0000i 0.0000 +0.0000i -0.0184 +0.0000i 0.1077 -0.0795i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i -0.0938 +0.0000i
The vector ALPHAR is
0.4295 -0.4295 0.0000 0.0000
The vector ALPHAI is
1.5363 1.5363 -1.4069 -0.7153
The vector BETA is
0.5000 0.5000 1.0000 1.0000
The deflating subspace corresponding to the eigenvalues with negative real part is
-0.2249 +0.4158i
-0.1984 -0.3100i
0.7286 -0.0427i
0.3282 -0.0251i
The companion subspace corresponding to the eigenvalues with negative real part is
-0.1542 -0.0712i
-0.4162 -0.3021i
-0.0806 -0.6946i
-0.4580 -0.0889i
The number of eigenvalues in the initial pencil with negative real part is 1
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB03GD.html 0000664 0000000 0000000 00000011142 14560147231 0020325 0 ustar 00root root 0000000 0000000
Purpose
To compute an orthogonal matrix Q and an orthogonal symplectic
matrix U for a real regular 2-by-2 or 4-by-4 skew-Hamiltonian/
Hamiltonian pencil a J B' J' B - b D with
( B11 B12 ) ( D11 D12 ) ( 0 I )
B = ( ), D = ( ), J = ( ),
( 0 B22 ) ( 0 -D11' ) ( -I 0 )
such that J Q' J' D Q and U' B Q keep block triangular form, but
the eigenvalues are reordered. The notation M' denotes the
transpose of the matrix M.
Specification
SUBROUTINE MB03GD( N, B, LDB, D, LDD, MACPAR, Q, LDQ, U, LDU,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDB, LDD, LDQ, LDU, LDWORK, N
C .. Array Arguments ..
DOUBLE PRECISION B( LDB, * ), D( LDD, * ), DWORK( * ),
$ MACPAR( * ), Q( LDQ, * ), U( LDU, * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the pencil a J B' J' B - b D. N = 2 or N = 4.
B (input) DOUBLE PRECISION array, dimension (LDB, N)
The leading N-by-N part of this array must contain the
non-trivial factor of the decomposition of the
skew-Hamiltonian input matrix J B' J' B. The (2,1) block
is not referenced.
LDB INTEGER
The leading dimension of the array B. LDB >= N.
D (input) DOUBLE PRECISION array, dimension (LDD, N)
The leading N/2-by-N part of this array must contain the
first block row of the second matrix of a J B' J' B - b D.
The matrix D has to be Hamiltonian. The strict lower
triangle of the (1,2) block is not referenced.
LDD INTEGER
The leading dimension of the array D. LDD >= N/2.
MACPAR (input) DOUBLE PRECISION array, dimension (2)
Machine parameters:
MACPAR(1) (machine precision)*base, DLAMCH( 'P' );
MACPAR(2) safe minimum, DLAMCH( 'S' ).
This argument is not used for N = 2.
Q (output) DOUBLE PRECISION array, dimension (LDQ, N)
The leading N-by-N part of this array contains the
orthogonal transformation matrix Q.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= N.
U (output) DOUBLE PRECISION array, dimension (LDU, N)
The leading N-by-N part of this array contains the
orthogonal symplectic transformation matrix U.
LDU INTEGER
The leading dimension of the array U. LDU >= N.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
If N = 2 then DWORK is not referenced.
LDWORK INTEGER
The length of the array DWORK.
If N = 2 then LDWORK >= 0; if N = 4 then LDWORK >= 12.
Error Indicator
INFO INTEGER
= 0: succesful exit;
= 1: B11 or B22 is a (numerically) singular matrix.
Method
The algorithm uses orthogonal transformations as described on page 22 in [1], but with an improved implementation.References
[1] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
Eigenproblems.
Tech. Rep., Technical University Chemnitz, Germany,
Nov. 2007.
Numerical Aspects
The algorithm is numerically backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute a unitary matrix Q and a unitary symplectic matrix U
for a complex regular 2-by-2 skew-Hamiltonian/Hamiltonian pencil
aS - bH with S = J Z' J' Z, where
( Z11 Z12 ) ( H11 H12 )
Z = ( ) and H = ( ),
( 0 Z22 ) ( 0 -H11' )
such that U' Z Q, (J Q J' )' H Q are both upper triangular, but the
eigenvalues of (J Q J')' ( aS - bH ) Q are in reversed order.
The matrices Q and U are represented by
( CO1 SI1 ) ( CO2 SI2 )
Q = ( ) and U = ( ), respectively.
( -SI1' CO1 ) ( -SI2' CO2 )
The notation M' denotes the conjugate transpose of the matrix M.
Specification
SUBROUTINE MB03GZ( Z11, Z12, Z22, H11, H12, CO1, SI1, CO2, SI2 )
C .. Scalar Arguments ..
DOUBLE PRECISION CO1, CO2
COMPLEX*16 H11, H12, SI1, SI2, Z11, Z12, Z22
Arguments
Input/Output Parameters
Z11 (input) COMPLEX*16
Upper left element of the non-trivial factor Z in the
factorization of S.
Z12 (input) COMPLEX*16
Upper right element of the non-trivial factor Z in the
factorization of S.
Z22 (input) COMPLEX*16
Lower right element of the non-trivial factor Z in the
factorization of S.
H11 (input) COMPLEX*16
Upper left element of the Hamiltonian matrix H.
H12 (input) COMPLEX*16
Upper right element of the Hamiltonian matrix H.
CO1 (output) DOUBLE PRECISION
Upper left element of Q.
SI1 (output) COMPLEX*16
Upper right element of Q.
CO2 (output) DOUBLE PRECISION
Upper left element of U.
SI2 (output) COMPLEX*16
Upper right element of U.
Method
The algorithm uses unitary and unitary symplectic transformations as described on page 37 in [1].References
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical Computation of Deflating Subspaces of Embedded
Hamiltonian Pencils.
Tech. Rep. SFB393/99-15, Technical University Chemnitz,
Germany, June 1999.
Numerical Aspects
The algorithm is numerically backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To determine an orthogonal matrix Q, for a real regular 2-by-2 or
4-by-4 skew-Hamiltonian/Hamiltonian pencil
( A11 A12 ) ( B11 B12 )
aA - bB = a ( ) - b ( )
( 0 A11' ) ( 0 -B11' )
in structured Schur form, such that J Q' J' (aA - bB) Q is still
in structured Schur form but the eigenvalues are exchanged. The
notation M' denotes the transpose of the matrix M.
Specification
SUBROUTINE MB03HD( N, A, LDA, B, LDB, MACPAR, Q, LDQ, DWORK,
$ INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDQ, N
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DWORK( * ),
$ MACPAR( * ), Q( LDQ, * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the pencil aA - bB. N = 2 or N = 4.
A (input) DOUBLE PRECISION array, dimension (LDA, N)
If N = 4, the leading N/2-by-N upper trapezoidal part of
this array must contain the first block row of the skew-
Hamiltonian matrix A of the pencil aA - bB in structured
Schur form. Only the entries (1,1), (1,2), (1,4), and
(2,2) are referenced.
If N = 2, this array is not referenced.
LDA INTEGER
The leading dimension of the array A. LDA >= N/2.
B (input) DOUBLE PRECISION array, dimension (LDB, N)
The leading N/2-by-N part of this array must contain the
first block row of the Hamiltonian matrix B of the
pencil aA - bB in structured Schur form. The entry (2,3)
is not referenced.
LDB INTEGER
The leading dimension of the array B. LDB >= N/2.
MACPAR (input) DOUBLE PRECISION array, dimension (2)
Machine parameters:
MACPAR(1) (machine precision)*base, DLAMCH( 'P' );
MACPAR(2) safe minimum, DLAMCH( 'S' ).
This argument is not used for N = 2.
Q (output) DOUBLE PRECISION array, dimension (LDQ, N)
The leading N-by-N part of this array contains the
orthogonal transformation matrix Q.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= N.
Workspace
DWORK DOUBLE PRECISION array, dimension (24)
If N = 2, then DWORK is not referenced.
Error Indicator
INFO INTEGER
= 0: succesful exit;
= 1: the leading N/2-by-N/2 block of the matrix B is
numerically singular, but slightly perturbed values
have been used. This is a warning.
Method
The algorithm uses orthogonal transformations as described on page 31 in [2]. The structure is exploited.References
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical computation of deflating subspaces of skew-
Hamiltonian/Hamiltonian pencils.
SIAM J. Matrix Anal. Appl., 24 (1), pp. 165-190, 2002.
[2] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
Eigenproblems.
Tech. Rep., Technical University Chemnitz, Germany,
Nov. 2007.
Numerical Aspects
The algorithm is numerically backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute a unitary matrix Q for a complex regular 2-by-2
skew-Hamiltonian/Hamiltonian pencil aS - bH with
( S11 S12 ) ( H11 H12 )
S = ( ), H = ( ),
( 0 S11' ) ( 0 -H11' )
such that J Q' J' (aS - bH) Q is upper triangular but the
eigenvalues are in reversed order. The matrix Q is represented by
( CO SI )
Q = ( ).
( -SI' CO )
The notation M' denotes the conjugate transpose of the matrix M.
Specification
SUBROUTINE MB03HZ( S11, S12, H11, H12, CO, SI )
C .. Scalar Arguments ..
DOUBLE PRECISION CO
COMPLEX*16 H11, H12, S11, S12, SI
Arguments
Input/Output Parameters
S11 (input) COMPLEX*16
Upper left element of the skew-Hamiltonian matrix S.
S12 (input) COMPLEX*16
Upper right element of the skew-Hamiltonian matrix S.
H11 (input) COMPLEX*16
Upper left element of the Hamiltonian matrix H.
H12 (input) COMPLEX*16
Upper right element of the Hamiltonian matrix H.
CO (output) DOUBLE PRECISION
Upper left element of Q.
SI (output) COMPLEX*16
Upper right element of Q.
Method
The algorithm uses unitary transformations as described on page 43 in [1].References
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical Computation of Deflating Subspaces of Embedded
Hamiltonian Pencils.
Tech. Rep. SFB393/99-15, Technical University Chemnitz,
Germany, June 1999.
Numerical Aspects
The algorithm is numerically backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To move the eigenvalues with strictly negative real parts of an
N-by-N real skew-Hamiltonian/Hamiltonian pencil aS - bH in
structured Schur form, with
( 0 I ) ( A D ) ( B F )
S = J Z' J' Z, J = ( ), Z = ( ), H = ( ),
( -I 0 ) ( 0 C ) ( 0 -B' )
to the leading principal subpencil, while keeping the triangular
form. Above, A is upper triangular, B is upper quasi-triangular,
and C is lower triangular.
The matrices Z and H are transformed by an orthogonal symplectic
matrix U and an orthogonal matrix Q such that
( Aout Dout )
Zout = U' Z Q = ( ), and
( 0 Cout )
(1)
( Bout Fout )
Hout = J Q' J' H Q = ( ),
( 0 -Bout' )
where Aout, Bout and Cout remain in triangular form. The notation
M' denotes the transpose of the matrix M.
Optionally, if COMPQ = 'I' or COMPQ = 'U', the orthogonal matrix Q
that fulfills (1) is computed.
Optionally, if COMPU = 'I' or COMPU = 'U', the orthogonal
symplectic matrix
( U1 U2 )
U = ( )
( -U2 U1 )
that fulfills (1) is computed.
Specification
SUBROUTINE MB03ID( COMPQ, COMPU, N, A, LDA, C, LDC, D, LDD, B,
$ LDB, F, LDF, Q, LDQ, U1, LDU1, U2, LDU2, NEIG,
$ IWORK, LIWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, COMPU
INTEGER INFO, LDA, LDB, LDC, LDD, LDF, LDQ, LDU1, LDU2,
$ LDWORK, LIWORK, N, NEIG
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), DWORK( * ), F( LDF, * ),
$ Q( LDQ, * ), U1( LDU1, * ), U2( LDU2, * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
Specifies whether or not the orthogonal transformations
should be accumulated in the array Q, as follows:
= 'N': Q is not computed;
= 'I': the array Q is initialized internally to the unit
matrix, and the orthogonal matrix Q is returned;
= 'U': the array Q contains an orthogonal matrix Q0 on
entry, and the matrix Q0*Q is returned, where Q
is the product of the orthogonal transformations
that are applied to the pencil aS - bH to reorder
the eigenvalues.
COMPU CHARACTER*1
Specifies whether or not the orthogonal symplectic
transformations should be accumulated in the arrays U1 and
U2, as follows:
= 'N': U1 and U2 are not computed;
= 'I': the arrays U1 and U2 are initialized internally,
and the submatrices U1 and U2 defining the
orthogonal symplectic matrix U are returned;
= 'U': the arrays U1 and U2 contain the corresponding
submatrices of an orthogonal symplectic matrix U0
on entry, and the updated submatrices U1 and U2
of the matrix product U0*U are returned, where U
is the product of the orthogonal symplectic
transformations that are applied to the pencil
aS - bH to reorder the eigenvalues.
Input/Output Parameters
N (input) INTEGER
The order of the pencil aS - bH. N >= 0, even.
A (input/output) DOUBLE PRECISION array, dimension
(LDA, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular matrix A. The elements of the
strictly lower triangular part of this array are not used.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Aout.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1, N/2).
C (input/output) DOUBLE PRECISION array, dimension
(LDC, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the lower triangular matrix C. The elements of the
strictly upper triangular part of this array are not used.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Cout.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1, N/2).
D (input/output) DOUBLE PRECISION array, dimension
(LDD, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix D.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Dout.
LDD INTEGER
The leading dimension of the array D. LDD >= MAX(1, N/2).
B (input/output) DOUBLE PRECISION array, dimension
(LDB, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper quasi-triangular matrix B.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed upper quasi-triangular part of
the matrix Bout.
The part below the first subdiagonal of this array is
not referenced.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N/2).
F (input/output) DOUBLE PRECISION array, dimension
(LDF, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular part of the symmetric matrix
F.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed upper triangular part of the
matrix Fout.
The strictly lower triangular part of this array is not
referenced, except for the element F(N/2,N/2-1), but its
initial value is preserved.
LDF INTEGER
The leading dimension of the array F. LDF >= MAX(1, N/2).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
On entry, if COMPQ = 'U', then the leading N-by-N part of
this array must contain a given matrix Q0, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q0 and the transformation matrix Q
used to transform the matrices Z and H.
On exit, if COMPQ = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q.
If COMPQ = 'N' this array is not referenced.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1, N), if COMPQ = 'I' or COMPQ = 'U'.
U1 (input/output) DOUBLE PRECISION array, dimension
(LDU1, N/2)
On entry, if COMPU = 'U', then the leading N/2-by-N/2 part
of this array must contain the upper left block of a
given matrix U0, and on exit, the leading N/2-by-N/2 part
of this array contains the updated upper left block U1 of
the product of the input matrix U0 and the transformation
matrix U used to transform the matrices Z and H.
On exit, if COMPU = 'I', then the leading N/2-by-N/2 part
of this array contains the upper left block U1 of the
orthogonal symplectic transformation matrix U.
If COMPU = 'N' this array is not referenced.
LDU1 INTEGER
The leading dimension of the array U1.
LDU1 >= 1, if COMPU = 'N';
LDU1 >= MAX(1, N/2), if COMPU = 'I' or COMPU = 'U'.
U2 (input/output) DOUBLE PRECISION array, dimension
(LDU2, N/2)
On entry, if COMPU = 'U', then the leading N/2-by-N/2 part
of this array must contain the upper right block of a
given matrix U0, and on exit, the leading N/2-by-N/2 part
of this array contains the updated upper right block U2 of
the product of the input matrix U0 and the transformation
matrix U used to transform the matrices Z and H.
On exit, if COMPU = 'I', then the leading N/2-by-N/2 part
of this array contains the upper right block U2 of the
orthogonal symplectic transformation matrix U.
If COMPU = 'N' this array is not referenced.
LDU2 INTEGER
The leading dimension of the array U2.
LDU2 >= 1, if COMPU = 'N';
LDU2 >= MAX(1, N/2), if COMPU = 'I' or COMPU = 'U'.
NEIG (output) INTEGER
The number of eigenvalues in aS - bH with strictly
negative real part.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK INTEGER
The dimension of the array IWORK.
LIWORK >= N+1.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The dimension of the array DWORK.
If COMPQ = 'N',
LDWORK >= MAX(2*N+48,171);
if COMPQ = 'I' or COMPQ = 'U',
LDWORK >= MAX(4*N+48,171).
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: the periodic QZ algorithm did not converge in SLICOT
Library routine MB03BB;
= 2: an error occured during the execution of MB03CD;
= 3: an error occured during the execution of MB03GD.
Method
The algorithm reorders the eigenvalues like the following scheme:
Step 1: Reorder the eigenvalues in the subpencil aC'*A - bB.
I. Reorder the eigenvalues with negative real parts to the
top.
II. Reorder the eigenvalues with positive real parts to the
bottom.
Step 2: Reorder the remaining eigenvalues with negative real
parts in the pencil aS - bH.
I. Exchange the eigenvalues between the last diagonal block
in aC'*A - bB and the last diagonal block in aS - bH.
II. Move the eigenvalues of the R-th block to the (MM+1)-th
block, where R denotes the number of upper quasi-
triangular blocks in aC'*A - bB and MM denotes the current
number of blocks in aC'*A - bB with eigenvalues with
negative real parts.
The algorithm uses a sequence of orthogonal transformations as
described on page 25 in [1]. To achieve those transformations the
elementary subroutines MB03CD and MB03GD are called for the
corresponding matrix structures.
References
[1] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
Eigenproblems.
Tech. Rep., Technical University Chemnitz, Germany,
Nov. 2007.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) real floating point operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To move the eigenvalues with strictly negative real parts of an
N-by-N complex skew-Hamiltonian/Hamiltonian pencil aS - bH in
structured Schur form, with
( 0 I )
S = J Z' J' Z, where J = ( ),
( -I 0 )
to the leading principal subpencil, while keeping the triangular
form. On entry, we have
( A D ) ( B F )
Z = ( ), H = ( ),
( 0 C ) ( 0 -B' )
where A and B are upper triangular and C is lower triangular.
Z and H are transformed by a unitary symplectic matrix U and a
unitary matrix Q such that
( Aout Dout )
Zout = U' Z Q = ( ), and
( 0 Cout )
(1)
( Bout Fout )
Hout = J Q' J' H Q = ( ),
( 0 -Bout' )
where Aout, Bout and Cout remain in triangular form. The notation
M' denotes the conjugate transpose of the matrix M.
Optionally, if COMPQ = 'I' or COMPQ = 'U', the unitary matrix Q
that fulfills (1) is computed.
Optionally, if COMPU = 'I' or COMPU = 'U', the unitary symplectic
matrix
( U1 U2 )
U = ( )
( -U2 U1 )
that fulfills (1) is computed.
Specification
SUBROUTINE MB03IZ( COMPQ, COMPU, N, A, LDA, C, LDC, D, LDD, B,
$ LDB, F, LDF, Q, LDQ, U1, LDU1, U2, LDU2, NEIG,
$ TOL, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, COMPU
INTEGER INFO, LDA, LDB, LDC, LDD, LDF, LDQ, LDU1, LDU2,
$ N, NEIG
DOUBLE PRECISION TOL
C .. Array Arguments ..
COMPLEX*16 A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), F( LDF, * ), Q( LDQ, * ),
$ U1( LDU1, * ), U2( LDU2, * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
Specifies whether or not the unitary transformations
should be accumulated in the array Q, as follows:
= 'N': Q is not computed;
= 'I': the array Q is initialized internally to the unit
matrix, and the unitary matrix Q is returned;
= 'U': the array Q contains a unitary matrix Q0 on
entry, and the matrix Q0*Q is returned, where Q
is the product of the unitary transformations
that are applied to the pencil aS - bH to reorder
the eigenvalues.
COMPU CHARACTER*1
Specifies whether or not the unitary symplectic
transformations should be accumulated in the arrays U1 and
U2, as follows:
= 'N': U1 and U2 are not computed;
= 'I': the arrays U1 and U2 are initialized internally,
and the submatrices U1 and U2 defining the
unitary symplectic matrix U are returned;
= 'U': the arrays U1 and U2 contain the corresponding
submatrices of a unitary symplectic matrix U0
on entry, and the updated submatrices U1 and U2
of the matrix product U0*U are returned, where U
is the product of the unitary symplectic
transformations that are applied to the pencil
aS - bH to reorder the eigenvalues.
Input/Output Parameters
N (input) INTEGER
The order of the pencil aS - bH. N >= 0, even.
A (input/output) COMPLEX*16 array, dimension (LDA, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular matrix A.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Aout.
The strictly lower triangular part of this array is not
referenced.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1, N/2).
C (input/output) COMPLEX*16 array, dimension (LDC, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the lower triangular matrix C.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Cout.
The strictly upper triangular part of this array is not
referenced.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1, N/2).
D (input/output) COMPLEX*16 array, dimension (LDD, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix D.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Dout.
LDD INTEGER
The leading dimension of the array D. LDD >= MAX(1, N/2).
B (input/output) COMPLEX*16 array, dimension (LDB, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular matrix B.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Bout.
The strictly lower triangular part of this array is not
referenced.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N/2).
F (input/output) COMPLEX*16 array, dimension (LDF, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular part of the Hermitian matrix
F.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Fout.
The strictly lower triangular part of this array is not
referenced.
LDF INTEGER
The leading dimension of the array F. LDF >= MAX(1, N/2).
Q (input/output) COMPLEX*16 array, dimension (LDQ, N)
On entry, if COMPQ = 'U', then the leading N-by-N part of
this array must contain a given matrix Q0, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q0 and the transformation matrix Q
used to transform the matrices S and H.
On exit, if COMPQ = 'I', then the leading N-by-N part of
this array contains the unitary transformation matrix Q.
If COMPQ = 'N' this array is not referenced.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1, N), if COMPQ = 'I' or COMPQ = 'U'.
U1 (input/output) COMPLEX*16 array, dimension (LDU1, N/2)
On entry, if COMPU = 'U', then the leading N/2-by-N/2 part
of this array must contain the upper left block of a
given matrix U0, and on exit, the leading N/2-by-N/2 part
of this array contains the updated upper left block U1 of
the product of the input matrix U0 and the transformation
matrix U used to transform the matrices S and H.
On exit, if COMPU = 'I', then the leading N/2-by-N/2 part
of this array contains the upper left block U1 of the
unitary symplectic transformation matrix U.
If COMPU = 'N' this array is not referenced.
LDU1 INTEGER
The leading dimension of the array U1.
LDU1 >= 1, if COMPU = 'N';
LDU1 >= MAX(1, N/2), if COMPU = 'I' or COMPU = 'U'.
U2 (input/output) COMPLEX*16 array, dimension (LDU2, N/2)
On entry, if COMPU = 'U', then the leading N/2-by-N/2 part
of this array must contain the upper right block of a
given matrix U0, and on exit, the leading N/2-by-N/2 part
of this array contains the updated upper right block U2 of
the product of the input matrix U0 and the transformation
matrix U used to transform the matrices S and H.
On exit, if COMPU = 'I', then the leading N/2-by-N/2 part
of this array contains the upper right block U2 of the
unitary symplectic transformation matrix U.
If COMPU = 'N' this array is not referenced.
LDU2 INTEGER
The leading dimension of the array U2.
LDU2 >= 1, if COMPU = 'N';
LDU2 >= MAX(1, N/2), if COMPU = 'I' or COMPU = 'U'.
NEIG (output) INTEGER
The number of eigenvalues in aS - bH with strictly
negative real part.
Tolerances
TOL DOUBLE PRECISION
The tolerance used to decide the sign of the eigenvalues.
If the user sets TOL > 0, then the given value of TOL is
used. If the user sets TOL <= 0, then an implicitly
computed, default tolerance, defined by MIN(N,10)*EPS, is
used instead, where EPS is the machine precision (see
LAPACK Library routine DLAMCH). A larger value might be
needed for pencils with multiple eigenvalues.
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value.
Method
The algorithm reorders the eigenvalues like the following scheme:
Step 1: Reorder the eigenvalues in the subpencil aC'*A - bB.
I. Reorder the eigenvalues with negative real parts to the
top.
II. Reorder the eigenvalues with positive real parts to the
bottom.
Step 2: Reorder the remaining eigenvalues with negative real
parts.
I. Exchange the eigenvalues between the last diagonal block
in aC'*A - bB and the last diagonal block in aS - bH.
II. Move the eigenvalues in the N/2-th place to the (MM+1)-th
place, where MM denotes the current number of eigenvalues
with negative real parts in aC'*A - bB.
The algorithm uses a sequence of unitary transformations as
described on page 38 in [1]. To achieve those transformations the
elementary SLICOT Library subroutines MB03CZ and MB03GZ are called
for the corresponding matrix structures.
References
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical Computation of Deflating Subspaces of Embedded
Hamiltonian Pencils.
Tech. Rep. SFB393/99-15, Technical University Chemnitz,
Germany, June 1999.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) complex floating point operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To move the eigenvalues with strictly negative real parts of an
N-by-N real skew-Hamiltonian/Hamiltonian pencil aS - bH in
structured Schur form,
( A D ) ( B F )
S = ( ), H = ( ),
( 0 A' ) ( 0 -B' )
with A upper triangular and B upper quasi-triangular, to the
leading principal subpencil, while keeping the triangular form.
The notation M' denotes the transpose of the matrix M.
The matrices S and H are transformed by an orthogonal matrix Q
such that
( Aout Dout )
Sout = J Q' J' S Q = ( ),
( 0 Aout' )
(1)
( Bout Fout ) ( 0 I )
Hout = J Q' J' H Q = ( ), with J = ( ),
( 0 -Bout' ) ( -I 0 )
where Aout is upper triangular and Bout is upper quasi-triangular.
Optionally, if COMPQ = 'I' or COMPQ = 'U', the orthogonal matrix Q
that fulfills (1), is computed.
Specification
SUBROUTINE MB03JD( COMPQ, N, A, LDA, D, LDD, B, LDB, F, LDF, Q,
$ LDQ, NEIG, IWORK, LIWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ
INTEGER INFO, LDA, LDB, LDD, LDF, LDQ, LDWORK, LIWORK,
$ N, NEIG
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), D( LDD, * ),
$ DWORK( * ), F( LDF, * ), Q( LDQ, * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
Specifies whether or not the orthogonal transformations
should be accumulated in the array Q, as follows:
= 'N': Q is not computed;
= 'I': the array Q is initialized internally to the unit
matrix, and the orthogonal matrix Q is returned;
= 'U': the array Q contains an orthogonal matrix Q0 on
entry, and the matrix Q0*Q is returned, where Q
is the product of the orthogonal transformations
that are applied to the pencil aS - bH to reorder
the eigenvalues.
Input/Output Parameters
N (input) INTEGER
The order of the pencil aS - bH. N >= 0, even.
A (input/output) DOUBLE PRECISION array, dimension
(LDA, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular matrix A. The elements of the
strictly lower triangular part of this array are not used.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Aout.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1, N/2).
D (input/output) DOUBLE PRECISION array, dimension
(LDD, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular part of the skew-symmetric
matrix D. The diagonal need not be set to zero.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed upper triangular part of the
matrix Dout.
The strictly lower triangular part of this array is
not referenced, except for the element D(N/2,N/2-1), but
its initial value is preserved.
LDD INTEGER
The leading dimension of the array D. LDD >= MAX(1, N/2).
B (input/output) DOUBLE PRECISION array, dimension
(LDB, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper quasi-triangular matrix B.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed upper quasi-triangular part of
the matrix Bout.
The part below the first subdiagonal of this array is
not referenced.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N/2).
F (input/output) DOUBLE PRECISION array, dimension
(LDF, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular part of the symmetric matrix
F.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed upper triangular part of the
matrix Fout.
The strictly lower triangular part of this array is not
referenced, except for the element F(N/2,N/2-1), but its
initial value is preserved.
LDF INTEGER
The leading dimension of the array F. LDF >= MAX(1, N/2).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
On entry, if COMPQ = 'U', then the leading N-by-N part of
this array must contain a given matrix Q0, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q0 and the transformation matrix Q
used to transform the matrices S and H.
On exit, if COMPQ = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q.
If COMPQ = 'N' this array is not referenced.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1, N), if COMPQ = 'I' or COMPQ = 'U'.
NEIG (output) INTEGER
The number of eigenvalues in aS - bH with strictly
negative real part.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK INTEGER
The dimension of the array IWORK.
LIWORK >= N+1.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The dimension of the array DWORK.
If COMPQ = 'N',
LDWORK >= MAX(2*N+32,108);
if COMPQ = 'I' or COMPQ = 'U',
LDWORK >= MAX(4*N+32,108).
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: error occured during execution of MB03DD;
= 2: error occured during execution of MB03HD.
Method
The algorithm reorders the eigenvalues like the following scheme:
Step 1: Reorder the eigenvalues in the subpencil aA - bB.
I. Reorder the eigenvalues with negative real parts to the
top.
II. Reorder the eigenvalues with positive real parts to the
bottom.
Step 2: Reorder the remaining eigenvalues with negative real
parts in the pencil aS - bH.
I. Exchange the eigenvalues between the last diagonal block
in aA - bB and the last diagonal block in aS - bH.
II. Move the eigenvalues of the R-th block to the (MM+1)-th
block, where R denotes the number of upper quasi-
triangular blocks in aA - bB and MM denotes the current
number of blocks in aA - bB with eigenvalues with negative
real parts.
The algorithm uses a sequence of orthogonal transformations as
described on page 33 in [1]. To achieve those transformations the
elementary subroutines MB03DD and MB03HD are called for the
corresponding matrix structures.
References
[1] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
Eigenproblems.
Tech. Rep., Technical University Chemnitz, Germany,
Nov. 2007.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) real floating point operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To move the eigenvalues with strictly negative real parts of an
N-by-N real skew-Hamiltonian/Hamiltonian pencil aS - bH in
structured Schur form,
( A D ) ( B F )
S = ( ), H = ( ),
( 0 A' ) ( 0 -B' )
with A upper triangular and B upper quasi-triangular, to the
leading principal subpencil, while keeping the triangular form.
The notation M' denotes the transpose of the matrix M.
The matrices S and H are transformed by an orthogonal matrix Q
such that
( Aout Dout )
Sout = J Q' J' S Q = ( ),
( 0 Aout' )
(1)
( Bout Fout ) ( 0 I )
Hout = J Q' J' H Q = ( ), with J = ( ),
( 0 -Bout' ) ( -I 0 )
where Aout is upper triangular and Bout is upper quasi-triangular.
Optionally, if COMPQ = 'I' or COMPQ = 'U', the orthogonal matrix Q
that fulfills (1), is computed.
Specification
SUBROUTINE MB03JP( COMPQ, N, A, LDA, D, LDD, B, LDB, F, LDF, Q,
$ LDQ, NEIG, IWORK, LIWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ
INTEGER INFO, LDA, LDB, LDD, LDF, LDQ, LDWORK, LIWORK,
$ N, NEIG
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), D( LDD, * ),
$ DWORK( * ), F( LDF, * ), Q( LDQ, * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
Specifies whether or not the orthogonal transformations
should be accumulated in the array Q, as follows:
= 'N': Q is not computed;
= 'I': the array Q is initialized internally to the unit
matrix, and the orthogonal matrix Q is returned;
= 'U': the array Q contains an orthogonal matrix Q0 on
entry, and the matrix Q0*Q is returned, where Q
is the product of the orthogonal transformations
that are applied to the pencil aS - bH to reorder
the eigenvalues.
Input/Output Parameters
N (input) INTEGER
The order of the pencil aS - bH. N >= 0, even.
A (input/output) DOUBLE PRECISION array, dimension
(LDA, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular matrix A. The elements of the
strictly lower triangular part of this array are not used.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Aout.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1, N/2).
D (input/output) DOUBLE PRECISION array, dimension
(LDD, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular part of the skew-symmetric
matrix D. The diagonal need not be set to zero.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed upper triangular part of the
matrix Dout.
The strictly lower triangular part of this array is
not referenced, except for the element D(N/2,N/2-1), but
its initial value is preserved.
LDD INTEGER
The leading dimension of the array D. LDD >= MAX(1, N/2).
B (input/output) DOUBLE PRECISION array, dimension
(LDB, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper quasi-triangular matrix B.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed upper quasi-triangular part of
the matrix Bout.
The part below the first subdiagonal of this array is
not referenced.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N/2).
F (input/output) DOUBLE PRECISION array, dimension
(LDF, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular part of the symmetric matrix
F.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed upper triangular part of the
matrix Fout.
The strictly lower triangular part of this array is not
referenced, except for the element F(N/2,N/2-1), but its
initial value is preserved.
LDF INTEGER
The leading dimension of the array F. LDF >= MAX(1, N/2).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
On entry, if COMPQ = 'U', then the leading N-by-N part of
this array must contain a given matrix Q0, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q0 and the transformation matrix Q
used to transform the matrices S and H.
On exit, if COMPQ = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q.
If COMPQ = 'N' this array is not referenced.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1, N), if COMPQ = 'I' or COMPQ = 'U'.
NEIG (output) INTEGER
The number of eigenvalues in aS - bH with strictly
negative real part.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK INTEGER
The dimension of the array IWORK.
LIWORK >= 3*N-3.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The dimension of the array DWORK.
If COMPQ = 'N',
LDWORK >= MAX(2*N+32,108)+5*N/2;
if COMPQ = 'I' or COMPQ = 'U',
LDWORK >= MAX(4*N+32,108)+5*N/2.
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: error occured during execution of MB03DD;
= 2: error occured during execution of MB03HD.
Method
The algorithm reorders the eigenvalues like the following scheme:
Step 1: Reorder the eigenvalues in the subpencil aA - bB.
I. Reorder the eigenvalues with negative real parts to the
top.
II. Reorder the eigenvalues with positive real parts to the
bottom.
Step 2: Reorder the remaining eigenvalues with negative real
parts in the pencil aS - bH.
I. Exchange the eigenvalues between the last diagonal block
in aA - bB and the last diagonal block in aS - bH.
II. Move the eigenvalues of the R-th block to the (MM+1)-th
block, where R denotes the number of upper quasi-
triangular blocks in aA - bB and MM denotes the current
number of blocks in aA - bB with eigenvalues with negative
real parts.
The algorithm uses a sequence of orthogonal transformations as
described on page 33 in [1]. To achieve those transformations the
elementary subroutines MB03DD and MB03HD are called for the
corresponding matrix structures.
References
[1] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
Eigenproblems.
Tech. Rep., Technical University Chemnitz, Germany,
Nov. 2007.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) real floating point operations.Further Comments
For large values of N, the routine applies the transformations on panels of columns. The user may specify in INFO the desired number of columns. If on entry INFO <= 0, then the routine estimates a suitable value of this number.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To move the eigenvalues with strictly negative real parts of an
N-by-N complex skew-Hamiltonian/Hamiltonian pencil aS - bH in
structured Schur form to the leading principal subpencil, while
keeping the triangular form. On entry, we have
( A D ) ( B F )
S = ( ), H = ( ),
( 0 A' ) ( 0 -B' )
where A and B are upper triangular.
S and H are transformed by a unitary matrix Q such that
( Aout Dout )
Sout = J Q' J' S Q = ( ), and
( 0 Aout' )
(1)
( Bout Fout ) ( 0 I )
Hout = J Q' J' H Q = ( ), with J = ( ),
( 0 -Bout' ) ( -I 0 )
where Aout and Bout remain in upper triangular form. The notation
M' denotes the conjugate transpose of the matrix M.
Optionally, if COMPQ = 'I' or COMPQ = 'U', the unitary matrix Q
that fulfills (1) is computed.
Specification
SUBROUTINE MB03JZ( COMPQ, N, A, LDA, D, LDD, B, LDB, F, LDF, Q,
$ LDQ, NEIG, TOL, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ
INTEGER INFO, LDA, LDB, LDD, LDF, LDQ, N, NEIG
DOUBLE PRECISION TOL
C .. Array Arguments ..
COMPLEX*16 A( LDA, * ), B( LDB, * ), D( LDD, * ),
$ F( LDF, * ), Q( LDQ, * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
Specifies whether or not the unitary transformations
should be accumulated in the array Q, as follows:
= 'N': Q is not computed;
= 'I': the array Q is initialized internally to the unit
matrix, and the unitary matrix Q is returned;
= 'U': the array Q contains a unitary matrix Q0 on
entry, and the matrix Q0*Q is returned, where Q
is the product of the unitary transformations
that are applied to the pencil aS - bH to reorder
the eigenvalues.
Input/Output Parameters
N (input) INTEGER
The order of the pencil aS - bH. N >= 0, even.
A (input/output) COMPLEX*16 array, dimension (LDA, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular matrix A.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Aout.
The strictly lower triangular part of this array is not
referenced.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1, N/2).
D (input/output) COMPLEX*16 array, dimension (LDD, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular part of the skew-Hermitian
matrix D.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Dout.
The strictly lower triangular part of this array is not
referenced.
LDD INTEGER
The leading dimension of the array D. LDD >= MAX(1, N/2).
B (input/output) COMPLEX*16 array, dimension (LDB, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular matrix B.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Bout.
The strictly lower triangular part of this array is not
referenced.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N/2).
F (input/output) COMPLEX*16 array, dimension (LDF, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular part of the Hermitian matrix
F.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Fout.
The strictly lower triangular part of this array is not
referenced.
LDF INTEGER
The leading dimension of the array F. LDF >= MAX(1, N/2).
Q (input/output) COMPLEX*16 array, dimension (LDQ, N)
On entry, if COMPQ = 'U', then the leading N-by-N part of
this array must contain a given matrix Q0, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q0 and the transformation matrix Q
used to transform the matrices S and H.
On exit, if COMPQ = 'I', then the leading N-by-N part of
this array contains the unitary transformation matrix Q.
If COMPQ = 'N' this array is not referenced.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1, N), if COMPQ = 'I' or COMPQ = 'U'.
NEIG (output) INTEGER
The number of eigenvalues in aS - bH with strictly
negative real part.
Tolerances
TOL DOUBLE PRECISION
The tolerance used to decide the sign of the eigenvalues.
If the user sets TOL > 0, then the given value of TOL is
used. If the user sets TOL <= 0, then an implicitly
computed, default tolerance, defined by MIN(N,10)*EPS, is
used instead, where EPS is the machine precision (see
LAPACK Library routine DLAMCH). A larger value might be
needed for pencils with multiple eigenvalues.
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value.
Method
The algorithm reorders the eigenvalues like the following scheme:
Step 1: Reorder the eigenvalues in the subpencil aA - bB.
I. Reorder the eigenvalues with negative real parts to the
top.
II. Reorder the eigenvalues with positive real parts to the
bottom.
Step 2: Reorder the remaining eigenvalues with negative real parts.
I. Exchange the eigenvalues between the last diagonal block
in aA - bB and the last diagonal block in aS - bH.
II. Move the eigenvalues in the N/2-th place to the (MM+1)-th
place, where MM denotes the current number of eigenvalues
with negative real parts in aA - bB.
The algorithm uses a sequence of unitary transformations as
described on page 43 in [1]. To achieve those transformations the
elementary SLICOT Library subroutines MB03DZ and MB03HZ are called
for the corresponding matrix structures.
References
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical Computation of Deflating Subspaces of Embedded
Hamiltonian Pencils.
Tech. Rep. SFB393/99-15, Technical University Chemnitz,
Germany, June 1999.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) complex floating point operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To reorder the diagonal blocks of the formal matrix product
T22_K^S(K) * T22_K-1^S(K-1) * ... * T22_1^S(1), (1)
of length K, in the generalized periodic Schur form
[ T11_k T12_k T13_k ]
T_k = [ 0 T22_k T23_k ], k = 1, ..., K, (2)
[ 0 0 T33_k ]
where
- the submatrices T11_k are NI(k+1)-by-NI(k), if S(k) = 1, or
NI(k)-by-NI(k+1), if S(k) = -1, and contain dimension-induced
infinite eigenvalues,
- the submatrices T22_k are NC-by-NC and contain core eigenvalues,
which are generically neither zero nor infinite,
- the submatrices T33_k contain dimension-induced zero
eigenvalues,
such that the block with starting row index IFST in (1) is moved
to row index ILST. The indices refer to the T22_k submatrices.
Optionally, the transformation matrices Q_1,...,Q_K from the
reduction into generalized periodic Schur form are updated with
respect to the performed reordering.
Specification
SUBROUTINE MB03KA( COMPQ, WHICHQ, WS, K, NC, KSCHUR, IFST, ILST,
$ N, NI, S, T, LDT, IXT, Q, LDQ, IXQ, TOL, IWORK,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ
LOGICAL WS
INTEGER IFST, ILST, INFO, K, KSCHUR, LDWORK, NC
C .. Array Arguments ..
INTEGER IWORK( * ), IXQ( * ), IXT( * ), LDQ( * ),
$ LDT( * ), N( * ), NI( * ), S( * ), WHICHQ( * )
DOUBLE PRECISION DWORK( * ), Q( * ), T( * ), TOL( * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
= 'N': do not compute any of the matrices Q_k;
= 'U': each coefficient of Q must contain an orthogonal
matrix Q1_k on entry, and the products Q1_k*Q_k are
returned, where Q_k, k = 1, ..., K, performed the
reordering;
= 'W': the computation of each Q_k is specified
individually in the array WHICHQ.
WHICHQ INTEGER array, dimension (K)
If COMPQ = 'W', WHICHQ(k) specifies the computation of Q_k
as follows:
= 0: do not compute Q_k;
> 0: the kth coefficient of Q must contain an orthogonal
matrix Q1_k on entry, and the product Q1_k*Q_k is
returned.
This array is not referenced if COMPQ <> 'W'.
WS LOGICAL
= .FALSE. : do not perform the strong stability tests;
= .TRUE. : perform the strong stability tests; often,
this is not needed, and omitting them can save
some computations.
Input/Output Parameters
K (input) INTEGER
The period of the periodic matrix sequences T and Q (the
number of factors in the matrix product). K >= 2.
(For K = 1, a standard eigenvalue reordering problem is
obtained.)
NC (input) INTEGER
The number of core eigenvalues. 0 <= NC <= min(N).
KSCHUR (input) INTEGER
The index for which the matrix T22_kschur is upper quasi-
triangular. All other T22 matrices are upper triangular.
IFST (input/output) INTEGER
ILST (input/output) INTEGER
Specify the reordering of the diagonal blocks, as follows:
The block with starting row index IFST in (1) is moved to
row index ILST by a sequence of direct swaps between adjacent
blocks in the product.
On exit, if IFST pointed on entry to the second row of a
2-by-2 block in the product, it is changed to point to the
first row; ILST always points to the first row of the block
in its final position in the product (which may differ from
its input value by +1 or -1).
1 <= IFST <= NC, 1 <= ILST <= NC.
N (input) INTEGER array, dimension (K)
The leading K elements of this array must contain the
dimensions of the factors of the formal matrix product T,
such that the k-th coefficient T_k is an N(k+1)-by-N(k)
matrix, if S(k) = 1, or an N(k)-by-N(k+1) matrix,
if S(k) = -1, k = 1, ..., K, where N(K+1) = N(1).
NI (input) INTEGER array, dimension (K)
The leading K elements of this array must contain the
dimensions of the factors of the matrix sequence T11_k.
N(k) >= NI(k) + NC >= 0.
S (input) INTEGER array, dimension (K)
The leading K elements of this array must contain the
signatures (exponents) of the factors in the K-periodic
matrix sequence. Each entry in S must be either 1 or -1;
the value S(k) = -1 corresponds to using the inverse of
the factor T_k.
T (input/output) DOUBLE PRECISION array, dimension (*)
On entry, this array must contain at position IXT(k) the
matrix T_k, which is at least N(k+1)-by-N(k), if S(k) = 1,
or at least N(k)-by-N(k+1), if S(k) = -1, in periodic
Schur form.
On exit, the matrices T_k are overwritten by the reordered
periodic Schur form.
LDT INTEGER array, dimension (K)
The leading dimensions of the matrices T_k in the one-
dimensional array T.
LDT(k) >= max(1,N(k+1)), if S(k) = 1,
LDT(k) >= max(1,N(k)), if S(k) = -1.
IXT INTEGER array, dimension (K)
Start indices of the matrices T_k in the one-dimensional
array T.
Q (input/output) DOUBLE PRECISION array, dimension (*)
On entry, this array must contain at position IXQ(k) a
matrix Q_k of size at least N(k)-by-N(k), provided that
COMPQ = 'U', or COMPQ = 'W' and WHICHQ(k) > 0.
On exit, if COMPQ = 'U', or COMPQ = 'W' and WHICHQ(k) > 0,
Q_k is post-multiplied with the orthogonal matrix that
performed the reordering.
This array is not referenced if COMPQ = 'N'.
LDQ INTEGER array, dimension (K)
The leading dimensions of the matrices Q_k in the one-
dimensional array Q.
LDQ(k) >= max(1,N(k)), if COMPQ = 'U', or COMPQ = 'W' and
WHICHQ(k) > 0;
This array is not referenced if COMPQ = 'N'.
IXQ INTEGER array, dimension (K)
Start indices of the matrices Q_k in the one-dimensional
array Q.
This array is not referenced if COMPQ = 'N'.
Tolerances
TOL DOUBLE PRECISION array, dimension (3)
This array contains tolerance parameters. The weak and
strong stability tests use a threshold computed by the
formula MAX( c*EPS*NRM, SMLNUM ), where c is a constant,
NRM is the Frobenius norm of the current matrix formed by
concatenating K pairs of adjacent diagonal blocks of sizes
1 and/or 2 in the T22_k submatrices from (2), which are
swapped, and EPS and SMLNUM are the machine precision and
safe minimum divided by EPS, respectively (see LAPACK
Library routine DLAMCH). The norm NRM is computed by this
routine; the other values are stored in the array TOL.
TOL(1), TOL(2), and TOL(3) contain c, EPS, and SMLNUM,
respectively. TOL(1) should normally be at least 10.
Workspace
IWORK INTEGER array, dimension (4*K)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= 10*K + MN, if all blocks between IFST and ILST
have order 1;
LDWORK >= 25*K + MN, if there is at least a block of
order 2, but no adjacent blocks of
order 2 can appear between IFST and
ILST during reordering;
LDWORK >= MAX(42*K + MN, 80*K - 48), if at least a pair of
adjacent blocks of order 2 can appear
between IFST and ILST during
reordering;
where MN = MXN, if MXN > 10, and MN = 0, otherwise, with
MXN = MAX(N(k),k=1,...,K).
If LDWORK = -1 a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -21, the LDWORK argument was too small;
= 1: the reordering of T failed because some eigenvalues
are too close to separate (the problem is very ill-
conditioned); T may have been partially reordered.
The returned value of ILST is the index where this
was detected.
Method
An adaptation of the LAPACK Library routine DTGEXC is used.Numerical Aspects
The implemented method is numerically backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To reorder the diagonal blocks of the formal matrix product
T22_K^S(K) * T22_K-1^S(K-1) * ... * T22_1^S(1) (1)
of length K in the generalized periodic Schur form
[ T11_k T12_k T13_k ]
T_k = [ 0 T22_k T23_k ], k = 1, ..., K, (2)
[ 0 0 T33_k ]
where
- the submatrices T11_k are NI(k+1)-by-NI(k), if S(k) = 1, or
NI(k)-by-NI(k+1), if S(k) = -1, and contain dimension-induced
infinite eigenvalues,
- the submatrices T22_k are NC-by-NC and contain core eigenvalues,
which are generically neither zero nor infinite,
- the submatrices T33_k contain dimension-induced zero
eigenvalues,
such that pairs of adjacent diagonal blocks of sizes 1 and/or 2 in
the product (1) are swapped.
Optionally, the transformation matrices Q_1,...,Q_K from the
reduction into generalized periodic Schur form are updated with
respect to the performed reordering.
Specification
SUBROUTINE MB03KB( COMPQ, WHICHQ, WS, K, NC, KSCHUR, J1, N1, N2,
$ N, NI, S, T, LDT, IXT, Q, LDQ, IXQ, TOL, IWORK,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ
LOGICAL WS
INTEGER INFO, J1, K, KSCHUR, LDWORK, N1, N2, NC
C .. Array Arguments ..
INTEGER IWORK( * ), IXQ( * ), IXT( * ), LDQ( * ),
$ LDT( * ), N( * ), NI( * ), S( * ), WHICHQ( * )
DOUBLE PRECISION DWORK( * ), Q( * ), T( * ), TOL( * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
= 'N': do not compute any of the matrices Q_k;
= 'U': each coefficient of Q must contain an orthogonal
matrix Q1_k on entry, and the products Q1_k*Q_k are
returned, where Q_k, k = 1, ..., K, performed the
reordering;
= 'W': the computation of each Q_k is specified
individually in the array WHICHQ.
WHICHQ INTEGER array, dimension (K)
If COMPQ = 'W', WHICHQ(k) specifies the computation of Q_k
as follows:
= 0: do not compute Q_k;
> 0: the kth coefficient of Q must contain an orthogonal
matrix Q1_k on entry, and the product Q1_k*Q_k is
returned.
This array is not referenced if COMPQ <> 'W'.
WS LOGICAL
= .FALSE. : do not perform the strong stability tests;
= .TRUE. : perform the strong stability tests; often,
this is not needed, and omitting them can save
some computations.
Input/Output Parameters
K (input) INTEGER
The period of the periodic matrix sequences T and Q (the
number of factors in the matrix product). K >= 2.
(For K = 1, a standard eigenvalue reordering problem is
obtained.)
NC (input) INTEGER
The number of core eigenvalues. 0 <= NC <= min(N).
KSCHUR (input) INTEGER
The index for which the matrix T22_kschur is upper quasi-
triangular.
J1 (input) INTEGER
The index of the first row and column of the first block
to swap in T22_k.
1 <= J1 <= NC-N1-N2+1.
N1 (input) INTEGER
The order of the first block to swap. N1 = 0, 1 or 2.
N2 (input) INTEGER
The order of the second block to swap. N2 = 0, 1 or 2.
N (input) INTEGER array, dimension (K)
The leading K elements of this array must contain the
dimensions of the factors of the formal matrix product T,
such that the k-th coefficient T_k is an N(k+1)-by-N(k)
matrix, if S(k) = 1, or an N(k)-by-N(k+1) matrix,
if S(k) = -1, k = 1, ..., K, where N(K+1) = N(1).
NI (input) INTEGER array, dimension (K)
The leading K elements of this array must contain the
dimensions of the factors of the matrix sequence T11_k.
N(k) >= NI(k) + NC >= 0.
S (input) INTEGER array, dimension (K)
The leading K elements of this array must contain the
signatures (exponents) of the factors in the K-periodic
matrix sequence. Each entry in S must be either 1 or -1;
the value S(k) = -1 corresponds to using the inverse of
the factor T_k.
T (input/output) DOUBLE PRECISION array, dimension (*)
On entry, this array must contain at position IXT(k) the
matrix T_k, which is at least N(k+1)-by-N(k), if S(k) = 1,
or at least N(k)-by-N(k+1), if S(k) = -1, in periodic
Schur form.
On exit, the matrices T_k are overwritten by the reordered
periodic Schur form.
LDT INTEGER array, dimension (K)
The leading dimensions of the matrices T_k in the one-
dimensional array T.
LDT(k) >= max(1,N(k+1)), if S(k) = 1,
LDT(k) >= max(1,N(k)), if S(k) = -1.
IXT INTEGER array, dimension (K)
Start indices of the matrices T_k in the one-dimensional
array T.
Q (input/output) DOUBLE PRECISION array, dimension (*)
On entry, this array must contain at position IXQ(k) a
matrix Q_k of size at least N(k)-by-N(k), provided that
COMPQ = 'U', or COMPQ = 'W' and WHICHQ(k) > 0.
On exit, if COMPQ = 'U', or COMPQ = 'W' and WHICHQ(k) > 0,
Q_k is post-multiplied with the orthogonal matrix that
performed the reordering.
This array is not referenced if COMPQ = 'N'.
LDQ INTEGER array, dimension (K)
The leading dimensions of the matrices Q_k in the one-
dimensional array Q. LDQ(k) >= 1, and
LDQ(k) >= max(1,N(k)), if COMPQ = 'U', or COMPQ = 'W' and
WHICHQ(k) > 0;
This array is not referenced if COMPQ = 'N'.
IXQ INTEGER array, dimension (K)
Start indices of the matrices Q_k in the one-dimensional
array Q.
This array is not referenced if COMPQ = 'N'.
Tolerances
TOL DOUBLE PRECISION array, dimension (3)
This array contains tolerance parameters. The weak and
strong stability tests use a threshold computed by the
formula MAX( c*EPS*NRM, SMLNUM ), where c is a constant,
NRM is the Frobenius norm of the matrix formed by
concatenating K pairs of adjacent diagonal blocks of sizes
1 and/or 2 in the T22_k submatrices from (2), which are
swapped, and EPS and SMLNUM are the machine precision and
safe minimum divided by EPS, respectively (see LAPACK
Library routine DLAMCH). The norm NRM is computed by this
routine; the other values are stored in the array TOL.
TOL(1), TOL(2), and TOL(3) contain c, EPS, and SMLNUM,
respectively. TOL(1) should normally be at least 10.
Workspace
IWORK INTEGER array, dimension (4*K)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= 10*K + MN, if N1 = 1, N2 = 1;
LDWORK >= 25*K + MN, if N1 = 1, N2 = 2;
LDWORK >= MAX(23*K + MN, 25*K - 12), if N1 = 2, N2 = 1;
LDWORK >= MAX(42*K + MN, 80*K - 48), if N1 = 2, N2 = 2;
where MN = MXN, if MXN > 10, and MN = 0, otherwise, with
MXN = MAX(N(k),k=1,...,K).
If LDWORK = -1 a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -22, then LDWORK is too small; appropriate
value for LDWORK is returned in DWORK(1); the other
arguments are not tested, for efficiency;
= 1: the swap was rejected from stability reasons; the
blocks are not swapped and T and Q are unchanged.
Method
The algorithm described in [1] is used. Both weak and strong stability tests are performed.References
[1] Granat, R., Kagstrom, B. and Kressner, D.
Computing periodic deflating subspaces associated with a
specified set of eigenvalues.
BIT Numerical Mathematics, vol. 47, 763-791, 2007.
Numerical Aspects
The implemented method is numerically backward stable.
3
The algorithm requires 0(K NC ) floating point operations.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To reduce a 2-by-2 general, formal matrix product A of length K,
A_K^s(K) * A_K-1^s(K-1) * ... * A_1^s(1),
to the periodic Hessenberg-triangular form using a K-periodic
sequence of elementary reflectors (Householder matrices). The
matrices A_k, k = 1, ..., K, are stored in the N-by-N-by-K array A
starting in the R-th row and column, and N can be 3 or 4.
Each elementary reflector H_k is represented as
H_k = I - tau_k * v_k * v_k', (1)
where I is the 2-by-2 identity, tau_k is a real scalar, and v_k is
a vector of length 2, k = 1,...,K, and it is constructed such that
the following holds for k = 1,...,K:
H_{k+1} * A_k * H_k = T_k, if s(k) = 1,
(2)
H_k * A_k * H_{k+1} = T_k, if s(k) = -1,
with H_{K+1} = H_1 and all T_k upper triangular except for
T_{khess} which is full. Clearly,
T_K^s(K) *...* T_1^s(1) = H_1 * A_K^s(K) *...* A_1^s(1) * H_1.
The reflectors are suitably applied to the whole, extended N-by-N
matrices Ae_k, not only to the submatrices A_k, k = 1, ..., K.
Specification
SUBROUTINE MB03KC( K, KHESS, N, R, S, A, LDA, V, TAU )
C .. Scalar Arguments ..
INTEGER K, KHESS, LDA, N, R
C .. Array Arguments ..
INTEGER S( * )
DOUBLE PRECISION A( * ), TAU( * ), V( * )
Arguments
Input/Output Parameters
K (input) INTEGER
The number of matrices in the sequence A_k. K >= 2.
KHESS (input) INTEGER
The index for which the returned matrix A_khess should be
in the Hessenberg form on output. 1 <= KHESS <= K.
N (input) INTEGER
The order of the extended matrices. N = 3 or N = 4.
R (input) INTEGER
The starting row and column index for the
2-by-2 submatrices. R = 1, or R = N-1.
S (input) INTEGER array, dimension (K)
The leading K elements of this array must contain the
signatures of the factors. Each entry in S must be either
1 or -1; the value S(k) = -1 corresponds to using the
inverse of the factor A_k.
A (input/output) DOUBLE PRECISION array, dimension (*)
On entry, this array must contain at position IXA(k) =
(k-1)*N*LDA+1 the N-by-N matrix Ae_k stored with leading
dimension LDA.
On exit, this array contains at position IXA(k) the
N-by-N matrix Te_k stored with leading dimension LDA.
LDA INTEGER
Leading dimension of the matrices Ae_k and Te_k in the
one-dimensional array A. LDA >= N.
V (output) DOUBLE PRECISION array, dimension (2*K)
On exit, this array contains the K vectors v_k,
k = 1,...,K, defining the elementary reflectors H_k as
in (1). The k-th reflector is stored in V(2*k-1:2*k).
TAU (output) DOUBLE PRECISION array, dimension (K)
On exit, this array contains the K values of tau_k,
k = 1,...,K, defining the elementary reflectors H_k
as in (1).
Method
A K-periodic sequence of elementary reflectors (Householder matrices) is used. The computations start for k = khess with the left reflector in (1), which is the identity matrix.Numerical Aspects
The implemented method is numerically backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To reorder the diagonal blocks of the formal matrix product
T22_K^S(K) * T22_K-1^S(K-1) * ... * T22_1^S(1), (1)
of length K, in the generalized periodic Schur form,
[ T11_k T12_k T13_k ]
T_k = [ 0 T22_k T23_k ], k = 1, ..., K, (2)
[ 0 0 T33_k ]
where
- the submatrices T11_k are NI(k+1)-by-NI(k), if S(k) = 1, or
NI(k)-by-NI(k+1), if S(k) = -1, and contain dimension-induced
infinite eigenvalues,
- the submatrices T22_k are NC-by-NC and contain core eigenvalues,
which are generically neither zero nor infinite,
- the submatrices T33_k contain dimension-induced zero
eigenvalues,
such that the M selected eigenvalues pointed to by the logical
vector SELECT end up in the leading part of the matrix sequence
T22_k.
Given that N(k) = N(k+1) for all k where S(k) = -1, the T11_k are
void and the first M columns of the updated orthogonal
transformation matrix sequence Q_1, ..., Q_K span a periodic
deflating subspace corresponding to the same eigenvalues.
Specification
SUBROUTINE MB03KD( COMPQ, WHICHQ, STRONG, K, NC, KSCHUR, N, NI, S,
$ SELECT, T, LDT, IXT, Q, LDQ, IXQ, M, TOL,
$ IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, STRONG
INTEGER INFO, K, KSCHUR, LDWORK, M, NC
DOUBLE PRECISION TOL
C .. Array Arguments ..
LOGICAL SELECT( * )
INTEGER IWORK( * ), IXQ( * ), IXT( * ), LDQ( * ),
$ LDT( * ), N( * ), NI( * ), S( * ), WHICHQ( * )
DOUBLE PRECISION DWORK( * ), Q( * ), T( * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
Specifies whether to compute the orthogonal transformation
matrices Q_k, as follows:
= 'N': do not compute any of the matrices Q_k;
= 'I': each coefficient of Q is initialized internally to
the identity matrix, and the orthogonal matrices
Q_k are returned, where Q_k, k = 1, ..., K,
performed the reordering;
= 'U': each coefficient of Q must contain an orthogonal
matrix Q1_k on entry, and the products Q1_k*Q_k are
returned;
= 'W': the computation of each Q_k is specified
individually in the array WHICHQ.
WHICHQ INTEGER array, dimension (K)
If COMPQ = 'W', WHICHQ(k) specifies the computation of Q_k
as follows:
= 0: do not compute Q_k;
= 1: the kth coefficient of Q is initialized to the
identity matrix, and the orthogonal matrix Q_k is
returned;
= 2: the kth coefficient of Q must contain an orthogonal
matrix Q1_k on entry, and the product Q1_k*Q_k is
returned.
This array is not referenced if COMPQ <> 'W'.
STRONG CHARACTER*1
Specifies whether to perform the strong stability tests,
as follows:
= 'N': do not perform the strong stability tests;
= 'S': perform the strong stability tests; often, this is
not needed, and omitting them can save some
computations.
Input/Output Parameters
K (input) INTEGER
The period of the periodic matrix sequences T and Q (the
number of factors in the matrix product). K >= 2.
(For K = 1, a standard eigenvalue reordering problem is
obtained.)
NC (input) INTEGER
The number of core eigenvalues. 0 <= NC <= min(N).
KSCHUR (input) INTEGER
The index for which the matrix T22_kschur is upper quasi-
triangular. All other T22 matrices are upper triangular.
N (input) INTEGER array, dimension (K)
The leading K elements of this array must contain the
dimensions of the factors of the formal matrix product T,
such that the k-th coefficient T_k is an N(k+1)-by-N(k)
matrix, if S(k) = 1, or an N(k)-by-N(k+1) matrix,
if S(k) = -1, k = 1, ..., K, where N(K+1) = N(1).
NI (input) INTEGER array, dimension (K)
The leading K elements of this array must contain the
dimensions of the factors of the matrix sequence T11_k.
N(k) >= NI(k) + NC >= 0.
S (input) INTEGER array, dimension (K)
The leading K elements of this array must contain the
signatures (exponents) of the factors in the K-periodic
matrix sequence. Each entry in S must be either 1 or -1;
the value S(k) = -1 corresponds to using the inverse of
the factor T_k.
SELECT (input) LOGICAL array, dimension (NC)
SELECT specifies the eigenvalues in the selected cluster.
To select a real eigenvalue w(j), SELECT(j) must be set to
.TRUE.. To select a complex conjugate pair of eigenvalues
w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
either SELECT(j) or SELECT(j+1) or both must be set to
.TRUE.; a complex conjugate pair of eigenvalues must be
either both included in the cluster or both excluded.
T (input/output) DOUBLE PRECISION array, dimension (*)
On entry, this array must contain at position IXT(k) the
matrix T_k, which is at least N(k+1)-by-N(k), if S(k) = 1,
or at least N(k)-by-N(k+1), if S(k) = -1, in periodic
Schur form.
On exit, the matrices T_k are overwritten by the reordered
periodic Schur form.
LDT INTEGER array, dimension (K)
The leading dimensions of the matrices T_k in the one-
dimensional array T.
LDT(k) >= max(1,N(k+1)), if S(k) = 1,
LDT(k) >= max(1,N(k)), if S(k) = -1.
IXT INTEGER array, dimension (K)
Start indices of the matrices T_k in the one-dimensional
array T.
Q (input/output) DOUBLE PRECISION array, dimension (*)
On entry, this array must contain at position IXQ(k) a
matrix Q_k of size at least N(k)-by-N(k), provided that
COMPQ = 'U', or COMPQ = 'W' and WHICHQ(k) = 2.
On exit, if COMPQ = 'I' or COMPQ = 'W' and WHICHQ(k) = 1,
Q_k contains the orthogonal matrix that performed the
reordering. If COMPQ = 'U', or COMPQ = 'W' and
WHICHQ(k) = 2, Q_k is post-multiplied with the orthogonal
matrix that performed the reordering.
This array is not referenced if COMPQ = 'N'.
LDQ INTEGER array, dimension (K)
The leading dimensions of the matrices Q_k in the one-
dimensional array Q.
LDQ(k) >= max(1,N(k)), if COMPQ = 'I', or COMPQ = 'U', or
COMPQ = 'W' and WHICHQ(k) > 0;
This array is not referenced if COMPQ = 'N'.
IXQ INTEGER array, dimension (K)
Start indices of the matrices Q_k in the one-dimensional
array Q.
This array is not referenced if COMPQ = 'N'.
M (output) INTEGER
The number of selected core eigenvalues which were
reordered to the top of T22_k.
Tolerances
TOL DOUBLE PRECISION
The tolerance parameter c. The weak and strong stability
tests performed for checking the reordering use a
threshold computed by the formula MAX(c*EPS*NRM, SMLNUM),
where NRM is the varying Frobenius norm of the matrices
formed by concatenating K pairs of adjacent diagonal
blocks of sizes 1 and/or 2 in the T22_k submatrices from
(2), which are swapped, and EPS and SMLNUM are the machine
precision and safe minimum divided by EPS, respectively
(see LAPACK Library routine DLAMCH). The value c should
normally be at least 10.
Workspace
IWORK INTEGER array, dimension (4*K)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= 10*K + MN, if all blocks involved in reordering
have order 1;
LDWORK >= 25*K + MN, if there is at least a block of
order 2, but no adjacent blocks of
order 2 are involved in reordering;
LDWORK >= MAX(42*K + MN, 80*K - 48), if there is at least
a pair of adjacent blocks of order 2
involved in reordering;
where MN = MXN, if MXN > 10, and MN = 0, otherwise, with
MXN = MAX(N(k),k=1,...,K).
If LDWORK = -1 a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reordering of T failed because some eigenvalues
are too close to separate (the problem is very ill-
conditioned); T may have been partially reordered.
Method
An adaptation of the LAPACK Library routine DTGSEN is used.Numerical Aspects
The implemented method is numerically backward stable.Further Comments
NoneExample
Program Text
* MB03KD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER KMAX, NMAX
PARAMETER ( KMAX = 6, NMAX = 50 )
INTEGER LDA1, LDA2, LDQ1, LDQ2, LDWORK, LIWORK
PARAMETER ( LDA1 = NMAX, LDA2 = NMAX, LDQ1 = NMAX,
$ LDQ2 = NMAX,
$ LDWORK = MAX( 42*KMAX + NMAX, 80*KMAX - 48,
$ KMAX + MAX( 2*NMAX, 8*KMAX ) ),
$ LIWORK = MAX( 4*KMAX, 2*KMAX + NMAX ) )
DOUBLE PRECISION HUND, ZERO
PARAMETER ( HUND = 1.0D2, ZERO = 0.0D0 )
*
* .. Local Scalars ..
CHARACTER COMPQ, DEFL, JOB, STRONG
INTEGER H, I, IHI, ILO, INFO, IWARN, J, K, L, M, N, P
DOUBLE PRECISION TOL
*
* .. Local Arrays ..
LOGICAL SELECT( NMAX )
INTEGER IWORK( LIWORK ), IXQ( KMAX ), IXT( KMAX ),
$ LDQ( KMAX ), LDT( KMAX ), ND( KMAX ),
$ NI( KMAX ), QIND( KMAX ), S( KMAX ),
$ SCAL( NMAX )
DOUBLE PRECISION A( LDA1, LDA2, KMAX ), ALPHAI( NMAX ),
$ ALPHAR( NMAX ), BETA( NMAX ), DWORK( LDWORK),
$ Q( LDQ1, LDQ2, KMAX ), QK( NMAX*NMAX*KMAX ),
$ T( NMAX*NMAX*KMAX )
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
*
* .. External Subroutines ..
EXTERNAL DLACPY, MB03BD, MB03KD
*
* .. Intrinsic Functions ..
INTRINSIC INT, MAX
*
* .. Executable Statements ..
*
WRITE( NOUT, FMT = 99999 )
* Skip the heading in the data file and read in the data.
READ( NIN, FMT = * )
READ( NIN, FMT = * ) JOB, DEFL, COMPQ, STRONG, K, N, H, ILO, IHI
IF( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE( NOUT, FMT = 99998 ) N
ELSE
TOL = HUND
READ( NIN, FMT = * ) ( S( I ), I = 1, K )
READ( NIN, FMT = * ) ( ( ( A( I, J, L ), J = 1, N ),
$ I = 1, N ), L = 1, K )
IF( LSAME( COMPQ, 'U' ) )
$ READ( NIN, FMT = * ) ( ( ( Q( I, J, L ), J = 1, N ),
$ I = 1, N ), L = 1, K )
IF( LSAME( COMPQ, 'P' ) ) THEN
READ( NIN, FMT = * ) ( QIND( I ), I = 1, K )
DO 10 L = 1, K
IF( QIND( L ).GT.0 )
$ READ( NIN, FMT = * ) ( ( Q( I, J, QIND( L ) ),
$ J = 1, N ), I = 1, N )
10 CONTINUE
END IF
IF( LSAME( JOB, 'E' ) )
$ JOB = 'S'
* Compute the eigenvalues and the transformed matrices.
CALL MB03BD( JOB, DEFL, COMPQ, QIND, K, N, H, ILO, IHI, S, A,
$ LDA1, LDA2, Q, LDQ1, LDQ2, ALPHAR, ALPHAI, BETA,
$ SCAL, IWORK, LIWORK, DWORK, LDWORK, IWARN, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99997 ) INFO
ELSE IF( IWARN.EQ.0 ) THEN
* Prepare the data for calling MB03KD, which uses different
* data structures and reverse ordering of the factors.
DO 20 L = 1, K
ND( L ) = MAX( 1, N )
NI( L ) = 0
LDT( L ) = MAX( 1, N )
IXT( L ) = ( L - 1 )*LDT( L )*N + 1
LDQ( L ) = MAX( 1, N )
IXQ( L ) = IXT( L )
IF( L.LE.INT( K/2 ) ) THEN
I = S( K - L + 1 )
S( K - L + 1 ) = S( L )
S( L ) = I
END IF
20 CONTINUE
DO 30 L = 1, K
CALL DLACPY( 'Full', N, N, A( 1, 1, K-L+1 ), LDA1,
$ T( IXT( L ) ), LDT( L ) )
30 CONTINUE
IF( LSAME( COMPQ, 'U' ) .OR. LSAME( COMPQ, 'I' ) ) THEN
COMPQ = 'U'
DO 40 L = 1, K
CALL DLACPY( 'Full', N, N, Q( 1, 1, K-L+1 ), LDQ1,
$ QK( IXQ( L ) ), LDQ( L ) )
40 CONTINUE
ELSE IF( LSAME( COMPQ, 'P' ) ) THEN
COMPQ = 'W'
DO 50 L = 1, K
IF( QIND( L ).LT.0 )
$ QIND( L ) = 2
P = QIND( L )
IF( P.NE.0 )
$ CALL DLACPY( 'Full', N, N, Q( 1, 1, K-P+1 ), LDQ1,
$ QK( IXQ( P ) ), LDQ( P ) )
50 CONTINUE
END IF
* Select eigenvalues with negative real part.
DO 60 I = 1, N
SELECT( I ) = ALPHAR( I ).LT.ZERO
60 CONTINUE
WRITE( NOUT, FMT = 99996 )
WRITE( NOUT, FMT = 99995 ) ( ALPHAR( I ), I = 1, N )
WRITE( NOUT, FMT = 99994 )
WRITE( NOUT, FMT = 99995 ) ( ALPHAI( I ), I = 1, N )
WRITE( NOUT, FMT = 99993 )
WRITE( NOUT, FMT = 99995 ) ( BETA( I ), I = 1, N )
WRITE( NOUT, FMT = 99992 )
WRITE( NOUT, FMT = 99991 ) ( SCAL( I ), I = 1, N )
* Compute the transformed matrices, after reordering the
* eigenvalues.
CALL MB03KD( COMPQ, QIND, STRONG, K, N, H, ND, NI, S,
$ SELECT, T, LDT, IXT, QK, LDQ, IXQ, M, TOL,
$ IWORK, DWORK, LDWORK, INFO )
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99990 ) INFO
ELSE
WRITE( NOUT, FMT = 99989 )
DO 80 L = 1, K
P = K - L + 1
WRITE( NOUT, FMT = 99988 ) L
DO 70 I = 1, N
WRITE( NOUT, FMT = 99995 )
$ ( T( IXT( P ) + I - 1 + ( J - 1 )*LDT( P ) ),
$ J = 1, N )
70 CONTINUE
80 CONTINUE
IF( LSAME( COMPQ, 'U' ) .OR. LSAME( COMPQ, 'I' ) ) THEN
WRITE( NOUT, FMT = 99987 )
DO 100 L = 1, K
P = K - L + 1
WRITE( NOUT, FMT = 99988 ) L
DO 90 I = 1, N
WRITE( NOUT, FMT = 99995 )
$ ( QK( IXQ( P ) + I - 1 +
$ ( J - 1 )*LDQ( P ) ), J = 1, N )
90 CONTINUE
100 CONTINUE
ELSE IF( LSAME( COMPQ, 'W' ) ) THEN
WRITE( NOUT, FMT = 99987 )
DO 120 L = 1, K
IF( QIND( L ).GT.0 ) THEN
P = K - QIND( L ) + 1
WRITE( NOUT, FMT = 99988 ) QIND( L )
DO 110 I = 1, N
WRITE( NOUT, FMT = 99995 )
$ ( QK( IXQ( P ) + I - 1 +
$ ( J - 1 )*LDQ( P ) ), J = 1, N )
110 CONTINUE
END IF
120 CONTINUE
END IF
END IF
ELSE
WRITE( NOUT, FMT = 99979 ) IWARN
END IF
END IF
STOP
*
99999 FORMAT( 'MB03KD EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT( 'INFO on exit from MB03BD = ', I2 )
99996 FORMAT( 'The vector ALPHAR is ' )
99995 FORMAT( 50( 1X, F8.4 ) )
99994 FORMAT( 'The vector ALPHAI is ' )
99993 FORMAT( 'The vector BETA is ' )
99992 FORMAT( 'The vector SCAL is ' )
99991 FORMAT( 50( 1X, I5 ) )
99990 FORMAT( 'INFO on exit from MB03KD = ', I2 )
99989 FORMAT( 'The matrix A on exit is ' )
99988 FORMAT( 'The factor ', I2, ' is ' )
99987 FORMAT( 'The matrix Q on exit is ' )
99986 FORMAT( 'LDT', 3I5 )
99985 FORMAT( 'IXT', 3I5 )
99984 FORMAT( 'LDQ', 3I5 )
99983 FORMAT( 'IXQ', 3I5 )
99982 FORMAT( 'ND' , 3I5 )
99981 FORMAT( 'NI' , 3I5)
99980 FORMAT( 'SELECT', 3L5 )
99979 FORMAT( 'IWARN on exit from MB03BD = ', I2 )
END
Program Data
MB03KD EXAMPLE PROGRAM DATA S C I N 3 3 2 1 3 -1 1 -1 2.0 0.0 1.0 0.0 -2.0 -1.0 0.0 0.0 3.0 1.0 2.0 0.0 4.0 -1.0 3.0 0.0 3.0 1.0 1.0 0.0 1.0 0.0 4.0 -1.0 0.0 0.0 -2.0Program Results
MB03KD EXAMPLE PROGRAM RESULTS
The vector ALPHAR is
0.3230 0.3230 -0.8752
The vector ALPHAI is
0.5694 -0.5694 0.0000
The vector BETA is
1.0000 1.0000 1.0000
The vector SCAL is
0 0 -1
The matrix A on exit is
The factor 1 is
2.5997 -0.0087 1.6898
0.0000 1.9846 0.1942
0.0000 0.0000 2.3259
The factor 2 is
-2.0990 -1.0831 -2.5601
0.0000 3.4838 0.2950
0.0000 3.4552 -2.1690
The factor 3 is
1.8451 0.9260 1.2717
0.0000 1.3976 -2.3544
0.0000 0.0000 -3.1023
The matrix Q on exit is
The factor 1 is
-0.2052 0.4647 -0.8614
0.2033 0.8811 0.4270
-0.9574 0.0875 0.2753
The factor 2 is
-0.7743 -0.1384 0.6176
0.6070 -0.4386 0.6627
0.1791 0.8880 0.4236
The factor 3 is
-0.6714 0.7225 -0.1651
-0.3658 -0.5168 -0.7740
-0.6446 -0.4593 0.6112
Purpose
To solve small periodic Sylvester-like equations (PSLE) op(A(i))*X( i ) + isgn*X(i+1)*op(B(i)) = -scale*C(i), S(i) = 1, op(A(i))*X(i+1) + isgn*X( i )*op(B(i)) = -scale*C(i), S(i) = -1. i = 1, ..., K, where op(A) means A or A**T, for the K-periodic matrix sequence X(i) = X(i+K), where A, B and C are K-periodic matrix sequences and A and B are in periodic real Schur form. The matrices A(i) are M-by-M and B(i) are N-by-N, with 1 <= M, N <= 2.Specification
SUBROUTINE MB03KE( TRANA, TRANB, ISGN, K, M, N, PREC, SMIN, S, A,
$ B, C, SCALE, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
LOGICAL TRANA, TRANB
INTEGER INFO, ISGN, K, LDWORK, M, N
DOUBLE PRECISION PREC, SCALE, SMIN
C .. Array Arguments ..
INTEGER S( * )
DOUBLE PRECISION A( * ), B( * ), C( * ), DWORK( * )
Arguments
Mode Parameters
TRANA LOGICAL
Specifies the form of op(A) to be used, as follows:
= .FALSE.: op(A) = A,
= .TRUE. : op(A) = A**T.
TRANB LOGICAL
Specifies the form of op(B) to be used, as follows:
= .FALSE.: op(B) = B,
= .TRUE. : op(B) = B**T.
ISGN INTEGER
Specifies which sign variant of the equations to solve.
ISGN = 1 or ISGN = -1.
Input/Output Parameters
K (input) INTEGER
The period of the periodic matrix sequences A, B, C and X.
K >= 2. (For K = 1, a standard Sylvester equation is
obtained.)
M (input) INTEGER
The order of the matrices A(i) and the number of rows of
the matrices C(i) and X(i), i = 1, ..., K. 1 <= M <= 2.
N (input) INTEGER
The order of the matrices B(i) and the number of columns
of the matrices C(i) and X(i), i = 1, ..., K.
1 <= N <= 2.
PREC (input) DOUBLE PRECISION
The relative machine precision. See the LAPACK Library
routine DLAMCH.
SMIN (input) DOUBLE PRECISION
The machine safe minimum divided by PREC.
S (input) INTEGER array, dimension (K)
The leading K elements of this array must contain the
signatures (exponents) of the factors in the K-periodic
matrix sequences for A and B. Each entry in S must be
either 1 or -1. Notice that it is assumed that the same
exponents are tied to both A and B on reduction to the
periodic Schur form.
A (input) DOUBLE PRECISION array, dimension (M*M*K)
On entry, this array must contain the M-by-M matrices
A(i), for i = 1, ..., K, stored with the leading dimension
M. Matrix A(i) is stored starting at position M*M*(i-1)+1.
B (input) DOUBLE PRECISION array, dimension (N*N*K)
On entry, this array must contain the N-by-N matrices
B(i), for i = 1, ..., K, stored with the leading dimension
N. Matrix B(i) is stored starting at position N*N*(i-1)+1.
C (input/output) DOUBLE PRECISION array, dimension (M*N*K)
On entry, this array must contain the M-by-N matrices
C(i), for i = 1, ..., K, stored with the leading dimension
M. Matrix C(i) is stored starting at position M*N*(i-1)+1.
On exit, the matrices C(i) are overwritten by the solution
sequence X(i).
SCALE (output) DOUBLE PRECISION
The scale factor, scale, set less than or equal to 1 to
avoid overflow in X.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
On exit, if INFO = -21, DWORK(1) returns the minimum value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= (4*K-3) * (M*N)**2 + K * M*N.
If LDWORK = -1 a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -21, then LDWORK is too small; appropriate
value for LDWORK is returned in DWORK(1); the other
arguments are not tested, for efficiency;
= 1: the solution would overflow with scale = 1, so
SCALE was set less than 1. This is a warning, not
an error.
Method
A version of the algorithm described in [1] is used. The routine uses a sparse Kronecker product representation Z of the PSLE and solves for X(i) from an associated linear system Z*x = c using structured (overlapping) variants of QR factorization and backward substitution.References
[1] Granat, R., Kagstrom, B. and Kressner, D.
Computing periodic deflating subspaces associated with a
specified set of eigenvalues.
BIT Numerical Mathematics, vol. 47, 763-791, 2007.
Numerical Aspects
The implemented method is numerically backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the relevant eigenvalues of a real N-by-N skew-
Hamiltonian/Hamiltonian pencil aS - bH, with
( A D ) ( B F )
S = ( ) and H = ( ), (1)
( E A' ) ( G -B' )
where the notation M' denotes the transpose of the matrix M.
Optionally, if COMPQ = 'C', an orthogonal basis of the right
deflating subspace of aS - bH corresponding to the eigenvalues
with strictly negative real part is computed.
Specification
SUBROUTINE MB03LD( COMPQ, ORTH, N, A, LDA, DE, LDDE, B, LDB, FG,
$ LDFG, NEIG, Q, LDQ, ALPHAR, ALPHAI, BETA,
$ IWORK, LIWORK, DWORK, LDWORK, BWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, ORTH
INTEGER INFO, LDA, LDB, LDDE, LDFG, LDQ, LDWORK,
$ LIWORK, N, NEIG
C .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), DE( LDDE, * ),
$ DWORK( * ), FG( LDFG, * ), Q( LDQ, * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
Specifies whether to compute the right deflating subspace
corresponding to the eigenvalues of aS - bH with strictly
negative real part.
= 'N': do not compute the deflating subspace;
= 'C': compute the deflating subspace and store it in the
leading subarray of Q.
ORTH CHARACTER*1
If COMPQ = 'C', specifies the technique for computing an
orthogonal basis of the deflating subspace, as follows:
= 'P': QR factorization with column pivoting;
= 'S': singular value decomposition.
If COMPQ = 'N', the ORTH value is not used.
Input/Output Parameters
N (input) INTEGER
The order of the pencil aS - bH. N >= 0, even.
A (input/output) DOUBLE PRECISION array, dimension
(LDA, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix A.
On exit, if COMPQ = 'C', the leading N/2-by-N/2 part of
this array contains the upper triangular matrix Aout
(see METHOD); otherwise, it contains the upper triangular
matrix A obtained just before the application of the
periodic QZ algorithm (see SLICOT Library routine MB04BD).
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1, N/2).
DE (input/output) DOUBLE PRECISION array, dimension
(LDDE, N/2+1)
On entry, the leading N/2-by-N/2 lower triangular part of
this array must contain the lower triangular part of the
skew-symmetric matrix E, and the N/2-by-N/2 upper
triangular part of the submatrix in the columns 2 to N/2+1
of this array must contain the upper triangular part of the
skew-symmetric matrix D.
The entries on the diagonal and the first superdiagonal of
this array need not be set, but are assumed to be zero.
On exit, if COMPQ = 'C', the leading N/2-by-N/2 lower
triangular part and the first superdiagonal contain the
transpose of the upper quasi-triangular matrix C2out (see
METHOD), and the (N/2-1)-by-(N/2-1) upper triangular part
of the submatrix in the columns 3 to N/2+1 of this array
contains the strictly upper triangular part of the
skew-symmetric matrix Dout (see METHOD), without the main
diagonal, which is zero.
On exit, if COMPQ = 'N', the leading N/2-by-N/2 lower
triangular part and the first superdiagonal contain the
transpose of the upper Hessenberg matrix C2, and the
(N/2-1)-by-(N/2-1) upper triangular part of the submatrix
in the columns 3 to N/2+1 of this array contains the
strictly upper triangular part of the skew-symmetric
matrix D (without the main diagonal) just before the
application of the periodic QZ algorithm.
LDDE INTEGER
The leading dimension of the array DE.
LDDE >= MAX(1, N/2).
B (input/output) DOUBLE PRECISION array, dimension
(LDB, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix B.
On exit, if COMPQ = 'C', the leading N/2-by-N/2 part of
this array contains the upper triangular matrix C1out
(see METHOD); otherwise, it contains the upper triangular
matrix C1 obtained just before the application of the
periodic QZ algorithm.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N/2).
FG (input/output) DOUBLE PRECISION array, dimension
(LDFG, N/2+1)
On entry, the leading N/2-by-N/2 lower triangular part of
this array must contain the lower triangular part of the
symmetric matrix G, and the N/2-by-N/2 upper triangular
part of the submatrix in the columns 2 to N/2+1 of this
array must contain the upper triangular part of the
symmetric matrix F.
On exit, if COMPQ = 'C', the leading N/2-by-N/2 part of
the submatrix in the columns 2 to N/2+1 of this array
contains the matrix Vout (see METHOD); otherwise, it
contains the matrix V obtained just before the application
of the periodic QZ algorithm.
LDFG INTEGER
The leading dimension of the array FG.
LDFG >= MAX(1, N/2).
NEIG (output) INTEGER
If COMPQ = 'C', the number of eigenvalues in aS - bH with
strictly negative real part.
Q (output) DOUBLE PRECISION array, dimension (LDQ, 2*N)
On exit, if COMPQ = 'C', the leading N-by-NEIG part of
this array contains an orthogonal basis of the right
deflating subspace corresponding to the eigenvalues of
aA - bB with strictly negative real part. The remaining
part of this array is used as workspace.
If COMPQ = 'N', this array is not referenced.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1, 2*N), if COMPQ = 'C'.
ALPHAR (output) DOUBLE PRECISION array, dimension (N/2)
The real parts of each scalar alpha defining an eigenvalue
of the pencil aS - bH.
ALPHAI (output) DOUBLE PRECISION array, dimension (N/2)
The imaginary parts of each scalar alpha defining an
eigenvalue of the pencil aS - bH.
If ALPHAI(j) is zero, then the j-th eigenvalue is real.
BETA (output) DOUBLE PRECISION array, dimension (N/2)
The scalars beta that define the eigenvalues of the pencil
aS - bH.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent the j-th eigenvalue of the pencil
aS - bH, in the form lambda = alpha/beta. Since lambda may
overflow, the ratios should not, in general, be computed.
Due to the skew-Hamiltonian/Hamiltonian structure of the
pencil, for every eigenvalue lambda, -lambda is also an
eigenvalue, and thus it has only to be saved once in
ALPHAR, ALPHAI and BETA.
Specifically, only eigenvalues with imaginary parts
greater than or equal to zero are stored; their conjugate
eigenvalues are not stored. If imaginary parts are zero
(i.e., for real eigenvalues), only positive eigenvalues
are stored. The remaining eigenvalues have opposite signs.
Workspace
IWORK INTEGER array, dimension (LIWORK)
On exit, if INFO = -19, IWORK(1) returns the minimum value
of LIWORK.
LIWORK INTEGER
The dimension of the array IWORK. LIWORK = 1, if N = 0,
LIWORK >= MAX( N + 12, 2*N + 3 ), if COMPQ = 'N',
LIWORK >= MAX( 32, 2*N + 3 ), if COMPQ = 'C'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
On exit, if INFO = -21, DWORK(1) returns the minimum value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK. LDWORK = 1, if N = 0,
LDWORK >= 3*(N/2)**2 + N**2 + MAX( L, 36 ),
if COMPQ = 'N',
where L = 4*N + 4, if N/2 is even, and
L = 4*N , if N/2 is odd;
LDWORK >= 8*N**2 + MAX( 8*N + 32, 272 ), if COMPQ = 'C'.
For good performance LDWORK should be generally larger.
If LDWORK = -1 a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
BWORK LOGICAL array, dimension (N/2)
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: periodic QZ iteration failed in the SLICOT Library
routines MB04BD or MB04HD (QZ iteration did not
converge or computation of the shifts failed);
= 2: standard QZ iteration failed in the SLICOT Library
routines MB04HD or MB03DD (called by MB03JD);
= 3: a numerically singular matrix was found in the SLICOT
Library routine MB03HD (called by MB03JD);
= 4: the singular value decomposition failed in the LAPACK
routine DGESVD (for ORTH = 'S');
= 5: some eigenvalues might be inaccurate. This is a
warning.
Method
First, the decompositions of S and H are computed via orthogonal
transformations Q1 and Q2 as follows:
( Aout Dout )
Q1' S J Q1 J' = ( ),
( 0 Aout' )
( Bout Fout )
J' Q2' J S Q2 = ( ) =: T, (2)
( 0 Bout' )
( C1out Vout ) ( 0 I )
Q1' H Q2 = ( ), where J = ( ),
( 0 C2out' ) ( -I 0 )
and Aout, Bout, C1out are upper triangular, C2out is upper quasi-
triangular and Dout and Fout are skew-symmetric.
Then, orthogonal matrices Q3 and Q4 are found, for the extended
matrices
( Aout 0 ) ( 0 C1out )
Se = ( ) and He = ( ),
( 0 Bout ) ( -C2out 0 )
such that S11 := Q4' Se Q3 is upper triangular and
H11 := Q4' He Q3 is upper quasi-triangular. The following matrices
are computed:
( Dout 0 ) ( 0 Vout )
S12 := Q4' ( ) Q4 and H12 := Q4' ( ) Q4.
( 0 Fout ) ( Vout' 0 )
Then, an orthogonal matrix Q is found such that the eigenvalues
with strictly negative real parts of the pencil
( S11 S12 ) ( H11 H12 )
a ( ) - b ( )
( 0 S11' ) ( 0 -H11' )
are moved to the top of this pencil.
Finally, an orthogonal basis of the right deflating subspace
corresponding to the eigenvalues with strictly negative real part
is computed. See also page 12 in [1] for more details.
References
[1] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
Eigenproblems.
Tech. Rep., Technical University Chemnitz, Germany,
Nov. 2007.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) floating point operations.Further Comments
This routine does not perform any scaling of the matrices. Scaling might sometimes be useful, and it should be done externally.Example
Program Text
* MB03LD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 50 )
INTEGER LDA, LDB, LDDE, LDFG, LDQ, LDWORK, LIWORK
PARAMETER ( LDA = NMAX/2, LDB = NMAX/2, LDDE = NMAX/2,
$ LDFG = NMAX/2, LDQ = 2*NMAX,
$ LDWORK = 8*NMAX*NMAX +
$ MAX( 8*NMAX + 32, NMAX/2 + 168,
$ 272 ),
$ LIWORK = MAX( 32, NMAX + 12, NMAX*2 + 3 ) )
*
* .. Local Scalars ..
CHARACTER COMPQ, ORTH
INTEGER I, INFO, J, M, N, NEIG
*
* .. Local Arrays ..
LOGICAL BWORK( NMAX/2 )
INTEGER IWORK( LIWORK )
DOUBLE PRECISION A( LDA, NMAX/2 ), ALPHAI( NMAX/2 ),
$ ALPHAR( NMAX/2 ), B( LDB, NMAX/2 ),
$ BETA( NMAX/2 ), DE( LDDE, NMAX/2+1 ),
$ DWORK( LDWORK ), FG( LDFG, NMAX/2+1 ),
$ Q( LDQ, 2*NMAX )
*
* .. External Subroutines ..
EXTERNAL MB03LD
*
* .. Intrinsic Functions ..
INTRINSIC MAX
*
* .. Executable Statements ..
*
WRITE( NOUT, FMT = 99999 )
* Skip the heading in the data file and read in the data.
READ( NIN, FMT = * )
READ( NIN, FMT = * ) COMPQ, ORTH, N
IF( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE( NOUT, FMT = 99998 ) N
ELSE
M = N/2
READ( NIN, FMT = * ) ( ( A( I, J ), J = 1, M ), I = 1, M )
READ( NIN, FMT = * ) ( ( DE( I, J ), J = 1, M+1 ), I = 1, M )
READ( NIN, FMT = * ) ( ( B( I, J ), J = 1, M ), I = 1, M )
READ( NIN, FMT = * ) ( ( FG( I, J ), J = 1, M+1 ), I = 1, M )
* Compute the eigenvalues and an orthogonal basis of the right
* deflating subspace of a real skew-Hamiltonian/Hamiltonian
* pencil, corresponding to the eigenvalues with strictly negative
* real part.
CALL MB03LD( COMPQ, ORTH, N, A, LDA, DE, LDDE, B, LDB, FG,
$ LDFG, NEIG, Q, LDQ, ALPHAR, ALPHAI, BETA, IWORK,
$ LIWORK, DWORK, LDWORK, BWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99997 ) INFO
ELSE
WRITE( NOUT, FMT = 99996 )
DO 10 I = 1, M
WRITE( NOUT, FMT = 99995 ) ( A( I, J ), J = 1, M )
10 CONTINUE
WRITE( NOUT, FMT = 99994 )
DO 20 I = 1, M
WRITE( NOUT, FMT = 99995 ) ( DE( I, J ), J = 1, M+1 )
20 CONTINUE
WRITE( NOUT, FMT = 99993 )
DO 30 I = 1, M
WRITE( NOUT, FMT = 99995 ) ( B( I, J ), J = 1, M )
30 CONTINUE
WRITE( NOUT, FMT = 99992 )
DO 40 I = 1, M
WRITE( NOUT, FMT = 99995 ) ( FG( I, J ), J = 2, M+1 )
40 CONTINUE
WRITE( NOUT, FMT = 99991 )
WRITE( NOUT, FMT = 99995 ) ( ALPHAR( I ), I = 1, M )
WRITE( NOUT, FMT = 99990 )
WRITE( NOUT, FMT = 99995 ) ( ALPHAI( I ), I = 1, M )
WRITE( NOUT, FMT = 99989 )
WRITE( NOUT, FMT = 99995 ) ( BETA( I ), I = 1, M )
IF( LSAME( COMPQ, 'C' ) .AND. NEIG.GT.0 ) THEN
WRITE( NOUT, FMT = 99988 )
DO 50 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( Q( I, J ), J = 1, NEIG )
50 CONTINUE
END IF
END IF
END IF
STOP
*
99999 FORMAT( 'MB03LD EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT( 'INFO on exit from MB03LD = ', I2 )
99996 FORMAT( 'The matrix A on exit is ' )
99995 FORMAT( 50( 1X, F8.4 ) )
99994 FORMAT( 'The matrix DE on exit is ' )
99993 FORMAT( 'The matrix C1 on exit is ' )
99992 FORMAT( 'The matrix V on exit is ' )
99991 FORMAT( 'The vector ALPHAR is ' )
99990 FORMAT( 'The vector ALPHAI is ' )
99989 FORMAT( 'The vector BETA is ' )
99988 FORMAT( 'The matrix Q is ' )
END
Program Data
MB03LD EXAMPLE PROGRAM DATA C P 8 3.1472 1.3236 4.5751 4.5717 4.0579 -4.0246 4.6489 -0.1462 -3.7301 -2.2150 -3.4239 3.0028 4.1338 0.4688 4.7059 -3.5811 0.0000 0.0000 -1.5510 -4.5974 -2.5127 3.5071 0.0000 0.0000 1.5961 2.4490 -3.1428 2.5648 0.0000 0.0000 -0.0596 3.0340 2.4892 -1.1604 0.0000 0.0000 0.6882 -3.3782 -3.3435 1.8921 -0.3061 2.9428 1.0198 2.4815 -4.8810 -1.8878 -2.3703 -0.4946 -1.6288 0.2853 1.5408 -4.1618 -2.4013 -2.7102 0.3834 -3.9335 3.1730 -3.1815 -2.3620 4.9613 4.6190 3.6869 3.6929 0.7970 0.4986 -4.9537 -4.1556 3.5303 1.2206 -1.4905 0.1325 -1.0022Program Results
MB03LD EXAMPLE PROGRAM RESULTS The matrix A on exit is -4.7460 4.1855 3.2696 -0.2244 0.0000 6.4157 2.8287 1.4553 0.0000 0.0000 7.4626 1.5726 0.0000 0.0000 0.0000 8.8702 The matrix DE on exit is -5.4562 2.5550 -1.3137 -6.3615 -0.8940 -2.1348 -7.9616 0.0000 1.0704 -0.0659 4.9694 1.1516 4.8504 0.0000 -0.6922 -2.2744 3.4912 0.5046 4.4394 0.0000 The matrix C1 on exit is 6.9525 -4.9881 2.3661 4.2188 0.0000 8.5009 0.7182 5.5533 0.0000 0.0000 -4.6650 -2.8177 0.0000 0.0000 0.0000 1.5124 The matrix V on exit is 0.9136 4.1106 -0.0079 3.5789 -1.1553 -1.4785 -1.5155 -0.8018 -2.2167 4.8029 1.3645 2.5202 -1.0994 -0.6144 0.3970 2.0730 The vector ALPHAR is 0.8314 -0.8314 0.8131 0.0000 The vector ALPHAI is 0.4372 0.4372 0.0000 0.9164 The vector BETA is 0.7071 0.7071 1.4142 2.8284 The matrix Q is -0.5844 -0.2949 0.1692 0.5470 -0.2324 -0.4524 0.0382 0.4673 -0.3092 -0.0378 0.0904 -0.4451 -0.0255 0.1497 -0.1929 -0.1286 -0.6067 -0.2275 -0.2260 0.4901 0.0951 0.5367 -0.0430 0.6123
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB03LF.html 0000664 0000000 0000000 00000050077 14560147231 0020346 0 ustar 00root root 0000000 0000000
Purpose
To compute the relevant eigenvalues of a real N-by-N skew-
Hamiltonian/Hamiltonian pencil aS - bH, with
( B F ) ( 0 I )
S = T Z = J Z' J' Z and H = ( ), J = ( ), (1)
( G -B' ) ( -I 0 )
where the notation M' denotes the transpose of the matrix M.
Optionally, if COMPQ = 'C', an orthogonal basis of the right
deflating subspace of aS - bH corresponding to the eigenvalues
with strictly negative real part is computed. Optionally, if
COMPU = 'C', an orthonormal basis of the companion subspace,
range(P_U) [1], which corresponds to the eigenvalues with strictly
negative real part, is computed.
Specification
SUBROUTINE MB03LF( COMPQ, COMPU, ORTH, N, Z, LDZ, B, LDB, FG,
$ LDFG, NEIG, Q, LDQ, U, LDU, ALPHAR, ALPHAI,
$ BETA, IWORK, LIWORK, DWORK, LDWORK, BWORK,
$ IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, COMPU, ORTH
INTEGER INFO, IWARN, LDB, LDFG, LDQ, LDU, LDWORK, LDZ,
$ LIWORK, N, NEIG
C .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION ALPHAI( * ), ALPHAR( * ), B( LDB, * ),
$ BETA( * ), DWORK( * ), FG( LDFG, * ),
$ Q( LDQ, * ), U( LDU, * ), Z( LDZ, * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
Specifies whether to compute the right deflating subspace
corresponding to the eigenvalues of aS - bH with strictly
negative real part.
= 'N': do not compute the deflating subspace;
= 'C': compute the deflating subspace and store it in the
leading subarray of Q.
COMPU CHARACTER*1
Specifies whether to compute the companion subspace
corresponding to the eigenvalues of aS - bH with strictly
negative real part.
= 'N': do not compute the companion subspace;
= 'C': compute the companion subspace and store it in the
leading subarray of U.
ORTH CHARACTER*1
If COMPQ = 'C' and/or COMPU = 'C', specifies the technique
for computing the orthogonal basis of the deflating
subspace, and/or of the companion subspace, as follows:
= 'P': QR factorization with column pivoting;
= 'S': singular value decomposition.
If COMPQ = 'N' and COMPU = 'N', the ORTH value is not
used.
Input/Output Parameters
N (input) INTEGER
The order of the pencil aS - bH. N >= 0, even.
Z (input/output) DOUBLE PRECISION array, dimension (LDZ, N)
On entry, the leading N-by-N part of this array must
contain the non-trivial factor Z in the factorization
S = J Z' J' Z of the skew-Hamiltonian matrix S.
On exit, if COMPQ = 'C' or COMPU = 'C', the leading
N-by-N part of this array contains the transformed upper
~
triangular matrix Z11 (see METHOD), after moving the
eigenvalues with strictly negative real part to the top
of the pencil (3). The strictly lower triangular part is
not zeroed.
If COMPQ = 'N' and COMPU = 'N', the leading N-by-N part of
this array contains the matrix Z obtained by the SLICOT
Library routine MB04AD just before the application of the
periodic QZ algorithm. The elements of the (2,1) block,
i.e., in the rows N/2+1 to N and in the columns 1 to N/2
are not set to zero, but are unchanged on exit.
LDZ INTEGER
The leading dimension of the array Z. LDZ >= MAX(1, N).
B (input) DOUBLE PRECISION array, dimension (LDB, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix B.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N/2).
FG (input) DOUBLE PRECISION array, dimension (LDFG, N/2+1)
On entry, the leading N/2-by-N/2 lower triangular part of
this array must contain the lower triangular part of the
symmetric matrix G, and the N/2-by-N/2 upper triangular
part of the submatrix in the columns 2 to N/2+1 of this
array must contain the upper triangular part of the
symmetric matrix F.
LDFG INTEGER
The leading dimension of the array FG.
LDFG >= MAX(1, N/2).
NEIG (output) INTEGER
If COMPQ = 'C' or COMPU = 'C', the number of eigenvalues
in aS - bH with strictly negative real part.
Q (output) DOUBLE PRECISION array, dimension (LDQ, 2*N)
On exit, if COMPQ = 'C', the leading N-by-NEIG part of
this array contains an orthogonal basis of the right
deflating subspace corresponding to the eigenvalues of
aS - bH with strictly negative real part. The remaining
part of this array is used as workspace.
If COMPQ = 'N', this array is not referenced.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1, 2*N), if COMPQ = 'C'.
U (output) DOUBLE PRECISION array, dimension (LDU, 2*N)
On exit, if COMPU = 'C', the leading N-by-NEIG part of
this array contains an orthogonal basis of the companion
subspace corresponding to the eigenvalues of aS - bH with
strictly negative real part. The remaining part of this
array is used as workspace.
If COMPU = 'N', this array is not referenced.
LDU INTEGER
The leading dimension of the array U.
LDU >= 1, if COMPU = 'N';
LDU >= MAX(1, N), if COMPU = 'C'.
ALPHAR (output) DOUBLE PRECISION array, dimension (N/2)
The real parts of each scalar alpha defining an eigenvalue
of the pencil aS - bH.
ALPHAI (output) DOUBLE PRECISION array, dimension (N/2)
The imaginary parts of each scalar alpha defining an
eigenvalue of the pencil aS - bH.
If ALPHAI(j) is zero, then the j-th eigenvalue is real.
BETA (output) DOUBLE PRECISION array, dimension (N/2)
The scalars beta that define the eigenvalues of the pencil
aS - bH.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent the j-th eigenvalue of the pencil
aS - bH, in the form lambda = alpha/beta. Since lambda may
overflow, the ratios should not, in general, be computed.
Due to the skew-Hamiltonian/Hamiltonian structure of the
pencil, for every eigenvalue lambda, -lambda is also an
eigenvalue, and thus it has only to be saved once in
ALPHAR, ALPHAI and BETA.
Specifically, only eigenvalues with imaginary parts
greater than or equal to zero are stored; their conjugate
eigenvalues are not stored. If imaginary parts are zero
(i.e., for real eigenvalues), only positive eigenvalues
are stored. The remaining eigenvalues have opposite signs.
As a consequence, pairs of complex eigenvalues, stored in
consecutive locations, are not complex conjugate.
Workspace
IWORK INTEGER array, dimension (LIWORK)
On exit, if INFO = -20, IWORK(1) returns the minimum value
of LIWORK.
LIWORK INTEGER
The dimension of the array IWORK.
LIWORK >= N + 18, if COMPQ = 'N' and COMPU = 'N';
LIWORK >= MAX( 2*N+1, 48 ), otherwise.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
On exit, if INFO = -22, DWORK(1) returns the minimum value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= b*N*N + 3*N*N/2 + MAX( 6*N, 54 ),
if COMPQ = 'N' and COMPU = 'N';
LDWORK >= d*N*N + MAX( N/2+252, 432 ), otherwise, where
b = a, d = c, if COMPU = 'N',
b = a+1, d = c+1, if COMPU = 'C', and
a = 2, c = 7, if COMPQ = 'N',
a = 4, c = 10, if COMPQ = 'C'.
For good performance LDWORK should be generally larger.
If LDWORK = -1 a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
BWORK LOGICAL array, dimension (N/2)
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: some eigenvalues might be unreliable. More details
can be obtained by running the SLICOT routine MB04AD.
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: periodic QZ iteration failed in the SLICOT Library
routines MB04AD, MB04CD or MB03BB (QZ iteration did
not converge or computation of the shifts failed);
= 2: standard QZ iteration failed in the SLICOT Library
routines MB04CD or MB03CD (called by MB03ID);
= 3: a numerically singular matrix was found in the SLICOT
Library routine MB03GD (called by MB03ID);
= 4: the singular value decomposition failed in the LAPACK
routine DGESVD (for ORTH = 'S').
Method
First, the decompositions of S and H are computed via orthogonal
matrices Q1 and Q2 and orthogonal symplectic matrices U1 and U2,
such that
( T11 T12 )
Q1' T U1 = Q1' J Z' J' U1 = ( ),
( 0 T22 )
( Z11 Z12 )
U2' Z Q2 = ( ), (2)
( 0 Z22 )
( H11 H12 )
Q1' H Q2 = ( ),
( 0 H22 )
where T11, T22', Z11, Z22', H11 are upper triangular and H22' is
upper quasi-triangular.
Then, orthogonal matrices Q3, Q4 and U3 are found, for the
matrices
~ ( T22' 0 ) ~ ( T11' 0 ) ~ ( 0 H11 )
Z11 = ( ), Z22 = ( ), H = ( ),
( 0 Z11 ) ( 0 Z22 ) ( -H22' 0 )
~ ~ ~ ~
such that Z11 := U3' Z11 Q4, Z22 := U3' Z22 Q3 are upper
~ ~
triangular and H11 := Q3' H Q4 is upper quasi-triangular. The
following matrices are computed:
~ ( -T12' 0 ) ~ ( 0 H12 )
Z12 := U3' ( ) Q3 and H12 := Q3' ( ) Q3.
( 0 Z12 ) ( H12' 0 )
Then, an orthogonal matrix Q and an orthogonal symplectic matrix U
are found such that the eigenvalues with strictly negative real
parts of the pencil
~ ~ ~ ~ ~ ~
( Z11 Z12 )' ( Z11 Z12 ) ( H11 H12 )
a J ( ~ ) J' ( ~ ) - b ( ~ ) (3)
( 0 Z22 ) ( 0 Z22 ) ( 0 -H11' )
are moved to the top of this pencil.
Finally, an orthogonal basis of the right deflating subspace
and an orthogonal basis of the companion subspace corresponding to
the eigenvalues with strictly negative real part are computed.
See also page 11 in [1] for more details.
References
[1] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
Eigenproblems.
Tech. Rep., Technical University Chemnitz, Germany,
Nov. 2007.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) floating point operations.Further Comments
This routine does not perform any scaling of the matrices. Scaling might sometimes be useful, and it should be done externally.Example
Program Text
* MB03LF EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 50 )
INTEGER LDB, LDFG, LDQ, LDU, LDWORK, LDZ, LIWORK
PARAMETER ( LDB = NMAX/2, LDFG = NMAX/2, LDQ = 2*NMAX,
$ LDU = NMAX, LDZ = NMAX,
$ LDWORK = 10*NMAX*NMAX +
$ MAX( NMAX*NMAX +
$ MAX( NMAX/2 + 252, 432 ),
$ MAX( 8*NMAX + 48, 171 ) ),
$ LIWORK = MAX( NMAX + 18, NMAX/2 + 48,
$ 5*NMAX/2 + 1 ) )
*
* .. Local Scalars ..
CHARACTER COMPQ, COMPU, ORTH
INTEGER I, INFO, IWARN, J, M, N, NEIG
*
* .. Local Arrays ..
LOGICAL BWORK( NMAX/2 )
INTEGER IWORK( LIWORK )
DOUBLE PRECISION ALPHAI( NMAX/2 ), ALPHAR( NMAX/2 ),
$ B( LDB, NMAX/2 ), BETA( NMAX/2 ),
$ DWORK( LDWORK ), FG( LDFG, NMAX/2+1 ),
$ Q( LDQ, 2*NMAX ), U( LDU, 2*NMAX ),
$ Z( LDZ, NMAX )
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
*
* .. External Subroutines ..
EXTERNAL MB03LF
*
* .. Intrinsic Functions ..
INTRINSIC MAX, MOD
*
* .. Executable Statements ..
*
WRITE( NOUT, FMT = 99999 )
* Skip the heading in the data file and read in the data.
READ( NIN, FMT = * )
READ( NIN, FMT = * ) COMPQ, COMPU, ORTH, N
IF( N.LT.0 .OR. N.GT.NMAX .OR. MOD( N, 2 ).NE.0 ) THEN
WRITE( NOUT, FMT = 99998 ) N
ELSE
M = N/2
READ( NIN, FMT = * ) ( ( Z( I, J ), J = 1, N ), I = 1, N )
READ( NIN, FMT = * ) ( ( B( I, J ), J = 1, M ), I = 1, M )
READ( NIN, FMT = * ) ( ( FG( I, J ), J = 1, M+1 ), I = 1, M )
* Compute the eigenvalues and orthogonal bases of the right
* deflating subspace and companion subspace of a real
* skew-Hamiltonian/Hamiltonian pencil, corresponding to the
* eigenvalues with strictly negative real part.
CALL MB03LF( COMPQ, COMPU, ORTH, N, Z, LDZ, B, LDB, FG, LDFG,
$ NEIG, Q, LDQ, U, LDU, ALPHAR, ALPHAI, BETA, IWORK,
$ LIWORK, DWORK, LDWORK, BWORK, IWARN, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99997 ) INFO
ELSE
WRITE( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( Z( I, J ), J = 1, N )
10 CONTINUE
WRITE( NOUT, FMT = 99994 )
WRITE( NOUT, FMT = 99995 ) ( ALPHAR( I ), I = 1, M )
WRITE( NOUT, FMT = 99993 )
WRITE( NOUT, FMT = 99995 ) ( ALPHAI( I ), I = 1, M )
WRITE( NOUT, FMT = 99992 )
WRITE( NOUT, FMT = 99995 ) ( BETA( I ), I = 1, M )
IF( LSAME( COMPQ, 'C' ) .AND. NEIG.GT.0 ) THEN
WRITE( NOUT, FMT = 99991 )
DO 20 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( Q( I, J ), J = 1, NEIG )
20 CONTINUE
END IF
IF( LSAME( COMPU, 'C' ) .AND. NEIG.GT.0 ) THEN
WRITE( NOUT, FMT = 99990 )
DO 30 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( U( I, J ), J = 1, NEIG )
30 CONTINUE
END IF
IF( LSAME( COMPQ, 'C' ) .OR. LSAME( COMPU, 'C' ) )
$ WRITE( NOUT, FMT = 99989 ) NEIG
END IF
END IF
STOP
*
99999 FORMAT ( 'MB03LF EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT ( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT ( 'INFO on exit from MB03LF = ', I2 )
99996 FORMAT (/'The matrix Z on exit is ' )
99995 FORMAT ( 50( 1X, F8.4 ) )
99994 FORMAT (/'The vector ALPHAR is ' )
99993 FORMAT (/'The vector ALPHAI is ' )
99992 FORMAT (/'The vector BETA is ' )
99991 FORMAT (/'The deflating subspace corresponding to the ',
$ 'eigenvalues with negative real part is ' )
99990 FORMAT (/'The companion subspace corresponding to the ',
$ 'eigenvalues with negative real part is ' )
99989 FORMAT (/'The number of eigenvalues in the initial pencil with ',
$ 'negative real part is ', I2 )
END
Program Data
MB03LF EXAMPLE PROGRAM DATA C C P 8 3.1472 4.5751 -0.7824 1.7874 -2.2308 -0.6126 2.0936 4.5974 4.0579 4.6489 4.1574 2.5774 -4.5383 -1.1844 2.5469 -1.5961 -3.7301 -3.4239 2.9221 2.4313 -4.0287 2.6552 -2.2397 0.8527 4.1338 4.7059 4.5949 -1.0777 3.2346 2.9520 1.7970 -2.7619 1.3236 4.5717 1.5574 1.5548 1.9483 -3.1313 1.5510 2.5127 -4.0246 -0.1462 -4.6429 -3.2881 -1.8290 -0.1024 -3.3739 -2.4490 -2.2150 3.0028 3.4913 2.0605 4.5022 -0.5441 -3.8100 0.0596 0.4688 -3.5811 4.3399 -4.6817 -4.6555 1.4631 -0.0164 1.9908 0.6882 -3.3782 -3.3435 1.8921 -0.3061 2.9428 1.0198 2.4815 -4.8810 -1.8878 -2.3703 -0.4946 -1.6288 0.2853 1.5408 -4.1618 -2.4013 -2.7102 0.3834 -3.9335 3.1730 -3.1815 -2.3620 4.9613 4.6190 3.6869 3.6929 0.7970 0.4986 -4.9537 -4.1556 3.5303 1.2206 -1.4905 0.1325 -1.0022Program Results
MB03LF EXAMPLE PROGRAM RESULTS The matrix Z on exit is 4.4128 0.1059 -1.8709 1.2963 -4.3448 2.7633 2.3580 2.1931 0.0000 10.0337 -1.9797 1.8052 -1.0112 1.1335 1.2374 0.3107 0.0000 0.0000 8.9476 1.8523 -1.8578 -0.5807 -1.4157 1.3007 0.0000 0.0000 0.0000 -7.0889 -2.1193 -2.1634 -2.4393 0.1148 0.0765 1.0139 0.0000 -1.5390 -8.3187 -5.0172 0.7738 -2.8626 1.1884 -0.9225 0.0000 0.2905 0.0000 6.4090 2.1994 -2.5933 -0.5931 0.1981 0.0000 -0.5280 0.0000 0.0000 4.7155 2.3817 1.8591 -1.8416 0.0000 -0.0807 0.0000 0.0000 0.0000 -5.3153 The vector ALPHAR is 0.7353 0.0000 0.5168 -0.5168 The vector ALPHAI is 0.0000 0.7190 0.5610 0.5610 The vector BETA is 2.0000 2.8284 11.3137 11.3137 The deflating subspace corresponding to the eigenvalues with negative real part is -0.2509 0.3670 0.0416 -0.3267 -0.7968 -0.1019 0.0263 0.0338 -0.5795 -0.0139 -0.0491 -0.5217 -0.4637 0.2992 -0.4403 -0.1345 0.3071 -0.0917 -0.1364 0.2013 0.3447 -0.7601 -0.0495 0.2426 The companion subspace corresponding to the eigenvalues with negative real part is -0.3219 0.6590 0.1693 -0.5216 -0.1829 -0.0689 -0.0413 -0.4664 -0.1359 0.1310 -0.1702 0.4543 -0.3598 0.2660 0.3355 -0.5082 -0.0512 -0.6035 -0.3582 -0.4513 0.4649 0.2991 0.0932 -0.2207 The number of eigenvalues in the initial pencil with negative real part is 3
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB03LP.html 0000664 0000000 0000000 00000032744 14560147231 0020361 0 ustar 00root root 0000000 0000000
Purpose
To compute the relevant eigenvalues of a real N-by-N skew-
Hamiltonian/Hamiltonian pencil aS - bH, with
( A D ) ( B F )
S = ( ) and H = ( ), (1)
( E A' ) ( G -B' )
where the notation M' denotes the transpose of the matrix M.
Optionally, if COMPQ = 'C', an orthogonal basis of the right
deflating subspace of aS - bH corresponding to the eigenvalues
with strictly negative real part is computed.
Specification
SUBROUTINE MB03LP( COMPQ, ORTH, N, A, LDA, DE, LDDE, B, LDB, FG,
$ LDFG, NEIG, Q, LDQ, ALPHAR, ALPHAI, BETA,
$ IWORK, LIWORK, DWORK, LDWORK, BWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, ORTH
INTEGER INFO, LDA, LDB, LDDE, LDFG, LDQ, LDWORK,
$ LIWORK, N, NEIG
C .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), DE( LDDE, * ),
$ DWORK( * ), FG( LDFG, * ), Q( LDQ, * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
Specifies whether to compute the right deflating subspace
corresponding to the eigenvalues of aS - bH with strictly
negative real part.
= 'N': do not compute the deflating subspace;
= 'C': compute the deflating subspace and store it in the
leading subarray of Q.
ORTH CHARACTER*1
If COMPQ = 'C', specifies the technique for computing an
orthogonal basis of the deflating subspace, as follows:
= 'P': QR factorization with column pivoting;
= 'S': singular value decomposition.
If COMPQ = 'N', the ORTH value is not used.
Input/Output Parameters
N (input) INTEGER
The order of the pencil aS - bH. N >= 0, even.
A (input/output) DOUBLE PRECISION array, dimension
(LDA, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix A.
On exit, if COMPQ = 'C', the leading N/2-by-N/2 part of
this array contains the upper triangular matrix Aout
(see METHOD); otherwise, it contains the upper triangular
matrix A obtained just before the application of the
periodic QZ algorithm (see SLICOT Library routine MB04BP).
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1, N/2).
DE (input/output) DOUBLE PRECISION array, dimension
(LDDE, N/2+1)
On entry, the leading N/2-by-N/2 lower triangular part of
this array must contain the lower triangular part of the
skew-symmetric matrix E, and the N/2-by-N/2 upper
triangular part of the submatrix in the columns 2 to N/2+1
of this array must contain the upper triangular part of the
skew-symmetric matrix D.
The entries on the diagonal and the first superdiagonal of
this array need not be set, but are assumed to be zero.
On exit, if COMPQ = 'C', the leading N/2-by-N/2 lower
triangular part and the first superdiagonal contain the
transpose of the upper quasi-triangular matrix C2out (see
METHOD), and the (N/2-1)-by-(N/2-1) upper triangular part
of the submatrix in the columns 3 to N/2+1 of this array
contains the strictly upper triangular part of the
skew-symmetric matrix Dout (see METHOD), without the main
diagonal, which is zero.
On exit, if COMPQ = 'N', the leading N/2-by-N/2 lower
triangular part and the first superdiagonal contain the
transpose of the upper Hessenberg matrix C2, and the
(N/2-1)-by-(N/2-1) upper triangular part of the submatrix
in the columns 3 to N/2+1 of this array contains the
strictly upper triangular part of the skew-symmetric
matrix D (without the main diagonal) just before the
application of the periodic QZ algorithm.
LDDE INTEGER
The leading dimension of the array DE.
LDDE >= MAX(1, N/2).
B (input/output) DOUBLE PRECISION array, dimension
(LDB, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix B.
On exit, if COMPQ = 'C', the leading N/2-by-N/2 part of
this array contains the upper triangular matrix C1out
(see METHOD); otherwise, it contains the upper triangular
matrix C1 obtained just before the application of the
periodic QZ algorithm.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N/2).
FG (input/output) DOUBLE PRECISION array, dimension
(LDFG, N/2+1)
On entry, the leading N/2-by-N/2 lower triangular part of
this array must contain the lower triangular part of the
symmetric matrix G, and the N/2-by-N/2 upper triangular
part of the submatrix in the columns 2 to N/2+1 of this
array must contain the upper triangular part of the
symmetric matrix F.
On exit, if COMPQ = 'C', the leading N/2-by-N/2 part of
the submatrix in the columns 2 to N/2+1 of this array
contains the matrix Vout (see METHOD); otherwise, it
contains the matrix V obtained just before the application
of the periodic QZ algorithm.
LDFG INTEGER
The leading dimension of the array FG.
LDFG >= MAX(1, N/2).
NEIG (output) INTEGER
If COMPQ = 'C', the number of eigenvalues in aS - bH with
strictly negative real part.
Q (output) DOUBLE PRECISION array, dimension (LDQ, 2*N)
On exit, if COMPQ = 'C', the leading N-by-NEIG part of
this array contains an orthogonal basis of the right
deflating subspace corresponding to the eigenvalues of
aA - bB with strictly negative real part. The remaining
part of this array is used as workspace.
If COMPQ = 'N', this array is not referenced.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1, 2*N), if COMPQ = 'C'.
ALPHAR (output) DOUBLE PRECISION array, dimension (N/2)
The real parts of each scalar alpha defining an eigenvalue
of the pencil aS - bH.
ALPHAI (output) DOUBLE PRECISION array, dimension (N/2)
The imaginary parts of each scalar alpha defining an
eigenvalue of the pencil aS - bH.
If ALPHAI(j) is zero, then the j-th eigenvalue is real.
BETA (output) DOUBLE PRECISION array, dimension (N/2)
The scalars beta that define the eigenvalues of the pencil
aS - bH.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent the j-th eigenvalue of the pencil
aS - bH, in the form lambda = alpha/beta. Since lambda may
overflow, the ratios should not, in general, be computed.
Due to the skew-Hamiltonian/Hamiltonian structure of the
pencil, for every eigenvalue lambda, -lambda is also an
eigenvalue, and thus it has only to be saved once in
ALPHAR, ALPHAI and BETA.
Specifically, only eigenvalues with imaginary parts
greater than or equal to zero are stored; their conjugate
eigenvalues are not stored. If imaginary parts are zero
(i.e., for real eigenvalues), only positive eigenvalues
are stored. The remaining eigenvalues have opposite signs.
Workspace
IWORK INTEGER array, dimension (LIWORK)
On exit, if INFO = -19, IWORK(1) returns the minimum value
of LIWORK.
LIWORK INTEGER
The dimension of the array IWORK. LIWORK = 1, if N = 0,
LIWORK >= MAX( N + 12, 2*N + 3 ), if COMPQ = 'N',
LIWORK >= MAX( 32, 6*N - 3 ), if COMPQ = 'C'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
On exit, if INFO = -21, DWORK(1) returns the minimum value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK. LDWORK = 1, if N = 0,
LDWORK >= 3*(N/2)**2 + N**2 + MAX( L, 36 ),
if COMPQ = 'N',
where L = 4*N + 4, if N/2 is even, and
L = 4*N , if N/2 is odd;
LDWORK >= 8*N**2 + MAX( 13*N + 32, 272 ), if COMPQ = 'C'.
For good performance LDWORK should be generally larger.
If LDWORK = -1 a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
BWORK LOGICAL array, dimension (N/2)
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: periodic QZ iteration failed in the SLICOT Library
routines MB04BP or MB04HD (QZ iteration did not
converge or computation of the shifts failed);
= 2: standard QZ iteration failed in the SLICOT Library
routines MB04HD or MB03DD (called by MB03JP);
= 3: a numerically singular matrix was found in the SLICOT
Library routine MB03HD (called by MB03JP);
= 4: the singular value decomposition failed in the LAPACK
routine DGESVD (for ORTH = 'S');
= 5: some eigenvalues might be inaccurate. This is a
warning.
Method
First, the decompositions of S and H are computed via orthogonal
transformations Q1 and Q2 as follows:
( Aout Dout )
Q1' S J Q1 J' = ( ),
( 0 Aout' )
( Bout Fout )
J' Q2' J S Q2 = ( ) =: T, (2)
( 0 Bout' )
( C1out Vout ) ( 0 I )
Q1' H Q2 = ( ), where J = ( ),
( 0 C2out' ) ( -I 0 )
and Aout, Bout, C1out are upper triangular, C2out is upper quasi-
triangular and Dout and Fout are skew-symmetric.
Then, orthogonal matrices Q3 and Q4 are found, for the extended
matrices
( Aout 0 ) ( 0 C1out )
Se = ( ) and He = ( ),
( 0 Bout ) ( -C2out 0 )
such that S11 := Q4' Se Q3 is upper triangular and
H11 := Q4' He Q3 is upper quasi-triangular. The following matrices
are computed:
( Dout 0 ) ( 0 Vout )
S12 := Q4' ( ) Q4 and H12 := Q4' ( ) Q4.
( 0 Fout ) ( Vout' 0 )
Then, an orthogonal matrix Q is found such that the eigenvalues
with strictly negative real parts of the pencil
( S11 S12 ) ( H11 H12 )
a ( ) - b ( )
( 0 S11' ) ( 0 -H11' )
are moved to the top of this pencil.
Finally, an orthogonal basis of the right deflating subspace
corresponding to the eigenvalues with strictly negative real part
is computed. See also page 12 in [1] for more details.
References
[1] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
Eigenproblems.
Tech. Rep., Technical University Chemnitz, Germany,
Nov. 2007.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) floating point operations.Further Comments
This routine does not perform any scaling of the matrices. Scaling might sometimes be useful, and it should be done externally. For large values of N, the routine applies the transformations on panels of columns. The user may specify in INFO the desired number of columns. If on entry INFO <= 0, then the routine estimates a suitable value of this number.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB03LZ.html 0000664 0000000 0000000 00000050064 14560147231 0020366 0 ustar 00root root 0000000 0000000
Purpose
To compute the eigenvalues of a complex N-by-N skew-Hamiltonian/
Hamiltonian pencil aS - bH, with
( A D ) ( B F )
S = ( ) and H = ( ). (1)
( E A' ) ( G -B' )
The structured Schur form of the embedded real skew-Hamiltonian/
skew-Hamiltonian pencil aB_S - bB_T, defined as
( Re(A) -Im(A) | Re(D) -Im(D) )
( | )
( Im(A) Re(A) | Im(D) Re(D) )
( | )
B_S = (-----------------+-----------------) , and
( | )
( Re(E) -Im(E) | Re(A') Im(A') )
( | )
( Im(E) Re(E) | -Im(A') Re(A') )
(2)
( -Im(B) -Re(B) | -Im(F) -Re(F) )
( | )
( Re(B) -Im(B) | Re(F) -Im(F) )
( | )
B_T = (-----------------+-----------------) , T = i*H,
( | )
( -Im(G) -Re(G) | -Im(B') Re(B') )
( | )
( Re(G) -Im(G) | -Re(B') -Im(B') )
is determined and used to compute the eigenvalues. The notation M'
denotes the conjugate transpose of the matrix M. Optionally,
if COMPQ = 'C', an orthonormal basis of the right deflating
subspace of the pencil aS - bH, corresponding to the eigenvalues
with strictly negative real part, is computed. Namely, after
transforming aB_S - bB_H by unitary matrices, we have
( BA BD ) ( BB BF )
B_Sout = ( ) and B_Hout = ( ), (3)
( 0 BA' ) ( 0 -BB' )
and the eigenvalues with strictly negative real part of the
complex pencil aB_Sout - bB_Hout are moved to the top. The
embedding doubles the multiplicities of the eigenvalues of the
pencil aS - bH.
Specification
SUBROUTINE MB03LZ( COMPQ, ORTH, N, A, LDA, DE, LDDE, B, LDB, FG,
$ LDFG, NEIG, Q, LDQ, ALPHAR, ALPHAI, BETA,
$ IWORK, DWORK, LDWORK, ZWORK, LZWORK, BWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, ORTH
INTEGER INFO, LDA, LDB, LDDE, LDFG, LDQ, LDWORK,
$ LZWORK, N, NEIG
C .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION ALPHAI( * ), ALPHAR( * ), BETA( * ), DWORK( * )
COMPLEX*16 A( LDA, * ), B( LDB, * ), DE( LDDE, * ),
$ FG( LDFG, * ), Q( LDQ, * ), ZWORK( * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
Specifies whether to compute the deflating subspace
corresponding to the eigenvalues of aS - bH with strictly
negative real part.
= 'N': do not compute the deflating subspace; compute the
eigenvalues only;
= 'C': compute the deflating subspace and store it in the
leading subarray of Q.
ORTH CHARACTER*1
If COMPQ = 'C', specifies the technique for computing an
orthonormal basis of the deflating subspace, as follows:
= 'P': QR factorization with column pivoting;
= 'S': singular value decomposition.
If COMPQ = 'N', the ORTH value is not used.
Input/Output Parameters
N (input) INTEGER
The order of the pencil aS - bH. N >= 0, even.
A (input/output) COMPLEX*16 array, dimension (LDA, N)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix A.
On exit, if COMPQ = 'C', the leading N-by-N part of this
array contains the upper triangular matrix BA in (3) (see
also METHOD). The strictly lower triangular part is not
zeroed; it is preserved in the leading N/2-by-N/2 part.
If COMPQ = 'N', this array is unchanged on exit.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1, N).
DE (input/output) COMPLEX*16 array, dimension (LDDE, N)
On entry, the leading N/2-by-N/2 lower triangular part of
this array must contain the lower triangular part of the
skew-Hermitian matrix E, and the N/2-by-N/2 upper
triangular part of the submatrix in the columns 2 to N/2+1
of this array must contain the upper triangular part of
the skew-Hermitian matrix D.
On exit, if COMPQ = 'C', the leading N-by-N part of this
array contains the skew-Hermitian matrix BD in (3) (see
also METHOD). The strictly lower triangular part of the
input matrix is preserved.
If COMPQ = 'N', this array is unchanged on exit.
LDDE INTEGER
The leading dimension of the array DE. LDDE >= MAX(1, N).
B (input/output) COMPLEX*16 array, dimension (LDB, N)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix B.
On exit, if COMPQ = 'C', the leading N-by-N part of this
array contains the upper triangular matrix BB in (3) (see
also METHOD). The strictly lower triangular part is not
zeroed; the elements below the first subdiagonal of the
input matrix are preserved.
If COMPQ = 'N', this array is unchanged on exit.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N).
FG (input/output) COMPLEX*16 array, dimension (LDFG, N)
On entry, the leading N/2-by-N/2 lower triangular part of
this array must contain the lower triangular part of the
Hermitian matrix G, and the N/2-by-N/2 upper triangular
part of the submatrix in the columns 2 to N/2+1 of this
array must contain the upper triangular part of the
Hermitian matrix F.
On exit, if COMPQ = 'C', the leading N-by-N part of this
array contains the Hermitian matrix BF in (3) (see also
METHOD). The strictly lower triangular part of the input
matrix is preserved. The diagonal elements might have tiny
imaginary parts.
If COMPQ = 'N', this array is unchanged on exit.
LDFG INTEGER
The leading dimension of the array FG. LDFG >= MAX(1, N).
NEIG (output) INTEGER
If COMPQ = 'C', the number of eigenvalues in aS - bH with
strictly negative real part.
Q (output) COMPLEX*16 array, dimension (LDQ, 2*N)
On exit, if COMPQ = 'C', the leading N-by-NEIG part of
this array contains an orthonormal basis of the right
deflating subspace corresponding to the eigenvalues of the
pencil aS - bH with strictly negative real part.
The remaining entries are meaningless.
If COMPQ = 'N', this array is not referenced.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1, 2*N), if COMPQ = 'C'.
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
The real parts of each scalar alpha defining an eigenvalue
of the pencil aS - bH.
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
The imaginary parts of each scalar alpha defining an
eigenvalue of the pencil aS - bH.
If ALPHAI(j) is zero, then the j-th eigenvalue is real.
BETA (output) DOUBLE PRECISION array, dimension (N)
The scalars beta that define the eigenvalues of the pencil
aS - bH.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent the j-th eigenvalue of the pencil
aS - bH, in the form lambda = alpha/beta. Since lambda may
overflow, the ratios should not, in general, be computed.
Workspace
IWORK INTEGER array, dimension (N+1)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
On exit, if INFO = -20, DWORK(1) returns the minimum value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= MAX( 4*N*N + 2*N + MAX(3,N) ), if COMPQ = 'N';
LDWORK >= MAX( 1, 11*N*N + 2*N ), if COMPQ = 'C'.
For good performance LDWORK should be generally larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) returns the optimal LZWORK.
On exit, if INFO = -22, ZWORK(1) returns the minimum value
of LZWORK.
LZWORK INTEGER
The dimension of the array ZWORK.
LZWORK >= 1, if COMPQ = 'N';
LZWORK >= 8*N + 4, if COMPQ = 'C'.
For good performance LZWORK should be generally larger.
If LZWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
ZWORK array, returns this value as the first entry of
the ZWORK array, and no error message related to LZWORK
is issued by XERBLA.
BWORK LOGICAL array, dimension (LBWORK)
LBWORK >= 0, if COMPQ = 'N';
LBWORK >= N - 1, if COMPQ = 'C'.
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: QZ iteration failed in the SLICOT Library routine
MB04FD (QZ iteration did not converge or computation
of the shifts failed);
= 2: QZ iteration failed in the LAPACK routine ZHGEQZ when
trying to triangularize the 2-by-2 blocks;
= 3: the singular value decomposition failed in the LAPACK
routine ZGESVD (for ORTH = 'S');
= 4: warning: the pencil is numerically singular.
Method
First, T = i*H is set. Then, the embeddings, B_S and B_T, of the
matrices S and T, are determined and, subsequently, the SLICOT
Library routine MB04FD is applied to compute the structured Schur
form, i.e., the factorizations
~ ( S11 S12 )
B_S = J Q' J' B_S Q = ( ) and
( 0 S11' )
~ ( T11 T12 ) ( 0 I )
B_T = J Q' J' B_T Q = ( ), with J = ( ),
( 0 T11' ) ( -I 0 )
where Q is real orthogonal, S11 is upper triangular, and T11 is
upper quasi-triangular.
Second, the SLICOT Library routine MB03JZ is applied, to compute a
~
unitary matrix Q, such that
~ ~
~ ~ ~ ( S11 S12 )
J Q' J' B_S Q = ( ~ ) =: B_Sout,
( 0 S11' )
~ ~ ~ ( H11 H12 )
J Q' J'(-i*B_T) Q = ( ) =: B_Hout,
( 0 -H11' )
~ ~ ~
with S11, H11 upper triangular, and such that Spec_-(B_S, -i*B_T)
is contained in the spectrum of the 2*NEIG-by-2*NEIG leading
~
principal subpencil aS11 - bH11.
Finally, the right deflating subspace is computed.
See also page 22 in [1] for more details.
References
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical Computation of Deflating Subspaces of Embedded
Hamiltonian Pencils.
Tech. Rep. SFB393/99-15, Technical University Chemnitz,
Germany, June 1999.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) complex floating point operations.Further Comments
This routine does not perform any scaling of the matrices. Scaling might sometimes be useful, and it should be done externally.Example
Program Text
* MB03LZ EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 50 )
INTEGER LDA, LDB, LDDE, LDFG, LDQ, LDWORK, LZWORK
PARAMETER ( LDA = NMAX, LDB = NMAX, LDDE = NMAX,
$ LDFG = NMAX, LDQ = 2*NMAX,
$ LDWORK = 11*NMAX*NMAX + 2*NMAX,
$ LZWORK = 8*NMAX + 4 )
*
* .. Local Scalars ..
CHARACTER*1 COMPQ, ORTH
INTEGER I, INFO, J, N, NEIG
*
* .. Local Arrays ..
COMPLEX*16 A( LDA, NMAX ), B( LDB, NMAX ),
$ DE( LDDE, NMAX ), FG( LDFG, NMAX ),
$ Q( LDQ, 2*NMAX ), ZWORK( LZWORK )
DOUBLE PRECISION ALPHAI( NMAX ), ALPHAR( NMAX ), BETA( NMAX ),
$ DWORK( LDWORK )
INTEGER IWORK( NMAX + 1 )
LOGICAL BWORK( NMAX )
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
*
* .. External Subroutines ..
EXTERNAL MB03LZ
*
* .. Intrinsic Functions ..
INTRINSIC MOD
*
* .. Executable statements ..
*
WRITE( NOUT, FMT = 99999 )
*
* Skip first line in data file.
*
READ( NIN, FMT = * )
READ( NIN, FMT = * ) COMPQ, ORTH, N
READ( NIN, FMT = * ) ( ( A( I, J ), J = 1, N/2 ), I = 1, N/2 )
READ( NIN, FMT = * ) ( ( DE( I, J ), J = 1, N/2+1 ), I = 1, N/2 )
READ( NIN, FMT = * ) ( ( B( I, J ), J = 1, N/2 ), I = 1, N/2 )
READ( NIN, FMT = * ) ( ( FG( I, J ), J = 1, N/2+1 ), I = 1, N/2 )
IF( N.LT.0 .OR. N.GT.NMAX .OR. MOD( N, 2 ).NE.0 ) THEN
WRITE( NOUT, FMT = 99998 ) N
ELSE
* Compute the eigenvalues and an orthogonal basis of the right
* deflating subspace of a complex skew-Hamiltonian/Hamiltonian
* pencil, corresponding to the eigenvalues with strictly negative
* real part.
CALL MB03LZ( COMPQ, ORTH, N, A, LDA, DE, LDDE, B, LDB, FG,
$ LDFG, NEIG, Q, LDQ, ALPHAR, ALPHAI, BETA, IWORK,
$ DWORK, LDWORK, ZWORK, LZWORK, BWORK, INFO )
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99997 ) INFO
ELSE
IF( LSAME( COMPQ, 'C' ) ) THEN
WRITE( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( A( I, J ), J = 1, N )
10 CONTINUE
WRITE( NOUT, FMT = 99994 )
DO 20 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( DE( I, J ), J = 1, N )
20 CONTINUE
WRITE( NOUT, FMT = 99993 )
DO 30 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( B( I, J ), J = 1, N )
30 CONTINUE
WRITE( NOUT, FMT = 99992 )
DO 40 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( FG( I, J ), J = 1, N )
40 CONTINUE
END IF
WRITE( NOUT, FMT = 99991 )
WRITE( NOUT, FMT = 99990 ) ( ALPHAR( I ), I = 1, N )
WRITE( NOUT, FMT = 99989 )
WRITE( NOUT, FMT = 99990 ) ( ALPHAI( I ), I = 1, N )
WRITE( NOUT, FMT = 99988 )
WRITE( NOUT, FMT = 99990 ) ( BETA( I ), I = 1, N )
IF( LSAME( COMPQ, 'C' ) .AND. NEIG.GT.0 ) THEN
WRITE( NOUT, FMT = 99987 )
DO 50 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( Q( I, J ), J = 1, NEIG )
50 CONTINUE
END IF
IF( LSAME( COMPQ, 'C' ) )
$ WRITE( NOUT, FMT = 99986 ) NEIG
END IF
END IF
STOP
99999 FORMAT ( 'MB03LZ EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT ( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT ( 'INFO on exit from MB03LZ = ', I2 )
99996 FORMAT (/'The matrix A on exit is ' )
99995 FORMAT ( 20( 1X, F9.4, SP, F9.4, S, 'i ') )
99994 FORMAT (/'The matrix D on exit is ' )
99993 FORMAT (/'The matrix B on exit is ' )
99992 FORMAT (/'The matrix F on exit is ' )
99991 FORMAT ( 'The vector ALPHAR is ' )
99990 FORMAT ( 50( 1X, F8.4 ) )
99989 FORMAT (/'The vector ALPHAI is ' )
99988 FORMAT (/'The vector BETA is ' )
99987 FORMAT (/'The deflating subspace corresponding to the ',
$ 'eigenvalues with negative real part is ' )
99986 FORMAT (/'The number of eigenvalues in the initial pencil with ',
$ 'negative real part is ', I2 )
END
Program Data
MB03LZ EXAMPLE PROGRAM DATA
C P 4
(0.0604,0.6568) (0.5268,0.2919)
(0.3992,0.6279) (0.4167,0.4316)
(0,0.4896) (0,0.9516) (0.3724,0.0526)
(0.9840,0.3394) (0,0.9203) (0,0.7378)
(0.2691,0.4177) (0.5478,0.3014)
(0.4228,0.9830) (0.9427,0.7010)
0.6663 0.6981 (0.1781,0.8818)
(0.5391,0.1711) 0.6665 0.1280
Program Results
MB03LZ EXAMPLE PROGRAM RESULTS
The matrix A on exit is
0.7430 +0.0000i -0.1431 -0.1304i -0.4169 -0.0495i 0.0650 -0.0262i
0.3992 +0.6279i 0.7398 -1.2647i -0.0861 -0.1075i 0.2826 +0.7725i
0.0000 +0.0000i 0.0000 +0.0000i 1.4799 +0.1442i -0.1094 -0.1061i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.6816 +0.2278i
The matrix D on exit is
0.0000 -0.6858i -0.3122 -0.1018i -0.7813 -0.4163i -0.1343 +0.3259i
0.9840 +0.3394i 0.0000 +0.1465i -0.1678 +0.2971i -0.0728 -0.6524i
0.0000 +0.0000i 0.0000 +0.0000i -0.0000 +0.2979i -0.0728 +0.3971i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.2414i
The matrix B on exit is
-1.5832 +0.5069i -0.0819 -0.1073i 0.7749 -0.0519i 0.0635 -0.0052i
0.0000 +0.0000i -0.1916 -0.0106i -0.0074 +0.0165i -0.1546 -0.6817i
0.0000 +0.0000i 0.0000 +0.0000i -0.0716 -0.1811i 0.3146 +0.1558i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i -1.6078 -0.0203i
The matrix F on exit is
0.3382 0.0000i -0.0622 +0.8488i 0.0042 +0.9053i -0.1584 +0.0726i
0.5391 +0.1711i -0.5888 +0.0000i 0.4089 +0.2018i -0.6913 -0.5011i
0.0000 +0.0000i 0.0000 +0.0000i -0.2712 +0.0000i 0.5114 +0.3726i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.5218 +0.0000i
The vector ALPHAR is
-1.5832 1.5832 -0.0842 0.0842
The vector ALPHAI is
0.5069 0.5069 -0.1642 -0.1642
The vector BETA is
0.7430 0.7430 1.4085 1.4085
The deflating subspace corresponding to the eigenvalues with negative real part is
-0.0793 -0.1949i 0.4845 -0.5472i
0.4349 +0.1710i -0.2878 +0.0952i
-0.1266 +0.1505i 0.1364 -0.4776i
-0.5035 +0.6671i 0.1628 +0.3174i
The number of eigenvalues in the initial pencil with negative real part is 2
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB03MD.html 0000664 0000000 0000000 00000026672 14560147231 0020351 0 ustar 00root root 0000000 0000000
Purpose
To compute an upper bound THETA using a bisection method such that
the bidiagonal matrix
|q(1) e(1) 0 ... 0 |
| 0 q(2) e(2) . |
J = | . . |
| . e(N-1)|
| 0 ... ... q(N) |
has precisely L singular values less than or equal to THETA plus
a given tolerance TOL.
This routine is mainly intended to be called only by other SLICOT
routines.
Specification
SUBROUTINE MB03MD( N, L, THETA, Q, E, Q2, E2, PIVMIN, TOL, RELTOL,
$ IWARN, INFO )
C .. Scalar Arguments ..
INTEGER INFO, IWARN, L, N
DOUBLE PRECISION PIVMIN, RELTOL, THETA, TOL
C .. Array Arguments ..
DOUBLE PRECISION E(*), E2(*), Q(*), Q2(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the bidiagonal matrix J. N >= 0.
L (input/output) INTEGER
On entry, L must contain the number of singular values
of J which must be less than or equal to the upper bound
computed by the routine. 0 <= L <= N.
On exit, L may be increased if the L-th smallest singular
value of J has multiplicity greater than 1. In this case,
L is increased by the number of singular values of J which
are larger than its L-th smallest one and approach the
L-th smallest singular value of J within a distance less
than TOL.
If L has been increased, then the routine returns with
IWARN set to 1.
THETA (input/output) DOUBLE PRECISION
On entry, THETA must contain an initial estimate for the
upper bound to be computed. If THETA < 0.0 on entry, then
one of the following default values is used.
If L = 0, THETA is set to 0.0 irrespective of the input
value of THETA; if L = 1, then THETA is taken as
MIN(ABS(Q(i))), for i = 1,2,...,N; otherwise, THETA is
taken as ABS(Q(N-L+1)).
On exit, THETA contains the computed upper bound such that
the bidiagonal matrix J has precisely L singular values
less than or equal to THETA + TOL.
Q (input) DOUBLE PRECISION array, dimension (N)
This array must contain the diagonal elements q(1),
q(2),...,q(N) of the bidiagonal matrix J. That is,
Q(i) = J(i,i) for i = 1,2,...,N.
E (input) DOUBLE PRECISION array, dimension (N-1)
This array must contain the superdiagonal elements
e(1),e(2),...,e(N-1) of the bidiagonal matrix J. That is,
E(k) = J(k,k+1) for k = 1,2,...,N-1.
Q2 (input) DOUBLE PRECISION array, dimension (N)
This array must contain the squares of the diagonal
elements q(1),q(2),...,q(N) of the bidiagonal matrix J.
That is, Q2(i) = J(i,i)**2 for i = 1,2,...,N.
E2 (input) DOUBLE PRECISION array, dimension (N-1)
This array must contain the squares of the superdiagonal
elements e(1),e(2),...,e(N-1) of the bidiagonal matrix J.
That is, E2(k) = J(k,k+1)**2 for k = 1,2,...,N-1.
PIVMIN (input) DOUBLE PRECISION
The minimum absolute value of a "pivot" in the Sturm
sequence loop.
PIVMIN >= max( max( |q(i)|, |e(k)| )**2*sf_min, sf_min ),
where i = 1,2,...,N, k = 1,2,...,N-1, and sf_min is at
least the smallest number that can divide one without
overflow (see LAPACK Library routine DLAMCH).
Note that this condition is not checked by the routine.
Tolerances
TOL DOUBLE PRECISION
This parameter defines the multiplicity of singular values
by considering all singular values within an interval of
length TOL as coinciding. TOL is used in checking how many
singular values are less than or equal to THETA. Also in
computing an appropriate upper bound THETA by a bisection
method, TOL is used as a stopping criterion defining the
minimum (absolute) subinterval width. TOL >= 0.
RELTOL DOUBLE PRECISION
This parameter specifies the minimum relative width of an
interval. When an interval is narrower than TOL, or than
RELTOL times the larger (in magnitude) endpoint, then it
is considered to be sufficiently small and bisection has
converged.
RELTOL >= BASE * EPS, where BASE is machine radix and EPS
is machine precision (see LAPACK Library routine DLAMCH).
Warning Indicator
IWARN INTEGER
= 0: no warnings;
= 1: if the value of L has been increased as the L-th
smallest singular value of J coincides with the
(L+1)-th smallest one.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Let s(i), i = 1,2,...,N, be the N non-negative singular values of
the bidiagonal matrix J arranged so that s(1) >= ... >= s(N) >= 0.
The routine then computes an upper bound T such that s(N-L) > T >=
s(N-L+1) as follows (see [2]).
First, if the initial estimate of THETA is not specified by the
user then the routine initialises THETA to be an estimate which
is close to the requested value of THETA if s(N-L) >> s(N-L+1).
Second, a bisection method (see [1, 8.5]) is used which generates
a sequence of shrinking intervals [Y,Z] such that either THETA in
[Y,Z] was found (so that J has L singular values less than or
equal to THETA), or
(number of s(i) <= Y) < L < (number of s(i) <= Z).
This bisection method is applied to an associated 2N-by-2N
symmetric tridiagonal matrix T" whose eigenvalues (see [1]) are
given by s(1),s(2),...,s(N),-s(1),-s(2),...,-s(N). One of the
starting values for the bisection method is the initial value of
THETA. If this value is an upper bound, then the initial lower
bound is set to zero, else the initial upper bound is computed
from the Gershgorin Circle Theorem [1, Theorem 7.2-1], applied to
T". The computation of the "number of s(i) <= Y (or Z)" is
achieved by calling SLICOT Library routine MB03ND, which applies
Sylvester's Law of Inertia or equivalently Sturm sequences
[1, 8.5] to the associated matrix T". If
Z - Y <= MAX( TOL, PIVMIN, RELTOL*MAX( ABS( Y ), ABS( Z ) ) )
at some stage of the bisection method, then at least two singular
values of J lie in the interval [Y,Z] within a distance less than
TOL from each other. In this case, s(N-L) and s(N-L+1) are assumed
to coincide, the upper bound T is set to the value of Z, the value
of L is increased and IWARN is set to 1.
References
[1] Golub, G.H. and Van Loan, C.F.
Matrix Computations.
The Johns Hopkins University Press, Baltimore, Maryland, 1983.
[2] Van Huffel, S. and Vandewalle, J.
The Partial Total Least Squares Algorithm.
J. Comput. and Appl. Math., 21, pp. 333-341, 1988.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
* MB03MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
* .. Local Scalars ..
DOUBLE PRECISION PIVMIN, RELTOL, SAFMIN, THETA, TOL
INTEGER I, INFO, IWARN, L, N
* .. Local Arrays ..
DOUBLE PRECISION E(NMAX-1), E2(NMAX-1), Q(NMAX), Q2(NMAX)
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* .. External Subroutines ..
EXTERNAL MB03MD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, THETA, L, TOL, RELTOL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) N
ELSE IF ( L.LT.0 .OR. L.GT.N ) THEN
WRITE ( NOUT, FMT = 99990 ) L
ELSE
READ ( NIN, FMT = * ) ( Q(I), I = 1,N )
READ ( NIN, FMT = * ) ( E(I), I = 1,N-1 )
* Print out the bidiagonal matrix J.
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N - 1
WRITE ( NOUT, FMT = 99996 ) I, I, Q(I), I, (I+1), E(I)
20 CONTINUE
WRITE ( NOUT, FMT = 99995 ) N, N, Q(N)
* Compute Q**2, E**2, and PIVMIN.
Q2(N) = Q(N)**2
PIVMIN = Q2(N)
DO 40 I = 1, N - 1
Q2(I) = Q(I)**2
E2(I) = E(I)**2
PIVMIN = MAX( PIVMIN, Q2(I), E2(I) )
40 CONTINUE
SAFMIN = DLAMCH( 'Safe minimum' )
PIVMIN = MAX( PIVMIN*SAFMIN, SAFMIN )
TOL = MAX( TOL, ZERO )
IF ( RELTOL.LE.ZERO )
$ RELTOL = DLAMCH( 'Base' )*DLAMCH( 'Epsilon' )
* Compute an upper bound THETA such that J has 3 singular values
* < = THETA.
CALL MB03MD( N, L, THETA, Q, E, Q2, E2, PIVMIN, TOL, RELTOL,
$ IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( IWARN.NE.0 ) WRITE ( NOUT, FMT = 99994 ) IWARN
WRITE ( NOUT, FMT = 99993 ) THETA
WRITE ( NOUT, FMT = 99992 ) L
END IF
END IF
STOP
*
99999 FORMAT (' MB03MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB03MD = ',I2)
99997 FORMAT (' The Bidiagonal Matrix J is',/)
99996 FORMAT (2(' (',I1,',',I1,') = ',F7.4,2X))
99995 FORMAT (' (',I1,',',I1,') = ',F7.4)
99994 FORMAT (' IWARN on exit from MB03MD = ',I2,/)
99993 FORMAT (/' The computed value of THETA is ',F7.4)
99992 FORMAT (/' J has ',I2,' singular values < = THETA')
99991 FORMAT (/' N is out of range.',/' N = ',I5)
99990 FORMAT (/' L is out of range.',/' L = ',I5)
END
Program Data
MB03MD EXAMPLE PROGRAM DATA 5 -3.0 3 0.0 0.0 1.0 2.0 3.0 4.0 5.0 2.0 3.0 4.0 5.0Program Results
MB03MD EXAMPLE PROGRAM RESULTS The Bidiagonal Matrix J is (1,1) = 1.0000 (1,2) = 2.0000 (2,2) = 2.0000 (2,3) = 3.0000 (3,3) = 3.0000 (3,4) = 4.0000 (4,4) = 4.0000 (4,5) = 5.0000 (5,5) = 5.0000 The computed value of THETA is 4.7500 J has 3 singular values < = THETA
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB03ND.html 0000664 0000000 0000000 00000021014 14560147231 0020333 0 ustar 00root root 0000000 0000000
Purpose
To find the number of singular values of the bidiagonal matrix
|q(1) e(1) . ... 0 |
| 0 q(2) e(2) . |
J = | . . |
| . e(N-1)|
| 0 ... ... 0 q(N) |
which are less than or equal to a given bound THETA.
This routine is intended to be called only by other SLICOT
routines.
Specification
INTEGER FUNCTION MB03ND( N, THETA, Q2, E2, PIVMIN, INFO )
C .. Scalar Arguments ..
INTEGER INFO, N
DOUBLE PRECISION PIVMIN, THETA
C .. Array Arguments ..
DOUBLE PRECISION E2(*), Q2(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the bidiagonal matrix J. N >= 0.
THETA (input) DOUBLE PRECISION
Given bound.
Note: If THETA < 0.0 on entry, then MB03ND is set to 0
as the singular values of J are non-negative.
Q2 (input) DOUBLE PRECISION array, dimension (N)
This array must contain the squares of the diagonal
elements q(1),q(2),...,q(N) of the bidiagonal matrix J.
That is, Q2(i) = J(i,i)**2 for i = 1,2,...,N.
E2 (input) DOUBLE PRECISION array, dimension (N-1)
This array must contain the squares of the superdiagonal
elements e(1),e(2),...,e(N-1) of the bidiagonal matrix J.
That is, E2(k) = J(k,k+1)**2 for k = 1,2,...,N-1.
PIVMIN (input) DOUBLE PRECISION
The minimum absolute value of a "pivot" in the Sturm
sequence loop.
PIVMIN >= max( max( |q(i)|, |e(k)| )**2*sf_min, sf_min ),
where i = 1,2,...,N, k = 1,2,...,N-1, and sf_min is at
least the smallest number that can divide one without
overflow (see LAPACK Library routine DLAMCH).
Note that this condition is not checked by the routine.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The computation of the number of singular values s(i) of J which
are less than or equal to THETA is based on applying Sylvester's
Law of Inertia, or equivalently, Sturm sequences [1,p.52] to the
unreduced symmetric tridiagonal matrices associated with J as
follows. Let T be the following 2N-by-2N symmetric matrix
associated with J:
| 0 J'|
T = | |.
| J 0 |
(The eigenvalues of T are given by s(1),s(2),...,s(N),-s(1),-s(2),
...,-s(N)). Then, by permuting the rows and columns of T into the
order 1, N+1, 2, N+2, ..., N, 2N it follows that T is orthogonally
similar to the tridiagonal matrix T" with zeros on its diagonal
and q(1), e(1), q(2), e(2), ..., e(N-1), q(N) on its offdiagonals
[3,4]. If q(1),q(2),...,q(N) and e(1),e(2),...,e(N-1) are nonzero,
Sylvester's Law of Inertia may be applied directly to T".
Otherwise, T" is block diagonal and each diagonal block (which is
then unreduced) must be analysed separately by applying
Sylvester's Law of Inertia.
References
[1] Parlett, B.N.
The Symmetric Eigenvalue Problem.
Prentice Hall, Englewood Cliffs, New Jersey, 1980.
[2] Demmel, J. and Kahan, W.
Computing Small Singular Values of Bidiagonal Matrices with
Guaranteed High Relative Accuracy.
Technical Report, Courant Inst., New York, March 1988.
[3] Van Huffel, S. and Vandewalle, J.
The Partial Total Least-Squares Algorithm.
J. Comput. and Appl. Math., 21, pp. 333-341, 1988.
[4] Golub, G.H. and Kahan, W.
Calculating the Singular Values and Pseudo-inverse of a
Matrix.
SIAM J. Numer. Anal., Ser. B, 2, pp. 205-224, 1965.
[5] Demmel, J.W., Dhillon, I. and Ren, H.
On the Correctness of Parallel Bisection in Floating Point.
Computer Science Division Technical Report UCB//CSD-94-805,
University of California, Berkeley, CA 94720, March 1994.
Numerical Aspects
The singular values s(i) could also be obtained with the use of the symmetric tridiagonal matrix T = J'J, whose eigenvalues are the squared singular values of J [4,p.213]. However, the method actually used by the routine is more accurate and equally efficient (see [2]). To avoid overflow, matrix J should be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value (and not much smaller than that, for maximal accuracy). With respect to accuracy the following condition holds (see [2]): If the established value is denoted by p, then at least p singular values of J are less than or equal to THETA/(1 - (3 x N - 1.5) x EPS) and no more than p singular values are less than or equal to THETA x (1 - (6 x N-2) x EPS)/(1 - (3 x N - 1.5) x EPS).Further Comments
NoneExample
Program Text
* MB03ND EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
* .. Local Scalars ..
DOUBLE PRECISION PIVMIN, SAFMIN, THETA
INTEGER I, INFO, N, NUMSV
* .. Local Arrays ..
DOUBLE PRECISION E(NMAX-1), E2(NMAX-1), Q(NMAX), Q2(NMAX)
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* .. External Functions ..
INTEGER MB03ND
EXTERNAL MB03ND
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, THETA
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( Q(I), I = 1,N )
READ ( NIN, FMT = * ) ( E(I), I = 1,N-1 )
* Print out the bidiagonal matrix J.
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N - 1
WRITE ( NOUT, FMT = 99996 ) I, I, Q(I), I, (I+1), E(I)
20 CONTINUE
WRITE ( NOUT, FMT = 99995 ) N, N, Q(N)
* Compute Q**2, E**2, and PIVMIN.
Q2(N) = Q(N)**2
PIVMIN = Q2(N)
DO 40 I = 1, N - 1
Q2(I) = Q(I)**2
E2(I) = E(I)**2
PIVMIN = MAX( PIVMIN, Q2(I), E2(I) )
40 CONTINUE
SAFMIN = DLAMCH( 'Safe minimum' )
PIVMIN = MAX( PIVMIN*SAFMIN, SAFMIN )
* Compute the number of singular values of J < = THETA.
NUMSV = MB03ND( N, THETA, Q2, E2, PIVMIN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) NUMSV, THETA
END IF
END IF
STOP
*
99999 FORMAT (' MB03ND EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB03ND = ',I2)
99997 FORMAT (' The Bidiagonal Matrix J is',/)
99996 FORMAT (2(' (',I1,',',I1,') = ',F7.4,2X))
99995 FORMAT (' (',I1,',',I1,') = ',F7.4)
99994 FORMAT (/' J has ',I2,' singular values < = ',F7.4)
99993 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
MB03ND EXAMPLE PROGRAM DATA 5 5.0 0.0 0.0 1.0 2.0 3.0 4.0 5.0 2.0 3.0 4.0 5.0Program Results
MB03ND EXAMPLE PROGRAM RESULTS The Bidiagonal Matrix J is (1,1) = 1.0000 (1,2) = 2.0000 (2,2) = 2.0000 (2,3) = 3.0000 (3,3) = 3.0000 (3,4) = 4.0000 (4,4) = 4.0000 (4,5) = 5.0000 (5,5) = 5.0000 J has 3 singular values < = 5.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB03NY.html 0000664 0000000 0000000 00000007426 14560147231 0020373 0 ustar 00root root 0000000 0000000
Purpose
To compute the smallest singular value of A - jwI.Specification
DOUBLE PRECISION FUNCTION MB03NY( N, OMEGA, A, LDA, S, DWORK,
$ LDWORK, CWORK, LCWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LCWORK, LDA, LDWORK, N
DOUBLE PRECISION OMEGA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), S(*)
COMPLEX*16 CWORK(*)
Function Value
MB03NY DOUBLE PRECISION
The smallest singular value of A - jwI (if INFO = 0).
If N = 0, the function value is set to zero.
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the the matrix A. N >= 0.
OMEGA (input) DOUBLE PRECISION
The constant factor of A - jwI.
A (input/workspace) DOUBLE PRECISION array, dimension
(LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A.
On exit, if OMEGA = 0, the contents of this array are
destroyed. Otherwise, this array is unchanged.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
S (output) DOUBLE PRECISION array, dimension (N)
The singular values of A - jwI in decreasing order.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX( 1, 5*N ).
For optimum performance LDWORK should be larger.
CWORK COMPLEX*16 array, dimension (LCWORK)
On exit, if INFO = 0 and OMEGA <> 0, CWORK(1) returns the
optimal value of LCWORK.
If OMEGA is zero, this array is not referenced.
LCWORK INTEGER
The length of the array CWORK.
LCWORK >= 1, if OMEGA = 0;
LCWORK >= MAX( 1, N*N+3*N ), if OMEGA <> 0.
For optimum performance LCWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 2: The SVD algorithm (in either LAPACK Library routine
DGESVD or ZGESVD) fails to converge; this error is
very rare.
Method
This procedure simply constructs the matrix A - jwI, and calls ZGESVD if w is not zero, or DGESVD if w = 0.Further Comments
This routine is not very efficient because it computes all singular values, but it is very accurate. The routine is intended to be called only from the SLICOT Library routine AB13FD.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute (optionally) a rank-revealing QR factorization of a
real general M-by-N matrix A, which may be rank-deficient,
and estimate its effective rank using incremental condition
estimation.
The routine uses a QR factorization with column pivoting:
A * P = Q * R, where R = [ R11 R12 ],
[ 0 R22 ]
with R11 defined as the largest leading submatrix whose estimated
condition number is less than 1/RCOND. The order of R11, RANK,
is the effective rank of A.
MB03OD does not perform any scaling of the matrix A.
Specification
SUBROUTINE MB03OD( JOBQR, M, N, A, LDA, JPVT, RCOND, SVLMAX, TAU,
$ RANK, SVAL, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBQR
INTEGER INFO, LDA, LDWORK, M, N, RANK
DOUBLE PRECISION RCOND, SVLMAX
C .. Array Arguments ..
INTEGER JPVT( * )
DOUBLE PRECISION A( LDA, * ), SVAL( 3 ), TAU( * ), DWORK( * )
Arguments
Mode Parameters
JOBQR CHARACTER*1
= 'Q': Perform a QR factorization with column pivoting;
= 'N': Do not perform the QR factorization (but assume
that it has been done outside).
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension
( LDA, N )
On entry with JOBQR = 'Q', the leading M by N part of this
array must contain the given matrix A.
On exit with JOBQR = 'Q', the leading min(M,N) by N upper
triangular part of A contains the triangular factor R,
and the elements below the diagonal, with the array TAU,
represent the orthogonal matrix Q as a product of
min(M,N) elementary reflectors.
On entry and on exit with JOBQR = 'N', the leading
min(M,N) by N upper triangular part of A contains the
triangular factor R, as determined by the QR factorization
with pivoting. The elements below the diagonal of A are
not referenced.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
JPVT (input/output) INTEGER array, dimension ( N )
On entry with JOBQR = 'Q', if JPVT(i) <> 0, the i-th
column of A is an initial column, otherwise it is a free
column. Before the QR factorization of A, all initial
columns are permuted to the leading positions; only the
remaining free columns are moved as a result of column
pivoting during the factorization. For rank determination
it is preferable that all columns be free.
On exit with JOBQR = 'Q', if JPVT(i) = k, then the i-th
column of A*P was the k-th column of A.
Array JPVT is not referenced when JOBQR = 'N'.
RCOND (input) DOUBLE PRECISION
RCOND is used to determine the effective rank of A, which
is defined as the order of the largest leading triangular
submatrix R11 in the QR factorization with pivoting of A,
whose estimated condition number is less than 1/RCOND.
RCOND >= 0.
NOTE that when SVLMAX > 0, the estimated rank could be
less than that defined above (see SVLMAX).
SVLMAX (input) DOUBLE PRECISION
If A is a submatrix of another matrix B, and the rank
decision should be related to that matrix, then SVLMAX
should be an estimate of the largest singular value of B
(for instance, the Frobenius norm of B). If this is not
the case, the input value SVLMAX = 0 should work.
SVLMAX >= 0.
TAU (output) DOUBLE PRECISION array, dimension ( MIN( M, N ) )
On exit with JOBQR = 'Q', the leading min(M,N) elements of
TAU contain the scalar factors of the elementary
reflectors.
Array TAU is not referenced when JOBQR = 'N'.
RANK (output) INTEGER
The effective (estimated) rank of A, i.e. the order of
the submatrix R11.
SVAL (output) DOUBLE PRECISION array, dimension ( 3 )
The estimates of some of the singular values of the
triangular factor R:
SVAL(1): largest singular value of R(1:RANK,1:RANK);
SVAL(2): smallest singular value of R(1:RANK,1:RANK);
SVAL(3): smallest singular value of R(1:RANK+1,1:RANK+1),
if RANK < MIN( M, N ), or of R(1:RANK,1:RANK),
otherwise.
If the triangular factorization is a rank-revealing one
(which will be the case if the leading columns were well-
conditioned), then SVAL(1) will also be an estimate for
the largest singular value of A, and SVAL(2) and SVAL(3)
will be estimates for the RANK-th and (RANK+1)-st singular
values of A, respectively.
By examining these values, one can confirm that the rank
is well defined with respect to the chosen value of RCOND.
The ratio SVAL(1)/SVAL(2) is an estimate of the condition
number of R(1:RANK,1:RANK).
Workspace
DWORK DOUBLE PRECISION array, dimension ( LDWORK )
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 3*N + 1, if JOBQR = 'Q';
LDWORK >= max( 1, 2*min( M, N ) ), if JOBQR = 'N'.
For good performance when JOBQR = 'Q', LDWORK should be
larger. Specifically, LDWORK >= 2*N + ( N + 1 )*NB, where
NB is the optimal block size for the LAPACK Library
routine DGEQP3.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine computes or uses a QR factorization with column pivoting of A, A * P = Q * R, with R defined above, and then finds the largest leading submatrix whose estimated condition number is less than 1/RCOND, taking the possible positive value of SVLMAX into account. This is performed using the LAPACK incremental condition estimation scheme and a slightly modified rank decision test.Further Comments
NoneExample
Program Text
* MB03OD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX
PARAMETER ( NMAX = 10, MMAX = 10 )
INTEGER LDA
PARAMETER ( LDA = NMAX )
INTEGER LDTAU
PARAMETER ( LDTAU = MIN(MMAX,NMAX) )
INTEGER LDWORK
PARAMETER ( LDWORK = 3*NMAX + 1 )
* .. Local Scalars ..
CHARACTER*1 JOBQR
INTEGER I, INFO, J, M, N, RANK
DOUBLE PRECISION RCOND, SVAL(3), SVLMAX
* ..
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), TAU(LDTAU)
INTEGER JPVT(NMAX)
* .. External Subroutines ..
EXTERNAL MB03OD
* .. Intrinsic Functions ..
INTRINSIC MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, N, JOBQR, RCOND, SVLMAX
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99972 ) N
ELSE
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99971 ) M
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,M )
* QR with column pivoting.
DO 10 I = 1, N
JPVT(I) = 0
10 CONTINUE
CALL MB03OD( JOBQR, M, N, A, LDA, JPVT, RCOND, SVLMAX, TAU,
$ RANK, SVAL, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99995 ) RANK
WRITE ( NOUT, FMT = 99994 ) ( JPVT(I), I = 1,N )
WRITE ( NOUT, FMT = 99993 ) ( SVAL(I), I = 1,3 )
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' MB03OD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB03OD = ',I2)
99995 FORMAT (' The rank is ',I5)
99994 FORMAT (' Column permutations are ',/(20(I3,2X)))
99993 FORMAT (' SVAL vector is ',/(20(1X,F10.4)))
99972 FORMAT (/' N is out of range.',/' N = ',I5)
99971 FORMAT (/' M is out of range.',/' M = ',I5)
END
Program Data
MB03OD EXAMPLE PROGRAM DATA 6 5 Q 5.D-16 0.0 1. 2. 6. 3. 5. -2. -1. -1. 0. -2. 5. 5. 1. 5. 1. -2. -1. -1. 0. -2. 4. 8. 4. 20. 4. -2. -1. -1. 0. -2.Program Results
MB03OD EXAMPLE PROGRAM RESULTS
The rank is 4
Column permutations are
4 3 1 5 2
SVAL vector is
22.7257 1.4330 0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB03OY.html 0000664 0000000 0000000 00000016610 14560147231 0020367 0 ustar 00root root 0000000 0000000
Purpose
To compute a rank-revealing QR factorization of a real general
M-by-N matrix A, which may be rank-deficient, and estimate its
effective rank using incremental condition estimation.
The routine uses a truncated QR factorization with column pivoting
[ R11 R12 ]
A * P = Q * R, where R = [ ],
[ 0 R22 ]
with R11 defined as the largest leading upper triangular submatrix
whose estimated condition number is less than 1/RCOND. The order
of R11, RANK, is the effective rank of A. Condition estimation is
performed during the QR factorization process. Matrix R22 is full
(but of small norm), or empty.
MB03OY does not perform any scaling of the matrix A.
Specification
SUBROUTINE MB03OY( M, N, A, LDA, RCOND, SVLMAX, RANK, SVAL, JPVT,
$ TAU, DWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, M, N, RANK
DOUBLE PRECISION RCOND, SVLMAX
C .. Array Arguments ..
INTEGER JPVT( * )
DOUBLE PRECISION A( LDA, * ), DWORK( * ), SVAL( 3 ), TAU( * )
Arguments
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension
( LDA, N )
On entry, the leading M-by-N part of this array must
contain the given matrix A.
On exit, the leading RANK-by-RANK upper triangular part
of A contains the triangular factor R11, and the elements
below the diagonal in the first RANK columns, with the
array TAU, represent the orthogonal matrix Q as a product
of RANK elementary reflectors.
The remaining N-RANK columns contain the result of the
QR factorization process used.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
RCOND (input) DOUBLE PRECISION
RCOND is used to determine the effective rank of A, which
is defined as the order of the largest leading triangular
submatrix R11 in the QR factorization with pivoting of A,
whose estimated condition number is less than 1/RCOND.
0 <= RCOND <= 1.
NOTE that when SVLMAX > 0, the estimated rank could be
less than that defined above (see SVLMAX).
SVLMAX (input) DOUBLE PRECISION
If A is a submatrix of another matrix B, and the rank
decision should be related to that matrix, then SVLMAX
should be an estimate of the largest singular value of B
(for instance, the Frobenius norm of B). If this is not
the case, the input value SVLMAX = 0 should work.
SVLMAX >= 0.
RANK (output) INTEGER
The effective (estimated) rank of A, i.e., the order of
the submatrix R11.
SVAL (output) DOUBLE PRECISION array, dimension ( 3 )
The estimates of some of the singular values of the
triangular factor R:
SVAL(1): largest singular value of R(1:RANK,1:RANK);
SVAL(2): smallest singular value of R(1:RANK,1:RANK);
SVAL(3): smallest singular value of R(1:RANK+1,1:RANK+1),
if RANK < MIN( M, N ), or of R(1:RANK,1:RANK),
otherwise.
If the triangular factorization is a rank-revealing one
(which will be the case if the leading columns were well-
conditioned), then SVAL(1) will also be an estimate for
the largest singular value of A, and SVAL(2) and SVAL(3)
will be estimates for the RANK-th and (RANK+1)-st singular
values of A, respectively.
By examining these values, one can confirm that the rank
is well defined with respect to the chosen value of RCOND.
The ratio SVAL(1)/SVAL(2) is an estimate of the condition
number of R(1:RANK,1:RANK).
JPVT (output) INTEGER array, dimension ( N )
If JPVT(i) = k, then the i-th column of A*P was the k-th
column of A.
TAU (output) DOUBLE PRECISION array, dimension ( MIN( M, N ) )
The leading RANK elements of TAU contain the scalar
factors of the elementary reflectors.
Workspace
DWORK DOUBLE PRECISION array, dimension ( 3*N-1 )Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine computes a truncated QR factorization with column
pivoting of A, A * P = Q * R, with R defined above, and,
during this process, finds the largest leading submatrix whose
estimated condition number is less than 1/RCOND, taking the
possible positive value of SVLMAX into account. This is performed
using the LAPACK incremental condition estimation scheme and a
slightly modified rank decision test. The factorization process
stops when RANK has been determined.
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = rank <= min(m,n).
Each H(i) has the form
H = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
A(i+1:m,i), and tau in TAU(i).
The matrix P is represented in jpvt as follows: If
jpvt(j) = i
then the jth column of P is the ith canonical unit vector.
References
[1] Bischof, C.H. and P. Tang.
Generalizing Incremental Condition Estimation.
LAPACK Working Notes 32, Mathematics and Computer Science
Division, Argonne National Laboratory, UT, CS-91-132,
May 1991.
[2] Bischof, C.H. and P. Tang.
Robust Incremental Condition Estimation.
LAPACK Working Notes 33, Mathematics and Computer Science
Division, Argonne National Laboratory, UT, CS-91-133,
May 1991.
Numerical Aspects
The algorithm is backward stable.Further Comments
For a matrix with a small norm, the rank is set to zero if the largest column Euclidean norm is smaller than or equal to RCOND.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute (optionally) a rank-revealing RQ factorization of a
real general M-by-N matrix A, which may be rank-deficient,
and estimate its effective rank using incremental condition
estimation.
The routine uses an RQ factorization with row pivoting:
P * A = R * Q, where R = [ R11 R12 ],
[ 0 R22 ]
with R22 defined as the largest trailing submatrix whose estimated
condition number is less than 1/RCOND. The order of R22, RANK,
is the effective rank of A.
MB03PD does not perform any scaling of the matrix A.
Specification
SUBROUTINE MB03PD( JOBRQ, M, N, A, LDA, JPVT, RCOND, SVLMAX, TAU,
$ RANK, SVAL, DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBRQ
INTEGER INFO, LDA, M, N, RANK
DOUBLE PRECISION RCOND, SVLMAX
C .. Array Arguments ..
INTEGER JPVT( * )
DOUBLE PRECISION A( LDA, * ), SVAL( 3 ), TAU( * ), DWORK( * )
Arguments
Mode Parameters
JOBRQ CHARACTER*1
= 'R': Perform an RQ factorization with row pivoting;
= 'N': Do not perform the RQ factorization (but assume
that it has been done outside).
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension
( LDA, N )
On entry with JOBRQ = 'R', the leading M-by-N part of this
array must contain the given matrix A.
On exit with JOBRQ = 'R',
if M <= N, the upper triangle of the subarray
A(1:M,N-M+1:N) contains the M-by-M upper triangular
matrix R;
if M >= N, the elements on and above the (M-N)-th
subdiagonal contain the M-by-N upper trapezoidal matrix R;
the remaining elements, with the array TAU, represent the
orthogonal matrix Q as a product of min(M,N) elementary
reflectors (see METHOD).
On entry and on exit with JOBRQ = 'N',
if M <= N, the upper triangle of the subarray
A(1:M,N-M+1:N) must contain the M-by-M upper triangular
matrix R;
if M >= N, the elements on and above the (M-N)-th
subdiagonal must contain the M-by-N upper trapezoidal
matrix R;
the remaining elements are not referenced.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
JPVT (input/output) INTEGER array, dimension ( M )
On entry with JOBRQ = 'R', if JPVT(i) <> 0, the i-th row
of A is a final row, otherwise it is a free row. Before
the RQ factorization of A, all final rows are permuted
to the trailing positions; only the remaining free rows
are moved as a result of row pivoting during the
factorization. For rank determination it is preferable
that all rows be free.
On exit with JOBRQ = 'R', if JPVT(i) = k, then the i-th
row of P*A was the k-th row of A.
Array JPVT is not referenced when JOBRQ = 'N'.
RCOND (input) DOUBLE PRECISION
RCOND is used to determine the effective rank of A, which
is defined as the order of the largest trailing triangular
submatrix R22 in the RQ factorization with pivoting of A,
whose estimated condition number is less than 1/RCOND.
RCOND >= 0.
NOTE that when SVLMAX > 0, the estimated rank could be
less than that defined above (see SVLMAX).
SVLMAX (input) DOUBLE PRECISION
If A is a submatrix of another matrix B, and the rank
decision should be related to that matrix, then SVLMAX
should be an estimate of the largest singular value of B
(for instance, the Frobenius norm of B). If this is not
the case, the input value SVLMAX = 0 should work.
SVLMAX >= 0.
TAU (output) DOUBLE PRECISION array, dimension ( MIN( M, N ) )
On exit with JOBRQ = 'R', the leading min(M,N) elements of
TAU contain the scalar factors of the elementary
reflectors.
Array TAU is not referenced when JOBRQ = 'N'.
RANK (output) INTEGER
The effective (estimated) rank of A, i.e. the order of
the submatrix R22.
SVAL (output) DOUBLE PRECISION array, dimension ( 3 )
The estimates of some of the singular values of the
triangular factor R:
SVAL(1): largest singular value of
R(M-RANK+1:M,N-RANK+1:N);
SVAL(2): smallest singular value of
R(M-RANK+1:M,N-RANK+1:N);
SVAL(3): smallest singular value of R(M-RANK:M,N-RANK:N),
if RANK < MIN( M, N ), or of
R(M-RANK+1:M,N-RANK+1:N), otherwise.
If the triangular factorization is a rank-revealing one
(which will be the case if the trailing rows were well-
conditioned), then SVAL(1) will also be an estimate for
the largest singular value of A, and SVAL(2) and SVAL(3)
will be estimates for the RANK-th and (RANK+1)-st singular
values of A, respectively.
By examining these values, one can confirm that the rank
is well defined with respect to the chosen value of RCOND.
The ratio SVAL(1)/SVAL(2) is an estimate of the condition
number of R(M-RANK+1:M,N-RANK+1:N).
Workspace
DWORK DOUBLE PRECISION array, dimension ( LDWORK )
where LDWORK = max( 1, 3*M ), if JOBRQ = 'R';
LDWORK = max( 1, 3*min( M, N ) ), if JOBRQ = 'N'.
Error Indicator
INFO INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine computes or uses an RQ factorization with row
pivoting of A, P * A = R * Q, with R defined above, and then
finds the largest trailing submatrix whose estimated condition
number is less than 1/RCOND, taking the possible positive value of
SVLMAX into account. This is performed using an adaptation of the
LAPACK incremental condition estimation scheme and a slightly
modified rank decision test.
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit
in A(m-k+i,1:n-k+i-1), and tau in TAU(i).
The matrix P is represented in jpvt as follows: If
jpvt(j) = i
then the jth row of P is the ith canonical unit vector.
References
[1] Bischof, C.H. and P. Tang.
Generalizing Incremental Condition Estimation.
LAPACK Working Notes 32, Mathematics and Computer Science
Division, Argonne National Laboratory, UT, CS-91-132,
May 1991.
[2] Bischof, C.H. and P. Tang.
Robust Incremental Condition Estimation.
LAPACK Working Notes 33, Mathematics and Computer Science
Division, Argonne National Laboratory, UT, CS-91-133,
May 1991.
Numerical Aspects
The algorithm is backward stable.Further Comments
NoneExample
Program Text
* MB03PD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX
PARAMETER ( NMAX = 10, MMAX = 10 )
INTEGER LDA
PARAMETER ( LDA = NMAX )
INTEGER LDTAU
PARAMETER ( LDTAU = MIN(MMAX,NMAX) )
INTEGER LDWORK
PARAMETER ( LDWORK = 3*MMAX )
* .. Local Scalars ..
CHARACTER*1 JOBRQ
INTEGER I, INFO, J, M, N, RANK
DOUBLE PRECISION RCOND, SVAL(3), SVLMAX
* ..
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), TAU(LDTAU)
INTEGER JPVT(MMAX)
* .. External Subroutines ..
EXTERNAL MB03PD
* .. Intrinsic Functions ..
INTRINSIC MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, N, JOBRQ, RCOND, SVLMAX
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99972 ) N
ELSE
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99971 ) M
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,M )
* RQ with row pivoting.
DO 10 I = 1, M
JPVT(I) = 0
10 CONTINUE
CALL MB03PD( JOBRQ, M, N, A, LDA, JPVT, RCOND, SVLMAX, TAU,
$ RANK, SVAL, DWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99995 ) RANK
WRITE ( NOUT, FMT = 99994 ) ( JPVT(I), I = 1,M )
WRITE ( NOUT, FMT = 99993 ) ( SVAL(I), I = 1,3 )
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' MB03PD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB03PD = ',I2)
99995 FORMAT (' The rank is ',I5)
99994 FORMAT (' Row permutations are ',/(20(I3,2X)))
99993 FORMAT (' SVAL vector is ',/(20(1X,F10.4)))
99972 FORMAT (/' N is out of range.',/' N = ',I5)
99971 FORMAT (/' M is out of range.',/' M = ',I5)
END
Program Data
MB03PD EXAMPLE PROGRAM DATA 6 5 R 5.D-16 0.0 1. 2. 6. 3. 5. -2. -1. -1. 0. -2. 5. 5. 1. 5. 1. -2. -1. -1. 0. -2. 4. 8. 4. 20. 4. -2. -1. -1. 0. -2.Program Results
MB03PD EXAMPLE PROGRAM RESULTS
The rank is 4
Row permutations are
2 4 6 3 1 5
SVAL vector is
24.5744 0.9580 0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB03PY.html 0000664 0000000 0000000 00000016613 14560147231 0020373 0 ustar 00root root 0000000 0000000
Purpose
To compute a rank-revealing RQ factorization of a real general
M-by-N matrix A, which may be rank-deficient, and estimate its
effective rank using incremental condition estimation.
The routine uses a truncated RQ factorization with row pivoting:
[ R11 R12 ]
P * A = R * Q, where R = [ ],
[ 0 R22 ]
with R22 defined as the largest trailing upper triangular
submatrix whose estimated condition number is less than 1/RCOND.
The order of R22, RANK, is the effective rank of A. Condition
estimation is performed during the RQ factorization process.
Matrix R11 is full (but of small norm), or empty.
MB03PY does not perform any scaling of the matrix A.
Specification
SUBROUTINE MB03PY( M, N, A, LDA, RCOND, SVLMAX, RANK, SVAL, JPVT,
$ TAU, DWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, M, N, RANK
DOUBLE PRECISION RCOND, SVLMAX
C .. Array Arguments ..
INTEGER JPVT( * )
DOUBLE PRECISION A( LDA, * ), DWORK( * ), SVAL( 3 ), TAU( * )
Arguments
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension
( LDA, N )
On entry, the leading M-by-N part of this array must
contain the given matrix A.
On exit, the upper triangle of the subarray
A(M-RANK+1:M,N-RANK+1:N) contains the RANK-by-RANK upper
triangular matrix R22; the remaining elements in the last
RANK rows, with the array TAU, represent the orthogonal
matrix Q as a product of RANK elementary reflectors
(see METHOD). The first M-RANK rows contain the result
of the RQ factorization process used.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
RCOND (input) DOUBLE PRECISION
RCOND is used to determine the effective rank of A, which
is defined as the order of the largest trailing triangular
submatrix R22 in the RQ factorization with pivoting of A,
whose estimated condition number is less than 1/RCOND.
0 <= RCOND <= 1.
NOTE that when SVLMAX > 0, the estimated rank could be
less than that defined above (see SVLMAX).
SVLMAX (input) DOUBLE PRECISION
If A is a submatrix of another matrix B, and the rank
decision should be related to that matrix, then SVLMAX
should be an estimate of the largest singular value of B
(for instance, the Frobenius norm of B). If this is not
the case, the input value SVLMAX = 0 should work.
SVLMAX >= 0.
RANK (output) INTEGER
The effective (estimated) rank of A, i.e., the order of
the submatrix R22.
SVAL (output) DOUBLE PRECISION array, dimension ( 3 )
The estimates of some of the singular values of the
triangular factor R:
SVAL(1): largest singular value of
R(M-RANK+1:M,N-RANK+1:N);
SVAL(2): smallest singular value of
R(M-RANK+1:M,N-RANK+1:N);
SVAL(3): smallest singular value of R(M-RANK:M,N-RANK:N),
if RANK < MIN( M, N ), or of
R(M-RANK+1:M,N-RANK+1:N), otherwise.
If the triangular factorization is a rank-revealing one
(which will be the case if the trailing rows were well-
conditioned), then SVAL(1) will also be an estimate for
the largest singular value of A, and SVAL(2) and SVAL(3)
will be estimates for the RANK-th and (RANK+1)-st singular
values of A, respectively.
By examining these values, one can confirm that the rank
is well defined with respect to the chosen value of RCOND.
The ratio SVAL(1)/SVAL(2) is an estimate of the condition
number of R(M-RANK+1:M,N-RANK+1:N).
JPVT (output) INTEGER array, dimension ( M )
If JPVT(i) = k, then the i-th row of P*A was the k-th row
of A.
TAU (output) DOUBLE PRECISION array, dimension ( MIN( M, N ) )
The trailing RANK elements of TAU contain the scalar
factors of the elementary reflectors.
Workspace
DWORK DOUBLE PRECISION array, dimension ( 3*M-1 )Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine computes a truncated RQ factorization with row
pivoting of A, P * A = R * Q, with R defined above, and,
during this process, finds the largest trailing submatrix whose
estimated condition number is less than 1/RCOND, taking the
possible positive value of SVLMAX into account. This is performed
using an adaptation of the LAPACK incremental condition estimation
scheme and a slightly modified rank decision test. The
factorization process stops when RANK has been determined.
The matrix Q is represented as a product of elementary reflectors
Q = H(k-rank+1) H(k-rank+2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit
in A(m-k+i,1:n-k+i-1), and tau in TAU(i).
The matrix P is represented in jpvt as follows: If
jpvt(j) = i
then the jth row of P is the ith canonical unit vector.
References
[1] Bischof, C.H. and P. Tang.
Generalizing Incremental Condition Estimation.
LAPACK Working Notes 32, Mathematics and Computer Science
Division, Argonne National Laboratory, UT, CS-91-132,
May 1991.
[2] Bischof, C.H. and P. Tang.
Robust Incremental Condition Estimation.
LAPACK Working Notes 33, Mathematics and Computer Science
Division, Argonne National Laboratory, UT, CS-91-133,
May 1991.
Numerical Aspects
The algorithm is backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To reorder the diagonal blocks of a principal submatrix of an upper quasi-triangular matrix A together with their eigenvalues by constructing an orthogonal similarity transformation UT. After reordering, the leading block of the selected submatrix of A has eigenvalues in a suitably defined domain of interest, usually related to stability/instability in a continuous- or discrete-time sense.Specification
SUBROUTINE MB03QD( DICO, STDOM, JOBU, N, NLOW, NSUP, ALPHA,
$ A, LDA, U, LDU, NDIM, DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, JOBU, STDOM
INTEGER INFO, LDA, LDU, N, NDIM, NLOW, NSUP
DOUBLE PRECISION ALPHA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), U(LDU,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the spectrum separation to be
performed as follows:
= 'C': continuous-time sense;
= 'D': discrete-time sense.
STDOM CHARACTER*1
Specifies whether the domain of interest is of stability
type (left part of complex plane or inside of a circle)
or of instability type (right part of complex plane or
outside of a circle) as follows:
= 'S': stability type domain;
= 'U': instability type domain.
JOBU CHARACTER*1
Indicates how the performed orthogonal transformations UT
are accumulated, as follows:
= 'I': U is initialized to the unit matrix and the matrix
UT is returned in U;
= 'U': the given matrix U is updated and the matrix U*UT
is returned in U.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A and U. N >= 1.
NLOW, (input) INTEGER
NSUP NLOW and NSUP specify the boundary indices for the rows
and columns of the principal submatrix of A whose diagonal
blocks are to be reordered. 1 <= NLOW <= NSUP <= N.
ALPHA (input) DOUBLE PRECISION
The boundary of the domain of interest for the eigenvalues
of A. If DICO = 'C', ALPHA is the boundary value for the
real parts of eigenvalues, while for DICO = 'D',
ALPHA >= 0 represents the boundary value for the moduli of
eigenvalues.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain a matrix in a real Schur form whose 1-by-1 and
2-by-2 diagonal blocks between positions NLOW and NSUP
are to be reordered.
On exit, the leading N-by-N part contains the ordered
real Schur matrix UT' * A * UT with the elements below the
first subdiagonal set to zero.
The leading NDIM-by-NDIM part of the principal submatrix
D = A(NLOW:NSUP,NLOW:NSUP) has eigenvalues in the domain
of interest and the trailing part of this submatrix has
eigenvalues outside the domain of interest.
The domain of interest for lambda(D), the eigenvalues of
D, is defined by the parameters ALPHA, DICO and STDOM as
follows:
For DICO = 'C':
Real(lambda(D)) < ALPHA if STDOM = 'S';
Real(lambda(D)) > ALPHA if STDOM = 'U'.
For DICO = 'D':
Abs(lambda(D)) < ALPHA if STDOM = 'S';
Abs(lambda(D)) > ALPHA if STDOM = 'U'.
LDA INTEGER
The leading dimension of array A. LDA >= N.
U (input/output) DOUBLE PRECISION array, dimension (LDU,N)
On entry with JOBU = 'U', the leading N-by-N part of this
array must contain a transformation matrix (e.g. from a
previous call to this routine).
On exit, if JOBU = 'U', the leading N-by-N part of this
array contains the product of the input matrix U and the
orthogonal matrix UT used to reorder the diagonal blocks
of A.
On exit, if JOBU = 'I', the leading N-by-N part of this
array contains the matrix UT of the performed orthogonal
transformations.
Array U need not be set on entry if JOBU = 'I'.
LDU INTEGER
The leading dimension of array U. LDU >= N.
NDIM (output) INTEGER
The number of eigenvalues of the selected principal
submatrix lying inside the domain of interest.
If NLOW = 1, NDIM is also the dimension of the invariant
subspace corresponding to the eigenvalues of the leading
NDIM-by-NDIM submatrix. In this case, if U is the
orthogonal transformation matrix used to compute and
reorder the real Schur form of A, its first NDIM columns
form an orthonormal basis for the above invariant
subspace.
Workspace
DWORK DOUBLE PRECISION array, dimension (N)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: A(NLOW,NLOW-1) is nonzero, i.e. A(NLOW,NLOW) is not
the leading element of a 1-by-1 or 2-by-2 diagonal
block of A, or A(NSUP+1,NSUP) is nonzero, i.e.
A(NSUP,NSUP) is not the bottom element of a 1-by-1
or 2-by-2 diagonal block of A;
= 2: two adjacent blocks are too close to swap (the
problem is very ill-conditioned).
Method
Given an upper quasi-triangular matrix A with 1-by-1 or 2-by-2 diagonal blocks, the routine reorders its diagonal blocks along with its eigenvalues by performing an orthogonal similarity transformation UT' * A * UT. The column transformation UT is also performed on the given (initial) transformation U (resulted from a possible previous step or initialized as the identity matrix). After reordering, the eigenvalues inside the region specified by the parameters ALPHA, DICO and STDOM appear at the top of the selected diagonal block between positions NLOW and NSUP. In other words, lambda(A(NLOW:NSUP,NLOW:NSUP)) are ordered such that lambda(A(NLOW:NLOW+NDIM-1,NLOW:NLOW+NDIM-1)) are inside and lambda(A(NLOW+NDIM:NSUP,NLOW+NDIM:NSUP)) are outside the domain of interest. If NLOW = 1, the first NDIM columns of U*UT span the corresponding invariant subspace of A.References
[1] Stewart, G.W.
HQR3 and EXCHQZ: FORTRAN subroutines for calculating and
ordering the eigenvalues of a real upper Hessenberg matrix.
ACM TOMS, 2, pp. 275-280, 1976.
Numerical Aspects
3 The algorithm requires less than 4*N operations.Further Comments
NoneExample
Program Text
* MB03QD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 10 )
INTEGER LDA, LDU
PARAMETER ( LDA = NMAX, LDU = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 3*NMAX )
* .. Local Scalars ..
CHARACTER*1 DICO, JOBU, STDOM
INTEGER I, INFO, J, N, NDIM, NLOW, NSUP
DOUBLE PRECISION ALPHA
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), U(LDU,NMAX),
$ WI(NMAX), WR(NMAX)
LOGICAL BWORK(NMAX)
* .. External Functions ..
LOGICAL SELECT
* .. External Subroutines ..
EXTERNAL DGEES, MB03QD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, NLOW, NSUP, ALPHA, DICO, STDOM, JOBU
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
* Compute Schur form, eigenvalues and Schur vectors.
CALL DGEES( 'Vectors', 'Not sorted', SELECT, N, A, LDA, NDIM,
$ WR, WI, U, LDU, DWORK, LDWORK, BWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
* Block reordering.
CALL MB03QD( DICO, STDOM, JOBU, N, NLOW, NSUP, ALPHA,
$ A, LDA, U, LDU, NDIM, DWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99996 ) NDIM
WRITE ( NOUT, FMT = 99994 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( U(I,J), J = 1,N )
20 CONTINUE
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' MB03QD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from DGEES = ',I2)
99997 FORMAT (' INFO on exit from MB03QD = ',I2)
99996 FORMAT (' The number of eigenvalues in the domain is ',I5)
99995 FORMAT (8X,20(1X,F8.4))
99994 FORMAT (/' The ordered Schur form matrix is ')
99993 FORMAT (/' The transformation matrix is ')
99992 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
MB03QD EXAMPLE PROGRAM DATA 4 1 4 0.0 C S U -1.0 37.0 -12.0 -12.0 -1.0 -10.0 0.0 4.0 2.0 -4.0 7.0 -6.0 2.0 2.0 7.0 -9.0Program Results
MB03QD EXAMPLE PROGRAM RESULTS
The number of eigenvalues in the domain is 4
The ordered Schur form matrix is
-3.1300 -26.5066 27.2262 -16.2009
0.9070 -3.1300 13.6254 8.9206
0.0000 0.0000 -3.3700 0.3419
0.0000 0.0000 -1.7879 -3.3700
The transformation matrix is
0.9611 0.1784 0.2064 -0.0440
-0.1468 -0.2704 0.8116 -0.4965
-0.2224 0.7675 0.4555 0.3924
-0.0733 0.5531 -0.3018 -0.7730
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB03QG.html 0000664 0000000 0000000 00000036570 14560147231 0020356 0 ustar 00root root 0000000 0000000
Purpose
To reorder the diagonal blocks of a principal subpencil of an upper quasi-triangular matrix pencil A-lambda*E together with their generalized eigenvalues, by constructing orthogonal similarity transformations UT and VT. After reordering, the leading block of the selected subpencil of A-lambda*E has generalized eigenvalues in a suitably defined domain of interest, usually related to stability/instability in a continuous- or discrete-time sense.Specification
SUBROUTINE MB03QG( DICO, STDOM, JOBU, JOBV, N, NLOW, NSUP, ALPHA,
$ A, LDA, E, LDE, U, LDU, V, LDV, NDIM, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, JOBU, JOBV, STDOM
INTEGER INFO, LDA, LDE, LDU, LDV, LDWORK, N, NDIM, NLOW,
$ NSUP
DOUBLE PRECISION ALPHA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), E(LDE,*), U(LDU,*), V(LDV,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the spectrum separation to be
performed, as follows:
= 'C': continuous-time sense;
= 'D': discrete-time sense.
STDOM CHARACTER*1
Specifies whether the domain of interest is of stability
type (left part of complex plane or inside of a circle)
or of instability type (right part of complex plane or
outside of a circle), as follows:
= 'S': stability type domain;
= 'U': instability type domain.
JOBU CHARACTER*1
Indicates how the performed orthogonal transformations UT
are accumulated, as follows:
= 'I': U is initialized to the unit matrix and the matrix
UT is returned in U;
= 'U': the given matrix U is updated and the matrix U*UT
is returned in U.
JOBV CHARACTER*1
Indicates how the performed orthogonal transformations VT
are accumulated, as follows:
= 'I': V is initialized to the unit matrix and the matrix
VT is returned in V;
= 'U': the given matrix V is updated and the matrix V*VT
is returned in V.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, E, U, and V. N >= 0.
NLOW, (input) INTEGER
NSUP (input) INTEGER
NLOW and NSUP specify the boundary indices for the rows
and columns of the principal subpencil of A - lambda*E
whose diagonal blocks are to be reordered.
0 <= NLOW <= NSUP <= N.
ALPHA (input) DOUBLE PRECISION
The boundary of the domain of interest for the eigenvalues
of A. If DICO = 'C', ALPHA is the boundary value for the
real parts of the generalized eigenvalues, while for
DICO = 'D', ALPHA >= 0 represents the boundary value for
the moduli of the generalized eigenvalues.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain a matrix in a real Schur form whose 1-by-1 and
2-by-2 diagonal blocks between positions NLOW and NSUP
are to be reordered.
On exit, the leading N-by-N part of this array contains
a real Schur matrix UT' * A * VT, with the elements below
the first subdiagonal set to zero.
The leading NDIM-by-NDIM part of the principal subpencil
B - lambda*C, defined with B := A(NLOW:NSUP,NLOW:NSUP),
C := E(NLOW:NSUP,NLOW:NSUP), has generalized eigenvalues
in the domain of interest and the trailing part of this
subpencil has generalized eigenvalues outside the domain
of interest.
The domain of interest for eig(B,C), the generalized
eigenvalues of the pair (B,C), is defined by the
parameters ALPHA, DICO and STDOM as follows:
For DICO = 'C':
Real(eig(B,C)) < ALPHA if STDOM = 'S';
Real(eig(B,C)) > ALPHA if STDOM = 'U'.
For DICO = 'D':
Abs(eig(B,C)) < ALPHA if STDOM = 'S';
Abs(eig(B,C)) > ALPHA if STDOM = 'U'.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain a matrix in an upper triangular form.
On exit, the leading N-by-N part of this array contains an
upper triangular matrix UT' * E * VT, with the elements
below the diagonal set to zero.
The leading NDIM-by-NDIM part of the principal subpencil
B - lambda*C, defined with B := A(NLOW:NSUP,NLOW:NSUP)
C := E(NLOW:NSUP,NLOW:NSUP) has generalized eigenvalues
in the domain of interest and the trailing part of this
subpencil has generalized eigenvalues outside the domain
of interest (see description of A).
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,N).
U (input/output) DOUBLE PRECISION array, dimension (LDU,N)
On entry with JOBU = 'U', the leading N-by-N part of this
array must contain a transformation matrix (e.g., from a
previous call to this routine).
On exit, if JOBU = 'U', the leading N-by-N part of this
array contains the product of the input matrix U and the
orthogonal matrix UT used to reorder the diagonal blocks
of A - lambda*E.
On exit, if JOBU = 'I', the leading N-by-N part of this
array contains the matrix UT of the performed orthogonal
transformations.
Array U need not be set on entry if JOBU = 'I'.
LDU INTEGER
The leading dimension of the array U. LDU >= MAX(1,N).
V (input/output) DOUBLE PRECISION array, dimension (LDV,N)
On entry with JOBV = 'U', the leading N-by-N part of this
array must contain a transformation matrix (e.g., from a
previous call to this routine).
On exit, if JOBV = 'U', the leading N-by-N part of this
array contains the product of the input matrix V and the
orthogonal matrix VT used to reorder the diagonal blocks
of A - lambda*E.
On exit, if JOBV = 'I', the leading N-by-N part of this
array contains the matrix VT of the performed orthogonal
transformations.
Array V need not be set on entry if JOBV = 'I'.
LDV INTEGER
The leading dimension of the array V. LDV >= MAX(1,N).
NDIM (output) INTEGER
The number of generalized eigenvalues of the selected
principal subpencil lying inside the domain of interest.
If NLOW = 1, NDIM is also the dimension of the deflating
subspace corresponding to the generalized eigenvalues of
the leading NDIM-by-NDIM subpencil. In this case, if U and
V are the orthogonal transformation matrices used to
compute and reorder the generalized real Schur form of the
pair (A,E), then the first NDIM columns of V form an
orthonormal basis for the above deflating subspace.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= 1, and if N > 1,
LDWORK >= 4*N + 16.
If LDWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message related to LDWORK is issued by
XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: A(NLOW,NLOW-1) is nonzero, i.e., A(NLOW,NLOW) is not
the leading element of a 1-by-1 or 2-by-2 diagonal
block of A, or A(NSUP+1,NSUP) is nonzero, i.e.,
A(NSUP,NSUP) is not the bottom element of a 1-by-1
or 2-by-2 diagonal block of A;
= 2: two adjacent blocks are too close to swap (the
problem is very ill-conditioned).
Method
Given an upper quasi-triangular matrix pencil A - lambda*E with 1-by-1 or 2-by-2 diagonal blocks, the routine reorders its diagonal blocks along with its eigenvalues by performing an orthogonal equivalence transformation UT'*(A - lambda*E)* VT. The column transformations UT and VT are also performed on the given (initial) transformations U and V (resulted from a possible previous step or initialized as identity matrices). After reordering, the generalized eigenvalues inside the region specified by the parameters ALPHA, DICO and STDOM appear at the top of the selected diagonal subpencil between positions NLOW and NSUP. In other words, lambda(A(Select,Select),E(Select,Select)) are ordered such that lambda(A(Inside,Inside),E(Inside,Inside)) are inside, and lambda(A(Outside,Outside),E(Outside,Outside)) are outside the domain of interest, where Select = NLOW:NSUP, Inside = NLOW:NLOW+NDIM-1, and Outside = NLOW+NDIM:NSUP. If NLOW = 1, the first NDIM columns of V*VT span the corresponding right deflating subspace of (A,E).References
[1] Stewart, G.W.
HQR3 and EXCHQZ: FORTRAN subroutines for calculating and
ordering the eigenvalues of a real upper Hessenberg matrix.
ACM TOMS, 2, pp. 275-280, 1976.
Numerical Aspects
3 The algorithm requires less than 4*N operations.Further Comments
NoneExample
Program Text
* MB03QG EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 10 )
INTEGER LDA, LDE, LDU, LDV
PARAMETER ( LDA = NMAX, LDE = NMAX, LDU = NMAX, LDV = NMAX)
INTEGER LDWORK
PARAMETER ( LDWORK = 8*NMAX + 16 )
* .. Local Scalars ..
CHARACTER*1 DICO, JOBU, JOBV, STDOM
INTEGER I, INFO, J, N, NDIM, NLOW, NSUP
DOUBLE PRECISION ALPHA
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), BETA(NMAX), DWORK(LDWORK),
$ E(LDE,NMAX), U(LDU,NMAX), V(LDV,NMAX), WI(NMAX),
$ WR(NMAX)
LOGICAL BWORK(NMAX)
* .. External Functions ..
LOGICAL DELCTG
* .. External Subroutines ..
EXTERNAL DGGES, MB03QG
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, NLOW, NSUP, ALPHA, DICO, STDOM, JOBU,
$ JOBV
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
* Compute Schur form, eigenvalues and Schur vectors.
CALL DGGES( 'Vectors', 'Vectors', 'Not sorted', DELCTG, N,
$ A, LDA, E, LDE, NDIM, WR, WI, BETA, U, LDU, V, LDV,
$ DWORK, LDWORK, BWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
* Block reordering.
CALL MB03QG( DICO, STDOM, JOBU, JOBV, N, NLOW, NSUP, ALPHA,
$ A, LDA, E, LDE, U, LDU, V, LDV, NDIM, DWORK,
$ LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99996 ) NDIM
WRITE ( NOUT, FMT = 99994 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 30 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( U(I,J), J = 1,N )
30 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( V(I,J), J = 1,N )
40 CONTINUE
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' MB03QG EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from DGEES = ',I2)
99997 FORMAT (' INFO on exit from MB03QG = ',I2)
99996 FORMAT (' The number of eigenvalues in the domain is ',I5)
99995 FORMAT (8X,20(1X,F8.4))
99994 FORMAT (/' The ordered Schur form matrix is ')
99993 FORMAT (/' The ordered triangular matrix is ')
99992 FORMAT (/' The transformation matrix U is ')
99991 FORMAT (/' The transformation matrix V is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
MB03QG EXAMPLE PROGRAM DATA 4 1 4 0.0 C S U U -1.0 37.0 -12.0 -12.0 -1.0 -10.0 0.0 4.0 2.0 -4.0 7.0 -6.0 2.0 2.0 7.0 -9.0 1.0 3.0 2.0 -1.0 -2.0 5.0 3.0 2.0 2.0 4.0 5.0 6.0 3.0 7.0 6.0 9.0Program Results
MB03QG EXAMPLE PROGRAM RESULTS
The number of eigenvalues in the domain is 2
The ordered Schur form matrix is
-1.4394 2.5550 -12.5655 -4.0714
2.8887 -1.1242 9.2819 -2.6724
0.0000 0.0000 -19.7785 36.4447
0.0000 0.0000 0.0000 3.5537
The ordered triangular matrix is
-16.0178 0.0000 2.3850 4.7645
0.0000 3.2809 -1.5640 1.9954
0.0000 0.0000 -3.0652 0.3039
0.0000 0.0000 0.0000 1.1671
The transformation matrix U is
-0.1518 -0.0737 -0.9856 0.0140
-0.2865 -0.9466 0.1136 -0.0947
-0.5442 0.0924 0.0887 0.8292
-0.7738 0.3000 0.0890 -0.5508
The transformation matrix V is
0.2799 0.9041 0.2685 0.1794
0.4009 -0.0714 0.3780 -0.8315
0.7206 -0.4006 0.2628 0.5012
0.4917 0.1306 -0.8462 -0.1588
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB03QV.html 0000664 0000000 0000000 00000005446 14560147231 0020373 0 ustar 00root root 0000000 0000000
Purpose
To compute the eigenvalues of an upper quasi-triangular matrix pencil.Specification
SUBROUTINE MB03QV( N, S, LDS, T, LDT, ALPHAR, ALPHAI, BETA, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDS, LDT, N
C .. Array Arguments ..
DOUBLE PRECISION ALPHAI(*), ALPHAR(*), BETA(*), S(LDS,*), T(LDT,*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrices S and T. N >= 0.
S (input) DOUBLE PRECISION array, dimension(LDS,N)
The upper quasi-triangular matrix S.
LDS INTEGER
The leading dimension of the array S. LDS >= max(1,N).
T (input) DOUBLE PRECISION array, dimension(LDT,N)
The upper triangular matrix T.
LDT INTEGER
The leading dimension of the array T. LDT >= max(1,N).
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
BETA (output) DOUBLE PRECISION array, dimension (N)
On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N,
are the generalized eigenvalues.
ALPHAR(j) + ALPHAI(j)*i, and BETA(j),j=1,...,N, are the
diagonals of the complex Schur form (S,T) that would
result if the 2-by-2 diagonal blocks of the real Schur
form of (A,B) were further reduced to triangular form
using 2-by-2 complex unitary transformations.
If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
positive, then the j-th and (j+1)-st eigenvalues are a
complex conjugate pair, with ALPHAI(j+1) negative.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the eigenvalues of a selected 2-by-2 diagonal block pair of an upper quasi-triangular pencil, to reduce the selected block pair to the standard form and to split it in the case of real eigenvalues, by constructing orthogonal matrices UT and VT. The transformations UT and VT are applied to the pair (A,E) by computing (UT'*A*VT, UT'*E*VT ), to the matrices U and V, by computing U*UT and V*VT.Specification
SUBROUTINE MB03QW( N, L, A, LDA, E, LDE, U, LDU, V, LDV,
$ ALPHAR, ALPHAI, BETA, INFO )
C .. Scalar Arguments ..
INTEGER INFO, L, LDA, LDE, LDU, LDV, N
DOUBLE PRECISION ALPHAI(*), ALPHAR(*), BETA(*)
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), E(LDE,*), U(LDU,*), V(LDV,*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, E, UT and VT. N >= 2.
L (input) INTEGER
Specifies the position of the block. 1 <= L < N.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the upper quasi-triangular matrix A whose
selected 2-by-2 diagonal block is to be processed.
On exit, the leading N-by-N part of this array contains
the upper quasi-triangular matrix A after its selected
block has been split and/or put in the LAPACK standard
form.
LDA INTEGER
The leading dimension of the array A. LDA >= N.
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain the upper triangular matrix E whose selected
2-by-2 diagonal block is to be processed.
On exit, the leading N-by-N part of this array contains
the transformed upper triangular matrix E (in the LAPACK
standard form).
LDE INTEGER
The leading dimension of the array E. LDE >= N.
U (input/output) DOUBLE PRECISION array, dimension (LDU,N)
On entry, the leading N-by-N part of this array must
contain a transformation matrix U.
On exit, the leading N-by-N part of this array contains
U*UT, where UT is the transformation matrix used to
split and/or standardize the selected block pair.
LDU INTEGER
The leading dimension of the array U. LDU >= N.
V (input/output) DOUBLE PRECISION array, dimension (LDV,N)
On entry, the leading N-by-N part of this array must
contain a transformation matrix V.
On exit, the leading N-by-N part of this array contains
V*VT, where VT is the transformation matrix used to
split and/or standardize the selected block pair.
LDV INTEGER
The leading dimension of the array V. LDV >= N.
ALPHAR (output) DOUBLE PRECISION array, dimension (2)
ALPHAI (output) DOUBLE PRECISION array, dimension (2)
BETA (output) DOUBLE PRECISION array, dimension (2)
(ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the
block pair of the pencil (A,B), k=1,2, i = sqrt(-1).
Note that BETA(k) may be zero.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Let A1 = ( A(L,L) A(L,L+1) ), E1 = ( E(L,L) E(L,L+1) ),
( A(L+1,L) A(L+1,L+1) ) ( E(L+1,L) E(L+1,L+1) )
be the specified 2-by-2 diagonal block pair of the pencil (A,B).
If the eigenvalues of (A1,E1) are complex, then they are computed
and stored in ER and EI, where the real part is stored in ER and
the positive imaginary part in EI. The 2-by-2 block pair is
reduced if necessary to the standard form, such that
A(L,L) = A(L+1,L+1), and A(L,L+1) and A(L+1,L) have oposite signs.
If the eigenvalues are real, the 2-by-2 block pair is reduced to
upper triangular form such that ABS(A(L,L)) >= ABS(A(L+1,L+1)).
In both cases, an orthogonal rotation U1' is constructed such that
U1'*A1*U1 has the appropriate form. Let UT be an extension of U1
to an N-by-N orthogonal matrix, using identity submatrices. Then A
and E are replaced by UT'*A*UT and UT'*E*UT, and the contents of
the arrays U and V are U * UT and V * VT, respectively.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the eigenvalues of an upper quasi-triangular matrix.Specification
SUBROUTINE MB03QX( N, T, LDT, WR, WI, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDT, N
C .. Array Arguments ..
DOUBLE PRECISION T(LDT, *), WI(*), WR(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix T. N >= 0.
T (input) DOUBLE PRECISION array, dimension(LDT,N)
The upper quasi-triangular matrix T.
LDT INTEGER
The leading dimension of the array T. LDT >= max(1,N).
WR, WI (output) DOUBLE PRECISION arrays, dimension (N)
The real and imaginary parts, respectively, of the
eigenvalues of T. The eigenvalues are stored in the same
order as on the diagonal of T. If T(i:i+1,i:i+1) is a
2-by-2 diagonal block with complex conjugated eigenvalues
then WI(i) > 0 and WI(i+1) = -WI(i).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the eigenvalues of a selected 2-by-2 diagonal block of an upper quasi-triangular matrix, to reduce the selected block to the standard form and to split the block in the case of real eigenvalues by constructing an orthogonal transformation UT. This transformation is applied to A (by similarity) and to another matrix U from the right.Specification
SUBROUTINE MB03QY( N, L, A, LDA, U, LDU, E1, E2, INFO )
C .. Scalar Arguments ..
INTEGER INFO, L, LDA, LDU, N
DOUBLE PRECISION E1, E2
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), U(LDU,*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrices A and UT. N >= 2.
L (input) INTEGER
Specifies the position of the block. 1 <= L < N.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the upper quasi-triangular matrix A whose
selected 2-by-2 diagonal block is to be processed.
On exit, the leading N-by-N part of this array contains
the upper quasi-triangular matrix A after its selected
block has been splitt and/or put in the LAPACK standard
form.
LDA INTEGER
The leading dimension of array A. LDA >= N.
U (input/output) DOUBLE PRECISION array, dimension (LDU,N)
On entry, the leading N-by-N part of this array must
contain a transformation matrix U.
On exit, the leading N-by-N part of this array contains
U*UT, where UT is the transformation matrix used to
split and/or standardize the selected block.
LDU INTEGER
The leading dimension of array U. LDU >= N.
E1, E2 (output) DOUBLE PRECISION
E1 and E2 contain either the real eigenvalues or the real
and positive imaginary parts, respectively, of the complex
eigenvalues of the selected 2-by-2 diagonal block of A.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Let A1 = ( A(L,L) A(L,L+1) )
( A(L+1,L) A(L+1,L+1) )
be the specified 2-by-2 diagonal block of matrix A.
If the eigenvalues of A1 are complex, then they are computed and
stored in E1 and E2, where the real part is stored in E1 and the
positive imaginary part in E2. The 2-by-2 block is reduced if
necessary to the standard form, such that A(L,L) = A(L+1,L+1), and
A(L,L+1) and A(L+1,L) have oposite signs. If the eigenvalues are
real, the 2-by-2 block is reduced to an upper triangular form such
that ABS(A(L,L)) >= ABS(A(L+1,L+1)).
In both cases, an orthogonal rotation U1' is constructed such that
U1'*A1*U1 has the appropriate form. Let UT be an extension of U1
to an N-by-N orthogonal matrix, using identity submatrices. Then A
is replaced by UT'*A*UT and the contents of array U is U * UT.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To reduce a matrix A in real Schur form to a block-diagonal form using well-conditioned non-orthogonal similarity transformations. The condition numbers of the transformations used for reduction are roughly bounded by PMAX, where PMAX is a given value. The transformations are optionally postmultiplied in a given matrix X. The real Schur form is optionally ordered, so that clustered eigenvalues are grouped in the same block.Specification
SUBROUTINE MB03RD( JOBX, SORT, N, PMAX, A, LDA, X, LDX, NBLCKS,
$ BLSIZE, WR, WI, TOL, DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBX, SORT
INTEGER INFO, LDA, LDX, N, NBLCKS
DOUBLE PRECISION PMAX, TOL
C .. Array Arguments ..
INTEGER BLSIZE(*)
DOUBLE PRECISION A(LDA,*), DWORK(*), WI(*), WR(*), X(LDX,*)
Arguments
Mode Parameters
JOBX CHARACTER*1
Specifies whether or not the transformations are
accumulated, as follows:
= 'N': The transformations are not accumulated;
= 'U': The transformations are accumulated in X (the
given matrix X is updated).
SORT CHARACTER*1
Specifies whether or not the diagonal blocks of the real
Schur form are reordered, as follows:
= 'N': The diagonal blocks are not reordered;
= 'S': The diagonal blocks are reordered before each
step of reduction, so that clustered eigenvalues
appear in the same block;
= 'C': The diagonal blocks are not reordered, but the
"closest-neighbour" strategy is used instead of
the standard "closest to the mean" strategy
(see METHOD);
= 'B': The diagonal blocks are reordered before each
step of reduction, and the "closest-neighbour"
strategy is used (see METHOD).
Input/Output Parameters
N (input) INTEGER
The order of the matrices A and X. N >= 0.
PMAX (input) DOUBLE PRECISION
An upper bound for the infinity norm of elementary
submatrices of the individual transformations used for
reduction (see METHOD). PMAX >= 1.0D0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A to be block-diagonalized, in real
Schur form.
On exit, the leading N-by-N part of this array contains
the computed block-diagonal matrix, in real Schur
canonical form. The non-diagonal blocks are set to zero.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
X (input/output) DOUBLE PRECISION array, dimension (LDX,N)
On entry, if JOBX = 'U', the leading N-by-N part of this
array must contain a given matrix X.
On exit, if JOBX = 'U', the leading N-by-N part of this
array contains the product of the given matrix X and the
transformation matrix that reduced A to block-diagonal
form. The transformation matrix is itself a product of
non-orthogonal similarity transformations having elements
with magnitude less than or equal to PMAX.
If JOBX = 'N', this array is not referenced.
LDX INTEGER
The leading dimension of array X.
LDX >= 1, if JOBX = 'N';
LDX >= MAX(1,N), if JOBX = 'U'.
NBLCKS (output) INTEGER
The number of diagonal blocks of the matrix A.
BLSIZE (output) INTEGER array, dimension (N)
The first NBLCKS elements of this array contain the orders
of the resulting diagonal blocks of the matrix A.
WR, (output) DOUBLE PRECISION arrays, dimension (N)
WI These arrays contain the real and imaginary parts,
respectively, of the eigenvalues of the matrix A.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in the ordering of the diagonal
blocks of the real Schur form matrix.
If the user sets TOL > 0, then the given value of TOL is
used as an absolute tolerance: a block i and a temporarily
fixed block 1 (the first block of the current trailing
submatrix to be reduced) are considered to belong to the
same cluster if their eigenvalues satisfy
| lambda_1 - lambda_i | <= TOL.
If the user sets TOL < 0, then the given value of TOL is
used as a relative tolerance: a block i and a temporarily
fixed block 1 are considered to belong to the same cluster
if their eigenvalues satisfy, for j = 1, ..., N,
| lambda_1 - lambda_i | <= | TOL | * max | lambda_j |.
If the user sets TOL = 0, then an implicitly computed,
default tolerance, defined by TOL = SQRT( SQRT( EPS ) )
is used instead, as a relative tolerance, where EPS is
the machine precision (see LAPACK Library routine DLAMCH).
If SORT = 'N' or 'C', this parameter is not referenced.
Workspace
DWORK DOUBLE PRECISION array, dimension (N)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Consider first that SORT = 'N'. Let
( A A )
( 11 12 )
A = ( ),
( 0 A )
( 22 )
be the given matrix in real Schur form, where initially A is the
11
first diagonal block of dimension 1-by-1 or 2-by-2. An attempt is
made to compute a transformation matrix X of the form
( I P )
X = ( ) (1)
( 0 I )
(partitioned as A), so that
( A 0 )
-1 ( 11 )
X A X = ( ),
( 0 A )
( 22 )
and the elements of P do not exceed the value PMAX in magnitude.
An adaptation of the standard method for solving Sylvester
equations [1], which controls the magnitude of the individual
elements of the computed solution [2], is used to obtain matrix P.
When this attempt failed, an 1-by-1 (or 2-by-2) diagonal block of
A , whose eigenvalue(s) is (are) the closest to the mean of those
22
of A is selected, and moved by orthogonal similarity
11
transformations in the leading position of A ; the moved diagonal
22
block is then added to the block A , increasing its order by 1
11
(or 2). Another attempt is made to compute a suitable
transformation matrix X with the new definitions of the blocks A
11
and A . After a successful transformation matrix X has been
22
obtained, it postmultiplies the current transformation matrix
(if JOBX = 'U'), and the whole procedure is repeated for the
matrix A .
22
When SORT = 'S', the diagonal blocks of the real Schur form are
reordered before each step of the reduction, so that each cluster
of eigenvalues, defined as specified in the definition of TOL,
appears in adjacent blocks. The blocks for each cluster are merged
together, and the procedure described above is applied to the
larger blocks. Using the option SORT = 'S' will usually provide
better efficiency than the standard option (SORT = 'N'), proposed
in [2], because there could be no or few unsuccessful attempts
to compute individual transformation matrices X of the form (1).
However, the resulting dimensions of the blocks are usually
larger; this could make subsequent calculations less efficient.
When SORT = 'C' or 'B', the procedure is similar to that for
SORT = 'N' or 'S', respectively, but the block of A whose
22
eigenvalue(s) is (are) the closest to those of A (not to their
11
mean) is selected and moved to the leading position of A . This
22
is called the "closest-neighbour" strategy.
References
[1] Bartels, R.H. and Stewart, G.W. T
Solution of the matrix equation A X + XB = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
[2] Bavely, C. and Stewart, G.W.
An Algorithm for Computing Reducing Subspaces by Block
Diagonalization.
SIAM J. Numer. Anal., 16, pp. 359-367, 1979.
[3] Demmel, J.
The Condition Number of Equivalence Transformations that
Block Diagonalize Matrix Pencils.
SIAM J. Numer. Anal., 20, pp. 599-610, 1983.
Numerical Aspects
3 4 The algorithm usually requires 0(N ) operations, but 0(N ) are possible in the worst case, when all diagonal blocks in the real Schur form of A are 1-by-1, and the matrix cannot be diagonalized by well-conditioned transformations.Further Comments
The individual non-orthogonal transformation matrices used in the reduction of A to a block-diagonal form have condition numbers of the order PMAX. This does not guarantee that their product is well-conditioned enough. The routine can be easily modified to provide estimates for the condition numbers of the clusters of eigenvalues.Example
Program Text
* MB03RD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 10 )
INTEGER LDA, LDX
PARAMETER ( LDA = NMAX, LDX = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 3*NMAX )
* .. Local Scalars ..
CHARACTER*1 JOBX, SORT
INTEGER I, INFO, J, N, NBLCKS, SDIM
DOUBLE PRECISION PMAX, TOL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), WI(NMAX), WR(NMAX),
$ X(LDX,NMAX)
INTEGER BLSIZE(NMAX)
LOGICAL BWORK(NMAX)
* .. External Functions ..
LOGICAL SELECT
* .. External Subroutines ..
EXTERNAL DGEES, MB03RD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, PMAX, TOL, JOBX, SORT
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99972 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
* Compute Schur form, eigenvalues and Schur vectors.
CALL DGEES( 'Vectors', 'Not sorted', SELECT, N, A, LDA, SDIM,
$ WR, WI, X, LDX, DWORK, LDWORK, BWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
* Block-diagonalization.
CALL MB03RD( JOBX, SORT, N, PMAX, A, LDA, X, LDX, NBLCKS,
$ BLSIZE, WR, WI, TOL, DWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99995 ) NBLCKS
WRITE ( NOUT, FMT = 99994 ) ( BLSIZE(I), I = 1,NBLCKS )
WRITE ( NOUT, FMT = 99993 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99992 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99992 ) ( X(I,J), J = 1,N )
20 CONTINUE
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' MB03RD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from DGEES = ',I2)
99997 FORMAT (' INFO on exit from MB03RD = ',I2)
99995 FORMAT (' The number of blocks is ',I5)
99994 FORMAT (' The orders of blocks are ',/(20(I3,2X)))
99993 FORMAT (' The block-diagonal matrix is ')
99992 FORMAT (8X,20(1X,F8.4))
99991 FORMAT (' The transformation matrix is ')
99972 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
MB03RD EXAMPLE PROGRAM DATA 8 1.D03 1.D-2 U S 1. -1. 1. 2. 3. 1. 2. 3. 1. 1. 3. 4. 2. 3. 4. 2. 0. 0. 1. -1. 1. 5. 4. 1. 0. 0. 0. 1. -1. 3. 1. 2. 0. 0. 0. 1. 1. 2. 3. -1. 0. 0. 0. 0. 0. 1. 5. 1. 0. 0. 0. 0. 0. 0. 0.99999999 -0.99999999 0. 0. 0. 0. 0. 0. 0.99999999 0.99999999Program Results
MB03RD EXAMPLE PROGRAM RESULTS
The number of blocks is 2
The orders of blocks are
6 2
The block-diagonal matrix is
1.0000 -1.0000 -1.2247 -0.7071 -3.4186 1.4577 0.0000 0.0000
1.0000 1.0000 0.0000 1.4142 -5.1390 3.1637 0.0000 0.0000
0.0000 0.0000 1.0000 -1.7321 -0.0016 2.0701 0.0000 0.0000
0.0000 0.0000 0.5774 1.0000 0.7516 1.1379 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 1.0000 -5.8606 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.1706 1.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 -0.8850
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000
The transformation matrix is
1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9045 0.1957
0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 -0.3015 0.9755
0.0000 0.0000 0.8165 0.0000 -0.5768 -0.0156 -0.3015 0.0148
0.0000 0.0000 -0.4082 0.7071 -0.5768 -0.0156 0.0000 -0.0534
0.0000 0.0000 -0.4082 -0.7071 -0.5768 -0.0156 0.0000 0.0801
0.0000 0.0000 0.0000 0.0000 -0.0276 0.9805 0.0000 0.0267
0.0000 0.0000 0.0000 0.0000 0.0332 -0.0066 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0011 0.1948 0.0000 0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB03RW.html 0000664 0000000 0000000 00000013406 14560147231 0020370 0 ustar 00root root 0000000 0000000
Purpose
To solve the Sylvester equation -AX + XB = C, where A and B are complex M-by-M and N-by-N matrices, respectively, in Schur form. This routine is intended to be called only by SLICOT Library routine MB03RZ. For efficiency purposes, the computations are aborted when the absolute value of an element of X is greater than a given value PMAX.Specification
SUBROUTINE MB03RW( M, N, PMAX, A, LDA, B, LDB, C, LDC, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, M, N
DOUBLE PRECISION PMAX
C .. Array Arguments ..
COMPLEX*16 A(LDA,*), B(LDB,*), C(LDC,*)
Arguments
Input/Output Parameters
M (input) INTEGER
The order of the matrix A and the number of rows of the
matrices C and X. M >= 0.
N (input) INTEGER
The order of the matrix B and the number of columns of the
matrices C and X. N >= 0.
PMAX (input) DOUBLE PRECISION
An upper bound for the absolute value of the elements of X
(see METHOD).
A (input) COMPLEX*16 array, dimension (LDA,M)
The leading M-by-M upper triangular part of this array
must contain the matrix A of the Sylvester equation.
The elements below the diagonal are not referenced.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,M).
B (input) COMPLEX*16 array, dimension (LDB,N)
The leading N-by-N upper triangular part of this array
must contain the matrix B of the Sylvester equation.
The elements below the diagonal are not referenced.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) COMPLEX*16 array, dimension (LDC,N)
On entry, the leading M-by-N part of this array must
contain the matrix C of the Sylvester equation.
On exit, if INFO = 0, the leading M-by-N part of this
array contains the solution matrix X of the Sylvester
equation, and each element of X (see METHOD) has the
absolute value less than or equal to PMAX.
On exit, if INFO = 1, the solution matrix X has not been
computed completely, because an element of X had the
absolute value greater than PMAX. Part of the matrix C has
possibly been overwritten with the corresponding part
of X.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,M).
Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: an element of X had the absolute value greater than
the given value PMAX.
= 2: A and B have common or very close eigenvalues;
perturbed values were used to solve the equation
(but the matrices A and B are unchanged). This is a
warning.
Method
The routine uses an adaptation of the standard method for solving
Sylvester equations [1], which controls the magnitude of the
individual elements of the computed solution [2]. The equation
-AX + XB = C can be rewritten as
m l-1
-A X + X B = C + sum A X - sum X B
kk kl kl ll kl i=k+1 ki il j=1 kj jl
for l = 1:n, and k = m:-1:1, where A , B , C , and X , are the
kk ll kl kl
elements defined by the partitioning induced by the Schur form
of A and B. So, the elements of X are found column by column,
starting from the bottom. If any such element has the absolute
value greater than the given value PMAX, the calculations are
ended.
References
[1] Bartels, R.H. and Stewart, G.W. T
Solution of the matrix equation A X + XB = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
[2] Bavely, C. and Stewart, G.W.
An Algorithm for Computing Reducing Subspaces by Block
Diagonalization.
SIAM J. Numer. Anal., 16, pp. 359-367, 1979.
Numerical Aspects
2 2 The algorithm requires 0(M N + MN ) operations.Further Comments
Let
( A C ) ( I X )
M = ( ), Y = ( ).
( 0 B ) ( 0 I )
Then
-1 ( A 0 )
Y M Y = ( ),
( 0 B )
hence Y is a non-unitary transformation matrix which performs the
reduction of M to a block-diagonal form. Bounding a norm of X is
equivalent to setting an upper bound to the condition number of
the transformation matrix Y.
Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To reorder the diagonal blocks of the principal submatrix between the indices KL and KU (KU >= KL) of a real Schur form matrix A together with their eigenvalues, using orthogonal similarity transformations, such that the block specified by KU is moved in the position KL. The transformations are optionally postmultiplied in a given matrix X.Specification
SUBROUTINE MB03RX( JOBV, N, KL, KU, A, LDA, X, LDX, WR, WI,
$ DWORK )
C .. Scalar Arguments ..
CHARACTER JOBV
INTEGER KL, KU, LDA, LDX, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), WI(*), WR(*), X(LDX,*)
Arguments
Mode Parameters
JOBV CHARACTER*1
Specifies whether or not the transformations are
accumulated, as follows:
= 'N': The transformations are not accumulated;
= 'V': The transformations are accumulated in X (the
given matrix X is updated).
Input/Output Parameters
N (input) INTEGER
The order of the matrices A and X. N >= 0.
KL (input) INTEGER
The lower boundary index for the rows and columns of the
principal submatrix of A whose diagonal blocks are to be
reordered, and also the target position for the block to
be moved. 1 <= KL <= KU <= N.
KU (input/output) INTEGER
On entry, KU specifies the upper boundary index for the
rows and columns of the principal submatrix of A whose
diagonal blocks are to be reordered, and also the original
position for the block to be moved. 1 <= KL <= KU <= N.
On exit, KU specifies the upper boundary index for the
rows and columns of the principal submatrix of A whose
diagonal blocks have been reordered. The given value will
be increased by 1 if the moved block was 2-by-2 and it has
been replaced by two 1-by-1 blocks. Otherwise, its input
value is preserved.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A in real Schur canonical form.
On exit, the leading N-by-N part of this array contains
the ordered real Schur canonical form.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
X (input/output) DOUBLE PRECISION array, dimension (LDX,N)
On entry, if JOBV = 'V', the leading N-by-N part of this
array must contain a given matrix X.
On exit, if JOBV = 'V', the leading N-by-N part of this
array contains the product of the given matrix X and the
transformation matrix that performed the reordering of A.
If JOBV = 'N', this array is not referenced.
LDX INTEGER
The leading dimension of array X.
LDX >= 1, if JOBV = 'N';
LDX >= MAX(1,N), if JOBV = 'V'.
WR, (input/output) DOUBLE PRECISION arrays, dimension (N)
WI On entry, these arrays must contain the real and imaginary
parts, respectively, of the eigenvalues of the matrix A.
On exit, these arrays contain the real and imaginary
parts, respectively, of the eigenvalues of the matrix A,
possibly reordered.
Workspace
DWORK DOUBLE PRECISION array, dimension (N)Method
An attempt is made to move the block in the position (KU,KU) to the position (KL,KL) by a sequence of orthogonal similarity transformations, each swapping two consecutive blocks. The standard algorithm [1], [2] usually succeeds to perform this reordering. A failure of this algorithm means that two consecutive blocks (one of them being the desired block possibly moved) are too close to swap. In such a case, the leading block of the two is tried to be moved in the position (KL,KL) and the procedure is repeated.References
[1] Stewart, G.W.
HQR3 and EXCHQZ: FORTRAN subroutines for calculating and
ordering the eigenvalues of a real upper Hessenberg matrix.
ACM TOMS, 2, pp. 275-280, 1976.
[2] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
Numerical Aspects
The algorithm is numerically stable. If some eigenvalues are ill-conditioned, their returned values could differ much from their input values.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve the Sylvester equation -AX + XB = C, where A and B are M-by-M and N-by-N matrices, respectively, in real Schur form. This routine is intended to be called only by SLICOT Library routine MB03RD. For efficiency purposes, the computations are aborted when the infinity norm of an elementary submatrix of X is greater than a given value PMAX.Specification
SUBROUTINE MB03RY( M, N, PMAX, A, LDA, B, LDB, C, LDC, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, M, N
DOUBLE PRECISION PMAX
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*)
Arguments
Input/Output Parameters
M (input) INTEGER
The order of the matrix A and the number of rows of the
matrices C and X. M >= 0.
N (input) INTEGER
The order of the matrix B and the number of columns of the
matrices C and X. N >= 0.
PMAX (input) DOUBLE PRECISION
An upper bound for the infinity norm of an elementary
submatrix of X (see METHOD).
A (input) DOUBLE PRECISION array, dimension (LDA,M)
The leading M-by-M part of this array must contain the
matrix A of the Sylvester equation, in real Schur form.
The elements below the real Schur form are not referenced.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,M).
B (input) DOUBLE PRECISION array, dimension (LDB,N)
The leading N-by-N part of this array must contain the
matrix B of the Sylvester equation, in real Schur form.
The elements below the real Schur form are not referenced.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading M-by-N part of this array must
contain the matrix C of the Sylvester equation.
On exit, if INFO = 0, the leading M-by-N part of this
array contains the solution matrix X of the Sylvester
equation, and each elementary submatrix of X (see METHOD)
has the infinity norm less than or equal to PMAX.
On exit, if INFO = 1, the solution matrix X has not been
computed completely, because an elementary submatrix of X
had the infinity norm greater than PMAX. Part of the
matrix C has possibly been overwritten with the
corresponding part of X.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,M).
Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: an elementary submatrix of X had the infinity norm
greater than the given value PMAX.
Method
The routine uses an adaptation of the standard method for solving
Sylvester equations [1], which controls the magnitude of the
individual elements of the computed solution [2]. The equation
-AX + XB = C can be rewritten as
p l-1
-A X + X B = C + sum A X - sum X B
kk kl kl ll kl i=k+1 ki il j=1 kj jl
for l = 1:q, and k = p:-1:1, where A , B , C , and X , are
kk ll kl kl
block submatrices defined by the partitioning induced by the Schur
form of A and B, and p and q are the numbers of the diagonal
blocks of A and B, respectively. So, the elementary submatrices of
X are found block column by block column, starting from the
bottom. If any such elementary submatrix has the infinity norm
greater than the given value PMAX, the calculations are ended.
References
[1] Bartels, R.H. and Stewart, G.W. T
Solution of the matrix equation A X + XB = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
[2] Bavely, C. and Stewart, G.W.
An Algorithm for Computing Reducing Subspaces by Block
Diagonalization.
SIAM J. Numer. Anal., 16, pp. 359-367, 1979.
Numerical Aspects
2 2 The algorithm requires 0(M N + MN ) operations.Further Comments
Let
( A C ) ( I X )
M = ( ), Y = ( ).
( 0 B ) ( 0 I )
Then
-1 ( A 0 )
Y M Y = ( ),
( 0 B )
hence Y is an non-orthogonal transformation matrix which performs
the reduction of M to a block-diagonal form. Bounding a norm of
X is equivalent to setting an upper bound to the condition number
of the transformation matrix Y.
Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To reduce an upper triangular complex matrix A (Schur form) to a block-diagonal form using well-conditioned non-unitary similarity transformations. The condition numbers of the transformations used for reduction are roughly bounded by PMAX, where PMAX is a given value. The transformations are optionally postmultiplied in a given matrix X. The Schur form is optionally ordered, so that clustered eigenvalues are grouped in the same block.Specification
SUBROUTINE MB03RZ( JOBX, SORT, N, PMAX, A, LDA, X, LDX, NBLCKS,
$ BLSIZE, W, TOL, INFO )C .. Scalar Arguments ..
CHARACTER JOBX, SORT
INTEGER INFO, LDA, LDX, N, NBLCKS
DOUBLE PRECISION PMAX, TOL
C .. Array Arguments ..
INTEGER BLSIZE(*)
COMPLEX*16 A(LDA,*), W(*), X(LDX,*)
Arguments
Mode Parameters
JOBX CHARACTER*1
Specifies whether or not the transformations are
accumulated, as follows:
= 'N': The transformations are not accumulated;
= 'U': The transformations are accumulated in X (the
given matrix X is updated).
SORT CHARACTER*1
Specifies whether or not the diagonal elements of the
Schur form are reordered, as follows:
= 'N': The diagonal elements are not reordered;
= 'S': The diagonal elements are reordered before each
step of reduction, so that clustered eigenvalues
appear in the same block;
= 'C': The diagonal elements are not reordered, but the
"closest-neighbour" strategy is used instead of
the standard "closest to the mean" strategy (see
METHOD);
= 'B': The diagonal elements are reordered before each
step of reduction, and the "closest-neighbour"
strategy is used (see METHOD).
Input/Output Parameters
N (input) INTEGER
The order of the matrices A and X. N >= 0.
PMAX (input) DOUBLE PRECISION
An upper bound for the absolute value of the elements of
the individual transformations used for reduction
(see METHOD). PMAX >= 1.0D0.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix A to be
block-diagonalized.
On exit, the leading N-by-N upper triangular part of this
array contains the computed block-diagonal matrix, in
Schur form.
The strictly lower triangular part is used as workspace,
but it is set to zero before exit.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
X (input/output) COMPLEX*16 array, dimension (LDX,*)
On entry, if JOBX = 'U', the leading N-by-N part of this
array must contain a given matrix X.
On exit, if JOBX = 'U', the leading N-by-N part of this
array contains the product of the given matrix X and the
transformation matrix that reduced A to block-diagonal
form. The transformation matrix is itself a product of
non-unitary similarity transformations having elements
with magnitude less than or equal to PMAX.
If JOBX = 'N', this array is not referenced.
LDX INTEGER
The leading dimension of array X.
LDX >= 1, if JOBX = 'N';
LDX >= MAX(1,N), if JOBX = 'U'.
NBLCKS (output) INTEGER
The number of diagonal blocks of the matrix A.
BLSIZE (output) INTEGER array, dimension (N)
The first NBLCKS elements of this array contain the orders
of the resulting diagonal blocks of the matrix A.
W (output) COMPLEX*16 array, dimension (N)
This array contains the eigenvalues of the matrix A.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in the ordering of the diagonal
elements of the upper triangular matrix.
If the user sets TOL > 0, then the given value of TOL is
used as an absolute tolerance: an eigenvalue i and a
temporarily fixed eigenvalue 1 (the first element of the
current trailing submatrix to be reduced) are considered
to belong to the same cluster if they satisfy
| lambda_1 - lambda_i | <= TOL.
If the user sets TOL < 0, then the given value of TOL is
used as a relative tolerance: an eigenvalue i and a
temporarily fixed eigenvalue 1 are considered to belong to
the same cluster if they satisfy, for j = 1, ..., N,
| lambda_1 - lambda_i | <= | TOL | * max | lambda_j |.
If the user sets TOL = 0, then an implicitly computed,
default tolerance, defined by TOL = SQRT( SQRT( EPS ) )
is used instead, as a relative tolerance, where EPS is
the machine precision (see LAPACK Library routine DLAMCH).
If SORT = 'N' or 'C', this parameter is not referenced.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Consider first that SORT = 'N'. Let
( A A )
( 11 12 )
A = ( ),
( 0 A )
( 22 )
be the given matrix in Schur form, where initially A is the
11
first diagonal element. An attempt is made to compute a
transformation matrix X of the form
( I P )
X = ( ) (1)
( 0 I )
(partitioned as A), so that
( A 0 )
-1 ( 11 )
X A X = ( ),
( 0 A )
( 22 )
and the elements of P do not exceed the value PMAX in magnitude.
An adaptation of the standard method for solving Sylvester
equations [1], which controls the magnitude of the individual
elements of the computed solution [2], is used to obtain matrix P.
When this attempt failed, a diagonal element of A , closest to
22
the mean of those of A is selected, and moved by unitary
11
similarity transformations in the leading position of A ; the
22
moved diagonal element is then added to the block A , increasing
11
its order by 1. Another attempt is made to compute a suitable
transformation matrix X with the new definitions of the blocks A
11
and A . After a successful transformation matrix X has been
22
obtained, it postmultiplies the current transformation matrix
(if JOBX = 'U'), and the whole procedure is repeated for the
block A .
22
When SORT = 'S', the diagonal elements of the Schur form are
reordered before each step of the reduction, so that each cluster
of eigenvalues, defined as specified in the definition of TOL,
appears in adjacent elements. The elements for each cluster are
merged together, and the procedure described above is applied to
the larger blocks. Using the option SORT = 'S' will usually
provide better efficiency than the standard option (SORT = 'N'),
proposed in [2], because there could be no or few unsuccessful
attempts to compute individual transformation matrices X of the
form (1). However, the resulting dimensions of the blocks are
usually larger; this could make subsequent calculations less
efficient.
When SORT = 'C' or 'B', the procedure is similar to that for
SORT = 'N' or 'S', respectively, but the block of A whose
22
eigenvalue(s) is (are) the closest to those of A (not to their
11
mean) is selected and moved to the leading position of A . This
22
is called the "closest-neighbour" strategy.
References
[1] Bartels, R.H. and Stewart, G.W. T
Solution of the matrix equation A X + XB = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
[2] Bavely, C. and Stewart, G.W.
An Algorithm for Computing Reducing Subspaces by Block
Diagonalization.
SIAM J. Numer. Anal., 16, pp. 359-367, 1979.
[3] Demmel, J.
The Condition Number of Equivalence Transformations that
Block Diagonalize Matrix Pencils.
SIAM J. Numer. Anal., 20, pp. 599-610, 1983.
Numerical Aspects
3 4 The algorithm usually requires 0(N ) operations, but 0(N ) are possible in the worst case, when the matrix cannot be diagonalized by well-conditioned transformations.Further Comments
The individual non-unitary transformation matrices used in the reduction of A to a block-diagonal form have condition numbers of the order PMAX. This does not guarantee that their product is well-conditioned enough. The routine can be easily modified to provide estimates for the condition numbers of the clusters of eigenvalues.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB03SD.html 0000664 0000000 0000000 00000022220 14560147231 0020340 0 ustar 00root root 0000000 0000000
Purpose
To compute the eigenvalues of an N-by-N square-reduced Hamiltonian
matrix
( A' G' )
H' = ( T ). (1)
( Q' -A' )
Here, A' is an N-by-N matrix, and G' and Q' are symmetric N-by-N
matrices. It is assumed without a check that H' is square-
reduced, i.e., that
2 ( A'' G'' )
H' = ( T ) with A'' upper Hessenberg. (2)
( 0 A'' )
T 2
(Equivalently, Q'A'- A' Q' = 0, A'' = A' + G'Q', and for i > j+1,
A''(i,j) = 0.) Ordinarily, H' is the output from SLICOT Library
routine MB04ZD. The eigenvalues of H' are computed as the square
roots of the eigenvalues of A''.
Specification
SUBROUTINE MB03SD( JOBSCL, N, A, LDA, QG, LDQG, WR, WI, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDQG, LDWORK, N
CHARACTER JOBSCL
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), QG(LDQG,*), WI(*), WR(*)
Arguments
Mode Parameters
JOBSCL CHARACTER*1
Specifies whether or not balancing operations should
be performed by the LAPACK subroutine DGEBAL on the
Hessenberg matrix A'' in (2), as follows:
= 'N': do not use balancing;
= 'S': do scaling in order to equilibrate the rows
and columns of A''.
See LAPACK subroutine DGEBAL and Section METHOD below.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, G, and Q. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
upper left block A' of the square-reduced Hamiltonian
matrix H' in (1), as produced by SLICOT Library routine
MB04ZD.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
QG (input) DOUBLE PRECISION array, dimension (LDQG,N+1)
The leading N-by-N lower triangular part of this array
must contain the lower triangle of the lower left
symmetric block Q' of the square-reduced Hamiltonian
matrix H' in (1), and the N-by-N upper triangular part of
the submatrix in the columns 2 to N+1 of this array must
contain the upper triangle of the upper right symmetric
block G' of the square-reduced Hamiltonian matrix H'
in (1), as produced by SLICOT Library routine MB04ZD.
So, if i >= j, then Q'(i,j) is stored in QG(i,j) and
G'(i,j) is stored in QG(j,i+1).
LDQG INTEGER
The leading dimension of the array QG. LDQG >= MAX(1,N).
WR (output) DOUBLE PRECISION array, dimension (N)
WI (output) DOUBLE PRECISION array, dimension (N)
The arrays WR and WI contain the real and imaginary parts,
respectively, of the N eigenvalues of H' with non-negative
real part. The remaining N eigenvalues are the negatives
of these eigenvalues.
Eigenvalues are stored in WR and WI in decreasing order of
magnitude of the real parts, i.e., WR(I) >= WR(I+1).
(In particular, an eigenvalue closest to the imaginary
axis is WR(N)+WI(N)i.)
In addition, eigenvalues with zero real part are sorted in
decreasing order of magnitude of imaginary parts. Note
that non-real eigenvalues with non-zero real part appear
in complex conjugate pairs, but eigenvalues with zero real
part do not, in general, appear in complex conjugate
pairs.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= MAX(1,N*(N+1)).
For good performance, LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, then the i-th argument had an illegal
value;
> 0: if INFO = i, i <= N, then LAPACK subroutine DHSEQR
failed to converge while computing the i-th
eigenvalue.
Method
The routine forms the upper Hessenberg matrix A'' in (2) and calls LAPACK subroutines to calculate its eigenvalues. The eigenvalues of H' are the square roots of the eigenvalues of A''.References
[1] Van Loan, C. F.
A Symplectic Method for Approximating All the Eigenvalues of
a Hamiltonian Matrix.
Linear Algebra and its Applications, 61, pp. 233-251, 1984.
[2] Byers, R.
Hamiltonian and Symplectic Algorithms for the Algebraic
Riccati Equation.
Ph. D. Thesis, Cornell University, Ithaca, NY, January 1983.
[3] Benner, P., Byers, R., and Barth, E.
Fortran 77 Subroutines for Computing the Eigenvalues of
Hamiltonian Matrices. I: The Square-Reduced Method.
ACM Trans. Math. Software, 26, 1, pp. 49-77, 2000.
Numerical Aspects
The algorithm requires (32/3)*N**3 + O(N**2) floating point
operations.
Eigenvalues computed by this subroutine are exact eigenvalues
of a perturbed Hamiltonian matrix H' + E where
|| E || <= c sqrt(eps) || H' ||,
c is a modest constant depending on the dimension N and eps is the
machine precision. Moreover, if the norm of H' and an eigenvalue
are of roughly the same magnitude, the computed eigenvalue is
essentially as accurate as the computed eigenvalue obtained by
traditional methods. See [1] or [2].
Further Comments
NoneExample
Program Text
* MB03SD EXAMPLE PROGRAM TEXT.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA, LDQG
PARAMETER ( LDA = NMAX, LDQG = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX*( NMAX+1 ) )
* .. Local Scalars ..
INTEGER I, INFO, J, N
CHARACTER*1 JOBSCL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), QG(LDQG,NMAX+1),
$ WI(NMAX), WR(NMAX)
* .. External Subroutines ..
EXTERNAL MB03SD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
* NOTE: input must define a square-reduced Hamiltonian matrix.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, JOBSCL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99998 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( QG(J,I+1), I = J,N ), J = 1,N )
READ ( NIN, FMT = * ) ( ( QG(I,J), I = J,N ), J = 1,N )
* Compute the eigenvalues.
CALL MB03SD( JOBSCL, N, A, LDA, QG, LDQG, WR, WI, DWORK,
$ LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
* Show the computed eigenvalues.
WRITE ( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99995 ) WR(I), ' + (', WI(I), ')i'
10 CONTINUE
DO 20 I = N, 1, -1
WRITE ( NOUT, FMT = 99995 ) -WR(I), ' + (', -WI(I), ')i'
20 CONTINUE
END IF
END IF
STOP
*
99999 FORMAT (' MB03SD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (/' N is out of range.',/' N = ',I5)
99997 FORMAT (' INFO on exit from MB03SD = ',I2)
99996 FORMAT (/' The eigenvalues are ')
99995 FORMAT (1X,F8.4,A,F8.4,A)
END
Program Data
MB03SD EXAMPLE PROGRAM DATA 3 S 2.0 0.0 0.0 0.0 1.0 2.0 0.0 -1.0 3.0 1.0 0.0 0.0 2.0 3.0 4.0 -2.0 0.0 0.0 0.0 0.0 0.0Program Results
MB03SD EXAMPLE PROGRAM RESULTS The eigenvalues are 2.0000 + ( 1.0000)i 2.0000 + ( -1.0000)i 1.4142 + ( 0.0000)i -1.4142 + ( 0.0000)i -2.0000 + ( 1.0000)i -2.0000 + ( -1.0000)i
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB03TD.html 0000664 0000000 0000000 00000037052 14560147231 0020352 0 ustar 00root root 0000000 0000000
Purpose
To reorder a matrix X in skew-Hamiltonian Schur form:
[ A G ] T
X = [ T ], G = -G,
[ 0 A ]
or in Hamiltonian Schur form:
[ A G ] T
X = [ T ], G = G,
[ 0 -A ]
where A is in upper quasi-triangular form, so that a selected
cluster of eigenvalues appears in the leading diagonal blocks
of the matrix A (in X) and the leading columns of [ U1; -U2 ] form
an orthonormal basis for the corresponding right invariant
subspace.
If X is skew-Hamiltonian, then each eigenvalue appears twice; one
copy corresponds to the j-th diagonal element and the other to the
(n+j)-th diagonal element of X. The logical array LOWER controls
which copy is to be reordered to the leading part of A.
If X is Hamiltonian then the eigenvalues appear in pairs
(lambda,-lambda); lambda corresponds to the j-th diagonal
element and -lambda to the (n+j)-th diagonal element of X.
The logical array LOWER controls whether lambda or -lambda is to
be reordered to the leading part of A.
The matrix A must be in Schur canonical form (as returned by the
LAPACK routine DHSEQR), that is, block upper triangular with
1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has
its diagonal elements equal and its off-diagonal elements of
opposite sign.
Specification
SUBROUTINE MB03TD( TYP, COMPU, SELECT, LOWER, N, A, LDA, G, LDG,
$ U1, LDU1, U2, LDU2, WR, WI, M, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER COMPU, TYP
INTEGER INFO, LDA, LDG, LDU1, LDU2, LDWORK, M, N
C .. Array Arguments ..
LOGICAL LOWER(*), SELECT(*)
DOUBLE PRECISION A(LDA,*), DWORK(*), G(LDG,*), U1(LDU1,*),
$ U2(LDU2,*), WI(*), WR(*)
Arguments
Mode Parameters
TYP CHARACTER*1
Specifies the type of the input matrix X:
= 'S': X is skew-Hamiltonian;
= 'H': X is Hamiltonian.
COMPU CHARACTER*1
= 'U': update the matrices U1 and U2 containing the
Schur vectors;
= 'N': do not update U1 and U2.
SELECT (input/output) LOGICAL array, dimension (N)
SELECT specifies the eigenvalues in the selected cluster.
To select a real eigenvalue w(j), SELECT(j) must be set
to .TRUE.. To select a complex conjugate pair of
eigenvalues w(j) and w(j+1), corresponding to a 2-by-2
diagonal block, both SELECT(j) and SELECT(j+1) must be set
to .TRUE.; a complex conjugate pair of eigenvalues must be
either both included in the cluster or both excluded.
LOWER (input/output) LOGICAL array, dimension (N)
LOWER controls which copy of a selected eigenvalue is
included in the cluster. If SELECT(j) is set to .TRUE.
for a real eigenvalue w(j); then LOWER(j) must be set to
.TRUE. if the eigenvalue corresponding to the (n+j)-th
diagonal element of X is to be reordered to the leading
part; and LOWER(j) must be set to .FALSE. if the
eigenvalue corresponding to the j-th diagonal element of
X is to be reordered to the leading part. Similarly, for
a complex conjugate pair of eigenvalues w(j) and w(j+1),
both LOWER(j) and LOWER(j+1) must be set to .TRUE. if the
eigenvalues corresponding to the (n+j:n+j+1,n+j:n+j+1)
diagonal block of X are to be reordered to the leading
part.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the upper quasi-triangular matrix A in Schur
canonical form.
On exit, the leading N-by-N part of this array contains
the reordered matrix A, again in Schur canonical form,
with the selected eigenvalues in the diagonal blocks.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
G (input/output) DOUBLE PRECISION array, dimension (LDG,N)
On entry, if TYP = 'S', the leading N-by-N part of this
array must contain the strictly upper triangular part of
the skew-symmetric matrix G. The rest of this array is not
referenced.
On entry, if TYP = 'H', the leading N-by-N part of this
array must contain the upper triangular part of the
symmetric matrix G. The rest of this array is not
referenced.
On exit, if TYP = 'S', the leading N-by-N part of this
array contains the strictly upper triangular part of the
skew-symmetric matrix G, updated by the orthogonal
symplectic transformation which reorders X.
On exit, if TYP = 'H', the leading N-by-N part of this
array contains the upper triangular part of the symmetric
matrix G, updated by the orthogonal symplectic
transformation which reorders X.
LDG INTEGER
The leading dimension of the array G. LDG >= MAX(1,N).
U1 (input/output) DOUBLE PRECISION array, dimension (LDU1,N)
On entry, if COMPU = 'U', the leading N-by-N part of this
array must contain U1, the (1,1) block of an orthogonal
symplectic matrix U = [ U1, U2; -U2, U1 ].
On exit, if COMPU = 'U', the leading N-by-N part of this
array contains the (1,1) block of the matrix U,
postmultiplied by the orthogonal symplectic transformation
which reorders X. The leading M columns of U form an
orthonormal basis for the specified invariant subspace.
If COMPU = 'N', this array is not referenced.
LDU1 INTEGER
The leading dimension of the array U1.
LDU1 >= MAX(1,N), if COMPU = 'U';
LDU1 >= 1, otherwise.
U2 (input/output) DOUBLE PRECISION array, dimension (LDU2,N)
On entry, if COMPU = 'U', the leading N-by-N part of this
array must contain U2, the (1,2) block of an orthogonal
symplectic matrix U = [ U1, U2; -U2, U1 ].
On exit, if COMPU = 'U', the leading N-by-N part of this
array contains the (1,2) block of the matrix U,
postmultiplied by the orthogonal symplectic transformation
which reorders X.
If COMPU = 'N', this array is not referenced.
LDU2 INTEGER
The leading dimension of the array U2.
LDU2 >= MAX(1,N), if COMPU = 'U';
LDU2 >= 1, otherwise.
WR (output) DOUBLE PRECISION array, dimension (N)
WI (output) DOUBLE PRECISION array, dimension (N)
The real and imaginary parts, respectively, of the
reordered eigenvalues of A. The eigenvalues are stored
in the same order as on the diagonal of A, with
WR(i) = A(i,i) and, if A(i:i+1,i:i+1) is a 2-by-2 diagonal
block, WI(i) > 0 and WI(i+1) = -WI(i). Note that if an
eigenvalue is sufficiently ill-conditioned, then its value
may differ significantly from its value before reordering.
M (output) INTEGER
The dimension of the specified invariant subspace.
0 <= M <= N.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -18, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,N).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
= 1: reordering of X failed because some eigenvalue pairs
are too close to separate (the problem is very
ill-conditioned); X may have been partially
reordered, and WR and WI contain the eigenvalues in
the same order as in X.
References
[1] Bai, Z. and Demmel, J.W.
On Swapping Diagonal Blocks in Real Schur Form.
Linear Algebra Appl., 186, pp. 73-95, 1993.
[2] Benner, P., Kressner, D., and Mehrmann, V.
Skew-Hamiltonian and Hamiltonian Eigenvalue Problems: Theory,
Algorithms and Applications. Techn. Report, TU Berlin, 2003.
Further Comments
NoneExample
Program Text
* MB03TD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 100 )
INTEGER LDA, LDG, LDRES, LDU1, LDU2, LDWORK
PARAMETER ( LDA = NMAX, LDG = NMAX, LDRES = NMAX,
$ LDU1 = NMAX, LDU2 = NMAX, LDWORK = 8*NMAX )
* .. Local Scalars ..
CHARACTER*1 COMPU, TYP
INTEGER I, INFO, J, N, M
* .. Local Arrays ..
LOGICAL LOWER(NMAX), SELECT(NMAX)
DOUBLE PRECISION A(LDA, NMAX), DWORK(LDWORK), G(LDG, NMAX),
$ RES(LDRES,NMAX), U1(LDU1,NMAX), U2(LDU2,NMAX),
$ WR(NMAX), WI(NMAX)
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION MA02JD
EXTERNAL LSAME, MA02JD
* .. External Subroutines ..
EXTERNAL MB03TD
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, TYP, COMPU
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( SELECT(J), J = 1,N )
READ ( NIN, FMT = * ) ( LOWER(J), J = 1,N )
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( G(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( COMPU, 'U' ) ) THEN
READ ( NIN, FMT = * ) ( ( U1(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( U2(I,J), J = 1,N ), I = 1,N )
END IF
CALL MB03TD( TYP, COMPU, SELECT, LOWER, N, A, LDA, G, LDG, U1,
$ LDU1, U2, LDU2, WR, WI, M, DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( LSAME( COMPU, 'U' ) ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99994 )
$ ( U1(I,J), J = 1,N ), ( U2(I,J), J = 1,N )
10 CONTINUE
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99994 )
$ ( -U2(I,J), J = 1,N ), ( U1(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99992 ) MA02JD( .FALSE., .FALSE., N,
$ U1, LDU1, U2, LDU2, RES, LDRES )
END IF
*
WRITE ( NOUT, FMT = 99996 )
DO 30 I = 1, N
WRITE ( NOUT, FMT = 99994 ) ( A(I,J), J = 1,N )
30 CONTINUE
*
WRITE ( NOUT, FMT = 99995 )
IF ( LSAME( TYP, 'S' ) ) THEN
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99994 )
$ ( -G(J,I), J = 1,I-1 ), ZERO, ( G(I,J), J = I+1,N )
40 CONTINUE
ELSE
DO 50 I = 1, N
WRITE ( NOUT, FMT = 99994 )
$ ( G(J,I), J = 1,I-1 ), ( G(I,J), J = I,N )
50 CONTINUE
END IF
END IF
END IF
*
99999 FORMAT (' MB03TD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB03TD = ',I2)
99997 FORMAT (' The orthogonal symplectic factor U is ')
99996 FORMAT (/' The matrix A in reordered Schur canonical form is ')
99995 FORMAT (/' The matrix G is ')
99994 FORMAT (20(1X,F9.4))
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' Orthogonality of U: || U''*U - I ||_F = ',G7.2)
END
Program Data
MB03TD EXAMPLE PROGRAM DATA
5 S U
.F. .T. .T. .F. .F.
.F. .T. .T. .F. .F.
0.9501 0.7621 0.6154 0.4057 0.0579
0 0.4565 0.7919 0.9355 0.3529
0 -0.6822 0.4565 0.9169 0.8132
0 0 0 0.4103 0.0099
0 0 0 0 0.1389
0 -0.1834 -0.1851 0.5659 0.3040
0 0 0.4011 -0.9122 0.2435
0 0 0 0.4786 -0.2432
0 0 0 0 -0.5272
0 0 0 0 0
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
Program Results
MB03TD EXAMPLE PROGRAM RESULTS The orthogonal symplectic factor U is 0.0407 0.4847 0.8737 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1245 -0.3866 0.2087 0.4509 -0.1047 0.3229 0.1248 -0.0843 0.1967 0.6415 -0.0933 0.4089 -0.2225 -0.4085 0.0709 -0.2171 0.2156 -0.1095 0.4348 0.5551 -0.1059 -0.5250 0.2962 -0.0295 0.2207 -0.6789 0.1133 -0.0312 0.2979 -0.1112 0.3937 0.3071 -0.1887 0.5332 -0.4351 -0.4423 0.0600 -0.0127 0.1679 -0.1179 0.0000 0.0000 0.0000 0.0000 0.0000 0.0407 0.4847 0.8737 0.0000 0.0000 -0.3229 -0.1248 0.0843 -0.1967 -0.6415 0.1245 -0.3866 0.2087 0.4509 -0.1047 0.2171 -0.2156 0.1095 -0.4348 -0.5551 -0.0933 0.4089 -0.2225 -0.4085 0.0709 0.6789 -0.1133 0.0312 -0.2979 0.1112 -0.1059 -0.5250 0.2962 -0.0295 0.2207 0.4423 -0.0600 0.0127 -0.1679 0.1179 0.3937 0.3071 -0.1887 0.5332 -0.4351 Orthogonality of U: || U'*U - I ||_F = .21E-14 The matrix A in reordered Schur canonical form is 0.4565 -0.4554 0.2756 -0.8651 -1.2050 1.1863 0.4565 0.2186 -0.0233 0.8293 0.0000 0.0000 0.9501 0.0625 -0.0064 0.0000 0.0000 0.0000 0.4103 0.5597 0.0000 0.0000 0.0000 0.0000 0.1389 The matrix G is 0.0000 0.3298 -0.0292 -0.1571 0.1751 -0.3298 0.0000 -0.0633 -0.2951 0.2396 0.0292 0.0633 0.0000 0.9567 0.7485 0.1571 0.2951 -0.9567 0.0000 0.2960 -0.1751 -0.2396 -0.7485 -0.2960 0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB03TS.html 0000664 0000000 0000000 00000014314 14560147231 0020365 0 ustar 00root root 0000000 0000000
Purpose
To swap diagonal blocks A11 and A22 of order 1 or 2 in the upper
quasi-triangular matrix A contained in a skew-Hamiltonian matrix
[ A G ] T
X = [ T ], G = -G,
[ 0 A ]
or in a Hamiltonian matrix
[ A G ] T
X = [ T ], G = G.
[ 0 -A ]
This routine is a modified version of the LAPACK subroutine
DLAEX2.
The matrix A must be in Schur canonical form (as returned by the
LAPACK routine DHSEQR), that is, block upper triangular with
1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has
its diagonal elements equal and its off-diagonal elements of
opposite sign.
Specification
SUBROUTINE MB03TS( ISHAM, WANTU, N, A, LDA, G, LDG, U1, LDU1, U2,
$ LDU2, J1, N1, N2, DWORK, INFO )
C .. Scalar Arguments ..
LOGICAL ISHAM, WANTU
INTEGER INFO, J1, LDA, LDG, LDU1, LDU2, N, N1, N2
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), G(LDG,*), U1(LDU1,*),
$ U2(LDU2,*)
Arguments
Mode Parameters
ISHAM LOGIGAL
Specifies the type of X:
= .TRUE.: X is a Hamiltonian matrix;
= .FALSE.: X is a skew-Hamiltonian matrix.
WANTU LOGIGAL
= .TRUE.: update the matrices U1 and U2 containing the
Schur vectors;
= .FALSE.: do not update U1 and U2.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the upper quasi-triangular matrix A, in Schur
canonical form.
On exit, the leading N-by-N part of this array contains
the reordered matrix A, again in Schur canonical form.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
G (input/output) DOUBLE PRECISION array, dimension (LDG,N)
On entry, the leading N-by-N part of this array must
contain the upper triangular part of the symmetric
matrix G, if ISHAM = .TRUE., or the strictly upper
triangular part of the skew-symmetric matrix G, otherwise.
The rest of this array is not referenced.
On exit, the leading N-by-N part of this array contains
the upper or strictly upper triangular part of the
symmetric or skew-symmetric matrix G, respectively,
updated by the orthogonal transformation which reorders A.
LDG INTEGER
The leading dimension of the array G. LDG >= MAX(1,N).
U1 (input/output) DOUBLE PRECISION array, dimension (LDU1,N)
On entry, if WANTU = .TRUE., the leading N-by-N part of
this array must contain the matrix U1.
On exit, if WANTU = .TRUE., the leading N-by-N part of
this array contains U1, postmultiplied by the orthogonal
transformation which reorders A. See the description in
the SLICOT subroutine MB03TD for further details.
If WANTU = .FALSE., this array is not referenced.
LDU1 INTEGER
The leading dimension of the array U1.
LDU1 >= MAX(1,N), if WANTU = .TRUE.;
LDU1 >= 1, otherwise.
U2 (input/output) DOUBLE PRECISION array, dimension (LDU2,N)
On entry, if WANTU = .TRUE., the leading N-by-N part of
this array must contain the matrix U2.
On exit, if WANTU = .TRUE., the leading N-by-N part of
this array contains U2, postmultiplied by the orthogonal
transformation which reorders A.
If WANTU = .FALSE., this array is not referenced.
LDU2 INTEGER
The leading dimension of the array U2.
LDU2 >= MAX(1,N), if WANTU = .TRUE.;
LDU2 >= 1, otherwise.
J1 (input) INTEGER
The index of the first row of the first block A11.
If J1+N1 < N, then A11 is swapped with the block starting
at (J1+N1+1)-th diagonal element.
If J1+N1 > N, then A11 is the last block in A and swapped
with -A11', if ISHAM = .TRUE.,
or A11', if ISHAM = .FALSE..
N1 (input) INTEGER
The order of the first block A11. N1 = 0, 1 or 2.
N2 (input) INTEGER
The order of the second block A22. N2 = 0, 1 or 2.
Workspace
DWORK DOUBLE PRECISION array, dimension (N)Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: the transformed matrix A would be too far from Schur
form; the blocks are not swapped and A, G, U1 and
U2 are unchanged.
References
[1] Bai, Z., and Demmel, J.W.
On swapping diagonal blocks in real Schur form.
Linear Algebra Appl., 186, pp. 73-95, 1993.
[2] Benner, P., Kressner, D., and Mehrmann, V.
Skew-Hamiltonian and Hamiltonian Eigenvalue Problems: Theory,
Algorithms and Applications. Techn. Report, TU Berlin, 2003.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute all, or part, of the singular value decomposition of a
real upper triangular matrix.
The N-by-N upper triangular matrix A is factored as A = Q*S*P',
where Q and P are N-by-N orthogonal matrices and S is an
N-by-N diagonal matrix with non-negative diagonal elements,
SV(1), SV(2), ..., SV(N), ordered such that
SV(1) >= SV(2) >= ... >= SV(N) >= 0.
The columns of Q are the left singular vectors of A, the diagonal
elements of S are the singular values of A and the columns of P
are the right singular vectors of A.
Either or both of Q and P' may be requested.
When P' is computed, it is returned in A.
Specification
SUBROUTINE MB03UD( JOBQ, JOBP, N, A, LDA, Q, LDQ, SV, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBP, JOBQ
INTEGER INFO, LDA, LDQ, LDWORK, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), Q(LDQ,*), SV(*)
Arguments
Mode Parameters
JOBQ CHARACTER*1
Specifies whether the user wishes to compute the matrix Q
of left singular vectors as follows:
= 'V': Left singular vectors are computed;
= 'N': No left singular vectors are computed.
JOBP CHARACTER*1
Specifies whether the user wishes to compute the matrix P'
of right singular vectors as follows:
= 'V': Right singular vectors are computed;
= 'N': No right singular vectors are computed.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix A.
On exit, if JOBP = 'V', the leading N-by-N part of this
array contains the N-by-N orthogonal matrix P'; otherwise
the N-by-N upper triangular part of A is used as internal
workspace. The strictly lower triangular part of A is set
internally to zero before the reduction to bidiagonal form
is performed.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
Q (output) DOUBLE PRECISION array, dimension (LDQ,N)
If JOBQ = 'V', the leading N-by-N part of this array
contains the orthogonal matrix Q.
If JOBQ = 'N', Q is not referenced.
LDQ INTEGER
The leading dimension of array Q.
LDQ >= 1, and when JOBQ = 'V', LDQ >= MAX(1,N).
SV (output) DOUBLE PRECISION array, dimension (N)
The N singular values of the matrix A, sorted in
descending order.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK;
if INFO > 0, DWORK(2:N) contains the unconverged
superdiagonal elements of an upper bidiagonal matrix B
whose diagonal is in SV (not necessarily sorted).
B satisfies A = Q*B*P', so it has the same singular
values as A, and singular vectors related by Q and P'.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,5*N).
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: the QR algorithm has failed to converge. In this
case INFO specifies how many superdiagonals did not
converge (see the description of DWORK).
This failure is not likely to occur.
Method
The routine reduces A to bidiagonal form by means of elementary reflectors and then uses the QR algorithm on the bidiagonal form.Further Comments
NoneExample
Program Text
* MB03UD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 10 )
INTEGER LDA, LDQ
PARAMETER ( LDA = NMAX, LDQ = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( 1, 5*NMAX ) )
* .. Local Scalars ..
CHARACTER*1 JOBQ, JOBP
INTEGER I, INFO, J, N
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), Q(LDQ,NMAX),
$ SV(NMAX)
* .. External Functions ..
LOGICAL LSAME
* .. External Subroutines ..
EXTERNAL MB03UD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, JOBQ, JOBP
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
* Compute the singular values and vectors.
CALL MB03UD( JOBQ, JOBP, N, A, LDA, Q, LDQ, SV, DWORK,
$ LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
WRITE ( NOUT, FMT = 99995 ) ( SV(I), I = 1,N )
IF ( LSAME( JOBP, 'V' ) ) THEN
WRITE ( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
10 CONTINUE
END IF
IF ( LSAME( JOBQ, 'V' ) ) THEN
WRITE ( NOUT, FMT = 99994 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,N )
20 CONTINUE
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' MB03UD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB03UD = ',I2)
99997 FORMAT (' Singular values are ',I5)
99996 FORMAT (/' The transpose of the right singular vectors matrix is '
$ )
99995 FORMAT (8X,20(1X,F8.4))
99994 FORMAT (/' The left singular vectors matrix is ')
99993 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
MB03UD EXAMPLE PROGRAM DATA 4 V V -1.0 37.0 -12.0 -12.0 0.0 -10.0 0.0 4.0 0.0 0.0 7.0 -6.0 0.0 0.0 0.0 -9.0Program Results
MB03UD EXAMPLE PROGRAM RESULTS
Singular values are
42.0909 11.7764 5.4420 0.2336
The transpose of the right singular vectors matrix is
0.0230 -0.9084 0.2759 0.3132
0.0075 -0.1272 0.5312 -0.8376
0.0092 0.3978 0.8009 0.4476
0.9997 0.0182 -0.0177 -0.0050
The left singular vectors matrix is
-0.9671 -0.0882 -0.0501 -0.2335
0.2456 -0.1765 -0.4020 -0.8643
0.0012 0.7425 0.5367 -0.4008
-0.0670 0.6401 -0.7402 0.1945
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB03VD.html 0000664 0000000 0000000 00000032152 14560147231 0020350 0 ustar 00root root 0000000 0000000
Purpose
To reduce a product of p real general matrices A = A_1*A_2*...*A_p
to upper Hessenberg form, H = H_1*H_2*...*H_p, where H_1 is
upper Hessenberg, and H_2, ..., H_p are upper triangular, by using
orthogonal similarity transformations on A,
Q_1' * A_1 * Q_2 = H_1,
Q_2' * A_2 * Q_3 = H_2,
...
Q_p' * A_p * Q_1 = H_p.
Specification
SUBROUTINE MB03VD( N, P, ILO, IHI, A, LDA1, LDA2, TAU, LDTAU,
$ DWORK, INFO )
C .. Scalar Arguments ..
INTEGER IHI, ILO, INFO, LDA1, LDA2, LDTAU, N, P
C .. Array Arguments ..
DOUBLE PRECISION A( LDA1, LDA2, * ), DWORK( * ), TAU( LDTAU, * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the square matrices A_1, A_2, ..., A_p.
N >= 0.
P (input) INTEGER
The number of matrices in the product A_1*A_2*...*A_p.
P >= 1.
ILO (input) INTEGER
IHI (input) INTEGER
It is assumed that all matrices A_j, j = 2, ..., p, are
already upper triangular in rows and columns 1:ILO-1 and
IHI+1:N, and A_1 is upper Hessenberg in rows and columns
1:ILO-1 and IHI+1:N, with A_1(ILO,ILO-1) = 0 (unless
ILO = 1), and A_1(IHI+1,IHI) = 0 (unless IHI = N).
If this is not the case, ILO and IHI should be set to 1
and N, respectively.
1 <= ILO <= max(1,N); min(ILO,N) <= IHI <= N.
A (input/output) DOUBLE PRECISION array, dimension
(LDA1,LDA2,P)
On entry, the leading N-by-N-by-P part of this array must
contain the matrices of factors to be reduced;
specifically, A(*,*,j) must contain A_j, j = 1, ..., p.
On exit, the leading N-by-N upper triangle and the first
subdiagonal of A(*,*,1) contain the upper Hessenberg
matrix H_1, and the elements below the first subdiagonal,
with the first column of the array TAU represent the
orthogonal matrix Q_1 as a product of elementary
reflectors. See FURTHER COMMENTS.
For j > 1, the leading N-by-N upper triangle of A(*,*,j)
contains the upper triangular matrix H_j, and the elements
below the diagonal, with the j-th column of the array TAU
represent the orthogonal matrix Q_j as a product of
elementary reflectors. See FURTHER COMMENTS.
LDA1 INTEGER
The first leading dimension of the array A.
LDA1 >= max(1,N).
LDA2 INTEGER
The second leading dimension of the array A.
LDA2 >= max(1,N).
TAU (output) DOUBLE PRECISION array, dimension (LDTAU,P)
The leading N-1 elements in the j-th column contain the
scalar factors of the elementary reflectors used to form
the matrix Q_j, j = 1, ..., P. See FURTHER COMMENTS.
LDTAU INTEGER
The leading dimension of the array TAU.
LDTAU >= max(1,N-1).
Workspace
DWORK DOUBLE PRECISION array, dimension (N)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The algorithm consists in ihi-ilo major steps. In each such step i, ilo <= i <= ihi-1, the subdiagonal elements in the i-th column of A_j are annihilated using a Householder transformation from the left, which is also applied to A_(j-1) from the right, for j = p:-1:2. Then, the elements below the subdiagonal of the i-th column of A_1 are annihilated, and the Householder transformation is also applied to A_p from the right. See FURTHER COMMENTS.References
[1] Bojanczyk, A.W., Golub, G. and Van Dooren, P.
The periodic Schur decomposition: algorithms and applications.
Proc. of the SPIE Conference (F.T. Luk, Ed.), 1770, pp. 31-42,
1992.
[2] Sreedhar, J. and Van Dooren, P.
Periodic Schur form and some matrix equations.
Proc. of the Symposium on the Mathematical Theory of Networks
and Systems (MTNS'93), Regensburg, Germany (U. Helmke,
R. Mennicken and J. Saurer, Eds.), Vol. 1, pp. 339-362, 1994.
Numerical Aspects
The algorithm is numerically stable.Further Comments
Each matrix Q_j is represented as a product of (ihi-ilo)
elementary reflectors,
Q_j = H_j(ilo) H_j(ilo+1) . . . H_j(ihi-1).
Each H_j(i), i = ilo, ..., ihi-1, has the form
H_j(i) = I - tau_j * v_j * v_j',
where tau_j is a real scalar, and v_j is a real vector with
v_j(1:i) = 0, v_j(i+1) = 1 and v_j(ihi+1:n) = 0; v_j(i+2:ihi)
is stored on exit in A_j(i+2:ihi,i), and tau_j in TAU(i,j).
The contents of A_1 are illustrated by the following example
for n = 7, ilo = 2, and ihi = 6:
on entry on exit
( a a a a a a a ) ( a h h h h h a )
( 0 a a a a a a ) ( 0 h h h h h a )
( 0 a a a a a a ) ( 0 h h h h h h )
( 0 a a a a a a ) ( 0 v2 h h h h h )
( 0 a a a a a a ) ( 0 v2 v3 h h h h )
( 0 a a a a a a ) ( 0 v2 v3 v4 h h h )
( 0 0 0 0 0 0 a ) ( 0 0 0 0 0 0 a )
where a denotes an element of the original matrix A_1, h denotes
a modified element of the upper Hessenberg matrix H_1, and vi
denotes an element of the vector defining H_1(i).
The contents of A_j, j > 1, are illustrated by the following
example for n = 7, ilo = 2, and ihi = 6:
on entry on exit
( a a a a a a a ) ( a h h h h h a )
( 0 a a a a a a ) ( 0 h h h h h h )
( 0 a a a a a a ) ( 0 v2 h h h h h )
( 0 a a a a a a ) ( 0 v2 v3 h h h h )
( 0 a a a a a a ) ( 0 v2 v3 v4 h h h )
( 0 a a a a a a ) ( 0 v2 v3 v4 v5 h h )
( 0 0 0 0 0 0 a ) ( 0 0 0 0 0 0 a )
where a denotes an element of the original matrix A_j, h denotes
a modified element of the upper triangular matrix H_j, and vi
denotes an element of the vector defining H_j(i). (The element
(1,2) in A_p is also unchanged for this example.)
Note that for P = 1, the LAPACK Library routine DGEHRD could be
more efficient on some computer architectures than this routine
(a BLAS 2 version).
Example
Program Text
* MB03VD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, PMAX
PARAMETER ( NMAX = 20, PMAX = 20 )
INTEGER LDA1, LDA2, LDQ1, LDQ2, LDTAU
PARAMETER ( LDA1 = NMAX, LDA2 = NMAX, LDQ1 = NMAX,
$ LDQ2 = NMAX, LDTAU = NMAX-1 )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* .. Local Scalars ..
DOUBLE PRECISION SSQ
INTEGER I, IHI, ILO, INFO, J, K, KP1, N, P
* .. Local Arrays ..
DOUBLE PRECISION A(LDA1,LDA2,PMAX), AS(LDA1,LDA2,PMAX),
$ DWORK(LDWORK), Q(LDQ1,LDQ2,PMAX),
$ QTA(LDQ1,NMAX), TAU(LDTAU,PMAX)
* .. External Functions ..
DOUBLE PRECISION DLANGE, DLAPY2
EXTERNAL DLANGE, DLAPY2
* .. External Subroutines ..
EXTERNAL DGEMM, DLACPY, DLASET, MB03VD, MB03VY
* .. Intrinsic Functions ..
INTRINSIC MIN
* .. Executable Statements ..
WRITE (NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, P, ILO, IHI
IF ( N.LT.0 .OR. N.GT.MIN( LDA1, LDA2 ) ) THEN
WRITE ( NOUT, FMT = 99991 ) N
ELSE
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) P
ELSE
* Read matrices A_1, ..., A_p from the input file.
DO 10 K = 1, P
READ ( NIN, FMT = * )
$ ( ( A(I,J,K), J = 1, N ), I = 1, N )
CALL DLACPY( 'F', N, N, A(1,1,K), LDA1, AS(1,1,K), LDA1 )
10 CONTINUE
* Reduce to the periodic Hessenberg form.
CALL MB03VD( N, P, ILO, IHI, A, LDA1, LDA2, TAU, LDTAU,
$ DWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99996 )
DO 30 K = 1, P
CALL DLACPY( 'L', N, N, A(1,1,K), LDA1, Q(1,1,K),
$ LDQ1 )
IF ( N.GT.1 ) THEN
IF ( N.GT.2 .AND. K.EQ.1 ) THEN
CALL DLASET( 'L', N-2, N-2, ZERO, ZERO,
$ A(3,1,K), LDA1 )
ELSE IF ( K.GT.1 ) THEN
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ A(2,1,K), LDA1 )
END IF
END IF
WRITE ( NOUT, FMT = 99995 ) K
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99994 ) ( A(I,J,K), J = 1, N )
20 CONTINUE
30 CONTINUE
* Accumulate the transformations.
CALL MB03VY( N, P, ILO, IHI, Q, LDQ1, LDQ2, TAU, LDTAU,
$ DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99993 )
DO 50 K = 1, P
WRITE ( NOUT, FMT = 99995 ) K
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99994 )
$ ( Q(I,J,K), J = 1, N )
40 CONTINUE
50 CONTINUE
* Compute error.
SSQ = ZERO
DO 60 K = 1, P
KP1 = K+1
IF( KP1.GT.P ) KP1 = 1
* Compute NORM (Z' * A * Z - Aout)
CALL DGEMM( 'T', 'N', N, N, N, ONE, Q(1,1,K), LDQ1,
$ AS(1,1,K), LDA1, ZERO, QTA, LDQ1 )
CALL DGEMM( 'N', 'N', N, N, N, ONE, QTA, LDQ1,
$ Q(1,1,KP1), LDQ1, -ONE, A(1,1,K),
$ LDA1 )
SSQ = DLAPY2( SSQ,
$ DLANGE( 'Frobenius', N, N, A(1,1,K),
$ LDA1, DWORK ) )
60 CONTINUE
WRITE ( NOUT, FMT = 99992 ) SSQ
END IF
END IF
END IF
END IF
STOP
99999 FORMAT (' MB03VD EXAMPLE PROGRAM RESULTS', /1X)
99998 FORMAT (' INFO on exit from MB03VD = ', I2)
99997 FORMAT (' INFO on exit from MB03VY = ', I2)
99996 FORMAT (' Reduced matrices')
99995 FORMAT (/' K = ', I5)
99994 FORMAT (8F8.4)
99993 FORMAT (/' Transformation matrices')
99992 FORMAT (/,' NORM (Q''*A*Q - Aout) = ', 1PD12.5)
99991 FORMAT (/, ' N is out of range.',/' N = ', I5)
99990 FORMAT (/, ' P is out of range.',/' P = ', I5)
END
Program Data
MB03VD EXAMPLE PROGRAM DATA 4 2 1 4 1.5 -.7 3.5 -.7 1. 0. 2. 3. 1.5 -.7 2.5 -.3 1. 0. 2. 1. 1.5 -.7 3.5 -.7 1. 0. 2. 3. 1.5 -.7 2.5 -.3 1. 0. 2. 1.Program Results
MB03VD EXAMPLE PROGRAM RESULTS Reduced matrices K = 1 -2.3926 2.7042 -0.9598 -1.2335 4.1417 -1.7046 1.3001 -1.3120 0.0000 -1.6247 -0.2534 1.6453 0.0000 0.0000 -0.0169 -0.4451 K = 2 -2.5495 2.3402 4.7021 0.2329 0.0000 1.9725 -0.2483 -2.3493 0.0000 0.0000 -0.6290 -0.5975 0.0000 0.0000 0.0000 -0.4426 Transformation matrices K = 1 1.0000 0.0000 0.0000 0.0000 0.0000 -0.7103 0.5504 -0.4388 0.0000 -0.4735 -0.8349 -0.2807 0.0000 -0.5209 0.0084 0.8536 K = 2 -0.5883 0.2947 0.7528 -0.0145 -0.3922 -0.8070 0.0009 -0.4415 -0.5883 0.4292 -0.6329 -0.2630 -0.3922 -0.2788 -0.1809 0.8577 NORM (Q'*A*Q - Aout) = 2.93760D-15
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB03VW.html 0000664 0000000 0000000 00000020674 14560147231 0020401 0 ustar 00root root 0000000 0000000
Purpose
To reduce the generalized matrix product
S(1) S(2) S(K)
A(:,:,1) * A(:,:,2) * ... * A(:,:,K)
to upper Hessenberg-triangular form, where A is N-by-N-by-K and S
is the signature array with values 1 or -1. The H-th matrix of A
is reduced to upper Hessenberg form while the other matrices are
triangularized. Unblocked version.
If COMPQ = 'U' or COMPZ = 'I', then the orthogonal factors are
computed and stored in the array Q so that for S(I) = 1,
T
Q(:,:,I)(in) A(:,:,I)(in) Q(:,:,MOD(I,K)+1)(in)
T (1)
= Q(:,:,I)(out) A(:,:,I)(out) Q(:,:,MOD(I,K)+1)(out) ,
and for S(I) = -1,
T
Q(:,:,MOD(I,K)+1)(in) A(:,:,I)(in) Q(:,:,I)(in)
T (2)
= Q(:,:,MOD(I,K)+1)(out) A(:,:,I)(out) Q(:,:,I)(out) .
A partial generation of the orthogonal factors can be realized via
the array QIND.
Specification
SUBROUTINE MB03VW( COMPQ, QIND, TRIU, N, K, H, ILO, IHI, S, A,
$ LDA1, LDA2, Q, LDQ1, LDQ2, IWORK, LIWORK,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, TRIU
INTEGER H, IHI, ILO, INFO, K, LDA1, LDA2, LDQ1, LDQ2,
$ LDWORK, LIWORK, N
C .. Array Arguments ..
INTEGER IWORK(*), QIND(*), S(*)
DOUBLE PRECISION A(LDA1,LDA2,*), DWORK(LDWORK), Q(LDQ1,LDQ2,*)
Arguments
Mode Parameters
COMPQ CHARACTER*1
Specifies whether or not the orthogonal transformations
should be accumulated in the array Q, as follows:
= 'N': do not modify Q;
= 'U': modify (update) the array Q by the orthogonal
transformations that are applied to the matrices in
the array A to reduce them to periodic Hessenberg-
triangular form;
= 'I': like COMPQ = 'U', except that each matrix in the
array Q will be first initialized to the identity
matrix;
= 'P': use the parameters as encoded in QIND.
QIND INTEGER array, dimension (K)
If COMPQ = 'P', then this array describes the generation
of the orthogonal factors as follows:
If QIND(I) > 0, then the array Q(:,:,QIND(I)) is
modified by the transformations corresponding to the
i-th orthogonal factor in (1) and (2).
If QIND(I) < 0, then the array Q(:,:,-QIND(I)) is
initialized to the identity and modified by the
transformations corresponding to the i-th orthogonal
factor in (1) and (2).
If QIND(I) = 0, then the transformations corresponding
to the i-th orthogonal factor in (1), (2) are not applied.
TRIU CHARACTER*1
Indicates how many matrices are reduced to upper
triangular form in the first stage of the algorithm,
as follows
= 'N': only matrices with negative signature;
= 'A': all possible N - 1 matrices.
The first choice minimizes the computational costs of the
algorithm, whereas the second is more cache efficient and
therefore faster on modern architectures.
Input/Output Parameters
N (input) INTEGER
The order of each factor in the array A. N >= 0.
K (input) INTEGER
The number of factors. K >= 0.
H (input/output) INTEGER
On entry, if H is in the interval [1,K] then the H-th
factor of A will be transformed to upper Hessenberg form.
Otherwise the most efficient H is chosen.
On exit, H indicates the factor of A which is in upper
Hessenberg form.
ILO (input) INTEGER
IHI (input) INTEGER
It is assumed that each factor in A is already upper
triangular in rows and columns 1:ILO-1 and IHI+1:N.
1 <= ILO <= IHI <= N, if N > 0;
ILO = 1 and IHI = 0, if N = 0.
If ILO = IHI, all factors are upper triangular.
S (input) INTEGER array, dimension (K)
The leading K elements of this array must contain the
signatures of the factors. Each entry in S must be either
1 or -1.
A (input/output) DOUBLE PRECISION array, dimension
(LDA1,LDA2,K)
On entry, the leading N-by-N-by-K part of this array must
contain the factors of the general product to be reduced.
On exit, A(:,:,H) is overwritten by an upper Hessenberg
matrix and each A(:,:,I), for I not equal to H, is
overwritten by an upper triangular matrix.
LDA1 INTEGER
The first leading dimension of the array A.
LDA1 >= MAX(1,N).
LDA2 INTEGER
The second leading dimension of the array A.
LDA2 >= MAX(1,N).
Q (input/output) DOUBLE PRECISION array, dimension
(LDQ1,LDQ2,K)
On entry, if COMPQ = 'U', the leading N-by-N-by-K part
of this array must contain the initial orthogonal factors
as described in (1) and (2).
On entry, if COMPQ = 'P', only parts of the leading
N-by-N-by-K part of this array must contain some
orthogonal factors as described by the parameters QIND.
If COMPQ = 'I', this array should not be set on entry.
On exit, if COMPQ = 'U' or COMPQ = 'I', the leading
N-by-N-by-K part of this array contains the modified
orthogonal factors as described in (1) and (2).
On exit, if COMPQ = 'P', only parts of the leading
N-by-N-by-K part contain some modified orthogonal factors
as described by the parameters QIND.
This array is not referenced if COMPQ = 'N'.
LDQ1 INTEGER
The first leading dimension of the array Q. LDQ1 >= 1,
and, if COMPQ <> 'N', LDQ1 >= MAX(1,N).
LDQ2 INTEGER
The second leading dimension of the array Q. LDQ2 >= 1,
and, if COMPQ <> 'N', LDQ2 >= MAX(1,N).
Workspace
IWORK INTEGER array, dimension (LIWORK)
On exit, if INFO = -17, IWORK(1) returns the needed
value of LIWORK.
LIWORK INTEGER
The length of the array IWORK. LIWORK >= MAX(1,3*K).
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
On exit, if INFO = -19, DWORK(1) returns the minimum
value of LIWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 1, if MIN(N,K) = 0, or N = 1 or ILO = IHI;
LDWORK >= M+MAX(IHI,N-ILO+1)), otherwise, where
M = IHI-ILO+1.
For optimum performance LDWORK should be larger.
If LDWORK = -1 a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB03VY.html 0000664 0000000 0000000 00000011461 14560147231 0020375 0 ustar 00root root 0000000 0000000
Purpose
To generate the real orthogonal matrices Q_1, Q_2, ..., Q_p,
which are defined as the product of ihi-ilo elementary reflectors
of order n, as returned by SLICOT Library routine MB03VD:
Q_j = H_j(ilo) H_j(ilo+1) . . . H_j(ihi-1).
Specification
SUBROUTINE MB03VY( N, P, ILO, IHI, A, LDA1, LDA2, TAU, LDTAU,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER IHI, ILO, INFO, LDA1, LDA2, LDTAU, LDWORK, N, P
C .. Array Arguments ..
DOUBLE PRECISION A( LDA1, LDA2, * ), DWORK( * ), TAU( LDTAU, * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrices Q_1, Q_2, ..., Q_p. N >= 0.
P (input) INTEGER
The number p of transformation matrices. P >= 1.
ILO (input) INTEGER
IHI (input) INTEGER
The values of the indices ilo and ihi, respectively, used
in the previous call of the SLICOT Library routine MB03VD.
1 <= ILO <= max(1,N); min(ILO,N) <= IHI <= N.
A (input/output) DOUBLE PRECISION array, dimension
(LDA1,LDA2,N)
On entry, the leading N-by-N strictly lower triangular
part of A(*,*,j) must contain the vectors which define the
elementary reflectors used for reducing A_j, as returned
by SLICOT Library routine MB03VD, j = 1, ..., p.
On exit, the leading N-by-N part of A(*,*,j) contains the
N-by-N orthogonal matrix Q_j, j = 1, ..., p.
LDA1 INTEGER
The first leading dimension of the array A.
LDA1 >= max(1,N).
LDA2 INTEGER
The second leading dimension of the array A.
LDA2 >= max(1,N).
TAU (input) DOUBLE PRECISION array, dimension (LDTAU,P)
The leading N-1 elements in the j-th column must contain
the scalar factors of the elementary reflectors used to
form the matrix Q_j, as returned by SLICOT Library routine
MB03VD.
LDTAU INTEGER
The leading dimension of the array TAU.
LDTAU >= max(1,N-1).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,N).
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Each matrix Q_j is generated as the product of the elementary reflectors used for reducing A_j. Standard LAPACK routines for Hessenberg and QR decompositions are used.References
[1] Bojanczyk, A.W., Golub, G. and Van Dooren, P.
The periodic Schur decomposition: algorithms and applications.
Proc. of the SPIE Conference (F.T. Luk, Ed.), 1770, pp. 31-42,
1992.
[2] Sreedhar, J. and Van Dooren, P.
Periodic Schur form and some matrix equations.
Proc. of the Symposium on the Mathematical Theory of Networks
and Systems (MTNS'93), Regensburg, Germany (U. Helmke,
R. Mennicken and J. Saurer, Eds.), Vol. 1, pp. 339-362, 1994.
Numerical Aspects
The algorithm is numerically stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To swap adjacent diagonal blocks A11*B11 and A22*B22 of size
1-by-1 or 2-by-2 in an upper (quasi) triangular matrix product
A*B by an orthogonal equivalence transformation.
(A, B) must be in periodic real Schur canonical form (as returned
by SLICOT Library routine MB03XP), i.e., A is block upper
triangular with 1-by-1 and 2-by-2 diagonal blocks, and B is upper
triangular.
Optionally, the matrices Q and Z of generalized Schur vectors are
updated.
Q(in) * A(in) * Z(in)' = Q(out) * A(out) * Z(out)',
Z(in) * B(in) * Q(in)' = Z(out) * B(out) * Q(out)'.
This routine is largely based on the LAPACK routine DTGEX2
developed by Bo Kagstrom and Peter Poromaa.
Specification
SUBROUTINE MB03WA( WANTQ, WANTZ, N1, N2, A, LDA, B, LDB, Q, LDQ,
$ Z, LDZ, INFO )
C .. Scalar Arguments ..
LOGICAL WANTQ, WANTZ
INTEGER INFO, LDA, LDB, LDQ, LDZ, N1, N2
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), Q(LDQ,*), Z(LDZ,*)
Arguments
Mode Parameters
WANTQ LOGICAL
Indicates whether or not the user wishes to accumulate
the matrix Q as follows:
= .TRUE. : The matrix Q is updated;
= .FALSE.: the matrix Q is not required.
WANTZ LOGICAL
Indicates whether or not the user wishes to accumulate
the matrix Z as follows:
= .TRUE. : The matrix Z is updated;
= .FALSE.: the matrix Z is not required.
Input/Output Parameters
N1 (input) INTEGER
The order of the first block A11*B11. N1 = 0, 1 or 2.
N2 (input) INTEGER
The order of the second block A22*B22. N2 = 0, 1 or 2.
A (input/output) DOUBLE PRECISION array, dimension
(LDA,N1+N2)
On entry, the leading (N1+N2)-by-(N1+N2) part of this
array must contain the matrix A.
On exit, the leading (N1+N2)-by-(N1+N2) part of this array
contains the matrix A of the reordered pair.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N1+N2).
B (input/output) DOUBLE PRECISION array, dimension
(LDB,N1+N2)
On entry, the leading (N1+N2)-by-(N1+N2) part of this
array must contain the matrix B.
On exit, the leading (N1+N2)-by-(N1+N2) part of this array
contains the matrix B of the reordered pair.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N1+N2).
Q (input/output) DOUBLE PRECISION array, dimension
(LDQ,N1+N2)
On entry, if WANTQ = .TRUE., the leading
(N1+N2)-by-(N1+N2) part of this array must contain the
orthogonal matrix Q.
On exit, the leading (N1+N2)-by-(N1+N2) part of this array
contains the updated matrix Q. Q will be a rotation
matrix for N1=N2=1.
This array is not referenced if WANTQ = .FALSE..
LDQ INTEGER
The leading dimension of the array Q. LDQ >= 1.
If WANTQ = .TRUE., LDQ >= N1+N2.
Z (input/output) DOUBLE PRECISION array, dimension
(LDZ,N1+N2)
On entry, if WANTZ = .TRUE., the leading
(N1+N2)-by-(N1+N2) part of this array must contain the
orthogonal matrix Z.
On exit, the leading (N1+N2)-by-(N1+N2) part of this array
contains the updated matrix Z. Z will be a rotation
matrix for N1=N2=1.
This array is not referenced if WANTZ = .FALSE..
LDZ INTEGER
The leading dimension of the array Z. LDZ >= 1.
If WANTZ = .TRUE., LDZ >= N1+N2.
Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: the transformed matrix (A, B) would be
too far from periodic Schur form; the blocks are
not swapped and (A,B) and (Q,Z) are unchanged.
Method
In the current code both weak and strong stability tests are performed. The user can omit the strong stability test by changing the internal logical parameter WANDS to .FALSE.. See ref. [2] for details.References
[1] Kagstrom, B.
A direct method for reordering eigenvalues in the generalized
real Schur form of a regular matrix pair (A,B), in M.S. Moonen
et al (eds.), Linear Algebra for Large Scale and Real-Time
Applications, Kluwer Academic Publ., 1993, pp. 195-218.
[2] Kagstrom, B., and Poromaa, P.
Computing eigenspaces with specified eigenvalues of a regular
matrix pair (A, B) and condition estimation: Theory,
algorithms and software, Numer. Algorithms, 1996, vol. 12,
pp. 369-407.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the Schur decomposition and the eigenvalues of a
product of matrices, H = H_1*H_2*...*H_p, with H_1 an upper
Hessenberg matrix and H_2, ..., H_p upper triangular matrices,
without evaluating the product. Specifically, the matrices Z_i
are computed, such that
Z_1' * H_1 * Z_2 = T_1,
Z_2' * H_2 * Z_3 = T_2,
...
Z_p' * H_p * Z_1 = T_p,
where T_1 is in real Schur form, and T_2, ..., T_p are upper
triangular.
The routine works primarily with the Hessenberg and triangular
submatrices in rows and columns ILO to IHI, but optionally applies
the transformations to all the rows and columns of the matrices
H_i, i = 1,...,p. The transformations can be optionally
accumulated.
Specification
SUBROUTINE MB03WD( JOB, COMPZ, N, P, ILO, IHI, ILOZ, IHIZ, H,
$ LDH1, LDH2, Z, LDZ1, LDZ2, WR, WI, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPZ, JOB
INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH1, LDH2, LDWORK,
$ LDZ1, LDZ2, N, P
C .. Array Arguments ..
DOUBLE PRECISION DWORK( * ), H( LDH1, LDH2, * ), WI( * ),
$ WR( * ), Z( LDZ1, LDZ2, * )
Arguments
Mode Parameters
JOB CHARACTER*1
Indicates whether the user wishes to compute the full
Schur form or the eigenvalues only, as follows:
= 'E': Compute the eigenvalues only;
= 'S': Compute the factors T_1, ..., T_p of the full
Schur form, T = T_1*T_2*...*T_p.
COMPZ CHARACTER*1
Indicates whether or not the user wishes to accumulate
the matrices Z_1, ..., Z_p, as follows:
= 'N': The matrices Z_1, ..., Z_p are not required;
= 'I': Z_i is initialized to the unit matrix and the
orthogonal transformation matrix Z_i is returned,
i = 1, ..., p;
= 'V': Z_i must contain an orthogonal matrix Q_i on
entry, and the product Q_i*Z_i is returned,
i = 1, ..., p.
Input/Output Parameters
N (input) INTEGER
The order of the matrix H. N >= 0.
P (input) INTEGER
The number of matrices in the product H_1*H_2*...*H_p.
P >= 1.
ILO (input) INTEGER
IHI (input) INTEGER
It is assumed that all matrices H_j, j = 2, ..., p, are
already upper triangular in rows and columns 1:ILO-1 and
IHI+1:N, and H_1 is upper quasi-triangular in rows and
columns 1:ILO-1 and IHI+1:N, with H_1(ILO,ILO-1) = 0
(unless ILO = 1), and H_1(IHI+1,IHI) = 0 (unless IHI = N).
The routine works primarily with the Hessenberg submatrix
in rows and columns ILO to IHI, but applies the
transformations to all the rows and columns of the
matrices H_i, i = 1,...,p, if JOB = 'S'.
1 <= ILO <= max(1,N); min(ILO,N) <= IHI <= N.
ILOZ (input) INTEGER
IHIZ (input) INTEGER
Specify the rows of Z to which the transformations must be
applied if COMPZ = 'I' or COMPZ = 'V'.
1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
H (input/output) DOUBLE PRECISION array, dimension
(LDH1,LDH2,P)
On entry, the leading N-by-N part of H(*,*,1) must contain
the upper Hessenberg matrix H_1 and the leading N-by-N
part of H(*,*,j) for j > 1 must contain the upper
triangular matrix H_j, j = 2, ..., p.
On exit, if JOB = 'S', the leading N-by-N part of H(*,*,1)
is upper quasi-triangular in rows and columns ILO:IHI,
with any 2-by-2 diagonal blocks corresponding to a pair of
complex conjugated eigenvalues, and the leading N-by-N
part of H(*,*,j) for j > 1 contains the resulting upper
triangular matrix T_j.
If JOB = 'E', the contents of H are unspecified on exit.
LDH1 INTEGER
The first leading dimension of the array H.
LDH1 >= max(1,N).
LDH2 INTEGER
The second leading dimension of the array H.
LDH2 >= max(1,N).
Z (input/output) DOUBLE PRECISION array, dimension
(LDZ1,LDZ2,P)
On entry, if COMPZ = 'V', the leading N-by-N-by-P part of
this array must contain the current matrix Q of
transformations accumulated by SLICOT Library routine
MB03VY.
If COMPZ = 'I', Z need not be set on entry.
On exit, if COMPZ = 'V', or COMPZ = 'I', the leading
N-by-N-by-P part of this array contains the transformation
matrices which produced the Schur form; the
transformations are applied only to the submatrices
Z_j(ILOZ:IHIZ,ILO:IHI), j = 1, ..., P.
If COMPZ = 'N', Z is not referenced.
LDZ1 INTEGER
The first leading dimension of the array Z.
LDZ1 >= 1, if COMPZ = 'N';
LDZ1 >= max(1,N), if COMPZ = 'I' or COMPZ = 'V'.
LDZ2 INTEGER
The second leading dimension of the array Z.
LDZ2 >= 1, if COMPZ = 'N';
LDZ2 >= max(1,N), if COMPZ = 'I' or COMPZ = 'V'.
WR (output) DOUBLE PRECISION array, dimension (N)
WI (output) DOUBLE PRECISION array, dimension (N)
The real and imaginary parts, respectively, of the
computed eigenvalues ILO to IHI are stored in the
corresponding elements of WR and WI. If two eigenvalues
are computed as a complex conjugate pair, they are stored
in consecutive elements of WR and WI, say the i-th and
(i+1)th, with WI(i) > 0 and WI(i+1) < 0. If JOB = 'S', the
eigenvalues are stored in the same order as on the
diagonal of the Schur form returned in H.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK. LDWORK >= IHI-ILO+P-1.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, ILO <= i <= IHI, the QR algorithm
failed to compute all the eigenvalues ILO to IHI
in a total of 30*(IHI-ILO+1) iterations;
the elements i+1:IHI of WR and WI contain those
eigenvalues which have been successfully computed.
Method
A refined version of the QR algorithm proposed in [1] and [2] is used. The elements of the subdiagonal, diagonal, and the first supradiagonal of current principal submatrix of H are computed in the process.References
[1] Bojanczyk, A.W., Golub, G. and Van Dooren, P.
The periodic Schur decomposition: algorithms and applications.
Proc. of the SPIE Conference (F.T. Luk, Ed.), 1770, pp. 31-42,
1992.
[2] Sreedhar, J. and Van Dooren, P.
Periodic Schur form and some matrix equations.
Proc. of the Symposium on the Mathematical Theory of Networks
and Systems (MTNS'93), Regensburg, Germany (U. Helmke,
R. Mennicken and J. Saurer, Eds.), Vol. 1, pp. 339-362, 1994.
Numerical Aspects
The algorithm is numerically stable.Further Comments
Note that for P = 1, the LAPACK Library routine DHSEQR could be more efficient on some computer architectures than this routine, because DHSEQR uses a block multishift QR algorithm. When P is large and JOB = 'S', it could be more efficient to compute the product matrix H, and use the LAPACK Library routines.Example
Program Text
* MB03WD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, PMAX
PARAMETER ( NMAX = 20, PMAX = 20 )
INTEGER LDA1, LDA2, LDTAU, LDZ1, LDZ2, LDZTA
PARAMETER ( LDA1 = NMAX, LDA2 = NMAX, LDTAU = NMAX-1,
$ LDZ1 = NMAX, LDZ2 = NMAX, LDZTA = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( NMAX, NMAX + PMAX - 2 ) )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* .. Local Scalars ..
DOUBLE PRECISION SSQ
INTEGER I, IHI, IHIZ, ILO, ILOZ, INFO, J, K, KP1, N, P
CHARACTER COMPZ, JOB
* .. Local Arrays ..
DOUBLE PRECISION A(LDA1,LDA2,PMAX), AS(LDA1,LDA2,PMAX),
$ DWORK(LDWORK), TAU(LDTAU,PMAX), WI(NMAX),
$ WR(NMAX), Z(LDZ1,LDZ2,PMAX), ZTA(LDZTA,NMAX)
* .. External Functions ..
DOUBLE PRECISION DLANGE, DLAPY2
LOGICAL LSAME
EXTERNAL DLANGE, DLAPY2, LSAME
* .. External Subroutines ..
EXTERNAL DGEMM, DLACPY, MB03VD, MB03VY, MB03WD, MB03WX
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
WRITE (NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, P, ILO, IHI, ILOZ, IHIZ, JOB, COMPZ
IF ( N.LT.0 .OR. N.GT.MIN( LDA1, LDA2 ) ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) P
ELSE
* Read matrices A_1, ..., A_p from the input file.
DO 10 K = 1, P
READ ( NIN, FMT = * )
$ ( ( A(I,J,K), J = 1, N ), I = 1, N )
CALL DLACPY( 'F', N, N, A(1,1,K), LDA1, AS(1,1,K), LDA1 )
10 CONTINUE
* Reduce to the periodic Hessenberg form.
CALL MB03VD( N, P, ILO, IHI, A, LDA1, LDA2, TAU, LDTAU,
$ DWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
IF ( LSAME( COMPZ, 'V' ) ) THEN
DO 20 K = 1, P
CALL DLACPY( 'L', N, N, A(1,1,K), LDA1, Z(1,1,K),
$ LDZ1 )
20 CONTINUE
* Accumulate the transformations.
CALL MB03VY( N, P, ILO, IHI, Z, LDZ1, LDZ2, TAU,
$ LDTAU, DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99996 ) INFO
STOP
ELSE
* Reduce to the periodic Schur form.
CALL MB03WD( JOB, COMPZ, N, P, ILO, IHI, ILOZ,
$ IHIZ, A, LDA1, LDA2, Z, LDZ1, LDZ2,
$ WR, WI, DWORK, LDWORK, INFO )
IF ( INFO.GT.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
WRITE ( NOUT, FMT = 99991 )
DO 30 I = MAX( ILO, INFO + 1 ), IHI
WRITE ( NOUT, FMT = 99990 ) WR(I), WI(I)
30 CONTINUE
STOP
END IF
IF ( INFO.LT.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
* Store the isolated eigenvalues.
CALL MB03WX( ILO-1, P, A, LDA1, LDA2, WR, WI,
$ INFO )
IF ( IHI.LT.N )
$ CALL MB03WX( N-IHI, P, A(IHI+1,IHI+1,1),
$ LDA1, LDA2, WR(IHI+1),
$ WI(IHI+1), INFO )
WRITE ( NOUT, FMT = 99991 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99990 ) WR(I), WI(I)
40 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 60 K = 1, P
WRITE ( NOUT, FMT = 99994 ) K
DO 50 I = 1, N
WRITE ( NOUT, FMT = 99993 )
$ ( A(I,J,K), J = 1, N )
50 CONTINUE
60 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 80 K = 1, P
WRITE ( NOUT, FMT = 99994 ) K
DO 70 I = 1, N
WRITE ( NOUT, FMT = 99993 )
$ ( Z(I,J,K), J = 1, N )
70 CONTINUE
80 CONTINUE
* Compute error.
SSQ = ZERO
DO 90 K = 1, P
KP1 = K+1
IF( KP1.GT.P ) KP1 = 1
* Compute NORM (Z' * A * Z - Aout)
CALL DGEMM( 'T', 'N', N, N, N, ONE, Z(1,1,K),
$ LDZ1, AS(1,1,K), LDA1, ZERO, ZTA,
$ LDZTA )
CALL DGEMM( 'N', 'N', N, N, N, ONE, ZTA,
$ LDZTA, Z(1,1,KP1), LDZ1, -ONE,
$ A(1,1,K), LDA1 )
SSQ = DLAPY2( SSQ,
$ DLANGE( 'Frobenius', N, N,
$ A(1,1,K), LDA1,
$ DWORK ) )
90 CONTINUE
WRITE ( NOUT, FMT = 99989 ) SSQ
END IF
END IF
END IF
END IF
END IF
END IF
STOP
99999 FORMAT (' MB03WD EXAMPLE PROGRAM RESULTS', /1X)
99998 FORMAT (' INFO on exit from MB03WD = ', I2)
99997 FORMAT (' INFO on exit from MB03VD = ', I2)
99996 FORMAT (' INFO on exit from MB03VY = ', I2)
99995 FORMAT (/' Reduced matrices')
99994 FORMAT (/' K = ', I5)
99993 FORMAT (8F8.4)
99992 FORMAT (/' Transformation matrices')
99991 FORMAT ( ' Computed eigenvalues'/)
99990 FORMAT (4X,'( ', F17.6,' ,', F17.6,' )')
99989 FORMAT (/,' NORM (Z''*A*Z - Aout) = ', 1PD12.5)
99988 FORMAT (/, ' N is out of range.',/' N = ', I5)
99987 FORMAT (/, ' P is out of range.',/' P = ', I5)
END
Program Data
MB03WD EXAMPLE PROGRAM DATA 4 2 1 4 1 4 S V 1.5 -.7 3.5 -.7 1. 0. 2. 3. 1.5 -.7 2.5 -.3 1. 0. 2. 1. 1.5 -.7 3.5 -.7 1. 0. 2. 3. 1.5 -.7 2.5 -.3 1. 0. 2. 1.Program Results
MB03WD EXAMPLE PROGRAM RESULTS
Computed eigenvalues
( 6.449861 , 7.817717 )
( 6.449861 , -7.817717 )
( 0.091315 , 0.000000 )
( 0.208964 , 0.000000 )
Reduced matrices
K = 1
2.2112 4.3718 -2.3362 0.8907
-0.9179 2.7688 -0.6570 -2.2426
0.0000 0.0000 0.3022 0.1932
0.0000 0.0000 0.0000 -0.4571
K = 2
2.9169 3.4539 2.2016 1.2367
0.0000 3.4745 1.0209 -2.0720
0.0000 0.0000 0.3022 -0.1932
0.0000 0.0000 0.0000 -0.4571
Transformation matrices
K = 1
0.3493 0.6751 -0.6490 0.0327
0.7483 -0.4863 -0.1249 -0.4336
0.2939 0.5504 0.7148 -0.3158
0.4813 -0.0700 0.2286 0.8433
K = 2
0.2372 0.7221 0.6490 0.0327
0.8163 -0.3608 0.1249 -0.4336
0.2025 0.5902 -0.7148 -0.3158
0.4863 0.0076 -0.2286 0.8433
NORM (Z'*A*Z - Aout) = 7.18432D-15
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB03WX.html 0000664 0000000 0000000 00000005524 14560147231 0020400 0 ustar 00root root 0000000 0000000
Purpose
To compute the eigenvalues of a product of matrices, T = T_1*T_2*...*T_p, where T_1 is an upper quasi-triangular matrix and T_2, ..., T_p are upper triangular matrices.Specification
SUBROUTINE MB03WX( N, P, T, LDT1, LDT2, WR, WI, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDT1, LDT2, N, P
C .. Array Arguments ..
DOUBLE PRECISION T( LDT1, LDT2, * ), WI( * ), WR( * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix T. N >= 0.
P (input) INTEGER
The number of matrices in the product T_1*T_2*...*T_p.
P >= 1.
T (input) DOUBLE PRECISION array, dimension (LDT1,LDT2,P)
The leading N-by-N part of T(*,*,1) must contain the upper
quasi-triangular matrix T_1 and the leading N-by-N part of
T(*,*,j) for j > 1 must contain the upper-triangular
matrix T_j, j = 2, ..., p.
The elements below the subdiagonal of T(*,*,1) and below
the diagonal of T(*,*,j), j = 2, ..., p, are not
referenced.
LDT1 INTEGER
The first leading dimension of the array T.
LDT1 >= max(1,N).
LDT2 INTEGER
The second leading dimension of the array T.
LDT2 >= max(1,N).
WR, WI (output) DOUBLE PRECISION arrays, dimension (N)
The real and imaginary parts, respectively, of the
eigenvalues of T. The eigenvalues are stored in the same
order as on the diagonal of T_1. If T(i:i+1,i:i+1,1) is a
2-by-2 diagonal block with complex conjugated eigenvalues
then WI(i) > 0 and WI(i+1) = -WI(i).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the eigenvalues of a Hamiltonian matrix,
[ A G ] T T
H = [ T ], G = G, Q = Q, (1)
[ Q -A ]
where A, G and Q are real n-by-n matrices.
Due to the structure of H all eigenvalues appear in pairs
(lambda,-lambda). This routine computes the eigenvalues of H
using an algorithm based on the symplectic URV and the periodic
Schur decompositions as described in [1],
T [ T G ]
U H V = [ T ], (2)
[ 0 S ]
where U and V are 2n-by-2n orthogonal symplectic matrices,
S is in real Schur form and T is upper triangular.
The algorithm is backward stable and preserves the eigenvalue
pairings in finite precision arithmetic.
Optionally, a symplectic balancing transformation to improve the
conditioning of eigenvalues is computed (see MB04DD). In this
case, the matrix H in decomposition (2) must be replaced by the
balanced matrix.
The SLICOT Library routine MB03ZD can be used to compute invariant
subspaces of H from the output of this routine.
Specification
SUBROUTINE MB03XD( BALANC, JOB, JOBU, JOBV, N, A, LDA, QG, LDQG,
$ T, LDT, U1, LDU1, U2, LDU2, V1, LDV1, V2, LDV2,
$ WR, WI, ILO, SCALE, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER BALANC, JOB, JOBU, JOBV
INTEGER ILO, INFO, LDA, LDQG, LDT, LDU1, LDU2, LDV1,
$ LDV2, LDWORK, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), QG(LDQG,*), SCALE(*),
$ T(LDT,*), U1(LDU1,*), U2(LDU2,*), V1(LDV1,*),
$ V2(LDV2,*), WI(*), WR(*)
Arguments
Mode Parameters
BALANC CHARACTER*1
Indicates how H should be diagonally scaled and/or
permuted to reduce its norm.
= 'N': Do not diagonally scale or permute;
= 'P': Perform symplectic permutations to make the matrix
closer to Hamiltonian Schur form. Do not diagonally
scale;
= 'S': Diagonally scale the matrix, i.e., replace A, G and
Q by D*A*D**(-1), D*G*D and D**(-1)*Q*D**(-1) where
D is a diagonal matrix chosen to make the rows and
columns of H more equal in norm. Do not permute;
= 'B': Both diagonally scale and permute A, G and Q.
Permuting does not change the norm of H, but scaling does.
JOB CHARACTER*1
Indicates whether the user wishes to compute the full
decomposition (2) or the eigenvalues only, as follows:
= 'E': compute the eigenvalues only;
= 'S': compute matrices T and S of (2);
= 'G': compute matrices T, S and G of (2).
JOBU CHARACTER*1
Indicates whether or not the user wishes to compute the
orthogonal symplectic matrix U of (2) as follows:
= 'N': the matrix U is not computed;
= 'U': the matrix U is computed.
JOBV CHARACTER*1
Indicates whether or not the user wishes to compute the
orthogonal symplectic matrix V of (2) as follows:
= 'N': the matrix V is not computed;
= 'V': the matrix V is computed.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A.
On exit, this array is overwritten. If JOB = 'S' or
JOB = 'G', the leading N-by-N part of this array contains
the matrix S in real Schur form of decomposition (2).
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
QG (input/output) DOUBLE PRECISION array, dimension
(LDQG,N+1)
On entry, the leading N-by-N+1 part of this array must
contain in columns 1:N the lower triangular part of the
matrix Q and in columns 2:N+1 the upper triangular part
of the matrix G.
On exit, this array is overwritten. If JOB = 'G', the
leading N-by-N+1 part of this array contains in columns
2:N+1 the matrix G of decomposition (2).
LDQG INTEGER
The leading dimension of the array QG. LDQG >= max(1,N).
T (output) DOUBLE PRECISION array, dimension (LDT,N)
On exit, if JOB = 'S' or JOB = 'G', the leading N-by-N
part of this array contains the upper triangular matrix T
of the decomposition (2). Otherwise, this array is used as
workspace.
LDT INTEGER
The leading dimension of the array T. LDT >= MAX(1,N).
U1 (output) DOUBLE PRECISION array, dimension (LDU1,N)
On exit, if JOBU = 'U', the leading N-by-N part of this
array contains the (1,1) block of the orthogonal
symplectic matrix U of decomposition (2).
LDU1 INTEGER
The leading dimension of the array U1. LDU1 >= 1.
LDU1 >= N, if JOBU = 'U'.
U2 (output) DOUBLE PRECISION array, dimension (LDU2,N)
On exit, if JOBU = 'U', the leading N-by-N part of this
array contains the (2,1) block of the orthogonal
symplectic matrix U of decomposition (2).
LDU2 INTEGER
The leading dimension of the array U2. LDU2 >= 1.
LDU2 >= N, if JOBU = 'U'.
V1 (output) DOUBLE PRECISION array, dimension (LDV1,N)
On exit, if JOBV = 'V', the leading N-by-N part of this
array contains the (1,1) block of the orthogonal
symplectic matrix V of decomposition (2).
LDV1 INTEGER
The leading dimension of the array V1. LDV1 >= 1.
LDV1 >= N, if JOBV = 'V'.
V2 (output) DOUBLE PRECISION array, dimension (LDV2,N)
On exit, if JOBV = 'V', the leading N-by-N part of this
array contains the (2,1) block of the orthogonal
symplectic matrix V of decomposition (2).
LDV2 INTEGER
The leading dimension of the array V2. LDV2 >= 1.
LDV2 >= N, if JOBV = 'V'.
WR (output) DOUBLE PRECISION array, dimension (N)
WI (output) DOUBLE PRECISION array, dimension (N)
On exit, the leading N elements of WR and WI contain the
real and imaginary parts, respectively, of N eigenvalues
that have nonnegative imaginary part. Their complex
conjugate eigenvalues are not stored. If imaginary parts
are zero (i.e., for real eigenvalues), only positive
eigenvalues are stored.
ILO (output) INTEGER
ILO is an integer value determined when H was balanced.
The balanced A(i,j) = 0 if I > J and J = 1,...,ILO-1.
The balanced Q(i,j) = 0 if J = 1,...,ILO-1 or
I = 1,...,ILO-1.
SCALE (output) DOUBLE PRECISION array, dimension (N)
On exit, if BALANC <> 'N', the leading N elements of this
array contain details of the permutation and/or scaling
factors applied when balancing H, see MB04DD.
This array is not referenced if BALANC = 'N'.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK, and DWORK(2) returns the 1-norm of the
scaled (if BALANC = 'S' or 'B') Hamiltonian matrix.
On exit, if INFO = -25, DWORK(1) returns the minimum
value of LDWORK.
LDWORK (input) INTEGER
The dimension of the array DWORK. LDWORK >= max( 2, 8*N ).
Moreover:
If JOB = 'E' or 'S' and JOBU = 'N' and JOBV = 'N',
LDWORK >= 7*N+N*N.
If JOB = 'G' and JOBU = 'N' and JOBV = 'N',
LDWORK >= max( 7*N+N*N, 2*N+3*N*N ).
If JOB = 'G' and JOBU = 'U' and JOBV = 'N',
LDWORK >= 7*N+2*N*N.
If JOB = 'G' and JOBU = 'N' and JOBV = 'V',
LDWORK >= 7*N+2*N*N.
If JOB = 'G' and JOBU = 'U' and JOBV = 'V',
LDWORK >= 7*N+N*N.
For good performance, LDWORK must generally be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO (output) INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, the periodic QR algorithm failed to
compute all the eigenvalues, elements i+1:N of WR
and WI contain eigenvalues which have converged.
References
[1] Benner, P., Mehrmann, V., and Xu, H.
A numerically stable, structure preserving method for
computing the eigenvalues of real Hamiltonian or symplectic
pencils.
Numer. Math., Vol. 78(3), pp. 329-358, 1998.
[2] Benner, P., Mehrmann, V., and Xu, H.
A new method for computing the stable invariant subspace of a
real Hamiltonian matrix, J. Comput. Appl. Math., vol. 86,
pp. 17-43, 1997.
Further Comments
NoneExample
Program Text
* MB03XD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 100 )
INTEGER LDA, LDQG, LDRES, LDT, LDU1, LDU2, LDV1, LDV2,
$ LDWORK
PARAMETER ( LDA = NMAX, LDQG = NMAX, LDRES = NMAX,
$ LDT = NMAX, LDU1 = NMAX, LDU2 = NMAX,
$ LDV1 = NMAX, LDV2 = NMAX,
$ LDWORK = 3*NMAX*NMAX + 7*NMAX )
* .. Local Scalars ..
CHARACTER*1 BALANC, JOB, JOBU, JOBV
INTEGER I, ILO, INFO, J, N
DOUBLE PRECISION TEMP
* .. Local Arrays ..
DOUBLE PRECISION A(LDA, NMAX), DWORK(LDWORK), QG(LDQG, NMAX+1),
$ RES(LDRES,3*NMAX+1), SCALE(NMAX), T(LDT,NMAX),
$ U1(LDU1,NMAX), U2(LDU2, NMAX), V1(LDV1,NMAX),
$ V2(LDV2, NMAX), WI(NMAX), WR(NMAX)
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLANGE, DLAPY2, MA02JD
EXTERNAL DLANGE, DLAPY2, LSAME, MA02JD
* .. External Subroutines ..
EXTERNAL DGEMM, DLACPY, MB03XD, MB04DD
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, BALANC, JOB, JOBU, JOBV
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
CALL DLACPY( 'All', N, N, A, LDA, RES(1,N+1), LDRES )
READ ( NIN, FMT = * ) ( ( QG(I,J), J = 1,N+1 ), I = 1,N )
CALL DLACPY( 'All', N, N+1, QG, LDQG, RES(1,2*N+1), LDRES )
INFO = 0
CALL MB03XD( BALANC, JOB, JOBU, JOBV, N, A, LDA, QG, LDQG,
$ T, LDT, U1, LDU1, U2, LDU2, V1, LDV1, V2, LDV2,
$ WR, WI, ILO, SCALE, DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) I, WR(I), WI(I)
20 CONTINUE
IF ( LSAME( JOB, 'S' ).OR.LSAME( JOB, 'G' ) ) THEN
WRITE ( NOUT, FMT = 99995 )
DO 30 I = 1, N
WRITE ( NOUT, FMT = 99990 ) ( A(I,J), J = 1,N )
30 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99990 ) ( T(I,J), J = 1,N )
40 CONTINUE
END IF
IF ( LSAME( JOB, 'G' ) ) THEN
WRITE ( NOUT, FMT = 99993 )
DO 50 I = 1, N
WRITE ( NOUT, FMT = 99990 ) ( QG(I,J+1), J = 1,N )
50 CONTINUE
END IF
C
IF ( LSAME( JOB, 'G' ).AND.LSAME( JOBU, 'U' ).AND.
$ LSAME( JOBV, 'V' ) ) THEN
CALL MB04DD( BALANC, N, RES(1,N+1), LDRES, RES(1,2*N+1),
$ LDRES, I, DWORK, INFO )
CALL DGEMM( 'No Transpose', 'No Transpose', N, N, N, ONE,
$ RES(1,N+1), LDRES, V1, LDV1, ZERO, RES,
$ LDRES )
CALL DSYMM ( 'Left', 'Upper', N, N, -ONE, RES(1,2*N+2),
$ LDRES, V2, LDV2, ONE, RES, LDRES )
CALL DGEMM( 'No Transpose', 'No Transpose', N, N, N,
$ -ONE, U1, LDU1, T, LDT, ONE, RES, LDRES )
TEMP = DLANGE( 'Frobenius', N, N, RES, LDRES, DWORK )
CALL DGEMM( 'No Transpose', 'No Transpose', N, N, N, ONE,
$ RES(1,N+1), LDRES, V2, LDV2, ZERO, RES,
$ LDRES )
CALL DSYMM( 'Left', 'Upper', N, N, ONE, RES(1,2*N+2),
$ LDRES, V1, LDV1, ONE, RES, LDRES )
CALL DGEMM( 'No Transpose', 'No Transpose', N, N, N,
$ -ONE, U1, LDU1, QG(1,2), LDQG, ONE, RES,
$ LDRES )
CALL DGEMM( 'No Transpose', 'Transpose', N, N, N,
$ -ONE, U2, LDU2, A, LDA, ONE, RES, LDRES )
TEMP = DLAPY2( TEMP, DLANGE( 'Frobenius', N, N, RES,
$ LDRES, DWORK ) )
CALL DSYMM( 'Left', 'Lower', N, N, ONE, RES(1,2*N+1),
$ LDRES, V1, LDV1, ZERO, RES, LDRES )
CALL DGEMM( 'Transpose', 'No Transpose', N, N, N, ONE,
$ RES(1,N+1), LDRES, V2, LDV2, ONE, RES,
$ LDRES )
CALL DGEMM( 'No Transpose', 'No Transpose', N, N, N, ONE,
$ U2, LDU2, T, LDT, ONE, RES, LDRES )
TEMP = DLAPY2( TEMP, DLANGE( 'Frobenius', N, N, RES,
$ LDRES, DWORK ) )
CALL DSYMM( 'Left', 'Lower', N, N, ONE, RES(1,2*N+1),
$ LDRES, V2, LDV2, ZERO, RES, LDRES )
CALL DGEMM( 'Transpose', 'No Transpose', N, N, N, -ONE,
$ RES(1,N+1), LDRES, V1, LDV1, ONE, RES,
$ LDRES )
CALL DGEMM( 'No Transpose', 'No Transpose', N, N, N, ONE,
$ U2, LDU2, QG(1,2), LDQG, ONE, RES, LDRES )
CALL DGEMM( 'No Transpose', 'Transpose', N, N, N,
$ -ONE, U1, LDU1, A, LDA, ONE, RES, LDRES )
TEMP = DLAPY2( TEMP, DLANGE( 'Frobenius', N, N, RES,
$ LDRES, DWORK ) )
WRITE ( NOUT, FMT = 99987 ) TEMP
END IF
C
IF ( LSAME( JOBU, 'U' ) ) THEN
WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99990 )
$ ( U1(I,J), J = 1,N ), ( U2(I,J), J = 1,N )
60 CONTINUE
DO 70 I = 1, N
WRITE ( NOUT, FMT = 99990 )
$ ( -U2(I,J), J = 1,N ), ( U1(I,J), J = 1,N )
70 CONTINUE
WRITE ( NOUT, FMT = 99986 ) MA02JD( .FALSE., .FALSE., N,
$ U1, LDU1, U2, LDU2, RES, LDRES )
END IF
IF ( LSAME( JOBV, 'V' ) ) THEN
WRITE ( NOUT, FMT = 99991 )
DO 80 I = 1, N
WRITE ( NOUT, FMT = 99990 )
$ ( V1(I,J), J = 1,N ), ( V2(I,J), J = 1,N )
80 CONTINUE
DO 90 I = 1, N
WRITE ( NOUT, FMT = 99990 )
$ ( -V2(I,J), J = 1,N ), ( V1(I,J), J = 1,N )
90 CONTINUE
WRITE ( NOUT, FMT = 99985 ) MA02JD( .FALSE., .FALSE., N,
$ V1, LDV1, V2, LDV2, RES, LDRES )
END IF
IF ( LSAME( BALANC, 'S' ).OR.LSAME( BALANC, 'B' ) ) THEN
WRITE ( NOUT, FMT = 99989 )
DO 100 I = 1, N
WRITE ( NOUT, FMT = 99996 ) I, SCALE(I)
100 CONTINUE
END IF
END IF
END IF
*
99999 FORMAT (' MB03XD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB03XD = ',I2)
99997 FORMAT (' The eigenvalues are',//' i',6X,
$ 'WR(i)',6X,'WI(i)',/)
99996 FORMAT (I4,3X,F8.4,3X,F8.4)
99995 FORMAT (/' The matrix S of the reduced matrix is')
99994 FORMAT (/' The matrix T of the reduced matrix is')
99993 FORMAT (/' The matrix G of the reduced matrix is')
99992 FORMAT (/' The orthogonal symplectic factor U is')
99991 FORMAT (/' The orthogonal symplectic factor V is')
99990 FORMAT (20(1X,F19.16))
99989 FORMAT (/' The diagonal scaling factors are ',//' i',6X,
$ 'SCALE(i)',/)
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' Residual: || H*V - U*R ||_F = ',G7.2)
99986 FORMAT (/' Orthogonality of U: || U^T U - I ||_F = ',G7.2)
99985 FORMAT (/' Orthogonality of V: || V^T V - I ||_F = ',G7.2)
END
Program Data
MB03XD EXAMPLE PROGRAM DATA 5 N G U V 3.7588548168313685e-001 9.1995720669587144e-001 1.9389317998466821e-001 5.4878212553858818e-001 6.2731478808399666e-001 9.8764628987858052e-003 8.4472150190817474e-001 9.0481233416635698e-001 9.3158335257969060e-001 6.9908013774533750e-001 4.1985780631021896e-001 3.6775288246828447e-001 5.6920574967174709e-001 3.3519743020639464e-001 3.9718395379261456e-001 7.5366962581358721e-001 6.2080133182114383e-001 6.3178992922175603e-001 6.5553105501201447e-001 4.1362889533818031e-001 7.9387177473231862e-001 7.3127726446634478e-001 2.3441295540825388e-001 3.9190420688900335e-001 6.5521294635567051e-001 1.8015558545989005e-001 4.1879254941592853e-001 2.7203760737317784e-001 2.8147214090719214e-001 1.7731904815580199e-001 3.4718672159409536e-001 2.7989257702981651e-001 3.5042861661866559e-001 2.5565572408444881e-001 4.3977750345993827e-001 2.8855026075967616e-001 2.1496327083014577e-001 1.7341073886969158e-001 3.9913855375815932e-001 4.0151317011596516e-001 4.0331887464437133e-001 2.6723538667317948e-001 3.7110275606849241e-001 3.7832182695699140e-001 3.3812641389556752e-001 8.4360396433341395e-002 4.3672540277019672e-001 7.0022228267365608e-002 3.8210230186291916e-001 1.9548216143135175e-001 2.9055490787446736e-001 4.7670819669167425e-001 1.4636498713707141e-001 2.7670398401519275e-001 2.9431082727794898e-002Program Results
MB03XD EXAMPLE PROGRAM RESULTS The eigenvalues are i WR(i) WI(i) 1 3.1941 0.0000 2 0.1350 0.3179 3 -0.1350 0.3179 4 0.0595 0.2793 5 -0.0595 0.2793 The matrix S of the reduced matrix is -3.1844761777714723 0.1612357243439330 -0.0628592203751083 0.2449004200921966 0.1974400149992633 0.0000000000000000 -0.1510667773167795 0.4260444411622876 -0.1775026035208666 0.3447278421198404 0.0000000000000000 -0.1386140422054281 -0.3006779624777419 0.2944143257134117 0.3456440339120381 0.0000000000000000 0.0000000000000000 0.0000000000000000 -0.2710128384740589 0.0933189808067083 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.4844146572359630 0.2004347508746724 The matrix T of the reduced matrix is 3.2038208121776366 0.1805955192510636 0.2466389119377562 -0.2539149302433404 -0.0359238844381174 0.0000000000000000 -0.7196686433290822 0.0000000000000000 0.2428659121580382 -0.0594190100670709 0.0000000000000000 0.0000000000000000 -0.1891741194498124 -0.3309578443491325 -0.0303520731950498 0.0000000000000000 0.0000000000000000 0.0000000000000000 -0.4361574461961496 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.1530894573304220 The matrix G of the reduced matrix is -0.0370982242678458 0.0917788436945730 -0.0560402416315236 0.1345152517579191 0.0256668227276665 0.0652183678916926 -0.0700457231988328 0.0350041175858816 -0.2233868768749277 -0.1171980260782820 -0.0626428681377074 0.2327575351902838 -0.1251515732208133 -0.0177816046663209 0.3696921118421109 0.0746042309265569 -0.0828007611045243 0.0217427473546003 -0.1157775118548850 -0.3161183681200607 0.1374372236164838 0.1002727885506978 0.4021556774753987 -0.0431072263235625 0.1067394572547804 Residual: || H*V - U*R ||_F = .38E-14 The orthogonal symplectic factor U is 0.3806883009357249 -0.0347810363019666 -0.5014665065895627 0.5389691288472425 0.2685446895251499 -0.1795922007470744 0.1908329820840935 0.0868799433942041 0.3114741142062469 -0.2579907627915116 0.4642712665555327 -0.5942766860716397 0.4781179763952658 0.2334370556238072 0.0166790369048881 -0.2447897730222852 -0.1028403314750053 -0.1157840914576285 -0.1873268885694422 0.1700708002861556 0.2772789197782788 -0.0130145392695854 -0.2123817030594153 -0.2550292626960007 -0.5049268366774478 -0.2243335325285329 0.3180998613802498 0.3315380214794935 0.1977859924739816 0.5072476567310036 0.4209268575081797 0.1499593172661210 -0.1925590746592206 -0.5472292877802408 0.4543329704184014 -0.2128397588651423 -0.2740560593051884 0.1941418870268831 -0.3096684962457407 -0.0581576193198820 0.3969669479129447 0.6321903535930841 0.3329156356041941 0.0163533225344391 -0.2638879466190056 -0.2002027567371933 -0.0040094115506845 -0.3979373387545256 0.1520881534833996 -0.2010804514091296 0.1795922007470744 -0.1908329820840935 -0.0868799433942041 -0.3114741142062469 0.2579907627915116 0.3806883009357249 -0.0347810363019666 -0.5014665065895627 0.5389691288472425 0.2685446895251499 0.2447897730222852 0.1028403314750053 0.1157840914576285 0.1873268885694422 -0.1700708002861556 0.4642712665555327 -0.5942766860716397 0.4781179763952658 0.2334370556238072 0.0166790369048881 0.2243335325285329 -0.3180998613802498 -0.3315380214794935 -0.1977859924739816 -0.5072476567310036 0.2772789197782788 -0.0130145392695854 -0.2123817030594153 -0.2550292626960007 -0.5049268366774478 0.2128397588651423 0.2740560593051884 -0.1941418870268831 0.3096684962457407 0.0581576193198820 0.4209268575081797 0.1499593172661210 -0.1925590746592206 -0.5472292877802408 0.4543329704184014 0.2002027567371933 0.0040094115506845 0.3979373387545256 -0.1520881534833996 0.2010804514091296 0.3969669479129447 0.6321903535930841 0.3329156356041941 0.0163533225344391 -0.2638879466190056 Orthogonality of U: || U^T U - I ||_F = .28E-14 The orthogonal symplectic factor V is 0.4447147692018332 -0.6830166755147445 -0.0002576861753461 0.5781954611783305 -0.0375091627893695 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.5121756358795815 0.0297197140254867 0.4332229148788674 -0.3240527006890555 0.5330850295256576 0.0299719306696789 -0.2322624725320732 -0.0280846899680330 -0.3044255686880015 -0.1077641482535463 0.3664711365265602 0.3288511296455134 0.0588396016404466 0.1134221597062252 0.1047567336850027 -0.0069083614679702 0.3351358347080169 -0.4922707032978909 0.4293545450291777 0.4372821269061838 0.4535357098437908 0.1062866148880810 -0.3964092656837799 -0.2211800890450648 0.0350667323996154 0.0167847133528844 0.2843629278945263 0.5958979805231206 0.3097336757510830 -0.2086733033047175 0.4450432900616098 0.2950206358263727 -0.1617837757183794 -0.0376369332204945 -0.6746752660482708 0.0248567764822071 -0.2810759958040465 -0.1653113624869875 -0.3528780198620394 -0.0254898556119200 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.4447147692018332 -0.6830166755147445 -0.0002576861753461 0.5781954611783305 -0.0375091627893695 -0.0299719306696789 0.2322624725320732 0.0280846899680330 0.3044255686880015 0.1077641482535463 0.5121756358795815 0.0297197140254867 0.4332229148788674 -0.3240527006890555 0.5330850295256576 0.0069083614679702 -0.3351358347080169 0.4922707032978909 -0.4293545450291777 -0.4372821269061838 0.3664711365265602 0.3288511296455134 0.0588396016404466 0.1134221597062252 0.1047567336850027 -0.0167847133528844 -0.2843629278945263 -0.5958979805231206 -0.3097336757510830 0.2086733033047175 0.4535357098437908 0.1062866148880810 -0.3964092656837799 -0.2211800890450648 0.0350667323996154 -0.0248567764822071 0.2810759958040465 0.1653113624869875 0.3528780198620394 0.0254898556119200 0.4450432900616098 0.2950206358263727 -0.1617837757183794 -0.0376369332204945 -0.6746752660482708 Orthogonality of V: || V^T V - I ||_F = .25E-14
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB03XP.html 0000664 0000000 0000000 00000043472 14560147231 0020375 0 ustar 00root root 0000000 0000000
Purpose
To compute the periodic Schur decomposition and the eigenvalues of
a product of matrices, H = A*B, with A upper Hessenberg and B
upper triangular without evaluating any part of the product.
Specifically, the matrices Q and Z are computed, so that
Q' * A * Z = S, Z' * B * Q = T
where S is in real Schur form, and T is upper triangular.
Specification
SUBROUTINE MB03XP( JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB,
$ Q, LDQ, Z, LDZ, ALPHAR, ALPHAI, BETA, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, COMPZ, JOB
INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDWORK, LDZ, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), ALPHAI(*), ALPHAR(*), B(LDB,*),
$ BETA(*), DWORK(*), Q(LDQ,*), Z(LDZ,*)
Arguments
Mode Parameters
JOB CHARACTER*1
Indicates whether the user wishes to compute the full
Schur form or the eigenvalues only, as follows:
= 'E': Compute the eigenvalues only;
= 'S': compute the factors S and T of the full
Schur form.
COMPQ CHARACTER*1
Indicates whether or not the user wishes to accumulate
the matrix Q as follows:
= 'N': The matrix Q is not required;
= 'I': Q is initialized to the unit matrix and the
orthogonal transformation matrix Q is returned;
= 'V': Q must contain an orthogonal matrix U on entry,
and the product U*Q is returned.
COMPZ CHARACTER*1
Indicates whether or not the user wishes to accumulate
the matrix Z as follows:
= 'N': The matrix Z is not required;
= 'I': Z is initialized to the unit matrix and the
orthogonal transformation matrix Z is returned;
= 'V': Z must contain an orthogonal matrix U on entry,
and the product U*Z is returned.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A and B. N >= 0.
ILO (input) INTEGER
IHI (input) INTEGER
It is assumed that the matrices A and B are already upper
triangular in rows and columns 1:ILO-1 and IHI+1:N.
The routine works primarily with the submatrices in rows
and columns ILO to IHI, but applies the transformations to
all the rows and columns of the matrices A and B, if
JOB = 'S'.
1 <= ILO <= max(1,N+1); min(ILO,N) <= IHI <= N.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array A must
contain the upper Hessenberg matrix A.
On exit, if JOB = 'S', the leading N-by-N part of this
array is upper quasi-triangular with any 2-by-2 diagonal
blocks corresponding to a pair of complex conjugated
eigenvalues.
If JOB = 'E', the diagonal elements and 2-by-2 diagonal
blocks of A will be correct, but the remaining parts of A
are unspecified on exit.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the leading N-by-N part of this array B must
contain the upper triangular matrix B.
On exit, if JOB = 'S', the leading N-by-N part of this
array contains the transformed upper triangular matrix.
2-by-2 blocks in B corresponding to 2-by-2 blocks in A
will be reduced to positive diagonal form. (I.e., if
A(j+1,j) is non-zero, then B(j+1,j)=B(j,j+1)=0 and B(j,j)
and B(j+1,j+1) will be positive.)
If JOB = 'E', the elements corresponding to diagonal
elements and 2-by-2 diagonal blocks in A will be correct,
but the remaining parts of B are unspecified on exit.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if COMPQ = 'V', then the leading N-by-N part of
this array must contain a matrix Q which is assumed to be
equal to the unit matrix except for the submatrix
Q(ILO:IHI,ILO:IHI).
If COMPQ = 'I', Q need not be set on entry.
On exit, if COMPQ = 'V' or COMPQ = 'I' the leading N-by-N
part of this array contains the transformation matrix
which produced the Schur form.
If COMPQ = 'N', Q is not referenced.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= 1.
If COMPQ <> 'N', LDQ >= MAX(1,N).
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
On entry, if COMPZ = 'V', then the leading N-by-N part of
this array must contain a matrix Z which is assumed to be
equal to the unit matrix except for the submatrix
Z(ILO:IHI,ILO:IHI).
If COMPZ = 'I', Z need not be set on entry.
On exit, if COMPZ = 'V' or COMPZ = 'I' the leading N-by-N
part of this array contains the transformation matrix
which produced the Schur form.
If COMPZ = 'N', Z is not referenced.
LDZ INTEGER
The leading dimension of the array Z. LDZ >= 1.
If COMPZ <> 'N', LDZ >= MAX(1,N).
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
BETA (output) DOUBLE PRECISION array, dimension (N)
The i-th (1 <= i <= N) computed eigenvalue is given by
BETA(I) * ( ALPHAR(I) + sqrt(-1)*ALPHAI(I) ). If two
eigenvalues are computed as a complex conjugate pair,
they are stored in consecutive elements of ALPHAR, ALPHAI
and BETA. If JOB = 'S', the eigenvalues are stored in the
same order as on the diagonales of the Schur forms of A
and B.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -19, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,N).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, then MB03XP failed to compute the Schur
form in a total of 30*(IHI-ILO+1) iterations;
elements 1:ilo-1 and i+1:n of ALPHAR, ALPHAI and
BETA contain successfully computed eigenvalues.
Method
The implemented algorithm is a multi-shift version of the periodic QR algorithm described in [1,3] with some minor modifications proposed in [2].References
[1] Bojanczyk, A.W., Golub, G.H., and Van Dooren, P.
The periodic Schur decomposition: Algorithms and applications.
Proc. of the SPIE Conference (F.T. Luk, Ed.), 1770, pp. 31-42,
1992.
[2] Kressner, D.
An efficient and reliable implementation of the periodic QZ
algorithm. Proc. of the IFAC Workshop on Periodic Control
Systems, pp. 187-192, 2001.
[3] Van Loan, C.
Generalized Singular Values with Algorithms and Applications.
Ph. D. Thesis, University of Michigan, 1973.
Numerical Aspects
The algorithm requires O(N**3) floating point operations and is backward stable.Further Comments
NoneExample
Program Text
* MB03XP EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 200 )
INTEGER LDA, LDB, LDQ, LDRES, LDZ, LDWORK
PARAMETER ( LDA = NMAX, LDB = NMAX, LDQ = NMAX,
$ LDRES = NMAX, LDWORK = NMAX, LDZ = NMAX )
* .. Local Scalars ..
INTEGER I, IHI, ILO, INFO, J, N
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), ALPHAI(NMAX), ALPHAR(NMAX),
$ B(LDA,NMAX), BETA(NMAX), DWORK(LDWORK),
$ Q(LDQ,NMAX), RES(LDRES,3*NMAX), Z(LDZ,NMAX)
* .. External Functions ..
DOUBLE PRECISION DLANGE
EXTERNAL DLANGE
* .. External Subroutines ..
EXTERNAL DGEMM, MB03XP
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, ILO, IHI
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
CALL DLACPY( 'All', N, N, A, LDA, RES(1,N+1), LDRES )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,N ), I = 1,N )
CALL DLACPY( 'All', N, N, B, LDB, RES(1,2*N+1), LDRES )
CALL MB03XP( 'S', 'I', 'I', N, ILO, IHI, A, LDA, B, LDB, Q,
$ LDQ, Z, LDZ, ALPHAR, ALPHAI, BETA, DWORK, LDWORK,
$ INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE (NOUT, FMT = 99991) ( A(I,J), J = 1,N )
10 CONTINUE
CALL DGEMM( 'No Transpose', 'No Transpose', N, N, N, ONE,
$ RES(1,N+1), LDRES, Z, LDZ, ZERO, RES, LDRES )
CALL DGEMM( 'No Transpose', 'No Transpose', N, N, N, -ONE,
$ Q, LDQ, A, LDA, ONE, RES, LDRES )
WRITE ( NOUT, FMT = 99989 ) DLANGE( 'Frobenius', N, N, RES,
$ LDRES, DWORK )
WRITE ( NOUT, FMT = 99995 )
DO 20 I = 1, N
WRITE (NOUT, FMT = 99991) ( B(I,J), J = 1,N )
20 CONTINUE
CALL DGEMM( 'No Transpose', 'No Transpose', N, N, N, ONE,
$ RES(1,2*N+1), LDRES, Q, LDQ, ZERO, RES, LDRES )
CALL DGEMM( 'No Transpose', 'No Transpose', N, N, N, -ONE,
$ Z, LDZ, B, LDB, ONE, RES, LDRES )
WRITE ( NOUT, FMT = 99988 ) DLANGE( 'Frobenius', N, N, RES,
$ LDRES, DWORK )
WRITE ( NOUT, FMT = 99994 )
DO 30 I = 1, N
WRITE (NOUT, FMT = 99991) ( Q(I,J), J = 1,N )
30 CONTINUE
CALL DGEMM( 'Transpose', 'No Transpose', N, N, N, ONE, Q,
$ LDQ, Q, LDQ, ONE, RES, LDRES )
DO 40 I = 1, N
RES(I,I) = RES(I,I) - ONE
40 CONTINUE
WRITE ( NOUT, FMT = 99987 ) DLANGE( 'Frobenius', N, N, RES,
$ LDRES, DWORK )
WRITE ( NOUT, FMT = 99993 )
DO 50 I = 1, N
WRITE (NOUT, FMT = 99991) ( Z(I,J), J = 1,N )
50 CONTINUE
CALL DGEMM( 'Transpose', 'No Transpose', N, N, N, ONE, Z,
$ LDZ, Z, LDZ, ONE, RES, LDRES )
DO 60 I = 1, N
RES(I,I) = RES(I,I) - ONE
60 CONTINUE
WRITE ( NOUT, FMT = 99986 ) DLANGE( 'Frobenius', N, N, RES,
$ LDRES, DWORK )
WRITE ( NOUT, FMT = 99992 )
DO 70 I = 1, N
WRITE ( NOUT, FMT = 99991 )
$ ALPHAR(I), ALPHAI(I), BETA(I)
70 CONTINUE
END IF
END IF
*
STOP
*
99999 FORMAT (' MB03XP EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB03XP = ',I2)
99996 FORMAT (' The reduced matrix A is ')
99995 FORMAT (/' The reduced matrix B is ')
99994 FORMAT (/' The orthogonal factor Q is ')
99993 FORMAT (/' The orthogonal factor Z is ')
99992 FORMAT (/4X,'ALPHAR',4X,'ALPHAI',4X,'BETA')
99991 FORMAT (1000(1X,F9.4))
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' Residual: || A*Z - Q*S ||_F = ',G7.2)
99988 FORMAT (/' Residual: || B*Q - Z*T ||_F = ',G7.2)
99987 FORMAT (/' Orthogonality of Q: || Q''*Q - I ||_F = ',G7.2)
99986 FORMAT (/' Orthogonality of Z: || Z''*Z - I ||_F = ',G7.2)
END
Program Data
MB03XP EXAMPLE PROGRAM DATA
8 1 8
0.9708 -1.1156 -0.0884 -0.2684 0.2152 0.0402 0.0333 0.5141
-1.6142 2.8635 1.0420 -0.2295 -0.3560 0.4885 0.1026 -0.0164
0 1.1138 0.3509 -0.0963 0.0875 0.2158 0.2444 -0.2838
0 0 -0.5975 0.1021 -0.1026 -0.0062 -0.2646 -0.0745
0 0 0 0.6181 0.1986 0.3612 -0.1750 0.3332
0 0 0 0 -0.7387 -0.5201 0.0713 0.0501
0 0 0 0 0 -0.2677 -0.4918 -0.2838
0 0 0 0 0 0 0.3011 0.3389
0.9084 0.1739 0.5915 0.8729 0.8188 0.1911 0.4122 0.5527
0 0.1708 0.1197 0.2379 0.4302 0.4225 0.9016 0.4001
0 0 0.0381 0.6458 0.8903 0.8560 0.0056 0.1988
0 0 0 0.9669 0.7349 0.4902 0.2974 0.6252
0 0 0 0 0.6873 0.8159 0.0492 0.7334
0 0 0 0 0 0.4608 0.6932 0.3759
0 0 0 0 0 0 0.6501 0.0099
0 0 0 0 0 0 0 0.4199
Program Results
MB03XP EXAMPLE PROGRAM RESULTS
The reduced matrix A is
-0.6290 -0.1397 -0.0509 0.1603 -0.3248 0.2381 0.0694 0.0103
1.5112 -3.4273 -0.4485 -0.4357 -0.3456 0.4619 0.5998 0.5654
0.0000 0.0000 0.0547 -0.4360 0.1714 -0.2103 -0.0900 -0.4011
0.0000 0.0000 0.6623 0.2038 0.2796 -0.2629 0.3837 0.2382
0.0000 0.0000 0.0000 0.0000 -0.6315 0.2071 -0.0174 -0.3538
0.0000 0.0000 0.0000 0.0000 0.0000 -0.5850 -0.1813 0.2435
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.7884 0.1535
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2832
Residual: || A*Z - Q*S ||_F = .60E-14
The reduced matrix B is
-0.9231 0.0000 -0.9834 0.1805 0.4428 0.3655 -0.4300 0.8498
0.0000 -0.1837 -0.1873 0.0681 0.8412 -0.0556 0.0538 0.6113
0.0000 0.0000 -1.8997 0.0000 0.5651 -0.2785 0.2882 1.0458
0.0000 0.0000 0.0000 -0.2602 0.3527 -0.0020 -0.3396 0.2739
0.0000 0.0000 0.0000 0.0000 0.8521 -0.0164 0.2115 0.5446
0.0000 0.0000 0.0000 0.0000 0.0000 0.0283 -0.5128 0.0153
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.4153 0.4587
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.5894
Residual: || B*Q - Z*T ||_F = .55E-14
The orthogonal factor Q is
-0.5333 0.3661 -0.1179 0.0264 0.0026 0.7527 0.0018 0.0189
0.0583 -0.8833 -0.0666 -0.0007 0.0017 0.4603 0.0050 0.0092
-0.8414 -0.2927 0.0347 0.0452 -0.0005 -0.4498 -0.0269 0.0001
0.0077 0.0046 -0.5687 -0.4810 0.0227 -0.0708 -0.6500 0.1312
0.0598 0.0059 -0.6128 0.7656 0.1348 -0.0863 0.0038 0.0954
-0.0242 -0.0016 -0.4295 -0.4163 0.3871 -0.0709 0.6964 -0.0417
0.0027 0.0001 0.3109 0.0620 0.8615 0.0378 -0.2267 0.3231
0.0012 0.0000 0.0188 -0.0514 -0.2987 -0.0172 0.2010 0.9312
Orthogonality of Q: || Q'*Q - I ||_F = .63E-14
The orthogonal factor Z is
0.9957 -0.0786 0.0397 -0.0032 0.0006 0.0227 0.0104 0.0123
0.0764 0.9956 0.0200 0.0073 -0.0009 0.0389 0.0263 0.0193
-0.0062 0.0235 0.6714 -0.0229 0.0271 -0.4461 -0.5354 -0.2486
-0.0445 -0.0437 0.6098 0.4197 -0.0656 0.6125 0.1248 0.2302
-0.0242 -0.0148 0.4049 -0.6041 0.2808 -0.1328 0.5972 0.1311
0.0096 0.0037 -0.0183 0.6539 0.5114 -0.4136 0.3620 -0.0913
-0.0019 -0.0004 -0.1055 -0.1544 0.7891 0.2944 -0.4436 0.2426
-0.0005 0.0000 -0.0039 0.0826 -0.1786 -0.3853 -0.1119 0.8946
Orthogonality of Z: || Z'*Z - I ||_F = .78E-14
ALPHAR ALPHAI BETA
0.4723 0.1464 1.2811
0.4723 -0.1464 1.2811
-0.0318 0.1527 2.4691
-0.0318 -0.1527 2.4691
-0.6315 0.0000 0.8521
-0.5850 0.0000 0.0283
-0.7884 0.0000 0.4153
0.2832 0.0000 0.5894
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB03XS.html 0000664 0000000 0000000 00000016631 14560147231 0020375 0 ustar 00root root 0000000 0000000
Purpose
To compute the eigenvalues and real skew-Hamiltonian Schur form of
a skew-Hamiltonian matrix,
[ A G ]
W = [ T ],
[ Q A ]
where A is an N-by-N matrix and G, Q are N-by-N skew-symmetric
matrices. Specifically, an orthogonal symplectic matrix U is
computed so that
T [ Aout Gout ]
U W U = [ T ] ,
[ 0 Aout ]
where Aout is in Schur canonical form (as returned by the LAPACK
routine DHSEQR). That is, Aout is block upper triangular with
1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has
its diagonal elements equal and its off-diagonal elements of
opposite sign.
Optionally, the matrix U is returned in terms of its first N/2
rows
[ U1 U2 ]
U = [ ].
[ -U2 U1 ]
Specification
SUBROUTINE MB03XS( JOBU, N, A, LDA, QG, LDQG, U1, LDU1, U2, LDU2,
$ WR, WI, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBU
INTEGER INFO, LDA, LDQG, LDU1, LDU2, LDWORK, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), QG(LDQG,*), U1(LDU1,*),
$ U2(LDU2,*), WI(*), WR(*)
Arguments
Mode Parameters
JOBU CHARACTER*1
Specifies whether matrix U is computed or not, as follows:
= 'N': transformation matrix U is not computed;
= 'U': transformation matrix U is computed.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A.
On exit, the leading N-by-N part of this array contains
the matrix Aout in Schur canonical form.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
QG (input/output) DOUBLE PRECISION array, dimension
(LDQG,N+1)
On entry, the leading N-by-N+1 part of this array must
contain in columns 1:N the strictly lower triangular part
of the matrix Q and in columns 2:N+1 the strictly upper
triangular part of the matrix G.
On exit, the leading N-by-N+1 part of this array contains
in columns 2:N+1 the strictly upper triangular part of the
skew-symmetric matrix Gout. The part which contained the
matrix Q is set to zero.
Note that the parts containing the diagonal and the first
superdiagonal of this array are not overwritten by zeros
only if JOBU = 'U' or LDWORK >= 2*N*N - N.
LDQG INTEGER
The leading dimension of the array QG. LDQG >= MAX(1,N).
U1 (output) DOUBLE PRECISION array, dimension (LDU1,N)
On exit, if JOBU = 'U', the leading N-by-N part of this
array contains the matrix U1.
If JOBU = 'N', this array is not referenced.
LDU1 INTEGER
The leading dimension of the array U1.
LDU1 >= MAX(1,N), if JOBU = 'U';
LDU1 >= 1, if JOBU = 'N'.
U2 (output) DOUBLE PRECISION array, dimension (LDU2,N)
On exit, if JOBU = 'U', the leading N-by-N part of this
array contains the matrix U2.
If JOBU = 'N', this array is not referenced.
LDU2 INTEGER
The leading dimension of the array U2.
LDU2 >= MAX(1,N), if JOBU = 'U';
LDU2 >= 1, if JOBU = 'N'.
WR (output) DOUBLE PRECISION array, dimension (N)
WI (output) DOUBLE PRECISION array, dimension (N)
The real and imaginary parts, respectively, of the
eigenvalues of Aout, which are half of the eigenvalues
of W. The eigenvalues are stored in the same order as on
the diagonal of Aout, with WR(i) = Aout(i,i) and, if
Aout(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0
and WI(i+1) = -WI(i).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
On exit, if INFO = -14, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,(N+5)*N), if JOBU = 'U';
LDWORK >= MAX(1,5*N,(N+1)*N), if JOBU = 'N'.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, DHSEQR failed to compute all of the
eigenvalues. Elements 1:ILO-1 and i+1:N of WR
and WI contain those eigenvalues which have been
successfully computed. The matrix A (and QG) has
been partially reduced; namely, A is upper
Hessenberg in the rows and columns ILO through i.
(See DHSEQR for details.)
Method
First, using the SLICOT Library routine MB04RB, an orthogonal
symplectic matrix UP is computed so that
T [ AP GP ]
UP W UP = [ T ]
[ 0 AP ]
is in Paige/Van Loan form. Next, the LAPACK routine DHSEQR is
applied to the matrix AP to compute an orthogonal matrix V so
that Aout = V'*AP*V is in Schur canonical form.
Finally, the transformations
[ V 0 ]
U = UP * [ ], Gout = V'*G*V,
[ 0 V ]
using the SLICOT Library routine MB01LD for the latter, are
performed.
References
[1] Van Loan, C.F.
A symplectic method for approximating all the eigenvalues of
a Hamiltonian matrix.
Linear Algebra and its Applications, 61, pp. 233-251, 1984.
[2] Kressner, D.
Block algorithms for orthogonal symplectic factorizations.
BIT, 43 (4), pp. 775-790, 2003.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To reduce 2*nb columns and rows of a real (k+2n)-by-(k+2n)
matrix H:
[ op(A) G ]
H = [ ],
[ Q op(B) ]
so that elements in the first nb columns below the k-th
subdiagonal of the (k+n)-by-n matrix op(A), in the first nb
columns and rows of the n-by-n matrix Q and in the first nb rows
above the diagonal of the n-by-(k+n) matrix op(B) are zero.
The reduction is performed by orthogonal symplectic
transformations UU'*H*VV and matrices U, V, YA, YB, YG, YQ, XA,
XB, XG, and XQ are returned so that
[ op(Aout)+U*YA'+XA*V' G+U*YG'+XG*V' ]
UU' H VV = [ ].
[ Qout+U*YQ'+XQ*V' op(Bout)+U*YB'+XB*V' ]
This is an auxiliary routine called by MB04TB.
Specification
SUBROUTINE MB03XU( LTRA, LTRB, N, K, NB, A, LDA, B, LDB, G, LDG,
$ Q, LDQ, XA, LDXA, XB, LDXB, XG, LDXG, XQ, LDXQ,
$ YA, LDYA, YB, LDYB, YG, LDYG, YQ, LDYQ, CSL,
$ CSR, TAUL, TAUR, DWORK )
C .. Scalar Arguments ..
LOGICAL LTRA, LTRB
INTEGER K, LDA, LDB, LDG, LDQ, LDXA, LDXB, LDXG, LDXQ,
$ LDYA, LDYB, LDYG, LDYQ, N, NB
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), CSL(*), CSR(*), DWORK(*),
$ G(LDG,*), Q(LDQ,*), TAUL(*), TAUR(*),
$ XA(LDXA,*), XB(LDXB,*), XG(LDXG,*), XQ(LDXQ,*),
$ YA(LDYA,*), YB(LDYB,*), YG(LDYG,*), YQ(LDYQ,*)
Arguments
Mode Parameters
LTRA LOGICAL
Specifies the form of op( A ) as follows:
= .FALSE.: op( A ) = A;
= .TRUE.: op( A ) = A'.
LTRB LOGICAL
Specifies the form of op( B ) as follows:
= .FALSE.: op( B ) = B;
= .TRUE.: op( B ) = B'.
Input/Output Parameters
N (input) INTEGER
The order of the matrix Q. N >= 0.
K (input) INTEGER
The offset of the reduction. Elements below the K-th
subdiagonal in the first NB columns of op(A) are
reduced to zero. K >= 0.
NB (input) INTEGER
The number of columns/rows to be reduced. N > NB >= 0.
A (input/output) DOUBLE PRECISION array, dimension
(LDA,N) if LTRA = .FALSE.
(LDA,K+N) if LTRA = .TRUE.
On entry with LTRA = .FALSE., the leading (K+N)-by-N part
of this array must contain the matrix A.
On entry with LTRA = .TRUE., the leading N-by-(K+N) part
of this array must contain the matrix A.
On exit with LTRA = .FALSE., the leading (K+N)-by-N part
of this array contains the matrix Aout and, in the zero
parts, information about the elementary reflectors used to
compute the reduction.
On exit with LTRA = .TRUE., the leading N-by-(K+N) part of
this array contains the matrix Aout and in the zero parts
information about the elementary reflectors.
LDA INTEGER
The leading dimension of the array A.
LDA >= MAX(1,K+N), if LTRA = .FALSE.;
LDA >= MAX(1,N), if LTRA = .TRUE..
B (input/output) DOUBLE PRECISION array, dimension
(LDB,K+N) if LTRB = .FALSE.
(LDB,N) if LTRB = .TRUE.
On entry with LTRB = .FALSE., the leading N-by-(K+N) part
of this array must contain the matrix B.
On entry with LTRB = .TRUE., the leading (K+N)-by-N part
of this array must contain the matrix B.
On exit with LTRB = .FALSE., the leading N-by-(K+N) part
of this array contains the matrix Bout and, in the zero
parts, information about the elementary reflectors used to
compute the reduction.
On exit with LTRB = .TRUE., the leading (K+N)-by-N part of
this array contains the matrix Bout and in the zero parts
information about the elementary reflectors.
LDB INTEGER
The leading dimension of the array B.
LDB >= MAX(1,N), if LTRB = .FALSE.;
LDB >= MAX(1,K+N), if LTRB = .TRUE..
G (input/output) DOUBLE PRECISION array, dimension (LDG,N)
On entry, the leading N-by-N part of this array must
contain the matrix G.
On exit, the leading N-by-N part of this array contains
the matrix Gout.
LDG INTEGER
The leading dimension of the array G. LDG >= MAX(1,N).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, the leading N-by-N part of this array must
contain the matrix Q.
On exit, the leading N-by-N part of this array contains
the matrix Qout and in the zero parts information about
the elementary reflectors used to compute the reduction.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= MAX(1,N).
XA (output) DOUBLE PRECISION array, dimension (LDXA,2*NB)
On exit, the leading N-by-(2*NB) part of this array
contains the matrix XA.
LDXA INTEGER
The leading dimension of the array XA. LDXA >= MAX(1,N).
XB (output) DOUBLE PRECISION array, dimension (LDXB,2*NB)
On exit, the leading (K+N)-by-(2*NB) part of this array
contains the matrix XB.
LDXB INTEGER
The leading dimension of the array XB. LDXB >= MAX(1,K+N).
XG (output) DOUBLE PRECISION array, dimension (LDXG,2*NB)
On exit, the leading (K+N)-by-(2*NB) part of this array
contains the matrix XG.
LDXG INTEGER
The leading dimension of the array XG. LDXG >= MAX(1,K+N).
XQ (output) DOUBLE PRECISION array, dimension (LDXQ,2*NB)
On exit, the leading N-by-(2*NB) part of this array
contains the matrix XQ.
LDXQ INTEGER
The leading dimension of the array XQ. LDXQ >= MAX(1,N).
YA (output) DOUBLE PRECISION array, dimension (LDYA,2*NB)
On exit, the leading (K+N)-by-(2*NB) part of this array
contains the matrix YA.
LDYA INTEGER
The leading dimension of the array YA. LDYA >= MAX(1,K+N).
YB (output) DOUBLE PRECISION array, dimension (LDYB,2*NB)
On exit, the leading N-by-(2*NB) part of this array
contains the matrix YB.
LDYB INTEGER
The leading dimension of the array YB. LDYB >= MAX(1,N).
YG (output) DOUBLE PRECISION array, dimension (LDYG,2*NB)
On exit, the leading (K+N)-by-(2*NB) part of this array
contains the matrix YG.
LDYG INTEGER
The leading dimension of the array YG. LDYG >= MAX(1,K+N).
YQ (output) DOUBLE PRECISION array, dimension (LDYQ,2*NB)
On exit, the leading N-by-(2*NB) part of this array
contains the matrix YQ.
LDYQ INTEGER
The leading dimension of the array YQ. LDYQ >= MAX(1,N).
CSL (output) DOUBLE PRECISION array, dimension (2*NB)
On exit, the first 2NB elements of this array contain the
cosines and sines of the symplectic Givens rotations
applied from the left-hand side used to compute the
reduction.
CSR (output) DOUBLE PRECISION array, dimension (2*NB)
On exit, the first 2NB-2 elements of this array contain
the cosines and sines of the symplectic Givens rotations
applied from the right-hand side used to compute the
reduction.
TAUL (output) DOUBLE PRECISION array, dimension (NB)
On exit, the first NB elements of this array contain the
scalar factors of some of the elementary reflectors
applied form the left-hand side.
TAUR (output) DOUBLE PRECISION array, dimension (NB)
On exit, the first NB-1 elements of this array contain the
scalar factors of some of the elementary reflectors
applied form the right-hand side.
Workspace
DWORK DOUBLE PRECISION array, dimension (5*NB)Method
For details regarding the representation of the orthogonal
symplectic matrices UU and VV within the arrays A, B, CSL, CSR, Q,
TAUL and TAUR see the description of MB04TB.
The contents of A, B, G and Q on exit are illustrated by the
following example with op(A) = A, op(B) = B, n = 5, k = 2 and
nb = 2:
( a r r a a ) ( g g g r r g g )
( a r r a a ) ( g g g r r g g )
( r r r r r ) ( r r r r r r r )
A = ( u2 r r r r ), G = ( r r r r r r r ),
( u2 u2 r a a ) ( g g g r r g g )
( u2 u2 r a a ) ( g g g r r g g )
( u2 u2 r a a ) ( g g g r r g g )
( t t v1 v1 v1 ) ( r r r r r v2 v2 )
( u1 t t v1 v1 ) ( r r r r r r v2 )
Q = ( u1 u1 r q q ), B = ( b b b r r b b ).
( u1 u1 r q q ) ( b b b r r b b )
( u1 u1 r q q ) ( b b b r r b b )
where a, b, g and q denote elements of the original matrices, r
denotes a modified element, t denotes a scalar factor of an
applied elementary reflector, ui and vi denote elements of the
matrices U and V, respectively.
Numerical Aspects
The algorithm requires ( 16*K + 32*N + 42 )*N*NB + ( 16*K + 112*N - 208/3*NB - 69 )*NB*NB - 29/3*NB floating point operations and is numerically backward stable.References
[1] Benner, P., Mehrmann, V., and Xu, H.
A numerically stable, structure preserving method for
computing the eigenvalues of real Hamiltonian or symplectic
pencils.
Numer. Math., Vol. 78 (3), pp. 329-358, 1998.
[2] Kressner, D.
Block algorithms for orthogonal symplectic factorizations.
BIT Numerical Mathematics, 43 (4), pp. 775-790, 2003.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the eigenvalues of a Hamiltonian matrix,
[ A G ] H H
H = [ H ], G = G , Q = Q , (1)
[ Q -A ]
where A, G and Q are complex n-by-n matrices.
Due to the structure of H, if lambda is an eigenvalue, then
-conjugate(lambda) is also an eigenvalue. This does not mean that
purely imaginary eigenvalues are necessarily multiple. The routine
computes the eigenvalues of H using an embedding to a real skew-
Hamiltonian matrix He,
[ Ae Ge ] T T
He = [ T ], Ge = -Ge , Qe = -Qe , (2)
[ Qe Ae ]
where Ae, Ge, and Qe are real 2*n-by-2*n matrices, defined by
[ Im(A) Re(A) ]
Ae = [ ],
[ -Re(A) Im(A) ]
[ triu(Im(G)) Re(G) ]
triu(Ge) = [ ],
[ 0 triu(Im(G)) ]
[ tril(Im(Q)) 0 ]
tril(Qe) = [ ],
[ -Re(Q) tril(Im(Q)) ]
and triu and tril denote the upper and lower triangle,
respectively. Then, an orthogonal symplectic matrix Ue is used to
reduce He to the structured real Schur form
T [ Se De ] T
Ue He Ue = [ T ], De = -De , (3)
[ 0 Se ]
where Ue is a 4n-by-4n real symplectic matrix, and Se is upper
quasi-triangular (real Schur form).
Optionally, if JOB = 'S', or JOB = 'G', the matrix i*He is further
transformed to the structured complex Schur form
H [ Sc Gc ] H
U (i*He) U = [ H ], Gc = Gc , (4)
[ 0 -Sc ]
where U is a 4n-by-4n unitary symplectic matrix, and Sc is upper
triangular (Schur form).
The algorithm is backward stable and preserves the spectrum
structure in finite precision arithmetic.
Optionally, a symplectic balancing transformation to improve the
conditioning of eigenvalues is computed (see the SLICOT Library
routine MB04DZ). In this case, the matrix He in decompositions (3)
and (4) must be replaced by the balanced matrix.
Specification
SUBROUTINE MB03XZ( BALANC, JOB, JOBU, N, A, LDA, QG, LDQG, U1,
$ LDU1, U2, LDU2, WR, WI, ILO, SCALE, DWORK,
$ LDWORK, ZWORK, LZWORK, BWORK, INFO )
C .. Scalar Arguments ..
CHARACTER BALANC, JOB, JOBU
INTEGER ILO, INFO, LDA, LDQG, LDU1, LDU2, LDWORK,
$ LZWORK, N
C .. Array Arguments ..
LOGICAL BWORK( * )
DOUBLE PRECISION DWORK( * ), SCALE( * ), WI( * ), WR( * )
COMPLEX*16 A( LDA, * ), QG( LDQG, * ), U1( LDU1, * ),
$ U2( LDU2, * ), ZWORK( * )
Arguments
Mode Parameters
BALANC CHARACTER*1
Indicates how H should be diagonally scaled and/or
permuted to reduce its norm.
= 'N': Do not diagonally scale or permute;
= 'P': Perform symplectic permutations to make the matrix
closer to skew-Hamiltonian Schur form. Do not
diagonally scale;
= 'S': Diagonally scale the matrix, i.e., replace A, G and
Q by D*A*D**(-1), D*G*D and D**(-1)*Q*D**(-1) where
D is a diagonal matrix chosen to make the rows and
columns of H more equal in norm. Do not permute;
= 'B': Both diagonally scale and permute A, G and Q.
Permuting does not change the norm of H, but scaling does.
JOB CHARACTER*1
Indicates whether the user wishes to compute the full
decomposition (4) or the eigenvalues only, as follows:
= 'E': compute the eigenvalues only;
= 'S': compute the matrix Sc of (4);
= 'G': compute the matrices Sc and Gc of (4).
JOBU CHARACTER*1
Indicates whether or not the user wishes to compute the
symplectic matrix Ue of (3), if JOB = 'E', or U of (4),
if JOB = 'S' or JOB = 'G', as follows:
= 'N': the matrix Ue or U is not computed;
= 'U': the matrix Ue or U is computed.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) COMPLEX*16 array, dimension (LDA,K)
where K = N, if JOB = 'E', and K = 2*N, if JOB <> 'E'.
On entry, the leading N-by-N part of this array must
contain the matrix A.
On exit, if JOB = 'E', the leading N-by-N part of this
array is unchanged, if BALANC = 'N', or it contains the
balanced (permuted and/or scaled) matrix A, if
BALANC <> 'N'.
On exit, if JOB = 'S' or JOB = 'G', the leading 2*N-by-2*N
upper triangular part of this array contains the matrix Sc
(complex Schur form) of decomposition (4).
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,K).
QG (input/output) COMPLEX*16 array, dimension
(LDQG,min(K+1,2*N))
On entry, the leading N-by-N+1 part of this array must
contain in columns 1:N the lower triangular part of the
matrix Q and in columns 2:N+1 the upper triangular part
of the matrix G.
On exit, if JOB <> 'G', the leading N-by-N+1 part of this
array is unchanged, if BALANC = 'N', or it contains the
balanced (permuted and/or scaled) parts of the matrices
Q and G (as above), if BALANC <> 'N'.
On exit, JOB = 'G', the leading 2*N-by-2*N upper
triangular part of this array contains the upper
triangular part of the matrix Gc in the decomposition (4).
LDQG INTEGER
The leading dimension of the array QG. LDQG >= max(1,K).
U1 (output) COMPLEX*16 array, dimension (LDU1,2*N)
On exit, if JOB = 'S' or JOB = 'G', and JOBU = 'U', the
leading 2*N-by-2*N part of this array contains the (1,1)
block of the unitary symplectic matrix U of the
decomposition (4).
If JOB = 'E' or JOBU = 'N', this array is not referenced.
LDU1 INTEGER
The leading dimension of the array U1. LDU1 >= 1.
LDU1 >= 2*N, if JOBU = 'U'.
U2 (output) COMPLEX*16 array, dimension (LDU2,2*N)
On exit, if JOB = 'S' or JOB = 'G', and JOBU = 'U', the
leading 2*N-by-2*N part of this array contains the (1,2)
block of the unitary symplectic matrix U of the
decomposition (4).
If JOB = 'E' or JOBU = 'N', this array is not referenced.
LDU2 INTEGER
The leading dimension of the array U2. LDU2 >= 1.
LDU2 >= 2*N, if JOBU = 'U'.
WR (output) DOUBLE PRECISION array, dimension (2*N)
WI (output) DOUBLE PRECISION array, dimension (2*N)
On exit, the leading 2*N elements of WR and WI contain the
real and imaginary parts, respectively, of the eigenvalues
of the Hamiltonian matrix H.
ILO (output) INTEGER
ILO is an integer value determined when H was balanced.
The balanced A(I,J) = 0 if I > J and J = 1,...,ILO-1.
The balanced Q(I,J) = 0 if J = 1,...,ILO-1 or
I = 1,...,ILO-1.
SCALE (output) DOUBLE PRECISION array, dimension (N)
On exit, if BALANC <> 'N', the leading N elements of this
array contain details of the permutation and/or scaling
factors applied when balancing H, see MB04DZ.
This array is not referenced if BALANC = 'N'.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK, and DWORK(2) returns the 1-norm of the
(scaled, if BALANC = 'S' or 'B') Hamiltonian matrix.
Moreover, the next locations of this array have the
following content:
- The leading 2*N-by-2*N upper Hessenberg part in the
locations 3:2+4*N*N contains the upper Hessenberg part of
the real Schur matrix Se in the decomposition (3);
- the leading 2*N-by-2*N upper triangular part in the
locations 3+4*N*N+2*N:2+8*N*N+2*N contains the upper
triangular part of the skew-symmetric matrix De in the
decomposition (3).
- If JOBU = 'U', the leading 2*N-by-2*N part in the
locations 3+8*N*N+2*N:2+12*N*N+2*N contains the (1,1)
block of the orthogonal symplectic matrix Ue of
decomposition (3).
- the leading 2*N-by-2*N part in the locations
3+12*N*N+2*N:2+16*N*N+2*N contains the (2,1) block of the
orthogonal symplectic matrix Ue.
On exit, if INFO = -18, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= MAX( 12*N**2 + 4*N, 8*N**2 + 12*N ) + 2,
if JOB = 'E' and JOBU = 'N';
LDWORK >= MAX( 2, 12*N**2 + 4*N, 8*N**2 + 12*N ),
if JOB = 'S' or 'G' and JOBU = 'N';
LDWORK >= 20*N**2 + 12*N + 2,
if JOB = 'E' and JOBU = 'U';
LDWORK >= MAX( 2, 20*N**2 + 12*N ),
if JOB = 'S' or 'G' and JOBU = 'U'.
For good performance, LDWORK must generally be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) returns the optimal
value of LZWORK.
On exit, if INFO = -20, ZWORK(1) returns the minimum
value of LZWORK.
LZWORK INTEGER
The dimension of the array ZWORK.
LZWORK >= 1, if JOB = 'E';
LZWORK >= MAX( 1, 12*N - 6 ), if JOB = 'S' and JOBU = 'N';
LZWORK >= MAX( 1, 12*N - 2 ), if JOB = 'G' or JOBU = 'U'.
If LZWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
ZWORK array, returns this value as the first entry of
the ZWORK array, and no error message related to LZWORK
is issued by XERBLA.
BWORK LOGICAL array, dimension (LBWORK)
LBWORK >= 0, if JOB = 'E';
LBWORK >= 2*N-1, if JOB = 'S' or JOB = 'G'.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, the QR algorithm failed to compute
all the eigenvalues; elements i+1:2*N of WR and
WI contain eigenvalues which have converged;
= 2*N+1: the QR algorithm failed to compute the
eigenvalues of a 2-by-2 real block.
Method
First, the extended matrix He in (2) is built. Then, the structured real Schur form in (3) is computed, using the SLICOT Library routine MB03XS. The eigenvalues of Se immediately give the eigenvalues of H. Finally, if required, Se is further transformed by using the complex QR algorithm to triangularize its 2-by-2 blocks, and Ge and U are updated, to obtain (4).References
[1] Benner, P., Mehrmann, V. and Xu, H.
A note on the numerical solution of complex Hamiltonian and
skew-Hamiltonian eigenvalue problems.
Electr. Trans. Num. Anal., 8, pp. 115-126, 1999.
[2] Van Loan, C.F.
A symplectic method for approximating all the eigenvalues of
a Hamiltonian matrix.
Linear Algebra and its Applications, 61, pp. 233-251, 1984.
Further Comments
NoneExample
Program Text
* MB03XZ EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZER
PARAMETER ( ZER = 0.0D0 )
COMPLEX*16 ZERO, ONE
PARAMETER ( ZERO = (0.0D0,0.0D0), ONE = (1.0D0,0.0D0) )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 100 )
INTEGER LDA, LDAE, LDAS, LDGE, LDQG, LDQGE, LDQGS, LDRES,
$ LDU1, LDU2, LDWORK, LZWORK
PARAMETER ( LDA = 2*NMAX, LDAE = 2*NMAX, LDAS = NMAX,
$ LDGE = 2*NMAX, LDQG = 2*NMAX, LDQGE = 2*NMAX,
$ LDQGS = NMAX, LDRES = 2*NMAX, LDU1 = 2*NMAX,
$ LDU2 = 2*NMAX,
$ LDWORK = 20*NMAX*NMAX + 12*NMAX + 2,
$ LZWORK = 12*NMAX - 2 )
* .. Local Scalars ..
CHARACTER*1 BALANC, JOB, JOBU
INTEGER I, ILO, INFO, J, M, N
DOUBLE PRECISION TEMP
* .. Local Arrays ..
COMPLEX*16 A(LDA, 2*NMAX), AE(LDAE, 2*NMAX), AS(LDAS, NMAX),
$ GE(LDGE, 2*NMAX), QG(LDQG, 2*NMAX+1),
$ QGE(LDQGE, 2*NMAX+1), QGS(LDQGS, NMAX+1),
$ RES(LDRES,2*NMAX), U1(LDU1,2*NMAX),
$ U2(LDU2, 2*NMAX), ZWORK(LZWORK)
DOUBLE PRECISION DWORK(LDWORK), SCALE(NMAX), WI(2*NMAX),
$ WR(2*NMAX)
LOGICAL BWORK(2*NMAX)
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAPY2, MA02JZ, ZLANGE
EXTERNAL DLAPY2, LSAME, MA02JZ, ZLANGE
* .. External Subroutines ..
EXTERNAL MA02EZ, MB03XZ, MB04DZ, ZCOPY, ZGEMM, ZLACPY,
$ ZLASET
* ..Intrinsic Functions..
INTRINSIC DBLE, DCMPLX, DIMAG
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, BALANC, JOB, JOBU
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
M = 2*N
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
CALL ZLACPY( 'All', N, N, A, LDA, AS, LDAS )
READ ( NIN, FMT = * ) ( ( QG(I,J), J = 1,N+1 ), I = 1,N )
CALL ZLACPY( 'All', N, N+1, QG, LDQG, QGS, LDQGS )
* Compute the eigenvalues and the transformed Hamiltonian matrix.
CALL MB03XZ( BALANC, JOB, JOBU, N, A, LDA, QG, LDQG, U1, LDU1,
$ U2, LDU2, WR, WI, ILO, SCALE, DWORK, LDWORK,
$ ZWORK, LZWORK, BWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, M
WRITE ( NOUT, FMT = 99996 ) I, WR(I), WI(I)
10 CONTINUE
IF ( LSAME( JOB, 'S' ).OR.LSAME( JOB, 'G' ) ) THEN
WRITE ( NOUT, FMT = 99995 )
DO 20 I = 1, M
WRITE ( NOUT, FMT = 99992 ) ( A(I,J), J = 1,M )
20 CONTINUE
END IF
IF ( LSAME( JOB, 'G' ) ) THEN
WRITE ( NOUT, FMT = 99994 )
DO 30 I = 1, M
WRITE ( NOUT, FMT = 99992 ) ( QG(I,J), J = 1,M )
30 CONTINUE
END IF
*
IF ( LSAME( JOB, 'G' ).AND.LSAME( JOBU, 'U' ) ) THEN
* Compute the residual of the formula (4) in MB03XZ.
CALL MB04DZ( BALANC, N, AS, LDAS, QGS, LDQGS, I, DWORK,
$ INFO )
CALL ZLASET( 'Lower', M-1, M-1, ZERO, ZERO, A(2,1), LDA )
CALL MA02EZ( 'Upper', 'Conjugate', 'Not skew', M, QG,
$ LDQG )
* Compute Ae, Ge, and Qe.
DO 60 J = 1, N
DO 40 I = 1, N
AE(I,J) = DCMPLX( ZER, DIMAG( AS(I,J) ) )
40 CONTINUE
DO 50 I = 1, N
AE(I+N,J) = -DCMPLX( ZER, DBLE( AS(I,J) ) )
50 CONTINUE
60 CONTINUE
*
DO 90 J = 1, N
DO 70 I = 1, N
AE(I,J+N) = -AE(I+N,J)
70 CONTINUE
DO 80 I = 1, N
AE(I+N,J+N) = AE(I,J)
80 CONTINUE
90 CONTINUE
*
DO 120 J = 1, N+1
DO 100 I = 1, N
QGE(I,J) = DCMPLX( ZER, DIMAG( QGS(I,J) ) )
100 CONTINUE
DO 110 I = J, N
QGE(I+N,J) = -DCMPLX( ZER, DBLE( QGS(I,J) ) )
110 CONTINUE
120 CONTINUE
*
DO 150 J = 1, N
DO 130 I = 1, J
QGE(I,J+N+1) = DCMPLX( ZER, DBLE( QGS(I,J+1) ) )
130 CONTINUE
DO 140 I = 1, N
QGE(I+N,J+N+1) = QGE(I,J+1)
140 CONTINUE
150 CONTINUE
CALL ZCOPY( N, QGE, 1, QGE(N+1,N+1), 1 )
CALL MA02EZ( 'Lower', 'Transpose', 'Not Skew', N,
$ QGE(N+1,1), LDQGE )
CALL MA02EZ( 'Upper', 'Transpose', 'Not Skew', N,
$ QGE(1,N+2), LDQGE )
*
CALL ZLACPY( 'Upper', M, M, QGE(1,2), LDQGE, GE, LDGE )
CALL MA02EZ( 'Upper', 'Transpose', 'Skew', M, GE, LDGE )
CALL MA02EZ( 'Lower', 'Transpose', 'Skew', M, QGE,
$ LDQGE )
* Compute the residual of the (1,1) block in (4).
CALL ZGEMM( 'No Transpose', 'No Transpose', M, M, M, ONE,
$ AE, LDAE, U1, LDU1, ZERO, RES, LDRES )
CALL ZGEMM( 'No Transpose', 'No Transpose', M, M, M,
$ -ONE, GE, LDGE, U2, LDU2, ONE, RES, LDRES )
CALL ZGEMM( 'No Transpose', 'No Transpose', M, M, M,
$ -ONE, U1, LDU1, A, LDA, ONE, RES, LDRES )
TEMP = ZLANGE( 'Frobenius', M, M, RES, LDRES, DWORK )
* Compute the residual of the (1,2) block in (4).
CALL ZGEMM( 'No Transpose', 'No Transpose', M, M, M, ONE,
$ AE, LDAE, U2, LDU2, ZERO, RES, LDRES )
CALL ZGEMM( 'No Transpose', 'No Transpose', M, M, M, ONE,
$ GE, LDGE, U1, LDU1, ONE, RES, LDRES )
CALL ZGEMM( 'No Transpose', 'No Transpose', M, M, M,
$ -ONE, U1, LDU1, QG, LDQG, ONE, RES, LDRES )
CALL ZGEMM( 'No Transpose', 'Conj Transpose', M, M, M,
$ ONE, U2, LDU2, A, LDA, ONE, RES, LDRES )
TEMP = DLAPY2( TEMP, ZLANGE( 'Frobenius', M, M, RES,
$ LDRES, DWORK ) )
* Compute the residual of the (2,1) block in (4).
CALL ZGEMM( 'No Transpose', 'No Transpose', M, M, M, ONE,
$ QGE, LDQGE, U1, LDU1, ZERO, RES, LDRES )
CALL ZGEMM( 'Transpose', 'No Transpose', M, M, M,
$ -ONE, AE, LDAE, U2, LDU2, ONE, RES, LDRES )
CALL ZGEMM( 'No Transpose', 'No Transpose', M, M, M, ONE,
$ U2, LDU2, A, LDA, ONE, RES, LDRES )
TEMP = DLAPY2( TEMP, ZLANGE( 'Frobenius', M, M, RES,
$ LDRES, DWORK ) )
* Compute the residual of the (2,2) block in (4).
CALL ZGEMM( 'No Transpose', 'No Transpose', M, M, M, ONE,
$ QGE, LDQGE, U2, LDU2, ZERO, RES, LDRES )
CALL ZGEMM( 'Transpose', 'No Transpose', M, M, M, ONE,
$ AE, LDAE, U1, LDU1, ONE, RES, LDRES )
CALL ZGEMM( 'No Transpose', 'No Transpose', M, M, M, ONE,
$ U2, LDU2, QG, LDQG, ONE, RES, LDRES )
CALL ZGEMM( 'No Transpose', 'Conj Transpose', M, M, M,
$ ONE, U1, LDU1, A, LDA, ONE, RES, LDRES )
TEMP = DLAPY2( TEMP, ZLANGE( 'Frobenius', M, M, RES,
$ LDRES, DWORK ) )
WRITE ( NOUT, FMT = 99989 ) TEMP
END IF
*
IF ( .NOT.LSAME( JOB, 'E' ).AND.LSAME( JOBU, 'U' ) ) THEN
WRITE ( NOUT, FMT = 99993 )
DO 160 I = 1, M
WRITE ( NOUT, FMT = 99992 )
$ ( U1(I,J), J = 1,M ), ( U2(I,J), J = 1,M )
160 CONTINUE
DO 170 I = 1, M
WRITE ( NOUT, FMT = 99992 )
$ ( -U2(I,J), J = 1,M ), ( U1(I,J), J = 1,M )
170 CONTINUE
WRITE ( NOUT, FMT = 99988 ) MA02JZ( .FALSE., .FALSE., M,
$ U1, LDU1, U2, LDU2, RES, LDRES )
END IF
IF ( LSAME( BALANC, 'S' ).OR.LSAME( BALANC, 'B' ) ) THEN
WRITE ( NOUT, FMT = 99991 )
DO 180 I = 1, N
WRITE ( NOUT, FMT = 99996 ) I, SCALE(I)
180 CONTINUE
END IF
END IF
END IF
*
99999 FORMAT (' MB03XZ EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB03XZ = ',I2)
99997 FORMAT (' The eigenvalues are',//' i',6X,
$ 'WR(i)',6X,'WI(i)',/)
99996 FORMAT (I4,3X,F8.4,3X,F8.4)
99995 FORMAT (/' The transformed matrix S is')
99994 FORMAT (/' The transformed matrix G is')
99993 FORMAT (/' The unitary symplectic factor U is')
99992 FORMAT (20(1X,F9.4,SP,F9.4,S,'i '))
99991 FORMAT (/' The diagonal scaling factors are ',//' i',6X,
$ 'SCALE(i)',/)
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' Residual: || i*He*U - U*Hc ||_F = ',G9.2)
99988 FORMAT (/' Orthogonality of U: || U^H U - I ||_F = ',G9.2)
END
Program Data
MB03XZ EXAMPLE PROGRAM DATA
4 N G U
(0.8147,0.4217) (0.6323,0.6557) (0.9575,0.6787) (0.9571,0.6554)
(0.9057,0.9157) (0.0975,0.0357) (0.9648,0.7577) (0.4853,0.1711)
(0.1269,0.7922) (0.2784,0.8491) (0.1576,0.7431) (0.8002,0.7060)
(0.9133,0.9594) (0.5468,0.9339) (0.9705,0.3922) (0.1418,0.0318)
0.2769 0.6948 (0.4387,0.7513) (0.1869,0.8909) (0.7094,0.1493)
(0.0462,0.1626) 0.3171 0.3816 (0.4898,0.9593) (0.7547,0.2575)
(0.0971,0.1190) (0.9502,0.5853) 0.7655 0.4456 (0.2760,0.8407)
(0.8235,0.4984) (0.0344,0.2238) (0.7952,0.6991) 0.6463 0.6797
Program Results
MB03XZ EXAMPLE PROGRAM RESULTS
The eigenvalues are
i WR(i) WI(i)
1 3.0844 2.7519
2 -3.0844 2.7519
3 0.5241 -1.3026
4 -0.5241 -1.3026
5 0.8824 -0.6918
6 -0.8824 -0.6918
7 0.4459 0.4748
8 -0.4459 0.4748
The transformed matrix S is
3.0844 +2.7519i 0.0618 +0.0000i -0.1952 +0.1977i 0.0439 +0.0628i 0.0599 -0.0344i -0.1543 -0.7126i -0.3906 +0.3615i 0.2877 +0.5766i
0.0000 +0.0000i -3.0844 +2.7519i -0.0458 -0.0727i -0.2607 +0.0867i 0.1505 -0.7137i -0.0717 +0.0066i -0.4008 +0.4356i 0.2819 +0.5317i
0.1269 +0.7922i 0.0000 +0.0000i 0.5241 -1.3026i -0.0175 +0.0350i -0.0676 +0.1183i 0.3695 -0.0335i -0.3138 -0.4268i 0.2973 -0.0042i
0.9133 +0.9594i 0.5468 +0.9339i 0.0000 +0.0000i -0.5241 -1.3026i 0.1453 +0.3375i 0.0590 -0.1483i 0.2795 +0.3002i -0.4594 -0.0099i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.8824 -0.6918i -0.1193 +0.0000i -0.1672 -0.0189i -0.1008 -0.2026i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i -0.8824 -0.6918i 0.0539 -0.1852i 0.1978 -0.0688i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.4459 +0.4748i 0.2987 +0.0000i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i -0.4459 +0.4748i
The transformed matrix G is
-0.2169 +0.0000i -0.0022 +0.0000i -0.2082 -0.0281i 0.0691 +0.0092i 0.0362 -0.0426i -0.2374 +0.4904i -0.2678 -0.5870i -0.2567 +0.3258i
0.0462 +0.1626i 0.2169 +0.0000i 0.0128 -0.0651i -0.0654 +0.2006i 0.2348 +0.4861i -0.0545 -0.0706i 0.1557 +0.4895i 0.3329 -0.4499i
0.0971 +0.1190i 0.9502 +0.5853i 0.1341 +0.0000i -0.0022 +0.0045i -0.0232 +0.0320i 0.3395 -0.3847i -0.0646 -0.2900i -0.0920 -0.1605i
0.8235 +0.4984i 0.0344 +0.2238i 0.7952 +0.6991i -0.1341 +0.0000i -0.1893 +0.4728i 0.0127 -0.0720i -0.0923 -0.0258i -0.3361 +0.0706i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i -0.5145 +0.0000i 0.0348 +0.0000i -0.0662 -0.2049i 0.2987 -0.0488i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.5145 +0.0000i -0.3004 +0.0327i -0.0524 -0.2070i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0241 +0.0000i 0.0081 +0.0000i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i -0.0241 +0.0000i
Residual: || i*He*U - U*Hc ||_F = .11E-13
The unitary symplectic factor U is
-0.3728 +0.1313i -0.3766 -0.1300i 0.1039 +0.2714i 0.1856 +0.2134i 0.4599 -0.0392i 0.4298 +0.0419i 0.1896 +0.1017i 0.2634 -0.0732i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i
-0.3079 +0.0989i -0.3110 -0.0979i -0.0062 +0.0288i 0.0277 +0.0067i -0.2041 +0.1907i -0.1908 -0.2040i -0.1357 -0.0260i -0.1886 +0.0187i 0.0570 -0.0194i 0.0576 +0.0192i -0.1622 +0.3576i 0.3834 +0.0034i 0.0436 +0.0676i 0.0408 -0.0723i -0.2802 +0.1272i -0.3893 -0.0915i
-0.2929 -0.0333i -0.2958 +0.0330i 0.1727 -0.3337i -0.3677 +0.0166i 0.1022 +0.0500i 0.0956 -0.0535i -0.2588 -0.0951i -0.3596 +0.0684i 0.0887 -0.0452i 0.0896 +0.0448i 0.0357 +0.0168i -0.0021 +0.0404i -0.0614 +0.0720i -0.0574 -0.0770i 0.2713 -0.1920i 0.3769 +0.1382i
-0.3061 +0.1212i -0.3092 -0.1200i -0.1737 -0.1180i -0.0210 -0.2121i -0.3424 -0.1782i -0.3201 +0.1906i 0.1698 +0.1487i 0.2359 -0.1070i 0.1763 -0.0047i 0.1781 +0.0046i 0.1017 -0.2162i -0.2335 +0.0013i 0.1817 +0.0174i 0.1699 -0.0186i 0.0902 +0.1296i 0.1253 -0.0933i
0.1153 +0.3771i 0.1164 -0.3734i 0.3105 +0.0106i -0.1350 +0.2928i -0.0741 +0.3559i -0.0692 -0.3808i 0.0178 +0.0002i 0.0247 -0.0002i -0.0148 -0.0576i -0.0150 +0.0570i 0.0145 -0.2255i -0.2011 -0.0837i 0.0523 +0.0093i 0.0488 -0.0099i 0.0689 +0.2345i 0.0957 -0.1688i
0.0856 +0.3022i 0.0864 -0.2992i 0.1935 +0.0888i -0.0133 +0.2179i -0.1613 -0.2850i -0.1508 +0.3049i -0.0548 +0.2795i -0.0761 -0.2011i -0.0226 -0.1038i -0.0228 +0.1027i 0.1693 +0.2699i 0.1539 +0.2735i -0.1978 -0.0842i -0.1849 +0.0901i 0.0400 -0.1732i 0.0556 +0.1246i
-0.0241 +0.2666i -0.0243 -0.2639i -0.3169 -0.0146i 0.1346 -0.3005i 0.0045 +0.1961i 0.0042 -0.2098i 0.0873 -0.3068i 0.1213 +0.2208i -0.0669 -0.1453i -0.0676 +0.1439i 0.1604 +0.0320i -0.0469 +0.1627i -0.0078 -0.1319i -0.0073 +0.1411i -0.0086 -0.4118i -0.0120 +0.2963i
0.1102 +0.2727i 0.1113 -0.2699i -0.2494 -0.0928i 0.0359 -0.2715i 0.1998 -0.2115i 0.1867 +0.2263i -0.0988 -0.1334i -0.1373 +0.0960i -0.0318 -0.2136i -0.0321 +0.2115i -0.1467 +0.0498i 0.1110 -0.1148i -0.0279 +0.2616i -0.0261 -0.2799i 0.0611 +0.3196i 0.0849 -0.2300i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i -0.3728 +0.1313i -0.3766 -0.1300i 0.1039 +0.2714i 0.1856 +0.2134i 0.4599 -0.0392i 0.4298 +0.0419i 0.1896 +0.1017i 0.2634 -0.0732i
-0.0570 +0.0194i -0.0576 -0.0192i 0.1622 -0.3576i -0.3834 -0.0034i -0.0436 -0.0676i -0.0408 +0.0723i 0.2802 -0.1272i 0.3893 +0.0915i -0.3079 +0.0989i -0.3110 -0.0979i -0.0062 +0.0288i 0.0277 +0.0067i -0.2041 +0.1907i -0.1908 -0.2040i -0.1357 -0.0260i -0.1886 +0.0187i
-0.0887 +0.0452i -0.0896 -0.0448i -0.0357 -0.0168i 0.0021 -0.0404i 0.0614 -0.0720i 0.0574 +0.0770i -0.2713 +0.1920i -0.3769 -0.1382i -0.2929 -0.0333i -0.2958 +0.0330i 0.1727 -0.3337i -0.3677 +0.0166i 0.1022 +0.0500i 0.0956 -0.0535i -0.2588 -0.0951i -0.3596 +0.0684i
-0.1763 +0.0047i -0.1781 -0.0046i -0.1017 +0.2162i 0.2335 -0.0013i -0.1817 -0.0174i -0.1699 +0.0186i -0.0902 -0.1296i -0.1253 +0.0933i -0.3061 +0.1212i -0.3092 -0.1200i -0.1737 -0.1180i -0.0210 -0.2121i -0.3424 -0.1782i -0.3201 +0.1906i 0.1698 +0.1487i 0.2359 -0.1070i
0.0148 +0.0576i 0.0150 -0.0570i -0.0145 +0.2255i 0.2011 +0.0837i -0.0523 -0.0093i -0.0488 +0.0099i -0.0689 -0.2345i -0.0957 +0.1688i 0.1153 +0.3771i 0.1164 -0.3734i 0.3105 +0.0106i -0.1350 +0.2928i -0.0741 +0.3559i -0.0692 -0.3808i 0.0178 +0.0002i 0.0247 -0.0002i
0.0226 +0.1038i 0.0228 -0.1027i -0.1693 -0.2699i -0.1539 -0.2735i 0.1978 +0.0842i 0.1849 -0.0901i -0.0400 +0.1732i -0.0556 -0.1246i 0.0856 +0.3022i 0.0864 -0.2992i 0.1935 +0.0888i -0.0133 +0.2179i -0.1613 -0.2850i -0.1508 +0.3049i -0.0548 +0.2795i -0.0761 -0.2011i
0.0669 +0.1453i 0.0676 -0.1439i -0.1604 -0.0320i 0.0469 -0.1627i 0.0078 +0.1319i 0.0073 -0.1411i 0.0086 +0.4118i 0.0120 -0.2963i -0.0241 +0.2666i -0.0243 -0.2639i -0.3169 -0.0146i 0.1346 -0.3005i 0.0045 +0.1961i 0.0042 -0.2098i 0.0873 -0.3068i 0.1213 +0.2208i
0.0318 +0.2136i 0.0321 -0.2115i 0.1467 -0.0498i -0.1110 +0.1148i 0.0279 -0.2616i 0.0261 +0.2799i -0.0611 -0.3196i -0.0849 +0.2300i 0.1102 +0.2727i 0.1113 -0.2699i -0.2494 -0.0928i 0.0359 -0.2715i 0.1998 -0.2115i 0.1867 +0.2263i -0.0988 -0.1334i -0.1373 +0.0960i
Orthogonality of U: || U^H U - I ||_F = .39E-14
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB03YA.html 0000664 0000000 0000000 00000016264 14560147231 0020356 0 ustar 00root root 0000000 0000000
Purpose
To annihilate one or two entries on the subdiagonal of the Hessenberg matrix A for dealing with zero elements on the diagonal of the triangular matrix B. MB03YA is an auxiliary routine called by SLICOT Library routines MB03XP and MB03YD.Specification
SUBROUTINE MB03YA( WANTT, WANTQ, WANTZ, N, ILO, IHI, ILOQ, IHIQ,
$ POS, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO )
C .. Scalar Arguments ..
LOGICAL WANTQ, WANTT, WANTZ
INTEGER IHI, IHIQ, ILO, ILOQ, INFO, LDA, LDB, LDQ, LDZ,
$ N, POS
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), Q(LDQ,*), Z(LDZ,*)
Arguments
Mode Parameters
WANTT LOGICAL
Indicates whether the user wishes to compute the full
Schur form or the eigenvalues only, as follows:
= .TRUE. : Compute the full Schur form;
= .FALSE.: compute the eigenvalues only.
WANTQ LOGICAL
Indicates whether or not the user wishes to accumulate
the matrix Q as follows:
= .TRUE. : The matrix Q is updated;
= .FALSE.: the matrix Q is not required.
WANTZ LOGICAL
Indicates whether or not the user wishes to accumulate
the matrix Z as follows:
= .TRUE. : The matrix Z is updated;
= .FALSE.: the matrix Z is not required.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A and B. N >= 0.
ILO (input) INTEGER
IHI (input) INTEGER
It is assumed that the matrices A and B are already
(quasi) upper triangular in rows and columns 1:ILO-1 and
IHI+1:N. The routine works primarily with the submatrices
in rows and columns ILO to IHI, but applies the
transformations to all the rows and columns of the
matrices A and B, if WANTT = .TRUE..
1 <= ILO <= max(1,N); min(ILO,N) <= IHI <= N.
ILOQ (input) INTEGER
IHIQ (input) INTEGER
Specify the rows of Q and Z to which transformations
must be applied if WANTQ = .TRUE. and WANTZ = .TRUE.,
respectively.
1 <= ILOQ <= ILO; IHI <= IHIQ <= N.
POS (input) INTEGER
The position of the zero element on the diagonal of B.
ILO <= POS <= IHI.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the upper Hessenberg matrix A.
On exit, the leading N-by-N part of this array contains
the updated matrix A where A(POS,POS-1) = 0, if POS > ILO,
and A(POS+1,POS) = 0, if POS < IHI.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the leading N-by-N part of this array must
contain an upper triangular matrix B with B(POS,POS) = 0.
On exit, the leading N-by-N part of this array contains
the updated upper triangular matrix B.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if WANTQ = .TRUE., then the leading N-by-N part
of this array must contain the current matrix Q of
transformations accumulated by MB03XP.
On exit, if WANTQ = .TRUE., then the leading N-by-N part
of this array contains the matrix Q updated in the
submatrix Q(ILOQ:IHIQ,ILO:IHI).
If WANTQ = .FALSE., Q is not referenced.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= 1.
If WANTQ = .TRUE., LDQ >= MAX(1,N).
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
On entry, if WANTZ = .TRUE., then the leading N-by-N part
of this array must contain the current matrix Z of
transformations accumulated by MB03XP.
On exit, if WANTZ = .TRUE., then the leading N-by-N part
of this array contains the matrix Z updated in the
submatrix Z(ILOQ:IHIQ,ILO:IHI).
If WANTZ = .FALSE., Z is not referenced.
LDZ INTEGER
The leading dimension of the array Z. LDZ >= 1.
If WANTZ = .TRUE., LDZ >= MAX(1,N).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The method is illustrated by Wilkinson diagrams for N = 5,
POS = 3:
[ x x x x x ] [ x x x x x ]
[ x x x x x ] [ o x x x x ]
A = [ o x x x x ], B = [ o o o x x ].
[ o o x x x ] [ o o o x x ]
[ o o o x x ] [ o o o o x ]
First, a QR factorization is applied to A(1:3,1:3) and the
resulting nonzero in the updated matrix B is immediately
annihilated by a Givens rotation acting on columns 1 and 2:
[ x x x x x ] [ x x x x x ]
[ x x x x x ] [ o x x x x ]
A = [ o o x x x ], B = [ o o o x x ].
[ o o x x x ] [ o o o x x ]
[ o o o x x ] [ o o o o x ]
Secondly, an RQ factorization is applied to A(4:5,4:5) and the
resulting nonzero in the updated matrix B is immediately
annihilated by a Givens rotation acting on rows 4 and 5:
[ x x x x x ] [ x x x x x ]
[ x x x x x ] [ o x x x x ]
A = [ o o x x x ], B = [ o o o x x ].
[ o o o x x ] [ o o o x x ]
[ o o o x x ] [ o o o o x ]
References
[1] Bojanczyk, A.W., Golub, G.H., and Van Dooren, P.
The periodic Schur decomposition: Algorithms and applications.
Proc. of the SPIE Conference (F.T. Luk, Ed.), 1770, pp. 31-42,
1992.
Numerical Aspects
The algorithm requires O(N**2) floating point operations and is backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To deal with small subtasks of the product eigenvalue problem. MB03YD is an auxiliary routine called by SLICOT Library routine MB03XP.Specification
SUBROUTINE MB03YD( WANTT, WANTQ, WANTZ, N, ILO, IHI, ILOQ, IHIQ,
$ A, LDA, B, LDB, Q, LDQ, Z, LDZ, ALPHAR, ALPHAI,
$ BETA, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
LOGICAL WANTQ, WANTT, WANTZ
INTEGER IHI, IHIQ, ILO, ILOQ, INFO, LDA, LDB, LDQ,
$ LDWORK, LDZ, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), ALPHAI(*), ALPHAR(*), B(LDB,*),
$ BETA(*), DWORK(*), Q(LDQ,*), Z(LDZ,*)
Arguments
Mode Parameters
WANTT LOGICAL
Indicates whether the user wishes to compute the full
Schur form or the eigenvalues only, as follows:
= .TRUE. : Compute the full Schur form;
= .FALSE.: compute the eigenvalues only.
WANTQ LOGICAL
Indicates whether or not the user wishes to accumulate
the matrix Q as follows:
= .TRUE. : The matrix Q is updated;
= .FALSE.: the matrix Q is not required.
WANTZ LOGICAL
Indicates whether or not the user wishes to accumulate
the matrix Z as follows:
= .TRUE. : The matrix Z is updated;
= .FALSE.: the matrix Z is not required.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A and B. N >= 0.
ILO (input) INTEGER
IHI (input) INTEGER
It is assumed that the matrices A and B are already
(quasi) upper triangular in rows and columns 1:ILO-1 and
IHI+1:N. The routine works primarily with the submatrices
in rows and columns ILO to IHI, but applies the
transformations to all the rows and columns of the
matrices A and B, if WANTT = .TRUE..
1 <= ILO <= max(1,N); min(ILO,N) <= IHI <= N.
ILOQ (input) INTEGER
IHIQ (input) INTEGER
Specify the rows of Q and Z to which transformations
must be applied if WANTQ = .TRUE. and WANTZ = .TRUE.,
respectively.
1 <= ILOQ <= ILO; IHI <= IHIQ <= N.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the upper Hessenberg matrix A.
On exit, if WANTT = .TRUE., the leading N-by-N part of
this array is upper quasi-triangular in rows and columns
ILO:IHI.
If WANTT = .FALSE., the diagonal elements and 2-by-2
diagonal blocks of A will be correct, but the remaining
parts of A are unspecified on exit.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the leading N-by-N part of this array must
contain the upper triangular matrix B.
On exit, if WANTT = .TRUE., the leading N-by-N part of
this array contains the transformed upper triangular
matrix. 2-by-2 blocks in B corresponding to 2-by-2 blocks
in A will be reduced to positive diagonal form. (I.e., if
A(j+1,j) is non-zero, then B(j+1,j)=B(j,j+1)=0 and B(j,j)
and B(j+1,j+1) will be positive.)
If WANTT = .FALSE., the elements corresponding to diagonal
elements and 2-by-2 diagonal blocks in A will be correct,
but the remaining parts of B are unspecified on exit.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if WANTQ = .TRUE., then the leading N-by-N part
of this array must contain the current matrix Q of
transformations accumulated by MB03XP.
On exit, if WANTQ = .TRUE., then the leading N-by-N part
of this array contains the matrix Q updated in the
submatrix Q(ILOQ:IHIQ,ILO:IHI).
If WANTQ = .FALSE., Q is not referenced.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= 1.
If WANTQ = .TRUE., LDQ >= MAX(1,N).
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
On entry, if WANTZ = .TRUE., then the leading N-by-N part
of this array must contain the current matrix Z of
transformations accumulated by MB03XP.
On exit, if WANTZ = .TRUE., then the leading N-by-N part
of this array contains the matrix Z updated in the
submatrix Z(ILOQ:IHIQ,ILO:IHI).
If WANTZ = .FALSE., Z is not referenced.
LDZ INTEGER
The leading dimension of the array Z. LDZ >= 1.
If WANTZ = .TRUE., LDZ >= MAX(1,N).
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
BETA (output) DOUBLE PRECISION array, dimension (N)
The i-th (ILO <= i <= IHI) computed eigenvalue is given
by BETA(I) * ( ALPHAR(I) + sqrt(-1)*ALPHAI(I) ). If two
eigenvalues are computed as a complex conjugate pair,
they are stored in consecutive elements of ALPHAR, ALPHAI
and BETA. If WANTT = .TRUE., the eigenvalues are stored in
the same order as on the diagonals of the Schur forms of
A and B.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = -19, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,N).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, then MB03YD failed to compute the Schur
form in a total of 30*(IHI-ILO+1) iterations;
elements i+1:n of ALPHAR, ALPHAI and BETA contain
successfully computed eigenvalues.
Method
The implemented algorithm is a double-shift version of the periodic QR algorithm described in [1,3] with some minor modifications [2]. The eigenvalues are computed via an implicit complex single shift algorithm.References
[1] Bojanczyk, A.W., Golub, G.H., and Van Dooren, P.
The periodic Schur decomposition: Algorithms and applications.
Proc. of the SPIE Conference (F.T. Luk, Ed.), 1770, pp. 31-42,
1992.
[2] Kressner, D.
An efficient and reliable implementation of the periodic QZ
algorithm. Proc. of the IFAC Workshop on Periodic Control
Systems, pp. 187-192, 2001.
[3] Van Loan, C.
Generalized Singular Values with Algorithms and Applications.
Ph. D. Thesis, University of Michigan, 1973.
Numerical Aspects
The algorithm requires O(N**3) floating point operations and is backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB03YT.html 0000664 0000000 0000000 00000007747 14560147231 0020407 0 ustar 00root root 0000000 0000000
Purpose
To compute the periodic Schur factorization of a real 2-by-2
matrix pair (A,B) where B is upper triangular. This routine
computes orthogonal (rotation) matrices given by CSL, SNL and CSR,
SNR such that
1) if the pair (A,B) has two real eigenvalues, then
[ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ]
[ 0 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ]
[ b11 b12 ] := [ CSR SNR ] [ b11 b12 ] [ CSL -SNL ]
[ 0 b22 ] [ -SNR CSR ] [ 0 b22 ] [ SNL CSL ],
2) if the pair (A,B) has a pair of complex conjugate eigenvalues,
then
[ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ]
[ a21 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ]
[ b11 0 ] := [ CSR SNR ] [ b11 b12 ] [ CSL -SNL ]
[ 0 b22 ] [ -SNR CSR ] [ 0 b22 ] [ SNL CSL ].
This is a modified version of the LAPACK routine DLAGV2 for
computing the real, generalized Schur decomposition of a
two-by-two matrix pencil.
Specification
SUBROUTINE MB03YT( A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL,
$ CSR, SNR )
C .. Scalar Arguments ..
INTEGER LDA, LDB
DOUBLE PRECISION CSL, CSR, SNL, SNR
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), ALPHAI(2), ALPHAR(2), B(LDB,*),
$ BETA(2)
Arguments
Input/Output Parameters
A (input/output) DOUBLE PRECISION array, dimension (LDA,2)
On entry, the leading 2-by-2 part of this array must
contain the matrix A.
On exit, the leading 2-by-2 part of this array contains
the matrix A of the pair in periodic Schur form.
LDA INTEGER
The leading dimension of the array A. LDA >= 2.
B (input/output) DOUBLE PRECISION array, dimension (LDB,2)
On entry, the leading 2-by-2 part of this array must
contain the upper triangular matrix B.
On exit, the leading 2-by-2 part of this array contains
the matrix B of the pair in periodic Schur form.
LDB INTEGER
The leading dimension of the array B. LDB >= 2.
ALPHAR (output) DOUBLE PRECISION array, dimension (2)
ALPHAI (output) DOUBLE PRECISION array, dimension (2)
BETA (output) DOUBLE PRECISION array, dimension (2)
(ALPHAR(k)+i*ALPHAI(k))*BETA(k) are the eigenvalues of the
pair (A,B), k=1,2, i = sqrt(-1). ALPHAI(1) >= 0.
CSL (output) DOUBLE PRECISION
The cosine of the first rotation matrix.
SNL (output) DOUBLE PRECISION
The sine of the first rotation matrix.
CSR (output) DOUBLE PRECISION
The cosine of the second rotation matrix.
SNR (output) DOUBLE PRECISION
The sine of the second rotation matrix.
References
[1] Van Loan, C.
Generalized Singular Values with Algorithms and Applications.
Ph. D. Thesis, University of Michigan, 1973.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
1. To compute, for a given matrix pair (A,B) in periodic Schur
form, orthogonal matrices Ur and Vr so that
T [ A11 A12 ] T [ B11 B12 ]
Vr * A * Ur = [ ], Ur * B * Vr = [ ], (1)
[ 0 A22 ] [ 0 B22 ]
is in periodic Schur form, and the eigenvalues of A11*B11
form a selected cluster of eigenvalues.
2. To compute an orthogonal matrix W so that
T [ 0 -A11 ] [ R11 R12 ]
W * [ ] * W = [ ], (2)
[ B11 0 ] [ 0 R22 ]
where the eigenvalues of R11 and -R22 coincide and have
positive real part.
Optionally, the matrix C is overwritten by Ur'*C*Vr.
All eigenvalues of A11*B11 must either be complex or real and
negative.
Specification
SUBROUTINE MB03ZA( COMPC, COMPU, COMPV, COMPW, WHICH, SELECT, N,
$ A, LDA, B, LDB, C, LDC, U1, LDU1, U2, LDU2, V1,
$ LDV1, V2, LDV2, W, LDW, WR, WI, M, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPC, COMPU, COMPV, COMPW, WHICH
INTEGER INFO, LDA, LDB, LDC, LDU1, LDU2, LDV1, LDV2,
$ LDW, LDWORK, M, N
C .. Array Arguments ..
LOGICAL SELECT(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
$ U1(LDU1,*), U2(LDU2,*), V1(LDV1,*), V2(LDV2,*),
$ W(LDW,*), WI(*), WR(*)
Arguments
Mode Parameters
COMPC CHARACTER*1
= 'U': update the matrix C;
= 'N': do not update C.
COMPU CHARACTER*1
= 'U': update the matrices U1 and U2;
= 'N': do not update U1 and U2.
See the description of U1 and U2.
COMPV CHARACTER*1
= 'U': update the matrices V1 and V2;
= 'N': do not update V1 and V2.
See the description of V1 and V2.
COMPW CHARACTER*1
Indicates whether or not the user wishes to accumulate
the matrix W as follows:
= 'N': the matrix W is not required;
= 'I': W is initialized to the unit matrix and the
orthogonal transformation matrix W is returned;
= 'V': W must contain an orthogonal matrix Q on entry,
and the product Q*W is returned.
WHICH CHARACTER*1
= 'A': select all eigenvalues, this effectively means
that Ur and Vr are identity matrices and A11 = A,
B11 = B;
= 'S': select a cluster of eigenvalues specified by
SELECT.
SELECT LOGICAL array, dimension (N)
If WHICH = 'S', then SELECT specifies the eigenvalues of
A*B in the selected cluster. To select a real eigenvalue
w(j), SELECT(j) must be set to .TRUE.. To select a complex
conjugate pair of eigenvalues w(j) and w(j+1),
corresponding to a 2-by-2 diagonal block in A, both
SELECT(j) and SELECT(j+1) must be set to .TRUE.; a complex
conjugate pair of eigenvalues must be either both included
in the cluster or both excluded.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the upper quasi-triangular matrix A of the matrix
pair (A,B) in periodic Schur form.
On exit, the leading M-by-M part of this array contains
the matrix R22 in (2).
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the leading N-by-N part of this array must
contain the upper triangular matrix B of the matrix pair
(A,B) in periodic Schur form.
On exit, the leading N-by-N part of this array is
overwritten.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, if COMPC = 'U', the leading N-by-N part of this
array must contain a general matrix C.
On exit, if COMPC = 'U', the leading N-by-N part of this
array contains the updated matrix Ur'*C*Vr.
If COMPC = 'N' or WHICH = 'A', this array is not
referenced.
LDC INTEGER
The leading dimension of the array C. LDC >= 1.
LDC >= N, if COMPC = 'U' and WHICH = 'S'.
U1 (input/output) DOUBLE PRECISION array, dimension (LDU1,N)
On entry, if COMPU = 'U' and WHICH = 'S', the leading
N-by-N part of this array must contain U1, the (1,1)
block of an orthogonal symplectic matrix
U = [ U1, U2; -U2, U1 ].
On exit, if COMPU = 'U' and WHICH = 'S', the leading
N-by-N part of this array contains U1*Ur.
If COMPU = 'N' or WHICH = 'A', this array is not
referenced.
LDU1 INTEGER
The leading dimension of the array U1. LDU1 >= 1.
LDU1 >= N, if COMPU = 'U' and WHICH = 'S'.
U2 (input/output) DOUBLE PRECISION array, dimension (LDU2,N)
On entry, if COMPU = 'U' and WHICH = 'S', the leading
N-by-N part of this array must contain U2, the (1,2)
block of an orthogonal symplectic matrix
U = [ U1, U2; -U2, U1 ].
On exit, if COMPU = 'U' and WHICH = 'S', the leading
N-by-N part of this array contains U2*Ur.
If COMPU = 'N' or WHICH = 'A', this array is not
referenced.
LDU2 INTEGER
The leading dimension of the array U2. LDU2 >= 1.
LDU2 >= N, if COMPU = 'U' and WHICH = 'S'.
V1 (input/output) DOUBLE PRECISION array, dimension (LDV1,N)
On entry, if COMPV = 'U' and WHICH = 'S', the leading
N-by-N part of this array must contain V1, the (1,1)
block of an orthogonal symplectic matrix
V = [ V1, V2; -V2, V1 ].
On exit, if COMPV = 'U' and WHICH = 'S', the leading
N-by-N part of this array contains V1*Vr.
If COMPV = 'N' or WHICH = 'A', this array is not
referenced.
LDV1 INTEGER
The leading dimension of the array V1. LDV1 >= 1.
LDV1 >= N, if COMPV = 'U' and WHICH = 'S'.
V2 (input/output) DOUBLE PRECISION array, dimension (LDV2,N)
On entry, if COMPV = 'U' and WHICH = 'S', the leading
N-by-N part of this array must contain V2, the (1,2)
block of an orthogonal symplectic matrix
V = [ V1, V2; -V2, V1 ].
On exit, if COMPV = 'U' and WHICH = 'S', the leading
N-by-N part of this array contains V2*Vr.
If COMPV = 'N' or WHICH = 'A', this array is not
referenced.
LDV2 INTEGER
The leading dimension of the array V2. LDV2 >= 1.
LDV2 >= N, if COMPV = 'U' and WHICH = 'S'.
W (input/output) DOUBLE PRECISION array, dimension (LDW,2*M)
On entry, if COMPW = 'V', then the leading 2*M-by-2*M part
of this array must contain a matrix W.
If COMPW = 'I', then W need not be set on entry, W is set
to the identity matrix.
On exit, if COMPW = 'I' or 'V' the leading 2*M-by-2*M part
of this array is post-multiplied by the transformation
matrix that produced (2).
If COMPW = 'N', this array is not referenced.
LDW INTEGER
The leading dimension of the array W. LDW >= 1.
LDW >= 2*M, if COMPW = 'I' or COMPW = 'V'.
WR (output) DOUBLE PRECISION array, dimension (M)
WI (output) DOUBLE PRECISION array, dimension (M)
The real and imaginary parts, respectively, of the
eigenvalues of R11. The eigenvalues are stored in the same
order as on the diagonal of R22, with
WR(i) = -R22(i,i) and, if R22(i:i+1,i:i+1) is a 2-by-2
diagonal block, WI(i) > 0 and WI(i+1) = -WI(i).
In exact arithmetic, these eigenvalue are the positive
square roots of the selected eigenvalues of the product
A*B. However, if an eigenvalue is sufficiently
ill-conditioned, then its value may differ significantly.
M (output) INTEGER
The number of selected eigenvalues.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = -28, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( 1, 4*N, 8*M ).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: reordering of the product A*B in Step 1 failed
because some eigenvalues are too close to separate;
= 2: reordering of some submatrix in Step 2 failed
because some eigenvalues are too close to separate;
= 3: the QR algorithm failed to compute the Schur form
of some submatrix in Step 2;
= 4: the condition that all eigenvalues of A11*B11 must
either be complex or real and negative is
numerically violated.
Method
Step 1 is performed using a reordering technique analogous to the LAPACK routine DTGSEN for reordering matrix pencils [1,2]. Step 2 is an implementation of Algorithm 2 in [3]. It requires O(M*N*N) floating point operations.References
[1] Kagstrom, B.
A direct method for reordering eigenvalues in the generalized
real Schur form of a regular matrix pair (A,B), in M.S. Moonen
et al (eds), Linear Algebra for Large Scale and Real-Time
Applications, Kluwer Academic Publ., 1993, pp. 195-218.
[2] Kagstrom, B. and Poromaa P.:
Computing eigenspaces with specified eigenvalues of a regular
matrix pair (A, B) and condition estimation: Theory,
algorithms and software, Numer. Algorithms, 1996, vol. 12,
pp. 369-407.
[3] Benner, P., Mehrmann, V., and Xu, H.
A new method for computing the stable invariant subspace of a
real Hamiltonian matrix, J. Comput. Appl. Math., 86,
pp. 17-43, 1997.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the stable and unstable invariant subspaces for a Hamiltonian matrix with no eigenvalues on the imaginary axis, using the output of the SLICOT Library routine MB03XD.Specification
SUBROUTINE MB03ZD( WHICH, METH, STAB, BALANC, ORTBAL, SELECT, N,
$ MM, ILO, SCALE, S, LDS, T, LDT, G, LDG, U1,
$ LDU1, U2, LDU2, V1, LDV1, V2, LDV2, M, WR, WI,
$ US, LDUS, UU, LDUU, LWORK, IWORK, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER BALANC, METH, ORTBAL, STAB, WHICH
INTEGER ILO, INFO, LDG, LDS, LDT, LDU1, LDU2, LDUS,
$ LDUU, LDV1, LDV2, LDWORK, M, MM, N
C .. Array Arguments ..
LOGICAL LWORK(*), SELECT(*)
INTEGER IWORK(*)
DOUBLE PRECISION DWORK(*), G(LDG,*), S(LDS,*), SCALE(*),
$ T(LDT,*), U1(LDU1,*), U2(LDU2,*), US(LDUS,*),
$ UU(LDUU,*), V1(LDV1,*), V2(LDV2,*), WI(*),
$ WR(*)
Arguments
Mode Parameters
WHICH CHARACTER*1
Specifies the cluster of eigenvalues for which the
invariant subspaces are computed:
= 'A': select all n eigenvalues;
= 'S': select a cluster of eigenvalues specified by
SELECT.
METH CHARACTER*1
If WHICH = 'A' this parameter specifies the method to be
used for computing bases of the invariant subspaces:
= 'S': compute the n-dimensional basis from a set of
n vectors;
= 'L': compute the n-dimensional basis from a set of
2*n vectors;
= 'Q': quick return of the set of n vectors;
= 'R': quick return of the set of 2*n vectors.
When in doubt, use METH = 'S'. In some cases, METH = 'L'
may result in more accurately computed invariant
subspaces, see [1].
Options METH = 'Q' or METH = 'R' return the range vectors
Y = [ Y1; Y2 ], where Y1 and Y2 have 2*n rows and n or 2*n
columns, respectively, which can be directly used, e.g.,
for finding the (stabilizing) solution of a Riccati
equation, by solving X*Y1 = Y2. Note that Y1 might be
singular when METH = 'Q'.
STAB CHARACTER*1
Specifies the type of invariant subspaces to be computed:
= 'S': compute the stable invariant subspace, i.e., the
invariant subspace belonging to those selected
eigenvalues that have negative real part;
= 'U': compute the unstable invariant subspace, i.e.,
the invariant subspace belonging to those
selected eigenvalues that have positive real
part;
= 'B': compute both the stable and unstable invariant
subspaces.
BALANC CHARACTER*1
Specifies the type of inverse balancing transformation
required:
= 'N': do nothing;
= 'P': do inverse transformation for permutation only;
= 'S': do inverse transformation for scaling only;
= 'B': do inverse transformations for both permutation
and scaling.
BALANC must be the same as the argument BALANC supplied to
MB03XD. Note that if the data is further post-processed,
e.g., for solving an algebraic Riccati equation, it is
recommended to delay inverse balancing (in particular the
scaling part) and apply it to the final result only,
see [2]. Inverse balancing is not used by this routine
if METH = 'Q' or METH = 'R'.
ORTBAL CHARACTER*1
If BALANC <> 'N', this option specifies how inverse
balancing is applied to the computed invariant subspaces:
= 'B': apply inverse balancing before orthogonal bases
for the invariant subspaces are computed;
= 'A': apply inverse balancing after orthogonal bases
for the invariant subspaces have been computed;
this may yield non-orthogonal bases if
BALANC = 'S' or BALANC = 'B'.
SELECT (input) LOGICAL array, dimension (N)
If WHICH = 'S', SELECT specifies the eigenvalues
corresponding to the positive and negative square
roots of the eigenvalues of S*T in the selected cluster.
To select a real eigenvalue w(j), SELECT(j) must be set
to .TRUE.. To select a complex conjugate pair of
eigenvalues w(j) and w(j+1), corresponding to a 2-by-2
diagonal block, both SELECT(j) and SELECT(j+1) must be set
to .TRUE.; a complex conjugate pair of eigenvalues must be
either both included in the cluster or both excluded.
This array is not referenced if WHICH = 'A'.
Input/Output Parameters
N (input) INTEGER
The order of the matrices S, T and G. N >= 0.
MM (input) INTEGER
The number of columns in the arrays US and/or UU.
If WHICH = 'A' and (METH = 'S' or METH = 'Q'), MM = N;
if WHICH = 'A' and (METH = 'L' or METH = 'R'), MM = 2*N;
if WHICH = 'S', MM = M.
The values above for MM give the numbers of vectors to be
returned, if METH = 'Q' or METH = 'R', or the numbers of
vectors to be used for computing a basis for the invariant
subspace(s), if METH = 'S' or METH = 'L', or WHICH = 'S'.
ILO (input) INTEGER
If BALANC <> 'N', then ILO is the integer returned by
MB03XD. 1 <= ILO <= N+1.
SCALE (input) DOUBLE PRECISION array, dimension (N)
If BALANC <> 'N', the leading N elements of this array
must contain details of the permutation and scaling
factors, as returned by MB03XD.
This array is not referenced if BALANC = 'N'.
S (input/output) DOUBLE PRECISION array, dimension (LDS,N)
On entry, the leading N-by-N part of this array must
contain the matrix S in real Schur form.
On exit, the leading N-by-N part of this array is
overwritten.
LDS INTEGER
The leading dimension of the array S. LDS >= max(1,N).
T (input/output) DOUBLE PRECISION array, dimension (LDT,N)
On entry, the leading N-by-N part of this array must
contain the upper triangular matrix T.
On exit, the leading N-by-N part of this array is
overwritten.
LDT INTEGER
The leading dimension of the array T. LDT >= max(1,N).
G (input/output) DOUBLE PRECISION array, dimension (LDG,N)
On entry, if METH = 'L' or METH = 'R', the leading N-by-N
part of this array must contain a general matrix G.
On exit, if METH = 'L' or METH = 'R', the leading N-by-N
part of this array is overwritten.
This array is not referenced if METH = 'S' or METH = 'Q'.
LDG INTEGER
The leading dimension of the array G. LDG >= 1.
LDG >= max(1,N) if METH = 'L' or METH = 'R'.
U1 (input/output) DOUBLE PRECISION array, dimension (LDU1,N)
On entry, the leading N-by-N part of this array must
contain the (1,1) block of an orthogonal symplectic
matrix U.
On exit, this array is overwritten.
LDU1 INTEGER
The leading dimension of the array U1. LDU1 >= MAX(1,N).
U2 (input/output) DOUBLE PRECISION array, dimension (LDU2,N)
On entry, the leading N-by-N part of this array must
contain the (2,1) block of an orthogonal symplectic
matrix U.
On exit, this array is overwritten.
LDU2 INTEGER
The leading dimension of the array U2. LDU2 >= MAX(1,N).
V1 (input/output) DOUBLE PRECISION array, dimension (LDV1,N)
On entry, the leading N-by-N part of this array must
contain the (1,1) block of an orthogonal symplectic
matrix V.
On exit, this array is overwritten.
LDV1 INTEGER
The leading dimension of the array V1. LDV1 >= MAX(1,N).
V2 (input/output) DOUBLE PRECISION array, dimension (LDV1,N)
On entry, the leading N-by-N part of this array must
contain the (2,1) block of an orthogonal symplectic
matrix V.
On exit, this array is overwritten.
LDV2 INTEGER
The leading dimension of the array V2. LDV2 >= MAX(1,N).
M (output) INTEGER
The number of selected eigenvalues.
WR (output) DOUBLE PRECISION array, dimension (M)
WI (output) DOUBLE PRECISION array, dimension (M)
On exit, the leading M elements of WR and WI contain the
real and imaginary parts, respectively, of the selected
eigenvalues that have nonpositive real part. Complex
conjugate pairs of eigenvalues with real part not equal
to zero will appear consecutively with the eigenvalue
having the positive imaginary part first. Note that, due
to roundoff errors, these numbers may differ from the
eigenvalues computed by MB03XD.
US (output) DOUBLE PRECISION array, dimension (LDUS,MM)
On exit, if STAB = 'S' or STAB = 'B', the leading
2*N-by-MM part of this array contains a basis for the
stable invariant subspace belonging to the selected
eigenvalues, if METH = 'S' or METH = 'L', or the range
vectors Y, if METH = 'Q' or METH = 'R' (see parameter
METH). This basis is orthogonal unless ORTBAL = 'A'.
LDUS INTEGER
The leading dimension of the array US. LDUS >= 1.
If STAB = 'S' or STAB = 'B', LDUS >= 2*N.
UU (output) DOUBLE PRECISION array, dimension (LDUU,MM)
On exit, if STAB = 'U' or STAB = 'B', the leading
2*N-by-MM part of this array contains a basis for the
unstable invariant subspace belonging to the selected
eigenvalues, if METH = 'S' or METH = 'L', or the range
vectors Y, if METH = 'Q' or METH = 'R' (see parameter
METH). This basis is orthogonal unless ORTBAL = 'A'.
LDUU INTEGER
The leading dimension of the array UU. LDUU >= 1.
If STAB = 'U' or STAB = 'B', LDUU >= 2*N.
Workspace
LWORK LOGICAL array, dimension (2*N)
This array is only referenced if WHICH = 'A' and
(METH = 'L' or METH = 'R').
IWORK INTEGER array, dimension (LIWORK)
LIWORK = 2*N, if WHICH = 'A' and METH = 'L';
LIWORK = N, if WHICH = 'A' and METH = 'S';
LIWORK = 0, if WHICH = 'A' and METH = 'Q' or METH = 'R';
LIWORK = M, if WHICH = 'S'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -35, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
If WHICH = 'S' or METH = 'S' or METH = 'Q':
LDWORK >= MAX( 1, 4*M*M + MAX( 8*M, 4*N ) ).
If WHICH = 'A' and (METH = 'L' or METH = 'R') and
( STAB = 'U' or STAB = 'S' ):
LDWORK >= MAX( 1, 2*N*N + 2*N, 8*N ).
If WHICH = 'A' and (METH = 'L' or METH = 'R') and
STAB = 'B':
LDWORK >= 8*N + 1.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: some of the selected eigenvalues are on or too close
to the imaginary axis;
= 2: reordering of the product S*T in routine MB03ZA
failed because some eigenvalues are too close to
separate;
= 3: the QR algorithm failed to compute some Schur form
in MB03ZA;
= 4: reordering of the Hamiltonian Schur form in routine
MB03TD failed because some eigenvalues are too close
to separate;
= 5: the computed stable invariant subspace for
METH = 'S' is inaccurate. This may be taken as a
warning and a suggestion to try METH = 'L';
= 6: the computed unstable invariant subspace for
METH = 'S' is inaccurate. This may be taken as a
warning and a suggestion to try METH = 'L'.
Method
This is an implementation of Algorithm 1 in [1].Numerical Aspects
The method is strongly backward stable for an embedded (skew-)Hamiltonian matrix, see [1]. Although good results have been reported if the eigenvalues are not too close to the imaginary axis, the method is not backward stable for the original Hamiltonian matrix itself.References
[1] Benner, P., Mehrmann, V., and Xu, H.
A new method for computing the stable invariant subspace of a
real Hamiltonian matrix, J. Comput. Appl. Math., 86,
pp. 17-43, 1997.
[2] Benner, P.
Symplectic balancing of Hamiltonian matrices.
SIAM J. Sci. Comput., 22 (5), pp. 1885-1904, 2001.
Further Comments
NoneExample
Program Text
* MB03ZD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 200 )
INTEGER LDG, LDRES, LDS, LDT, LDU1, LDU2, LDUS, LDUU,
$ LDV1, LDV2, LDWORK
PARAMETER ( LDG = NMAX, LDRES = 2*NMAX, LDS = NMAX,
$ LDT = NMAX, LDU1 = NMAX, LDU2 = NMAX,
$ LDUS = 2*NMAX, LDUU = 2*NMAX, LDV1 = NMAX,
$ LDV2 = NMAX, LDWORK = 3*NMAX*NMAX + 7*NMAX )
* .. Local Scalars ..
CHARACTER*1 BALANC, METH, ORTBAL, STAB, WHICH
INTEGER I, ILO, INFO, J, M, N
* .. Local Arrays ..
LOGICAL LWORK(2*NMAX), SELECT(NMAX)
INTEGER IWORK(2*NMAX)
DOUBLE PRECISION DWORK(LDWORK), G(LDG, NMAX), RES(LDRES,NMAX),
$ S(LDS, NMAX), SCALE(NMAX), T(LDT,NMAX),
$ U1(LDU1,NMAX), U2(LDU2, NMAX), US(LDUS,2*NMAX),
$ UU(LDUU,2*NMAX), V1(LDV1,NMAX), V2(LDV2, NMAX),
$ WI(NMAX), WR(NMAX)
* .. External Functions ..
EXTERNAL DLANGE, LSAME
LOGICAL LSAME
DOUBLE PRECISION DLANGE
* .. External Subroutines ..
EXTERNAL MB03ZD
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, ILO, WHICH, METH, STAB, BALANC, ORTBAL
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N
ELSE
*
IF ( LSAME( WHICH, 'S' ) )
$ READ ( NIN, FMT = * ) ( SELECT(I), I = 1,N )
READ ( NIN, FMT = * ) ( ( S(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( T(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( WHICH, 'A' ).AND.LSAME( METH, 'L' ) )
$ READ ( NIN, FMT = * ) ( ( G(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( BALANC, 'P' ).OR.LSAME( BALANC, 'S' ).OR.
$ LSAME( BALANC, 'B' ) )
$ READ ( NIN, FMT = * ) ( SCALE(I), I = 1,N )
READ ( NIN, FMT = * ) ( ( U1(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( U2(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( V1(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( V2(I,J), J = 1,N ), I = 1,N )
*
CALL MB03ZD( WHICH, METH, STAB, BALANC, ORTBAL, SELECT, N, 2*N,
$ ILO, SCALE, S, LDS, T, LDT, G, LDG, U1, LDU1, U2,
$ LDU2, V1, LDV1, V2, LDV2, M, WR, WI, US, LDUS,
$ UU, LDUU, LWORK, IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) I, WR(I), WI(I)
20 CONTINUE
*
IF ( LSAME( STAB, 'S' ).OR.LSAME( STAB, 'B' ) ) THEN
WRITE ( NOUT, FMT = 99995 )
DO 30 I = 1, 2*N
WRITE ( NOUT, FMT = 99993 ) ( US(I,J), J = 1,M )
30 CONTINUE
IF ( LSAME( ORTBAL, 'B' ).OR.LSAME( BALANC, 'N' ).OR.
$ LSAME( BALANC, 'P' ) ) THEN
CALL DGEMM( 'Transpose', 'No Transpose', M, M, 2*N,
$ ONE, US, LDUS, US, LDUS, ZERO, RES,
$ LDRES )
DO 40 I = 1, M
RES(I,I) = RES(I,I) - ONE
40 CONTINUE
WRITE ( NOUT, FMT = 99991 ) DLANGE( 'Frobenius', M, M,
$ RES, LDRES, DWORK )
END IF
CALL DGEMM( 'Transpose', 'No Transpose', M, M, N, ONE,
$ US, LDUS, US(N+1,1), LDUS, ZERO, RES, LDRES )
CALL DGEMM( 'Transpose', 'No Transpose', M, M, N, -ONE,
$ US(N+1,1), LDUS, US, LDUS, ONE, RES, LDRES )
WRITE ( NOUT, FMT = 99990 ) DLANGE( 'Frobenius', M, M,
$ RES, LDRES, DWORK )
END IF
*
IF ( LSAME( STAB, 'U' ).OR.LSAME( STAB, 'B' ) ) THEN
WRITE ( NOUT, FMT = 99994 )
DO 50 I = 1, 2*N
WRITE ( NOUT, FMT = 99993 ) ( UU(I,J), J = 1,M )
50 CONTINUE
IF ( LSAME( ORTBAL, 'B' ).OR.LSAME( BALANC, 'N' ).OR.
$ LSAME( BALANC, 'P' ) ) THEN
CALL DGEMM( 'Transpose', 'No Transpose', M, M, 2*N,
$ ONE, UU, LDUU, UU, LDUU, ZERO, RES,
$ LDRES )
DO 60 I = 1, M
RES(I,I) = RES(I,I) - ONE
60 CONTINUE
WRITE ( NOUT, FMT = 99989 ) DLANGE( 'Frobenius', M, M,
$ RES, LDRES, DWORK )
END IF
CALL DGEMM( 'Transpose', 'No Transpose', M, M, N, ONE,
$ UU, LDUU, UU(N+1,1), LDUU, ZERO, RES, LDRES )
CALL DGEMM( 'Transpose', 'No Transpose', M, M, N, -ONE,
$ UU(N+1,1), LDUU, UU, LDUU, ONE, RES, LDRES )
WRITE ( NOUT, FMT = 99988 ) DLANGE( 'Frobenius', M, M,
$ RES, LDRES, DWORK )
END IF
END IF
END IF
*
99999 FORMAT (' MB03ZD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB03ZD = ',I2)
99997 FORMAT (' The stable eigenvalues are',//' i',6X,
$ 'WR(i)',6X,'WI(i)',/)
99996 FORMAT (I4,3X,F8.4,3X,F8.4)
99995 FORMAT (/' A basis for the stable invariant subspace is')
99994 FORMAT (/' A basis for the unstable invariant subspace is')
99993 FORMAT (20(1X,F9.3))
99992 FORMAT (/' N is out of range.',/' N = ',I5)
99991 FORMAT (/' Orthogonality of US: || US''*US - I ||_F = ',G7.2)
99990 FORMAT (/' Symplecticity of US: || US''*J*US ||_F = ',G7.2)
99989 FORMAT (/' Orthogonality of UU: || UU''*UU - I ||_F = ',G7.2)
99988 FORMAT (/' Symplecticity of UU: || UU''*J*UU ||_F = ',G7.2)
END
Program Data
MB03ZD EXAMPLE PROGRAM DATA 5 1 A L B N B -3.1844761777714732 0.1612357243439331 -0.0628592203751138 0.2449004200921981 0.1974400149992579 0.0000000000000000 -0.1510667773167784 0.4260444411622838 -0.1775026035208615 0.3447278421198472 0.0000000000000000 -0.1386140422054264 -0.3006779624777515 0.2944143257134196 0.3456440339120323 0.0000000000000000 0.0000000000000000 0.0000000000000000 -0.2710128384740570 0.0933189808067138 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.4844146572359603 0.2004347508746697 3.2038208121776366 0.1805955192510651 0.2466389119377561 -0.2539149302433368 -0.0359238844381195 0.0000000000000000 -0.7196686433290816 0.0000000000000000 0.2428659121580384 -0.0594190100670832 0.0000000000000000 0.0000000000000000 -0.1891741194498107 -0.3309578443491266 -0.0303520731950515 0.0000000000000000 0.0000000000000000 0.0000000000000000 -0.4361574461961550 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.1530894573304220 -0.0370982242678464 0.0917788436945724 -0.0560402416315252 0.1345152517579192 0.0256668227276700 0.0652183678916931 -0.0700457231988297 0.0350041175858839 -0.2233868768749268 -0.1171980260782843 -0.0626428681377119 0.2327575351902772 -0.1251515732208170 -0.0177816046663201 0.3696921118421182 0.0746042309265599 -0.0828007611045140 0.0217427473546043 -0.1157775118548851 -0.3161183681200527 0.1374372236164812 0.1002727885506992 0.4021556774753973 -0.0431072263235579 0.1067394572547867 0.3806883009357247 -0.0347810363019649 -0.5014665065895758 0.5389691288472394 0.2685446895251367 0.4642712665555326 -0.5942766860716395 0.4781179763952615 0.2334370556238151 0.0166790369048933 0.2772789197782788 -0.0130145392695876 -0.2123817030594055 -0.2550292626960107 -0.5049268366774490 0.4209268575081796 0.1499593172661228 -0.1925590746592156 -0.5472292877802402 0.4543329704184054 0.3969669479129449 0.6321903535930828 0.3329156356041961 0.0163533225344433 -0.2638879466190024 -0.1795922007470742 0.1908329820840911 0.0868799433942070 0.3114741142062388 -0.2579907627915167 -0.2447897730222852 -0.1028403314750045 -0.1157840914576285 -0.1873268885694406 0.1700708002861580 -0.2243335325285328 0.3180998613802520 0.3315380214794822 0.1977859924739963 0.5072476567310013 -0.2128397588651423 -0.2740560593051881 0.1941418870268881 -0.3096684962457369 -0.0581576193198714 -0.2002027567371932 -0.0040094115506855 -0.3979373387545264 0.1520881534833910 -0.2010804514091372 0.4447147692018334 -0.6830166755147440 -0.0002576861753487 0.5781954611783305 -0.0375091627893805 0.5121756358795817 0.0297197140254773 0.4332229148788766 -0.3240527006890552 0.5330850295256511 0.3664711365265602 0.3288511296455119 0.0588396016404451 0.1134221597062257 0.1047567336850078 0.4535357098437908 0.1062866148880792 -0.3964092656837774 -0.2211800890450674 0.0350667323996222 0.4450432900616097 0.2950206358263853 -0.1617837757183893 -0.0376369332204927 -0.6746752660482623 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0299719306696789 -0.2322624725320701 -0.0280846899680325 -0.3044255686880000 -0.1077641482535519 -0.0069083614679702 0.3351358347080056 -0.4922707032978891 0.4293545450291714 0.4372821269062001 0.0167847133528843 0.2843629278945327 0.5958979805231146 0.3097336757510886 -0.2086733033047188 0.0248567764822071 -0.2810759958040470 -0.1653113624869834 -0.3528780198620412 -0.0254898556119252Program Results
MB03ZD EXAMPLE PROGRAM RESULTS
The stable eigenvalues are
i WR(i) WI(i)
1 -3.1941 0.0000
2 -0.1350 0.3179
3 -0.1350 -0.3179
4 -0.0595 0.2793
5 -0.0595 -0.2793
A basis for the stable invariant subspace is
-0.102 -0.116 0.627 0.118 -0.605
-0.100 -0.510 -0.266 0.504 0.124
-0.179 0.015 -0.112 -0.142 0.413
-0.055 0.252 0.182 -0.134 0.100
-0.078 0.576 -0.271 -0.252 -0.177
0.340 -0.135 0.053 -0.234 -0.110
0.528 0.108 -0.205 0.219 -0.096
0.397 -0.429 0.161 -0.598 0.199
0.444 0.342 0.447 0.406 0.440
0.434 0.014 -0.383 0.072 -0.391
Orthogonality of US: || US'*US - I ||_F = .62E-15
Symplecticity of US: || US'*J*US ||_F = .23E-14
A basis for the unstable invariant subspace is
-0.428 0.383 0.048 0.105 0.187
-0.506 -0.100 0.541 0.245 0.223
-0.334 -0.524 -0.044 -0.153 0.126
-0.453 0.076 0.103 -0.525 -0.268
-0.436 0.098 -0.752 0.209 -0.251
-0.093 -0.089 0.258 -0.114 -0.725
-0.112 -0.196 -0.186 -0.302 0.394
-0.120 -0.286 0.027 0.680 -0.119
-0.102 0.630 0.079 0.040 0.127
-0.091 -0.171 -0.136 -0.136 0.231
Orthogonality of UU: || UU'*UU - I ||_F = .69E-15
Symplecticity of UU: || UU'*J*UU ||_F = .10E-13
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04AD.html 0000664 0000000 0000000 00000075132 14560147231 0020331 0 ustar 00root root 0000000 0000000
Purpose
To compute the eigenvalues of a real N-by-N skew-Hamiltonian/
Hamiltonian pencil aS - bH with
( 0 I )
S = T Z = J Z' J' Z, where J = ( ), (1)
( -I 0 )
via generalized symplectic URV decomposition. That is, orthogonal
matrices Q1 and Q2 and orthogonal symplectic matrices U1 and U2
are computed such that
( T11 T12 )
Q1' T U1 = Q1' J Z' J' U1 = ( ) = Tout,
( 0 T22 )
( Z11 Z12 )
U2' Z Q2 = ( ) = Zout, (2)
( 0 Z22 )
( H11 H12 )
Q1' H Q2 = ( ) = Hout,
( 0 H22 )
where T11, T22', Z11, Z22', H11 are upper triangular and H22' is
upper quasi-triangular. The notation M' denotes the transpose of
the matrix M.
Optionally, if COMPQ1 = 'I' or COMPQ1 = 'U', the orthogonal
transformation matrix Q1 will be computed.
Optionally, if COMPQ2 = 'I' or COMPQ2 = 'U', the orthogonal
transformation matrix Q2 will be computed.
Optionally, if COMPU1 = 'I' or COMPU1 = 'U', the orthogonal
symplectic transformation matrix
( U11 U12 )
U1 = ( )
( -U12 U11 )
will be computed.
Optionally, if COMPU2 = 'I' or COMPU2 = 'U', the orthogonal
symplectic transformation matrix
( U21 U22 )
U2 = ( )
( -U22 U21 )
will be computed.
Specification
SUBROUTINE MB04AD( JOB, COMPQ1, COMPQ2, COMPU1, COMPU2, N, Z, LDZ,
$ H, LDH, Q1, LDQ1, Q2, LDQ2, U11, LDU11, U12,
$ LDU12, U21, LDU21, U22, LDU22, T, LDT, ALPHAR,
$ ALPHAI, BETA, IWORK, LIWORK, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ1, COMPQ2, COMPU1, COMPU2, JOB
INTEGER INFO, LDH, LDQ1, LDQ2, LDT, LDU11, LDU12,
$ LDU21, LDU22, LDWORK, LDZ, LIWORK, N
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION ALPHAI( * ), ALPHAR( * ), BETA( * ),
$ DWORK( * ), H( LDH, * ), Q1( LDQ1, * ),
$ Q2( LDQ2, * ), T( LDT, * ), U11( LDU11, * ),
$ U12( LDU12, * ), U21( LDU21, * ),
$ U22( LDU22, * ), Z( LDZ, * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'E': compute the eigenvalues only; Z, T, and H will not
necessarily be put into the forms in (2); H22' is
upper Hessenberg;
= 'T': put Z, T, and H into the forms in (2), and return
the eigenvalues in ALPHAR, ALPHAI and BETA.
COMPQ1 CHARACTER*1
Specifies whether to compute the orthogonal transformation
matrix Q1, as follows:
= 'N': Q1 is not computed;
= 'I': the array Q1 is initialized internally to the unit
matrix, and the orthogonal matrix Q1 is returned;
= 'U': the array Q1 contains an orthogonal matrix Q01 on
entry, and the product Q01*Q1 is returned, where Q1
is the product of the orthogonal transformations
that are applied to the pencil aT*Z - bH on the
left to reduce T, Z, and H to the forms in (2),
for COMPQ1 = 'I'.
COMPQ2 CHARACTER*1
Specifies whether to compute the orthogonal transformation
matrix Q2, as follows:
= 'N': Q2 is not computed;
= 'I': the array Q2 is initialized internally to the unit
matrix, and the orthogonal matrix Q2 is returned;
= 'U': the array Q2 contains an orthogonal matrix Q02 on
entry, and the product Q02*Q2 is returned, where Q2
is the product of the orthogonal transformations
that are applied to the pencil aT*Z - bH on the
right to reduce T, Z, and H to the forms in (2),
for COMPQ2 = 'I'.
COMPU1 CHARACTER*1
Specifies whether to compute the orthogonal symplectic
transformation matrix U1, as follows:
= 'N': U1 is not computed;
= 'I': the arrays U11 and U12 are initialized internally
to the unit and zero matrices, respectively, and
the corresponding submatrices of the orthogonal
symplectic matrix U1 are returned;
= 'U': the arrays U11 and U12 contain the corresponding
submatrices of an orthogonal symplectic matrix U01
on entry, and the updated submatrices U11 and U12
of the matrix product U01*U1 are returned, where U1
is the product of the orthogonal symplectic
transformations that are applied to the pencil
aT*Z - bH to reduce T, Z, and H to the forms in
(2), for COMPU1 = 'I'.
COMPU2 CHARACTER*1
Specifies whether to compute the orthogonal symplectic
transformation matrix U2, as follows:
= 'N': U2 is not computed;
= 'I': the arrays U21 and U22 are initialized internally
to the unit and zero matrices, respectively, and
the corresponding submatrices of the orthogonal
symplectic matrix U2 are returned;
= 'U': the arrays U21 and U22 contain the corresponding
submatrices of an orthogonal symplectic matrix U02
on entry, and the updated submatrices U21 and U22
of the matrix product U02*U2 are returned, where U2
is the product of the orthogonal symplectic
transformations that are applied to the pencil
aT*Z - bH to reduce T, Z, and H to the forms in
(2), for COMPU2 = 'I'.
Input/Output Parameters
N (input) INTEGER
The order of the pencil aS - bH. N >= 0, even.
Z (input/output) DOUBLE PRECISION array, dimension (LDZ, N)
On entry, the leading N-by-N part of this array must
contain the matrix Z.
On exit, if JOB = 'T', the leading N-by-N part of this
array contains the matrix Zout; otherwise, it contains the
matrix Z obtained just before the application of the
periodic QZ algorithm.
The elements of the (2,1) block, i.e., in the rows N/2+1
to N and in the columns 1 to N/2 are not set to zero, but
are unchanged on exit.
LDZ INTEGER
The leading dimension of the array Z. LDZ >= MAX(1, N).
H (input/output) DOUBLE PRECISION array, dimension (LDH, N)
On entry, the leading N-by-N part of this array must
contain the Hamiltonian matrix H (H22 = -H11', H12 = H12',
H21 = H21').
On exit, if JOB = 'T', the leading N-by-N part of this
array contains the matrix Hout; otherwise, it contains the
matrix H obtained just before the application of the
periodic QZ algorithm.
LDH INTEGER
The leading dimension of the array H. LDH >= MAX(1, N).
Q1 (input/output) DOUBLE PRECISION array, dimension (LDQ1, N)
On entry, if COMPQ1 = 'U', then the leading N-by-N part of
this array must contain a given matrix Q01, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q01 and the transformation matrix Q1
used to transform the matrices Z, T and H.
On exit, if COMPQ1 = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q1.
If COMPQ1 = 'N', this array is not referenced.
LDQ1 INTEGER
The leading dimension of the array Q1.
LDQ1 >= 1, if COMPQ1 = 'N';
LDQ1 >= MAX(1, N), if COMPQ1 = 'I' or COMPQ1 = 'U'.
Q2 (input/output) DOUBLE PRECISION array, dimension (LDQ2, N)
On entry, if COMPQ2 = 'U', then the leading N-by-N part of
this array must contain a given matrix Q02, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q02 and the transformation matrix Q2
used to transform the matrices Z, T and H.
On exit, if COMPQ2 = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q2.
If COMPQ2 = 'N', this array is not referenced.
LDQ2 INTEGER
The leading dimension of the array Q2.
LDQ2 >= 1, if COMPQ2 = 'N';
LDQ2 >= MAX(1, N), if COMPQ2 = 'I' or COMPQ2 = 'U'.
U11 (input/output) DOUBLE PRECISION array, dimension
(LDU11, N/2)
On entry, if COMPU1 = 'U', then the leading N/2-by-N/2
part of this array must contain the upper left block of a
given matrix U01, and on exit, the leading N/2-by-N/2 part
of this array contains the updated upper left block U11 of
the product of the input matrix U01 and the transformation
matrix U1 used to transform the matrices Z, T, and H.
On exit, if COMPU1 = 'I', then the leading N/2-by-N/2 part
of this array contains the upper left block U11 of the
orthogonal symplectic transformation matrix U1.
If COMPU1 = 'N' this array is not referenced.
LDU11 INTEGER
The leading dimension of the array U11.
LDU11 >= 1, if COMPU1 = 'N';
LDU11 >= MAX(1, N/2), if COMPU1 = 'I' or COMPU1 = 'U'.
U12 (input/output) DOUBLE PRECISION array, dimension
(LDU12, N/2)
On entry, if COMPU1 = 'U', then the leading N/2-by-N/2
part of this array must contain the upper right block of a
given matrix U01, and on exit, the leading N/2-by-N/2 part
of this array contains the updated upper right block U12
of the product of the input matrix U01 and the
transformation matrix U1 used to transform the matrices
Z, T, and H.
On exit, if COMPU1 = 'I', then the leading N/2-by-N/2 part
of this array contains the upper right block U12 of the
orthogonal symplectic transformation matrix U1.
If COMPU1 = 'N' this array is not referenced.
LDU12 INTEGER
The leading dimension of the array U12.
LDU12 >= 1, if COMPU1 = 'N';
LDU12 >= MAX(1, N/2), if COMPU1 = 'I' or COMPU1 = 'U'.
U21 (input/output) DOUBLE PRECISION array, dimension
(LDU21, N/2)
On entry, if COMPU2 = 'U', then the leading N/2-by-N/2
part of this array must contain the upper left block of a
given matrix U02, and on exit, the leading N/2-by-N/2 part
of this array contains the updated upper left block U21 of
the product of the input matrix U02 and the transformation
matrix U2 used to transform the matrices Z, T, and H.
On exit, if COMPU2 = 'I', then the leading N/2-by-N/2 part
of this array contains the upper left block U21 of the
orthogonal symplectic transformation matrix U2.
If COMPU2 = 'N' this array is not referenced.
LDU21 INTEGER
The leading dimension of the array U21.
LDU21 >= 1, if COMPU2 = 'N';
LDU21 >= MAX(1, N/2), if COMPU2 = 'I' or COMPU2 = 'U'.
U22 (input/output) DOUBLE PRECISION array, dimension
(LDU22, N/2)
On entry, if COMPU2 = 'U', then the leading N/2-by-N/2
part of this array must contain the upper right block of a
given matrix U02, and on exit, the leading N/2-by-N/2 part
of this array contains the updated upper right block U22
of the product of the input matrix U02 and the
transformation matrix U2 used to transform the matrices
Z, T, and H.
On exit, if COMPU2 = 'I', then the leading N/2-by-N/2 part
of this array contains the upper right block U22 of the
orthogonal symplectic transformation matrix U2.
If COMPU2 = 'N' this array is not referenced.
LDU22 INTEGER
The leading dimension of the array U22.
LDU22 >= 1, if COMPU2 = 'N';
LDU22 >= MAX(1, N/2), if COMPU2 = 'I' or COMPU2 = 'U'.
T (output) DOUBLE PRECISION array, dimension (LDT, N)
If JOB = 'T', the leading N-by-N part of this array
contains the matrix Tout; otherwise, it contains the
matrix T obtained just before the application of the
periodic QZ algorithm.
LDT INTEGER
The leading dimension of the array T. LDT >= MAX(1, N).
ALPHAR (output) DOUBLE PRECISION array, dimension (N/2)
The real parts of each scalar alpha defining an eigenvalue
of the pencil aS - bH.
ALPHAI (output) DOUBLE PRECISION array, dimension (N/2)
The imaginary parts of each scalar alpha defining an
eigenvalue of the pencil aS - bH.
If ALPHAI(j) is zero, then the j-th eigenvalue is real.
BETA (output) DOUBLE PRECISION array, dimension (N/2)
The scalars beta defining the eigenvalues of the pencil
aS - bH.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent the j-th eigenvalue of the pencil
aS - bH, in the form lambda = alpha/beta. Since lambda may
overflow, the ratios should not, in general, be computed.
Due to the skew-Hamiltonian/Hamiltonian structure of the
pencil, for every eigenvalue lambda, -lambda is also an
eigenvalue, and thus it has only to be saved once in
ALPHAR, ALPHAI and BETA.
Specifically, only eigenvalues with imaginary parts
greater than or equal to zero are stored; their conjugate
eigenvalues are not stored. If imaginary parts are zero
(i.e., for real eigenvalues), only positive eigenvalues
are stored. The remaining eigenvalues have opposite signs.
As a consequence, pairs of complex eigenvalues, stored in
consecutive locations, are not complex conjugate.
Workspace
IWORK INTEGER array, dimension (LIWORK)
On exit, if INFO = 3, IWORK(1) contains the number of
(pairs of) possibly inaccurate eigenvalues, q <= N/2, and
IWORK(2), ..., IWORK(q+1) indicate their indices.
Specifically, a positive value is an index of a real or
purely imaginary eigenvalue, corresponding to a 1-by-1
block, while the absolute value of a negative entry in
IWORK is an index to the first eigenvalue in a pair of
consecutively stored eigenvalues, corresponding to a
2-by-2 block. A 2-by-2 block may have two complex, two
real, two purely imaginary, or one real and one purely
imaginary eigenvalue.
For i = q+2, ..., 2*q+1, IWORK(i) contains a pointer to
the starting location in DWORK of the i-th quadruple of
1-by-1 blocks, if IWORK(i-q) > 0, or 2-by-2 blocks,
if IWORK(i-q) < 0, defining unreliable eigenvalues.
IWORK(2*q+2) contains the number of the 1-by-1 blocks, and
IWORK(2*q+3) contains the number of the 2-by-2 blocks,
corresponding to unreliable eigenvalues. IWORK(2*q+4)
contains the total number t of the 2-by-2 blocks.
If INFO = 0, then q = 0, therefore IWORK(1) = 0.
LIWORK INTEGER
The dimension of the array IWORK. LIWORK >= N+18.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or INFO = 3, DWORK(1) returns the
optimal LDWORK, and DWORK(2), ..., DWORK(7) contain the
Frobenius norms of the factors of the formal matrix
product used by the algorithm. In addition, DWORK(8), ...,
DWORK(7+6*s) contain the s sextuple values corresponding
to the 1-by-1 blocks. Their eigenvalues are real or purely
imaginary. Such an eigenvalue is obtained from
-i*sqrt((a1/a2/a3)*(a4/a5/a6)), but always taking a
positive sign, where a1, ..., a6 are the corresponding
sextuple values.
Moreover, DWORK(8+6*s), ..., DWORK(7+6*s+24*t) contain the
t groups of sextuple 2-by-2 matrices corresponding to the
2-by-2 blocks. Their eigenvalue pairs are either complex,
or placed on the real and imaginary axes. Such an
eigenvalue pair is obtained as -1i*sqrt(ev), but taking
positive imaginary parts, where ev are the eigenvalues of
the product A1*inv(A2)*inv(A3)*A4*inv(A5)*inv(A6), where
A1, ..., A6 define the corresponding 2-by-2 matrix
sextuple.
On exit, if INFO = -31, DWORK(1) returns the minimum value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
If JOB = 'E' and COMPQ1 = 'N' and COMPQ2 = 'N' and
COMPU1 = 'N' and COMPU2 = 'N', then
LDWORK >= 3/2*N**2 + MAX(6*N,54);
else, LDWORK >= 3*N**2 + MAX(6*N,54).
For good performance LDWORK should generally be larger.
If LDWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message related to LDWORK is issued by
XERBLA.
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: the periodic QZ algorithm was not able to reveal
information about the eigenvalues from the 2-by-2
blocks in the SLICOT Library routine MB03BD;
= 2: the periodic QZ algorithm did not converge in the
SLICOT Library routine MB03BD;
= 3: some eigenvalues might be inaccurate, and details can
be found in IWORK and DWORK. This is a warning.
Method
The algorithm uses Givens rotations and Householder reflections to annihilate elements in T, Z, and H such that T11, T22', Z11, Z22', and H11 are upper triangular and H22' is upper Hessenberg. Finally the periodic QZ algorithm is applied to transform H22' to upper quasi-triangular form while T11, T22', Z11, Z22', and H11 stay in upper triangular form. See also page 17 in [1] for more details.References
[1] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
Eigenproblems.
Tech. Rep., Technical University Chemnitz, Germany,
Nov. 2007.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) real floating point operations.Further Comments
NoneExample
Program Text
* MB04AD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 50 )
INTEGER LDH, LDQ1, LDQ2, LDT, LDU11, LDU12, LDU21,
$ LDU22, LDWORK, LDZ, LIWORK
PARAMETER ( LDH = NMAX, LDQ1 = NMAX, LDQ2 = NMAX,
$ LDT = NMAX, LDU11 = NMAX/2, LDU12 = NMAX/2,
$ LDU21 = NMAX/2, LDU22 = NMAX/2,
$ LDWORK = 3*NMAX*NMAX + MAX( NMAX, 48 ) + 6,
$ LDZ = NMAX, LIWORK = NMAX + 18 )
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
*
* .. Local Scalars ..
CHARACTER COMPQ1, COMPQ2, COMPU1, COMPU2, JOB
INTEGER I, INFO, J, M, N
*
* .. Local Arrays ..
INTEGER IWORK( LIWORK )
DOUBLE PRECISION ALPHAI( NMAX/2 ), ALPHAR( NMAX/2 ),
$ BETA( NMAX/2 ), DWORK( LDWORK ),
$ H( LDH, NMAX ), Q1( LDQ1, NMAX ),
$ Q2( LDQ2, NMAX ), T( LDT, NMAX ),
$ U11( LDU11, NMAX/2 ), U12( LDU12, NMAX/2 ),
$ U21( LDU21, NMAX/2 ), U22( LDU22, NMAX/2 ),
$ Z( LDZ, NMAX )
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
*
* .. External Subroutines ..
EXTERNAL DLASET, MB04AD
*
* .. Intrinsic Functions ..
INTRINSIC MAX
*
* .. Executable Statements ..
*
WRITE( NOUT, FMT = 99999 )
* Skip the heading in the data file and read in the data.
READ( NIN, FMT = * )
READ( NIN, FMT = * ) JOB, COMPQ1, COMPQ2, COMPU1, COMPU2, N
IF( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE( NOUT, FMT = 99998 ) N
ELSE
READ( NIN, FMT = * ) ( ( Z( I, J ), J = 1, N ), I = 1, N )
READ( NIN, FMT = * ) ( ( H( I, J ), J = 1, N ), I = 1, N )
* Compute the eigenvalues of a real skew-Hamiltonian/Hamiltonian
* pencil (factored version).
CALL MB04AD( JOB, COMPQ1, COMPQ2, COMPU1, COMPU2, N, Z, LDZ, H,
$ LDH, Q1, LDQ1, Q2, LDQ2, U11, LDU11, U12, LDU12,
$ U21, LDU21, U22, LDU22, T, LDT, ALPHAR, ALPHAI,
$ BETA, IWORK, LIWORK, DWORK, LDWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99997 ) INFO
ELSE
M = N/2
CALL DLASET( 'Full', M, M, ZERO, ZERO, Z( M+1, 1 ), LDZ )
WRITE( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( T( I, J ), J = 1, N )
10 CONTINUE
WRITE( NOUT, FMT = 99994 )
DO 20 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( Z( I, J ), J = 1, N )
20 CONTINUE
WRITE( NOUT, FMT = 99993 )
DO 30 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( H( I, J ), J = 1, N )
30 CONTINUE
IF( LSAME( COMPQ1, 'I' ) ) THEN
WRITE( NOUT, FMT = 99992 )
DO 40 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( Q1( I, J ), J = 1, N )
40 CONTINUE
END IF
IF( LSAME( COMPQ2, 'I' ) ) THEN
WRITE( NOUT, FMT = 99991 )
DO 50 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( Q2( I, J ), J = 1, N )
50 CONTINUE
END IF
IF( LSAME( COMPU1, 'I' ) ) THEN
WRITE( NOUT, FMT = 99990 )
DO 60 I = 1, M
WRITE( NOUT, FMT = 99995 ) ( U11( I, J ), J = 1, M )
60 CONTINUE
WRITE( NOUT, FMT = 99989 )
DO 70 I = 1, M
WRITE( NOUT, FMT = 99995 ) ( U12( I, J ), J = 1, M )
70 CONTINUE
END IF
IF( LSAME( COMPU2, 'I' ) ) THEN
WRITE( NOUT, FMT = 99988 )
DO 80 I = 1, M
WRITE( NOUT, FMT = 99995 ) ( U21( I, J ), J = 1, M )
80 CONTINUE
WRITE( NOUT, FMT = 99987 )
DO 90 I = 1, M
WRITE( NOUT, FMT = 99995 ) ( U22( I, J ), J = 1, M )
90 CONTINUE
END IF
WRITE( NOUT, FMT = 99986 )
WRITE( NOUT, FMT = 99995 ) ( ALPHAR( I ), I = 1, M )
WRITE( NOUT, FMT = 99985 )
WRITE( NOUT, FMT = 99995 ) ( ALPHAI( I ), I = 1, M )
WRITE( NOUT, FMT = 99984 )
WRITE( NOUT, FMT = 99995 ) ( BETA( I ), I = 1, M )
END IF
END IF
STOP
*
99999 FORMAT( 'MB04AD EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT( 'INFO on exit from MB04AD = ', I2 )
99996 FORMAT( 'The matrix T on exit is ' )
99995 FORMAT( 50( 1X, F8.4 ) )
99994 FORMAT( 'The matrix Z on exit is ' )
99993 FORMAT( 'The matrix H is ' )
99992 FORMAT( 'The matrix Q1 is ' )
99991 FORMAT( 'The matrix Q2 is ' )
99990 FORMAT( 'The upper left block of the matrix U1 is ' )
99989 FORMAT( 'The upper right block of the matrix U1 is ' )
99988 FORMAT( 'The upper left block of the matrix U2 is ' )
99987 FORMAT( 'The upper right block of the matrix U2 is ' )
99986 FORMAT( 'The vector ALPHAR is ' )
99985 FORMAT( 'The vector ALPHAI is ' )
99984 FORMAT( 'The vector BETA is ' )
END
Program Data
MB04AD EXAMPLE PROGRAM DATA T I I I I 8 3.1472 4.5751 -0.7824 1.7874 -2.2308 -0.6126 2.0936 4.5974 4.0579 4.6489 4.1574 2.5774 -4.5383 -1.1844 2.5469 -1.5961 -3.7301 -3.4239 2.9221 2.4313 -4.0287 2.6552 -2.2397 0.8527 4.1338 4.7059 4.5949 -1.0777 3.2346 2.9520 1.7970 -2.7619 1.3236 4.5717 1.5574 1.5548 1.9483 -3.1313 1.5510 2.5127 -4.0246 -0.1462 -4.6429 -3.2881 -1.8290 -0.1024 -3.3739 -2.4490 -2.2150 3.0028 3.4913 2.0605 4.5022 -0.5441 -3.8100 0.0596 0.4688 -3.5811 4.3399 -4.6817 -4.6555 1.4631 -0.0164 1.9908 3.9090 -3.5071 3.1428 -3.0340 -1.4834 3.7401 -0.1715 0.4026 4.5929 -2.4249 -2.5648 -2.4892 3.7401 -2.1416 1.6251 2.6645 0.4722 3.4072 4.2926 1.1604 -0.1715 1.6251 -4.2415 -0.0602 -3.6138 -2.4572 -1.5002 -0.2671 0.4026 2.6645 -0.0602 -3.7009 0.6882 -1.8421 -4.1122 0.1317 -3.9090 -4.5929 -0.4722 3.6138 -1.8421 2.9428 -0.4340 1.3834 3.5071 2.4249 -3.4072 2.4572 -4.1122 -0.4340 -2.3703 0.5231 -3.1428 2.5648 -4.2926 1.5002 0.1317 1.3834 0.5231 -4.1618 3.0340 2.4892 -1.1604 0.2671Program Results
MB04AD EXAMPLE PROGRAM RESULTS The matrix T on exit is -3.9699 3.7658 5.5815 -1.7750 -0.8818 -0.0511 -4.2158 1.9054 0.0000 5.3686 -5.9166 4.9163 1.3839 0.8870 3.9458 -4.9167 0.0000 0.0000 5.9641 1.9432 -2.0680 2.4402 -1.4091 5.8512 0.0000 0.0000 0.0000 5.9983 -3.8172 4.0147 -2.0739 -1.2570 0.0000 0.0000 0.0000 0.0000 8.2005 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.5732 8.0098 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.6017 2.4397 5.9751 0.0000 0.0000 0.0000 0.0000 0.0000 -2.5869 0.5598 0.2544 5.2129 The matrix Z on exit is -6.4705 -2.5511 -4.0551 -1.9895 -2.7642 0.7532 -4.1047 -2.2046 0.0000 7.3589 -4.4480 -2.7491 -1.5465 -1.4345 -0.9272 1.3121 0.0000 0.0000 4.9125 -0.4968 5.3574 3.8579 5.2547 -1.7324 0.0000 0.0000 0.0000 9.0822 0.0460 -0.3382 3.9302 3.1084 0.0000 0.0000 0.0000 0.0000 6.1869 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 5.5573 6.6549 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 2.7456 -3.5789 4.3432 0.0000 0.0000 0.0000 0.0000 0.0000 0.1549 3.5335 3.1346 4.1062 The matrix H is -7.4834 0.4404 2.3558 1.6724 -0.4630 1.9533 1.5724 -2.7254 0.0000 -7.3500 3.7414 3.7466 0.2837 0.6849 0.7727 -4.2140 0.0000 0.0000 -2.3493 -3.7994 -0.6872 1.1773 -2.6901 -5.1494 0.0000 0.0000 0.0000 -3.4719 5.3322 0.4182 1.9779 1.5175 0.0000 0.0000 0.0000 0.0000 -6.1880 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -3.3324 9.0833 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -1.8703 0.0799 -2.8180 0.0000 0.0000 0.0000 0.0000 0.0000 -2.3477 3.3110 0.6561 0.7281 The matrix Q1 is -0.2489 -0.1409 0.3615 0.6458 0.0113 0.6063 -0.0470 0.0238 -0.2436 0.1294 -0.0874 -0.4103 0.3408 0.3628 -0.3267 0.6272 -0.4316 -0.2352 0.5553 -0.2811 -0.2198 -0.2880 -0.4564 -0.1773 0.1992 -0.2176 -0.5198 0.1561 -0.1523 0.1299 -0.7281 -0.2197 0.0161 0.7390 0.1125 -0.2226 -0.1003 0.3608 -0.1118 -0.4886 -0.5824 0.0984 -0.3052 0.1996 0.5889 -0.2442 0.0060 -0.3341 -0.3246 0.4661 -0.1835 0.3523 -0.5153 -0.3034 -0.0865 0.3931 -0.4559 -0.2961 -0.3790 -0.3127 -0.4356 0.3452 0.3642 -0.1467 The matrix Q2 is 0.0288 -0.1842 -0.6791 -0.2115 -0.4790 0.4212 -0.0417 -0.2253 -0.0666 -0.0787 -0.3711 0.1737 -0.0482 -0.5770 -0.6785 0.1607 0.1506 0.6328 0.0518 -0.6266 0.0652 -0.0790 -0.2854 -0.2994 -0.2900 -0.2737 -0.0076 -0.3671 -0.2017 -0.6241 0.4521 -0.2675 0.3353 0.4107 0.0326 0.1400 -0.6447 -0.2043 0.2561 0.4187 0.0905 -0.1648 -0.2363 -0.5323 0.3180 0.0286 0.1252 0.7126 -0.7246 0.0468 0.3328 -0.1794 -0.3639 0.2257 -0.2623 0.2786 0.4922 -0.5353 0.4803 -0.2501 -0.2723 0.0199 -0.3194 -0.0371 The upper left block of the matrix U1 is 0.4144 0.2249 0.6015 -0.1964 -0.0198 0.5131 -0.2823 -0.3058 -0.6620 0.1508 0.2237 0.0240 -0.0743 -0.4323 -0.0332 -0.7263 The upper right block of the matrix U1 is -0.3474 0.1306 -0.3391 -0.3530 -0.3760 0.1550 0.6087 -0.1646 0.1707 0.6553 -0.1262 -0.1177 0.3048 -0.0773 0.0767 -0.4173 The upper left block of the matrix U2 is 0.1403 -0.6447 -0.6536 -0.3707 0.7069 0.2609 -0.0091 -0.1702 -0.1218 -0.1120 0.3766 -0.5154 0.0773 0.6349 -0.5070 -0.1810 The upper right block of the matrix U2 is 0.0000 0.0000 0.0000 0.0000 0.1182 0.1587 0.1930 -0.5716 0.6051 -0.2720 0.3364 0.1089 0.2823 -0.0386 -0.1529 0.4434 The vector ALPHAR is 0.0000 0.7122 0.0000 0.7450 The vector ALPHAI is 0.7540 0.0000 0.7465 0.0000 The vector BETA is 4.0000 4.0000 8.0000 16.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04AZ.html 0000664 0000000 0000000 00000064203 14560147231 0020354 0 ustar 00root root 0000000 0000000
Purpose
To compute the eigenvalues of a complex N-by-N skew-Hamiltonian/
Hamiltonian pencil aS - bH, with
H T ( B F ) ( Z11 Z12 )
S = J Z J Z and H = ( H ), Z =: ( ). (1)
( G -B ) ( Z21 Z22 )
The structured Schur form of the embedded real skew-Hamiltonian/
H T
skew-Hamiltonian pencil, aB_S - bB_T, with B_S = J B_Z J B_Z,
( Re(Z11) -Im(Z11) | Re(Z12) -Im(Z12) )
( | )
( Im(Z11) Re(Z11) | Im(Z12) Re(Z12) )
( | )
B_Z = (---------------------+---------------------) ,
( | )
( Re(Z21) -Im(Z21) | Re(Z22) -Im(Z22) )
( | )
( Im(Z21) Re(Z21) | Im(Z22) Re(Z22) )
(2)
( -Im(B) -Re(B) | -Im(F) -Re(F) )
( | )
( Re(B) -Im(B) | Re(F) -Im(F) )
( | )
B_T = (-----------------+-----------------) , T = i*H,
( | T T )
( -Im(G) -Re(G) | -Im(B ) Re(B ) )
( | T T )
( Re(G) -Im(G) | -Re(B ) -Im(B ) )
is determined and used to compute the eigenvalues. Optionally,
if JOB = 'T', the pencil aB_S - bB_H is transformed by a unitary
matrix Q and a unitary symplectic matrix U to the structured Schur
H T
form aB_Sout - bB_Hout, with B_Sout = J B_Zout J B_Zout,
( BA BD ) ( BB BF )
B_Zout = ( ) and B_Hout = ( H ), (3)
( 0 BC ) ( 0 -BB )
where BA and BB are upper triangular, BC is lower triangular,
and BF is Hermitian. B_H above is defined as B_H = -i*B_T.
The embedding doubles the multiplicities of the eigenvalues of
the pencil aS - bH.
Optionally, if COMPQ = 'C', the unitary matrix Q is computed.
Optionally, if COMPU = 'C', the unitary symplectic matrix U is
computed.
Specification
SUBROUTINE MB04AZ( JOB, COMPQ, COMPU, N, Z, LDZ, B, LDB, FG,
$ LDFG, D, LDD, C, LDC, Q, LDQ, U, LDU, ALPHAR,
$ ALPHAI, BETA, IWORK, LIWORK, DWORK, LDWORK,
$ ZWORK, LZWORK, BWORK, INFO )C .. Scalar Arguments ..
CHARACTER COMPQ, COMPU, JOB
INTEGER INFO, LDB, LDC, LDD, LDFG, LDQ, LDU, LDWORK,
$ LDZ, LIWORK, LZWORK, N
C .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION ALPHAI( * ), ALPHAR( * ), BETA( * ), DWORK( * )
COMPLEX*16 B( LDB, * ), C( LDC, * ), D( LDD, * ),
$ FG( LDFG, * ), Q( LDQ, * ), U( LDU, * ),
$ Z( LDZ, * ), ZWORK( * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'E': compute the eigenvalues only; S and H will not
necessarily be transformed as in (3).
= 'T': put S and H into the forms in (3) and return the
eigenvalues in ALPHAR, ALPHAI and BETA.
COMPQ CHARACTER*1
Specifies whether to compute the unitary transformation
matrix Q, as follows:
= 'N': do not compute the unitary matrix Q;
= 'C': the array Q is initialized internally to the unit
matrix, and the unitary matrix Q is returned.
COMPU CHARACTER*1
Specifies whether to compute the unitary symplectic
transformation matrix U, as follows:
= 'N': do not compute the unitary symplectic matrix U;
= 'C': the array U is initialized internally to the unit
matrix, and the unitary symplectic matrix U is
returned.
Input/Output Parameters
N (input) INTEGER
Order of the pencil aS - bH. N >= 0, even.
Z (input/output) COMPLEX*16 array, dimension (LDZ, N)
On entry, the leading N-by-N part of this array must
contain the non-trivial factor Z in the factorization
H T
S = J Z J Z of the skew-Hamiltonian matrix S.
On exit, if JOB = 'T', the leading N-by-N part of this
array contains the upper triangular matrix BA in (3)
(see also METHOD). The strictly lower triangular part is
not zeroed. The submatrix in the rows N/2+1 to N and the
first N/2 columns is unchanged, except possibly for the
entry (N/2+1,N/2), which might be set to zero.
If JOB = 'E', this array is unchanged on exit.
LDZ INTEGER
The leading dimension of the array Z. LDZ >= MAX(1, N).
B (input/output) COMPLEX*16 array, dimension (LDB, K), where
K = N, if JOB = 'T', and K = M, if JOB = 'E'.
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix B.
On exit, if JOB = 'T', the leading N-by-N part of this
array contains the upper triangular matrix BB in (3)
(see also METHOD).
The strictly lower triangular part is not zeroed.
If JOB = 'E', this array is unchanged on exit.
LDB INTEGER
The leading dimension of the array B.
LDB >= MAX(1, M), if JOB = 'E';
LDB >= MAX(1, N), if JOB = 'T'.
FG (input/output) COMPLEX*16 array, dimension (LDFG, P),
where P = MAX(M+1,N), if JOB = 'T', and
P = M+1, if JOB = 'E'.
On entry, the leading N/2-by-N/2 lower triangular part of
this array must contain the lower triangular part of the
Hermitian matrix G, and the N/2-by-N/2 upper triangular
part of the submatrix in the columns 2 to N/2+1 of this
array must contain the upper triangular part of the
Hermitian matrix F. Accidental nonzero imaginary parts on
the main diagonals of F and G do not perturb the results.
On exit, if JOB = 'T', the leading N-by-N part of this
array contains the Hermitian matrix BF in (3) (see also
METHOD). The strictly lower triangular part of the input
matrix is preserved. The diagonal elements might have tiny
imaginary parts, since they have not been annihilated.
If JOB = 'E', this array is unchanged on exit.
LDFG INTEGER
The leading dimension of the array FG.
LDFG >= MAX(1, M), if JOB = 'E';
LDFG >= MAX(1, N), if JOB = 'T'.
D (output) COMPLEX*16 array, dimension (LDD, N)
If JOB = 'T', the leading N-by-N part of this array
contains the matrix BD in (3) (see also METHOD).
If JOB = 'E', this array is not referenced.
LDD INTEGER
The leading dimension of the array D.
LDD >= 1, if JOB = 'E';
LDD >= MAX(1, N), if JOB = 'T'.
C (output) COMPLEX*16 array, dimension (LDC, N)
If JOB = 'T', the leading N-by-N part of this array
contains the lower triangular matrix BC in (3) (see also
METHOD). The part over the first superdiagonal is not set.
If JOB = 'E', this array is not referenced.
LDC INTEGER
The leading dimension of the array C.
LDC >= 1, if JOB = 'E';
LDC >= MAX(1, N), if JOB = 'T'.
Q (output) COMPLEX*16 array, dimension (LDQ, 2*N)
On exit, if COMPQ = 'C' and JOB = 'T', then the leading
2*N-by-2*N part of this array contains the unitary
transformation matrix Q.
If COMPQ = 'C' and JOB = 'E', this array contains the
orthogonal transformation which reduced B_Z and B_T
in the first step of the algorithm (see METHOD).
If COMPQ = 'N', this array is not referenced.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1, 2*N), if COMPQ = 'C'.
U (output) COMPLEX*16 array, dimension (LDU, 2*N)
On exit, if COMPU = 'C' and JOB = 'T', then the leading
N-by-2*N part of this array contains the leading N-by-2*N
part of the unitary symplectic transformation matrix U.
If COMPU = 'C' and JOB = 'E', this array contains the
first N rows of the transformation U which reduced B_Z
and B_T in the first step of the algorithm (see METHOD).
If COMPU = 'N', this array is not referenced.
LDU INTEGER
The leading dimension of the array U.
LDU >= 1, if COMPU = 'N';
LDU >= MAX(1, N), if COMPU = 'C'.
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
The real parts of each scalar alpha defining an eigenvalue
of the pencil aS - bH.
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
The imaginary parts of each scalar alpha defining an
eigenvalue of the pencil aS - bH.
If ALPHAI(j) is zero, then the j-th eigenvalue is real.
BETA (output) DOUBLE PRECISION array, dimension (N)
The scalars beta that define the eigenvalues of the pencil
aS - bH.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent the j-th eigenvalue of the pencil
aS - bH, in the form lambda = alpha/beta. Since lambda may
overflow, the ratios should not, in general, be computed.
Workspace
IWORK INTEGER array, dimension (LIWORK)
On exit, if INFO = 3, IWORK(1) contains the number of
(pairs of) possibly inaccurate eigenvalues, q <= N/2, and
IWORK(2), ..., IWORK(q+1) indicate their indices.
Specifically, a positive value is an index of a real or
purely imaginary eigenvalue, corresponding to a 1-by-1
block, while the absolute value of a negative entry in
IWORK is an index to the first eigenvalue in a pair of
consecutively stored eigenvalues, corresponding to a
2-by-2 block. A 2-by-2 block may have two complex, two
real, two purely imaginary, or one real and one purely
imaginary eigenvalue. The blocks are those in B_T and B_S.
For i = q+2, ..., 2*q+1, IWORK(i) contains a pointer to
the starting location in DWORK of the i-th triplet of
1-by-1 blocks, if IWORK(i-q) > 0, or 2-by-2 blocks,
if IWORK(i-q) < 0, defining unreliable eigenvalues.
IWORK(2*q+2) contains the number of the 1-by-1 blocks, and
IWORK(2*q+3) contains the number of the 2-by-2 blocks,
corresponding to unreliable eigenvalues. IWORK(2*q+4)
contains the total number t of the 2-by-2 blocks.
If INFO = 0, then q = 0, therefore IWORK(1) = 0.
LIWORK INTEGER
The dimension of the array IWORK. LIWORK >= 2*N+9.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or INFO = 3, DWORK(1) returns the
optimal LDWORK, and DWORK(2), ..., DWORK(4) contain the
Frobenius norms of the factors of the formal matrix
product used by the algorithm. In addition, DWORK(5), ...,
DWORK(4+3*s) contain the s triplet values corresponding
to the 1-by-1 blocks. Their eigenvalues are real or purely
imaginary. Such an eigenvalue is obtained as -a1/a2/a3*i,
where a1, ..., a3 are the corresponding triplet values,
and i is the purely imaginary unit.
Moreover, DWORK(5+3*s), ..., DWORK(4+3*s+12*t) contain the
t groups of triplet 2-by-2 matrices corresponding to the
2-by-2 blocks. Their eigenvalue pairs are either complex,
or placed on the real and imaginary axes. Such an
eigenvalue pair is given by imag( ev ) - real( ev )*i,
where ev is the spectrum of the matrix product
A1*inv(A2)*inv(A3), and A1, ..., A3 define the
corresponding 2-by-2 matrix triplet.
On exit, if INFO = -25, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= c*N**2 + N + MAX(6*N, 27), where
c = 18, if COMPU = 'C';
c = 16, if COMPQ = 'C' and COMPU = 'N';
c = 13, if COMPQ = 'N' and COMPU = 'N'.
For good performance LDWORK should be generally larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) returns the optimal LZWORK.
On exit, if INFO = -27, ZWORK(1) returns the minimum
value of LZWORK.
LZWORK INTEGER
The dimension of the array ZWORK.
LZWORK >= 8*N + 28, if JOB = 'T' and COMPQ = 'C';
LZWORK >= 6*N + 28, if JOB = 'T' and COMPQ = 'N';
LZWORK >= 1, if JOB = 'E'.
For good performance LZWORK should be generally larger.
If LZWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
ZWORK array, returns this value as the first entry of
the ZWORK array, and no error message related to LZWORK
is issued by XERBLA.
BWORK LOGICAL array, dimension (LBWORK)
LBWORK >= 0, if JOB = 'E';
LBWORK >= N, if JOB = 'T'.
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: the algorithm was not able to reveal information
about the eigenvalues from the 2-by-2 blocks in the
SLICOT Library routine MB03BD (called by MB04ED);
= 2: periodic QZ iteration failed in the SLICOT Library
routines MB03BD or MB03BZ when trying to
triangularize the 2-by-2 blocks;
= 3: some eigenvalues might be inaccurate. This is a
warning.
Method
First T = i*H is set. Then, the embeddings, B_Z and B_T, of the
matrices S and T, are determined and, subsequently, the SLICOT
Library routine MB04ED is applied to compute the structured Schur
form, i.e., the factorizations
~ T ( BZ11 BZ12 )
B_Z = U B_Z Q = ( ) and
( 0 BZ22 )
~ T T ( T11 T12 )
B_T = J Q J B_T Q = ( T ),
( 0 T11 )
where Q is real orthogonal, U is real orthogonal symplectic, BZ11,
BZ22' are upper triangular and T11 is upper quasi-triangular.
If JOB = 'T', the 2-by-2 blocks are triangularized using the
periodic QZ algorithm.
References
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical Computation of Deflating Subspaces of Embedded
Hamiltonian Pencils.
Tech. Rep. SFB393/99-15, Technical University Chemnitz,
Germany, June 1999.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) complex floating point operations.Further Comments
This routine does not perform any scaling of the matrices. Scaling might sometimes be useful, and it should be done externally.Example
Program Text
* MB04AZ EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 50 )
INTEGER LDB, LDC, LDD, LDFG, LDQ, LDU, LDWORK, LDZ,
$ LIWORK, LZWORK
PARAMETER ( LDB = NMAX, LDC = NMAX, LDD = NMAX,
$ LDFG = NMAX, LDQ = 2*NMAX, LDU = NMAX,
$ LDWORK = 18*NMAX*NMAX + NMAX + MAX( 2*NMAX,
$ 24 ) + 3,
$ LDZ = NMAX, LIWORK = 2*NMAX + 9,
$ LZWORK = 8*NMAX + 28 )
*
* .. Local Scalars ..
CHARACTER COMPQ, COMPU, JOB
INTEGER I, INFO, J, M, N
*
* .. Local Arrays ..
COMPLEX*16 B( LDB, NMAX ), C( LDC, NMAX ),
$ D( LDD, NMAX ), FG( LDFG, NMAX ),
$ Q( LDQ, 2*NMAX ), U( LDU, 2*NMAX ),
$ Z( LDZ, NMAX ), ZWORK( LZWORK )
DOUBLE PRECISION ALPHAI( NMAX ), ALPHAR( NMAX ),
$ BETA( NMAX ), DWORK(LDWORK )
INTEGER IWORK( LIWORK )
LOGICAL BWORK( NMAX )
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
*
* .. External Subroutines ..
EXTERNAL MB04AZ
*
* .. Intrinsic Functions ..
INTRINSIC MAX, MOD
*
* .. Executable Statements ..
*
WRITE( NOUT, FMT = 99999 )
* Skip the heading in the data file and read in the data.
READ( NIN, FMT = * )
READ( NIN, FMT = * ) JOB, COMPQ, COMPU, N
IF( N.LT.0 .OR. N.GT.NMAX .OR. MOD( N, 2 ).NE.0 ) THEN
WRITE( NOUT, FMT = 99998 ) N
ELSE
READ( NIN, FMT = * ) ( ( Z( I, J ), J = 1, N ), I=1, N )
READ( NIN, FMT = * ) ( ( B( I, J ), J = 1, N/2 ), I=1, N/2 )
READ( NIN, FMT = * ) ( ( FG( I, J ), J = 1, N/2+1 ), I=1, N/2 )
* Compute the eigenvalues of a complex skew-Hamiltonian/
* Hamiltonian pencil (factored version).
CALL MB04AZ( JOB, COMPQ, COMPU, N, Z, LDZ, B, LDB, FG, LDFG,
$ D, LDD, C, LDC, Q, LDQ, U, LDU, ALPHAR, ALPHAI,
$ BETA, IWORK, LIWORK, DWORK, LDWORK, ZWORK, LZWORK,
$ BWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99997 ) INFO
ELSE
M = N/2
IF( LSAME( JOB, 'T' ) ) THEN
WRITE( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( Z( I, J ), J = 1, N )
10 CONTINUE
WRITE( NOUT, FMT = 99994 )
DO 20 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( B( I, J ), J = 1, N )
20 CONTINUE
WRITE( NOUT, FMT = 99993 )
DO 30 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( FG( I, J ), J = 1, N )
30 CONTINUE
WRITE( NOUT, FMT = 99992 )
DO 40 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( D( I, J ), J = 1, N )
40 CONTINUE
WRITE( NOUT, FMT = 99991 )
DO 50 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( C( I, J ), J = 1, N )
50 CONTINUE
END IF
IF( LSAME( COMPQ, 'C' ) ) THEN
WRITE( NOUT, FMT = 99990 )
DO 60 I = 1, 2*N
WRITE( NOUT, FMT = 99995 ) ( Q( I, J ), J = 1, 2*N )
60 CONTINUE
END IF
IF( LSAME( COMPU, 'C' ) ) THEN
WRITE( NOUT, FMT = 99989 )
DO 70 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( U( I, J ), J = 1, 2*N )
70 CONTINUE
END IF
WRITE( NOUT, FMT = 99988 )
WRITE( NOUT, FMT = 99987 ) ( ALPHAR( I ), I = 1, N )
WRITE( NOUT, FMT = 99986 )
WRITE( NOUT, FMT = 99987 ) ( ALPHAI( I ), I = 1, N )
WRITE( NOUT, FMT = 99985 )
WRITE( NOUT, FMT = 99987 ) ( BETA( I ), I = 1, N )
END IF
END IF
STOP
*
99999 FORMAT ( 'MB04AZ EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT ( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT ( 'INFO on exit from MB04AZ = ', I2 )
99996 FORMAT (/' The transformed matrix Z is' )
99995 FORMAT (20(1X,F9.4,SP,F9.4,S,'i '))
99994 FORMAT (/' The transformed matrix B is' )
99993 FORMAT (/' The transformed matrix FG is' )
99992 FORMAT (/' The matrix D is' )
99991 FORMAT (/' The matrix C is' )
99990 FORMAT (/' The matrix Q is' )
99989 FORMAT (/' The upper part of the matrix U is' )
99988 FORMAT (/' The vector ALPHAR is ' )
99987 FORMAT ( 50( 1X, F8.4 ) )
99986 FORMAT (/' The vector ALPHAI is ' )
99985 FORMAT (/' The vector BETA is ' )
END
Program Data
MB04AZ EXAMPLE PROGRAM DATA
T C C 4
(0.4941,0.8054) (0.8909,0.8865) (0.0305,0.9786) (0.9047,0.0596)
(0.7790,0.5767) (0.3341,0.0286) (0.7440,0.7126) (0.6098,0.6819)
(0.7150,0.1829) (0.6987,0.4899) (0.5000,0.5004) (0.6176,0.0424)
(0.9037,0.2399) (0.1978,0.1679) (0.4799,0.4710) (0.8594,0.0714)
(0.5216,0.7224) (0.8181,0.6596)
(0.0967,0.1498) (0.8175,0.5185)
0.9729 0.8003 (0.4323,0.8313)
(0.6489,0.1331) 0.4537 0.8253
Program Results
MB04AZ EXAMPLE PROGRAM RESULTS
The transformed matrix Z is
0.4545 +0.0000i 0.7904 +0.0000i -0.1601 +0.0000i -0.2691 +0.0000i
0.7790 +0.5767i 0.4273 +0.0000i 0.1459 +0.0000i 0.1298 +0.0000i
0.7150 +0.1829i 0.6987 +0.4899i 0.6715 +0.0000i -0.3001 +0.0000i
0.9037 +0.2399i 0.1978 +0.1679i 0.4799 +0.4710i 0.7924 +0.0000i
The transformed matrix B is
0.0000 -1.7219i 0.0000 +0.7762i 0.0000 +0.5342i 0.0000 +0.0845i
0.0000 +0.0000i 0.0000 +0.8862i 0.0000 +0.5186i 0.0000 -0.1429i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +1.1122i 0.0000 +0.2898i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.4889i
The transformed matrix FG is
0.0000 +0.0000i 0.0000 +0.4145i 0.0000 -0.7921i 0.0000 +0.5630i
0.6489 +0.1331i 0.0000 +0.0000i 0.0000 +1.5982i 0.0000 +0.5818i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.5819i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i
The matrix D is
2.2139 +0.0000i 0.0402 +0.0000i -0.2787 +0.0000i 1.0465 +0.0000i
-0.5021 +0.0000i 0.5502 +0.0000i -0.2771 +0.0000i -0.4521 +0.0000i
-0.0398 +0.0000i 0.4046 +0.0000i 0.0149 +0.0000i 0.7577 +0.0000i
-0.1550 +0.0000i 2.0660 +0.0000i 1.6075 +0.0000i 0.5836 +0.0000i
The matrix C is
0.3159 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i
0.7819 +0.0000i -0.7575 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i
-0.3494 +0.0000i 0.8622 +0.0000i 0.7539 +0.0000i 0.0000 +0.0000i
1.1178 +0.0000i 0.3133 +0.0000i 0.4638 +0.0000i 1.2348 +0.0000i
The matrix Q is
-0.4983 +0.0000i 0.3694 +0.0000i -0.4754 +0.0000i 0.2791 +0.0000i 0.1950 +0.0000i 0.2416 +0.0000i 0.1869 +0.0000i 0.4242 +0.0000i
-0.1045 +0.0000i 0.3309 +0.0000i 0.5730 +0.0000i -0.5566 +0.0000i 0.0877 +0.0000i 0.3543 +0.0000i 0.1761 +0.0000i 0.2779 +0.0000i
-0.2586 +0.0000i -0.4457 +0.0000i -0.2838 +0.0000i -0.5436 +0.0000i 0.5524 +0.0000i -0.1064 +0.0000i -0.2040 +0.0000i 0.0205 +0.0000i
-0.0040 +0.0000i 0.3845 +0.0000i 0.2469 +0.0000i 0.2965 +0.0000i 0.5799 +0.0000i 0.0840 +0.0000i -0.4788 +0.0000i -0.3614 +0.0000i
0.7958 +0.0000i 0.1597 +0.0000i -0.3420 +0.0000i -0.1047 +0.0000i 0.3370 +0.0000i 0.1813 +0.0000i 0.2275 +0.0000i 0.1229 +0.0000i
-0.0600 +0.0000i -0.4599 +0.0000i 0.3487 +0.0000i 0.3910 +0.0000i 0.4005 +0.0000i 0.1010 +0.0000i 0.5819 +0.0000i -0.0345 +0.0000i
-0.0539 +0.0000i 0.3817 +0.0000i 0.0501 +0.0000i -0.1114 +0.0000i 0.1751 +0.0000i -0.8212 +0.0000i 0.3606 +0.0000i -0.0380 +0.0000i
0.1846 +0.0000i -0.1577 +0.0000i 0.2510 +0.0000i 0.2293 +0.0000i 0.0911 +0.0000i -0.2834 +0.0000i -0.3779 +0.0000i 0.7708 +0.0000i
The upper part of the matrix U is
-0.2544 +0.0000i -0.1844 +0.0000i 0.7632 +0.0000i 0.5646 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i
0.4272 +0.0000i -0.0454 +0.0000i -0.2918 +0.0000i 0.5721 +0.0000i 0.1446 +0.0000i -0.1611 +0.0000i 0.3609 +0.0000i -0.4752 +0.0000i
0.6936 +0.0000i 0.3812 +0.0000i 0.2606 +0.0000i 0.0848 +0.0000i -0.3359 +0.0000i -0.1592 +0.0000i -0.3242 +0.0000i 0.2348 +0.0000i
0.2320 +0.0000i -0.4483 +0.0000i -0.1629 +0.0000i 0.1784 +0.0000i -0.2898 +0.0000i 0.7525 +0.0000i -0.0511 +0.0000i 0.1841 +0.0000i
The vector ALPHAR is
0.0000 0.0000 0.0000 0.0000
The vector ALPHAI is
-1.4991 -1.3690 1.0985 0.9993
The vector BETA is
0.1250 0.5000 0.5000 2.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04BD.html 0000664 0000000 0000000 00000062130 14560147231 0020324 0 ustar 00root root 0000000 0000000
Purpose
To compute the eigenvalues of a real N-by-N skew-Hamiltonian/
Hamiltonian pencil aS - bH with
( A D ) ( C V )
S = ( ) and H = ( ). (1)
( E A' ) ( W -C' )
Optionally, if JOB = 'T', decompositions of S and H will be
computed via orthogonal transformations Q1 and Q2 as follows:
( Aout Dout )
Q1' S J Q1 J' = ( ),
( 0 Aout' )
( Bout Fout )
J' Q2' J S Q2 = ( ) =: T, (2)
( 0 Bout' )
( C1out Vout ) ( 0 I )
Q1' H Q2 = ( ), where J = ( )
( 0 C2out' ) ( -I 0 )
and Aout, Bout, C1out are upper triangular, C2out is upper quasi-
triangular and Dout and Fout are skew-symmetric. The notation M'
denotes the transpose of the matrix M.
Optionally, if COMPQ1 = 'I' or COMPQ1 = 'U', then the orthogonal
transformation matrix Q1 will be computed.
Optionally, if COMPQ2 = 'I' or COMPQ2 = 'U', then the orthogonal
transformation matrix Q2 will be computed.
Specification
SUBROUTINE MB04BD( JOB, COMPQ1, COMPQ2, N, A, LDA, DE, LDDE, C1,
$ LDC1, VW, LDVW, Q1, LDQ1, Q2, LDQ2, B, LDB, F,
$ LDF, C2, LDC2, ALPHAR, ALPHAI, BETA, IWORK,
$ LIWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ1, COMPQ2, JOB
INTEGER INFO, LDA, LDB, LDC1, LDC2, LDDE, LDF, LDQ1,
$ LDQ2, LDVW, LDWORK, LIWORK, N
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), C1( LDC1, * ),
$ C2( LDC2, * ), DE( LDDE, * ), DWORK( * ),
$ F( LDF, * ), Q1( LDQ1, * ), Q2( LDQ2, * ),
$ VW( LDVW, * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'E': compute the eigenvalues only; S and H will not
necessarily be transformed as in (2).
= 'T': put S and H into the forms in (2) and return the
eigenvalues in ALPHAR, ALPHAI and BETA.
COMPQ1 CHARACTER*1
Specifies whether to compute the orthogonal transformation
matrix Q1, as follows:
= 'N': Q1 is not computed;
= 'I': the array Q1 is initialized internally to the unit
matrix, and the orthogonal matrix Q1 is returned;
= 'U': the array Q1 contains an orthogonal matrix Q on
entry, and the product Q*Q1 is returned, where Q1
is the product of the orthogonal transformations
that are applied to the pencil aS - bH to reduce
S and H to the forms in (2), for COMPQ1 = 'I'.
COMPQ2 CHARACTER*1
Specifies whether to compute the orthogonal transformation
matrix Q2, as follows:
= 'N': Q2 is not computed;
= 'I': on exit, the array Q2 contains the orthogonal
matrix Q2;
= 'U': on exit, the array Q2 contains the matrix product
J*Q*J'*Q2, where Q2 is the product of the
orthogonal transformations that are applied to
the pencil aS - bH to reduce S and H to the forms
in (2), for COMPQ2 = 'I'.
Setting COMPQ2 <> 'N' assumes COMPQ2 = COMPQ1.
Input/Output Parameters
N (input) INTEGER
The order of the pencil aS - bH. N >= 0, even.
A (input/output) DOUBLE PRECISION array, dimension
(LDA, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix A.
On exit, if JOB = 'T', the leading N/2-by-N/2 part of this
array contains the matrix Aout; otherwise, it contains the
upper triangular matrix A obtained just before the
application of the periodic QZ algorithm.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1, N/2).
DE (input/output) DOUBLE PRECISION array, dimension
(LDDE, N/2+1)
On entry, the leading N/2-by-N/2 strictly lower triangular
part of this array must contain the strictly lower
triangular part of the skew-symmetric matrix E, and the
N/2-by-N/2 strictly upper triangular part of the submatrix
in the columns 2 to N/2+1 of this array must contain the
strictly upper triangular part of the skew-symmetric
matrix D.
The entries on the diagonal and the first superdiagonal of
this array need not be set, but are assumed to be zero.
On exit, if JOB = 'T', the leading N/2-by-N/2 strictly
upper triangular part of the submatrix in the columns 2 to
N/2+1 of this array contains the strictly upper triangular
part of the skew-symmetric matrix Dout.
If JOB = 'E', the leading N/2-by-N/2 strictly upper
triangular part of the submatrix in the columns 2 to N/2+1
of this array contains the strictly upper triangular part
of the skew-symmetric matrix D just before the application
of the periodic QZ algorithm. The remaining entries are
meaningless.
LDDE INTEGER
The leading dimension of the array DE.
LDDE >= MAX(1, N/2).
C1 (input/output) DOUBLE PRECISION array, dimension
(LDC1, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix C1 = C.
On exit, if JOB = 'T', the leading N/2-by-N/2 part of this
array contains the matrix C1out; otherwise, it contains
the upper triangular matrix C1 obtained just before the
application of the periodic QZ algorithm.
LDC1 INTEGER
The leading dimension of the array C1.
LDC1 >= MAX(1, N/2).
VW (input/output) DOUBLE PRECISION array, dimension
(LDVW, N/2+1)
On entry, the leading N/2-by-N/2 lower triangular part of
this array must contain the lower triangular part of the
symmetric matrix W, and the N/2-by-N/2 upper triangular
part of the submatrix in the columns 2 to N/2+1 of this
array must contain the upper triangular part of the
symmetric matrix V.
On exit, if JOB = 'T', the N/2-by-N/2 part in the columns
2 to N/2+1 of this array contains the matrix Vout.
If JOB = 'E', the N/2-by-N/2 part in the columns 2 to
N/2+1 of this array contains the matrix V just before the
application of the periodic QZ algorithm.
LDVW INTEGER
The leading dimension of the array VW.
LDVW >= MAX(1, N/2).
Q1 (input/output) DOUBLE PRECISION array, dimension (LDQ1, N)
On entry, if COMPQ1 = 'U', then the leading N-by-N part of
this array must contain a given matrix Q, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q and the transformation matrix Q1
used to transform the matrices S and H.
On exit, if COMPQ1 = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q1.
If COMPQ1 = 'N', this array is not referenced.
LDQ1 INTEGER
The leading dimension of the array Q1.
LDQ1 >= 1, if COMPQ1 = 'N';
LDQ1 >= MAX(1, N), if COMPQ1 = 'I' or COMPQ1 = 'U'.
Q2 (output) DOUBLE PRECISION array, dimension (LDQ2, N)
On exit, if COMPQ2 = 'U', then the leading N-by-N part of
this array contains the product of the matrix J*Q*J' and
the transformation matrix Q2 used to transform the
matrices S and H.
On exit, if COMPQ2 = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q2.
If COMPQ2 = 'N', this array is not referenced.
LDQ2 INTEGER
The leading dimension of the array Q2.
LDQ2 >= 1, if COMPQ2 = 'N';
LDQ2 >= MAX(1, N), if COMPQ2 = 'I' or COMPQ2 = 'U'.
B (output) DOUBLE PRECISION array, dimension (LDB, N/2)
On exit, if JOB = 'T', the leading N/2-by-N/2 part of this
array contains the matrix Bout; otherwise, it contains the
upper triangular matrix B obtained just before the
application of the periodic QZ algorithm.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N/2).
F (output) DOUBLE PRECISION array, dimension (LDF, N/2)
On exit, if JOB = 'T', the leading N/2-by-N/2 strictly
upper triangular part of this array contains the strictly
upper triangular part of the skew-symmetric matrix Fout.
If JOB = 'E', the leading N/2-by-N/2 strictly upper
triangular part of this array contains the strictly upper
triangular part of the skew-symmetric matrix F just before
the application of the periodic QZ algorithm.
The entries on the leading N/2-by-N/2 lower triangular
part of this array are not referenced.
LDF INTEGER
The leading dimension of the array F. LDF >= MAX(1, N/2).
C2 (output) DOUBLE PRECISION array, dimension (LDC2, N/2)
On exit, if JOB = 'T', the leading N/2-by-N/2 part of this
array contains the matrix C2out; otherwise, it contains
the upper Hessenberg matrix C2 obtained just before the
application of the periodic QZ algorithm.
LDC2 INTEGER
The leading dimension of the array C2.
LDC2 >= MAX(1, N/2).
ALPHAR (output) DOUBLE PRECISION array, dimension (N/2)
The real parts of each scalar alpha defining an eigenvalue
of the pencil aS - bH.
ALPHAI (output) DOUBLE PRECISION array, dimension (N/2)
The imaginary parts of each scalar alpha defining an
eigenvalue of the pencil aS - bH.
If ALPHAI(j) is zero, then the j-th eigenvalue is real.
BETA (output) DOUBLE PRECISION array, dimension (N/2)
The scalars beta that define the eigenvalues of the pencil
aS - bH.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent the j-th eigenvalue of the pencil
aS - bH, in the form lambda = alpha/beta. Since lambda may
overflow, the ratios should not, in general, be computed.
Due to the skew-Hamiltonian/Hamiltonian structure of the
pencil, for every eigenvalue lambda, -lambda is also an
eigenvalue, and thus it has only to be saved once in
ALPHAR, ALPHAI and BETA.
Specifically, only eigenvalues with imaginary parts
greater than or equal to zero are stored; their conjugate
eigenvalues are not stored. If imaginary parts are zero
(i.e., for real eigenvalues), only positive eigenvalues
are stored. The remaining eigenvalues have opposite signs.
As a consequence, pairs of complex eigenvalues, stored in
consecutive locations, are not complex conjugate.
Workspace
IWORK INTEGER array, dimension (LIWORK)
On exit, if INFO = 3, IWORK(1) contains the number of
(pairs of) possibly inaccurate eigenvalues, q <= N/2, and
IWORK(2), ..., IWORK(q+1) indicate their indices.
Specifically, a positive value is an index of a real or
purely imaginary eigenvalue, corresponding to a 1-by-1
block, while the absolute value of a negative entry in
IWORK is an index to the first eigenvalue in a pair of
consecutively stored eigenvalues, corresponding to a
2-by-2 block. A 2-by-2 block may have two complex, two
real, two purely imaginary, or one real and one purely
imaginary eigenvalue.
For i = q+2, ..., 2*q+1, IWORK(i) contains a pointer to
the starting location in DWORK of the (i-q-1)-th quadruple
of 1-by-1 blocks, if IWORK(i-q) > 0, or 2-by-2 blocks,
if IWORK(i-q) < 0, defining unreliable eigenvalues.
IWORK(2*q+2) contains the number of the 1-by-1 blocks, and
IWORK(2*q+3) contains the number of the 2-by-2 blocks,
corresponding to unreliable eigenvalues. IWORK(2*q+4)
contains the total number t of the 2-by-2 blocks.
If INFO = 0, then q = 0, therefore IWORK(1) = 0.
LIWORK INTEGER
The dimension of the array IWORK. LIWORK >= N+12.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or INFO = 3, DWORK(1) returns the
optimal LDWORK, and DWORK(2), ..., DWORK(5) contain the
Frobenius norms of the factors of the formal matrix
product used by the algorithm. In addition, DWORK(6), ...,
DWORK(5+4*s) contain the s quadruple values corresponding
to the 1-by-1 blocks. Their eigenvalues are real or purely
imaginary. Such an eigenvalue is obtained from
-i*sqrt(a1*a3/a2/a4), but always taking a positive sign,
where a1, ..., a4 are the corresponding quadruple values.
Moreover, DWORK(6+4*s), ..., DWORK(5+4*s+16*t) contain the
t groups of quadruple 2-by-2 matrices corresponding to the
2-by-2 blocks. Their eigenvalue pairs are either complex,
or placed on the real and imaginary axes. Such an
eigenvalue pair is obtained as -1i*sqrt(ev), but taking
positive imaginary parts, where ev are the eigenvalues of
the product A1*inv(A2)*A3*inv(A4), where A1, ..., A4
define the corresponding 2-by-2 matrix quadruple.
On exit, if INFO = -27, DWORK(1) returns the minimum value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
If JOB = 'E' and COMPQ1 = 'N' and COMPQ2 = 'N',
LDWORK >= N**2 + MAX(L,36);
if JOB = 'T' or COMPQ1 <> 'N' or COMPQ2 <> 'N',
LDWORK >= 2*N**2 + MAX(L,36);
where
L = 4*N + 4, if N/2 is even, and
L = 4*N, if N/2 is odd.
For good performance LDWORK should generally be larger.
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: problem during computation of the eigenvalues;
= 2: periodic QZ algorithm did not converge in the SLICOT
Library subroutine MB03BD;
= 3: some eigenvalues might be inaccurate, and details can
be found in IWORK and DWORK. This is a warning.
Method
The algorithm uses Givens rotations and Householder reflections to annihilate elements in S, T, and H such that A, B, and C1 are upper triangular and C2 is upper Hessenberg. Finally, the periodic QZ algorithm is applied to transform C2 to upper quasi-triangular form while A, B, and C1 stay in upper triangular form. See also page 27 in [1] for more details.References
[1] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
Eigenproblems.
Tech. Rep., Technical University Chemnitz, Germany,
Nov. 2007.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) real floating point operations.Further Comments
NoneExample
Program Text
* MB04BD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 50 )
INTEGER LDA, LDB, LDC1, LDC2, LDDE, LDF, LDQ1, LDQ2,
$ LDVW, LDWORK, LIWORK
PARAMETER ( LDA = NMAX/2, LDB = NMAX/2, LDC1 = NMAX/2,
$ LDC2 = NMAX/2, LDDE = NMAX/2, LDF = NMAX/2,
$ LDQ1 = NMAX, LDQ2 = NMAX, LDVW = NMAX/2,
$ LDWORK = 2*NMAX*NMAX + MAX( 4*NMAX, 36 ),
$ LIWORK = MAX( NMAX + 12, 2*NMAX + 3 ) )
*
* .. Local Scalars ..
CHARACTER COMPQ1, COMPQ2, JOB
INTEGER I, INFO, J, M, N
*
* .. Local Arrays ..
INTEGER IWORK( LIWORK )
DOUBLE PRECISION A( LDA, NMAX/2 ), ALPHAI( NMAX/2 ),
$ ALPHAR( NMAX/2 ), B( LDB, NMAX/2 ),
$ BETA( NMAX/2 ), C1( LDC1, NMAX/2 ),
$ C2( LDC2, NMAX/2 ), DE( LDDE, NMAX/2+1 ),
$ DWORK( LDWORK ), F( LDF, NMAX/2 ),
$ Q1( LDQ1, NMAX ), Q2( LDQ2, NMAX ),
$ VW( LDVW, NMAX/2+1 )
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
*
* .. External Subroutines ..
EXTERNAL MB04BD
*
* .. Intrinsic Functions ..
INTRINSIC MAX
*
* .. Executable Statements ..
*
WRITE( NOUT, FMT = 99999 )
* Skip the heading in the data file and read in the data.
READ( NIN, FMT = * )
READ( NIN, FMT = * ) JOB, COMPQ1, COMPQ2, N
IF( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE( NOUT, FMT = 99998 ) N
ELSE
M = N/2
READ( NIN, FMT = * ) ( ( A( I, J ), J = 1, M ), I = 1, M )
READ( NIN, FMT = * ) ( ( DE( I, J ), J = 1, M+1 ), I = 1, M )
READ( NIN, FMT = * ) ( ( C1( I, J ), J = 1, M ), I = 1, M )
READ( NIN, FMT = * ) ( ( VW( I, J ), J = 1, M+1 ), I = 1, M )
* Compute the eigenvalues of a real skew-Hamiltonian/Hamiltonian
* pencil.
CALL MB04BD( JOB, COMPQ1, COMPQ2, N, A, LDA, DE, LDDE, C1,
$ LDC1, VW, LDVW, Q1, LDQ1, Q2, LDQ2, B, LDB, F,
$ LDF, C2, LDC2, ALPHAR, ALPHAI, BETA, IWORK,
$ LIWORK, DWORK, LDWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99997 ) INFO
ELSE
WRITE( NOUT, FMT = 99996 )
DO 10 I = 1, M
WRITE( NOUT, FMT = 99995 ) ( A( I, J ), J = 1, M )
10 CONTINUE
WRITE( NOUT, FMT = 99994 )
DO 20 I = 1, M
WRITE( NOUT, FMT = 99995 ) ( DE( I, J ), J = 2, M+1 )
20 CONTINUE
WRITE( NOUT, FMT = 99993 )
DO 30 I = 1, M
WRITE( NOUT, FMT = 99995 ) ( B( I, J ), J = 1, M )
30 CONTINUE
WRITE( NOUT, FMT = 99992 )
DO 40 I = 1, M
WRITE( NOUT, FMT = 99995 ) ( F( I, J ), J = 1, M )
40 CONTINUE
WRITE( NOUT, FMT = 99991 )
DO 50 I = 1, M
WRITE( NOUT, FMT = 99995 ) ( C1( I, J ), J = 1, M )
50 CONTINUE
WRITE( NOUT, FMT = 99990 )
DO 60 I = 1, M
WRITE( NOUT, FMT = 99995 ) ( C2( I, J ), J = 1, M )
60 CONTINUE
WRITE( NOUT, FMT = 99989 )
DO 70 I = 1, M
WRITE( NOUT, FMT = 99995 ) ( VW( I, J ), J = 2, M+1 )
70 CONTINUE
WRITE( NOUT, FMT = 99988 )
WRITE( NOUT, FMT = 99995 ) ( ALPHAR( I ), I = 1, M )
WRITE( NOUT, FMT = 99987 )
WRITE( NOUT, FMT = 99995 ) ( ALPHAI( I ), I = 1, M )
WRITE( NOUT, FMT = 99986 )
WRITE( NOUT, FMT = 99995 ) ( BETA( I ), I = 1, M )
WRITE( NOUT, FMT = 99985 )
IF( .NOT.LSAME( COMPQ1, 'N' ) ) THEN
DO 80 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( Q1( I, J ), J = 1, N )
80 CONTINUE
END IF
IF( .NOT.LSAME( COMPQ2, 'N' ) ) THEN
WRITE( NOUT, FMT = 99984 )
DO 90 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( Q2( I, J ), J = 1, N )
90 CONTINUE
END IF
END IF
END IF
STOP
*
99999 FORMAT( 'MB04BD EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT( 'INFO on exit from MB04BD = ', I2 )
99996 FORMAT( 'The matrix A on exit is ' )
99995 FORMAT( 50( 1X, F8.4 ) )
99994 FORMAT( 'The matrix D on exit is ' )
99993 FORMAT( 'The matrix B on exit is ' )
99992 FORMAT( 'The matrix F on exit is ' )
99991 FORMAT( 'The matrix C1 on exit is ' )
99990 FORMAT( 'The matrix C2 on exit is ' )
99989 FORMAT( 'The matrix V on exit is ' )
99988 FORMAT( 'The vector ALPHAR is ' )
99987 FORMAT( 'The vector ALPHAI is ' )
99986 FORMAT( 'The vector BETA is ' )
99985 FORMAT( 'The matrix Q1 is ' )
99984 FORMAT( 'The matrix Q2 is ' )
END
Program Data
MB04BD EXAMPLE PROGRAM DATA T I I 8 3.1472 1.3236 4.5751 4.5717 4.0579 -4.0246 4.6489 -0.1462 -3.7301 -2.2150 -3.4239 3.0028 4.1338 0.4688 4.7059 -3.5811 0.0000 0.0000 -1.5510 -4.5974 -2.5127 3.5071 0.0000 0.0000 1.5961 2.4490 -3.1428 2.5648 0.0000 0.0000 -0.0596 3.0340 2.4892 -1.1604 0.0000 0.0000 0.6882 -3.3782 -3.3435 1.8921 -0.3061 2.9428 1.0198 2.4815 -4.8810 -1.8878 -2.3703 -0.4946 -1.6288 0.2853 1.5408 -4.1618 -2.4013 -2.7102 0.3834 -3.9335 3.1730 -3.1815 -2.3620 4.9613 4.6190 3.6869 3.6929 0.7970 0.4986 -4.9537 -4.1556 3.5303 1.2206 -1.4905 0.1325 -1.0022Program Results
MB04BD EXAMPLE PROGRAM RESULTS The matrix A on exit is -4.7460 4.1855 3.2696 -0.2244 0.0000 6.4157 2.8287 1.4553 0.0000 0.0000 7.4626 1.5726 0.0000 0.0000 0.0000 8.8702 The matrix D on exit is 0.0000 -1.3137 -6.3615 -0.8940 0.0000 0.0000 1.0704 -0.0659 4.4324 0.0000 0.0000 -0.6922 0.5254 1.6653 0.0000 0.0000 The matrix B on exit is -6.4937 -2.1982 -1.3881 1.3477 0.0000 4.6929 0.6650 -4.1191 0.0000 0.0000 9.1725 3.4721 0.0000 0.0000 0.0000 7.2106 The matrix F on exit is 0.0000 -1.1367 2.2966 -1.0744 0.0000 0.0000 3.7875 0.9427 0.0000 0.0000 0.0000 -4.7136 0.0000 0.0000 0.0000 0.0000 The matrix C1 on exit is 6.9525 -4.9881 2.3661 4.2188 0.0000 8.5009 0.7182 5.5533 0.0000 0.0000 -4.6650 -2.8177 0.0000 0.0000 0.0000 1.5124 The matrix C2 on exit is -5.4562 -2.1348 4.9694 -2.2744 2.5550 -7.9616 1.1516 3.4912 0.0000 0.0000 4.8504 0.5046 0.0000 0.0000 0.0000 4.4394 The matrix V on exit is 0.9136 4.1106 -0.0079 3.5789 -1.1553 -1.4785 -1.5155 -0.8018 -2.2167 4.8029 1.3645 2.5202 -1.0994 -0.6144 0.3970 2.0730 The vector ALPHAR is 0.8314 -0.8314 0.8131 0.0000 The vector ALPHAI is 0.4372 0.4372 0.0000 0.9164 The vector BETA is 0.7071 0.7071 1.4142 2.8284 The matrix Q1 is -0.0098 0.1978 0.2402 0.5274 0.1105 -0.0149 -0.1028 0.7759 -0.6398 0.2356 0.2765 -0.1301 -0.5351 -0.3078 0.2435 0.0373 0.1766 -0.4781 0.2657 -0.5415 0.0968 -0.4663 -0.0983 0.3741 0.3207 -0.1980 0.1141 0.0240 -0.1712 0.2630 0.8513 0.1451 -0.6551 -0.2956 -0.0288 -0.1169 0.5593 0.3381 0.1753 0.1055 -0.0246 -0.2759 0.2470 -0.1408 -0.4837 0.6567 -0.4042 0.1172 -0.0772 -0.0121 -0.8394 -0.1852 -0.2673 0.0046 0.0159 0.4282 0.1442 0.6884 0.1257 -0.5860 0.2110 0.2699 0.0363 0.1657 The matrix Q2 is -0.2891 0.3096 0.6312 0.6498 0.0000 0.0000 0.0000 0.0000 0.1887 0.1936 -0.3857 0.3664 0.5660 0.1238 -0.2080 -0.5148 -0.2492 -0.2877 -0.0874 0.1110 -0.1081 -0.2999 0.6800 -0.5207 -0.7430 -0.0646 -0.4689 0.1556 -0.2401 0.0181 -0.3724 0.0562 -0.0999 -0.2026 -0.0355 0.0866 0.5587 -0.6625 -0.0114 0.4349 -0.4357 0.1209 0.0489 -0.2990 0.5094 0.5191 0.3837 0.1661 -0.2429 0.4131 0.2549 -0.5525 0.0749 -0.3829 -0.2690 -0.4190 0.0889 0.7439 -0.3960 0.0697 -0.1821 -0.1988 0.3687 0.2616
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04BP.html 0000664 0000000 0000000 00000042273 14560147231 0020346 0 ustar 00root root 0000000 0000000
Purpose
To compute the eigenvalues of a real N-by-N skew-Hamiltonian/
Hamiltonian pencil aS - bH with
( A D ) ( C V )
S = ( ) and H = ( ). (1)
( E A' ) ( W -C' )
Optionally, if JOB = 'T', decompositions of S and H will be
computed via orthogonal transformations Q1 and Q2 as follows:
( Aout Dout )
Q1' S J Q1 J' = ( ),
( 0 Aout' )
( Bout Fout )
J' Q2' J S Q2 = ( ) =: T, (2)
( 0 Bout' )
( C1out Vout ) ( 0 I )
Q1' H Q2 = ( ), where J = ( )
( 0 C2out' ) ( -I 0 )
and Aout, Bout, C1out are upper triangular, C2out is upper quasi-
triangular and Dout and Fout are skew-symmetric. The notation M'
denotes the transpose of the matrix M.
Optionally, if COMPQ1 = 'I' or COMPQ1 = 'U', then the orthogonal
transformation matrix Q1 will be computed.
Optionally, if COMPQ2 = 'I' or COMPQ2 = 'U', then the orthogonal
transformation matrix Q2 will be computed.
Specification
SUBROUTINE MB04BP( JOB, COMPQ1, COMPQ2, N, A, LDA, DE, LDDE, C1,
$ LDC1, VW, LDVW, Q1, LDQ1, Q2, LDQ2, B, LDB, F,
$ LDF, C2, LDC2, ALPHAR, ALPHAI, BETA, IWORK,
$ LIWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ1, COMPQ2, JOB
INTEGER INFO, LDA, LDB, LDC1, LDC2, LDDE, LDF, LDQ1,
$ LDQ2, LDVW, LDWORK, LIWORK, N
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), C1( LDC1, * ),
$ C2( LDC2, * ), DE( LDDE, * ), DWORK( * ),
$ F( LDF, * ), Q1( LDQ1, * ), Q2( LDQ2, * ),
$ VW( LDVW, * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'E': compute the eigenvalues only; S and H will not
necessarily be transformed as in (2).
= 'T': put S and H into the forms in (2) and return the
eigenvalues in ALPHAR, ALPHAI and BETA.
COMPQ1 CHARACTER*1
Specifies whether to compute the orthogonal transformation
matrix Q1, as follows:
= 'N': Q1 is not computed;
= 'I': the array Q1 is initialized internally to the unit
matrix, and the orthogonal matrix Q1 is returned;
= 'U': the array Q1 contains an orthogonal matrix Q on
entry, and the product Q*Q1 is returned, where Q1
is the product of the orthogonal transformations
that are applied to the pencil aS - bH to reduce
S and H to the forms in (2), for COMPQ1 = 'I'.
COMPQ2 CHARACTER*1
Specifies whether to compute the orthogonal transformation
matrix Q2, as follows:
= 'N': Q2 is not computed;
= 'I': on exit, the array Q2 contains the orthogonal
matrix Q2;
= 'U': on exit, the array Q2 contains the matrix product
J*Q*J'*Q2, where Q2 is the product of the
orthogonal transformations that are applied to
the pencil aS - bH to reduce S and H to the forms
in (2), for COMPQ2 = 'I'.
Setting COMPQ2 <> 'N' assumes COMPQ2 = COMPQ1.
Input/Output Parameters
N (input) INTEGER
The order of the pencil aS - bH. N >= 0, even.
A (input/output) DOUBLE PRECISION array, dimension
(LDA, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix A.
On exit, if JOB = 'T', the leading N/2-by-N/2 part of this
array contains the matrix Aout; otherwise, it contains the
upper triangular matrix A obtained just before the
application of the periodic QZ algorithm.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1, N/2).
DE (input/output) DOUBLE PRECISION array, dimension
(LDDE, N/2+1)
On entry, the leading N/2-by-N/2 strictly lower triangular
part of this array must contain the strictly lower
triangular part of the skew-symmetric matrix E, and the
N/2-by-N/2 strictly upper triangular part of the submatrix
in the columns 2 to N/2+1 of this array must contain the
strictly upper triangular part of the skew-symmetric
matrix D.
The entries on the diagonal and the first superdiagonal of
this array need not be set, but are assumed to be zero.
On exit, if JOB = 'T', the leading N/2-by-N/2 strictly
upper triangular part of the submatrix in the columns 2 to
N/2+1 of this array contains the strictly upper triangular
part of the skew-symmetric matrix Dout.
If JOB = 'E', the leading N/2-by-N/2 strictly upper
triangular part of the submatrix in the columns 2 to N/2+1
of this array contains the strictly upper triangular part
of the skew-symmetric matrix D just before the application
of the periodic QZ algorithm. The remaining entries are
meaningless.
LDDE INTEGER
The leading dimension of the array DE.
LDDE >= MAX(1, N/2).
C1 (input/output) DOUBLE PRECISION array, dimension
(LDC1, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix C1 = C.
On exit, if JOB = 'T', the leading N/2-by-N/2 part of this
array contains the matrix C1out; otherwise, it contains
the upper triangular matrix C1 obtained just before the
application of the periodic QZ algorithm.
LDC1 INTEGER
The leading dimension of the array C1.
LDC1 >= MAX(1, N/2).
VW (input/output) DOUBLE PRECISION array, dimension
(LDVW, N/2+1)
On entry, the leading N/2-by-N/2 lower triangular part of
this array must contain the lower triangular part of the
symmetric matrix W, and the N/2-by-N/2 upper triangular
part of the submatrix in the columns 2 to N/2+1 of this
array must contain the upper triangular part of the
symmetric matrix V.
On exit, if JOB = 'T', the N/2-by-N/2 part in the columns
2 to N/2+1 of this array contains the matrix Vout.
If JOB = 'E', the N/2-by-N/2 part in the columns 2 to
N/2+1 of this array contains the matrix V just before the
application of the periodic QZ algorithm.
LDVW INTEGER
The leading dimension of the array VW.
LDVW >= MAX(1, N/2).
Q1 (input/output) DOUBLE PRECISION array, dimension (LDQ1, N)
On entry, if COMPQ1 = 'U', then the leading N-by-N part of
this array must contain a given matrix Q, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q and the transformation matrix Q1
used to transform the matrices S and H.
On exit, if COMPQ1 = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q1.
If COMPQ1 = 'N', this array is not referenced.
LDQ1 INTEGER
The leading dimension of the array Q1.
LDQ1 >= 1, if COMPQ1 = 'N';
LDQ1 >= MAX(1, N), if COMPQ1 = 'I' or COMPQ1 = 'U'.
Q2 (output) DOUBLE PRECISION array, dimension (LDQ2, N)
On exit, if COMPQ2 = 'U', then the leading N-by-N part of
this array contains the product of the matrix J*Q*J' and
the transformation matrix Q2 used to transform the
matrices S and H.
On exit, if COMPQ2 = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q2.
If COMPQ2 = 'N', this array is not referenced.
LDQ2 INTEGER
The leading dimension of the array Q2.
LDQ2 >= 1, if COMPQ2 = 'N';
LDQ2 >= MAX(1, N), if COMPQ2 = 'I' or COMPQ2 = 'U'.
B (output) DOUBLE PRECISION array, dimension (LDB, N/2)
On exit, if JOB = 'T', the leading N/2-by-N/2 part of this
array contains the matrix Bout; otherwise, it contains the
upper triangular matrix B obtained just before the
application of the periodic QZ algorithm.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N/2).
F (output) DOUBLE PRECISION array, dimension (LDF, N/2)
On exit, if JOB = 'T', the leading N/2-by-N/2 strictly
upper triangular part of this array contains the strictly
upper triangular part of the skew-symmetric matrix Fout.
If JOB = 'E', the leading N/2-by-N/2 strictly upper
triangular part of this array contains the strictly upper
triangular part of the skew-symmetric matrix F just before
the application of the periodic QZ algorithm.
The entries on the leading N/2-by-N/2 lower triangular
part of this array are not referenced.
LDF INTEGER
The leading dimension of the array F. LDF >= MAX(1, N/2).
C2 (output) DOUBLE PRECISION array, dimension (LDC2, N/2)
On exit, if JOB = 'T', the leading N/2-by-N/2 part of this
array contains the matrix C2out; otherwise, it contains
the upper Hessenberg matrix C2 obtained just before the
application of the periodic QZ algorithm.
LDC2 INTEGER
The leading dimension of the array C2.
LDC2 >= MAX(1, N/2).
ALPHAR (output) DOUBLE PRECISION array, dimension (N/2)
The real parts of each scalar alpha defining an eigenvalue
of the pencil aS - bH.
ALPHAI (output) DOUBLE PRECISION array, dimension (N/2)
The imaginary parts of each scalar alpha defining an
eigenvalue of the pencil aS - bH.
If ALPHAI(j) is zero, then the j-th eigenvalue is real.
BETA (output) DOUBLE PRECISION array, dimension (N/2)
The scalars beta that define the eigenvalues of the pencil
aS - bH.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent the j-th eigenvalue of the pencil
aS - bH, in the form lambda = alpha/beta. Since lambda may
overflow, the ratios should not, in general, be computed.
Due to the skew-Hamiltonian/Hamiltonian structure of the
pencil, for every eigenvalue lambda, -lambda is also an
eigenvalue, and thus it has only to be saved once in
ALPHAR, ALPHAI and BETA.
Specifically, only eigenvalues with imaginary parts
greater than or equal to zero are stored; their conjugate
eigenvalues are not stored. If imaginary parts are zero
(i.e., for real eigenvalues), only positive eigenvalues
are stored. The remaining eigenvalues have opposite signs.
As a consequence, pairs of complex eigenvalues, stored in
consecutive locations, are not complex conjugate.
Workspace
IWORK INTEGER array, dimension (LIWORK)
On exit, if INFO = 3, IWORK(1) contains the number of
(pairs of) possibly inaccurate eigenvalues, q <= N/2, and
IWORK(2), ..., IWORK(q+1) indicate their indices.
Specifically, a positive value is an index of a real or
purely imaginary eigenvalue, corresponding to a 1-by-1
block, while the absolute value of a negative entry in
IWORK is an index to the first eigenvalue in a pair of
consecutively stored eigenvalues, corresponding to a
2-by-2 block. A 2-by-2 block may have two complex, two
real, two purely imaginary, or one real and one purely
imaginary eigenvalue.
For i = q+2, ..., 2*q+1, IWORK(i) contains a pointer to
the starting location in DWORK of the i-th quadruple of
1-by-1 blocks, if IWORK(i-q) > 0, or 2-by-2 blocks,
if IWORK(i-q) < 0, defining unreliable eigenvalues.
IWORK(2*q+2) contains the number of the 1-by-1 blocks, and
IWORK(2*q+3) contains the number of the 2-by-2 blocks,
corresponding to unreliable eigenvalues. IWORK(2*q+4)
contains the total number t of the 2-by-2 blocks.
If INFO = 0, then q = 0, therefore IWORK(1) = 0.
LIWORK INTEGER
The dimension of the array IWORK. LIWORK >= N+12.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or INFO = 3, DWORK(1) returns the
optimal LDWORK, and DWORK(2), ..., DWORK(5) contain the
Frobenius norms of the factors of the formal matrix
product used by the algorithm. In addition, DWORK(6), ...,
DWORK(5+4*s) contain the s quadruple values corresponding
to the 1-by-1 blocks. Their eigenvalues are real or purely
imaginary. Such an eigenvalue is obtained from
-i*sqrt(a1*a3/a2/a4), but always taking a positive sign,
where a1, ..., a4 are the corresponding quadruple values.
Moreover, DWORK(6+4*s), ..., DWORK(5+4*s+16*t) contain the
t groups of quadruple 2-by-2 matrices corresponding to the
2-by-2 blocks. Their eigenvalue pairs are either complex,
or placed on the real and imaginary axes. Such an
eigenvalue pair is obtained as -1i*sqrt(ev), but taking
positive imaginary parts, where ev are the eigenvalues of
the product A1*inv(A2)*A3*inv(A4), where A1, ..., A4
define the corresponding 2-by-2 matrix quadruple.
On exit, if INFO = -27, DWORK(1) returns the minimum value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
If JOB = 'E' and COMPQ1 = 'N' and COMPQ2 = 'N',
LDWORK >= N**2 + MAX(L,36);
if JOB = 'T' or COMPQ1 <> 'N' or COMPQ2 <> 'N',
LDWORK >= 2*N**2 + MAX(L,36);
where
L = 4*N + 4, if N/2 is even, and
L = 4*N, if N/2 is odd.
For good performance LDWORK should generally be larger.
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: problem during computation of the eigenvalues;
= 2: periodic QZ algorithm did not converge in the SLICOT
Library subroutine MB03BD;
= 3: some eigenvalues might be inaccurate, and details can
be found in IWORK and DWORK. This is a warning.
Method
The algorithm uses Givens rotations and Householder reflections to annihilate elements in S, T, and H such that A, B, and C1 are upper triangular and C2 is upper Hessenberg. Finally, the periodic QZ algorithm is applied to transform C2 to upper quasi-triangular form while A, B, and C1 stay in upper triangular form. See also page 27 in [1] for more details.References
[1] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
Eigenproblems.
Tech. Rep., Technical University Chemnitz, Germany,
Nov. 2007.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) real floating point operations.Further Comments
For large values of N, the routine applies the transformations for reducing T on panels of columns. The user may specify in INFO the desired number of columns. If on entry INFO < 0, then the routine estimates a suitable value of this number. If INFO = 0, the routine MB04BD is directly called.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04BZ.html 0000664 0000000 0000000 00000054555 14560147231 0020366 0 ustar 00root root 0000000 0000000
Purpose
To compute the eigenvalues of a complex N-by-N skew-Hamiltonian/
Hamiltonian pencil aS - bH, with
( A D ) ( B F )
S = ( H ) and H = ( H ). (1)
( E A ) ( G -B )
This routine computes the eigenvalues using an embedding to a real
skew-Hamiltonian/skew-Hamiltonian pencil aB_S - bB_T, defined as
( Re(A) -Im(A) | Re(D) -Im(D) )
( | )
( Im(A) Re(A) | Im(D) Re(D) )
( | )
B_S = (-----------------+-----------------) , and
( | T T )
( Re(E) -Im(E) | Re(A ) Im(A ) )
( | T T )
( Im(E) Re(E) | -Im(A ) Re(A ) )
(2)
( -Im(B) -Re(B) | -Im(F) -Re(F) )
( | )
( Re(B) -Im(B) | Re(F) -Im(F) )
( | )
B_T = (-----------------+-----------------) , T = i*H.
( | T T )
( -Im(G) -Re(G) | -Im(B ) Re(B ) )
( | T T )
( Re(G) -Im(G) | -Re(B ) -Im(B ) )
Optionally, if JOB = 'T', the pencil aB_S - bB_H (B_H = -i*B_T) is
transformed by a unitary matrix Q to the structured Schur form
( BA BD ) ( BB BF )
B_Sout = ( H ) and B_Hout = ( H ), (3)
( 0 BA ) ( 0 -BB )
where BA and BB are upper triangular, BD is skew-Hermitian, and
BF is Hermitian. The embedding doubles the multiplicities of the
eigenvalues of the pencil aS - bH. Optionally, if COMPQ = 'C', the
unitary matrix Q is computed.
Specification
SUBROUTINE MB04BZ( JOB, COMPQ, N, A, LDA, DE, LDDE, B, LDB, FG,
$ LDFG, Q, LDQ, ALPHAR, ALPHAI, BETA, IWORK,
$ DWORK, LDWORK, ZWORK, LZWORK, BWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, JOB
INTEGER INFO, LDA, LDB, LDDE, LDFG, LDQ, LDWORK,
$ LZWORK, N
C .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION ALPHAI( * ), ALPHAR( * ), BETA( * ), DWORK( * )
COMPLEX*16 A( LDA, * ), B( LDB, * ), DE( LDDE, * ),
$ FG( LDFG, * ), Q( LDQ, * ), ZWORK( * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'E': compute the eigenvalues only; S and H will not
necessarily be transformed as in (3).
= 'T': put S and H into the forms in (3) and return the
eigenvalues in ALPHAR, ALPHAI and BETA.
COMPQ CHARACTER*1
Specifies whether to compute the unitary transformation
matrix Q, as follows:
= 'N': Q is not computed;
= 'C': compute the unitary transformation matrix Q.
Input/Output Parameters
N (input) INTEGER
The order of the pencil aS - bH. N >= 0, even.
A (input/output) COMPLEX*16 array, dimension (LDA, K)
where K = N/2, if JOB = 'E', and K = N, if JOB = 'T'.
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix A.
On exit, if JOB = 'T', the leading N-by-N part of this
array contains the upper triangular matrix BA in (3) (see
also METHOD). The strictly lower triangular part is not
zeroed, but it is preserved.
If JOB = 'E', this array is unchanged on exit.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1, K).
DE (input/output) COMPLEX*16 array, dimension
(LDDE, MIN(K+1,N))
On entry, the leading N/2-by-N/2 lower triangular part of
this array must contain the lower triangular part of the
skew-Hermitian matrix E, and the N/2-by-N/2 upper
triangular part of the submatrix in the columns 2 to N/2+1
of this array must contain the upper triangular part of
the skew-Hermitian matrix D.
On exit, if JOB = 'T', the leading N-by-N part of this
array contains the skew-Hermitian matrix BD in (3) (see
also METHOD). The strictly lower triangular part of the
input matrix is preserved.
If JOB = 'E', this array is unchanged on exit.
LDDE INTEGER
The leading dimension of the array DE. LDDE >= MAX(1, K).
B (input/output) COMPLEX*16 array, dimension (LDB, K)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix B.
On exit, if JOB = 'T', the leading N-by-N part of this
array contains the upper triangular matrix BB in (3) (see
also METHOD). The strictly lower triangular part is not
zeroed; the elements below the first subdiagonal of the
input matrix are preserved.
If JOB = 'E', this array is unchanged on exit.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, K).
FG (input/output) COMPLEX*16 array, dimension
(LDFG, MIN(K+1,N))
On entry, the leading N/2-by-N/2 lower triangular part of
this array must contain the lower triangular part of the
Hermitian matrix G, and the N/2-by-N/2 upper triangular
part of the submatrix in the columns 2 to N/2+1 of this
array must contain the upper triangular part of the
Hermitian matrix F.
On exit, if JOB = 'T', the leading N-by-N part of this
array contains the Hermitian matrix BF in (3) (see also
METHOD). The strictly lower triangular part of the input
matrix is preserved. The diagonal elements might have tiny
imaginary parts.
If JOB = 'E', this array is unchanged on exit.
LDFG INTEGER
The leading dimension of the array FG. LDFG >= MAX(1, K).
Q (output) COMPLEX*16 array, dimension (LDQ, 2*N)
On exit, if COMPQ = 'C', the leading 2*N-by-2*N part of
this array contains the unitary transformation matrix Q
that reduced the matrices B_S and B_H to the form in (3).
However, if JOB = 'E', the reduction was possibly not
completed: the matrix B_H may have 2-by-2 diagonal blocks,
and the array Q returns the orthogonal matrix that
performed the partial reduction.
If COMPQ = 'N', this array is not referenced.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1, 2*N), if COMPQ = 'C'.
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
The real parts of each scalar alpha defining an eigenvalue
of the pencil aS - bH.
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
The imaginary parts of each scalar alpha defining an
eigenvalue of the pencil aS - bH.
If ALPHAI(j) is zero, then the j-th eigenvalue is real.
BETA (output) DOUBLE PRECISION array, dimension (N)
The scalars beta that define the eigenvalues of the pencil
aS - bH.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent the j-th eigenvalue of the pencil
aS - bH, in the form lambda = alpha/beta. Since lambda may
overflow, the ratios should not, in general, be computed.
Workspace
IWORK INTEGER array, dimension (2*N+4)
On exit, IWORK(1) contains the number, q, of unreliable,
possibly inaccurate (pairs of) eigenvalues, and the
absolute values in IWORK(2), ..., IWORK(q+1) are their
indices, as well as of the corresponding 1-by-1 and 2-by-2
diagonal blocks of the arrays B and A on exit, if
JOB = 'T'. Specifically, a positive value is an index of
a real or purely imaginary eigenvalue, corresponding to a
1-by-1 block, while the absolute value of a negative entry
in IWORK is an index to the first eigenvalue in a pair of
consecutively stored eigenvalues, corresponding to a
2-by-2 block. Moreover, IWORK(q+2),..., IWORK(2*q+1)
contain pointers to the starting elements in DWORK where
each block pair is stored. Specifically, if IWORK(i+1) > 0
then DWORK(r) and DWORK(r+1) store corresponding diagonal
elements of T11 and S11, respectively, and if
IWORK(i+1) < 0, then DWORK(r:r+3) and DWORK(r+4:r+7) store
the elements of the block in T11 and S11, respectively
(see Section METHOD), where r = IWORK(q+1+i). Moreover,
IWORK(2*q+2) contains the number of the 1-by-1 blocks, and
IWORK(2*q+3) contains the number of the 2-by-2 blocks,
corresponding to unreliable eigenvalues. IWORK(2*q+4)
contains the total number t of the 2-by-2 blocks.
If INFO = 0, then q = 0, therefore IWORK(1) = 0.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or INFO = 3, DWORK(1) returns the
optimal LDWORK, and DWORK(2) and DWORK(3) contain the
Frobenius norms of the matrices B_S and B_T. These norms
are used in the tests to decide that some eigenvalues are
considered as numerically unreliable. Moreover, DWORK(4),
..., DWORK(3+2*s) contain the s pairs of values of the
1-by-1 diagonal elements of T11 and S11. The eigenvalue of
such a block pair is obtained from -i*T11(i,i)/S11(i,i).
Similarly, DWORK(4+2*s), ..., DWORK(3+2*s+8*t) contain the
t groups of pairs of 2-by-2 diagonal submatrices of T11
and S11, stored column-wise. The spectrum of such a block
pair is obtained from -i*ev, where ev are the eigenvalues
of (T11(i:i+1,i:i+1),S11(i:i+1,i:i+1)).
On exit, if INFO = -19, DWORK(1) returns the minimum value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK. If COMPQ = 'N',
LDWORK >= MAX( 3, 4*N*N + 3*N ), if JOB = 'E';
LDWORK >= MAX( 3, 5*N*N + 3*N ), if JOB = 'T';
LDWORK >= MAX( 3, 11*N*N + 2*N ), if COMPQ = 'C'.
For good performance LDWORK should be generally larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) returns the optimal LZWORK.
On exit, if INFO = -21, ZWORK(1) returns the minimum value
of LZWORK.
LZWORK INTEGER
The dimension of the array ZWORK.
LZWORK >= 1, if JOB = 'E'; otherwise,
LZWORK >= 6*N + 4, if COMPQ = 'N';
LZWORK >= 8*N + 4, if COMPQ = 'C'.
If LZWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
ZWORK array, returns this value as the first entry of
the ZWORK array, and no error message related to LZWORK
is issued by XERBLA.
BWORK LOGICAL array, dimension (LBWORK)
LBWORK >= 0, if JOB = 'E';
LBWORK >= N, if JOB = 'T'.
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: QZ iteration failed in the SLICOT Library routine
MB04FD (QZ iteration did not converge or computation
of the shifts failed);
= 2: QZ iteration failed in the LAPACK routine ZHGEQZ when
trying to triangularize the 2-by-2 blocks;
= 3: warning: the pencil is numerically singular.
Method
First, T = i*H is set. Then, the embeddings, B_S and B_T, of the
matrices S and T, are determined and, subsequently, the SLICOT
Library routine MB04FD is applied to compute the structured Schur
form, i.e., the factorizations
~ T T ( S11 S12 )
B_S = J Q J B_S Q = ( T ) and
( 0 S11 )
~ T T ( T11 T12 )
B_T = J Q J B_T Q = ( T ),
( 0 T11 )
where Q is real orthogonal, S11 is upper triangular, and T11 is
upper quasi-triangular. If JOB = 'T', then the matrices above are
~
further transformed so that the 2-by-2 blocks in i*B_T are split
into 1-by-1 blocks. If COMPQ = 'C', the transformations are
accumulated in the unitary matrix Q.
See also page 22 in [1] for more details.
References
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical Computation of Deflating Subspaces of Embedded
Hamiltonian Pencils.
Tech. Rep. SFB393/99-15, Technical University Chemnitz,
Germany, June 1999.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) complex floating point operations.Further Comments
The returned eigenvalues are those of the pencil (-i*T11,S11), where i is the purely imaginary unit. If JOB = 'E', the returned matrix T11 is quasi-triangular. Note that the off-diagonal elements of the 2-by-2 blocks of S11 are zero by construction. If JOB = 'T', the returned eigenvalues correspond to the diagonal elements of BB and BA. This routine does not perform any scaling of the matrices. Scaling might sometimes be useful, and it should be done externally.Example
Program Text
* MB04BZ EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 50 )
INTEGER LDA, LDB, LDDE, LDFG, LDQ, LDWORK, LZWORK
PARAMETER ( LDA = NMAX, LDB = NMAX, LDDE = NMAX,
$ LDFG = NMAX, LDQ = 2*NMAX,
$ LDWORK = 11*NMAX*NMAX + 2*NMAX,
$ LZWORK = 8*NMAX + 4 )
*
* .. Local Scalars ..
CHARACTER COMPQ, JOB
INTEGER I, INFO, J, M, N
*
* .. Local Arrays ..
COMPLEX*16 A( LDA, NMAX ), B( LDB, NMAX ),
$ DE( LDDE, NMAX ), FG( LDFG, NMAX ),
$ Q( LDQ, 2*NMAX ), ZWORK( LZWORK )
DOUBLE PRECISION ALPHAI( NMAX ), ALPHAR( NMAX ),
$ BETA( NMAX ), DWORK( LDWORK )
INTEGER IWORK( 2*NMAX+3 )
LOGICAL BWORK( NMAX )
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
*
* .. External Subroutines ..
EXTERNAL MB04BZ
*
* .. Intrinsic Functions ..
INTRINSIC MOD
*
* .. Executable Statements ..
*
WRITE( NOUT, FMT = 99999 )
* Skip the heading in the data file and read in the data.
READ( NIN, FMT = * )
READ( NIN, FMT = * ) JOB, COMPQ, N
IF( N.LT.0 .OR. N.GT.NMAX .OR. MOD( N, 2 ).NE.0 ) THEN
WRITE( NOUT, FMT = 99998 ) N
ELSE
M = N/2
READ( NIN, FMT = * ) ( ( A( I, J ), J = 1, M ), I = 1, M )
READ( NIN, FMT = * ) ( ( DE( I, J ), J = 1, M+1 ), I = 1, M )
READ( NIN, FMT = * ) ( ( B( I, J ), J = 1, M ), I = 1, M )
READ( NIN, FMT = * ) ( ( FG( I, J ), J = 1, M+1 ), I = 1, M )
* Compute the eigenvalues of a complex skew-Hamiltonian/
* Hamiltonian pencil.
CALL MB04BZ( JOB, COMPQ, N, A, LDA, DE, LDDE, B, LDB, FG, LDFG,
$ Q, LDQ, ALPHAR, ALPHAI, BETA, IWORK, DWORK,
$ LDWORK, ZWORK, LZWORK, BWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99997 ) INFO
ELSE
IF( LSAME( JOB, 'T' ) ) THEN
WRITE( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( A( I, J ), J = 1, N )
10 CONTINUE
WRITE( NOUT, FMT = 99994 )
DO 20 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( DE( I, J ), J = 1, N )
20 CONTINUE
WRITE( NOUT, FMT = 99993 )
DO 30 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( B( I, J ), J = 1, N )
30 CONTINUE
WRITE( NOUT, FMT = 99992 )
DO 40 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( FG( I, J ), J = 1, N )
40 CONTINUE
END IF
IF( LSAME( COMPQ, 'C' ) ) THEN
WRITE( NOUT, FMT = 99991 )
DO 50 I = 1, 2*N
WRITE( NOUT, FMT = 99995 ) ( Q( I, J ), J = 1, 2*N )
50 CONTINUE
END IF
WRITE( NOUT, FMT = 99990 )
WRITE( NOUT, FMT = 99989 ) ( ALPHAR( I ), I = 1, N )
WRITE( NOUT, FMT = 99988 )
WRITE( NOUT, FMT = 99989 ) ( ALPHAI( I ), I = 1, N )
WRITE( NOUT, FMT = 99987 )
WRITE( NOUT, FMT = 99989 ) ( BETA( I ), I = 1, N )
END IF
END IF
STOP
*
99999 FORMAT ( 'MB04BZ EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT ( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT ( 'INFO on exit from MB04BZ = ', I2 )
99996 FORMAT (/'The matrix A on exit is ' )
99995 FORMAT (20( 1X, F9.4, SP, F9.4, S, 'i ') )
99994 FORMAT (/'The matrix D on exit is ' )
99993 FORMAT (/'The matrix B on exit is ' )
99992 FORMAT (/'The matrix F on exit is ' )
99991 FORMAT (/'The matrix Q is ' )
99990 FORMAT (/'The vector ALPHAR is ' )
99989 FORMAT ( 50( 1X, F8.4 ) )
99988 FORMAT (/'The vector ALPHAI is ' )
99987 FORMAT (/'The vector BETA is ' )
END
Program Data
MB04BZ EXAMPLE PROGRAM DATA
T C 4
(0.0604,0.6568) (0.5268,0.2919)
(0.3992,0.6279) (0.4167,0.4316)
(0,0.4896) (0,0.9516) (0.3724,0.0526)
(0.9840,0.3394) (0,0.9203) (0,0.7378)
(0.2691,0.4177) (0.5478,0.3014)
(0.4228,0.9830) (0.9427,0.7010)
0.6663 0.6981 (0.1781,0.8818)
(0.5391,0.1711) 0.6665 0.1280
Program Results
MB04BZ EXAMPLE PROGRAM RESULTS
The matrix A on exit is
0.7430 +0.0000i 0.0389 -0.4330i -0.1155 -0.1366i -0.6586 -0.3210i
0.3992 +0.6279i 0.7548 +0.0000i 0.6099 -0.2308i 0.2140 +0.1260i
0.0000 +0.0000i 0.0000 +0.0000i 1.4085 +0.0000i 0.0848 +0.4972i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 1.4725 +0.0000i
The matrix D on exit is
0.0000 -0.6858i 0.1839 -0.0474i -0.4428 -0.1290i 0.4759 +0.0380i
0.9840 +0.3394i 0.0000 +0.6858i -0.6339 +0.1358i 0.4204 -0.2140i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 -0.2110i -0.0159 -0.0338i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.2110i
The matrix B on exit is
-1.5832 +0.5069i -0.0097 +0.0866i 0.1032 -0.1431i -0.0426 +0.7942i
0.0000 +0.0000i 1.6085 +0.5150i -0.1342 -0.8180i 0.5143 +0.0178i
0.0000 +0.0000i 0.0000 +0.0000i -0.0842 -0.1642i 0.0246 -0.0264i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0880 -0.1716i
The matrix F on exit is
0.3382 0.0000i 0.0234 +0.0907i -0.1619 +0.9033i -0.8227 +0.0204i
0.5391 +0.1711i -0.3382 +0.0000i -0.6525 +0.2455i -0.3532 -0.6409i
0.0000 +0.0000i 0.0000 +0.0000i 0.0120 0.0000i 0.0019 -0.0009i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i -0.0120 +0.0000i
The matrix Q is
0.1422 +0.5446i -0.3877 -0.1273i -0.4363 +0.1705i 0.0348 -0.5440i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i
0.1594 -0.2382i 0.1967 -0.2467i -0.1376 -0.0961i -0.1070 -0.2058i -0.1273 +0.0585i -0.0852 +0.1020i 0.6125 -0.1059i -0.0172 +0.5589i
-0.3659 -0.0211i -0.0291 +0.4967i -0.0729 +0.4236i 0.3169 -0.0008i 0.2947 -0.1080i 0.1614 -0.2342i 0.2867 -0.0578i -0.0170 +0.2603i
0.1846 +0.4089i -0.2815 -0.2018i 0.3220 -0.1600i -0.0526 +0.3937i 0.2747 -0.0655i 0.1045 -0.2159i 0.2085 -0.3104i -0.3052 +0.1463i
-0.0201 -0.2898i 0.2131 -0.0081i -0.2165 -0.1055i -0.1324 -0.3133i 0.1660 -0.1635i 0.2250 -0.1390i -0.1590 -0.4634i -0.5310 -0.2239i
0.1342 -0.1295i 0.1128 -0.1990i -0.0712 -0.1686i -0.1490 -0.1336i 0.6198 +0.0113i 0.0281 -0.4762i -0.0462 +0.3244i 0.3464 +0.0086i
0.2305 -0.1358i 0.1292 -0.3311i -0.0106 +0.4992i 0.3906 +0.0997i 0.1429 +0.3376i -0.4310 -0.0866i -0.0894 -0.1336i -0.1601 -0.1055i
-0.2601 +0.0835i -0.0940 +0.3652i -0.0213 -0.3116i -0.2502 -0.0995i 0.1361 +0.4589i -0.5898 -0.0730i 0.0294 -0.1192i -0.1253 +0.0085i
The vector ALPHAR is
-1.5832 1.5832 -0.0842 0.0842
The vector ALPHAI is
0.5069 0.5069 -0.1642 -0.1642
The vector BETA is
0.7430 0.7430 1.4085 1.4085
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04CD.html 0000664 0000000 0000000 00000034015 14560147231 0020326 0 ustar 00root root 0000000 0000000
Purpose
To compute the transformed matrices A, B and D, using orthogonal
matrices Q1, Q2 and Q3 for a real N-by-N regular pencil
( A11 0 ) ( B11 0 ) ( 0 D12 )
aA*B - bD = a ( ) ( ) - b ( ), (1)
( 0 A22 ) ( 0 B22 ) ( D21 0 )
where A11, A22, B11, B22 and D12 are upper triangular, D21 is
upper quasi-triangular and the generalized matrix product
-1 -1 -1 -1
A11 D12 B22 A22 D21 B11 is upper quasi-triangular, such
that Q3' A Q2, Q2' B Q1 are upper triangular, Q3' D Q1 is upper
quasi-triangular and the transformed pencil
a(Q3' A B Q1) - b(Q3' D Q1) is in generalized Schur form. The
notation M' denotes the transpose of the matrix M.
Specification
SUBROUTINE MB04CD( COMPQ1, COMPQ2, COMPQ3, N, A, LDA, B, LDB, D,
$ LDD, Q1, LDQ1, Q2, LDQ2, Q3, LDQ3, IWORK,
$ LIWORK, DWORK, LDWORK, BWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ1, COMPQ2, COMPQ3
INTEGER INFO, LDA, LDB, LDD, LDQ1, LDQ2, LDQ3, LDWORK,
$ LIWORK, N
C .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), D( LDD, * ),
$ DWORK( * ), Q1( LDQ1, * ), Q2( LDQ2, * ),
$ Q3( LDQ3, * )
Arguments
Mode Parameters
COMPQ1 CHARACTER*1
Specifies whether to compute the orthogonal transformation
matrix Q1, as follows:
= 'N': Q1 is not computed;
= 'I': the array Q1 is initialized internally to the unit
matrix, and the orthogonal matrix Q1 is returned;
= 'U': the array Q1 contains an orthogonal matrix Q01 on
entry, and the matrix Q01*Q1 is returned, where Q1
is the product of the orthogonal transformations
that are applied on the right to the pencil
aA*B - bD in (1).
COMPQ2 CHARACTER*1
Specifies whether to compute the orthogonal transformation
matrix Q2, as follows:
= 'N': Q2 is not computed;
= 'I': the array Q2 is initialized internally to the unit
matrix, and the orthogonal matrix Q2 is returned;
= 'U': the array Q2 contains an orthogonal matrix Q02 on
entry, and the matrix Q02*Q2 is returned, where Q2
is the product of the orthogonal transformations
that are applied on the left to the pencil
aA*B - bD in (1).
COMPQ3 CHARACTER*1
Specifies whether to compute the orthogonal transformation
matrix Q3, as follows:
= 'N': Q3 is not computed;
= 'I': the array Q3 is initialized internally to the unit
matrix, and the orthogonal matrix Q3 is returned;
= 'U': the array Q3 contains an orthogonal matrix Q01 on
entry, and the matrix Q03*Q3 is returned, where Q3
is the product of the orthogonal transformations
that are applied on the right to the pencil
aA*B - bD in (1).
Input/Output Parameters
N (input) INTEGER
Order of the pencil aA*B - bD. N >= 0, even.
A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
On entry, the leading N-by-N block diagonal part of this
array must contain the matrix A in (1). The off-diagonal
blocks need not be set to zero.
On exit, the leading N-by-N part of this array contains
the transformed upper triangular matrix.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1, N).
B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
On entry, the leading N-by-N block diagonal part of this
array must contain the matrix B in (1). The off-diagonal
blocks need not be set to zero.
On exit, the leading N-by-N part of this array contains
the transformed upper triangular matrix.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N).
D (input/output) DOUBLE PRECISION array, dimension (LDD, N)
On entry, the leading N-by-N block anti-diagonal part of
this array must contain the matrix D in (1). The diagonal
blocks need not be set to zero.
On exit, the leading N-by-N part of this array contains
the transformed upper quasi-triangular matrix.
LDD INTEGER
The leading dimension of the array D. LDD >= MAX(1, N).
Q1 (input/output) DOUBLE PRECISION array, dimension (LDQ1, N)
On entry, if COMPQ1 = 'U', then the leading N-by-N part of
this array must contain a given matrix Q01, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q01 and the transformation matrix Q1
used to transform the matrices A, B, and D.
On exit, if COMPQ1 = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q1.
If COMPQ1 = 'N' this array is not referenced.
LDQ1 INTEGER
LDQ1 >= 1, if COMPQ1 = 'N';
LDQ1 >= MAX(1, N), if COMPQ1 = 'I' or COMPQ1 = 'U'.
Q2 (input/output) DOUBLE PRECISION array, dimension (LDQ2, N)
On entry, if COMPQ2 = 'U', then the leading N-by-N part of
this array must contain a given matrix Q02, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q02 and the transformation matrix Q2
used to transform the matrices A, B, and D.
On exit, if COMPQ2 = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q2.
If COMPQ2 = 'N' this array is not referenced.
LDQ2 INTEGER
The leading dimension of the array Q2.
LDQ2 >= 1, if COMPQ2 = 'N';
LDQ2 >= MAX(1, N), if COMPQ2 = 'I' or COMPQ2 = 'U'.
Q3 (input/output) DOUBLE PRECISION array, dimension (LDQ3, N)
On entry, if COMPQ3 = 'U', then the leading N-by-N part of
this array must contain a given matrix Q03, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q03 and the transformation matrix Q3
used to transform the matrices A, B and D.
On exit, if COMPQ3 = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q3.
If COMPQ3 = 'N' this array is not referenced.
LDQ3 INTEGER
The leading dimension of the array Q3.
LDQ3 >= 1, if COMPQ3 = 'N';
LDQ3 >= MAX(1, N), if COMPQ3 = 'I' or COMPQ3 = 'U'.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK INTEGER
The dimension of the array IWORK.
LIWORK >= MAX( N/2+1, 48 ).
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
On exit, if INFO = -20, DWORK(1) returns the minimum value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= 3*N*N + MAX( N/2 + 252, 432 ).
For good performance LDWORK should be generally larger.
If LDWORK = -1 a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
BWORK LOGICAL array, dimension (N/2)
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: the periodic QZ algorithm failed to reorder the
eigenvalues (the problem is very ill-conditioned) in
the SLICOT Library routine MB03KD;
= 2: the standard QZ algorithm failed in the LAPACK
routine DGGEV, called by the SLICOT routine MB03CD;
= 3: the standard QZ algorithm failed in the LAPACK
routines DGGES, called by the SLICOT routines MB03CD
or MB03ED;
= 4: the standard QZ algorithm failed to reorder the
eigenvalues in the LAPACK routine DTGSEN, called by
the SLICOT routine MB03CD.
Method
First, the periodic QZ algorithm (see also [2] and [3]) is applied
-1 -1 -1 -1
to the formal matrix product A11 D12 B22 A22 D21 B11 to
reorder the eigenvalues, i.e., orthogonal matrices V1, V2, V3, V4,
V5 and V6 are computed such that V2' A11 V1, V2' D12 V3,
V4' B22 V3, V5' A22 V4, V5' D21 V6 and V1' B11 V6 keep the
triangular form, but they can be partitioned into 2-by-2 block
forms and the last diagonal blocks correspond to all nonpositive
real eigenvalues of the formal product, and the first diagonal
blocks correspond to the remaining eigenvalues.
Second, Q1 = diag(V6, V3), Q2 = diag(V1, V4), Q3 = diag(V2, V5)
and
( AA11 AA12 0 0 )
( )
( 0 AA22 0 0 )
A := Q3' A Q2 =: ( ),
( 0 0 AA33 AA34 )
( )
( 0 0 0 AA44 )
( BB11 BB12 0 0 )
( )
( 0 BB22 0 0 )
B := Q2' B Q1 =: ( ),
( 0 0 BB33 BB34 )
( )
( 0 0 0 BB44 )
( 0 0 DD13 DD14 )
( )
( 0 0 0 DD24 )
D := Q3' D Q1 =: ( ),
( DD31 DD32 0 0 )
( )
( 0 DD42 0 0 )
-1 -1 -1 -1
are set, such that AA22 DD24 BB44 AA44 DD42 BB22 has only
nonpositive real eigenvalues.
Third, the permutation matrix
( I 0 0 0 )
( )
( 0 0 I 0 )
P = ( ),
( 0 I 0 0 )
( )
( 0 0 0 I )
where I denotes the identity matrix of appropriate size is used to
transform aA*B - bD to block upper triangular form
( AA11 0 | AA12 0 )
( | )
( 0 AA33 | 0 AA34 ) ( AA1 * )
A := P' A P = (-----------+-----------) = ( ),
( 0 0 | AA22 0 ) ( 0 AA2 )
( | )
( 0 0 | 0 AA44 )
( BB11 0 | BB12 0 )
( | )
( 0 BB33 | 0 BB34 ) ( BB1 * )
B := P' B P = (-----------+-----------) = ( ),
( 0 0 | BB22 0 ) ( 0 BB2 )
( | )
( 0 0 | 0 BB44 )
( 0 DD13 | 0 DD14 )
( | )
( DD31 0 | DD32 0 ) ( DD1 * )
D := P' D P = (-----------+-----------) = ( ).
( 0 0 | 0 DD24 ) ( 0 DD2 )
( | )
( 0 0 | DD42 0 )
Then, further orthogonal transformations that are provided by the
SLICOT Library routines MB03ED and MB03CD are used to
triangularize the subpencil aAA1 BB1 - bDD1.
Finally, the subpencil aAA2 BB2 - bDD2 is triangularized by
applying a special permutation matrix.
See also page 22 in [1] for more details.
References
[1] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
Eigenproblems.
Tech. Rep., Technical University Chemnitz, Germany,
Nov. 2007.
[2] Bojanczyk, A., Golub, G. H. and Van Dooren, P.
The periodic Schur decomposition: algorithms and applications.
In F.T. Luk (editor), Advanced Signal Processing Algorithms,
Architectures, and Implementations III, Proc. SPIE Conference,
vol. 1770, pp. 31-42, 1992.
[3] Hench, J. J. and Laub, A. J.
Numerical Solution of the discrete-time periodic Riccati
equation. IEEE Trans. Automat. Control, 39, 1197-1210, 1994.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) real floating point operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To apply from the left the inverse of a balancing transformation,
computed by the SLICOT Library routine MB04DP, to the matrix
[ V1 ]
[ ],
[ sgn*V2 ]
where sgn is either +1 or -1.
Specification
SUBROUTINE MB04DB( JOB, SGN, N, ILO, LSCALE, RSCALE, M, V1, LDV1,
$ V2, LDV2, INFO )
C .. Scalar Arguments ..
CHARACTER JOB, SGN
INTEGER ILO, INFO, LDV1, LDV2, M, N
C .. Array Arguments ..
DOUBLE PRECISION LSCALE(*), RSCALE(*), V1(LDV1,*), V2(LDV2,*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the type of inverse transformation required:
= 'N': do nothing, return immediately;
= 'P': do inverse transformation for permutation only;
= 'S': do inverse transformation for scaling only;
= 'B': do inverse transformations for both permutation
and scaling.
JOB must be the same as the argument JOB supplied to
MB04DP.
SGN CHARACTER*1
Specifies the sign to use for V2:
= 'P': sgn = +1;
= 'N': sgn = -1.
Input/Output Parameters
N (input) INTEGER
The number of rows of the matrices V1 and V2. N >= 0.
ILO (input) INTEGER
The integer ILO determined by MB04DP.
1 <= ILO <= N+1.
LSCALE (input) DOUBLE PRECISION array, dimension (N)
Details of the permutation and scaling factors applied
from the left, as returned by MB04DP.
RSCALE (input) DOUBLE PRECISION array, dimension (N)
Details of the permutation and scaling factors applied
from the right, as returned by MB04DP.
M (input) INTEGER
The number of columns of the matrices V1 and V2. M >= 0.
V1 (input/output) DOUBLE PRECISION array, dimension (LDV1,M)
On entry, the leading N-by-M part of this array must
contain the matrix V1.
On exit, the leading N-by-M part of this array is
overwritten by the updated matrix V1 of the transformed
matrix.
LDV1 INTEGER
The leading dimension of the array V1. LDV1 >= max(1,N).
V2 (input/output) DOUBLE PRECISION array, dimension (LDV2,M)
On entry, the leading N-by-M part of this array must
contain the matrix V2.
On exit, the leading N-by-M part of this array is
overwritten by the updated matrix V2 of the transformed
matrix.
LDV2 INTEGER
The leading dimension of the array V2. LDV2 >= max(1,N).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
References
[1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
[2] Benner, P.
Symplectic balancing of Hamiltonian matrices.
SIAM J. Sci. Comput., 22 (5), pp. 1885-1904, 2001.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To balance a real Hamiltonian matrix,
[ A G ]
H = [ T ] ,
[ Q -A ]
where A is an N-by-N matrix and G, Q are N-by-N symmetric
matrices. This involves, first, permuting H by a symplectic
similarity transformation to isolate eigenvalues in the first
1:ILO-1 elements on the diagonal of A; and second, applying a
diagonal similarity transformation to rows and columns
ILO:N, N+ILO:2*N to make the rows and columns as close in 1-norm
as possible. Both steps are optional.
Specification
SUBROUTINE MB04DD( JOB, N, A, LDA, QG, LDQG, ILO, SCALE, INFO )
C .. Scalar Arguments ..
CHARACTER JOB
INTEGER ILO, INFO, LDA, LDQG, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), QG(LDQG,*), SCALE(*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the operations to be performed on H:
= 'N': none, set ILO = 1, SCALE(I) = 1.0, I = 1 .. N;
= 'P': permute only;
= 'S': scale only;
= 'B': both permute and scale.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A.
On exit, the leading N-by-N part of this array contains
the matrix A of the balanced Hamiltonian. In particular,
the strictly lower triangular part of the first ILO-1
columns of A is zero.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
QG (input/output) DOUBLE PRECISION array, dimension
(LDQG,N+1)
On entry, the leading N-by-N+1 part of this array must
contain the lower triangular part of the matrix Q and
the upper triangular part of the matrix G.
On exit, the leading N-by-N+1 part of this array contains
the lower and upper triangular parts of the matrices Q and
G, respectively, of the balanced Hamiltonian. In
particular, the lower triangular part of the first ILO-1
columns of QG is zero.
LDQG INTEGER
The leading dimension of the array QG. LDQG >= MAX(1,N).
ILO (output) INTEGER
ILO-1 is the number of deflated eigenvalues in the
balanced Hamiltonian matrix.
SCALE (output) DOUBLE PRECISION array of dimension (N)
Details of the permutations and scaling factors applied to
H. For j = 1,...,ILO-1 let P(j) = SCALE(j). If P(j) <= N,
then rows and columns P(j) and P(j)+N are interchanged
with rows and columns j and j+N, respectively. If
P(j) > N, then row and column P(j)-N are interchanged with
row and column j+N by a generalized symplectic
permutation. For j = ILO,...,N the j-th element of SCALE
contains the factor of the scaling applied to row and
column j.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
References
[1] Benner, P.
Symplectic balancing of Hamiltonian matrices.
SIAM J. Sci. Comput., 22 (5), pp. 1885-1904, 2001.
Further Comments
NoneExample
Program Text
* MB04DD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 100 )
INTEGER LDA, LDQG
PARAMETER ( LDA = NMAX, LDQG = NMAX )
* .. Local Scalars ..
CHARACTER*1 JOB
INTEGER I, ILO, INFO, J, N
* .. Local Arrays ..
DOUBLE PRECISION A(LDA, NMAX), DUMMY(1), QG(LDQG, NMAX+1),
$ SCALE(NMAX)
* .. External Functions ..
DOUBLE PRECISION DLANTR, DLAPY2
EXTERNAL DLANTR, DLAPY2
* .. External Subroutines ..
EXTERNAL MB04DD
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, JOB
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( QG(I,J), J = 1,N+1 ), I = 1,N )
CALL MB04DD( JOB, N, A, LDA, QG, LDQG, ILO, SCALE, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 30 I = 1, N
WRITE (NOUT, FMT = 99995) ( A(I,J), J = 1,N )
30 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 40 I = 1, N
WRITE (NOUT, FMT = 99995) ( QG(I,J), J = 1,N+1 )
40 CONTINUE
WRITE (NOUT, FMT = 99993) ILO
IF ( ILO.GT.1 ) THEN
WRITE (NOUT, FMT = 99992) DLAPY2( DLANTR( 'Frobenius',
$ 'Lower', 'No Unit', N-1, ILO-1, A(2,1), LDA,
$ DUMMY ), DLANTR( 'Frobenius', 'Lower', 'No Unit',
$ N, ILO-1, QG(1,1), LDQG, DUMMY ) )
END IF
END IF
END IF
*
99999 FORMAT (' MB04DD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB04DD = ',I2)
99997 FORMAT (' The balanced matrix A is ')
99996 FORMAT (/' The balanced matrix QG is ')
99995 FORMAT (20(1X,F12.4))
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' ILO = ',I4)
99992 FORMAT (/' Norm of subdiagonal blocks: ',G7.2)
END
Program Data
MB04DD EXAMPLE PROGRAM DATA
6 B
0 0 0 0 0 0
0.0994 0 0 0 0 0.9696
0.3248 0 0 0 0.4372 0.8308
0 0 0 0.0717 0 0
0 0 0 0 0 0.1976
0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0.0651 0 0
0 0 0 0 0 0 0
0 0 0.0444 0 0 0.1957 0
0.8144 0 0 0 0.3652 0 0.9121
0.9023 0 0 0 0 0 1.0945
Program Results
MB04DD EXAMPLE PROGRAM RESULTS
The balanced matrix A is
0.0000 0.0000 0.0000 0.0000 0.0000 0.9696
0.0000 0.0000 0.0000 0.0000 -0.8144 -0.9023
0.0000 0.0000 0.0000 0.0000 0.1093 0.2077
0.0000 0.0000 0.0000 0.0717 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.1976
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
The balanced matrix QG is
0.0000 0.0000 0.0994 0.0000 0.0651 0.0000 0.0000
0.0000 0.0000 0.0000 0.0812 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.1776 0.0000 0.0000 0.1957 0.0000
0.0000 0.0000 0.0000 0.0000 0.3652 0.0000 0.9121
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0945
ILO = 3
Norm of subdiagonal blocks: 0.0
Purpose
To apply the inverse of a balancing transformation, computed by
the SLICOT Library routines MB04DD or MB04DS, to a 2*N-by-M matrix
[ V1 ]
[ ],
[ sgn*V2 ]
where sgn is either +1 or -1.
Specification
SUBROUTINE MB04DI( JOB, SGN, N, ILO, SCALE, M, V1, LDV1, V2, LDV2,
$ INFO )
C .. Scalar Arguments ..
CHARACTER JOB, SGN
INTEGER ILO, INFO, LDV1, LDV2, M, N
C .. Array Arguments ..
DOUBLE PRECISION SCALE(*), V1(LDV1,*), V2(LDV2,*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the type of inverse transformation required:
= 'N': do nothing, return immediately;
= 'P': do inverse transformation for permutation only;
= 'S': do inverse transformation for scaling only;
= 'B': do inverse transformations for both permutation
and scaling.
JOB must be the same as the argument JOB supplied to
MB04DD or MB04DS.
SGN CHARACTER*1
Specifies the sign to use for V2:
= 'P': sgn = +1;
= 'N': sgn = -1.
Input/Output Parameters
N (input) INTEGER
The number of rows of the matrices V1 and V2. N >= 0.
ILO (input) INTEGER
The integer ILO determined by MB04DD or MB04DS.
1 <= ILO <= N+1.
SCALE (input) DOUBLE PRECISION array, dimension (N)
Details of the permutation and scaling factors, as
returned by MB04DD or MB04DS.
M (input) INTEGER
The number of columns of the matrices V1 and V2. M >= 0.
V1 (input/output) DOUBLE PRECISION array, dimension (LDV1,M)
On entry, the leading N-by-M part of this array must
contain the matrix V1.
On exit, the leading N-by-M part of this array is
overwritten by the updated matrix V1 of the transformed
matrix.
LDV1 INTEGER
The leading dimension of the array V1. LDV1 >= max(1,N).
V2 (input/output) DOUBLE PRECISION array, dimension (LDV2,M)
On entry, the leading N-by-M part of this array must
contain the matrix V2.
On exit, the leading N-by-M part of this array is
overwritten by the updated matrix V2 of the transformed
matrix.
LDV2 INTEGER
The leading dimension of the array V2. LDV2 >= max(1,N).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
References
[1] Benner, P.
Symplectic balancing of Hamiltonian matrices.
SIAM J. Sci. Comput., 22 (5), pp. 1885-1904, 2001.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To balance a pair of N-by-N real matrices (A,B). This involves, first, permuting A and B by equivalence transformations to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal of A and B; and second, applying a diagonal equivalence transformation to rows and columns ILO to IHI to make the rows and columns as close in 1-norm as possible. Both steps are optional. Balancing may reduce the 1-norms of the matrices, and improve the accuracy of the computed eigenvalues and/or eigenvectors in the generalized eigenvalue problem A*x = lambda*B*x. This routine may optionally improve the conditioning of the scaling transformation compared to the LAPACK routine DGGBAL.Specification
SUBROUTINE MB04DL( JOB, N, THRESH, A, LDA, B, LDB, ILO, IHI,
$ LSCALE, RSCALE, DWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER JOB
INTEGER IHI, ILO, INFO, IWARN, LDA, LDB, N
DOUBLE PRECISION THRESH
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), LSCALE(*),
$ RSCALE(*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the operations to be performed on A and B:
= 'N': none: simply set ILO = 1, LSCALE(I) = 1.0 and
RSCALE(I) = 1.0 for I = 1,...,N.
= 'P': permute only;
= 'S': scale only;
= 'B': both permute and scale.
Input/Output Parameters
N (input) INTEGER
The order of matrices A and B. N >= 0.
THRESH (input) DOUBLE PRECISION
If JOB = 'S' or JOB = 'B', and THRESH >= 0, threshold
value for magnitude of the elements to be considered in
the scaling process: elements with magnitude less than or
equal to THRESH*MXNORM are ignored for scaling, where
MXNORM is the maximum of the 1-norms of the original
submatrices A(s,s) and B(s,s), with s = ILO:IHI.
If THRESH < 0, the subroutine finds the scaling factors
for which some conditions, detailed below, are fulfilled.
A sequence of increasing strictly positive threshold
values is used.
If THRESH = -1, the condition is that
max( norm(A(s,s),1)/norm(B(s,s),1),
norm(B(s,s),1)/norm(S(s,s),1) ) (1)
has the smallest value, for the threshold values used,
where A(s,s) and B(s,s) are the scaled submatrices.
If THRESH = -2, the norm ratio reduction (1) is tried, but
the subroutine may return IWARN = 1 and reset the scaling
factors to 1, if this seems suitable. See the description
of the argument IWARN and FURTHER COMMENTS.
If THRESH = -3, the condition is that
norm(A(s,s),1)*norm(B(s,s),1) (2)
has the smallest value for the scaled submatrices.
If THRESH = -4, the norm reduction in (2) is tried, but
the subroutine may return IWARN = 1 and reset the scaling
factors to 1, as for THRESH = -2 above.
If THRESH = -VALUE, with VALUE >= 10, the condition
numbers of the left and right scaling transformations will
be bounded by VALUE, i.e., the ratios between the largest
and smallest entries in LSCALE(s) and RSCALE(s), will be
at most VALUE. VALUE should be a power of 10.
If JOB = 'N' or JOB = 'P', the value of THRESH is
irrelevant.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A.
On exit, the leading N-by-N part of this array contains
the balanced matrix A.
In particular, the strictly lower triangular part of the
first ILO-1 columns and the last N-IHI rows of A is zero.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
On entry, the leading N-by-N part of this array must
contain the matrix B.
On exit, the leading N-by-N part of this array contains
the balanced matrix B.
In particular, the strictly lower triangular part of the
first ILO-1 columns and the last N-IHI rows of B is zero.
If JOB = 'N', the arrays A and B are not referenced.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N).
ILO (output) INTEGER
IHI (output) INTEGER
ILO and IHI are set to integers such that on exit
A(i,j) = 0 and B(i,j) = 0 if i > j and
j = 1,...,ILO-1 or i = IHI+1,...,N.
If JOB = 'N' or 'S', ILO = 1 and IHI = N.
LSCALE (output) DOUBLE PRECISION array, dimension (N)
Details of the permutations and scaling factors applied
to the left side of A and B. If P(j) is the index of the
row interchanged with row j, and D(j) is the scaling
factor applied to row j, then
LSCALE(j) = P(j) for j = 1,...,ILO-1
= D(j) for j = ILO,...,IHI
= P(j) for j = IHI+1,...,N.
The order in which the interchanges are made is N to
IHI+1, then 1 to ILO-1.
RSCALE (output) DOUBLE PRECISION array, dimension (N)
Details of the permutations and scaling factors applied
to the right side of A and B. If P(j) is the index of the
column interchanged with column j, and D(j) is the scaling
factor applied to column j, then
RSCALE(j) = P(j) for j = 1,...,ILO-1
= D(j) for j = ILO,...,IHI
= P(j) for j = IHI+1,...,N.
The order in which the interchanges are made is N to
IHI+1, then 1 to ILO-1.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK) where
LDWORK = 0, if JOB = 'N' or JOB = 'P', or N = 0;
LDWORK = 6*N, if (JOB = 'S' or JOB = 'B') and THRESH >= 0;
LDWORK = 8*N, if (JOB = 'S' or JOB = 'B') and THRESH < 0.
On exit, if JOB = 'S' or JOB = 'B', DWORK(1) and DWORK(2)
contain the initial 1-norms of A(s,s) and B(s,s), and
DWORK(3) and DWORK(4) contain their final 1-norms,
respectively. Moreover, DWORK(5) contains the THRESH value
used (irrelevant if IWARN = 1 or ILO = IHI).
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: scaling has been requested, for THRESH = -2 or
THRESH = -4, but it most probably would not improve
the accuracy of the computed solution for a related
eigenproblem (since maximum norm increased
significantly compared to the original pencil
matrices and (very) high and/or small scaling
factors occurred). The returned scaling factors have
been reset to 1, but information about permutations,
if requested, has been preserved.
Error Indicator
INFO INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Balancing consists of applying an equivalence transformation
to isolate eigenvalues and/or to make the 1-norms of the rows
and columns ILO,...,IHI of A and B nearly equal. If THRESH < 0,
a search is performed to find those scaling factors giving the
smallest norm ratio or product defined above (see the description
of the parameter THRESH).
Assuming JOB = 'S', let Dl and Dr be diagonal matrices containing
the vectors LSCALE and RSCALE, respectively. The returned matrices
are obtained using the equivalence transformation
Dl*A*Dr and Dl*B*Dr.
For THRESH = 0, the routine returns essentially the same results
as the LAPACK subroutine DGGBAL [1]. Setting THRESH < 0, usually
gives better results than DGGBAL for badly scaled matrix pencils.
References
[1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
Numerical Aspects
No rounding errors appear if JOB = 'P'.Further Comments
If THRESH = -2, the increase of the maximum norm of the scaled submatrices, compared to the maximum norm of the initial submatrices, is bounded by MXGAIN = 100. If THRESH = -2, or THRESH = -4, the maximum condition number of the scaling transformations is bounded by MXCOND = 1/SQRT(EPS), where EPS is the machine precision (see LAPACK Library routine DLAMCH).Example
Program Text
* MB04DL EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 10 )
INTEGER LDA, LDB
PARAMETER ( LDA = NMAX, LDB = NMAX )
* .. Local Scalars ..
CHARACTER*1 JOB
INTEGER I, ILO, INFO, IWARN, J, N
DOUBLE PRECISION THRESH
* .. Local Arrays ..
DOUBLE PRECISION A(LDA, NMAX), B(LDB, NMAX), DWORK(8*NMAX),
$ LSCALE(NMAX), RSCALE(NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL MB04DL
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, JOB, THRESH
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99985 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,N ), I = 1,N )
CALL MB04DL( JOB, N, THRESH, A, LDA, B, LDB, ILO, IHI, LSCALE,
$ RSCALE, DWORK, IWARN, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,N )
40 CONTINUE
WRITE ( NOUT, FMT = 99994 ) ILO
WRITE ( NOUT, FMT = 99993 ) IHI
WRITE ( NOUT, FMT = 99991 )
WRITE ( NOUT, FMT = 99995 ) ( LSCALE(I), I = 1,N )
WRITE ( NOUT, FMT = 99990 )
WRITE ( NOUT, FMT = 99995 ) ( RSCALE(I), I = 1,N )
IF ( LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'B' ) ) THEN
IF ( .NOT.( THRESH.EQ.-2 .OR. THRESH.EQ.-4 ) ) THEN
WRITE ( NOUT, FMT = 99989 )
WRITE ( NOUT, FMT = 99995 ) ( DWORK(I), I = 1,2 )
WRITE ( NOUT, FMT = 99988 )
WRITE ( NOUT, FMT = 99995 ) ( DWORK(I), I = 3,4 )
WRITE ( NOUT, FMT = 99987 )
WRITE ( NOUT, FMT = 99995 ) ( DWORK(5) )
ELSE
WRITE ( NOUT, FMT = 99986 ) IWARN
END IF
END IF
END IF
END IF
*
99999 FORMAT (' MB04DL EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB04DL = ',I2)
99997 FORMAT (' The balanced matrix A is ')
99996 FORMAT (/' The balanced matrix B is ')
99995 FORMAT (20(1X,G12.4))
99994 FORMAT (/' ILO = ',I4)
99993 FORMAT (/' IHI = ',I4)
99991 FORMAT (/' The permutations and left scaling factors are ')
99990 FORMAT (/' The permutations and right scaling factors are ')
99989 FORMAT (/' The initial 1-norms of the (sub)matrices are ')
99988 FORMAT (/' The final 1-norms of the (sub)matrices are ')
99987 FORMAT (/' The threshold value finally used is ')
99986 FORMAT (/' IWARN on exit from MB04DL = ',I2)
99985 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
MB04DL EXAMPLE PROGRAM DATA
4 B -3
1 0 -1e-12 0
0 -2 0 0
-1 -1 -1 0
-1 -1 0 2
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
Program Results
MB04DL EXAMPLE PROGRAM RESULTS
The balanced matrix A is
2.000 -1.000 0.000 -1.000
0.000 1.000 -0.1000E-11 0.000
0.000 -1.000 -1.000 -1.000
0.000 0.000 0.000 -2.000
The balanced matrix B is
1.000 0.000 0.000 0.000
0.000 1.000 0.000 0.000
0.000 0.000 1.000 0.000
0.000 0.000 0.000 1.000
ILO = 2
IHI = 3
The permutations and left scaling factors are
2.000 1.000 1.000 2.000
The permutations and right scaling factors are
2.000 1.000 1.000 2.000
The initial 1-norms of the (sub)matrices are
2.000 1.000
The final 1-norms of the (sub)matrices are
2.000 1.000
The threshold value finally used is
0.2500E-12
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04DP.html 0000664 0000000 0000000 00000043136 14560147231 0020347 0 ustar 00root root 0000000 0000000
Purpose
To balance the 2*N-by-2*N skew-Hamiltonian/Hamiltonian pencil
aS - bH, with
( A D ) ( C V )
S = ( ) and H = ( ), A, C N-by-N, (1)
( E A' ) ( W -C' )
where D and E are skew-symmetric, and V and W are symmetric
matrices. This involves, first, permuting aS - bH by a symplectic
equivalence transformation to isolate eigenvalues in the first
1:ILO-1 elements on the diagonal of A and C; and second, applying
a diagonal equivalence transformation to make the pairs of rows
and columns ILO:N and N+ILO:2*N as close in 1-norm as possible.
Both steps are optional. Balancing may reduce the 1-norms of the
matrices S and H.
Specification
SUBROUTINE MB04DP( JOB, N, THRESH, A, LDA, DE, LDDE, C, LDC, VW,
$ LDVW, ILO, LSCALE, RSCALE, DWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER JOB
INTEGER ILO, INFO, IWARN, LDA, LDC, LDDE, LDVW, N
DOUBLE PRECISION THRESH
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), C(LDC,*), DE(LDDE,*), DWORK(*),
$ LSCALE(*), RSCALE(*), VW(LDVW,*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the operations to be performed on S and H:
= 'N': none: simply set ILO = 1, LSCALE(I) = 1.0 and
RSCALE(I) = 1.0 for i = 1,...,N.
= 'P': permute only;
= 'S': scale only;
= 'B': both permute and scale.
Input/Output Parameters
N (input) INTEGER
The order of matrices A, D, E, C, V, and W. N >= 0.
THRESH (input) DOUBLE PRECISION
If JOB = 'S' or JOB = 'B', and THRESH >= 0, threshold
value for magnitude of the elements to be considered in
the scaling process: elements with magnitude less than or
equal to THRESH*MXNORM are ignored for scaling, where
MXNORM is the maximum of the 1-norms of the original
submatrices S(s,s) and H(s,s), with s = [ILO:N,N+ILO:2*N].
If THRESH < 0, the subroutine finds the scaling factors
for which some conditions, detailed below, are fulfilled.
A sequence of increasing strictly positive threshold
values is used.
If THRESH = -1, the condition is that
max( norm(H(s,s),1)/norm(S(s,s),1),
norm(S(s,s),1)/norm(H(s,s),1) ) (1)
has the smallest value, for the threshold values used,
where S(s,s) and H(s,s) are the scaled submatrices.
If THRESH = -2, the norm ratio reduction (1) is tried, but
the subroutine may return IWARN = 1 and reset the scaling
factors to 1, if this seems suitable. See the description
of the argument IWARN and FURTHER COMMENTS.
If THRESH = -3, the condition is that
norm(H(s,s),1)*norm(S(s,s),1) (2)
has the smallest value for the scaled submatrices.
If THRESH = -4, the norm reduction in (2) is tried, but
the subroutine may return IWARN = 1 and reset the scaling
factors to 1, as for THRESH = -2 above.
If THRESH = -VALUE, with VALUE >= 10, the condition
numbers of the left and right scaling transformations will
be bounded by VALUE, i.e., the ratios between the largest
and smallest entries in [LSCALE(ILO:N); RSCALE(ILO:N)]
will be at most VALUE. VALUE should be a power of 10.
If JOB = 'N' or JOB = 'P', the value of THRESH is
irrelevant.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A.
On exit, the leading N-by-N part of this array contains
the matrix A of the balanced skew-Hamiltonian matrix S.
In particular, the strictly lower triangular part of the
first ILO-1 columns of A is zero.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
DE (input/output) DOUBLE PRECISION array, dimension
(LDDE, N+1)
On entry, the leading N-by-N strictly lower triangular
part of this array must contain the strictly lower
triangular part of the skew-symmetric matrix E, and the
N-by-N strictly upper triangular part of the submatrix
in the columns 2 to N+1 of this array must contain the
strictly upper triangular part of the skew-symmetric
matrix D.
The entries on the diagonal and the first superdiagonal of
this array need not be set, but are assumed to be zero.
On exit, the leading N-by-N strictly lower triangular
part of this array contains the strictly lower triangular
part of the balanced matrix E, and the N-by-N strictly
upper triangular part of the submatrix in the columns 2 to
N+1 of this array contains the strictly upper triangular
part of the balanced matrix D. In particular, the strictly
lower triangular part of the first ILO-1 columns of DE is
zero.
LDDE INTEGER
The leading dimension of the array DE. LDDE >= MAX(1, N).
C (input/output) DOUBLE PRECISION array, dimension (LDC, N)
On entry, the leading N-by-N part of this array must
contain the matrix C.
On exit, the leading N-by-N part of this array contains
the matrix C of the balanced Hamiltonian matrix H.
In particular, the strictly lower triangular part of the
first ILO-1 columns of C is zero.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1, N).
VW (input/output) DOUBLE PRECISION array, dimension
(LDVW, N+1)
On entry, the leading N-by-N lower triangular part of
this array must contain the lower triangular part of the
symmetric matrix W, and the N-by-N upper triangular
part of the submatrix in the columns 2 to N+1 of this
array must contain the upper triangular part of the
symmetric matrix V.
On exit, the leading N-by-N lower triangular part of this
array contains the lower triangular part of the balanced
matrix W, and the N-by-N upper triangular part of the
submatrix in the columns 2 to N+1 of this array contains
the upper triangular part of the balanced matrix V. In
particular, the lower triangular part of the first ILO-1
columns of VW is zero.
LDVW INTEGER
The leading dimension of the array VW. LDVW >= MAX(1, N).
ILO (output) INTEGER
ILO-1 is the number of deflated eigenvalues in the
balanced skew-Hamiltonian/Hamiltonian matrix pencil.
ILO is set to 1 if JOB = 'N' or JOB = 'S'.
LSCALE (output) DOUBLE PRECISION array, dimension (N)
Details of the permutations of S and H and scaling applied
to A, D, C, and V from the left. For j = 1,...,ILO-1 let
P(j) = LSCALE(j). If P(j) <= N, then rows and columns P(j)
and P(j)+N are interchanged with rows and columns j and
j+N, respectively. If P(j) > N, then row and column P(j)-N
are interchanged with row and column j+N by a generalized
symplectic permutation. For j = ILO,...,N the j-th element
of LSCALE contains the factor of the scaling applied to
row j of the matrices A, D, C, and V.
RSCALE (output) DOUBLE PRECISION array, dimension (N)
Details of the permutations of S and H and scaling applied
to A, E, C, and W from the right. For j = 1,...,ILO-1 let
P(j) = RSCALE(j). If P(j) <= N, then rows and columns P(j)
and P(j)+N are interchanged with rows and columns j and
j+N, respectively. If P(j) > N, then row and column P(j)-N
are interchanged with row and column j+N by a generalized
symplectic permutation. For j = ILO,...,N the j-th element
of RSCALE contains the factor of the scaling applied to
column j of the matrices A, E, C, and W.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK) where
LDWORK = 0, if JOB = 'N' or JOB = 'P', or N = 0;
LDWORK = 6*N, if (JOB = 'S' or JOB = 'B') and THRESH >= 0;
LDWORK = 8*N, if (JOB = 'S' or JOB = 'B') and THRESH < 0.
On exit, if JOB = 'S' or JOB = 'B', DWORK(1) and DWORK(2)
contain the initial 1-norms of S(s,s) and H(s,s), and
DWORK(3) and DWORK(4) contain their final 1-norms,
respectively. Moreover, DWORK(5) contains the THRESH value
used (irrelevant if IWARN = 1 or ILO = N).
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: scaling has been requested, for THRESH = -2 or
THRESH = -4, but it most probably would not improve
the accuracy of the computed solution for a related
eigenproblem (since maximum norm increased
significantly compared to the original pencil
matrices and (very) high and/or small scaling
factors occurred). The returned scaling factors have
been reset to 1, but information about permutations,
if requested, has been preserved.
Error Indicator
INFO INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Balancing consists of applying a (symplectic) equivalence
transformation to isolate eigenvalues and/or to make the 1-norms
of each pair of rows and columns indexed by s of S and H nearly
equal. If THRESH < 0, a search is performed to find those scaling
factors giving the smallest norm ratio or product defined above
(see the description of the parameter THRESH).
Assuming JOB = 'S', let Dl and Dr be diagonal matrices containing
the vectors LSCALE and RSCALE, respectively. The returned matrices
are obtained using the equivalence transformation
( Dl 0 ) ( A D ) ( Dr 0 ) ( Dl 0 ) ( C V ) ( Dr 0 )
( ) ( ) ( ), ( ) ( ) ( ).
( 0 Dr ) ( E A' ) ( 0 Dl ) ( 0 Dr ) ( W -C' ) ( 0 Dl )
For THRESH = 0, the routine returns essentially the same results
as the LAPACK subroutine DGGBAL [1]. Setting THRESH < 0, usually
gives better results than DGGBAL for badly scaled matrix pencils.
References
[1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
[2] Benner, P.
Symplectic balancing of Hamiltonian matrices.
SIAM J. Sci. Comput., 22 (5), pp. 1885-1904, 2001.
Numerical Aspects
The transformations used preserve the skew-Hamiltonian/Hamiltonian structure and do not introduce significant rounding errors. No rounding errors appear if JOB = 'P'. If T is the global transformation matrix applied to the right, then J'*T*J is the global transformation matrix applied to the left, where J = [ 0 I; -I 0 ], with blocks of order N.Further Comments
If THRESH = -2, the increase of the maximum norm of the scaled submatrices, compared to the maximum norm of the initial submatrices, is bounded by MXGAIN = 100. If THRESH = -2, or THRESH = -4, the maximum condition number of the scaling transformations is bounded by MXCOND = 1/SQRT(EPS), where EPS is the machine precision (see LAPACK Library routine DLAMCH).Example
Program Text
* MB04DP EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 10 )
INTEGER LDA, LDC, LDDE, LDVW
PARAMETER ( LDA = NMAX, LDC = NMAX, LDDE = NMAX,
$ LDVW = NMAX )
* .. Local Scalars ..
CHARACTER*1 JOB
INTEGER I, ILO, INFO, IWARN, J, N
DOUBLE PRECISION THRESH
* .. Local Arrays ..
DOUBLE PRECISION A(LDA, NMAX), DWORK(8*NMAX), C(LDC, NMAX),
$ DE(LDDE, NMAX+1), LSCALE(NMAX), RSCALE(NMAX),
$ VW(LDVW, NMAX+1)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL MB04DP
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, JOB, THRESH
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99985 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( DE(I,J), J = 1,N+1 ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( VW(I,J), J = 1,N+1 ), I = 1,N )
CALL MB04DP( JOB, N, THRESH, A, LDA, DE, LDDE, C, LDC, VW,
$ LDVW, ILO, LSCALE, RSCALE, DWORK, IWARN, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99993 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99993 ) ( DE(I,J), J = 1,N+1 )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 30 I = 1, N
WRITE ( NOUT, FMT = 99993 ) ( C(I,J), J = 1,N )
30 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99993 ) ( VW(I,J), J = 1,N+1 )
40 CONTINUE
WRITE ( NOUT, FMT = 99992 ) ILO
WRITE ( NOUT, FMT = 99991 )
WRITE ( NOUT, FMT = 99993 ) ( LSCALE(I), I = 1,N )
WRITE ( NOUT, FMT = 99990 )
WRITE ( NOUT, FMT = 99993 ) ( RSCALE(I), I = 1,N )
IF ( LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'B' ) ) THEN
IF ( .NOT.( THRESH.EQ.-2 .OR. THRESH.EQ.-4 ) ) THEN
WRITE ( NOUT, FMT = 99989 )
WRITE ( NOUT, FMT = 99993 ) ( DWORK(I), I = 1,2 )
WRITE ( NOUT, FMT = 99988 )
WRITE ( NOUT, FMT = 99993 ) ( DWORK(I), I = 3,4 )
WRITE ( NOUT, FMT = 99987 )
WRITE ( NOUT, FMT = 99993 ) ( DWORK(5) )
ELSE
WRITE ( NOUT, FMT = 99986 ) IWARN
END IF
END IF
END IF
END IF
*
99999 FORMAT (' MB04DP EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB04DP = ',I2)
99997 FORMAT (' The balanced matrix A is ')
99996 FORMAT (/' The balanced matrix DE is ')
99995 FORMAT (' The balanced matrix C is ')
99994 FORMAT (/' The balanced matrix VW is ')
99993 FORMAT (20(1X,G12.4))
99992 FORMAT (/' ILO = ',I4)
99991 FORMAT (/' The permutations and left scaling factors are ')
99990 FORMAT (/' The permutations and right scaling factors are ')
99989 FORMAT (/' The initial 1-norms of the (sub)matrices are ')
99988 FORMAT (/' The final 1-norms of the (sub)matrices are ')
99987 FORMAT (/' The threshold value finally used is ')
99986 FORMAT (/' IWARN on exit from MB04DP = ',I2)
99985 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
MB04DP EXAMPLE PROGRAM DATA
2 B -3
1 0
0 1
0 0 0
0 0 0
1 0
0 -2
-1 -1.0e-12 0
-1 -1 0
Program Results
MB04DP EXAMPLE PROGRAM RESULTS
The balanced matrix A is
1.000 0.000
0.000 1.000
The balanced matrix DE is
0.000 0.000 0.000
0.000 0.000 0.000
The balanced matrix C is
2.000 1.000
0.000 1.000
The balanced matrix VW is
0.000 1.000 0.000
0.000 -1.000 -0.1000E-11
ILO = 2
The permutations and left scaling factors are
4.000 1.000
The permutations and right scaling factors are
4.000 1.000
The initial 1-norms of the (sub)matrices are
1.000 2.000
The final 1-norms of the (sub)matrices are
1.000 2.000
The threshold value finally used is
-3.000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04DS.html 0000664 0000000 0000000 00000021107 14560147231 0020344 0 ustar 00root root 0000000 0000000
Purpose
To balance a real skew-Hamiltonian matrix
[ A G ]
S = [ T ] ,
[ Q A ]
where A is an N-by-N matrix and G, Q are N-by-N skew-symmetric
matrices. This involves, first, permuting S by a symplectic
similarity transformation to isolate eigenvalues in the first
1:ILO-1 elements on the diagonal of A; and second, applying a
diagonal similarity transformation to rows and columns
ILO:N, N+ILO:2*N to make the rows and columns as close in 1-norm
as possible. Both steps are optional.
Specification
SUBROUTINE MB04DS( JOB, N, A, LDA, QG, LDQG, ILO, SCALE, INFO )
C .. Scalar Arguments ..
CHARACTER JOB
INTEGER ILO, INFO, LDA, LDQG, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), QG(LDQG,*), SCALE(*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the operations to be performed on S:
= 'N': none, set ILO = 1, SCALE(I) = 1.0, I = 1 .. N;
= 'P': permute only;
= 'S': scale only;
= 'B': both permute and scale.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A.
On exit, the leading N-by-N part of this array contains
the matrix A of the balanced skew-Hamiltonian. In
particular, the strictly lower triangular part of the
first ILO-1 columns of A is zero.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
QG (input/output) DOUBLE PRECISION array, dimension
(LDQG,N+1)
On entry, the leading N-by-N+1 part of this array must
contain in columns 1:N the strictly lower triangular part
of the matrix Q and in columns 2:N+1 the strictly upper
triangular part of the matrix G. The parts containing the
diagonal and the first supdiagonal of this array are not
referenced.
On exit, the leading N-by-N+1 part of this array contains
the strictly lower and strictly upper triangular parts of
the matrices Q and G, respectively, of the balanced
skew-Hamiltonian. In particular, the strictly lower
triangular part of the first ILO-1 columns of QG is zero.
LDQG INTEGER
The leading dimension of the array QG. LDQG >= MAX(1,N).
ILO (output) INTEGER
ILO-1 is the number of deflated eigenvalues in the
balanced skew-Hamiltonian matrix.
SCALE (output) DOUBLE PRECISION array of dimension (N)
Details of the permutations and scaling factors applied to
S. For j = 1,...,ILO-1 let P(j) = SCALE(j). If P(j) <= N,
then rows and columns P(j) and P(j)+N are interchanged
with rows and columns j and j+N, respectively. If
P(j) > N, then row and column P(j)-N are interchanged with
row and column j+N by a generalized symplectic
permutation. For j = ILO,...,N the j-th element of SCALE
contains the factor of the scaling applied to row and
column j.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
References
[1] Benner, P.
Symplectic balancing of Hamiltonian matrices.
SIAM J. Sci. Comput., 22 (5), pp. 1885-1904, 2001.
Further Comments
NoneExample
Program Text
* MB04DS EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 100 )
INTEGER LDA, LDQG
PARAMETER ( LDA = NMAX, LDQG = NMAX )
* .. Local Scalars ..
CHARACTER*1 JOB
INTEGER I, ILO, INFO, J, N
* .. Local Arrays ..
DOUBLE PRECISION A(LDA, NMAX), DUMMY(1), QG(LDQG, NMAX+1),
$ SCALE(NMAX)
* .. External Functions ..
DOUBLE PRECISION DLANTR, DLAPY2
EXTERNAL DLANTR, DLAPY2
* .. External Subroutines ..
EXTERNAL MB04DS
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, JOB
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( QG(I,J), J = 1,N+1 ), I = 1,N )
CALL MB04DS( JOB, N, A, LDA, QG, LDQG, ILO, SCALE, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 30 I = 1, N
WRITE (NOUT, FMT = 99995) ( A(I,J), J = 1,N )
30 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 40 I = 1, N
WRITE (NOUT, FMT = 99995) ( QG(I,J), J = 1,N+1 )
40 CONTINUE
WRITE (NOUT, FMT = 99993) ILO
IF ( ILO.GT.1 ) THEN
WRITE (NOUT, FMT = 99992) DLAPY2( DLANTR( 'Frobenius',
$ 'Lower', 'No Unit', N-1, ILO-1, A(2,1), LDA,
$ DUMMY ), DLANTR( 'Frobenius', 'Lower', 'No Unit',
$ N-1, ILO-1, QG(2,1), LDQG, DUMMY ) )
END IF
END IF
END IF
*
99999 FORMAT (' MB04DS EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB04DS = ',I2)
99997 FORMAT (' The balanced matrix A is ')
99996 FORMAT (/' The balanced matrix QG is ')
99995 FORMAT (20(1X,F9.4))
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' ILO = ',I4)
99992 FORMAT (/' Norm of subdiagonal blocks: ',G7.2)
END
Program Data
MB04DS EXAMPLE PROGRAM DATA
6 B
0.0576 0 0.5208 0 0.7275 -0.7839
0.1901 0.0439 0.1663 0.0928 0.6756 -0.5030
0.5962 0 0.4418 0 -0.5955 0.7176
0.5869 0 0.3939 0.0353 0.6992 -0.0147
0.2222 0 -0.3663 0 0.5548 -0.4608
0 0 0 0 0 0.1338
0 0 -0.9862 -0.4544 -0.4733 0.4435 0
0 0 0 -0.6927 0.6641 0.4453 0
-0.3676 0 0 0 0.0841 0.3533 0
0 0 0 0 0 0.0877 0
0.9561 0 0.4784 0 0 0 0
-0.0164 -0.4514 -0.8289 -0.6831 -0.1536 0 0
Program Results
MB04DS EXAMPLE PROGRAM RESULTS
The balanced matrix A is
0.1338 0.4514 0.6831 0.8289 0.1536 0.0164
0.0000 0.0439 0.0928 0.1663 0.6756 0.1901
0.0000 0.0000 0.0353 0.3939 0.6992 0.5869
0.0000 0.0000 0.0000 0.4418 -0.5955 0.5962
0.0000 0.0000 0.0000 -0.3663 0.5548 0.2222
0.0000 0.0000 0.0000 0.5208 0.7275 0.0576
The balanced matrix QG is
0.0000 0.0000 0.5030 0.0147 -0.7176 0.4608 0.7839
0.0000 0.0000 0.0000 0.6641 -0.6927 0.4453 0.9862
0.0000 0.0000 0.0000 0.0000 -0.0841 0.0877 0.4733
0.0000 0.0000 0.0000 0.0000 0.0000 0.3533 0.4544
0.0000 0.0000 0.0000 0.4784 0.0000 0.0000 -0.4435
0.0000 0.0000 0.0000 0.3676 -0.9561 0.0000 0.0000
ILO = 4
Norm of subdiagonal blocks: 0.0
Purpose
To perform a symplectic scaling on the Hamiltonian matrix
( A G )
H = ( T ), (1)
( Q -A )
i.e., perform either the symplectic scaling transformation
-1
( A' G' ) ( D 0 ) ( A G ) ( D 0 )
H' <-- ( T ) = ( ) ( T ) ( -1 ), (2)
( Q' -A' ) ( 0 D ) ( Q -A ) ( 0 D )
where D is a diagonal scaling matrix, or the symplectic norm
scaling transformation
( A'' G'' ) 1 ( A G/tau )
H'' <-- ( T ) = --- ( T ), (3)
( Q'' -A'' ) tau ( tau Q -A )
where tau is a real scalar. Note that if tau is not equal to 1,
then (3) is NOT a similarity transformation. The eigenvalues
of H are then tau times the eigenvalues of H''.
For symplectic scaling (2), D is chosen to give the rows and
columns of A' approximately equal 1-norms and to give Q' and G'
approximately equal norms. (See METHOD below for details.) For
norm scaling, tau = MAX(1, ||A||, ||G||, ||Q||) where ||.||
denotes the 1-norm (column sum norm).
Specification
SUBROUTINE MB04DY( JOBSCL, N, A, LDA, QG, LDQG, D, DWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDQG, N
CHARACTER JOBSCL
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), D(*), DWORK(*), QG(LDQG,*)
Arguments
Mode Parameters
JOBSCL CHARACTER*1
Indicates which scaling strategy is used, as follows:
= 'S' : do the symplectic scaling (2);
= '1' or 'O': do the 1-norm scaling (3);
= 'N' : do nothing; set INFO and return.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, G, and Q. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On input, if JOBSCL <> 'N', the leading N-by-N part of
this array must contain the upper left block A of the
Hamiltonian matrix H in (1).
On output, if JOBSCL <> 'N', the leading N-by-N part of
this array contains the leading N-by-N part of the scaled
Hamiltonian matrix H' in (2) or H'' in (3), depending on
the setting of JOBSCL.
If JOBSCL = 'N', this array is not referenced.
LDA INTEGER
The leading dimension of the array A.
LDA >= MAX(1,N), if JOBSCL <> 'N';
LDA >= 1, if JOBSCL = 'N'.
QG (input/output) DOUBLE PRECISION array, dimension
(LDQG,N+1)
On input, if JOBSCL <> 'N', the leading N-by-N lower
triangular part of this array must contain the lower
triangle of the lower left symmetric block Q of the
Hamiltonian matrix H in (1), and the N-by-N upper
triangular part of the submatrix in the columns 2 to N+1
of this array must contain the upper triangle of the upper
right symmetric block G of H in (1).
So, if i >= j, then Q(i,j) = Q(j,i) is stored in QG(i,j)
and G(i,j) = G(j,i) is stored in QG(j,i+1).
On output, if JOBSCL <> 'N', the leading N-by-N lower
triangular part of this array contains the lower triangle
of the lower left symmetric block Q' or Q'', and the
N-by-N upper triangular part of the submatrix in the
columns 2 to N+1 of this array contains the upper triangle
of the upper right symmetric block G' or G'' of the scaled
Hamiltonian matrix H' in (2) or H'' in (3), depending on
the setting of JOBSCL.
If JOBSCL = 'N', this array is not referenced.
LDQG INTEGER
The leading dimension of the array QG.
LDQG >= MAX(1,N), if JOBSCL <> 'N';
LDQG >= 1, if JOBSCL = 'N'.
D (output) DOUBLE PRECISION array, dimension (nd)
If JOBSCL = 'S', then nd = N and D contains the diagonal
elements of the diagonal scaling matrix in (2).
If JOBSCL = '1' or 'O', then nd = 1 and D(1) is set to tau
from (3). In this case, no other elements of D are
referenced.
If JOBSCL = 'N', this array is not referenced.
Workspace
DWORK DOUBLE PRECISION array, dimension (N)
If JOBSCL = 'N', this array is not referenced.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, then the i-th argument had an illegal
value.
Method
1. Symplectic scaling (JOBSCL = 'S'):
First, LAPACK subroutine DGEBAL is used to equilibrate the 1-norms
of the rows and columns of A using a diagonal scaling matrix D_A.
Then, H is similarily transformed by the symplectic diagonal
matrix D1 = diag(D_A,D_A**(-1)). Next, the off-diagonal blocks of
the resulting Hamiltonian matrix are equilibrated in the 1-norm
using the symplectic diagonal matrix D2 of the form
( I/rho 0 )
D2 = ( )
( 0 rho*I )
where rho is a real scalar. Thus, in (2), D = D1*D2.
2. Norm scaling (JOBSCL = '1' or 'O'):
The norm of the matrices A and G of (1) is reduced by setting
A := A/tau and G := G/(tau**2) where tau is the power of the
base of the arithmetic closest to MAX(1, ||A||, ||G||, ||Q||) and
||.|| denotes the 1-norm.
References
[1] Benner, P., Byers, R., and Barth, E.
Fortran 77 Subroutines for Computing the Eigenvalues of
Hamiltonian Matrices. I: The Square-Reduced Method.
ACM Trans. Math. Software, 26, 1, pp. 49-77, 2000.
Numerical Aspects
For symplectic scaling, the complexity of the used algorithms is hard to estimate and depends upon how well the rows and columns of A in (1) are equilibrated. In one sweep, each row/column of A is scaled once, i.e., the cost of one sweep is N**2 multiplications. Usually, 3-6 sweeps are enough to equilibrate the norms of the rows and columns of a matrix. Roundoff errors are possible as LAPACK routine DGEBAL does NOT use powers of the machine base for scaling. The second stage (equilibrating ||G|| and ||Q||) requires N**2 multiplications. For norm scaling, 3*N**2 + O(N) multiplications are required and NO rounding errors occur as all multiplications are performed with powers of the machine base.Further Comments
NoneExample
Program Text
* MB04DY EXAMPLE PROGRAM TEXT.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA, LDQG
PARAMETER ( LDA = NMAX, LDQG = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, N
CHARACTER*1 JOBSCL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), D(NMAX), DWORK(LDWORK),
$ QG(LDQG,NMAX+1)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL MB04DY
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, JOBSCL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99998 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( QG(J,I+1), I = J,N ), J = 1,N )
READ ( NIN, FMT = * ) ( ( QG(I,J), I = J,N ), J = 1,N )
* Scale the Hamiltonian matrix.
CALL MB04DY( JOBSCL, N, A, LDA, QG, LDQG, D, DWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
* Show the scaled Hamiltonian matrix.
WRITE ( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99993 ) ( A(I,J), J = 1,N ),
$ ( QG(J,I+1), J = 1,I-1 ), ( QG(I,J+1), J = I,N )
10 CONTINUE
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99993 ) ( QG(I,J), J = 1,I-1 ),
$ ( QG(J,I), J = I,N ), ( -A(J,I), J = 1,N )
20 CONTINUE
* Show the scaling factors.
IF ( LSAME( JOBSCL, 'S' ) ) THEN
WRITE ( NOUT, FMT = 99995 )
WRITE ( NOUT, FMT = 99993 ) ( D(I), I = 1,N )
ELSE IF ( LSAME( JOBSCL, '1' ) .OR. LSAME( JOBSCL, 'O' ) )
$ THEN
WRITE ( NOUT, FMT = 99994 )
WRITE ( NOUT, FMT = 99993 ) D(1)
END IF
ENDIF
END IF
STOP
*
99999 FORMAT (' MB04DY EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (/' N is out of range.',/' N = ',I5)
99997 FORMAT (' INFO on exit from MB04DY = ',I2)
99996 FORMAT (/' The scaled Hamiltonian is ')
99995 format (/' The scaling factors are ')
99994 format (/' The scaling factor tau is ')
99993 FORMAT (1X,8(F10.4))
END
Program Data
MB04DY EXAMPLE PROGRAM DATA 3 S -0.4 0.05 0.0007 -4.7 0.8 0.025 81.0 29.0 -0.9 0.0034 0.0014 0.00077 -0.005 0.0004 0.003 -18.0 -12.0 43.0 99.0 420.0 -200.0Program Results
MB04DY EXAMPLE PROGRAM RESULTS
The scaled Hamiltonian is
-0.4000 0.4000 0.3584 418.4403 21.5374 0.1851
-0.5875 0.8000 1.6000 21.5374 -9.6149 0.0120
0.1582 0.4531 -0.9000 0.1851 0.0120 0.0014
-0.0001 -0.0008 0.1789 0.4000 0.5875 -0.1582
-0.0008 0.0515 13.9783 -0.4000 -0.8000 -0.4531
0.1789 13.9783 -426.0056 -0.3584 -1.6000 0.9000
The scaling factors are
0.0029 0.0228 1.4595
Purpose
To balance a complex Hamiltonian matrix,
[ A G ]
H = [ H ] ,
[ Q -A ]
where A is an N-by-N matrix and G, Q are N-by-N Hermitian
matrices. This involves, first, permuting H by a symplectic
similarity transformation to isolate eigenvalues in the first
1:ILO-1 elements on the diagonal of A; and second, applying a
diagonal similarity transformation to rows and columns
ILO:N, N+ILO:2*N to make the rows and columns as close in 1-norm
as possible. Both steps are optional. Assuming ILO = 1, let D be a
diagonal matrix of order N with the scaling factors on the
diagonal. The scaled Hamiltonian is defined by
[ D**-1*A*D D**-1*G*D**-1 ]
Hs = [ H ] .
[ D*Q*D -D*A *D**-1 ]
Specification
SUBROUTINE MB04DZ( JOB, N, A, LDA, QG, LDQG, ILO, SCALE, INFO )
C .. Scalar Arguments ..
CHARACTER JOB
INTEGER ILO, INFO, LDA, LDQG, N
C .. Array Arguments ..
DOUBLE PRECISION SCALE(*)
COMPLEX*16 A(LDA,*), QG(LDQG,*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the operations to be performed on H:
= 'N': none, set ILO = 1, SCALE(I) = 1.0, I = 1 .. N;
= 'P': permute only;
= 'S': scale only;
= 'B': both permute and scale.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A.
On exit, the leading N-by-N part of this array contains
the matrix A of the balanced Hamiltonian. In particular,
the strictly lower triangular part of the first ILO-1
columns of A is zero.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
QG (input/output) COMPLEX*16 array, dimension
(LDQG,N+1)
On entry, the leading N-by-N+1 part of this array must
contain the lower triangular part of the matrix Q and
the upper triangular part of the matrix G.
On exit, the leading N-by-N+1 part of this array contains
the lower and upper triangular parts of the matrices Q and
G, respectively, of the balanced Hamiltonian. In
particular, the lower triangular part of the first ILO-1
columns of QG is zero.
LDQG INTEGER
The leading dimension of the array QG. LDQG >= MAX(1,N).
ILO (output) INTEGER
ILO-1 is the number of deflated eigenvalues in the
balanced Hamiltonian matrix.
SCALE (output) DOUBLE PRECISION array of dimension (N)
Details of the permutations and scaling factors applied to
H. For j = 1,...,ILO-1 let P(j) = SCALE(j). If P(j) <= N,
then rows and columns P(j) and P(j)+N are interchanged
with rows and columns j and j+N, respectively. If
P(j) > N, then row and column P(j)-N are interchanged with
row and column j+N by a generalized symplectic
permutation. For j = ILO,...,N the j-th element of SCALE
contains the factor of the scaling applied to row and
column j.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
References
[1] Benner, P.
Symplectic balancing of Hamiltonian matrices.
SIAM J. Sci. Comput., 22 (5), pp. 1885-1904, 2001.
Further Comments
NoneExample
Program Text
* MB04DZ EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 100 )
INTEGER LDA, LDQG
PARAMETER ( LDA = NMAX, LDQG = NMAX )
* .. Local Scalars ..
CHARACTER*1 JOB
INTEGER I, ILO, INFO, J, N
* .. Local Arrays ..
COMPLEX*16 A(LDA, NMAX), QG(LDQG, NMAX+1)
DOUBLE PRECISION DUMMY(1), SCALE(NMAX)
* .. External Functions ..
DOUBLE PRECISION DLAPY2, ZLANTR
EXTERNAL DLAPY2, ZLANTR
* .. External Subroutines ..
EXTERNAL MB04DZ
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, JOB
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( QG(I,J), J = 1,N+1 ), I = 1,N )
CALL MB04DZ( JOB, N, A, LDA, QG, LDQG, ILO, SCALE, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE (NOUT, FMT = 99995) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE (NOUT, FMT = 99995) ( QG(I,J), J = 1,N+1 )
20 CONTINUE
WRITE (NOUT, FMT = 99993) ILO
IF ( ILO.GT.1 ) THEN
WRITE (NOUT, FMT = 99992) DLAPY2( ZLANTR( 'Frobenius',
$ 'Lower', 'No Unit', N-1, ILO-1, A(2,1), LDA,
$ DUMMY ), ZLANTR( 'Frobenius', 'Lower', 'No Unit',
$ N, ILO-1, QG(1,1), LDQG, DUMMY ) )
END IF
END IF
END IF
*
99999 FORMAT (' MB04DZ EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB04DZ = ',I2)
99997 FORMAT (' The balanced matrix A is ')
99996 FORMAT (/' The balanced matrix QG is ')
99995 FORMAT (20(1X,F9.4,SP,F9.4,S,'i '))
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' ILO = ',I4)
99992 FORMAT (/' Norm of subdiagonal blocks: ',G7.2)
END
Program Data
MB04DZ EXAMPLE PROGRAM DATA
6 B
(0,0) (0,0) (0,0) (0,0) (0,0) (0,0)
(.0994,0) (0,0) (0,0) (0,0) (0,0) (.9696,0)
(.3248,0) (0,0) (0,0) (0,0) (.4372,0) (.8308,0)
(0,0) (0,0) (0,0) (.0717,0) (0,0) (0,0)
(0,0) (0,0) (0,0) (0,0) (0,0) (.1976,0)
(0,0) (0,0) (0,0) (0,0) (0,0) (0,0)
(0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0)
(0,0) (0,0) (0,0) (0,0) (.0651,0) (0,0) (0,0)
(0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0)
(0,0) (0,0) (.0444,0) (0,0) (0,0) (.1957,0) (0,0)
(.8144,0) (0,0) (0,0) (0,0) (.3652,0) (0,0) (.9121,0)
(.9023,0) (0,0) (0,0) (0,0) (0,0) (0,0) 1.0945,0)
Program Results
MB04DZ EXAMPLE PROGRAM RESULTS
The balanced matrix A is
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.9696 +0.0000i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i -0.8144 +0.0000i -0.9023 +0.0000i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.1093 +0.0000i 0.2077 +0.0000i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0717 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.1976 +0.0000i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i
The balanced matrix QG is
0.0000 +0.0000i 0.0000 +0.0000i 0.0994 +0.0000i 0.0000 +0.0000i 0.0651 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0812 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i
0.0000 +0.0000i 0.0000 +0.0000i 0.1776 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.1957 +0.0000i 0.0000 +0.0000i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.3652 +0.0000i 0.0000 +0.0000i 0.9121 +0.0000i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 1.0945 +0.0000i
ILO = 3
Norm of subdiagonal blocks: 0.0
Purpose
To compute the eigenvalues of a real N-by-N skew-Hamiltonian/
skew-Hamiltonian pencil aS - bT with
( B F ) ( 0 I )
S = J Z' J' Z and T = ( ), where J = ( ). (1)
( G B' ) ( -I 0 )
Optionally, if JOB = 'T', the pencil aS - bT will be transformed
to the structured Schur form: an orthogonal transformation matrix
Q and an orthogonal symplectic transformation matrix U are
computed, such that
( Z11 Z12 )
U' Z Q = ( ) = Zout, and
( 0 Z22 )
(2)
( Bout Fout )
J Q' J' T Q = ( ),
( 0 Bout' )
where Z11 and Z22' are upper triangular and Bout is upper quasi-
triangular. The notation M' denotes the transpose of the matrix M.
Optionally, if COMPQ = 'I', the orthogonal transformation matrix Q
will be computed.
Optionally, if COMPU = 'I' or COMPU = 'U', the orthogonal
symplectic transformation matrix
( U1 U2 )
U = ( )
( -U2 U1 )
will be computed.
Specification
SUBROUTINE MB04ED( JOB, COMPQ, COMPU, N, Z, LDZ, B, LDB, FG, LDFG,
$ Q, LDQ, U1, LDU1, U2, LDU2, ALPHAR, ALPHAI,
$ BETA, IWORK, LIWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, COMPU, JOB
INTEGER INFO, LDB, LDFG, LDQ, LDU1, LDU2, LDWORK, LDZ,
$ LIWORK, N
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION ALPHAI( * ), ALPHAR( * ), B( LDB, * ),
$ BETA( * ), DWORK( * ), FG( LDFG, * ),
$ Q( LDQ, * ), U1( LDU1, * ), U2( LDU2, * ),
$ Z( LDZ, * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'E': compute the eigenvalues only; Z and T will not
necessarily be put into the forms in (2);
= 'T': put Z and T into the forms in (2), and return the
eigenvalues in ALPHAR, ALPHAI and BETA.
COMPQ CHARACTER*1
Specifies whether to compute the orthogonal transformation
matrix Q as follows:
= 'N': Q is not computed;
= 'I': the array Q is initialized internally to the unit
matrix, and the orthogonal matrix Q is returned.
COMPU CHARACTER*1
Specifies whether to compute the orthogonal symplectic
transformation matrix U as follows:
= 'N': U is not computed;
= 'I': the array U is initialized internally to the unit
matrix, and the orthogonal matrix U is returned;
= 'U': the arrays U1 and U2 contain the corresponding
submatrices of an orthogonal symplectic matrix U0
on entry, and the updated submatrices U1 and U2
of the matrix product U0*U are returned, where U
is the product of the orthogonal symplectic
transformations that are applied to the pencil
aS - bT to reduce Z and T to the forms in (2), for
COMPU = 'I'.
Input/Output Parameters
N (input) INTEGER
The order of the pencil aS - bT. N >= 0, even.
Z (input/output) DOUBLE PRECISION array, dimension (LDZ, N)
On entry, the leading N-by-N part of this array must
contain the matrix Z.
On exit, if JOB = 'T', the leading N-by-N part of this
array contains the matrix Zout; otherwise, it contains the
matrix Z just before the application of the periodic QZ
algorithm. The entries in the rows N/2+1 to N and the
first N/2 columns are unchanged.
LDZ INTEGER
The leading dimension of the array Z. LDZ >= MAX(1, N).
B (input/output) DOUBLE PRECISION array, dimension
(LDB, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix B.
On exit, if JOB = 'T', the leading N/2-by-N/2 part of this
array contains the matrix Bout; otherwise, it contains the
matrix B just before the application of the periodic QZ
algorithm.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N/2).
FG (input/output) DOUBLE PRECISION array, dimension
(LDFG, N/2+1)
On entry, the leading N/2-by-N/2 strictly lower triangular
part of this array must contain the strictly lower
triangular part of the skew-symmetric matrix G, and the
N/2-by-N/2 strictly upper triangular part of the submatrix
in the columns 2 to N/2+1 of this array must contain the
strictly upper triangular part of the skew-symmetric
matrix F.
On exit, if JOB = 'T', the leading N/2-by-N/2 strictly
upper triangular part of the submatrix in the columns 2 to
N/2+1 of this array contains the strictly upper triangular
part of the skew-symmetric matrix Fout.
If JOB = 'E', the leading N/2-by-N/2 strictly upper
triangular part of the submatrix in the columns 2 to N/2+1
of this array contains the strictly upper triangular part
of the skew-symmetric matrix F just before the application
of the QZ algorithm.
The entries on the diagonal and the first superdiagonal of
this array are not referenced, but are assumed to be zero.
Moreover, the diagonal and the first subdiagonal of this
array on exit coincide to the corresponding diagonals of
this array on entry.
LDFG INTEGER
The leading dimension of the array FG.
LDFG >= MAX(1, N/2).
Q (output) DOUBLE PRECISION array, dimension (LDQ, N)
On exit, if COMPQ = 'I', the leading N-by-N part of this
array contains the orthogonal transformation matrix Q.
On exit, if COMPQ = 'N', the leading N-by-N part of this
array contains the orthogonal matrix Q1, such that
( Z11 Z12 )
Z*Q1 = ( ),
( 0 Z22 )
where Z11 and Z22' are upper triangular (the first step
of the algorithm).
LDQ INTEGER
The leading dimension of the array Q. LDQ >= MAX(1, N).
U1 (input/output) DOUBLE PRECISION array, dimension
(LDU1, N/2)
On entry, if COMPU = 'U', then the leading N/2-by-N/2 part
of this array must contain the upper left block of a
given matrix U0, and on exit, the leading N/2-by-N/2 part
of this array contains the updated upper left block U1 of
the product of the input matrix U0 and the transformation
matrix U used to transform the matrices Z and T.
On exit, if COMPU = 'I', then the leading N/2-by-N/2 part
of this array contains the upper left block U1 of the
orthogonal symplectic transformation matrix U.
If COMPU = 'N' this array is not referenced.
LDU1 INTEGER
The leading dimension of the array U1.
LDU1 >= 1, if COMPU = 'N';
LDU1 >= MAX(1, N/2), if COMPU = 'I' or COMPU = 'U'.
U2 (input/output) DOUBLE PRECISION array, dimension
(LDU2, N/2)
On entry, if COMPU = 'U', then the leading N/2-by-N/2 part
of this array must contain the upper right block of a
given matrix U0, and on exit, the leading N/2-by-N/2 part
of this array contains the updated upper right block U2 of
the product of the input matrix U0 and the transformation
matrix U used to transform the matrices Z and T.
On exit, if COMPU = 'I', then the leading N/2-by-N/2 part
of this array contains the upper right block U2 of the
orthogonal symplectic transformation matrix U.
If COMPU = 'N' this array is not referenced.
LDU2 INTEGER
The leading dimension of the array U2.
LDU2 >= 1, if COMPU = 'N';
LDU2 >= MAX(1, N/2), if COMPU = 'I' or COMPU = 'U'.
ALPHAR (output) DOUBLE PRECISION array, dimension (N/2)
The real parts of each scalar alpha defining an eigenvalue
of the pencil aS - bT.
ALPHAI (output) DOUBLE PRECISION array, dimension (N/2)
The imaginary parts of each scalar alpha defining an
eigenvalue of the pencil aS - bT.
If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
positive, then the j-th and (j+1)-st eigenvalues are a
complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).
BETA (output) DOUBLE PRECISION array, dimension (N/2)
The scalars beta that define the eigenvalues of the pencil
aS - bT.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent the j-th eigenvalue of the pencil
aS - bT, in the form lambda = alpha/beta. Since lambda may
overflow, the ratios should not, in general, be computed.
Due to the skew-Hamiltonian/skew-Hamiltonian structure of
the pencil, every eigenvalue occurs twice and thus it has
only to be saved once in ALPHAR, ALPHAI and BETA.
Workspace
IWORK INTEGER array, dimension (LIWORK)
On exit, if INFO = 3, IWORK(1) contains the number of
(pairs of) possibly inaccurate eigenvalues, q <= N/2, and
IWORK(2), ..., IWORK(q+1) indicate their indices.
Specifically, a positive value is an index of a real or
purely imaginary eigenvalue, corresponding to a 1-by-1
block, while the absolute value of a negative entry in
IWORK is an index to the first eigenvalue in a pair of
consecutively stored eigenvalues, corresponding to a
2-by-2 block. A 2-by-2 block may have two complex, two
real, two purely imaginary, or one real and one purely
imaginary eigenvalue.
For i = q+2, ..., 2*q+1, IWORK(i) contains a pointer to
the starting location in DWORK of the i-th triplet of
1-by-1 blocks, if IWORK(i-q) > 0, or 2-by-2 blocks,
if IWORK(i-q) < 0, defining unreliable eigenvalues.
IWORK(2*q+2) contains the number of the 1-by-1 blocks, and
IWORK(2*q+3) contains the number of the 2-by-2 blocks,
corresponding to unreliable eigenvalues. IWORK(2*q+4)
contains the total number t of the 2-by-2 blocks.
If INFO = 0, then q = 0, therefore IWORK(1) = 0.
LIWORK INTEGER
The dimension of the array IWORK. LIWORK >= N+9.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or INFO = 3, DWORK(1) returns the
optimal LDWORK, and DWORK(2), ..., DWORK(4) contain the
Frobenius norms of the factors of the formal matrix
product used by the algorithm. In addition, DWORK(5), ...,
DWORK(4+3*s) contain the s triplet values corresponding
to the 1-by-1 blocks. Their eigenvalues are real or purely
imaginary. Such an eigenvalue is obtained as a1/a2/a3,
where a1, ..., a3 are the corresponding triplet values.
Moreover, DWORK(5+3*s), ..., DWORK(4+3*s+12*t) contain the
t groups of triplet 2-by-2 matrices corresponding to the
2-by-2 blocks. Their eigenvalue pairs are either complex,
or placed on the real and imaginary axes. Such an
eigenvalue pair is the spectrum of the matrix product
A1*inv(A2)*inv(A3), where A1, ..., A3 define the
corresponding 2-by-2 matrix triplet.
On exit, if INFO = -23, DWORK(1) returns the minimum value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
If JOB = 'E' and COMPQ = 'N' and COMPU = 'N',
LDWORK >= 3/4*N**2+MAX(3*N, 27);
else, LDWORK >= 3/2*N**2+MAX(3*N, 27).
For good performance LDWORK should generally be larger.
If LDWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message related to LDWORK is issued by
XERBLA.
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: problem during computation of the eigenvalues;
= 2: periodic QZ algorithm did not converge in the SLICOT
Library subroutine MB03BD;
= 3: some eigenvalues might be inaccurate, and details can
be found in IWORK and DWORK. This is a warning.
Method
The algorithm uses Givens rotations and Householder reflections to
annihilate elements in Z and T such that Z is in a special block
triangular form and T is in skew-Hamiltonian Hessenberg form:
( Z11 Z12 ) ( B1 F1 )
Z = ( ), T = ( ),
( 0 Z22 ) ( 0 B1' )
with Z11 and Z22' upper triangular and B1 upper Hessenberg.
Subsequently, the periodic QZ algorithm is applied to the pencil
aZ22' Z11 - bB1 to determine orthogonal matrices Q1, Q2 and U such
that U' Z11 Q1, Q2' Z22' U are upper triangular and Q2' B1 Q1 is
upper quasi-triangular. See also page 35 in [1] for more details.
References
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical Computation of Deflating Subspaces of Embedded
Hamiltonian Pencils.
Tech. Rep. SFB393/99-15, Technical University Chemnitz,
Germany, June 1999.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) real floating point operations.Further Comments
NoneExample
Program Text
* MB04ED EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 60 )
INTEGER LDB, LDFG, LDQ, LDU1, LDU2, LDWORK, LDZ,
$ LIWORK
PARAMETER ( LDB = NMAX/2, LDFG = NMAX/2,
$ LDQ = NMAX, LDU1 = NMAX/2, LDU2 = NMAX/2,
$ LDWORK = 3*NMAX**2/2 + MAX( NMAX, 24 ) + 3,
$ LDZ = NMAX, LIWORK = NMAX + 9 )
*
* .. Local Scalars ..
CHARACTER COMPQ, COMPU, JOB
INTEGER I, INFO, J, N
*
* .. Local Arrays ..
INTEGER IWORK( LIWORK )
DOUBLE PRECISION ALPHAI( NMAX/2 ), ALPHAR( NMAX/2 ),
$ B( LDB, NMAX/2 ), BETA( NMAX/2 ),
$ DWORK( LDWORK ), FG( LDFG, NMAX/2+1 ),
$ Q( LDQ, NMAX ), U1( LDU1, NMAX/2 ),
$ U2( LDU2, NMAX/2 ), Z( LDZ, NMAX )
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
*
* .. External Subroutines ..
EXTERNAL MB04ED
*
* .. Intrinsic Functions ..
INTRINSIC MAX, MOD
*
* .. Executable statements ..
*
WRITE( NOUT, FMT = 99999 )
*
* Skip first line in data file.
*
READ( NIN, FMT = * )
READ( NIN, FMT = * ) JOB, COMPQ, COMPU, N
READ( NIN, FMT = * ) ( ( Z( I, J ), J = 1, N ), I = 1, N )
READ( NIN, FMT = * ) ( ( B( I, J ), J = 1, N/2 ), I = 1, N/2 )
READ( NIN, FMT = * ) ( ( FG( I, J ), J = 1, N/2+1 ), I = 1, N/2 )
IF( LSAME( COMPU, 'U' ) ) THEN
READ( NIN, FMT = * ) ( ( U1( I, J ), J = 1, N/2 ), I = 1, N/2 )
READ( NIN, FMT = * ) ( ( U2( I, J ), J = 1, N/2 ), I = 1, N/2 )
END IF
IF( N.LT.0 .OR. N.GT.NMAX .OR. MOD( N, 2 ).NE.0 ) THEN
WRITE( NOUT, FMT = 99998 ) N
ELSE
*
* Test of MB04ED.
*
CALL MB04ED( JOB, COMPQ, COMPU, N, Z, LDZ, B, LDB, FG, LDFG, Q,
$ LDQ, U1, LDU1, U2, LDU2, ALPHAR, ALPHAI, BETA,
$ IWORK, LIWORK, DWORK, LDWORK, INFO )
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99997 ) INFO
ELSE
WRITE( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( Z(I,J), J = 1, N )
10 CONTINUE
WRITE( NOUT, FMT = 99994 )
DO 20 I = 1, N/2
WRITE( NOUT, FMT = 99995 ) ( B(I,J), J = 1, N/2 )
20 CONTINUE
WRITE( NOUT, FMT = 99993 )
DO 30 I = 1, N/2
WRITE( NOUT, FMT = 99995 ) ( FG(I,J), J = 1, N/2+1 )
30 CONTINUE
WRITE( NOUT, FMT = 99992 )
WRITE( NOUT, FMT = 99995 ) ( ALPHAR(I), I = 1, N/2 )
WRITE( NOUT, FMT = 99995 ) ( ALPHAI(I), I = 1, N/2 )
WRITE( NOUT, FMT = 99995 ) ( BETA(I), I = 1, N/2 )
WRITE( NOUT, FMT = 99991 )
DO 40 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( Q( I, J ), J = 1, N )
40 CONTINUE
IF ( .NOT.LSAME( COMPU, 'N' ) ) THEN
WRITE( NOUT, FMT = 99990 )
DO 50 I = 1, N/2
WRITE( NOUT, FMT = 99995 ) ( U1( I, J ), J = 1, N/2 )
50 CONTINUE
WRITE( NOUT, FMT = 99989 )
DO 60 I = 1, N/2
WRITE( NOUT, FMT = 99995 ) ( U2( I, J ), J = 1, N/2 )
60 CONTINUE
END IF
END IF
END IF
STOP
99999 FORMAT ( 'MB04ED EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT ( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT ( 'INFO on exit from MB04ED = ', I2 )
99996 FORMAT (/' The transformed matrix Z is' )
99995 FORMAT ( 60( 1X, F8.4 ) )
99994 FORMAT (/' The transformed matrix B is' )
99993 FORMAT (/' The transformed matrix FG is' )
99992 FORMAT (/' The real, imaginary, and beta parts of eigenvalues are'
$ )
99991 FORMAT (/' The matrix Q is ' )
99990 FORMAT (/' The upper left block of the matrix U is ' )
99989 FORMAT (/' The upper right block of the matrix U is ' )
END
Program Data
MB04ED EXAMPLE PROGRAM DATA T I I 8 0.0949 3.3613 -4.7663 -0.5534 0.6408 -3.2793 3.4253 2.9654 0.1138 -1.5903 2.1837 -4.1648 -4.3775 -1.7454 0.1744 2.3262 2.7505 4.4048 4.4183 3.0478 2.7728 2.3048 -0.6451 -1.2045 3.6091 -4.1716 3.4461 3.6880 -0.0985 3.8458 0.2528 -1.3859 0.4352 -3.2829 3.7246 0.4794 -0.3690 -1.5562 -3.4817 -2.2902 1.3080 -3.9881 -3.5497 3.5020 2.2582 4.4764 -4.4080 -1.6818 1.1308 -1.5087 2.4730 2.1553 -1.7129 -4.8669 -2.4102 4.2274 4.7933 -4.3671 -0.0473 -2.0092 1.2439 -4.7385 3.4242 -0.2764 2.0936 1.5510 4.5974 2.5127 2.5469 -3.3739 -1.5961 -2.4490 -2.2397 -3.8100 0.8527 0.0596 1.7970 -0.0164 -2.7619 1.9908 1.0000 2.0000 -4.0500 1.3353 0.2899 -0.4318 2.0000 2.0000 -2.9860 -0.0160 1.0241 0.9469 2.0000 2.0000 1.3303 0.0946 -0.1272 -4.4003 2.0000 2.0000Program Results
MB04ED EXAMPLE PROGRAM RESULTS The transformed matrix Z is -2.5678 -2.9888 0.4304 -2.8719 2.7331 1.3072 1.7565 2.8246 0.0000 -3.8520 -6.0992 6.2935 -3.0386 -5.5317 -1.2189 3.9973 0.0000 0.0000 4.4560 4.4602 0.6080 -4.4326 3.7959 -0.6297 0.0000 0.0000 0.0000 7.0155 1.5557 2.1441 3.6649 -2.3864 0.4352 -3.2829 3.7246 0.4794 -5.3205 0.0000 0.0000 0.0000 1.3080 -3.9881 -3.5497 3.5020 2.2466 6.9633 0.0000 0.0000 1.1308 -1.5087 2.4730 2.1553 -1.7204 -0.8164 8.1468 0.0000 4.7933 -4.3671 -0.0473 -2.0092 -3.9547 0.2664 1.0382 5.5977 The transformed matrix B is 3.8629 -1.3266 0.1253 2.1882 0.0000 3.7258 -3.5913 -2.4583 0.0000 -3.6551 -2.5063 -0.8378 0.0000 0.0000 0.0000 -6.7384 The transformed matrix FG is 1.0000 2.0000 -0.7448 -1.2359 -1.3653 0.0158 2.0000 2.0000 3.4030 3.2344 -1.1665 2.5791 2.0000 2.0000 -0.4096 3.3823 -1.2344 3.9016 2.0000 2.0000 The real, imaginary, and beta parts of eigenvalues are 1.1310 -0.0697 -0.0697 -0.6864 0.0000 0.6035 -0.6035 0.0000 4.0000 4.0000 4.0000 4.0000 The matrix Q is -0.6042 -0.4139 -0.4742 0.1400 -0.2947 0.3462 -0.0980 0.0534 -0.3706 0.1367 0.4442 -0.1381 -0.1210 0.2913 0.7248 -0.0524 0.1325 -0.2735 -0.0515 -0.5084 -0.3163 -0.2855 0.1638 0.6619 0.2373 0.5514 -0.4988 0.3373 -0.3852 -0.0007 0.3329 0.1339 0.4777 -0.4517 0.2739 0.5172 -0.0775 0.3874 0.1088 0.2395 -0.0116 -0.4372 -0.1843 0.2474 0.1236 -0.6052 0.4772 -0.3228 0.1237 -0.0310 -0.4300 -0.2090 0.7209 0.3408 0.2898 0.1883 0.4245 -0.1871 -0.1803 -0.4655 -0.3304 0.2849 0.0623 -0.5843 The upper left block of the matrix U is 0.0154 -0.5058 -0.5272 0.6826 0.4829 -0.1519 -0.2921 -0.3491 0.4981 0.1532 0.1019 0.1810 -0.0188 0.6270 -0.5260 0.0587 The upper right block of the matrix U is 0.0000 0.0000 0.0000 0.0000 0.3179 -0.4312 -0.1802 -0.4659 0.5644 0.0873 0.4480 0.3979 -0.3137 -0.3330 0.3413 0.0239
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04FD.html 0000664 0000000 0000000 00000045266 14560147231 0020343 0 ustar 00root root 0000000 0000000
Purpose
To compute the eigenvalues of a real N-by-N skew-Hamiltonian/
skew-Hamiltonian pencil aS - bT with
( A D ) ( B F )
S = ( ) and T = ( ). (1)
( E A' ) ( G B' )
Optionally, if JOB = 'T', the pencil aS - bT will be transformed
to the structured Schur form: an orthogonal transformation matrix
Q is computed such that
( Aout Dout )
J Q' J' S Q = ( ), and
( 0 Aout' )
(2)
( Bout Fout ) ( 0 I )
J Q' J' T Q = ( ), where J = ( ),
( 0 Bout' ) ( -I 0 )
Aout is upper triangular, and Bout is upper quasi-triangular. The
notation M' denotes the transpose of the matrix M.
Optionally, if COMPQ = 'I' or COMPQ = 'U', the orthogonal
transformation matrix Q will be computed.
Specification
SUBROUTINE MB04FD( JOB, COMPQ, N, A, LDA, DE, LDDE, B, LDB, FG,
$ LDFG, Q, LDQ, ALPHAR, ALPHAI, BETA, IWORK,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, JOB
INTEGER INFO, LDA, LDB, LDDE, LDFG, LDQ, LDWORK, N
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), DE( LDDE, * ),
$ DWORK( * ), FG( LDFG, * ), Q( LDQ, * )
INTEGER IWORK( * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'E': compute the eigenvalues only; S and T will not
necessarily be put into skew-Hamiltonian
triangular form (2);
= 'T': put S and T into skew-Hamiltonian triangular form
(2), and return the eigenvalues in ALPHAR, ALPHAI
and BETA.
COMPQ CHARACTER*1
Specifies whether to compute the orthogonal transformation
matrix Q as follows:
= 'N': Q is not computed;
= 'I': the array Q is initialized internally to the unit
matrix, and the orthogonal matrix Q is returned;
= 'U': the array Q contains an orthogonal matrix Q0 on
entry, and the product Q0*Q is returned, where Q
is the product of the orthogonal transformations
that are applied to the pencil aS - bT to reduce
S and T to the forms in (2), for COMPQ = 'I'.
Input/Output Parameters
N (input) INTEGER
The order of the pencil aS - bT. N >= 0, even.
A (input/output) DOUBLE PRECISION array, dimension
(LDA, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix A.
On exit, if JOB = 'T', the leading N/2-by-N/2 part of this
array contains the matrix Aout; otherwise, it contains
meaningless elements, except for the diagonal blocks,
which are correctly set.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1, N/2).
DE (input/output) DOUBLE PRECISION array, dimension
(LDDE, N/2+1)
On entry, the leading N/2-by-N/2 strictly lower triangular
part of this array must contain the strictly lower
triangular part of the skew-symmetric matrix E, and the
N/2-by-N/2 strictly upper triangular part of the submatrix
in the columns 2 to N/2+1 of this array must contain the
strictly upper triangular part of the skew-symmetric
matrix D.
The entries on the diagonal and the first superdiagonal of
this array are not referenced, but are assumed to be zero.
On exit, if JOB = 'T', the leading N/2-by-N/2 strictly
upper triangular part of the submatrix in the columns
2 to N/2+1 of this array contains the strictly upper
triangular part of the skew-symmetric matrix Dout.
If JOB = 'E', the leading N/2-by-N/2 strictly upper
triangular part of the submatrix in the columns 2 to N/2+1
of this array contains the strictly upper triangular part
of the skew-symmetric matrix D just before the application
of the QZ algorithm. The remaining entries are
meaningless.
LDDE INTEGER
The leading dimension of the array DE.
LDDE >= MAX(1, N/2).
B (input/output) DOUBLE PRECISION array, dimension
(LDB, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix B.
On exit, if JOB = 'T', the leading N/2-by-N/2 part of this
array contains the matrix Bout; otherwise, it contains
meaningless elements, except for the diagonal 1-by-1 and
2-by-2 blocks, which are correctly set.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N/2).
FG (input/output) DOUBLE PRECISION array, dimension
(LDFG, N/2+1)
On entry, the leading N/2-by-N/2 strictly lower triangular
part of this array must contain the strictly lower
triangular part of the skew-symmetric matrix G, and the
N/2-by-N/2 strictly upper triangular part of the submatrix
in the columns 2 to N/2+1 of this array must contain the
strictly upper triangular part of the skew-symmetric
matrix F.
The entries on the diagonal and the first superdiagonal of
this array are not referenced, but are assumed to be zero.
On exit, if JOB = 'T', the leading N/2-by-N/2 strictly
upper triangular part of the submatrix in the columns
2 to N/2+1 of this array contains the strictly upper
triangular part of the skew-symmetric matrix Fout.
If JOB = 'E', the leading N/2-by-N/2 strictly upper
triangular part of the submatrix in the columns 2 to N/2+1
of this array contains the strictly upper triangular part
of the skew-symmetric matrix F just before the application
of the QZ algorithm. The remaining entries are
meaningless.
LDFG INTEGER
The leading dimension of the array FG.
LDFG >= MAX(1, N/2).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
On entry, if COMPQ = 'U', then the leading N-by-N part of
this array must contain a given matrix Q0, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q0 and the transformation matrix Q
used to transform the matrices S and T.
On exit, if COMPQ = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q.
If COMPQ = 'N', this array is not referenced.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1, N), if COMPQ = 'I' or COMPQ = 'U'.
ALPHAR (output) DOUBLE PRECISION array, dimension (N/2)
The real parts of each scalar alpha defining an eigenvalue
of the pencil aS - bT.
ALPHAI (output) DOUBLE PRECISION array, dimension (N/2)
The imaginary parts of each scalar alpha defining an
eigenvalue of the pencil aS - bT.
If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
positive, then the j-th and (j+1)-st eigenvalues are a
complex conjugate pair.
BETA (output) DOUBLE PRECISION array, dimension (N/2)
The scalars beta that define the eigenvalues of the pencil
aS - bT.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent the j-th eigenvalue of the pencil
aS - bT, in the form lambda = alpha/beta. Since lambda may
overflow, the ratios should not, in general, be computed.
Due to the skew-Hamiltonian/skew-Hamiltonian structure of
the pencil, every eigenvalue occurs twice and thus it has
only to be saved once in ALPHAR, ALPHAI and BETA.
Workspace
IWORK INTEGER array, dimension (N/2+1)
On exit, IWORK(1) contains the number of (pairs of)
possibly inaccurate eigenvalues, q <= N/2, and the
nonzero absolute values in IWORK(2), ..., IWORK(N/2+1) are
indices of the possibly inaccurate eigenvalues, as well as
of the corresponding 1-by-1 or 2-by-2 diagonal blocks in
the arrays A and B on exit. The 2-by-2 blocks correspond
to negative values in IWORK. One negative value is stored
for each such eigenvalue pair. Its modulus indicates the
starting index of a 2-by-2 block.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK;
DWORK(2) and DWORK(3) contain the Frobenius norms of the
matrices S and T on entry. These norms are used in the
tests to decide that some eigenvalues are considered as
unreliable.
On exit, if INFO = -19, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= MAX(3,N/2), if JOB = 'E' and COMPQ = 'N';
LDWORK >= MAX(3,N**2/4+N/2), if JOB = 'T' and COMPQ = 'N';
LDWORK >= MAX(1,3*N**2/4), if COMPQ<> 'N'.
For good performance LDWORK should generally be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: QZ iteration failed in the LAPACK Library routine
DHGEQZ. (QZ iteration did not converge or computation
of the shifts failed.)
= 2: warning: the pencil is numerically singular.
Method
The algorithm uses Givens rotations and Householder reflections to
annihilate elements in S and T such that S is in skew-Hamiltonian
triangular form and T is in skew-Hamiltonian Hessenberg form:
( A1 D1 ) ( B1 F1 )
S = ( ), T = ( ),
( 0 A1' ) ( 0 B1' )
where A1 is upper triangular and B1 is upper Hessenberg.
Subsequently, the QZ algorithm is applied to the pencil aA1 - bB1
to determine orthogonal matrices Q1 and Q2 such that
Q2' A1 Q1 is upper triangular and Q2' B1 Q1 is upper quasi-
triangular.
See also page 40 in [1] for more details.
References
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical Computation of Deflating Subspaces of Embedded
Hamiltonian Pencils.
Tech. Rep. SFB393/99-15, Technical University Chemnitz,
Germany, June 1999.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) real floating point operations.Further Comments
NoneExample
Program Text
* MB04FD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 50 )
INTEGER LDA, LDB, LDDE, LDFG, LDQ, LDWORK
PARAMETER ( LDA = NMAX/2, LDB = NMAX/2, LDDE = NMAX/2,
$ LDFG = NMAX/2, LDQ = NMAX, LDWORK =
$ 3*NMAX*NMAX/4 )
*
* .. Local Scalars ..
CHARACTER COMPQ, JOB
INTEGER I, INFO, J, N
*
* .. Local Arrays ..
DOUBLE PRECISION A( LDA, NMAX/2 ), ALPHAI( NMAX/2 ),
$ ALPHAR( NMAX/2 ), B( LDB, NMAX/2 ),
$ BETA( NMAX/2 ), DE( LDDE, NMAX/2+1 ),
$ DWORK( LDWORK ), FG( LDFG, NMAX/2+1 ),
$ Q( LDQ, NMAX )
INTEGER IWORK( NMAX/2+1 )
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
*
* .. External Subroutines ..
EXTERNAL MB04FD
*
* .. Intrinsic Functions ..
INTRINSIC MOD
*
* .. Executable statements ..
*
WRITE( NOUT, FMT = 99999 )
*
* Skip first line in data file.
*
READ( NIN, FMT = * )
READ( NIN, FMT = * ) JOB, COMPQ, N
READ( NIN, FMT = * ) ( ( A( I, J ), J = 1, N/2 ), I = 1, N/2 )
READ( NIN, FMT = * ) ( ( DE( I, J ), J = 1, N/2+1 ), I = 1, N/2 )
READ( NIN, FMT = * ) ( ( B( I, J ), J = 1, N/2 ), I = 1, N/2 )
READ( NIN, FMT = * ) ( ( FG( I, J ), J = 1, N/2+1 ), I = 1, N/2 )
IF( LSAME( COMPQ, 'U' ) )
$ READ( NIN, FMT = * ) ( ( Q( I, J ), J = 1, N ), I = 1, N )
IF( N.LT.0 .OR. N.GT.NMAX .OR. MOD( N, 2 ).NE.0 ) THEN
WRITE( NOUT, FMT = 99998 ) N
ELSE
*
* Test of MB04FD.
*
CALL MB04FD( JOB, COMPQ, N, A, LDA, DE, LDDE, B, LDB, FG, LDFG,
$ Q, LDQ, ALPHAR, ALPHAI, BETA, IWORK, DWORK,
$ LDWORK, INFO )
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99997 ) INFO
ELSE
IF( LSAME( JOB, 'T' ) ) THEN
WRITE( NOUT, FMT = 99996 )
DO 10 I = 1, N/2
WRITE( NOUT, FMT = 99995 ) ( A( I, J ), J = 1, N/2 )
10 CONTINUE
END IF
WRITE( NOUT, FMT = 99994 )
DO 20 I = 1, N/2
WRITE( NOUT, FMT = 99995 ) ( DE( I, J ), J = 1, N/2+1 )
20 CONTINUE
IF( LSAME( JOB, 'T' ) ) THEN
WRITE( NOUT, FMT = 99993 )
DO 30 I = 1, N/2
WRITE( NOUT, FMT = 99995 ) ( B( I, J ), J = 1, N/2 )
30 CONTINUE
END IF
WRITE( NOUT, FMT = 99992 )
DO 40 I = 1, N/2
WRITE( NOUT, FMT = 99995 ) ( FG( I, J ), J = 1, N/2+1 )
40 CONTINUE
IF( .NOT.LSAME( COMPQ, 'N' ) ) THEN
WRITE( NOUT, FMT = 99991 )
DO 50 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( Q( I, J ), J = 1, N )
50 CONTINUE
END IF
WRITE( NOUT, FMT = 99990 )
WRITE( NOUT, FMT = 99995 ) ( ALPHAR(I), I = 1, N/2 )
WRITE( NOUT, FMT = 99995 ) ( ALPHAI(I), I = 1, N/2 )
WRITE( NOUT, FMT = 99995 ) ( BETA(I), I = 1, N/2 )
END IF
END IF
STOP
99999 FORMAT ( 'MB04FD EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT ( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT ( 'INFO on exit from MB04FD = ', I2 )
99996 FORMAT (/' The transformed matrix A is' )
99995 FORMAT ( 51( 1X, F8.4 ) )
99994 FORMAT (/' The transformed matrix DE is' )
99993 FORMAT (/' The transformed matrix B is' )
99992 FORMAT (/' The transformed matrix FG is' )
99991 FORMAT (/' The transformed matrix Q is ' )
99990 FORMAT (/' The real, imaginary, and beta parts of eigenvalues are'
$ )
END
Program Data
MB04FD EXAMPLE PROGRAM DATA
T I 8
0.8147 0.6323 0.9575 0.9571
0.9057 0.0975 0.9648 0.4853
0.1269 0.2784 0.1576 0.8002
0.9133 0.5468 0.9705 0.1418
0.4217 0.6557 0.6787 0.6554 0.2769
0.9157 0.0357 0.7577 0.1711 0.0461
0.7922 0.8491 0.7431 0.7060 0.0971
0.9594 0.9339 0.3922 0.0318 0.8234
0.6948 0.4387 0.1868 0.7093
0.3170 0.3815 0.4897 0.7546
0.9502 0.7655 0.4455 0.2760
0.0344 0.7951 0.6463 0.6797
0.6550 0.9597 0.7512 0.8909 0.1492
0.1626 0.3403 0.2550 0.9592 0.2575
0.1189 0.5852 0.5059 0.5472 0.8407
0.4983 0.2238 0.6990 0.1386 0.2542
Program Results
MB04FD EXAMPLE PROGRAM RESULTS The transformed matrix A is 0.0550 -0.3064 0.1543 0.2170 0.0000 1.2189 0.3267 -1.3622 0.0000 0.0000 0.7734 -0.6215 0.0000 0.0000 0.0000 2.5172 The transformed matrix DE is 0.4217 0.6557 0.4277 -0.2877 -0.3980 -1.5448 0.0357 0.7577 -0.2582 -0.4775 0.3220 1.1004 0.7431 0.7060 -1.1102 0.3899 0.3463 0.4843 0.0318 0.8234 The transformed matrix B is 0.8482 -0.4425 -0.3643 0.8333 0.0000 -0.5919 -0.0987 -0.7923 0.0000 0.0000 1.1021 0.1926 0.0000 0.0000 0.0000 1.9788 The transformed matrix FG is 0.6550 0.9597 0.3266 -0.6059 -0.9618 -0.3151 0.3403 0.2550 -0.4771 -0.3005 0.1987 -0.0141 0.5059 0.5472 -0.4172 0.7914 0.0371 0.0000 0.1386 0.2542 The transformed matrix Q is 0.0762 0.7824 -0.2771 -0.5526 0.0000 0.0000 0.0000 0.0000 0.1786 -0.2450 0.4261 -0.5360 0.1843 -0.5857 0.2437 -0.0522 -0.2452 0.1425 0.4831 -0.0742 -0.1834 -0.0856 -0.6529 0.4620 -0.2123 -0.3870 -0.6431 -0.2548 0.3061 -0.1791 -0.2518 0.3707 -0.4085 0.0154 0.0784 -0.0738 0.3908 0.0160 -0.3950 -0.7157 0.6418 -0.2868 0.0258 -0.3305 -0.0286 0.4768 -0.3901 -0.1253 -0.0077 -0.1340 -0.2740 -0.0534 -0.7708 -0.3910 -0.1981 -0.3432 -0.5275 -0.2394 0.1131 -0.4684 -0.3018 0.4869 0.3216 0.0252 The real, imaginary, and beta parts of eigenvalues are 0.8482 -0.5919 1.1021 1.9788 0.0000 0.0000 0.0000 0.0000 0.0550 1.2189 0.7734 2.5172
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04FP.html 0000664 0000000 0000000 00000031527 14560147231 0020352 0 ustar 00root root 0000000 0000000
Purpose
To compute the eigenvalues of a real N-by-N skew-Hamiltonian/
skew-Hamiltonian pencil aS - bT with
( A D ) ( B F )
S = ( ) and T = ( ). (1)
( E A' ) ( G B' )
Optionally, if JOB = 'T', the pencil aS - bT will be transformed
to the structured Schur form: an orthogonal transformation matrix
Q is computed such that
( Aout Dout )
J Q' J' S Q = ( ), and
( 0 Aout' )
(2)
( Bout Fout ) ( 0 I )
J Q' J' T Q = ( ), where J = ( ),
( 0 Bout' ) ( -I 0 )
Aout is upper triangular, and Bout is upper quasi-triangular. The
notation M' denotes the transpose of the matrix M.
Optionally, if COMPQ = 'I' or COMPQ = 'U', the orthogonal
transformation matrix Q will be computed.
Specification
SUBROUTINE MB04FP( JOB, COMPQ, N, A, LDA, DE, LDDE, B, LDB, FG,
$ LDFG, Q, LDQ, ALPHAR, ALPHAI, BETA, IWORK,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, JOB
INTEGER INFO, LDA, LDB, LDDE, LDFG, LDQ, LDWORK, N
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), DE( LDDE, * ),
$ DWORK( * ), FG( LDFG, * ), Q( LDQ, * )
INTEGER IWORK( * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'E': compute the eigenvalues only; S and T will not
necessarily be put into skew-Hamiltonian
triangular form (2);
= 'T': put S and T into skew-Hamiltonian triangular form
(2), and return the eigenvalues in ALPHAR, ALPHAI
and BETA.
COMPQ CHARACTER*1
Specifies whether to compute the orthogonal transformation
matrix Q as follows:
= 'N': Q is not computed;
= 'I': the array Q is initialized internally to the unit
matrix, and the orthogonal matrix Q is returned;
= 'U': the array Q contains an orthogonal matrix Q0 on
entry, and the product Q0*Q is returned, where Q
is the product of the orthogonal transformations
that are applied to the pencil aS - bT to reduce
S and T to the forms in (2), for COMPQ = 'I'.
Input/Output Parameters
N (input) INTEGER
The order of the pencil aS - bT. N >= 0, even.
A (input/output) DOUBLE PRECISION array, dimension
(LDA, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix A.
On exit, if JOB = 'T', the leading N/2-by-N/2 part of this
array contains the matrix Aout; otherwise, it contains
meaningless elements, except for the diagonal blocks,
which are correctly set.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1, N/2).
DE (input/output) DOUBLE PRECISION array, dimension
(LDDE, N/2+1)
On entry, the leading N/2-by-N/2 strictly lower triangular
part of this array must contain the strictly lower
triangular part of the skew-symmetric matrix E, and the
N/2-by-N/2 strictly upper triangular part of the submatrix
in the columns 2 to N/2+1 of this array must contain the
strictly upper triangular part of the skew-symmetric
matrix D.
The entries on the diagonal and the first superdiagonal of
this array are not referenced, but are assumed to be zero.
On exit, if JOB = 'T', the leading N/2-by-N/2 strictly
upper triangular part of the submatrix in the columns
2 to N/2+1 of this array contains the strictly upper
triangular part of the skew-symmetric matrix Dout.
If JOB = 'E', the leading N/2-by-N/2 strictly upper
triangular part of the submatrix in the columns 2 to N/2+1
of this array contains the strictly upper triangular part
of the skew-symmetric matrix D just before the application
of the QZ algorithm. The remaining entries are
meaningless.
LDDE INTEGER
The leading dimension of the array DE.
LDDE >= MAX(1, N/2).
B (input/output) DOUBLE PRECISION array, dimension
(LDB, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix B.
On exit, if JOB = 'T', the leading N/2-by-N/2 part of this
array contains the matrix Bout; otherwise, it contains
meaningless elements, except for the diagonal 1-by-1 and
2-by-2 blocks, which are correctly set.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N/2).
FG (input/output) DOUBLE PRECISION array, dimension
(LDFG, N/2+1)
On entry, the leading N/2-by-N/2 strictly lower triangular
part of this array must contain the strictly lower
triangular part of the skew-symmetric matrix G, and the
N/2-by-N/2 strictly upper triangular part of the submatrix
in the columns 2 to N/2+1 of this array must contain the
strictly upper triangular part of the skew-symmetric
matrix F.
The entries on the diagonal and the first superdiagonal of
this array are not referenced, but are assumed to be zero.
On exit, if JOB = 'T', the leading N/2-by-N/2 strictly
upper triangular part of the submatrix in the columns
2 to N/2+1 of this array contains the strictly upper
triangular part of the skew-symmetric matrix Fout.
If JOB = 'E', the leading N/2-by-N/2 strictly upper
triangular part of the submatrix in the columns 2 to N/2+1
of this array contains the strictly upper triangular part
of the skew-symmetric matrix F just before the application
of the QZ algorithm. The remaining entries are
meaningless.
LDFG INTEGER
The leading dimension of the array FG.
LDFG >= MAX(1, N/2).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
On entry, if COMPQ = 'U', then the leading N-by-N part of
this array must contain a given matrix Q0, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q0 and the transformation matrix Q
used to transform the matrices S and T.
On exit, if COMPQ = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q.
If COMPQ = 'N', this array is not referenced.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1, N), if COMPQ = 'I' or COMPQ = 'U'.
ALPHAR (output) DOUBLE PRECISION array, dimension (N/2)
The real parts of each scalar alpha defining an eigenvalue
of the pencil aS - bT.
ALPHAI (output) DOUBLE PRECISION array, dimension (N/2)
The imaginary parts of each scalar alpha defining an
eigenvalue of the pencil aS - bT.
If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
positive, then the j-th and (j+1)-st eigenvalues are a
complex conjugate pair.
BETA (output) DOUBLE PRECISION array, dimension (N/2)
The scalars beta that define the eigenvalues of the pencil
aS - bT.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent the j-th eigenvalue of the pencil
aS - bT, in the form lambda = alpha/beta. Since lambda may
overflow, the ratios should not, in general, be computed.
Due to the skew-Hamiltonian/skew-Hamiltonian structure of
the pencil, every eigenvalue occurs twice and thus it has
only to be saved once in ALPHAR, ALPHAI and BETA.
Workspace
IWORK INTEGER array, dimension (N/2+1)
On exit, IWORK(1) contains the number of (pairs of)
possibly inaccurate eigenvalues, q <= N/2, and the
nonzero absolute values in IWORK(2), ..., IWORK(N/2+1) are
indices of the possibly inaccurate eigenvalues, as well as
of the corresponding 1-by-1 or 2-by-2 diagonal blocks in
the arrays A and B on exit. The 2-by-2 blocks correspond
to negative values in IWORK. One negative value is stored
for each such eigenvalue pair. Its modulus indicates the
starting index of a 2-by-2 block.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK;
DWORK(2) and DWORK(3) contain the Frobenius norms of the
matrices S and T on entry. These norms are used in the
tests to decide that some eigenvalues are considered as
unreliable.
On exit, if INFO = -19, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= MAX(3,N/2,2*N-6), if JOB = 'E' and COMPQ = 'N';
LDWORK >= MAX(3,N**2/4+N/2), if JOB = 'T' and COMPQ = 'N';
LDWORK >= MAX(1,3*N**2/4), if COMPQ<> 'N'.
For good performance LDWORK should generally be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: QZ iteration failed in the LAPACK Library routine
DHGEQZ. (QZ iteration did not converge or computation
of the shifts failed.)
= 2: warning: the pencil is numerically singular.
Method
The algorithm uses Givens rotations and Householder reflections to
annihilate elements in S and T such that S is in skew-Hamiltonian
triangular form and T is in skew-Hamiltonian Hessenberg form:
( A1 D1 ) ( B1 F1 )
S = ( ), T = ( ),
( 0 A1' ) ( 0 B1' )
where A1 is upper triangular and B1 is upper Hessenberg.
Subsequently, the QZ algorithm is applied to the pencil aA1 - bB1
to determine orthogonal matrices Q1 and Q2 such that
Q2' A1 Q1 is upper triangular and Q2' B1 Q1 is upper quasi-
triangular.
See also page 40 in [1] for more details.
References
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical Computation of Deflating Subspaces of Embedded
Hamiltonian Pencils.
Tech. Rep. SFB393/99-15, Technical University Chemnitz,
Germany, June 1999.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) real floating point operations.Further Comments
For large values of N, the routine applies the transformations for reducing T on panels of columns. The user may specify in INFO the desired number of columns. If on entry INFO <= 0, then the routine estimates a suitable value of this number.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04GD.html 0000664 0000000 0000000 00000015715 14560147231 0020340 0 ustar 00root root 0000000 0000000
Purpose
To compute an RQ factorization with row pivoting of a real m-by-n matrix A: P*A = R*Q.Specification
SUBROUTINE MB04GD( M, N, A, LDA, JPVT, TAU, DWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
C .. Array Arguments ..
INTEGER JPVT( * )
DOUBLE PRECISION A( LDA, * ), DWORK( * ), TAU( * )
Arguments
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the m-by-n matrix A.
On exit,
if m <= n, the upper triangle of the subarray
A(1:m,n-m+1:n) contains the m-by-m upper triangular
matrix R;
if m >= n, the elements on and above the (m-n)-th
subdiagonal contain the m-by-n upper trapezoidal matrix R;
the remaining elements, with the array TAU, represent the
orthogonal matrix Q as a product of min(m,n) elementary
reflectors (see METHOD).
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
JPVT (input/output) INTEGER array, dimension (M)
On entry, if JPVT(i) .ne. 0, the i-th row of A is permuted
to the bottom of P*A (a trailing row); if JPVT(i) = 0,
the i-th row of A is a free row.
On exit, if JPVT(i) = k, then the i-th row of P*A
was the k-th row of A.
TAU (output) DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors.
Workspace
DWORK DOUBLE PRECISION array, dimension (3*M)Error Indicator
INFO INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit
in A(m-k+i,1:n-k+i-1), and tau in TAU(i).
The matrix P is represented in jpvt as follows: If
jpvt(j) = i
then the jth row of P is the ith canonical unit vector.
References
[1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
Numerical Aspects
The algorithm is backward stable.Further Comments
NoneExample
Program Text
* MB04GD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX
PARAMETER ( NMAX = 10, MMAX = 10 )
INTEGER LDA
PARAMETER ( LDA = MMAX )
INTEGER LDTAU
PARAMETER ( LDTAU = MIN(MMAX,NMAX) )
INTEGER LDWORK
PARAMETER ( LDWORK = 3*MMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, M, N
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), TAU(LDTAU)
INTEGER JPVT(MMAX)
* .. External Subroutines ..
EXTERNAL DLASET, MB04GD
* .. Intrinsic Functions ..
INTRINSIC MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, N
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99972 ) N
ELSE
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99971 ) M
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,M )
READ ( NIN, FMT = * ) ( JPVT(I), I = 1,M )
* RQ with row pivoting.
CALL MB04GD( M, N, A, LDA, JPVT, TAU, DWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) ( JPVT(I), I = 1,M )
WRITE ( NOUT, FMT = 99990 )
IF ( M.GE.N ) THEN
IF ( N.GT.1 )
$ CALL DLASET( 'Lower', N-1, N-1, ZERO, ZERO,
$ A(M-N+2,1), LDA )
ELSE
CALL DLASET( 'Full', M, N-M-1, ZERO, ZERO, A, LDA )
CALL DLASET( 'Lower', M, M, ZERO, ZERO, A(1,N-M),
$ LDA )
END IF
DO 20 I = 1, M
WRITE ( NOUT, FMT = 99989 ) ( A(I,J), J = 1,N )
20 CONTINUE
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' MB04GD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB04GD = ',I2)
99994 FORMAT (' Row permutations are ',/(20(I3,2X)))
99990 FORMAT (/' The matrix A is ')
99989 FORMAT (20(1X,F8.4))
99972 FORMAT (/' N is out of range.',/' N = ',I5)
99971 FORMAT (/' M is out of range.',/' M = ',I5)
END
Program Data
MB04GD EXAMPLE PROGRAM DATA 6 5 1. 2. 6. 3. 5. -2. -1. -1. 0. -2. 5. 5. 1. 5. 1. -2. -1. -1. 0. -2. 4. 8. 4. 20. 4. -2. -1. -1. 0. -2. 0 0 0 0 0 0Program Results
MB04GD EXAMPLE PROGRAM RESULTS Row permutations are 2 4 6 3 1 5 The matrix A is 0.0000 -1.0517 -1.8646 -1.9712 1.2374 0.0000 -1.0517 -1.8646 -1.9712 1.2374 0.0000 -1.0517 -1.8646 -1.9712 1.2374 0.0000 0.0000 4.6768 0.0466 -7.4246 0.0000 0.0000 0.0000 6.7059 -5.4801 0.0000 0.0000 0.0000 0.0000 -22.6274
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04HD.html 0000664 0000000 0000000 00000026433 14560147231 0020340 0 ustar 00root root 0000000 0000000
Purpose
To compute the transformed matrices A and B, using orthogonal
matrices Q1 and Q2 for a real N-by-N regular pencil
( A11 0 ) ( 0 B12 )
aA - bB = a ( ) - b ( ), (1)
( 0 A22 ) ( B21 0 )
where A11, A22 and B12 are upper triangular, B21 is upper
quasi-triangular and the generalized matrix product
-1 -1
A11 B12 A22 B21 is in periodic Schur form, such that the
matrix Q2' A Q1 is upper triangular, Q2' B Q1 is upper
quasi-triangular and the transformed pencil
a(Q2' A Q1) - b(Q2' B Q1) is in generalized Schur form. The
notation M' denotes the transpose of the matrix M.
Specification
SUBROUTINE MB04HD( COMPQ1, COMPQ2, N, A, LDA, B, LDB, Q1, LDQ1,
$ Q2, LDQ2, IWORK, LIWORK, DWORK, LDWORK, BWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ1, COMPQ2
INTEGER INFO, LDA, LDB, LDQ1, LDQ2, LDWORK, LIWORK, N
C .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DWORK( * ),
$ Q1( LDQ1, * ), Q2( LDQ2, * )
Arguments
Mode Parameters
COMPQ1 CHARACTER*1
Specifies whether to compute the orthogonal transformation
matrix Q1, as follows:
= 'N': Q1 is not computed;
= 'I': the array Q1 is initialized internally to the unit
matrix, and the orthogonal matrix Q1 is returned;
= 'U': the array Q1 contains an orthogonal matrix Q01 on
entry, and the matrix Q01*Q1 is returned, where Q1
is the product of the orthogonal transformations
that are applied on the right to the pencil
aA - bB in (1).
COMPQ2 CHARACTER*1
Specifies whether to compute the orthogonal transformation
matrix Q2, as follows:
= 'N': Q2 is not computed;
= 'I': the array Q2 is initialized internally to the unit
matrix, and the orthogonal matrix Q2 is returned;
= 'U': the array Q2 contains an orthogonal matrix Q02 on
entry, and the matrix Q02*Q2 is returned, where Q2
is the product of the orthogonal transformations
that are applied on the left to the pencil aA - bB
in (1).
Input/Output Parameters
N (input) INTEGER
Order of the pencil aA - bB, N >= 0, even.
A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
On entry, the leading N-by-N block diagonal part of this
array must contain the matrix A in (1). The off-diagonal
blocks need not be set to zero.
On exit, the leading N-by-N part of this array contains
the transformed upper triangular matrix.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1, N).
B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
On entry, the leading N-by-N block anti-diagonal part of
this array must contain the matrix B in (1). The diagonal
blocks need not be set to zero.
On exit, the leading N-by-N part of this array contains
the transformed upper quasi-triangular matrix.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N).
Q1 (input/output) DOUBLE PRECISION array, dimension (LDQ1, N)
On entry, if COMPQ1 = 'U', then the leading N-by-N part of
this array must contain a given matrix Q01, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q01 and the transformation matrix Q1
used to transform the matrices A and B.
On exit, if COMPQ1 = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q1.
If COMPQ1 = 'N' this array is not referenced.
LDQ1 INTEGER
The leading dimension of the array Q1.
LDQ1 >= 1, if COMPQ1 = 'N';
LDQ1 >= MAX(1, N), if COMPQ1 = 'I' or COMPQ1 = 'U'.
Q2 (input/output) DOUBLE PRECISION array, dimension (LDQ2, N)
On entry, if COMPQ2 = 'U', then the leading N-by-N part of
this array must contain a given matrix Q02, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q02 and the transformation matrix Q2
used to transform the matrices A and B.
On exit, if COMPQ2 = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q2.
If COMPQ2 = 'N' this array is not referenced.
LDQ2 INTEGER
The leading dimension of the array Q2.
LDQ2 >= 1, if COMPQ2 = 'N';
LDQ2 >= MAX(1, N), if COMPQ2 = 'I' or COMPQ2 = 'U'.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK INTEGER
The dimension of the array IWORK.
LIWORK >= MAX( N/2+1, 32 ).
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
On exit, if INFO = -16, DWORK(1) returns the minimum value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= 2*N*N + MAX( N/2 + 168, 272 ).
For good performance LDWORK should be generally larger.
If LDWORK = -1 a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
BWORK LOGICAL array, dimension (N/2)
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: the periodic QZ algorithm failed to reorder the
eigenvalues (the problem is very ill-conditioned) in
the SLICOT Library routine MB03KD;
= 2: the standard QZ algorithm failed in the LAPACK
routines DGGES or DHGEQZ, called by the SLICOT
routines MB03DD or MB03FD;
= 3: the eigenvalue reordering failed in the LAPACK
routine DTGEX2, called by the SLICOT routine MB03FD;
= 4: the standard QZ algorithm failed to reorder the
eigenvalues in the LAPACK routine DTGSEN, called by
the SLICOT routine MB03DD.
Method
First, the periodic QZ algorithm (see also [2] and [3]) is applied
-1 -1
to the formal matrix product A11 B12 A22 B21 to reorder the
eigenvalues, i.e., orthogonal matrices V1, V2, V3 and V4 are
computed such that V2' A11 V1, V2' B12 V3, V4' A22 V3 and
V4' B21 V1 keep the triangular form, but they can be partitioned
into 2-by-2 block forms and the last diagonal blocks correspond to
all nonpositive real eigenvalues of the formal product, and the
first diagonal blocks correspond to the remaining eigenvalues.
Second, Q1 = diag(V1, V3), Q2 = diag(V2, V4) and
( AA11 AA12 0 0 )
( )
( 0 AA22 0 0 )
A := Q2' A Q1 =: ( ),
( 0 0 AA33 AA34 )
( )
( 0 0 0 AA44 )
( 0 0 BB13 BB14 )
( )
( 0 0 0 BB24 )
B := Q2' B Q1 =: ( ),
( BB31 BB32 0 0 )
( )
( 0 BB42 0 0 )
-1 -1
are set, such that AA22 BB24 AA44 BB42 has only nonpositive
real eigenvalues.
Third, the permutation matrix
( I 0 0 0 )
( )
( 0 0 I 0 )
P = ( ),
( 0 I 0 0 )
( )
( 0 0 0 I )
where I denotes the identity matrix of appropriate size, is used
to transform aA - bB to block upper triangular form
( AA11 0 | AA12 0 )
( | )
( 0 AA33 | 0 AA34 ) ( AA1 * )
A := P' A P = (-----------+-----------) = ( ),
( 0 0 | AA22 0 ) ( 0 AA2 )
( | )
( 0 0 | 0 AA44 )
( 0 BB13 | 0 BB14 )
( | )
( BB31 0 | BB32 0 ) ( BB1 * )
B := P' B P = (-----------+-----------) = ( ).
( 0 0 | 0 BB24 ) ( 0 BB2 )
( | )
( 0 0 | BB42 0 )
Then, further orthogonal transformations that are provided by
MB03FD and MB03DD are used to triangularize the subpencil
aAA1 - bBB1.
Finally, the subpencil aAA2 - bBB2 is triangularized by applying a
special permutation matrix.
See also page 31 in [1] for more details.
References
[1] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
Eigenproblems.
Tech. Rep., Technical University Chemnitz, Germany,
Nov. 2007.
[2] Bojanczyk, A., Golub, G. H. and Van Dooren, P.
The periodic Schur decomposition: algorithms and applications.
In F.T. Luk (editor), Advanced Signal Processing Algorithms,
Architectures, and Implementations III, Proc. SPIE Conference,
vol. 1770, pp. 31-42, 1992.
[3] Hench, J. J. and Laub, A. J.
Numerical Solution of the discrete-time periodic Riccati
equation. IEEE Trans. Automat. Control, 39, 1197-1210, 1994.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) real floating point operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute a QR factorization of an n-by-m matrix A (A = Q * R),
having a p-by-min(p,m) zero triangle in the lower left-hand side
corner, as shown below, for n = 8, m = 7, and p = 2:
[ x x x x x x x ]
[ x x x x x x x ]
[ x x x x x x x ]
[ x x x x x x x ]
A = [ x x x x x x x ],
[ x x x x x x x ]
[ 0 x x x x x x ]
[ 0 0 x x x x x ]
and optionally apply the transformations to an n-by-l matrix B
(from the left). The problem structure is exploited. This
computation is useful, for instance, in combined measurement and
time update of one iteration of the time-invariant Kalman filter
(square root information filter).
Specification
SUBROUTINE MB04ID( N, M, P, L, A, LDA, B, LDB, TAU, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
INTEGER INFO, L, LDA, LDB, LDWORK, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), TAU(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The number of rows of the matrix A. N >= 0.
M (input) INTEGER
The number of columns of the matrix A. M >= 0.
P (input) INTEGER
The order of the zero triagle. P >= 0.
L (input) INTEGER
The number of columns of the matrix B. L >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,M)
On entry, the leading N-by-M part of this array must
contain the matrix A. The elements corresponding to the
zero P-by-MIN(P,M) lower trapezoidal/triangular part
(if P > 0) are not referenced.
On exit, the elements on and above the diagonal of this
array contain the MIN(N,M)-by-M upper trapezoidal matrix
R (R is upper triangular, if N >= M) of the QR
factorization, and the relevant elements below the
diagonal contain the trailing components (the vectors v,
see Method) of the elementary reflectors used in the
factorization.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,L)
On entry, the leading N-by-L part of this array must
contain the matrix B.
On exit, the leading N-by-L part of this array contains
the updated matrix B.
If L = 0, this array is not referenced.
LDB INTEGER
The leading dimension of array B.
LDB >= MAX(1,N) if L > 0;
LDB >= 1 if L = 0.
TAU (output) DOUBLE PRECISION array, dimension MIN(N,M)
The scalar factors of the elementary reflectors used.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK The length of the array DWORK.
LDWORK >= MAX(1,M-1,M-P,L).
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine uses min(N,M) Householder transformations exploiting
the zero pattern of the matrix. A Householder matrix has the form
( 1 ),
H = I - tau *u *u', u = ( v )
i i i i i ( i)
where v is an (N-P+I-2)-vector. The components of v are stored
i i
in the i-th column of A, beginning from the location i+1, and
tau is stored in TAU(i).
i
Numerical Aspects
The algorithm is backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04IY.html 0000664 0000000 0000000 00000012121 14560147231 0020353 0 ustar 00root root 0000000 0000000
Purpose
To overwrite the real n-by-m matrix C with Q' * C, Q * C,
C * Q', or C * Q, according to the following table
SIDE = 'L' SIDE = 'R'
TRANS = 'N': Q * C C * Q
TRANS = 'T': Q'* C C * Q'
where Q is a real orthogonal matrix defined as the product of
k elementary reflectors
Q = H(1) H(2) . . . H(k)
as returned by SLICOT Library routine MB04ID. Q is of order n
if SIDE = 'L' and of order m if SIDE = 'R'.
Specification
SUBROUTINE MB04IY( SIDE, TRANS, N, M, K, P, A, LDA, TAU, C, LDC,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, K, LDA, LDC, LDWORK, M, N, P
CHARACTER SIDE, TRANS
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), DWORK( * ), TAU( * )
Arguments
Mode Parameters
SIDE CHARACTER*1
Specify if Q or Q' is applied from the left or right,
as follows:
= 'L': apply Q or Q' from the left;
= 'R': apply Q or Q' from the right.
TRANS CHARACTER*1
Specify if Q or Q' is to be applied, as follows:
= 'N': apply Q (No transpose);
= 'T': apply Q' (Transpose).
Input/Output Parameters
N (input) INTEGER
The number of rows of the matrix C. N >= 0.
M (input) INTEGER
The number of columns of the matrix C. M >= 0.
K (input) INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
N >= K >= 0, if SIDE = 'L';
M >= K >= 0, if SIDE = 'R'.
P (input) INTEGER
The order of the zero triagle (or the number of rows of
the zero trapezoid) in the matrix triangularized by SLICOT
Library routine MB04ID. P >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,K)
On input, the elements in the rows i+1:min(n,n-p-1+i) of
the i-th column, and TAU(i), represent the orthogonal
reflector H(i), so that matrix Q is the product of
elementary reflectors: Q = H(1) H(2) . . . H(k).
A is modified by the routine but restored on exit.
LDA INTEGER
The leading dimension of the array A.
LDA >= max(1,N), if SIDE = 'L';
LDA >= max(1,M), if SIDE = 'R'.
TAU (input) DOUBLE PRECISION array, dimension (K)
The scalar factors of the elementary reflectors.
C (input/output) DOUBLE PRECISION array, dimension (LDC,M)
On entry, the leading N-by-M part of this array must
contain the matrix C.
On exit, the leading N-by-M part of this array contains
the updated matrix C.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,N).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,M), if SIDE = 'L';
LDWORK >= MAX(1,N), if SIDE = 'R'.
For optimum performance LDWORK >= M*NB if SIDE = 'L',
or LDWORK >= N*NB if SIDE = 'R', where NB is the optimal
block size.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
If SIDE = 'L', each elementary reflector H(i) modifies n-p elements of each column of C, for i = 1:p+1, and n-i+1 elements, for i = p+2:k. If SIDE = 'R', each elementary reflector H(i) modifies m-p elements of each row of C, for i = 1:p+1, and m-i+1 elements, for i = p+2:k.Numerical Aspects
The implemented method is numerically stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute a QR factorization of an n-by-m matrix A (A = Q * R),
having a p-by-min(p,m) zero triangle in the lower left-hand side
corner, as shown below, for n = 8, m = 7, and p = 2:
[ x x x x x x x ]
[ x x x x x x x ]
[ x x x x x x x ]
[ x x x x x x x ]
A = [ x x x x x x x ],
[ x x x x x x x ]
[ 0 x x x x x x ]
[ 0 0 x x x x x ]
and optionally apply the transformations to an n-by-l matrix B
(from the left). The problem structure is exploited. This
computation is useful, for instance, in combined measurement and
time update of one iteration of the time-invariant Kalman filter
(square root information filter).
Specification
SUBROUTINE MB04IZ( N, M, P, L, A, LDA, B, LDB, TAU, ZWORK, LZWORK,
$ INFO )
C .. Scalar Arguments ..
INTEGER INFO, L, LDA, LDB, LZWORK, M, N, P
C .. Array Arguments ..
COMPLEX*16 A(LDA,*), B(LDB,*), TAU(*), ZWORK(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The number of rows of the matrix A. N >= 0.
M (input) INTEGER
The number of columns of the matrix A. M >= 0.
P (input) INTEGER
The order of the zero triagle. P >= 0.
L (input) INTEGER
The number of columns of the matrix B. L >= 0.
A (input/output) COMPLEX*16 array, dimension (LDA,M)
On entry, the leading N-by-M part of this array must
contain the matrix A. The elements corresponding to the
zero P-by-MIN(P,M) lower trapezoidal/triangular part
(if P > 0) are not referenced.
On exit, the elements on and above the diagonal of this
array contain the MIN(N,M)-by-M upper trapezoidal matrix
R (R is upper triangular, if N >= M) of the QR
factorization, and the relevant elements below the
diagonal contain the trailing components (the vectors v,
see Method) of the elementary reflectors used in the
factorization.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) COMPLEX*16 array, dimension (LDB,L)
On entry, the leading N-by-L part of this array must
contain the matrix B.
On exit, the leading N-by-L part of this array contains
the updated matrix B.
If L = 0, this array is not referenced.
LDB INTEGER
The leading dimension of array B.
LDB >= MAX(1,N) if L > 0;
LDB >= 1 if L = 0.
TAU (output) COMPLEX*16 array, dimension MIN(N,M)
The scalar factors of the elementary reflectors used.
Workspace
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) returns the optimal value
of LZWORK.
LZWORK The length of the array ZWORK.
LZWORK >= MAX(1,M-1,M-P,L).
For optimum performance LZWORK should be larger.
If LZWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
ZWORK array, returns this value as the first entry of
the ZWORK array, and no error message related to LZWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine uses min(N,M) Householder transformations exploiting
the zero pattern of the matrix. A Householder matrix has the form
( 1 ),
H = I - tau *u *u', u = ( v )
i i i i i ( i)
where v is an (N-P+I-2)-vector. The components of v are stored
i i
in the i-th column of A, beginning from the location i+1, and
tau is stored in TAU(i).
i
Numerical Aspects
The algorithm is backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04JD.html 0000664 0000000 0000000 00000011555 14560147231 0020341 0 ustar 00root root 0000000 0000000
Purpose
To compute an LQ factorization of an n-by-m matrix A (A = L * Q),
having a min(n,p)-by-p zero triangle in the upper right-hand side
corner, as shown below, for n = 8, m = 7, and p = 2:
[ x x x x x 0 0 ]
[ x x x x x x 0 ]
[ x x x x x x x ]
[ x x x x x x x ]
A = [ x x x x x x x ],
[ x x x x x x x ]
[ x x x x x x x ]
[ x x x x x x x ]
and optionally apply the transformations to an l-by-m matrix B
(from the right). The problem structure is exploited. This
computation is useful, for instance, in combined measurement and
time update of one iteration of the time-invariant Kalman filter
(square root covariance filter).
Specification
SUBROUTINE MB04JD( N, M, P, L, A, LDA, B, LDB, TAU, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
INTEGER INFO, L, LDA, LDB, LDWORK, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), TAU(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The number of rows of the matrix A. N >= 0.
M (input) INTEGER
The number of columns of the matrix A. M >= 0.
P (input) INTEGER
The order of the zero triagle. P >= 0.
L (input) INTEGER
The number of rows of the matrix B. L >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,M)
On entry, the leading N-by-M part of this array must
contain the matrix A. The elements corresponding to the
zero MIN(N,P)-by-P upper trapezoidal/triangular part
(if P > 0) are not referenced.
On exit, the elements on and below the diagonal of this
array contain the N-by-MIN(N,M) lower trapezoidal matrix
L (L is lower triangular, if N <= M) of the LQ
factorization, and the relevant elements above the
diagonal contain the trailing components (the vectors v,
see Method) of the elementary reflectors used in the
factorization.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading L-by-M part of this array must
contain the matrix B.
On exit, the leading L-by-M part of this array contains
the updated matrix B.
If L = 0, this array is not referenced.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,L).
TAU (output) DOUBLE PRECISION array, dimension MIN(N,M)
The scalar factors of the elementary reflectors used.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK The length of the array DWORK.
LDWORK >= MAX(1,N-1,N-P,L).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine uses min(N,M) Householder transformations exploiting
the zero pattern of the matrix. A Householder matrix has the form
( 1 ),
H = I - tau *u *u', u = ( v )
i i i i i ( i)
where v is an (M-P+I-2)-vector. The components of v are stored
i i
in the i-th row of A, beginning from the location i+1, and tau
i
is stored in TAU(i).
Numerical Aspects
The algorithm is backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04KD.html 0000664 0000000 0000000 00000012314 14560147231 0020334 0 ustar 00root root 0000000 0000000
Purpose
To calculate a QR factorization of the first block column and
apply the orthogonal transformations (from the left) also to the
second block column of a structured matrix, as follows
_
[ R 0 ] [ R C ]
Q' * [ ] = [ ]
[ A B ] [ 0 D ]
_
where R and R are upper triangular. The matrix A can be full or
upper trapezoidal/triangular. The problem structure is exploited.
This computation is useful, for instance, in combined measurement
and time update of one iteration of the Kalman filter (square
root information filter).
Specification
SUBROUTINE MB04KD( UPLO, N, M, P, R, LDR, A, LDA, B, LDB, C, LDC,
$ TAU, DWORK )
C .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, LDB, LDC, LDR, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
$ R(LDR,*), TAU(*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Indicates if the matrix A is or not triangular as follows:
= 'U': Matrix A is upper trapezoidal/triangular;
= 'F': Matrix A is full.
Input/Output Parameters
N (input) INTEGER _
The order of the matrices R and R. N >= 0.
M (input) INTEGER
The number of columns of the matrices B, C and D. M >= 0.
P (input) INTEGER
The number of rows of the matrices A, B and D. P >= 0.
R (input/output) DOUBLE PRECISION array, dimension (LDR,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix R.
On exit, the leading N-by-N upper triangular part of this
_
array contains the upper triangular matrix R.
The strict lower triangular part of this array is not
referenced.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, if UPLO = 'F', the leading P-by-N part of this
array must contain the matrix A. If UPLO = 'U', the
leading MIN(P,N)-by-N part of this array must contain the
upper trapezoidal (upper triangular if P >= N) matrix A,
and the elements below the diagonal are not referenced.
On exit, the leading P-by-N part (upper trapezoidal or
triangular, if UPLO = 'U') of this array contains the
trailing components (the vectors v, see Method) of the
elementary reflectors used in the factorization.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,P).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading P-by-M part of this array must
contain the matrix B.
On exit, the leading P-by-M part of this array contains
the computed matrix D.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,P).
C (output) DOUBLE PRECISION array, dimension (LDC,M)
The leading N-by-M part of this array contains the
computed matrix C.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,N).
TAU (output) DOUBLE PRECISION array, dimension (N)
The scalar factors of the elementary reflectors used.
Workspace
DWORK DOUBLE PRECISION array, dimension (N)Method
The routine uses N Householder transformations exploiting the zero
pattern of the block matrix. A Householder matrix has the form
( 1 ),
H = I - tau *u *u', u = ( v )
i i i i i ( i)
where v is a P-vector, if UPLO = 'F', or an min(i,P)-vector, if
i
UPLO = 'U'. The components of v are stored in the i-th column
i
of A, and tau is stored in TAU(i).
i
Numerical Aspects
The algorithm is backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04LD.html 0000664 0000000 0000000 00000012267 14560147231 0020344 0 ustar 00root root 0000000 0000000
Purpose
To calculate an LQ factorization of the first block row and apply
the orthogonal transformations (from the right) also to the second
block row of a structured matrix, as follows
_
[ L A ] [ L 0 ]
[ ]*Q = [ ]
[ 0 B ] [ C D ]
_
where L and L are lower triangular. The matrix A can be full or
lower trapezoidal/triangular. The problem structure is exploited.
This computation is useful, for instance, in combined measurement
and time update of one iteration of the Kalman filter (square
root covariance filter).
Specification
SUBROUTINE MB04LD( UPLO, N, M, P, L, LDL, A, LDA, B, LDB, C, LDC,
$ TAU, DWORK )
C .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, LDB, LDC, LDL, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
$ L(LDL,*), TAU(*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Indicates if the matrix A is or not triangular as follows:
= 'L': Matrix A is lower trapezoidal/triangular;
= 'F': Matrix A is full.
Input/Output Parameters
N (input) INTEGER _
The order of the matrices L and L. N >= 0.
M (input) INTEGER
The number of columns of the matrices A, B and D. M >= 0.
P (input) INTEGER
The number of rows of the matrices B, C and D. P >= 0.
L (input/output) DOUBLE PRECISION array, dimension (LDL,N)
On entry, the leading N-by-N lower triangular part of this
array must contain the lower triangular matrix L.
On exit, the leading N-by-N lower triangular part of this
_
array contains the lower triangular matrix L.
The strict upper triangular part of this array is not
referenced.
LDL INTEGER
The leading dimension of array L. LDL >= MAX(1,N).
A (input/output) DOUBLE PRECISION array, dimension (LDA,M)
On entry, if UPLO = 'F', the leading N-by-M part of this
array must contain the matrix A. If UPLO = 'L', the
leading N-by-MIN(N,M) part of this array must contain the
lower trapezoidal (lower triangular if N <= M) matrix A,
and the elements above the diagonal are not referenced.
On exit, the leading N-by-M part (lower trapezoidal or
triangular, if UPLO = 'L') of this array contains the
trailing components (the vectors v, see Method) of the
elementary reflectors used in the factorization.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading P-by-M part of this array must
contain the matrix B.
On exit, the leading P-by-M part of this array contains
the computed matrix D.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,P).
C (output) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array contains the
computed matrix C.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
TAU (output) DOUBLE PRECISION array, dimension (N)
The scalar factors of the elementary reflectors used.
Workspace
DWORK DOUBLE PRECISION array, dimension (N)Method
The routine uses N Householder transformations exploiting the zero
pattern of the block matrix. A Householder matrix has the form
( 1 ),
H = I - tau *u *u', u = ( v )
i i i i i ( i)
where v is an M-vector, if UPLO = 'F', or an min(i,M)-vector, if
i
UPLO = 'L'. The components of v are stored in the i-th row of A,
i
and tau is stored in TAU(i).
i
Numerical Aspects
The algorithm is backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04MD.html 0000664 0000000 0000000 00000013657 14560147231 0020351 0 ustar 00root root 0000000 0000000
Purpose
To reduce the 1-norm of a general real matrix A by balancing. This involves diagonal similarity transformations applied iteratively to A to make the rows and columns as close in norm as possible. This routine can be used instead LAPACK Library routine DGEBAL, when no reduction of the 1-norm of the matrix is possible with DGEBAL, as for upper triangular matrices. LAPACK Library routine DGEBAK, with parameters ILO = 1, IHI = N, and JOB = 'S', should be used to apply the backward transformation.Specification
SUBROUTINE MB04MD( N, MAXRED, A, LDA, SCALE, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, N
DOUBLE PRECISION MAXRED
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), SCALE( * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
MAXRED (input/output) DOUBLE PRECISION
On entry, the maximum allowed reduction in the 1-norm of
A (in an iteration) if zero rows or columns are
encountered.
If MAXRED > 0.0, MAXRED must be larger than one (to enable
the norm reduction).
If MAXRED <= 0.0, then the value 10.0 for MAXRED is
used.
On exit, if the 1-norm of the given matrix A is non-zero,
the ratio between the 1-norm of the given matrix and the
1-norm of the balanced matrix. Usually, this ratio will be
larger than one, but it can sometimes be one, or even less
than one (for instance, for some companion matrices).
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the input matrix A.
On exit, the leading N-by-N part of this array contains
the balanced matrix.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
SCALE (output) DOUBLE PRECISION array, dimension (N)
The scaling factors applied to A. If D(j) is the scaling
factor applied to row and column j, then SCALE(j) = D(j),
for j = 1,...,N.
Error Indicator
INFO INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Balancing consists of applying a diagonal similarity transformation inv(D) * A * D to make the 1-norms of each row of A and its corresponding column nearly equal. Information about the diagonal matrix D is returned in the vector SCALE.References
[1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
* MB04MD EXAMPLE PROGRAM TEXT.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA
PARAMETER ( LDA = NMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, N
DOUBLE PRECISION MAXRED
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), SCALE(NMAX)
* .. External Subroutines ..
EXTERNAL MB04MD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, MAXRED
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
* Balance matrix A.
CALL MB04MD( N, MAXRED, A, LDA, SCALE, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( A(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99994 ) ( SCALE(I), I = 1,N )
END IF
END IF
STOP
*
99999 FORMAT (' MB04MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB04MD = ',I2)
99997 FORMAT (' The balanced matrix is ')
99996 FORMAT (20(1X,F10.4))
99994 FORMAT (/' SCALE is ',/20(1X,F10.4))
99993 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
MB04MD EXAMPLE PROGRAM DATA 4 0.0 1.0 0.0 0.0 0.0 300.0 400.0 500.0 600.0 1.0 2.0 0.0 0.0 1.0 1.0 1.0 1.0Program Results
MB04MD EXAMPLE PROGRAM RESULTS
The balanced matrix is
1.0000 0.0000 0.0000 0.0000
30.0000 400.0000 50.0000 60.0000
1.0000 20.0000 0.0000 0.0000
1.0000 10.0000 1.0000 1.0000
SCALE is
1.0000 10.0000 1.0000 1.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04ND.html 0000664 0000000 0000000 00000013346 14560147231 0020345 0 ustar 00root root 0000000 0000000
Purpose
To calculate an RQ factorization of the first block row and
apply the orthogonal transformations (from the right) also to the
second block row of a structured matrix, as follows
_
[ A R ] [ 0 R ]
[ ] * Q' = [ _ _ ]
[ C B ] [ C B ]
_
where R and R are upper triangular. The matrix A can be full or
upper trapezoidal/triangular. The problem structure is exploited.
Specification
SUBROUTINE MB04ND( UPLO, N, M, P, R, LDR, A, LDA, B, LDB, C, LDC,
$ TAU, DWORK )
C .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, LDB, LDC, LDR, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
$ R(LDR,*), TAU(*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Indicates if the matrix A is or not triangular as follows:
= 'U': Matrix A is upper trapezoidal/triangular;
= 'F': Matrix A is full.
Input/Output Parameters
N (input) INTEGER _
The order of the matrices R and R. N >= 0.
M (input) INTEGER
The number of rows of the matrices B and C. M >= 0.
P (input) INTEGER
The number of columns of the matrices A and C. P >= 0.
R (input/output) DOUBLE PRECISION array, dimension (LDR,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix R.
On exit, the leading N-by-N upper triangular part of this
_
array contains the upper triangular matrix R.
The strict lower triangular part of this array is not
referenced.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
A (input/output) DOUBLE PRECISION array, dimension (LDA,P)
On entry, if UPLO = 'F', the leading N-by-P part of this
array must contain the matrix A. For UPLO = 'U', if
N <= P, the upper triangle of the subarray A(1:N,P-N+1:P)
must contain the N-by-N upper triangular matrix A, and if
N >= P, the elements on and above the (N-P)-th subdiagonal
must contain the N-by-P upper trapezoidal matrix A.
On exit, if UPLO = 'F', the leading N-by-P part of this
array contains the trailing components (the vectors v, see
METHOD) of the elementary reflectors used in the
factorization. If UPLO = 'U', the upper triangle of the
subarray A(1:N,P-N+1:P) (if N <= P), or the elements on
and above the (N-P)-th subdiagonal (if N >= P), contain
the trailing components (the vectors v, see METHOD) of the
elementary reflectors used in the factorization.
The remaining elements are not referenced.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the leading M-by-N part of this array must
contain the matrix B.
On exit, the leading M-by-N part of this array contains
_
the computed matrix B.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,M).
C (input/output) DOUBLE PRECISION array, dimension (LDC,P)
On entry, the leading M-by-P part of this array must
contain the matrix C.
On exit, the leading M-by-P part of this array contains
_
the computed matrix C.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,M).
TAU (output) DOUBLE PRECISION array, dimension (N)
The scalar factors of the elementary reflectors used.
Workspace
DWORK DOUBLE PRECISION array, dimension (MAX(N-1,M))Method
The routine uses N Householder transformations exploiting the zero
pattern of the block matrix. A Householder matrix has the form
( 1 )
H = I - tau *u *u', u = ( v ),
i i i i i ( i)
where v is a P-vector, if UPLO = 'F', or a min(N-i+1,P)-vector,
i
if UPLO = 'U'. The components of v are stored in the i-th row
i
of A, and tau is stored in TAU(i), i = N,N-1,...,1.
i
In-line code for applying Householder transformations is used
whenever possible (see MB04NY routine).
Numerical Aspects
The algorithm is backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04NY.html 0000664 0000000 0000000 00000006776 14560147231 0020403 0 ustar 00root root 0000000 0000000
Purpose
To apply a real elementary reflector H to a real m-by-(n+1)
matrix C = [ A B ], from the right, where A has one column. H is
represented in the form
( 1 )
H = I - tau * u *u', u = ( ),
( v )
where tau is a real scalar and v is a real n-vector.
If tau = 0, then H is taken to be the unit matrix.
In-line code is used if H has order < 11.
Specification
SUBROUTINE MB04NY( M, N, V, INCV, TAU, A, LDA, B, LDB, DWORK )
C .. Scalar Arguments ..
INTEGER INCV, LDA, LDB, M, N
DOUBLE PRECISION TAU
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DWORK( * ), V( * )
Arguments
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrices A and B. M >= 0.
N (input) INTEGER
The number of columns of the matrix B. N >= 0.
V (input) DOUBLE PRECISION array, dimension
(1+(N-1)*ABS( INCV ))
The vector v in the representation of H.
INCV (input) INTEGER
The increment between the elements of v. INCV <> 0.
TAU (input) DOUBLE PRECISION
The scalar factor of the elementary reflector H.
A (input/output) DOUBLE PRECISION array, dimension (LDA,1)
On entry, the leading M-by-1 part of this array must
contain the matrix A.
On exit, the leading M-by-1 part of this array contains
the updated matrix A (the first column of C * H).
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,M).
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the leading M-by-N part of this array must
contain the matrix B.
On exit, the leading M-by-N part of this array contains
the updated matrix B (the last n columns of C * H).
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,M).
Workspace
DWORK DOUBLE PRECISION array, dimension (M)
DWORK is not referenced if H has order less than 11.
Method
The routine applies the elementary reflector H, taking the special structure of C into account.Numerical Aspects
The algorithm is backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To calculate a QR factorization of the first block column and
apply the orthogonal transformations (from the left) also to the
second block column of a structured matrix, as follows
_ _
[ R B ] [ R B ]
Q' * [ ] = [ _ ]
[ A C ] [ 0 C ]
_
where R and R are upper triangular. The matrix A can be full or
upper trapezoidal/triangular. The problem structure is exploited.
Specification
SUBROUTINE MB04OD( UPLO, N, M, P, R, LDR, A, LDA, B, LDB, C, LDC,
$ TAU, DWORK )
C .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, LDB, LDC, LDR, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
$ R(LDR,*), TAU(*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Indicates if the matrix A is or not triangular as follows:
= 'U': Matrix A is upper trapezoidal/triangular;
= 'F': Matrix A is full.
Input/Output Parameters
N (input) INTEGER _
The order of the matrices R and R. N >= 0.
M (input) INTEGER
The number of columns of the matrices B and C. M >= 0.
P (input) INTEGER
The number of rows of the matrices A and C. P >= 0.
R (input/output) DOUBLE PRECISION array, dimension (LDR,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix R.
On exit, the leading N-by-N upper triangular part of this
_
array contains the upper triangular matrix R.
The strict lower triangular part of this array is not
referenced.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, if UPLO = 'F', the leading P-by-N part of this
array must contain the matrix A. If UPLO = 'U', the
leading MIN(P,N)-by-N part of this array must contain the
upper trapezoidal (upper triangular if P >= N) matrix A,
and the elements below the diagonal are not referenced.
On exit, the leading P-by-N part (upper trapezoidal or
triangular, if UPLO = 'U') of this array contains the
trailing components (the vectors v, see Method) of the
elementary reflectors used in the factorization.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,P).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the matrix B.
On exit, the leading N-by-M part of this array contains
_
the computed matrix B.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,M)
On entry, the leading P-by-M part of this array must
contain the matrix C.
On exit, the leading P-by-M part of this array contains
_
the computed matrix C.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
TAU (output) DOUBLE PRECISION array, dimension (N)
The scalar factors of the elementary reflectors used.
Workspace
DWORK DOUBLE PRECISION array, dimension (MAX(N-1,M))Method
The routine uses N Householder transformations exploiting the zero
pattern of the block matrix. A Householder matrix has the form
( 1 )
H = I - tau *u *u', u = ( v ),
i i i i i ( i)
where v is a P-vector, if UPLO = 'F', or a min(i,P)-vector, if
i
UPLO = 'U'. The components of v are stored in the i-th column
i
of A, and tau is stored in TAU(i).
i
In-line code for applying Householder transformations is used
whenever possible (see MB04OY routine).
Numerical Aspects
The algorithm is backward stable.Further Comments
NoneExample
Program Text
* MB04OD EXAMPLE PROGRAM TEXT.
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER (ZERO = 0.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, NMAX, PMAX
PARAMETER ( MMAX = 20, NMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDR
PARAMETER ( LDA = PMAX, LDB = NMAX, LDC = PMAX,
$ LDR = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( NMAX-1,MMAX ) )
* .. Local Scalars ..
CHARACTER*1 UPLO
INTEGER I, J, M, N, P
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,MMAX),
$ DWORK(LDWORK), R(LDR,NMAX), TAU(NMAX)
* .. External Subroutines ..
EXTERNAL MB04OD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, UPLO
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) M
ELSE
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) P
ELSE
READ ( NIN, FMT = * ) ( ( R(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,M ), I = 1,P )
* Compute and apply QR factorization.
CALL MB04OD( UPLO, N, M, P, R, LDR, A, LDA, B, LDB, C,
$ LDC, TAU, DWORK )
*
WRITE ( NOUT, FMT = 99997 )
DO 40 I = 1, N
DO 20 J = 1, I-1
R(I,J) = ZERO
20 CONTINUE
WRITE ( NOUT, FMT = 99996 ) ( R(I,J), J = 1,N )
40 CONTINUE
IF ( M.GT.0 ) THEN
WRITE ( NOUT, FMT = 99995 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( B(I,J), J = 1,M )
60 CONTINUE
IF ( P.GT.0 ) THEN
WRITE ( NOUT, FMT = 99994 )
DO 80 I = 1, P
WRITE ( NOUT, FMT = 99996 ) ( C(I,J), J = 1,M )
80 CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' MB04OD EXAMPLE PROGRAM RESULTS',/1X)
99997 FORMAT (' The updated matrix R is ')
99996 FORMAT (20(1X,F10.4))
99995 FORMAT (' The updated matrix B is ')
99994 FORMAT (' The updated matrix C is ')
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' M is out of range.',/' M = ',I5)
99991 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
MB04OD EXAMPLE PROGRAM DATA 3 2 2 F 3. 2. 1. 0. 2. 1. 0. 0. 1. 2. 3. 1. 4. 6. 5. 3. 2. 1. 3. 3. 2. 1. 3. 3. 2.Program Results
MB04OD EXAMPLE PROGRAM RESULTS
The updated matrix R is
-5.3852 -6.6850 -4.6424
0.0000 -2.8828 -2.0694
0.0000 0.0000 -1.7793
The updated matrix B is
-4.2710 -3.7139
-0.1555 -2.1411
-1.6021 0.9398
The updated matrix C is
0.5850 1.0141
-2.7974 -3.1162
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04OW.html 0000664 0000000 0000000 00000013143 14560147231 0020364 0 ustar 00root root 0000000 0000000
Purpose
To perform the QR factorization
( U ) = Q*( R ), where U = ( U1 U2 ), R = ( R1 R2 ),
( x' ) ( 0 ) ( 0 T ) ( 0 R3 )
where U and R are (m+n)-by-(m+n) upper triangular matrices, x is
an m+n element vector, U1 is m-by-m, T is n-by-n, stored
separately, and Q is an (m+n+1)-by-(m+n+1) orthogonal matrix.
The matrix ( U1 U2 ) must be supplied in the m-by-(m+n) upper
trapezoidal part of the array A and this is overwritten by the
corresponding part ( R1 R2 ) of R. The remaining upper triangular
part of R, R3, is overwritten on the array T.
The transformations performed are also applied to the (m+n+1)-by-p
matrix ( B' C' d )' (' denotes transposition), where B, C, and d'
are m-by-p, n-by-p, and 1-by-p matrices, respectively.
Specification
SUBROUTINE MB04OW( M, N, P, A, LDA, T, LDT, X, INCX, B, LDB,
$ C, LDC, D, INCD )
C .. Scalar Arguments ..
INTEGER INCD, INCX, LDA, LDB, LDC, LDT, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(*), T(LDT,*),
$ X(*)
Arguments
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix ( U1 U2 ). M >= 0.
N (input) INTEGER
The order of the matrix T. N >= 0.
P (input) INTEGER
The number of columns of the matrices B and C. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading M-by-(M+N) upper trapezoidal part of
this array must contain the upper trapezoidal matrix
( U1 U2 ).
On exit, the leading M-by-(M+N) upper trapezoidal part of
this array contains the upper trapezoidal matrix ( R1 R2 ).
The strict lower triangle of A is not referenced.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
T (input/output) DOUBLE PRECISION array, dimension (LDT,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix T.
On exit, the leading N-by-N upper triangular part of this
array contains the upper triangular matrix R3.
The strict lower triangle of T is not referenced.
LDT INTEGER
The leading dimension of the array T. LDT >= max(1,N).
X (input/output) DOUBLE PRECISION array, dimension
(1+(M+N-1)*INCX), if M+N > 0, or dimension (0), if M+N = 0.
On entry, the incremented array X must contain the
vector x. On exit, the content of X is changed.
INCX (input) INTEGER
Specifies the increment for the elements of X. INCX > 0.
B (input/output) DOUBLE PRECISION array, dimension (LDB,P)
On entry, the leading M-by-P part of this array must
contain the matrix B.
On exit, the leading M-by-P part of this array contains
the transformed matrix B.
If M = 0 or P = 0, this array is not referenced.
LDB INTEGER
The leading dimension of the array B.
LDB >= max(1,M), if P > 0;
LDB >= 1, if P = 0.
C (input/output) DOUBLE PRECISION array, dimension (LDC,P)
On entry, the leading N-by-P part of this array must
contain the matrix C.
On exit, the leading N-by-P part of this array contains
the transformed matrix C.
If N = 0 or P = 0, this array is not referenced.
LDC INTEGER
The leading dimension of the array C.
LDC >= max(1,N), if P > 0;
LDC >= 1, if P = 0.
D (input/output) DOUBLE PRECISION array, dimension
(1+(P-1)*INCD), if P > 0, or dimension (0), if P = 0.
On entry, the incremented array D must contain the
vector d.
On exit, this incremented array contains the transformed
vector d.
If P = 0, this array is not referenced.
INCD (input) INTEGER
Specifies the increment for the elements of D. INCD > 0.
Method
Let q = m+n. The matrix Q is formed as a sequence of plane rotations in planes (1, q+1), (2, q+1), ..., (q, q+1), the rotation in the (j, q+1)th plane, Q(j), being chosen to annihilate the jth element of x.Numerical Aspects
The algorithm requires 0((M+N)*(M+N+P)) operations and is backward stable.Further Comments
For P = 0, this routine produces the same result as SLICOT Library routine MB04OX, but matrix T may not be stored in the array A.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To perform the QR factorization
(U ) = Q*(R),
(x') (0)
where U and R are n-by-n upper triangular matrices, x is an
n element vector and Q is an (n+1)-by-(n+1) orthogonal matrix.
U must be supplied in the n-by-n upper triangular part of the
array A and this is overwritten by R.
Specification
SUBROUTINE MB04OX( N, A, LDA, X, INCX )
C .. Scalar Arguments ..
INTEGER INCX, LDA, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), X(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The number of elements of X and the order of the square
matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix U.
On exit, the leading N-by-N upper triangular part of this
array contains the upper triangular matrix R.
The strict lower triangle of A is not referenced.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
X (input/output) DOUBLE PRECISION array, dimension
(1+(N-1)*INCX)
On entry, the incremented array X must contain the
vector x. On exit, the content of X is changed.
INCX (input) INTEGER.
Specifies the increment for the elements of X. INCX > 0.
Method
The matrix Q is formed as a sequence of plane rotations in planes (1, n+1), (2, n+1), ..., (n, n+1), the rotation in the (j, n+1)th plane, Q(j), being chosen to annihilate the jth element of x.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To apply a real elementary reflector H to a real (m+1)-by-n
matrix C = [ A ], from the left, where A has one row. H is
[ B ]
represented in the form
( 1 )
H = I - tau * u *u', u = ( ),
( v )
where tau is a real scalar and v is a real m-vector.
If tau = 0, then H is taken to be the unit matrix.
In-line code is used if H has order < 11.
Specification
SUBROUTINE MB04OY( M, N, V, TAU, A, LDA, B, LDB, DWORK )
C .. Scalar Arguments ..
INTEGER LDA, LDB, M, N
DOUBLE PRECISION TAU
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DWORK( * ), V( * )
Arguments
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix B. M >= 0.
N (input) INTEGER
The number of columns of the matrices A and B. N >= 0.
V (input) DOUBLE PRECISION array, dimension (M)
The vector v in the representation of H.
TAU (input) DOUBLE PRECISION
The scalar factor of the elementary reflector H.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading 1-by-N part of this array must
contain the matrix A.
On exit, the leading 1-by-N part of this array contains
the updated matrix A (the first row of H * C).
LDA INTEGER
The leading dimension of array A. LDA >= 1.
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the leading M-by-N part of this array must
contain the matrix B.
On exit, the leading M-by-N part of this array contains
the updated matrix B (the last m rows of H * C).
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,M).
Workspace
DWORK DOUBLE PRECISION array, dimension (N)
DWORK is not referenced if H has order less than 11.
Method
The routine applies the elementary reflector H, taking the special structure of C into account.Numerical Aspects
The algorithm is backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To reduce a Hamiltonian like matrix
[ A G ] T T
H = [ T ] , G = G , Q = Q,
[ Q -A ]
or a skew-Hamiltonian like matrix
[ A G ] T T
W = [ T ] , G = -G , Q = -Q,
[ Q A ]
so that elements below the (k+1)-th subdiagonal in the first nb
columns of the (k+n)-by-n matrix A, and offdiagonal elements
in the first nb columns and rows of the n-by-n matrix Q are zero.
The reduction is performed by an orthogonal symplectic
transformation UU'*H*UU and matrices U, XA, XG, XQ, and YA are
returned so that
[ Aout + U*XA'+ YA*U' Gout + U*XG'+ XG*U' ]
UU'*H*UU = [ ].
[ Qout + U*XQ'+ XQ*U' -Aout'- XA*U'- U*YA' ]
Similarly,
[ Aout + U*XA'+ YA*U' Gout + U*XG'- XG*U' ]
UU'*W*UU = [ ].
[ Qout + U*XQ'- XQ*U' Aout'+ XA*U'+ U*YA' ]
This is an auxiliary routine called by MB04PB.
Specification
SUBROUTINE MB04PA( LHAM, N, K, NB, A, LDA, QG, LDQG, XA, LDXA,
$ XG, LDXG, XQ, LDXQ, YA, LDYA, CS, TAU, DWORK )
C .. Scalar Arguments ..
LOGICAL LHAM
INTEGER K, LDA, LDQG, LDXA, LDXG, LDXQ, LDYA, N, NB
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), CS(*), DWORK(*), QG(LDQG,*), TAU(*),
$ XA(LDXA,*), XG(LDXG,*), XQ(LDXQ,*), YA(LDYA,*)
Arguments
Mode Parameters
LHAM LOGICAL
Specifies the type of matrix to be reduced:
= .FALSE. : skew-Hamiltonian like W;
= .TRUE. : Hamiltonian like H.
Input/Output Parameters
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
K (input) INTEGER
The offset of the reduction. Elements below the (K+1)-th
subdiagonal in the first NB columns of A are reduced
to zero. K >= 0.
NB (input) INTEGER
The number of columns/rows to be reduced. N > NB >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading (K+N)-by-N part of this array must
contain the matrix A.
On exit, the leading (K+N)-by-N part of this array
contains the matrix Aout and in the zero part
information about the elementary reflectors used to
compute the reduction.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,K+N).
QG (input/output) DOUBLE PRECISION array, dimension
(LDQG,N+1)
On entry, the leading N+K-by-N+1 part of this array must
contain in the bottom left part the lower triangular part
of the N-by-N matrix Q and in the remainder the upper
trapezoidal part of the last N columns of the N+K-by-N+K
matrix G.
On exit, the leading N+K-by-N+1 part of this array
contains parts of the matrices Q and G in the same fashion
as on entry only that the zero parts of Q contain
information about the elementary reflectors used to
compute the reduction. Note that if LHAM = .FALSE. then
the (K-1)-th and K-th subdiagonals are not referenced.
LDQG INTEGER
The leading dimension of the array QG. LDQG >= MAX(1,N+K).
XA (output) DOUBLE PRECISION array, dimension (LDXA,2*NB)
On exit, the leading N-by-(2*NB) part of this array
contains the matrix XA.
LDXA INTEGER
The leading dimension of the array XA. LDXA >= MAX(1,N).
XG (output) DOUBLE PRECISION array, dimension (LDXG,2*NB)
On exit, the leading (K+N)-by-(2*NB) part of this array
contains the matrix XG.
LDXG INTEGER
The leading dimension of the array XG. LDXG >= MAX(1,K+N).
XQ (output) DOUBLE PRECISION array, dimension (LDXQ,2*NB)
On exit, the leading N-by-(2*NB) part of this array
contains the matrix XQ.
LDXQ INTEGER
The leading dimension of the array XQ. LDXQ >= MAX(1,N).
YA (output) DOUBLE PRECISION array, dimension (LDYA,2*NB)
On exit, the leading (K+N)-by-(2*NB) part of this array
contains the matrix YA.
LDYA INTEGER
The leading dimension of the array YA. LDYA >= MAX(1,K+N).
CS (output) DOUBLE PRECISION array, dimension (2*NB)
On exit, the first 2*NB elements of this array contain the
cosines and sines of the symplectic Givens rotations used
to compute the reduction.
TAU (output) DOUBLE PRECISION array, dimension (NB)
On exit, the first NB elements of this array contain the
scalar factors of some of the elementary reflectors.
Workspace
DWORK DOUBLE PRECISION array, dimension (3*NB)Method
For details regarding the representation of the orthogonal
symplectic matrix UU within the arrays A, QG, CS, TAU see the
description of MB04PU.
The contents of A and QG on exit are illustrated by the following
example with n = 5, k = 2 and nb = 2:
( a r r a a ) ( g g r r g g )
( a r r a a ) ( g g r r g g )
( a r r a a ) ( q g r r g g )
A = ( r r r r r ), QG = ( t r r r r r ),
( u2 r r r r ) ( u1 t r r r r )
( u2 u2 r a a ) ( u1 u1 r q g g )
( u2 u2 r a a ) ( u1 u1 r q q g )
where a, g and q denote elements of the original matrices, r
denotes a modified element, t denotes a scalar factor of an
applied elementary reflector and ui denote elements of the
matrix U.
References
[1] C. F. VAN LOAN:
A symplectic method for approximating all the eigenvalues of
a Hamiltonian matrix.
Linear Algebra and its Applications, 61, pp. 233-251, 1984.
[2] D. KRESSNER:
Block algorithms for orthogonal symplectic factorizations.
BIT, 43 (4), pp. 775-790, 2003.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To reduce a Hamiltonian matrix,
[ A G ]
H = [ T ] ,
[ Q -A ]
where A is an N-by-N matrix and G,Q are N-by-N symmetric matrices,
to Paige/Van Loan (PVL) form. That is, an orthogonal symplectic U
is computed so that
T [ Aout Gout ]
U H U = [ T ] ,
[ Qout -Aout ]
where Aout is upper Hessenberg and Qout is diagonal.
Blocked version.
Specification
SUBROUTINE MB04PB( N, ILO, A, LDA, QG, LDQG, CS, TAU, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER ILO, INFO, LDA, LDQG, LDWORK, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), CS(*), DWORK(*), QG(LDQG,*), TAU(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
ILO (input) INTEGER
It is assumed that A is already upper triangular and Q is
zero in rows and columns 1:ILO-1. ILO is normally set by a
previous call to MB04DD; otherwise it should be set to 1.
1 <= ILO <= N, if N > 0; ILO = 1, if N = 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A.
On exit, the leading N-by-N part of this array contains
the matrix Aout and, in the zero part of Aout,
information about the elementary reflectors used to
compute the PVL factorization.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
QG (input/output) DOUBLE PRECISION array, dimension
(LDQG,N+1)
On entry, the leading N-by-N+1 part of this array must
contain the lower triangular part of the matrix Q and
the upper triangular part of the matrix G.
On exit, the leading N-by-N+1 part of this array contains
the diagonal of the matrix Qout, the upper triangular part
of the matrix Gout and, in the zero parts of Qout,
information about the elementary reflectors used to
compute the PVL factorization.
LDQG INTEGER
The leading dimension of the array QG. LDQG >= MAX(1,N).
CS (output) DOUBLE PRECISION array, dimension (2N-2)
On exit, the first 2N-2 elements of this array contain the
cosines and sines of the symplectic Givens rotations used
to compute the PVL factorization.
TAU (output) DOUBLE PRECISION array, dimension (N-1)
On exit, the first N-1 elements of this array contain the
scalar factors of some of the elementary reflectors.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK, 8*N*NB + 3*NB, where NB is the optimal
block size determined by the function UE01MD.
On exit, if INFO = -10, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,N-1).
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The matrix U is represented as a product of symplectic reflectors
and Givens rotations
U = diag( H(1),H(1) ) G(1) diag( F(1),F(1) )
diag( H(2),H(2) ) G(2) diag( F(2),F(2) )
....
diag( H(n-1),H(n-1) ) G(n-1) diag( F(n-1),F(n-1) ).
Each H(i) has the form
H(i) = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in
QG(i+2:n,i), and tau in QG(i+1,i).
Each F(i) has the form
F(i) = I - nu * w * w'
where nu is a real scalar, and w is a real vector with
w(1:i) = 0 and w(i+1) = 1; w(i+2:n) is stored on exit in
A(i+2:n,i), and nu in TAU(i).
Each G(i) is a Givens rotation acting on rows i+1 and n+i+1,
where the cosine is stored in CS(2*i-1) and the sine in
CS(2*i).
Numerical Aspects
The algorithm requires O(N**3) floating point operations and is strongly backward stable.References
[1] C. F. VAN LOAN:
A symplectic method for approximating all the eigenvalues of
a Hamiltonian matrix.
Linear Algebra and its Applications, 61, pp. 233-251, 1984.
[2] D. KRESSNER:
Block algorithms for orthogonal symplectic factorizations.
BIT, 43 (4), pp. 775-790, 2003.
Further Comments
NoneExample
Program Text
* MB04PB/MB04WP EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, NBMAX
PARAMETER ( NMAX = 7, NBMAX = 3 )
INTEGER LDA, LDQG, LDRES, LDU1, LDU2, LDWORK
PARAMETER ( LDA = NMAX, LDQG = NMAX, LDRES = NMAX,
$ LDU1 = NMAX, LDU2 = NMAX,
$ LDWORK = 8*NBMAX*NMAX + 3*NBMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, N
* .. Local Arrays ..
DOUBLE PRECISION A(LDA, NMAX), CS(2*NMAX), DWORK(LDWORK),
$ QG(LDQG, NMAX+1), RES(LDRES,3*NMAX+1), TAU(NMAX),
$ U1(LDU1,NMAX), U2(LDU2, NMAX)
* .. External Functions ..
DOUBLE PRECISION MA02ID, MA02JD
EXTERNAL MA02ID, MA02JD
* .. External Subroutines ..
EXTERNAL DGEMM, DLACPY, DLASET, DSCAL, DSYMM, DSYR,
$ DSYR2K, DTRMM, MB04PB, MB04WP
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
CALL DLACPY( 'All', N, N, A, LDA, RES(1,N+1), LDRES )
READ ( NIN, FMT = * ) ( ( QG(I,J), J = 1,N+1 ), I = 1,N )
CALL DLACPY( 'All', N, N+1, QG, LDQG, RES(1,2*N+1), LDRES )
CALL MB04PB( N, 1, A, LDA, QG, LDQG, CS, TAU, DWORK, LDWORK,
$ INFO )
INFO = 0
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
CALL DLACPY( 'Lower', N, N, A, LDA, U1, LDU1 )
CALL DLACPY( 'Lower', N, N, QG, LDQG, U2, LDU2 )
CALL MB04WP( N, 1, U1, LDU1, U2, LDU2, CS, TAU, DWORK,
$ LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
IF ( N.GT.2 )
$ CALL DLASET( 'Lower', N-2, N-2, ZERO, ZERO, A(3,1),
$ LDA )
IF ( N.GT.1 )
$ CALL DLASET( 'Lower', N-1, N-1, ZERO, ZERO, QG(2,1),
$ LDQG )
WRITE ( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( U1(I,J), J = 1,N ), ( U2(I,J), J = 1,N )
10 CONTINUE
DO 20 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( -U2(I,J), J = 1,N ), ( U1(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99991 ) MA02JD( .FALSE., .FALSE., N,
$ U1, LDU1, U2, LDU2, RES, LDRES )
WRITE ( NOUT, FMT = 99995 )
DO 30 I = 1, N
WRITE (NOUT, FMT = 99993) ( A(I,J), J = 1,N )
30 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 40 I = 1, N
WRITE (NOUT, FMT = 99993) ( QG(I,J), J = 1,N+1 )
40 CONTINUE
C
CALL DGEMM( 'No Transpose', 'No Transpose', N, N, N, ONE,
$ U1, LDU1, A, LDA, ZERO, RES, LDRES )
CALL DGEMM( 'No Transpose', 'Transpose', N, N, N, -ONE,
$ RES, LDRES, U1, LDU1, ONE, RES(1,N+1),
$ LDRES )
CALL DGEMM( 'No Transpose', 'Transpose', N, N, N, ONE,
$ U2, LDU2, A, LDA, ZERO, RES, LDRES )
CALL DGEMM( 'No Transpose', 'Transpose', N, N, N, ONE,
$ RES, LDRES, U2, LDU2, ONE, RES(1,N+1),
$ LDRES )
CALL DSYMM ( 'Right', 'Upper', N, N, ONE, QG(1,2), LDQG,
$ U1, LDU1, ZERO, RES, LDRES )
CALL DGEMM( 'No Transpose', 'Transpose', N, N, N, -ONE,
$ RES, LDRES, U2, LDU2, ONE, RES(1,N+1),
$ LDRES )
CALL DLACPY( 'All', N, N, U2, LDU2, RES, LDRES )
DO 50 I = 1, N
CALL DSCAL( N, QG(I,I), RES(1,I), 1 )
50 CONTINUE
CALL DGEMM( 'No Transpose', 'Transpose', N, N, N, -ONE,
$ RES, LDRES, U1, LDU1, ONE, RES(1,N+1),
$ LDRES )
CALL DGEMM( 'No Transpose', 'No Transpose', N, N, N, ONE,
$ U2, LDU2, A, LDA, ZERO, RES, LDRES )
CALL DSYR2K( 'Lower', 'No Transpose', N, N, ONE, RES,
$ LDRES, U1, LDU1, ONE, RES(1,2*N+1), LDRES )
CALL DSCAL( N, ONE/TWO, QG(1,2), LDQG+1 )
CALL DLACPY( 'Full', N, N, U2, LDU2, RES, LDRES )
CALL DTRMM( 'Right', 'Upper' , 'No Transpose',
$ 'Not unit', N, N, ONE, QG(1,2), LDQG,
$ RES, LDRES )
CALL DSYR2K( 'Lower', 'No Transpose', N, N, ONE, RES,
$ LDRES, U2, LDU2, ONE, RES(1,2*N+1), LDRES )
DO 60 I = 1, N
CALL DSYR( 'Lower', N, -QG(I,I), U1(1,I), 1,
$ RES(1,2*N+1), LDRES )
60 CONTINUE
CALL DGEMM( 'No Transpose', 'No Transpose', N, N, N, ONE,
$ U1, LDU1, A, LDA, ZERO, RES, LDRES )
CALL DSYR2K( 'Upper', 'No Transpose', N, N, ONE, RES,
$ LDRES, U2, LDU2, ONE, RES(1,2*N+2), LDRES )
CALL DLACPY( 'Full', N, N, U1, LDU1, RES, LDRES )
CALL DTRMM( 'Right', 'Upper' , 'No Transpose',
$ 'Not unit', N, N, ONE, QG(1,2), LDQG,
$ RES, LDRES )
CALL DSYR2K( 'Upper', 'No Transpose', N, N, -ONE, RES,
$ LDRES, U1, LDU1, ONE, RES(1,2*N+2), LDRES )
DO 70 I = 1, N
CALL DSYR( 'Upper', N, QG(I,I), U2(1,I), 1,
$ RES(1,2*N+2), LDRES )
70 CONTINUE
C
WRITE ( NOUT, FMT = 99990 ) MA02ID( 'Hamiltonian',
$ 'Frobenius', N, RES(1,N+1), LDRES, RES(1,2*N+1),
$ LDRES, DWORK )
END IF
END IF
END IF
*
99999 FORMAT (' TMB04PB EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB04PB = ',I2)
99997 FORMAT (' INFO on exit from MB04WP = ',I2)
99996 FORMAT (' The symplectic orthogonal factor U is ')
99995 FORMAT (/' The reduced matrix A is ')
99994 FORMAT (/' The reduced matrix QG is ')
99993 FORMAT (20(1X,F9.4))
99992 FORMAT (/' N is out of range.',/' N = ',I5)
99991 FORMAT (/' Orthogonality of U: || U''*U - I ||_F = ',G7.2)
99990 FORMAT (/' Residual: || H - U*R*U'' ||_F = ',G7.2)
END
Program Data
MB04PB EXAMPLE PROGRAM DATA
5
0.9501 0.7621 0.6154 0.4057 0.0579
0.2311 0.4565 0.7919 0.9355 0.3529
0.6068 0.0185 0.9218 0.9169 0.8132
0.4860 0.8214 0.7382 0.4103 0.0099
0.8913 0.4447 0.1763 0.8936 0.1389
0.3869 0.4055 0.2140 1.0224 1.1103 0.7016
1.3801 0.7567 1.4936 1.2913 0.9515 1.1755
0.7993 1.7598 1.6433 1.0503 0.8839 1.1010
1.2019 1.1956 0.9346 0.6824 0.7590 1.1364
0.8780 0.9029 1.6565 1.1022 0.7408 0.3793
Program Results
TMB04PB EXAMPLE PROGRAM RESULTS
The symplectic orthogonal factor U is
1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 -0.0927 0.2098 0.5594 -0.0226 0.0000 0.5538 0.3184 0.2519 -0.4031
0.0000 -0.2435 0.4745 -0.6362 -0.2542 0.0000 0.3207 -0.2455 0.0595 -0.2819
0.0000 -0.1950 -0.1770 -0.1519 -0.2857 0.0000 0.4823 0.4122 -0.2060 0.6173
0.0000 -0.3576 -0.0480 0.2302 0.4512 0.0000 0.3523 -0.6047 -0.3110 0.1635
0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000
0.0000 -0.5538 -0.3184 -0.2519 0.4031 0.0000 -0.0927 0.2098 0.5594 -0.0226
0.0000 -0.3207 0.2455 -0.0595 0.2819 0.0000 -0.2435 0.4745 -0.6362 -0.2542
0.0000 -0.4823 -0.4122 0.2060 -0.6173 0.0000 -0.1950 -0.1770 -0.1519 -0.2857
0.0000 -0.3523 0.6047 0.3110 -0.1635 0.0000 -0.3576 -0.0480 0.2302 0.4512
Orthogonality of U: || U'*U - I ||_F = .77E-15
The reduced matrix A is
0.9501 -1.5494 0.5268 0.3187 -0.6890
-2.4922 2.0907 -1.3598 0.5682 0.5618
0.0000 -1.7723 0.3960 -0.2624 -0.3709
0.0000 0.0000 -0.2648 0.2136 -0.3226
0.0000 0.0000 0.0000 -0.2308 0.2319
The reduced matrix QG is
0.3869 0.4055 0.0992 0.5237 -0.4110 -0.4861
0.0000 -3.7784 -4.1609 0.3614 0.3606 -0.0696
0.0000 0.0000 1.2192 -0.0848 0.2007 0.3735
0.0000 0.0000 0.0000 -0.8646 0.1538 -0.1970
0.0000 0.0000 0.0000 0.0000 -0.4527 0.0743
Residual: || H - U*R*U' ||_F = .33E-14
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04PU.html 0000664 0000000 0000000 00000035303 14560147231 0020365 0 ustar 00root root 0000000 0000000
Purpose
To reduce a Hamiltonian matrix,
[ A G ]
H = [ T ] ,
[ Q -A ]
where A is an N-by-N matrix and G,Q are N-by-N symmetric matrices,
to Paige/Van Loan (PVL) form. That is, an orthogonal symplectic U
is computed so that
T [ Aout Gout ]
U H U = [ T ] ,
[ Qout -Aout ]
where Aout is upper Hessenberg and Qout is diagonal.
Unblocked version.
Specification
SUBROUTINE MB04PU( N, ILO, A, LDA, QG, LDQG, CS, TAU, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER ILO, INFO, LDA, LDQG, LDWORK, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), CS(*), DWORK(*), QG(LDQG,*), TAU(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
ILO (input) INTEGER
It is assumed that A is already upper triangular and Q is
zero in rows and columns 1:ILO-1. ILO is normally set by a
previous call to MB04DD; otherwise it should be set to 1.
1 <= ILO <= N, if N > 0; ILO = 1, if N = 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A.
On exit, the leading N-by-N part of this array contains
the matrix Aout and, in the zero part of Aout,
information about the elementary reflectors used to
compute the PVL factorization.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
QG (input/output) DOUBLE PRECISION array, dimension
(LDQG,N+1)
On entry, the leading N-by-N+1 part of this array must
contain the lower triangular part of the matrix Q and
the upper triangular part of the matrix G.
On exit, the leading N-by-N+1 part of this array contains
the diagonal of the matrix Qout, the upper triangular part
of the matrix Gout and, in the zero parts of Qout,
information about the elementary reflectors used to
compute the PVL factorization.
LDQG INTEGER
The leading dimension of the array QG. LDQG >= MAX(1,N).
CS (output) DOUBLE PRECISION array, dimension (2N-2)
On exit, the first 2N-2 elements of this array contain the
cosines and sines of the symplectic Givens rotations used
to compute the PVL factorization.
TAU (output) DOUBLE PRECISION array, dimension (N-1)
On exit, the first N-1 elements of this array contain the
scalar factors of some of the elementary reflectors.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -10, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,N-1).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The matrix U is represented as a product of symplectic reflectors
and Givens rotations
U = diag( H(1),H(1) ) G(1) diag( F(1),F(1) )
diag( H(2),H(2) ) G(2) diag( F(2),F(2) )
....
diag( H(n-1),H(n-1) ) G(n-1) diag( F(n-1),F(n-1) ).
Each H(i) has the form
H(i) = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in
QG(i+2:n,i), and tau in QG(i+1,i).
Each F(i) has the form
F(i) = I - nu * w * w'
where nu is a real scalar, and w is a real vector with
w(1:i) = 0 and w(i+1) = 1; w(i+2:n) is stored on exit in
A(i+2:n,i), and nu in TAU(i).
Each G(i) is a Givens rotation acting on rows i+1 and n+i+1,
where the cosine is stored in CS(2*i-1) and the sine in
CS(2*i).
Numerical Aspects
The algorithm requires 40/3 N**3 + O(N) floating point operations and is strongly backward stable.References
[1] C. F. VAN LOAN:
A symplectic method for approximating all the eigenvalues of
a Hamiltonian matrix.
Linear Algebra and its Applications, 61, pp. 233-251, 1984.
Further Comments
NoneExample
Program Text
* MB04PU/MB04WP EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 100 )
INTEGER LDA, LDQG, LDRES, LDU1, LDU2, LDWORK
PARAMETER ( LDA = NMAX, LDQG = NMAX, LDRES = NMAX,
$ LDU1 = NMAX, LDU2 = NMAX, LDWORK = 2*NMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, N
* .. Local Arrays ..
DOUBLE PRECISION A(LDA, NMAX), CS(2*NMAX), DWORK(LDWORK),
$ QG(LDQG, NMAX+1), RES(LDRES,3*NMAX+1), TAU(NMAX),
$ U1(LDU1,NMAX), U2(LDU2, NMAX)
* .. External Functions ..
DOUBLE PRECISION MA02ID, MA02JD
EXTERNAL MA02ID, MA02JD
* .. External Subroutines ..
EXTERNAL DGEMM, DLACPY, DLASET, DSCAL, DSYMM, DSYR,
$ DSYR2K, DTRMM, MB04PU, MB04WP
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
CALL DLACPY( 'All', N, N, A, LDA, RES(1,N+1), LDRES )
READ ( NIN, FMT = * ) ( ( QG(I,J), J = 1,N+1 ), I = 1,N )
CALL DLACPY( 'All', N, N+1, QG, LDQG, RES(1,2*N+1), LDRES )
CALL MB04PU( N, 1, A, LDA, QG, LDQG, CS, TAU, DWORK, LDWORK,
$ INFO )
INFO = 0
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
CALL DLACPY( 'Lower', N, N, A, LDA, U1, LDU1 )
CALL DLACPY( 'Lower', N, N, QG, LDQG, U2, LDU2 )
CALL MB04WP( N, 1, U1, LDU1, U2, LDU2, CS, TAU, DWORK,
$ LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
IF ( N.GT.2 )
$ CALL DLASET( 'Lower', N-2, N-2, ZERO, ZERO, A(3,1),
$ LDA )
IF ( N.GT.1 )
$ CALL DLASET( 'Lower', N-1, N-1, ZERO, ZERO, QG(2,1),
$ LDQG )
WRITE ( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( U1(I,J), J = 1,N ), ( U2(I,J), J = 1,N )
10 CONTINUE
DO 20 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( -U2(I,J), J = 1,N ), ( U1(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99991 ) MA02JD( .FALSE., .FALSE., N,
$ U1, LDU1, U2, LDU2, RES, LDRES )
WRITE ( NOUT, FMT = 99995 )
DO 30 I = 1, N
WRITE (NOUT, FMT = 99993) ( A(I,J), J = 1,N )
30 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 40 I = 1, N
WRITE (NOUT, FMT = 99993) ( QG(I,J), J = 1,N+1 )
40 CONTINUE
C
CALL DGEMM( 'No Transpose', 'No Transpose', N, N, N, ONE,
$ U1, LDU1, A, LDA, ZERO, RES, LDRES )
CALL DGEMM( 'No Transpose', 'Transpose', N, N, N, -ONE,
$ RES, LDRES, U1, LDU1, ONE, RES(1,N+1),
$ LDRES )
CALL DGEMM( 'No Transpose', 'Transpose', N, N, N, ONE,
$ U2, LDU2, A, LDA, ZERO, RES, LDRES )
CALL DGEMM( 'No Transpose', 'Transpose', N, N, N, ONE,
$ RES, LDRES, U2, LDU2, ONE, RES(1,N+1),
$ LDRES )
CALL DSYMM ( 'Right', 'Upper', N, N, ONE, QG(1,2), LDQG,
$ U1, LDU1, ZERO, RES, LDRES )
CALL DGEMM( 'No Transpose', 'Transpose', N, N, N, -ONE,
$ RES, LDRES, U2, LDU2, ONE, RES(1,N+1),
$ LDRES )
CALL DLACPY( 'All', N, N, U2, LDU2, RES, LDRES )
DO 50 I = 1, N
CALL DSCAL( N, QG(I,I), RES(1,I), 1 )
50 CONTINUE
CALL DGEMM( 'No Transpose', 'Transpose', N, N, N, -ONE,
$ RES, LDRES, U1, LDU1, ONE, RES(1,N+1),
$ LDRES )
CALL DGEMM( 'No Transpose', 'No Transpose', N, N, N, ONE,
$ U2, LDU2, A, LDA, ZERO, RES, LDRES )
CALL DSYR2K( 'Lower', 'No Transpose', N, N, ONE, RES,
$ LDRES, U1, LDU1, ONE, RES(1,2*N+1), LDRES )
CALL DSCAL( N, ONE/TWO, QG(1,2), LDQG+1 )
CALL DLACPY( 'Full', N, N, U2, LDU2, RES, LDRES )
CALL DTRMM( 'Right', 'Upper' , 'No Transpose',
$ 'Not unit', N, N, ONE, QG(1,2), LDQG,
$ RES, LDRES )
CALL DSYR2K( 'Lower', 'No Transpose', N, N, ONE, RES,
$ LDRES, U2, LDU2, ONE, RES(1,2*N+1), LDRES )
DO 60 I = 1, N
CALL DSYR( 'Lower', N, -QG(I,I), U1(1,I), 1,
$ RES(1,2*N+1), LDRES )
60 CONTINUE
CALL DGEMM( 'No Transpose', 'No Transpose', N, N, N, ONE,
$ U1, LDU1, A, LDA, ZERO, RES, LDRES )
CALL DSYR2K( 'Upper', 'No Transpose', N, N, ONE, RES,
$ LDRES, U2, LDU2, ONE, RES(1,2*N+2), LDRES )
CALL DLACPY( 'Full', N, N, U1, LDU1, RES, LDRES )
CALL DTRMM( 'Right', 'Upper' , 'No Transpose',
$ 'Not unit', N, N, ONE, QG(1,2), LDQG,
$ RES, LDRES )
CALL DSYR2K( 'Upper', 'No Transpose', N, N, -ONE, RES,
$ LDRES, U1, LDU1, ONE, RES(1,2*N+2), LDRES )
DO 70 I = 1, N
CALL DSYR( 'Upper', N, QG(I,I), U2(1,I), 1,
$ RES(1,2*N+2), LDRES )
70 CONTINUE
C
WRITE ( NOUT, FMT = 99990 ) MA02ID( 'Hamiltonian',
$ 'Frobenius', N, RES(1,N+1), LDRES, RES(1,2*N+1),
$ LDRES, DWORK )
END IF
END IF
END IF
*
99999 FORMAT (' TMB04PU EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB04PU = ',I2)
99997 FORMAT (' INFO on exit from MB04WP = ',I2)
99996 FORMAT (' The symplectic orthogonal factor U is ')
99995 FORMAT (/' The reduced matrix A is ')
99994 FORMAT (/' The reduced matrix QG is ')
99993 FORMAT (20(1X,F9.4))
99992 FORMAT (/' N is out of range.',/' N = ',I5)
99991 FORMAT (/' Orthogonality of U: || U''*U - I ||_F = ',G7.2)
99990 FORMAT (/' Residual: || H - U*R*U'' ||_F = ',G7.2)
END
Program Data
MB04PU EXAMPLE PROGRAM DATA
5
0.9501 0.7621 0.6154 0.4057 0.0579
0.2311 0.4565 0.7919 0.9355 0.3529
0.6068 0.0185 0.9218 0.9169 0.8132
0.4860 0.8214 0.7382 0.4103 0.0099
0.8913 0.4447 0.1763 0.8936 0.1389
0.4055 0.3869 1.3801 0.7993 1.2019 0.8780
0.2140 1.4936 0.7567 1.7598 1.1956 0.9029
1.0224 1.2913 1.0503 1.6433 0.9346 1.6565
1.1103 0.9515 0.8839 0.7590 0.6824 1.1022
0.7016 1.1755 1.1010 1.1364 0.3793 0.7408
Program Results
TMB04PU EXAMPLE PROGRAM RESULTS
The symplectic orthogonal factor U is
1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 -0.1119 0.7763 -0.2005 -0.0001 0.0000 0.1036 -0.2783 -0.2583 0.4356
0.0000 -0.2937 0.2320 0.4014 0.5541 0.0000 0.4949 0.1187 -0.0294 -0.3632
0.0000 -0.2352 -0.2243 -0.7056 -0.0500 0.0000 0.5374 0.3102 -0.0893 0.0318
0.0000 -0.4314 -0.0354 0.2658 -0.6061 0.0000 0.3396 -0.3230 0.3931 0.0207
0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000
0.0000 -0.1036 0.2783 0.2583 -0.4356 0.0000 -0.1119 0.7763 -0.2005 -0.0001
0.0000 -0.4949 -0.1187 0.0294 0.3632 0.0000 -0.2937 0.2320 0.4014 0.5541
0.0000 -0.5374 -0.3102 0.0893 -0.0318 0.0000 -0.2352 -0.2243 -0.7056 -0.0500
0.0000 -0.3396 0.3230 -0.3931 -0.0207 0.0000 -0.4314 -0.0354 0.2658 -0.6061
Orthogonality of U: || U'*U - I ||_F = .16E-14
The reduced matrix A is
0.9501 -1.8690 0.8413 -0.0344 -0.0817
-2.0660 2.7118 -1.6646 0.7606 -0.0285
0.0000 -2.4884 0.4115 -0.4021 0.3964
0.0000 0.0000 -0.5222 0.1767 -0.3081
0.0000 0.0000 0.0000 0.1915 -0.3426
The reduced matrix QG is
0.4055 0.3869 -0.4295 0.9242 -0.7990 -0.0268
0.0000 -3.0834 -2.5926 0.0804 0.1386 -0.1630
0.0000 0.0000 1.3375 0.9618 -0.0263 0.1829
0.0000 0.0000 0.0000 -0.3556 0.6662 0.2123
0.0000 0.0000 0.0000 0.0000 0.1337 -0.8622
Residual: || H - U*R*U' ||_F = .60E-14
Purpose
To apply a real elementary reflector H to a real m-by-n matrix
C, from either the left or the right. H is represented in the form
( 1 )
H = I - tau * u *u', u = ( ),
( v )
where tau is a real scalar and v is a real vector.
If tau = 0, then H is taken to be the unit matrix.
In-line code is used if H has order < 11.
Specification
SUBROUTINE MB04PY( SIDE, M, N, V, TAU, C, LDC, DWORK )
C .. Scalar Arguments ..
CHARACTER SIDE
INTEGER LDC, M, N
DOUBLE PRECISION TAU
C .. Array Arguments ..
DOUBLE PRECISION C( LDC, * ), DWORK( * ), V( * )
Arguments
Mode Parameters
SIDE CHARACTER*1
Indicates whether the elementary reflector should be
applied from the left or from the right, as follows:
= 'L': Compute H * C;
= 'R': Compute C * H.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix C. M >= 0.
N (input) INTEGER
The number of columns of the matrix C. N >= 0.
V (input) DOUBLE PRECISION array, dimension
(M-1), if SIDE = 'L', or
(N-1), if SIDE = 'R'.
The vector v in the representation of H.
TAU (input) DOUBLE PRECISION
The scalar factor of the elementary reflector H.
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading M-by-N part of this array must
contain the matrix C.
On exit, the leading M-by-N part of this array contains
the matrix H * C, if SIDE = 'L', or C * H, if SIDE = 'R'.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,M).
Workspace
DWORK DOUBLE PRECISION array, dimension (N), if SIDE = 'L', or
(M), if SIDE = 'R'.
DWORK is not referenced if H has order less than 11.
Method
The routine applies the elementary reflector H, taking its special structure into account. The multiplications by the first component of u (which is 1) are avoided, to increase the efficiency.Numerical Aspects
The algorithm is backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To overwrite general real m-by-n matrices C and D, or their
transposes, with
[ op(C) ]
Q * [ ] if TRANQ = 'N', or
[ op(D) ]
T [ op(C) ]
Q * [ ] if TRANQ = 'T',
[ op(D) ]
where Q is defined as the product of symplectic reflectors and
Givens rotations,
Q = diag( H(1),H(1) ) G(1) diag( F(1),F(1) )
diag( H(2),H(2) ) G(2) diag( F(2),F(2) )
....
diag( H(k),H(k) ) G(k) diag( F(k),F(k) ).
Blocked version.
Specification
SUBROUTINE MB04QB( TRANC, TRAND, TRANQ, STOREV, STOREW, M, N, K,
$ V, LDV, W, LDW, C, LDC, D, LDD, CS, TAU, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER STOREV, STOREW, TRANC, TRAND, TRANQ
INTEGER INFO, K, LDC, LDD, LDV, LDW, LDWORK, M, N
C .. Array Arguments ..
DOUBLE PRECISION C(LDC,*), CS(*), D(LDD,*), DWORK(*), TAU(*),
$ V(LDV,*), W(LDW,*)
Arguments
Mode Parameters
TRANC CHARACTER*1
Specifies the form of op( C ) as follows:
= 'N': op( C ) = C;
= 'T': op( C ) = C';
= 'C': op( C ) = C'.
TRAND CHARACTER*1
Specifies the form of op( D ) as follows:
= 'N': op( D ) = D;
= 'T': op( D ) = D';
= 'C': op( D ) = D'.
TRANQ CHARACTER*1
= 'N': apply Q;
= 'T': apply Q'.
STOREV CHARACTER*1
Specifies how the vectors which define the concatenated
Householder reflectors contained in V are stored:
= 'C': columnwise;
= 'R': rowwise.
STOREW CHARACTER*1
Specifies how the vectors which define the concatenated
Householder reflectors contained in W are stored:
= 'C': columnwise;
= 'R': rowwise.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrices op(C) and op(D).
M >= 0.
N (input) INTEGER
The number of columns of the matrices op(C) and op(D).
N >= 0.
K (input) INTEGER
The number of elementary reflectors whose product defines
the matrix Q. M >= K >= 0.
V (input) DOUBLE PRECISION array, dimension
(LDV,K) if STOREV = 'C',
(LDV,M) if STOREV = 'R'
On entry with STOREV = 'C', the leading M-by-K part of
this array must contain in its columns the vectors which
define the elementary reflectors F(i).
On entry with STOREV = 'R', the leading K-by-M part of
this array must contain in its rows the vectors which
define the elementary reflectors F(i).
LDV INTEGER
The leading dimension of the array V.
LDV >= MAX(1,M), if STOREV = 'C';
LDV >= MAX(1,K), if STOREV = 'R'.
W (input) DOUBLE PRECISION array, dimension
(LDW,K) if STOREW = 'C',
(LDW,M) if STOREW = 'R'
On entry with STOREW = 'C', the leading M-by-K part of
this array must contain in its columns the vectors which
define the elementary reflectors H(i).
On entry with STOREW = 'R', the leading K-by-M part of
this array must contain in its rows the vectors which
define the elementary reflectors H(i).
LDW INTEGER
The leading dimension of the array W.
LDW >= MAX(1,M), if STOREW = 'C';
LDW >= MAX(1,K), if STOREW = 'R'.
C (input/output) DOUBLE PRECISION array, dimension
(LDC,N) if TRANC = 'N',
(LDC,M) if TRANC = 'T' or TRANC = 'C'
On entry with TRANC = 'N', the leading M-by-N part of
this array must contain the matrix C.
On entry with TRANC = 'C' or TRANC = 'T', the leading
N-by-M part of this array must contain the transpose of
the matrix C.
On exit with TRANC = 'N', the leading M-by-N part of
this array contains the updated matrix C.
On exit with TRANC = 'C' or TRANC = 'T', the leading
N-by-M part of this array contains the transpose of the
updated matrix C.
LDC INTEGER
The leading dimension of the array C.
LDC >= MAX(1,M), if TRANC = 'N';
LDC >= MAX(1,N), if TRANC = 'T' or TRANC = 'C'.
D (input/output) DOUBLE PRECISION array, dimension
(LDD,N) if TRAND = 'N',
(LDD,M) if TRAND = 'T' or TRAND = 'C'
On entry with TRAND = 'N', the leading M-by-N part of
this array must contain the matrix D.
On entry with TRAND = 'C' or TRAND = 'T', the leading
N-by-M part of this array must contain the transpose of
the matrix D.
On exit with TRAND = 'N', the leading M-by-N part of
this array contains the updated matrix D.
On exit with TRAND = 'C' or TRAND = 'T', the leading
N-by-M part of this array contains the transpose of the
updated matrix D.
LDD INTEGER
The leading dimension of the array D.
LDD >= MAX(1,M), if TRAND = 'N';
LDD >= MAX(1,N), if TRAND = 'T' or TRAND = 'C'.
CS (input) DOUBLE PRECISION array, dimension (2*K)
On entry, the first 2*K elements of this array must
contain the cosines and sines of the symplectic Givens
rotations G(i).
TAU (input) DOUBLE PRECISION array, dimension (K)
On entry, the first K elements of this array must
contain the scalar factors of the elementary reflectors
F(i).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -20, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,N).
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
References
[1] Kressner, D.
Block algorithms for orthogonal symplectic factorizations.
BIT, 43 (4), pp. 775-790, 2003.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To apply the orthogonal symplectic block reflector
[ I+V*T*V' V*R*S*V' ]
Q = [ ]
[ -V*R*S*V' I+V*T*V' ]
or its transpose to a real 2m-by-n matrix [ op(A); op(B) ] from
the left.
The k-by-k upper triangular blocks of the matrices
[ S1 ] [ T11 T12 T13 ]
R = [ R1 R2 R3 ], S = [ S2 ], T = [ T21 T22 T23 ],
[ S3 ] [ T31 T32 T33 ]
with R2 unit and S1, R3, T21, T31, T32 strictly upper triangular,
are stored rowwise in the arrays RS and T, respectively.
Specification
SUBROUTINE MB04QC( STRAB, TRANA, TRANB, TRANQ, DIRECT, STOREV,
$ STOREW, M, N, K, V, LDV, W, LDW, RS, LDRS, T,
$ LDT, A, LDA, B, LDB, DWORK )
C .. Scalar Arguments ..
CHARACTER DIRECT, STOREV, STOREW, STRAB, TRANA, TRANB,
$ TRANQ
INTEGER K, LDA, LDB, LDRS, LDT, LDV, LDW, M, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), RS(LDRS,*),
$ T(LDT,*), V(LDV,*), W(LDW,*)
Arguments
Mode Parameters
STRAB CHARACTER*1
Specifies the structure of the first blocks of A and B:
= 'Z': the leading K-by-N submatrices of op(A) and op(B)
are (implicitly) assumed to be zero;
= 'N'; no structure to mention.
TRANA CHARACTER*1
Specifies the form of op( A ) as follows:
= 'N': op( A ) = A;
= 'T': op( A ) = A';
= 'C': op( A ) = A'.
TRANB CHARACTER*1
Specifies the form of op( B ) as follows:
= 'N': op( B ) = B;
= 'T': op( B ) = B';
= 'C': op( B ) = B'.
DIRECT CHARACTER*1
This is a dummy argument, which is reserved for future
extensions of this subroutine. Not referenced.
TRANQ CHARACTER*1
= 'N': apply Q;
= 'T': apply Q'.
STOREV CHARACTER*1
Specifies how the vectors which define the concatenated
Householder reflectors contained in V are stored:
= 'C': columnwise;
= 'R': rowwise.
STOREW CHARACTER*1
Specifies how the vectors which define the concatenated
Householder reflectors contained in W are stored:
= 'C': columnwise;
= 'R': rowwise.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrices op(A) and op(B).
M >= 0.
N (input) INTEGER
The number of columns of the matrices op(A) and op(B).
N >= 0.
K (input) INTEGER
The order of the triangular matrices defining R, S and T.
M >= K >= 0.
V (input) DOUBLE PRECISION array, dimension
(LDV,K) if STOREV = 'C',
(LDV,M) if STOREV = 'R'
On entry with STOREV = 'C', the leading M-by-K part of
this array must contain in its columns the vectors which
define the elementary reflector used to form parts of Q.
On entry with STOREV = 'R', the leading K-by-M part of
this array must contain in its rows the vectors which
define the elementary reflector used to form parts of Q.
LDV INTEGER
The leading dimension of the array V.
LDV >= MAX(1,M), if STOREV = 'C';
LDV >= MAX(1,K), if STOREV = 'R'.
W (input) DOUBLE PRECISION array, dimension
(LDW,K) if STOREW = 'C',
(LDW,M) if STOREW = 'R'
On entry with STOREW = 'C', the leading M-by-K part of
this array must contain in its columns the vectors which
define the elementary reflector used to form parts of Q.
On entry with STOREW = 'R', the leading K-by-M part of
this array must contain in its rows the vectors which
define the elementary reflector used to form parts of Q.
LDW INTEGER
The leading dimension of the array W.
LDW >= MAX(1,M), if STOREW = 'C';
LDW >= MAX(1,K), if STOREW = 'R'.
RS (input) DOUBLE PRECISION array, dimension (K,6*K)
On entry, the leading K-by-6*K part of this array must
contain the upper triangular matrices defining the factors
R and S of the symplectic block reflector Q. The
(strictly) lower portions of this array are not
referenced.
LDRS INTEGER
The leading dimension of the array RS. LDRS >= MAX(1,K).
T (input) DOUBLE PRECISION array, dimension (K,9*K)
On entry, the leading K-by-9*K part of this array must
contain the upper triangular matrices defining the factor
T of the symplectic block reflector Q. The (strictly)
lower portions of this array are not referenced.
LDT INTEGER
The leading dimension of the array T. LDT >= MAX(1,K).
A (input/output) DOUBLE PRECISION array, dimension
(LDA,N) if TRANA = 'N',
(LDA,M) if TRANA = 'C' or TRANA = 'T'
On entry with TRANA = 'N', the leading M-by-N part of this
array must contain the matrix A.
On entry with TRANA = 'T' or TRANA = 'C', the leading
N-by-M part of this array must contain the matrix A.
LDA INTEGER
The leading dimension of the array A.
LDA >= MAX(1,M), if TRANA = 'N';
LDA >= MAX(1,N), if TRANA = 'C' or TRANA = 'T'.
B (input/output) DOUBLE PRECISION array, dimension
(LDB,N) if TRANB = 'N',
(LDB,M) if TRANB = 'C' or TRANB = 'T'
On entry with TRANB = 'N', the leading M-by-N part of this
array must contain the matrix B.
On entry with TRANB = 'T' or TRANB = 'C', the leading
N-by-M part of this array must contain the matrix B.
LDB INTEGER
The leading dimension of the array B.
LDB >= MAX(1,M), if TRANB = 'N';
LDB >= MAX(1,N), if TRANB = 'C' or TRANB = 'T'.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK), where
LDWORK >= 8*N*K, if STRAB = 'Z',
LDWORK >= 9*N*K, if STRAB = 'N'.
References
[1] Kressner, D.
Block algorithms for orthogonal symplectic factorizations.
BIT, 43 (4), pp. 775-790, 2003.
Numerical Aspects
The algorithm requires 16*( M - K )*N + ( 26*K - 4 )*K*N floating point operations if STRAB = 'Z' and additional ( 12*K + 2 )*K*N floating point operations if STRAB = 'N'.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To form the triangular block factors R, S and T of a symplectic
block reflector SH, which is defined as a product of 2k
concatenated Householder reflectors and k Givens rotations,
SH = diag( H(1),H(1) ) G(1) diag( F(1),F(1) )
diag( H(2),H(2) ) G(2) diag( F(2),F(2) )
....
diag( H(k),H(k) ) G(k) diag( F(k),F(k) ).
The upper triangular blocks of the matrices
[ S1 ] [ T11 T12 T13 ]
R = [ R1 R2 R3 ], S = [ S2 ], T = [ T21 T22 T23 ],
[ S3 ] [ T31 T32 T33 ]
with R2 unit and S1, R3, T21, T31, T32 strictly upper triangular,
are stored rowwise in the arrays RS and T, respectively.
Specification
SUBROUTINE MB04QF( DIRECT, STOREV, STOREW, N, K, V, LDV, W, LDW,
$ CS, TAU, RS, LDRS, T, LDT, DWORK )
C .. Scalar Arguments ..
CHARACTER DIRECT, STOREV, STOREW
INTEGER K, LDRS, LDT, LDV, LDW, N
C .. Array Arguments ..
DOUBLE PRECISION CS(*), DWORK(*), RS(LDRS,*), T(LDT,*),
$ TAU(*), V(LDV,*), W(LDW,*)
Arguments
Mode Parameters
DIRECT CHARACTER*1
This is a dummy argument, which is reserved for future
extensions of this subroutine. Not referenced.
STOREV CHARACTER*1
Specifies how the vectors which define the concatenated
Householder F(i) reflectors are stored:
= 'C': columnwise;
= 'R': rowwise.
STOREW CHARACTER*1
Specifies how the vectors which define the concatenated
Householder H(i) reflectors are stored:
= 'C': columnwise;
= 'R': rowwise.
Input/Output Parameters
N (input) INTEGER
The order of the Householder reflectors F(i) and H(i).
N >= 0.
K (input) INTEGER
The number of Givens rotations. K >= 1.
V (input) DOUBLE PRECISION array, dimension
(LDV,K) if STOREV = 'C',
(LDV,N) if STOREV = 'R'
On entry with STOREV = 'C', the leading N-by-K part of
this array must contain in its i-th column the vector
which defines the elementary reflector F(i).
On entry with STOREV = 'R', the leading K-by-N part of
this array must contain in its i-th row the vector
which defines the elementary reflector F(i).
LDV INTEGER
The leading dimension of the array V.
LDV >= MAX(1,N), if STOREV = 'C';
LDV >= K, if STOREV = 'R'.
W (input) DOUBLE PRECISION array, dimension
(LDW,K) if STOREW = 'C',
(LDW,N) if STOREW = 'R'
On entry with STOREW = 'C', the leading N-by-K part of
this array must contain in its i-th column the vector
which defines the elementary reflector H(i).
On entry with STOREV = 'R', the leading K-by-N part of
this array must contain in its i-th row the vector
which defines the elementary reflector H(i).
LDW INTEGER
The leading dimension of the array W.
LDW >= MAX(1,N), if STOREW = 'C';
LDW >= K, if STOREW = 'R'.
CS (input) DOUBLE PRECISION array, dimension (2*K)
On entry, the first 2*K elements of this array must
contain the cosines and sines of the symplectic Givens
rotations G(i).
TAU (input) DOUBLE PRECISION array, dimension (K)
On entry, the first K elements of this array must
contain the scalar factors of the elementary reflectors
F(i).
RS (output) DOUBLE PRECISION array, dimension (K,6*K)
On exit, the leading K-by-6*K part of this array contains
the upper triangular matrices defining the factors R and
S of the symplectic block reflector SH. The (strictly)
lower portions of this array are not used.
LDRS INTEGER
The leading dimension of the array RS. LDRS >= K.
T (output) DOUBLE PRECISION array, dimension (K,9*K)
On exit, the leading K-by-9*K part of this array contains
the upper triangular matrices defining the factor T of the
symplectic block reflector SH. The (strictly) lower
portions of this array are not used.
LDT INTEGER
The leading dimension of the array T. LDT >= K.
Workspace
DWORK DOUBLE PRECISION array, dimension (3*K)References
[1] Kressner, D.
Block algorithms for orthogonal symplectic factorizations.
BIT, 43 (4), pp. 775-790, 2003.
Numerical Aspects
The algorithm requires ( 4*K - 2 )*K*N + 19/3*K*K*K + 1/2*K*K + 43/6*K - 4 floating point operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
Overwrites general real m-by-n/n-by-m matrices C and D with
[ op(C) ]
U * [ ] if TRANU = 'N', or
[ op(D) ]
T [ op(C) ]
U * [ ] if TRANU = 'T',
[ op(D) ]
where U is defined as the product of symplectic reflectors and
Givens rotations,
U = diag( H(1),H(1) ) G(1) diag( F(1),F(1) )
diag( H(2),H(2) ) G(2) diag( F(2),F(2) )
....
diag( H(k),H(k) ) G(k) diag( F(k),F(k) ),
with k = m-1, as returned by the SLICOT Library routines MB04PU
or MB04RU.
Specification
SUBROUTINE MB04QS( TRANC, TRAND, TRANU, M, N, ILO, V, LDV, W, LDW,
$ C, LDC, D, LDD, CS, TAU, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TRANC, TRAND, TRANU
INTEGER ILO, INFO, LDC, LDD, LDV, LDW, LDWORK, M, N
C .. Array Arguments ..
DOUBLE PRECISION C(LDC,*), CS(*), D(LDD,*), DWORK(*), TAU(*),
$ V(LDV,*), W(LDW,*)
Arguments
Mode Parameters
TRANC CHARACTER*1
Specifies the form of op( C ) as follows:
= 'N': op( C ) = C;
= 'T': op( C ) = C';
= 'C': op( C ) = C'.
TRAND CHARACTER*1
Specifies the form of op( D ) as follows:
= 'N': op( D ) = D;
= 'T': op( D ) = D';
= 'C': op( D ) = D'.
TRANU CHARACTER*1
Specifies whether U or U' is applied as follows:
= 'N': apply U;
= 'T': apply U'.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrices op(C) and op(D).
M >= 0.
N (input) INTEGER
The number of columns of the matrices op(C) and op(D).
N >= 0.
ILO (input) INTEGER
ILO must have the same value as in the previous call of
MB04PU or MB04RU. U is equal to the unit matrix except in
the submatrix
U([ilo+1:m m+ilo+1:2*m], [ilo+1:m m+ilo+1:2*m]).
1 <= ILO <= M+1.
V (input) DOUBLE PRECISION array, dimension (LDV,M)
On entry, the leading M-by-M part of this array must
contain in its columns the vectors which define the
elementary reflectors H(i).
LDV INTEGER
The leading dimension of the array V. LDV >= MAX(1,M).
W (input) DOUBLE PRECISION array, dimension (LDW,M)
On entry, the leading M-by-M part of this array must
contain in its columns the vectors which define the
elementary reflectors F(i).
LDW INTEGER
The leading dimension of the array W. LDW >= MAX(1,M).
C (input/output) DOUBLE PRECISION array, dimension
(LDC,N) if TRANC = 'N',
(LDC,M) if TRANC = 'T' or TRANC = 'C'.
On entry with TRANC = 'N', the leading M-by-N part of this
array must contain the matrix C.
On entry with TRANC = 'T' or TRANC = 'C', the leading
N-by-M part of this array must contain the transpose of
the matrix C.
On exit with TRANC = 'N', the leading M-by-N part of this
array contains the updated matrix C.
On exit with TRANC = 'T' or TRANC = 'C', the leading
N-by-M part of this array contains the transpose of the
updated matrix C.
LDC INTEGER
The leading dimension of the array C.
LDC >= MAX(1,M), if TRANC = 'N';
LDC >= MAX(1,N), if TRANC = 'T' or TRANC = 'C'.
D (input/output) DOUBLE PRECISION array, dimension
(LDD,N) if TRAND = 'N',
(LDD,M) if TRAND = 'T' or TRAND = 'C'.
On entry with TRAND = 'N', the leading M-by-N part of this
array must contain the matrix D.
On entry with TRAND = 'T' or TRAND = 'C', the leading
N-by-M part of this array must contain the transpose of
the matrix D.
On exit with TRAND = 'N', the leading M-by-N part of this
array contains the updated matrix D.
On exit with TRAND = 'T' or TRAND = 'C', the leading
N-by-M part of this array contains the transpose of the
updated matrix D.
LDD INTEGER
The leading dimension of the array D.
LDD >= MAX(1,M), if TRAND = 'N';
LDD >= MAX(1,N), if TRAND = 'T' or TRAND = 'C'.
CS (input) DOUBLE PRECISION array, dimension (2*N-2)
On entry, the first 2*N-2 elements of this array must
contain the cosines and sines of the symplectic Givens
rotations G(i), as returned by MB04PU or MB04RU.
TAU (input) DOUBLE PRECISION array, dimension (N-1)
On entry, the first N-1 elements of this array must
contain the scalar factors of the elementary reflectors
F(i), as returned by MB04PU or MB04RU.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
On exit, if INFO = -18, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,N).
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To overwrite general real m-by-n matrices C and D, or their
transposes, with
[ op(C) ]
Q * [ ] if TRANQ = 'N', or
[ op(D) ]
T [ op(C) ]
Q * [ ] if TRANQ = 'T',
[ op(D) ]
where Q is defined as the product of symplectic reflectors and
Givens rotations,
Q = diag( H(1),H(1) ) G(1) diag( F(1),F(1) )
diag( H(2),H(2) ) G(2) diag( F(2),F(2) )
....
diag( H(k),H(k) ) G(k) diag( F(k),F(k) ).
Unblocked version.
Specification
SUBROUTINE MB04QU( TRANC, TRAND, TRANQ, STOREV, STOREW, M, N, K,
$ V, LDV, W, LDW, C, LDC, D, LDD, CS, TAU, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER STOREV, STOREW, TRANC, TRAND, TRANQ
INTEGER INFO, K, LDC, LDD, LDV, LDW, LDWORK, M, N
C .. Array Arguments ..
DOUBLE PRECISION CS(*), DWORK(*), C(LDC,*), D(LDD,*), V(LDV,*),
$ W(LDW,*), TAU(*)
Arguments
Mode Parameters
TRANC CHARACTER*1
Specifies the form of op( C ) as follows:
= 'N': op( C ) = C;
= 'T': op( C ) = C';
= 'C': op( C ) = C'.
STOREV CHARACTER*1
Specifies how the vectors which define the concatenated
Householder reflectors contained in V are stored:
= 'C': columnwise;
= 'R': rowwise.
STOREW CHARACTER*1
Specifies how the vectors which define the concatenated
Householder reflectors contained in W are stored:
= 'C': columnwise;
= 'R': rowwise.
TRAND CHARACTER*1
Specifies the form of op( D ) as follows:
= 'N': op( D ) = D;
= 'T': op( D ) = D';
= 'C': op( D ) = D'.
TRANQ CHARACTER*1
= 'N': apply Q;
= 'T': apply Q'.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrices op(C) and op(D).
M >= 0.
N (input) INTEGER
The number of columns of the matrices op(C) and op(D).
N >= 0.
K (input) INTEGER
The number of elementary reflectors whose product defines
the matrix Q. M >= K >= 0.
V (input) DOUBLE PRECISION array, dimension
(LDV,K) if STOREV = 'C',
(LDV,M) if STOREV = 'R'
On entry with STOREV = 'C', the leading M-by-K part of
this array must contain in its columns the vectors which
define the elementary reflectors F(i).
On entry with STOREV = 'R', the leading K-by-M part of
this array must contain in its rows the vectors which
define the elementary reflectors F(i).
LDV INTEGER
The leading dimension of the array V.
LDV >= MAX(1,M), if STOREV = 'C';
LDV >= MAX(1,K), if STOREV = 'R'.
W (input) DOUBLE PRECISION array, dimension
(LDW,K) if STOREW = 'C',
(LDW,M) if STOREW = 'R'
On entry with STOREW = 'C', the leading M-by-K part of
this array must contain in its columns the vectors which
define the elementary reflectors H(i).
On entry with STOREW = 'R', the leading K-by-M part of
this array must contain in its rows the vectors which
define the elementary reflectors H(i).
LDW INTEGER
The leading dimension of the array W.
LDW >= MAX(1,M), if STOREW = 'C';
LDW >= MAX(1,K), if STOREW = 'R'.
C (input/output) DOUBLE PRECISION array, dimension
(LDC,N) if TRANC = 'N',
(LDC,M) if TRANC = 'T' or TRANC = 'C'
On entry with TRANC = 'N', the leading M-by-N part of
this array must contain the matrix C.
On entry with TRANC = 'C' or TRANC = 'T', the leading
N-by-M part of this array must contain the transpose of
the matrix C.
On exit with TRANC = 'N', the leading M-by-N part of
this array contains the updated matrix C.
On exit with TRANC = 'C' or TRANC = 'T', the leading
N-by-M part of this array contains the transpose of the
updated matrix C.
LDC INTEGER
The leading dimension of the array C.
LDC >= MAX(1,M), if TRANC = 'N';
LDC >= MAX(1,N), if TRANC = 'T' or TRANC = 'C'.
D (input/output) DOUBLE PRECISION array, dimension
(LDD,N) if TRAND = 'N',
(LDD,M) if TRAND = 'T' or TRAND = 'C'
On entry with TRAND = 'N', the leading M-by-N part of
this array must contain the matrix D.
On entry with TRAND = 'C' or TRAND = 'T', the leading
N-by-M part of this array must contain the transpose of
the matrix D.
On exit with TRAND = 'N', the leading M-by-N part of
this array contains the updated matrix D.
On exit with TRAND = 'C' or TRAND = 'T', the leading
N-by-M part of this array contains the transpose of the
updated matrix D.
LDD INTEGER
The leading dimension of the array D.
LDD >= MAX(1,M), if TRAND = 'N';
LDD >= MAX(1,N), if TRAND = 'T' or TRAND = 'C'.
CS (input) DOUBLE PRECISION array, dimension (2*K)
On entry, the first 2*K elements of this array must
contain the cosines and sines of the symplectic Givens
rotations G(i).
TAU (input) DOUBLE PRECISION array, dimension (K)
On entry, the first K elements of this array must
contain the scalar factors of the elementary reflectors
F(i).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -20, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,N).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To reduce a skew-Hamiltonian matrix,
[ A G ]
W = [ T ] ,
[ Q A ]
where A is an N-by-N matrix and G, Q are N-by-N skew-symmetric
matrices, to Paige/Van Loan (PVL) form. That is, an orthogonal
symplectic matrix U is computed so that
T [ Aout Gout ]
U W U = [ T ] ,
[ 0 Aout ]
where Aout is in upper Hessenberg form.
Blocked version.
Specification
SUBROUTINE MB04RB( N, ILO, A, LDA, QG, LDQG, CS, TAU, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER ILO, INFO, LDA, LDQG, LDWORK, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), CS(*), DWORK(*), QG(LDQG,*), TAU(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
ILO (input) INTEGER
It is assumed that A is already upper triangular and Q is
zero in rows and columns 1:ILO-1. ILO is normally set by a
previous call to the SLICOT Library routine MB04DS;
otherwise it should be set to 1.
1 <= ILO <= N+1, if N > 0; ILO = 1, if N = 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A.
On exit, the leading N-by-N part of this array contains
the matrix Aout and, in the zero part of Aout,
information about the elementary reflectors used to
compute the PVL factorization.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
QG (input/output) DOUBLE PRECISION array, dimension
(LDQG,N+1)
On entry, the leading N-by-N+1 part of this array must
contain in columns 1:N the strictly lower triangular part
of the matrix Q and in columns 2:N+1 the strictly upper
triangular part of the matrix G. The parts containing the
diagonal and the first superdiagonal of this array are not
referenced.
On exit, the leading N-by-N+1 part of this array contains
in its first N-1 columns information about the elementary
reflectors used to compute the PVL factorization and in
its last N columns the strictly upper triangular part of
the matrix Gout.
LDQG INTEGER
The leading dimension of the array QG. LDQG >= MAX(1,N).
CS (output) DOUBLE PRECISION array, dimension (2N-2)
On exit, the first 2N-2 elements of this array contain the
cosines and sines of the symplectic Givens rotations used
to compute the PVL factorization.
TAU (output) DOUBLE PRECISION array, dimension (N-1)
On exit, the first N-1 elements of this array contain the
scalar factors of some of the elementary reflectors.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK, 8*N*NB + 3*NB, where NB is the optimal
block size.
On exit, if INFO = -10, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,N-1).
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
An algorithm similar to the block algorithm for the symplectic
URV factorization described in [2] is used.
The matrix U is represented as a product of symplectic reflectors
and Givens rotations
U = diag( H(1),H(1) ) G(1) diag( F(1),F(1) )
diag( H(2),H(2) ) G(2) diag( F(2),F(2) )
....
diag( H(n-1),H(n-1) ) G(n-1) diag( F(n-1),F(n-1) ).
Each H(i) has the form
H(i) = I - tau * v * v'
where tau is a real scalar, and v is a real vector with v(1:i) = 0
and v(i+1) = 1; v(i+2:n) is stored on exit in QG(i+2:n,i), and
tau in QG(i+1,i).
Each F(i) has the form
F(i) = I - nu * w * w'
where nu is a real scalar, and w is a real vector with w(1:i) = 0
and w(i+1) = 1; w(i+2:n) is stored on exit in A(i+2:n,i), and
nu in TAU(i).
Each G(i) is a Givens rotation acting on rows i+1 and n+i+1, where
the cosine is stored in CS(2*i-1) and the sine in CS(2*i).
Numerical Aspects
The algorithm requires O(N**3) floating point operations and is strongly backward stable.References
[1] Van Loan, C.F.
A symplectic method for approximating all the eigenvalues of
a Hamiltonian matrix.
Linear Algebra and its Applications, 61, pp. 233-251, 1984.
[2] Kressner, D.
Block algorithms for orthogonal symplectic factorizations.
BIT, 43(4), pp. 775-790, 2003.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To reduce a real matrix pair (A,B) in generalized real Schur form to a block-diagonal form using well-conditioned non-orthogonal equivalence transformations. The condition numbers of the left and right transformations used for the reduction are roughly bounded by PMAX, where PMAX is a given value. The transformations are optionally postmultiplied in the given matrices X and Y. The generalized Schur form is optionally ordered, so that clustered eigenvalues are grouped in the same pair of blocks.Specification
SUBROUTINE MB04RD( JOBX, JOBY, SORT, N, PMAX, A, LDA, B, LDB, X,
$ LDX, Y, LDY, NBLCKS, BLSIZE, ALPHAR, ALPHAI,
$ BETA, TOL, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBX, JOBY, SORT
INTEGER INFO, LDA, LDB, LDWORK, LDX, LDY, N, NBLCKS
DOUBLE PRECISION PMAX, TOL
C .. Array Arguments ..
INTEGER BLSIZE(*), IWORK(*)
DOUBLE PRECISION A(LDA,*), ALPHAI(*), ALPHAR(*), B(LDB,*),
$ BETA(*), DWORK(*), X(LDX,*), Y(LDY,*)
Arguments
Mode Parameters
JOBX CHARACTER*1
Specifies whether or not the left transformations are
accumulated, as follows:
= 'N': The left transformations are not accumulated;
= 'U': The left transformations are accumulated in X
(the given matrix X is updated).
JOBY CHARACTER*1
Specifies whether or not the right transformations are
accumulated, as follows:
= 'N': The right transformations are not accumulated;
= 'U': The right transformations are accumulated in Y
(the given matrix Y is updated).
SORT CHARACTER*1
Specifies whether or not the diagonal blocks of the
generalized real Schur form are reordered, as follows:
= 'N': The diagonal blocks are not reordered;
= 'S': The diagonal blocks are reordered before each
step of reduction, so that clustered eigenvalues
appear in the same pair of blocks.
= 'C': The diagonal blocks are not reordered, but the
"closest-neighbour" strategy is used instead of
the standard "closest to the mean" strategy (see
METHOD);
= 'B': The diagonal blocks are reordered before each
step of reduction, and the "closest-neighbour"
strategy is used (see METHOD).
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, B, X and Y. N >= 0.
PMAX (input) DOUBLE PRECISION
An upper bound for the absolute value of the elements of
the individual transformations used for reduction
(see METHOD and FURTHER COMMENTS). PMAX >= 1.0D0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N upper quasi-triangular part
of this array must contain the upper quasi-triangular
matrix A in the generalized real Schur form, as returned
by the LAPACK Library routine DGGES. The lower triangular
part below the Schur matrix is used as workspace.
On exit, the leading N-by-N upper quasi-triangular part of
this array contains the computed block-diagonal matrix, in
real Schur canonical form, corresponding to the given
matrix A. The remaining part is set to zero.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix B in the
generalized real Schur form, as returned by the LAPACK
Library routine DGGES. The diagonal elements of B are
non-negative. The strictly lower triangular part is used
as workspace. The matrix B is assumed nonzero.
On exit, the leading N-by-N upper triangular part of this
array contains the computed upper triangular block-
diagonal matrix, corresponding to the given matrix B. The
remaining part is set to zero. The diagonal elements of B
are non-negative.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
X (input/output) DOUBLE PRECISION array, dimension (LDX,*)
On entry, if JOBX = 'U', the leading N-by-N part of this
array must contain a given matrix X, for instance the left
transformation matrix VSL returned by the LAPACK Library
routine DGGES.
On exit, if JOBX = 'U', the leading N-by-N part of this
array contains the product of the given matrix X and the
left transformation matrix that reduced (A,B) to block-
diagonal form. The local transformation matrix is itself a
product of non-orthogonal equivalence transformations
having elements with magnitude less than or equal to PMAX.
If JOBX = 'N', this array is not referenced.
LDX INTEGER
The leading dimension of the array X.
LDX >= 1, if JOBX = 'N';
LDX >= MAX(1,N), if JOBX = 'U'.
Y (input/output) DOUBLE PRECISION array, dimension (LDY,*)
On entry, if JOBY = 'U', the leading N-by-N part of this
array must contain a given matrix Y, for instance the
right transformation matrix VSR returned by the LAPACK
Library routine DGGES.
On exit, if JOBY = 'U', the leading N-by-N part of this
array contains the product of the given matrix Y and the
right transformation matrix that reduced (A,B) to block-
diagonal form. The local transformation matrix is itself a
product of non-orthogonal equivalence transformations
having elements with magnitude less than or equal to PMAX.
If JOBY = 'N', this array is not referenced.
LDY INTEGER
The leading dimension of the array Y.
LDY >= 1, if JOBY = 'N';
LDY >= MAX(1,N), if JOBY = 'U'.
NBLCKS (output) INTEGER
The number of diagonal blocks of the matrices A and B.
BLSIZE (output) INTEGER array, dimension (N)
The first NBLCKS elements of this array contain the orders
of the resulting diagonal blocks of the matrices A and B.
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
BETA (output) DOUBLE PRECISION array, dimension (N)
On exit, if INFO = 0, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j),
j = 1, ..., N, will be the generalized eigenvalues.
ALPHAR(j) + ALPHAI(j)*i, and BETA(j), j = 1, ..., N, are
the diagonals of the complex Schur form (S,T) that would
result if the 2-by-2 diagonal blocks of the real Schur
form of (A,B) were further reduced to triangular form
using 2-by-2 complex unitary transformations.
If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
positive, then the j-th and (j+1)-st eigenvalues are a
complex conjugate pair, with ALPHAI(j+1) negative.
All BETA(j) are non-negative real numbers.
The quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may
easily over- or underflow, and BETA(j) may even be zero.
Thus, the user should avoid naively computing the ratio.
If A and B are obtained from general matrices using DGGES,
ALPHA will be always less than and usually comparable with
norm(A) in magnitude, and BETA always less than and
usually comparable with norm(B).
Tolerances
TOL DOUBLE PRECISION
If SORT = 'S' or 'B', the tolerance to be used in the
ordering of the diagonal blocks of the upper triangular
matrix pair.
If the user sets TOL > 0, then the given value of TOL is
used as an absolute tolerance: a pair of blocks i and a
temporarily fixed pair of blocks 1 (the first pair of
blocks of the current trailing pair of submatrices to be
reduced) are considered to belong to the same cluster if
their eigenvalues satisfy the following "distance"
condition
| lambda_1 - lambda_i | <= TOL.
If the user sets TOL < 0, then the given value of TOL is
used as a relative tolerance: a pair of blocks i and a
temporarily fixed pair of blocks 1 are considered to
belong to the same cluster if their eigenvalues satisfy,
for finite lambda_j, j = 1, ..., N,
| lambda_1 - lambda_i | <= | TOL | * max | lambda_j |.
If the user sets TOL = 0, then an implicitly computed,
default tolerance, defined by TOL = SQRT( SQRT( EPS ) )
is used instead, as a relative tolerance, where EPS is
the machine precision (see LAPACK Library routine DLAMCH).
The approximate symmetric chordal metric is used as
"distance" of two complex, possibly infinite numbers, x
and y. This metric is given by the formula
d(x,y) = min( |x-y|, |1/x-1/y| ),
taking into account the special cases of infinite or NaN
values.
If SORT = 'N' or 'C', this parameter is not referenced.
Workspace
IWORK INTEGER array, dimension (N+6)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal value
of LDWORK.
On exit, if INFO = -21, DWORK(1) returns the minimum
value of LDWORK. When LDWORK = 0 is set on entry, the
routine will return this value for INFO, and also set
DWORK(1), but no error message related to LDWORK is
issued by XERBLA.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= 1, if N <= 1;
LDWORK >= 4*N + 16, if N > 1.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the matrix pencil defined by A and B is singular.
Method
Consider first that SORT = 'N'. Let
( A A ) ( B B )
( 11 12 ) ( 11 12 )
A = ( ), B = ( ),
( 0 A ) ( 0 B )
( 22 ) ( 22 )
be the given matrix pair in generalized real Schur form, where
initially A and B are the first pair of diagonal blocks of
11 11
dimension 1-by-1 or 2-by-2. An attempt is made to compute the
transformation matrices X and Y of the form
( I V ) ( I W )
X = ( ), Y = ( ) (1)
( 0 I ) ( 0 I )
(partitioned as A and B ), so that (' denotes the transpose)
( A 0 ) ( B 0 )
( 11 ) ( 11 )
X' A Y = ( ), X' B Y = ( ),
( 0 A ) ( 0 B )
( 22 ) ( 22 )
and the elements of V and W do not exceed the value PMAX in
magnitude. An adaptation of the standard method for solving
generalized Sylvester equations [1], which controls the magnitude
of the individual elements of the computed solution [2], is used
to obtain V and W. When this attempt fails, a 1-by-1 (or 2-by-2)
pair of diagonal blocks of (A , B ), whose eigenvalue(s) is
22 22
(are) the closest to the mean of those of (A , B ) is selected,
11 11
and moved by orthogonal equivalence transformations in the leading
position of (A , B ); the moved diagonal blocks in A and B are
22 22
then added to the blocks A and B , respectively, increasing
11 11
their order by 1 (or 2). Another attempt is made to compute
suitable transformation matrices X and Y with the new definitions
of the blocks A , A , B , and B . After successful
11 22 11 22
transformation matrices X and Y have been obtained, they
postmultiply the current transformation matrices (if JOBX = 'U'
and/or JOBY = 'U') and the whole procedure is repeated for the new
blocks A and B .
22 22
When SORT = 'S', the diagonal blocks of the generalized real Schur
form are reordered before each step of the reduction, so that each
cluster of generalized eigenvalues, defined as specified in the
definition of TOL, appears in adjacent blocks. The blocks for
each cluster are merged together, and the procedure described
above is applied to the larger blocks. Using the option SORT = 'S'
will usually provide better efficiency than the standard option
(SORT = 'N'), proposed in [2], because there could be no or few
unsuccessful attempts to compute individual transformation
matrices X and Y of the form (1). However, the resulting
dimensions of the blocks are usually larger; this could make
subsequent calculations less efficient.
When SORT = 'C' or 'B', the procedure is similar to that for
SORT = 'N' or 'S', respectively, but the blocks of A and B
22 22
whose eigenvalue(s) is (are) the closest to those of (A , B )
11 11
(not to their mean) are selected and moved to the leading position
of A and B . This is called the "closest-neighbour" strategy.
22 22
References
[1] Kagstrom, B. and Westin, L.
Generalized Schur Methods with Condition Estimators for
Solving the Generalized Sylvester Equation.
IEEE Trans. Auto. Contr., 34, pp. 745-751, 1989.
[2] Bavely, C. and Stewart, G.W.
An Algorithm for Computing Reducing Subspaces by Block
Diagonalization.
SIAM J. Numer. Anal., 16, pp. 359-367, 1979.
[3] Demmel, J.
The Condition Number of Equivalence Transformations that
Block Diagonalize Matrix Pencils.
SIAM J. Numer. Anal., 20, pp. 599-610, 1983.
Numerical Aspects
3 4 The algorithm usually requires 0(N ) operations, but 0(N ) are possible in the worst case, when the matrix pencil cannot be diagonalized by well-conditioned transformations.Further Comments
The individual non-orthogonal transformation matrices used in the reduction of A and B to a block-diagonal form have condition numbers of the order PMAX. This does not guarantee that their product is well-conditioned enough. The routine can be easily modified to provide estimates for the condition numbers of the clusters of generalized eigenvalues.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04RS.html 0000664 0000000 0000000 00000015733 14560147231 0020372 0 ustar 00root root 0000000 0000000
Purpose
To solve the generalized real Sylvester equation
A * R - L * B = scale * C, (1)
D * R - L * E = scale * F,
using Level 1 and 2 BLAS, where R and L are unknown real M-by-N
matrices, and (A, D), (B, E) and (C, F) are given matrix pairs of
size M-by-M, N-by-N and M-by-N, respectively. (A,D) and (B,E) must
be in generalized Schur canonical form, i.e., A, B are upper
quasi-triangular and D, E are upper triangular.
The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an
output scaling factor chosen to avoid overflow.
This routine is intended to be called only by SLICOT Library
routine MB04RT. For efficiency purposes, the computations are
aborted when the absolute value of an element of R or L is greater
than a given value PMAX.
Specification
SUBROUTINE MB04RS( M, N, PMAX, A, LDA, B, LDB, C, LDC, D, LDD, E,
$ LDE, F, LDF, SCALE, IWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N
DOUBLE PRECISION PMAX, SCALE
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), E( LDE, * ), F( LDF, * )
Arguments
Input/Output Parameters
M (input) INTEGER
The order of the matrices A and D, and the row dimension
of C, F, R and L. M >= 0.
N (input) INTEGER
The order of the matrices B and E, and the column
dimension of C, F, R and L. N >= 0.
PMAX (input) DOUBLE PRECISION
An upper bound for the absolute value of the elements of
the solution (R, L). PMAX >= 1.0D0.
A (input) DOUBLE PRECISION array, dimension (LDA, M)
On entry, the leading M-by-M upper quasi-triangular part
of this array must contain the matrix A in the generalized
real Schur form, as returned by LAPACK routine DGGES.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1, M).
B (input) DOUBLE PRECISION array, dimension (LDB, N)
On entry, the leading N-by-N upper quasi-triangular part
of this array must contain the matrix B in the generalized
real Schur form.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1, N).
C (input/output) DOUBLE PRECISION array, dimension (LDC, N)
On entry, the leading M-by-N part of this array must
contain the right-hand-side of the first matrix equation
in (1).
On exit, if INFO = 0, the leading M-by-N part of this
array contains the solution R.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1, M).
D (input) DOUBLE PRECISION array, dimension (LDD, M)
On entry, the leading M-by-M upper triangular part of this
array must contain the matrix D in the generalized real
Schur form. The diagonal elements are non-negative.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1, M).
E (input) DOUBLE PRECISION array, dimension (LDE, N)
On entry, the leading N-by-N upper triangular part of this
array must contain the matrix E in the generalized real
Schur form. The diagonal elements are non-negative.
LDE INTEGER
The leading dimension of the array E. LDE >= max(1, N).
F (input/output) DOUBLE PRECISION array, dimension (LDF, N)
On entry, the leading M-by-N part of this array must
contain the right-hand-side of the second matrix equation
in (1).
On exit, if INFO = 0, the leading M-by-N part of this
array contains the solution L.
LDF INTEGER
The leading dimension of the array F. LDF >= max(1, M).
SCALE (output) DOUBLE PRECISION
On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions
R and L (C and F on entry) will hold the solutions to a
slightly perturbed system but the input matrices A, B, D
and E have not been changed. If SCALE = 0, R and L will
hold the solutions to the homogeneous system with C = 0
and F = 0. Normally, SCALE = 1.
Workspace
IWORK INTEGER array, dimension (M+N+2)Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: an element of R or L had the absolute value greater
than the given value PMAX.
= 2: the matrix pairs (A, D) and (B, E) have common or
very close eigenvalues. The matrix Z in section
METHOD is (almost) singular.
Method
The routine uses an adaptation of the method for solving
generalized Sylvester equations [1], which controls the magnitude
of the individual elements of the computed solution [2].
In matrix notation, solving equation (1) corresponds to solve
Zx = scale * b, where Z is defined as
Z = [ kron(In, A) -kron(B', Im) ] (2)
[ kron(In, D) -kron(E', Im) ],
Ik is the identity matrix of size k and X' is the transpose of X.
kron(X, Y) is the Kronecker product between the matrices X and Y.
The routine solves a number of systems (2) with n and m at most 2.
References
[1] Kagstrom, B. and Westin, L.
Generalized Schur Methods with Condition Estimators for
Solving the Generalized Sylvester Equation.
IEEE Trans. Auto. Contr., 34, pp. 745-751, 1989.
[2] Kagstrom, B. and Westin, L.
GSYLV - Fortran Routines for the Generalized Schur Method with
Dif Estimators for Solving the Generalized Sylvester Equation.
Report UMINF-132.86, Institute of Information Processing,
Univ. of Umea, Sweden, July 1987.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve the generalized real Sylvester equation
A * R - L * B = scale * C, (1)
D * R - L * E = scale * F,
using Level 3 BLAS, where R and L are unknown M-by-N matrices, and
(A, D), (B, E) and (C, F) are given real matrix pairs of size
M-by-M, N-by-N and M-by-N, respectively. (A,D) and (B,E) must be
in generalized Schur canonical form, i.e., A, B are upper quasi-
triangular and D, E are upper triangular.
The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an
output scaling factor chosen to avoid overflow.
This routine is intended to be called only by SLICOT Library
routine MB04RD. For efficiency purposes, the computations are
aborted when the absolute value of an element of R or L is greater
than a given value PMAX.
Specification
SUBROUTINE MB04RT( M, N, PMAX, A, LDA, B, LDB, C, LDC, D, LDD, E,
$ LDE, F, LDF, SCALE, IWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N
DOUBLE PRECISION PMAX, SCALE
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), E( LDE, * ), F( LDF, * )
Arguments
Input/Output Parameters
M (input) INTEGER
The order of the matrices A and D, and the row dimension
of C, F, R and L. M >= 0.
N (input) INTEGER
The order of the matrices B and E, and the column
dimension of C, F, R and L. N >= 0.
PMAX (input) DOUBLE PRECISION
An upper bound for the absolute value of the elements of
the solution (R, L). PMAX >= 1.0D0.
A (input) DOUBLE PRECISION array, dimension (LDA, M)
On entry, the leading M-by-M upper quasi-triangular part
of this array must contain the matrix A in the generalized
real Schur form, as returned by LAPACK routine DGGES.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1, M).
B (input) DOUBLE PRECISION array, dimension (LDB, N)
On entry, the leading N-by-N upper quasi-triangular part
of this array must contain the matrix B in the generalized
real Schur form.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1, N).
C (input/output) DOUBLE PRECISION array, dimension (LDC, N)
On entry, the leading M-by-N part of this array must
contain the right-hand-side of the first matrix equation
in (1).
On exit, if INFO = 0, the leading M-by-N part of this
array contains the solution R.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1, M).
D (input) DOUBLE PRECISION array, dimension (LDD, M)
On entry, the leading M-by-M upper triangular part of this
array must contain the matrix D in the generalized real
Schur form. The diagonal elements are non-negative.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1, M).
E (input) DOUBLE PRECISION array, dimension (LDE, N)
On entry, the leading N-by-N upper triangular part of this
array must contain the matrix E in the generalized real
Schur form. The diagonal elements are non-negative.
LDE INTEGER
The leading dimension of the array E. LDE >= max(1, N).
F (input/output) DOUBLE PRECISION array, dimension (LDF, N)
On entry, the leading M-by-N part of this array must
contain the right-hand-side of the second matrix equation
in (1).
On exit, if INFO = 0, the leading M-by-N part of this
array contains the solution L.
LDF INTEGER
The leading dimension of the array F. LDF >= max(1, M).
SCALE (output) DOUBLE PRECISION
On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions
R and L (C and F on entry) will hold the solutions to a
slightly perturbed system but the input matrices A, B, D
and E have not been changed. If SCALE = 0, R and L will
hold the solutions to the homogeneous system with C = 0
and F = 0. Normally, SCALE = 1.
Workspace
IWORK INTEGER array, dimension (M+N+6)Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: an element of R or L had the absolute value greater
than the given value PMAX.
= 2: the matrix pairs (A, D) and (B, E) have common or
very close eigenvalues. The matrix Z in section
METHOD is (almost) singular.
Method
The routine uses an adaptation of the method for solving
generalized Sylvester equations [1], which controls the magnitude
of the individual elements of the computed solution [2].
In matrix notation, solving equation (1) corresponds to solve
Zx = scale * b, where Z is defined as
Z = [ kron(In, A) -kron(B', Im) ] (2)
[ kron(In, D) -kron(E', Im) ],
Ik is the identity matrix of size k and X' is the transpose of X.
kron(X, Y) is the Kronecker product between the matrices X and Y.
References
[1] Kagstrom, B. and Westin, L.
Generalized Schur Methods with Condition Estimators for
Solving the Generalized Sylvester Equation.
IEEE Trans. Auto. Contr., 34, pp. 745-751, 1989.
[2] Kagstrom, B. and Westin, L.
GSYLV - Fortran Routines for the Generalized Schur Method with
Dif Estimators for Solving the Generalized Sylvester Equation.
Report UMINF-132.86, Institute of Information Processing,
Univ. of Umea, Sweden, July 1987.
[3] Kagstrom, B.
A Perturbation Analysis of the Generalized Sylvester Equation
(AR - LB, DR - LE ) = (C, F).
SIAM J. Matrix Anal. Appl., 15(4), pp. 1045-1060, 1994.
[4] Kagstrom, B. and Poromaa, P.
LAPACK-Style Algorithms and Software for Solving the
Generalized Sylvester Equation and Estimating the Separation
between Regular Matrix Pairs.
ACM Trans. on Math. Software, 22(1), pp. 78103, 1996.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04RU.html 0000664 0000000 0000000 00000013502 14560147231 0020364 0 ustar 00root root 0000000 0000000
Purpose
To reduce a skew-Hamiltonian matrix,
[ A G ]
W = [ T ] ,
[ Q A ]
where A is an N-by-N matrix and G, Q are N-by-N skew-symmetric
matrices, to Paige/Van Loan (PVL) form. That is, an orthogonal
symplectic matrix U is computed so that
T [ Aout Gout ]
U W U = [ T ] ,
[ 0 Aout ]
where Aout is in upper Hessenberg form.
Unblocked version.
Specification
SUBROUTINE MB04RU( N, ILO, A, LDA, QG, LDQG, CS, TAU, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER ILO, INFO, LDA, LDQG, LDWORK, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), CS(*), DWORK(*), QG(LDQG,*), TAU(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
ILO (input) INTEGER
It is assumed that A is already upper triangular and Q is
zero in rows and columns 1:ILO-1. ILO is normally set by a
previous call to the SLICOT Library routine MB04DS;
otherwise it should be set to 1.
1 <= ILO <= N+1, if N > 0; ILO = 1, if N = 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A.
On exit, the leading N-by-N part of this array contains
the matrix Aout and, in the zero part of Aout, information
about the elementary reflectors used to compute the
PVL factorization.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
QG (input/output) DOUBLE PRECISION array, dimension
(LDQG,N+1)
On entry, the leading N-by-N+1 part of this array must
contain in columns 1:N the strictly lower triangular part
of the matrix Q and in columns 2:N+1 the strictly upper
triangular part of the matrix G. The parts containing the
diagonal and the first superdiagonal of this array are not
referenced.
On exit, the leading N-by-N+1 part of this array contains
in its first N-1 columns information about the elementary
reflectors used to compute the PVL factorization and in
its last N columns the strictly upper triangular part of
the matrix Gout.
LDQG INTEGER
The leading dimension of the array QG. LDQG >= MAX(1,N).
CS (output) DOUBLE PRECISION array, dimension (2N-2)
On exit, the first 2N-2 elements of this array contain the
cosines and sines of the symplectic Givens rotations used
to compute the PVL factorization.
TAU (output) DOUBLE PRECISION array, dimension (N-1)
On exit, the first N-1 elements of this array contain the
scalar factors of some of the elementary reflectors.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
On exit, if INFO = -10, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,N-1).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The matrix U is represented as a product of symplectic reflectors
and Givens rotations
U = diag( H(1),H(1) ) G(1) diag( F(1),F(1) )
diag( H(2),H(2) ) G(2) diag( F(2),F(2) )
....
diag( H(n-1),H(n-1) ) G(n-1) diag( F(n-1),F(n-1) ).
Each H(i) has the form
H(i) = I - tau * v * v'
where tau is a real scalar, and v is a real vector with v(1:i) = 0
and v(i+1) = 1; v(i+2:n) is stored on exit in QG(i+2:n,i), and
tau in QG(i+1,i).
Each F(i) has the form
F(i) = I - nu * w * w'
where nu is a real scalar, and w is a real vector with w(1:i) = 0
and w(i+1) = 1; w(i+2:n) is stored on exit in A(i+2:n,i), and
nu in TAU(i).
Each G(i) is a Givens rotation acting on rows i+1 and n+i+1, where
the cosine is stored in CS(2*i-1) and the sine in CS(2*i).
Numerical Aspects
The algorithm requires 40/3 N**3 + O(N) floating point operations and is strongly backward stable.References
[1] Van Loan, C.F.
A symplectic method for approximating all the eigenvalues of
a Hamiltonian matrix.
Linear Algebra and its Applications, 61, pp. 233-251, 1984.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve the generalized complex Sylvester equation
A * R - L * B = scale * C, (1)
D * R - L * E = scale * F,
using Level 1 and 2 BLAS, where R and L are unknown M-by-N
matrices, and (A, D), (B, E) and (C, F) are given matrix pairs of
size M-by-M, N-by-N and M-by-N, respectively. A, B, D and E are
complex upper triangular (i.e., (A,D) and (B,E) are in generalized
Schur form).
The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an
output scaling factor chosen to avoid overflow.
This routine is intended to be called only by SLICOT Library
routine MB04RW. For efficiency purposes, the computations are
aborted when the absolute value of an element of R or L is greater
than a given value PMAX.
Specification
SUBROUTINE MB04RV( M, N, PMAX, A, LDA, B, LDB, C, LDC, D, LDD, E,
$ LDE, F, LDF, SCALE, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N
DOUBLE PRECISION PMAX, SCALE
C .. Array Arguments ..
COMPLEX*16 A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), E( LDE, * ), F( LDF, * )
Arguments
Input/Output Parameters
M (input) INTEGER
The order of the matrices A and D, and the row dimension
of C, F, R and L. M >= 0.
N (input) INTEGER
The order of the matrices B and E, and the column
dimension of C, F, R and L. N >= 0.
PMAX (input) DOUBLE PRECISION
An upper bound for the "absolute value" of the elements of
the solution (R, L). (See FURTHER COMMENTS.)
PMAX >= 1.0D0.
A (input) COMPLEX*16 array, dimension (LDA, M)
On entry, the leading M-by-M upper triangular part of this
array must contain the matrix A in the generalized complex
Schur form, as returned by LAPACK routine ZGGES.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1, M).
B (input) COMPLEX*16 array, dimension (LDB, N)
On entry, the leading N-by-N upper triangular part of this
array must contain the matrix B in the generalized complex
Schur form.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1, N).
C (input/output) COMPLEX*16 array, dimension (LDC, N)
On entry, the leading M-by-N part of this array must
contain the right-hand-side of the first matrix equation
in (1).
On exit, if INFO = 0, the leading M-by-N part of this
array contains the solution R.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1, M).
D (input) COMPLEX*16 array, dimension (LDD, M)
On entry, the leading M-by-M upper triangular part of this
array must contain the matrix D in the generalized complex
Schur form. The diagonal elements are non-negative real.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1, M).
E (input) COMPLEX*16 array, dimension (LDE, N)
On entry, the leading N-by-N upper triangular part of this
array must contain the matrix E in the generalized complex
Schur form. The diagonal elements are non-negative real.
LDE INTEGER
The leading dimension of the array E. LDE >= max(1, N).
F (input/output) COMPLEX*16 array, dimension (LDF, N)
On entry, the leading M-by-N part of this array must
contain the right-hand-side of the second matrix equation
in (1).
On exit, if INFO = 0, the leading M-by-N part of this
array contains the solution L.
LDF INTEGER
The leading dimension of the array F. LDF >= max(1, M).
SCALE (output) DOUBLE PRECISION
On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions
R and L (C and F on entry) will hold the solutions to a
slightly perturbed system but the input matrices A, B, D
and E have not been changed. If SCALE = 0, R and L will
hold the solutions to the homogeneous system with C = 0
and F = 0. Normally, SCALE = 1.
Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: an element of R or L had the absolute value greater
than the given value PMAX.
= 2: the matrix pairs (A, D) and (B, E) have common or
very close eigenvalues. The matrix Z in section
METHOD is (almost) singular.
Method
The routine uses an adaptation of the method for solving
generalized Sylvester equations [1], which controls the magnitude
of the individual elements of the computed solution [2].
In matrix notation, solving equation (1) corresponds to solve
Zx = scale * b, where Z is defined as
Z = [ kron(In, A) -kron(B', Im) ] (2)
[ kron(In, D) -kron(E', Im) ],
Ik is the identity matrix of size k and X' is the transpose of X.
kron(X, Y) is the Kronecker product between the matrices X and Y.
References
[1] Kagstrom, B. and Westin, L.
Generalized Schur Methods with Condition Estimators for
Solving the Generalized Sylvester Equation.
IEEE Trans. Auto. Contr., 34, pp. 745-751, 1989.
[2] Kagstrom, B. and Westin, L.
GSYLV - Fortran Routines for the Generalized Schur Method with
Dif Estimators for Solving the Generalized Sylvester Equation.
Report UMINF-132.86, Institute of Information Processing,
Univ. of Umea, Sweden, July 1987.
Further Comments
For efficiency reasons, the "absolute value" of a complex number x is computed as |real(x)| + |imag(x)|.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve the generalized complex Sylvester equation
A * R - L * B = scale * C, (1)
D * R - L * E = scale * F,
using Level 3 BLAS, where R and L are unknown M-by-N matrices, and
(A, D), (B, E) and (C, F) are given matrix pairs of size M-by-M,
N-by-N and M-by-N, respectively. A, B, D and E are complex upper
triangular (i.e., (A,D) and (B,E) are in generalized Schur form).
The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an
output scaling factor chosen to avoid overflow.
This routine is intended to be called only by SLICOT Library
routine MB04RZ. For efficiency purposes, the computations are
aborted when the absolute value of an element of R or L is greater
than a given value PMAX.
Specification
SUBROUTINE MB04RW( M, N, PMAX, A, LDA, B, LDB, C, LDC, D, LDD, E,
$ LDE, F, LDF, SCALE, IWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N
DOUBLE PRECISION PMAX, SCALE
C .. Array Arguments ..
INTEGER IWORK( * )
COMPLEX*16 A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), E( LDE, * ), F( LDF, * )
Arguments
Input/Output Parameters
M (input) INTEGER
The order of the matrices A and D, and the row dimension
of C, F, R and L. M >= 0.
N (input) INTEGER
The order of the matrices B and E, and the column
dimension of C, F, R and L. N >= 0.
PMAX (input) DOUBLE PRECISION
An upper bound for the "absolute value" of the elements of
the solution (R, L). (See FURTHER COMMENTS.)
PMAX >= 1.0D0.
A (input) COMPLEX*16 array, dimension (LDA, M)
On entry, the leading M-by-M upper triangular part of this
array must contain the matrix A in the generalized complex
Schur form, as returned by LAPACK routine ZGGES.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1, M).
B (input) COMPLEX*16 array, dimension (LDB, N)
On entry, the leading N-by-N upper triangular part of this
array must contain the matrix B in the generalized complex
Schur form.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1, N).
C (input/output) COMPLEX*16 array, dimension (LDC, N)
On entry, the leading M-by-N part of this array must
contain the right-hand-side of the first matrix equation
in (1).
On exit, if INFO = 0, the leading M-by-N part of this
array contains the solution R.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1, M).
D (input) COMPLEX*16 array, dimension (LDD, M)
On entry, the leading M-by-M upper triangular part of this
array must contain the matrix D in the generalized complex
Schur form. The diagonal elements are non-negative real.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1, M).
E (input) COMPLEX*16 array, dimension (LDE, N)
On entry, the leading N-by-N upper triangular part of this
array must contain the matrix E in the generalized complex
Schur form. The diagonal elements are non-negative real.
LDE INTEGER
The leading dimension of the array E. LDE >= max(1, N).
F (input/output) COMPLEX*16 array, dimension (LDF, N)
On entry, the leading M-by-N part of this array must
contain the right-hand-side of the second matrix equation
in (1).
On exit, if INFO = 0, the leading M-by-N part of this
array contains the solution L.
LDF INTEGER
The leading dimension of the array F. LDF >= max(1, M).
SCALE (output) DOUBLE PRECISION
On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions
R and L (C and F on entry) will hold the solutions to a
slightly perturbed system but the input matrices A, B, D
and E have not been changed. If SCALE = 0, R and L will
hold the solutions to the homogeneous system with C = 0
and F = 0. Normally, SCALE = 1.
Workspace
IWORK INTEGER array, dimension (M+N+2)Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: an element of R or L had the absolute value greater
than the given value PMAX.
= 2: the matrix pairs (A, D) and (B, E) have common or
very close eigenvalues. The matrix Z in section
METHOD is (almost) singular.
Method
The routine uses an adaptation of the method for solving
generalized Sylvester equations [1], which controls the magnitude
of the individual elements of the computed solution [2].
In matrix notation, solving equation (1) corresponds to solve
Zx = scale * b, where Z is defined as
Z = [ kron(In, A) -kron(B', Im) ] (2)
[ kron(In, D) -kron(E', Im) ],
Ik is the identity matrix of size k and X' is the transpose of X.
kron(X, Y) is the Kronecker product between the matrices X and Y.
References
[1] Kagstrom, B. and Westin, L.
Generalized Schur Methods with Condition Estimators for
Solving the Generalized Sylvester Equation.
IEEE Trans. Auto. Contr., 34, pp. 745-751, 1989.
[2] Kagstrom, B. and Westin, L.
GSYLV - Fortran Routines for the Generalized Schur Method with
Dif Estimators for Solving the Generalized Sylvester Equation.
Report UMINF-132.86, Institute of Information Processing,
Univ. of Umea, Sweden, July 1987.
[3] Kagstrom, B.
A Perturbation Analysis of the Generalized Sylvester Equation
(AR - LB, DR - LE ) = (C, F).
SIAM J. Matrix Anal. Appl., 15(4), pp. 1045-1060, 1994.
[4] Kagstrom, B. and Poromaa, P.
LAPACK-Style Algorithms and Software for Solving the
Generalized Sylvester Equation and Estimating the Separation
between Regular Matrix Pairs.
ACM Trans. on Math. Software, 22(1), pp. 78103, 1996.
Further Comments
For efficiency reasons, the "absolute value" of a complex number x is computed as |real(x)| + |imag(x)|.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04RZ.html 0000664 0000000 0000000 00000035604 14560147231 0020400 0 ustar 00root root 0000000 0000000
Purpose
To reduce a complex matrix pair (A,B) in generalized complex Schur form to a block-diagonal form using well-conditioned non-unitary equivalence transformations. The condition numbers of the left and right transformations used for the reduction are roughly bounded by PMAX, where PMAX is a given value. The transformations are optionally postmultiplied in the given matrices X and Y. The generalized Schur form is optionally ordered, so that clustered eigenvalues are grouped in the same pair of blocks.Specification
SUBROUTINE MB04RZ( JOBX, JOBY, SORT, N, PMAX, A, LDA, B, LDB, X,
$ LDX, Y, LDY, NBLCKS, BLSIZE, ALPHA, BETA,
$ TOL, IWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBX, JOBY, SORT
INTEGER INFO, LDA, LDB, LDX, LDY, N, NBLCKS
DOUBLE PRECISION PMAX, TOL
C .. Array Arguments ..
INTEGER BLSIZE(*), IWORK(*)
COMPLEX*16 A(LDA,*), ALPHA(*), B(LDB,*), BETA(*), X(LDX,*),
$ Y(LDY,*)
Arguments
Mode Parameters
JOBX CHARACTER*1
Specifies whether or not the left transformations are
accumulated, as follows:
= 'N': The left transformations are not accumulated;
= 'U': The left transformations are accumulated in X
(the given matrix X is updated).
JOBY CHARACTER*1
Specifies whether or not the right transformations are
accumulated, as follows:
= 'N': The right transformations are not accumulated;
= 'U': The right transformations are accumulated in Y
(the given matrix Y is updated).
SORT CHARACTER*1
Specifies whether or not the diagonal elements of the
generalized Schur form are reordered, as follows:
= 'N': The diagonal elements are not reordered;
= 'S': The diagonal elements are reordered before each
step of reduction, so that clustered eigenvalues
appear in the same pair of blocks.
= 'C': The diagonal elements are not reordered, but the
"closest-neighbour" strategy is used instead of
the standard "closest to the mean" strategy (see
METHOD);
= 'B': The diagonal elements are reordered before each
step of reduction, and the "closest-neighbour"
strategy is used (see METHOD).
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, B, X and Y. N >= 0.
PMAX (input) DOUBLE PRECISION
An upper bound for the "absolute value" of the elements of
the individual transformations used for reduction
(see METHOD and FURTHER COMMENTS). PMAX >= 1.0D0.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix A in the
generalized complex Schur form, as returned by the LAPACK
Library routine ZGGES. The strictly lower triangular part
is used as workspace.
On exit, the leading N-by-N upper triangular part of
this array contains the computed block-diagonal matrix,
corresponding to the given matrix A. The remaining part
is set to zero.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) COMPLEX*16 array, dimension (LDB,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix B in the
generalized complex Schur form, as returned by the LAPACK
Library routine ZGGES. The diagonal elements of B are real
non-negative, hence the imaginary parts of these elements
are zero. The strictly lower triangular part is used as
workspace. The matrix B is assumed nonzero.
On exit, the leading N-by-N upper triangular part of this
array contains the computed upper triangular block-
diagonal matrix, corresponding to the given matrix B. The
remaining part is set to zero. The diagonal elements of B
are real non-negative.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
X (input/output) COMPLEX*16 array, dimension (LDX,*)
On entry, if JOBX = 'U', the leading N-by-N part of this
array must contain a given matrix X, for instance the left
transformation matrix VSL returned by the LAPACK Library
routine ZGGES.
On exit, if JOBX = 'U', the leading N-by-N part of this
array contains the product of the given matrix X and the
left transformation matrix that reduced (A,B) to block-
diagonal form. The local transformation matrix is itself a
product of non-unitary equivalence transformations having
elements with magnitude less than or equal to PMAX.
If JOBX = 'N', this array is not referenced.
LDX INTEGER
The leading dimension of the array X.
LDX >= 1, if JOBX = 'N';
LDX >= MAX(1,N), if JOBX = 'U'.
Y (input/output) COMPLEX*16 array, dimension (LDY,*)
On entry, if JOBY = 'U', the leading N-by-N part of this
array must contain a given matrix Y, for instance the
right transformation matrix VSR returned by the LAPACK
Library routine ZGGES.
On exit, if JOBY = 'U', the leading N-by-N part of this
array contains the product of the given matrix Y and the
right transformation matrix that reduced (A,B) to block-
diagonal form. The local transformation matrix is itself a
product of non-unitary equivalence transformations having
elements with magnitude less than or equal to PMAX.
If JOBY = 'N', this array is not referenced.
LDY INTEGER
The leading dimension of the array Y.
LDY >= 1, if JOBY = 'N';
LDY >= MAX(1,N), if JOBY = 'U'.
NBLCKS (output) INTEGER
The number of diagonal blocks of the matrices A and B.
BLSIZE (output) INTEGER array, dimension (N)
The first NBLCKS elements of this array contain the orders
of the resulting diagonal blocks of the matrices A and B.
ALPHA (output) COMPLEX*16 array, dimension (N)
BETA (output) COMPLEX*16 array, dimension (N)
If INFO = 0, then ALPHA(j)/BETA(j), j = 1, ..., N, are
the generalized eigenvalues of the matrix pair (A, B)
(the diagonals of the complex Schur form).
All BETA(j) are non-negative real numbers.
The quotients ALPHA(j)/BETA(j) may easily over- or
underflow, and BETA(j) may even be zero. Thus, the user
should avoid naively computing the ratio.
If A and B are obtained from general matrices using ZGGES,
ALPHA will be always less than and usually comparable with
norm(A) in magnitude, and BETA always less than and
usually comparable with norm(B).
Tolerances
TOL DOUBLE PRECISION
If SORT = 'S' or 'B', the tolerance to be used in the
ordering of the diagonal elements of the upper triangular
matrix pair.
If the user sets TOL > 0, then the given value of TOL is
used as an absolute tolerance: an eigenvalue i and a
temporarily fixed eigenvalue 1 (represented by the first
element of the current trailing pair of submatrices to be
reduced) are considered to belong to the same cluster if
they satisfy the following "distance" condition
| lambda_1 - lambda_i | <= TOL.
If the user sets TOL < 0, then the given value of TOL is
used as a relative tolerance: an eigenvalue i and a
temporarily fixed eigenvalue 1 are considered to belong to
the same cluster if they satisfy, for finite lambda_j,
j = 1, ..., N,
| lambda_1 - lambda_i | <= | TOL | * max | lambda_j |.
If the user sets TOL = 0, then an implicitly computed,
default tolerance, defined by TOL = SQRT( SQRT( EPS ) )
is used instead, as a relative tolerance, where EPS is
the machine precision (see LAPACK Library routine DLAMCH).
The approximate symmetric chordal metric is used as
"distance" of two complex, possibly infinite numbers, x
and y. This metric is given by the formula
d(x,y) = min( |x-y|, |1/x-1/y| ),
taking into account the special cases of infinite or NaN
values.
If SORT = 'N' or 'C', this parameter is not referenced.
Workspace
IWORK INTEGER array, dimension (N+2)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the matrix pencil defined by A and B is singular.
Method
Consider first that SORT = 'N'. Let
( A A ) ( B B )
( 11 12 ) ( 11 12 )
A = ( ), B = ( ),
( 0 A ) ( 0 B )
( 22 ) ( 22 )
be the given matrix pair in generalized Schur form, where
initially A and B are the first diagonal elements.
11 11
An attempt is made to compute the transformation matrices X and Y
of the form
( I V ) ( I W )
X = ( ), Y = ( ) (1)
( 0 I ) ( 0 I )
(partitioned as A and B ), so that (' denotes conjugate-transpose)
( A 0 ) ( B 0 )
( 11 ) ( 11 )
X' A Y = ( ), X' B Y = ( ),
( 0 A ) ( 0 B )
( 22 ) ( 22 )
and the elements of V and W do not exceed the value PMAX in
magnitude. An adaptation of the standard method for solving
generalized Sylvester equations [1], which controls the magnitude
of the individual elements of the computed solution [2], is used
to obtain V and W. When this attempt fails, an eigenvalue of
(A , B ) closest to the mean of those of (A , B ) is selected,
22 22 11 11
and moved by unitary equivalence transformations in the leading
position of (A , B ); the moved diagonal elements in A and B are
22 22
then added to the blocks A and B , respectively, increasing
11 11
their order by 1. Another attempt is made to compute suitable
transformation matrices X and Y with the new definitions of the
blocks A , A , B , and B . After successful transformation
11 22 11 22
matrices X and Y have been obtained, they postmultiply the current
transformation matrices (if JOBX = 'U' and/or JOBY = 'U') and the
whole procedure is repeated for the new blocks A and B .
22 22
When SORT = 'S', the diagonal elements of the generalized Schur
form are reordered before each step of the reduction, so that each
cluster of generalized eigenvalues, defined as specified in the
definition of TOL, appears in adjacent elements. The elements for
each cluster are merged together, and the procedure described
above is applied to the larger blocks. Using the option SORT = 'S'
will usually provide better efficiency than the standard option
(SORT = 'N'), proposed in [2], because there could be no or few
unsuccessful attempts to compute individual transformation
matrices X and Y of the form (1). However, the resulting
dimensions of the blocks are usually larger; this could make
subsequent calculations less efficient.
When SORT = 'C' or 'B', the procedure is similar to that for
SORT = 'N' or 'S', respectively, but the blocks of A and B
22 22
whose eigenvalue(s) is (are) the closest to those of (A , B )
11 11
(not to their mean) are selected and moved to the leading position
of A and B . This is called the "closest-neighbour" strategy.
22 22
References
[1] Kagstrom, B. and Westin, L.
Generalized Schur Methods with Condition Estimators for
Solving the Generalized Sylvester Equation.
IEEE Trans. Auto. Contr., 34, pp. 745-751, 1989.
[2] Bavely, C. and Stewart, G.W.
An Algorithm for Computing Reducing Subspaces by Block
Diagonalization.
SIAM J. Numer. Anal., 16, pp. 359-367, 1979.
[3] Demmel, J.
The Condition Number of Equivalence Transformations that
Block Diagonalize Matrix Pencils.
SIAM J. Numer. Anal., 20, pp. 599-610, 1983.
Numerical Aspects
3 4 The algorithm usually requires 0(N ) operations, but 0(N ) are possible in the worst case, when the matrix pencil cannot be diagonalized by well-conditioned transformations.Further Comments
The individual non-unitary transformation matrices used in the reduction of A and B to a block-diagonal form have condition numbers of the order PMAX. This does not guarantee that their product is well-conditioned enough. The routine can be easily modified to provide estimates for the condition numbers of the clusters of generalized eigenvalues. For efficiency reasons, the "absolute value" (or "magnitude") of the complex elements x of the transformation matrices is computed as |real(x)| + |imag(x)|.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04SU.html 0000664 0000000 0000000 00000012715 14560147231 0020372 0 ustar 00root root 0000000 0000000
Purpose
To compute a symplectic QR decomposition of a real 2M-by-N matrix
[A; B],
[ A ] [ R11 R12 ]
[ ] = Q * R = Q [ ],
[ B ] [ R21 R22 ]
where Q is a symplectic orthogonal matrix, R11 is upper triangular
and R21 is strictly upper triangular.
If [A; B] is symplectic then, theoretically, R21 = 0 and
R22 = inv(R11)^T. Unblocked version.
Specification
SUBROUTINE MB04SU( M, N, A, LDA, B, LDB, CS, TAU, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDWORK, M, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), CS(*), DWORK(*), TAU(*)
Arguments
Input/Output Parameters
M (input) INTEGER
The number of rows of A and B. M >= 0.
N (input) INTEGER
The number of columns of A and B. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading M-by-N part of this array must
contain the matrix A.
On exit, the leading M-by-N part of this array contains
the matrix [ R11 R12 ] and, in the zero parts of R,
information about the elementary reflectors used to
compute the symplectic QR decomposition.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,M).
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the leading M-by-N part of this array must
contain the matrix B.
On exit, the leading M-by-N part of this array contains
the matrix [ R21 R22 ] and, in the zero parts of B,
information about the elementary reflectors used to
compute the symplectic QR decomposition.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,M).
CS (output) DOUBLE PRECISION array, dimension (2 * min(M,N))
On exit, the first 2*min(M,N) elements of this array
contain the cosines and sines of the symplectic Givens
rotations used to compute the symplectic QR decomposition.
TAU (output) DOUBLE PRECISION array, dimension (min(M,N))
On exit, the first min(M,N) elements of this array
contain the scalar factors of some of the elementary
reflectors.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -10, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,N).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The matrix Q is represented as a product of symplectic reflectors
and Givens rotations
Q = diag( H(1),H(1) ) G(1) diag( F(1),F(1) )
diag( H(2),H(2) ) G(2) diag( F(2),F(2) )
....
diag( H(k),H(k) ) G(k) diag( F(k),F(k) ),
where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * w * w'
where tau is a real scalar, and w is a real vector with
w(1:i-1) = 0 and w(i) = 1; w(i+1:m) is stored on exit in
B(i+1:m,i), and tau in B(i,i).
Each F(i) has the form
F(i) = I - nu * v * v'
where nu is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
A(i+1:m,i), and nu in TAU(i).
Each G(i) is a Givens rotation acting on rows i of A and B,
where the cosine is stored in CS(2*i-1) and the sine in
CS(2*i).
References
[1] Bunse-Gerstner, A.
Matrix factorizations for symplectic QR-like methods.
Linear Algebra Appl., 83, pp. 49-77, 1986.
[2] Byers, R.
Hamiltonian and Symplectic Algorithms for the Algebraic
Riccati Equation.
Ph.D. Dissertation, Center for Applied Mathematics,
Cornell University, Ithaca, NY, 1983.
Numerical Aspects
The algorithm requires
8*M*N*N - 8/3*N*N*N + 2*M*N + 6*N*N + 8/3*N, if M >= N,
8*M*M*N - 8/3*M*M*M + 14*M*N - 6*M*M + 8/3*N, if M <= N,
floating point operations and is numerically backward stable.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute a symplectic URV (SURV) decomposition of a real
2N-by-2N matrix H,
[ op(A) G ] [ op(R11) R12 ]
H = [ ] = U R V' = U * [ ] * V' ,
[ Q op(B) ] [ 0 op(R22) ]
where A, B, G, Q, R12 are real N-by-N matrices, op(R11) is a real
N-by-N upper triangular matrix, op(R22) is a real N-by-N lower
Hessenberg matrix and U, V are 2N-by-2N orthogonal symplectic
matrices. Blocked version.
Specification
SUBROUTINE MB04TB( TRANA, TRANB, N, ILO, A, LDA, B, LDB, G, LDG,
$ Q, LDQ, CSL, CSR, TAUL, TAUR, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER TRANA, TRANB
INTEGER ILO, INFO, LDA, LDB, LDG, LDQ, LDWORK, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), CSL(*), CSR(*), DWORK(*),
$ G(LDG,*), Q(LDQ,*), TAUL(*), TAUR(*)
Arguments
Mode Parameters
TRANA CHARACTER*1
Specifies the form of op( A ) as follows:
= 'N': op( A ) = A;
= 'T': op( A ) = A';
= 'C': op( A ) = A'.
TRANB CHARACTER*1
Specifies the form of op( B ) as follows:
= 'N': op( B ) = B;
= 'T': op( B ) = B';
= 'C': op( B ) = B'.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
ILO (input) INTEGER
It is assumed that op(A) is already upper triangular,
op(B) is lower triangular and Q is zero in rows and
columns 1:ILO-1. ILO is normally set by a previous call
to MB04DD; otherwise it should be set to 1.
1 <= ILO <= N+1, if N > 0; ILO = 1, if N = 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A.
On exit, the leading N-by-N part of this array contains
the triangular matrix R11, and in the zero part
information about the elementary reflectors used to
compute the SURV decomposition.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the leading N-by-N part of this array must
contain the matrix B.
On exit, the leading N-by-N part of this array contains
the Hessenberg matrix R22, and in the zero part
information about the elementary reflectors used to
compute the SURV decomposition.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
G (input/output) DOUBLE PRECISION array, dimension (LDG,N)
On entry, the leading N-by-N part of this array must
contain the matrix G.
On exit, the leading N-by-N part of this array contains
the matrix R12.
LDG INTEGER
The leading dimension of the array G. LDG >= MAX(1,N).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, the leading N-by-N part of this array must
contain the matrix Q.
On exit, the leading N-by-N part of this array contains
information about the elementary reflectors used to
compute the SURV decomposition.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= MAX(1,N).
CSL (output) DOUBLE PRECISION array, dimension (2N)
On exit, the first 2N elements of this array contain the
cosines and sines of the symplectic Givens rotations
applied from the left-hand side used to compute the SURV
decomposition.
CSR (output) DOUBLE PRECISION array, dimension (2N-2)
On exit, the first 2N-2 elements of this array contain the
cosines and sines of the symplectic Givens rotations
applied from the right-hand side used to compute the SURV
decomposition.
TAUL (output) DOUBLE PRECISION array, dimension (N)
On exit, the first N elements of this array contain the
scalar factors of some of the elementary reflectors
applied form the left-hand side.
TAUR (output) DOUBLE PRECISION array, dimension (N-1)
On exit, the first N-1 elements of this array contain the
scalar factors of some of the elementary reflectors
applied form the right-hand side.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK, (16*N + 5)*NB, where NB is the optimal
block size determined by the function UE01MD.
On exit, if INFO = -16, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,N).
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The matrices U and V are represented as products of symplectic
reflectors and Givens rotations
U = diag( HU(1),HU(1) ) GU(1) diag( FU(1),FU(1) )
diag( HU(2),HU(2) ) GU(2) diag( FU(2),FU(2) )
....
diag( HU(n),HU(n) ) GU(n) diag( FU(n),FU(n) ),
V = diag( HV(1),HV(1) ) GV(1) diag( FV(1),FV(1) )
diag( HV(2),HV(2) ) GV(2) diag( FV(2),FV(2) )
....
diag( HV(n-1),HV(n-1) ) GV(n-1) diag( FV(n-1),FV(n-1) ).
Each HU(i) has the form
HU(i) = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in
Q(i+1:n,i), and tau in Q(i,i).
Each FU(i) has the form
FU(i) = I - nu * w * w'
where nu is a real scalar, and w is a real vector with
w(1:i-1) = 0 and w(i) = 1; w(i+1:n) is stored on exit in
A(i+1:n,i), if op(A) = 'N', and in A(i,i+1:n), otherwise. The
scalar nu is stored in TAUL(i).
Each GU(i) is a Givens rotation acting on rows i and n+i,
where the cosine is stored in CSL(2*i-1) and the sine in
CSL(2*i).
Each HV(i) has the form
HV(i) = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in
Q(i,i+2:n), and tau in Q(i,i+1).
Each FV(i) has the form
FV(i) = I - nu * w * w'
where nu is a real scalar, and w is a real vector with
w(1:i) = 0 and w(i+1) = 1; w(i+2:n) is stored on exit in
B(i,i+2:n), if op(B) = 'N', and in B(i+2:n,i), otherwise.
The scalar nu is stored in TAUR(i).
Each GV(i) is a Givens rotation acting on columns i+1 and n+i+1,
where the cosine is stored in CSR(2*i-1) and the sine in
CSR(2*i).
Numerical Aspects
The algorithm requires 80/3*N**3 + ( 64*NB + 77 )*N**2 + ( -16*NB + 48 )*NB*N + O(N) floating point operations, where NB is the used block size, and is numerically backward stable.References
[1] Benner, P., Mehrmann, V., and Xu, H.
A numerically stable, structure preserving method for
computing the eigenvalues of real Hamiltonian or symplectic
pencils. Numer. Math., Vol 78 (3), pp. 329-358, 1998.
[2] Kressner, D.
Block algorithms for orthogonal symplectic factorizations.
BIT, 43 (4), pp. 775-790, 2003.
Further Comments
NoneExample
Program Text
* MB04TB/MB04WR EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NBMAX, NMAX
PARAMETER ( NBMAX = 64, NMAX = 421 )
INTEGER LDA, LDB, LDG, LDQ, LDRES, LDU1, LDU2, LDV1,
$ LDV2, LDWORK
PARAMETER ( LDA = NMAX, LDB = NMAX, LDG = NMAX, LDQ = NMAX,
$ LDRES = NMAX, LDU1 = NMAX, LDU2 = NMAX,
$ LDV1 = NMAX, LDV2 = NMAX,
$ LDWORK = NBMAX*( 16*NMAX + 1 ) )
* .. Local Scalars ..
CHARACTER*1 TRANA, TRANB, TRANV1
INTEGER I, INFO, J, N
DOUBLE PRECISION TEMP
* .. Local Arrays ..
DOUBLE PRECISION A(LDA, NMAX), B(LDB, NMAX), CSL(2*NMAX),
$ CSR(2*NMAX), DWORK(LDWORK), G(LDG, NMAX),
$ Q(LDQ, NMAX), RES(LDRES,5*NMAX), TAUL(NMAX),
$ TAUR(NMAX), U1(LDU1, NMAX), U2(LDU2, NMAX),
$ V1(LDV1, NMAX), V2(LDV2, NMAX)
* .. External Functions ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLANGE, DLAPY2, MA02JD
EXTERNAL DLANGE, DLAPY2, LSAME, MA02JD
* .. External Subroutines ..
EXTERNAL DGEMM, DLACPY, DLASET, MB04TB, MB04WR
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, TRANA, TRANB
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
CALL DLACPY( 'All', N, N, A, LDA, RES, LDRES )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,N ), I = 1,N )
CALL DLACPY( 'All', N, N, B, LDB, RES(1,N+1), LDRES )
READ ( NIN, FMT = * ) ( ( G(I,J), J = 1,N ), I = 1,N )
CALL DLACPY( 'All', N, N, G, LDG, RES(1,2*N+1), LDRES )
READ ( NIN, FMT = * ) ( ( Q(I,J), J = 1,N ), I = 1,N )
CALL DLACPY( 'All', N, N, Q, LDQ, RES(1,3*N+1), LDRES )
CALL MB04TB( TRANA, TRANB, N, 1, A, LDA, B, LDB, G, LDG, Q,
$ LDQ, CSL, CSR, TAUL, TAUR, DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
CALL DLACPY( 'All', N, N, A, LDA, U1, LDU1 )
CALL DLACPY( 'All', N, N, Q, LDQ, U2, LDU2 )
CALL MB04WR( 'U', TRANA, N, 1, U1, LDU1, U2, LDU2, CSL,
$ TAUL, DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
CALL DLACPY( 'All', N, N, Q, LDQ, V2, LDV2 )
CALL DLACPY( 'All', N, N, B, LDB, V1, LDV1 )
CALL MB04WR( 'V', TRANB, N, 1, V1, LDV1, V2, LDV2,
$ CSR, TAUR, DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99996 )
IF ( LSAME( TRANA, 'N' ) ) THEN
DO 10 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( U1(I,J), J = 1,N ), ( U2(I,J), J = 1,N )
10 CONTINUE
DO 20 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( -U2(I,J), J = 1,N ), ( U1(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99991 ) MA02JD( .FALSE.,
$ .FALSE., N, U1, LDU1, U2, LDU2,
$ RES(1,4*N+1), LDRES )
ELSE
DO 30 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( U1(J,I), J = 1,N ), ( U2(I,J), J = 1,N )
30 CONTINUE
DO 40 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( -U2(I,J), J = 1,N ), ( U1(J,I), J = 1,N )
40 CONTINUE
WRITE ( NOUT, FMT = 99991 ) MA02JD( .TRUE.,
$ .FALSE., N, U1, LDU1, U2, LDU2,
$ RES(1,4*N+1), LDRES )
END IF
WRITE ( NOUT, FMT = 99995 )
CALL DLASET( 'All', N, N, ZERO, ZERO, Q, LDQ )
IF ( LSAME( TRANA, 'N' ) ) THEN
CALL DLASET( 'Lower', N-1, N-1, ZERO, ZERO,
$ A(2,1), LDA )
DO 50 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( A(I,J), J = 1,N ), ( G(I,J), J = 1,N )
50 CONTINUE
ELSE
CALL DLASET( 'Upper', N-1, N-1, ZERO, ZERO,
$ A(1,2), LDA )
DO 60 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( A(J,I), J = 1,N ), ( G(I,J), J = 1,N )
60 CONTINUE
END IF
IF ( LSAME( TRANB, 'N' ) ) THEN
IF ( N.GT.1 ) THEN
CALL DLASET( 'Upper', N-2, N-2, ZERO, ZERO,
$ B(1,3), LDB )
END IF
DO 70 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( Q(I,J), J = 1,N ), ( B(I,J), J = 1,N )
70 CONTINUE
ELSE
IF ( N.GT.1 ) THEN
C CALL DLASET( 'Lower', N-2, N-2, ZERO, ZERO,
C $ B(3,1), LDB )
END IF
DO 80 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( Q(I,J), J = 1,N ), ( B(J,I), J = 1,N )
80 CONTINUE
END IF
C
IF ( LSAME( TRANB, 'N' ) ) THEN
TRANV1 = 'T'
ELSE
TRANV1 = 'N'
END IF
CALL DGEMM( TRANA, TRANV1, N, N, N, ONE, RES, LDRES,
$ V1, LDV1, ZERO, RES(1,4*N+1), LDRES )
CALL DGEMM( 'No Transpose', 'Transpose', N, N, N,
$ -ONE, RES(1,2*N+1), LDRES, V2, LDV2, ONE,
$ RES(1,4*N+1), LDRES )
CALL DGEMM( TRANA, TRANA, N, N, N, -ONE, U1, LDU1,
$ A, LDA, ONE, RES(1,4*N+1), LDRES )
TEMP = DLANGE( 'Frobenius', N, N, RES(1,4*N+1),
$ LDRES, DWORK )
CALL DGEMM( TRANA, 'Transpose', N, N, N, ONE, RES,
$ LDRES, V2, LDV2, ZERO, RES(1,4*N+1),
$ LDRES )
CALL DGEMM( 'No Transpose', TRANV1, N, N, N, ONE,
$ RES(1,2*N+1), LDRES, V1, LDV1, ONE,
$ RES(1,4*N+1), LDRES )
CALL DGEMM( TRANA, 'No Transpose', N, N, N, -ONE,
$ U1, LDU1, G, LDG, ONE, RES(1,4*N+1),
$ LDRES )
CALL DGEMM( 'No Transpose', TRANB, N, N, N, -ONE,
$ U2, LDU2, B, LDB, ONE, RES(1,4*N+1),
$ LDRES )
TEMP = DLAPY2( TEMP, DLANGE( 'Frobenius', N, N,
$ RES(1,4*N+1), LDRES, DWORK ) )
CALL DGEMM( 'No Transpose', TRANV1, N, N, N, ONE,
$ RES(1,3*N+1), LDRES, V1, LDV1, ZERO,
$ RES(1,4*N+1), LDRES )
CALL DGEMM( TRANB, 'Transpose', N, N, N, -ONE,
$ RES(1,N+1), LDRES, V2, LDV2, ONE,
$ RES(1,4*N+1), LDRES )
CALL DGEMM( 'No Transpose', TRANA, N, N, N, ONE,
$ U2, LDU2, A, LDA, ONE, RES(1,4*N+1),
$ LDRES )
TEMP = DLAPY2( TEMP, DLANGE( 'Frobenius', N, N,
$ RES(1,4*N+1), LDRES, DWORK ) )
CALL DGEMM( 'No Transpose', 'Transpose', N, N, N, ONE,
$ RES(1,3*N+1), LDRES, V2, LDV2, ZERO,
$ RES(1,4*N+1), LDRES )
CALL DGEMM( TRANB, TRANV1, N, N, N, ONE, RES(1,N+1),
$ LDRES, V1, LDV1, ONE, RES(1,4*N+1),
$ LDRES )
CALL DGEMM( 'No Transpose', 'No Transpose', N, N, N,
$ ONE, U2, LDU2, G, LDG, ONE, RES(1,4*N+1),
$ LDRES )
CALL DGEMM( TRANA, TRANB, N, N, N, -ONE, U1, LDU1,
$ B, LDB, ONE, RES(1,4*N+1), LDRES )
TEMP = DLAPY2( TEMP, DLANGE( 'Frobenius', N, N,
$ RES(1,4*N+1), LDRES, DWORK ) )
WRITE ( NOUT, FMT = 99990 ) TEMP
C
WRITE ( NOUT, FMT = 99994 )
IF ( LSAME( TRANB, 'N' ) ) THEN
DO 90 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( V1(J,I), J = 1,N ), ( V2(J,I), J = 1,N )
90 CONTINUE
DO 100 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( -V2(J,I), J = 1,N ), ( V1(J,I), J = 1,N )
100 CONTINUE
WRITE ( NOUT, FMT = 99989 ) MA02JD( .TRUE.,
$ .TRUE., N, V1, LDV1, V2, LDV2,
$ RES(1,4*N+1), LDRES )
ELSE
DO 110 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( V1(I,J), J = 1,N ), ( V2(J,I), J = 1,N )
110 CONTINUE
DO 120 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( -V2(J,I), J = 1,N ), ( V1(I,J), J = 1,N )
120 CONTINUE
WRITE ( NOUT, FMT = 99989 ) MA02JD( .FALSE.,
$ .TRUE., N, V1, LDV1, V2, LDV2,
$ RES(1,4*N+1), LDRES )
END IF
END IF
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' MB04TB EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB04TB = ',I2)
99997 FORMAT (' INFO on exit from MB04WR = ',I2)
99996 FORMAT (' The orthogonal symplectic factor U is ')
99995 FORMAT (/' The factor R is ')
99994 FORMAT (/' The orthogonal symplectic factor V is ')
99993 FORMAT (20(1X,F9.4))
99992 FORMAT (/' N is out of range.',/' N = ',I5)
99991 FORMAT (/' Orthogonality of U: || U^T U - I ||_F = ',G7.2)
99990 FORMAT (/' Residual: || H*V - U*R ||_F = ',G7.2)
99989 FORMAT (/' Orthogonality of V: || V^T V - I ||_F = ',G7.2)
END
Program Data
MB04TB EXAMPLE PROGRAM DATA
5 N N
0.4643 0.3655 0.6853 0.5090 0.3718
0.3688 0.6460 0.4227 0.6798 0.5135
0.7458 0.5043 0.9419 0.9717 0.9990
0.7140 0.4941 0.7802 0.5272 0.1220
0.7418 0.0339 0.7441 0.0436 0.6564
-0.4643 -0.3688 -0.7458 -0.7140 -0.7418
-0.3655 -0.6460 -0.5043 -0.4941 -0.0339
-0.6853 -0.4227 -0.9419 -0.7802 -0.7441
-0.5090 -0.6798 -0.9717 -0.5272 -0.0436
-0.3718 -0.5135 -0.9990 -0.1220 -0.6564
0.7933 1.5765 1.0711 1.0794 0.8481
1.5765 0.1167 1.5685 0.8756 0.5037
1.0711 1.5685 0.9902 0.3858 0.2109
1.0794 0.8756 0.3858 1.8834 1.4338
0.8481 0.5037 0.2109 1.4338 0.1439
1.0786 1.5264 1.1721 1.5343 0.4756
1.5264 0.8644 0.6872 1.1379 0.6499
1.1721 0.6872 1.5194 1.1197 1.0158
1.5343 1.1379 1.1197 0.6612 0.2004
0.4756 0.6499 1.0158 0.2004 1.2188
Program Results
MB04TB EXAMPLE PROGRAM RESULTS
The orthogonal symplectic factor U is
-0.1513 0.0756 -0.0027 0.1694 -0.2999 0.3515 -0.4843 0.6545 -0.1995 -0.1627
-0.1202 0.2320 0.1662 -0.2835 -0.0508 0.4975 0.3319 -0.2686 -0.4186 -0.4649
-0.2431 0.2724 0.3439 0.3954 0.0236 0.3820 -0.2863 -0.4324 0.3706 0.1984
-0.2327 -0.1509 -0.3710 -0.1240 -0.0393 0.5000 0.3659 0.1429 0.0493 0.6015
-0.2418 -0.2928 -0.0836 -0.5549 0.4824 0.1550 -0.4441 -0.0396 0.2376 -0.1702
-0.3515 0.4843 -0.6545 0.1995 0.1627 -0.1513 0.0756 -0.0027 0.1694 -0.2999
-0.4975 -0.3319 0.2686 0.4186 0.4649 -0.1202 0.2320 0.1662 -0.2835 -0.0508
-0.3820 0.2863 0.4324 -0.3706 -0.1984 -0.2431 0.2724 0.3439 0.3954 0.0236
-0.5000 -0.3659 -0.1429 -0.0493 -0.6015 -0.2327 -0.1509 -0.3710 -0.1240 -0.0393
-0.1550 0.4441 0.0396 -0.2376 0.1702 -0.2418 -0.2928 -0.0836 -0.5549 0.4824
Orthogonality of U: || U^T U - I ||_F = .17E-14
The factor R is
-3.0684 4.6724 -0.2613 -0.1996 0.0208 -0.1071 -0.1355 -0.1400 0.4652 -0.5032
0.0000 -1.8037 -0.0301 -0.1137 0.1771 0.0277 0.3929 0.5424 0.5220 -0.4843
0.0000 0.0000 -0.7617 -0.1874 0.2557 0.1244 -0.0012 0.4091 0.5123 -0.3522
0.0000 0.0000 0.0000 -0.6931 -0.4293 -0.3718 0.1542 -0.3635 0.0336 -0.9832
0.0000 0.0000 0.0000 0.0000 0.6469 0.2074 0.0266 0.2028 0.1995 0.2517
0.0000 0.0000 0.0000 0.0000 0.0000 2.6325 -4.7377 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 -0.2702 0.9347 -1.1210 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 -0.3219 -0.5394 0.1748 -0.4788 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 -0.1431 -0.1021 0.4974 -0.3565 -0.6402
0.0000 0.0000 0.0000 0.0000 0.0000 -0.1622 -0.2368 0.6126 -0.7369 0.6915
Residual: || H*V - U*R ||_F = .76E-14
The orthogonal symplectic factor V is
1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 -0.4740 0.6013 -0.2299 -0.4282 0.0000 0.0061 -0.1732 0.3134 0.2220
0.0000 -0.5553 -0.2623 0.6622 -0.3042 0.0000 -0.0382 0.2453 -0.1662 0.0509
0.0000 -0.5563 0.0322 -0.1431 0.4461 0.0000 -0.0665 -0.4132 -0.3100 -0.4457
0.0000 -0.3872 -0.4022 -0.4194 0.3541 0.0000 -0.0406 0.3820 0.3006 0.3861
0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000
0.0000 -0.0061 0.1732 -0.3134 -0.2220 0.0000 -0.4740 0.6013 -0.2299 -0.4282
0.0000 0.0382 -0.2453 0.1662 -0.0509 0.0000 -0.5553 -0.2623 0.6622 -0.3042
0.0000 0.0665 0.4132 0.3100 0.4457 0.0000 -0.5563 0.0322 -0.1431 0.4461
0.0000 0.0406 -0.3820 -0.3006 -0.3861 0.0000 -0.3872 -0.4022 -0.4194 0.3541
Orthogonality of V: || V^T V - I ||_F = .17E-14
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04TS.html 0000664 0000000 0000000 00000056430 14560147231 0020373 0 ustar 00root root 0000000 0000000
Purpose
To compute a symplectic URV (SURV) decomposition of a real
2N-by-2N matrix H:
[ op(A) G ] T [ op(R11) R12 ] T
H = [ ] = U R V = U * [ ] * V ,
[ Q op(B) ] [ 0 op(R22) ]
where A, B, G, Q, R12 are real N-by-N matrices, op(R11) is a real
N-by-N upper triangular matrix, op(R22) is a real N-by-N lower
Hessenberg matrix and U, V are 2N-by-2N orthogonal symplectic
matrices. Unblocked version.
Specification
SUBROUTINE MB04TS( TRANA, TRANB, N, ILO, A, LDA, B, LDB, G, LDG,
$ Q, LDQ, CSL, CSR, TAUL, TAUR, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER TRANA, TRANB
INTEGER ILO, INFO, LDA, LDB, LDG, LDQ, LDWORK, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), CSL(*), CSR(*), DWORK(*),
$ G(LDG,*), Q(LDQ,*), TAUL(*), TAUR(*)
Arguments
Mode Parameters
TRANA CHARACTER*1
Specifies the form of op( A ) as follows:
= 'N': op( A ) = A;
= 'T': op( A ) = A';
= 'C': op( A ) = A'.
TRANB CHARACTER*1
Specifies the form of op( B ) as follows:
= 'N': op( B ) = B;
= 'T': op( B ) = B';
= 'C': op( B ) = B'.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
ILO (input) INTEGER
It is assumed that op(A) is already upper triangular,
op(B) is lower triangular and Q is zero in rows and
columns 1:ILO-1. ILO is normally set by a previous call
to MB04DD; otherwise it should be set to 1.
1 <= ILO <= N, if N > 0; ILO=1, if N=0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A.
On exit, the leading N-by-N part of this array contains
the triangular matrix R11, and in the zero part
information about the elementary reflectors used to
compute the SURV decomposition.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the leading N-by-N part of this array must
contain the matrix B.
On exit, the leading N-by-N part of this array contains
the Hessenberg matrix R22, and in the zero part
information about the elementary reflectors used to
compute the SURV decomposition.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
G (input/output) DOUBLE PRECISION array, dimension (LDG,N)
On entry, the leading N-by-N part of this array must
contain the matrix G.
On exit, the leading N-by-N part of this array contains
the matrix R12.
LDG INTEGER
The leading dimension of the array G. LDG >= MAX(1,N).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, the leading N-by-N part of this array must
contain the matrix Q.
On exit, the leading N-by-N part of this array contains
information about the elementary reflectors used to
compute the SURV decomposition.
LDQ INTEGER
The leading dimension of the array Q. LDG >= MAX(1,N).
CSL (output) DOUBLE PRECISION array, dimension (2N)
On exit, the first 2N elements of this array contain the
cosines and sines of the symplectic Givens rotations
applied from the left-hand side used to compute the SURV
decomposition.
CSR (output) DOUBLE PRECISION array, dimension (2N-2)
On exit, the first 2N-2 elements of this array contain the
cosines and sines of the symplectic Givens rotations
applied from the right-hand side used to compute the SURV
decomposition.
TAUL (output) DOUBLE PRECISION array, dimension (N)
On exit, the first N elements of this array contain the
scalar factors of some of the elementary reflectors
applied from the left-hand side.
TAUR (output) DOUBLE PRECISION array, dimension (N-1)
On exit, the first N-1 elements of this array contain the
scalar factors of some of the elementary reflectors
applied from the right-hand side.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -16, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,N).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The matrices U and V are represented as products of symplectic
reflectors and Givens rotations
U = diag( HU(1),HU(1) ) GU(1) diag( FU(1),FU(1) )
diag( HU(2),HU(2) ) GU(2) diag( FU(2),FU(2) )
....
diag( HU(n),HU(n) ) GU(n) diag( FU(n),FU(n) ),
V = diag( HV(1),HV(1) ) GV(1) diag( FV(1),FV(1) )
diag( HV(2),HV(2) ) GV(2) diag( FV(2),FV(2) )
....
diag( HV(n-1),HV(n-1) ) GV(n-1) diag( FV(n-1),FV(n-1) ).
Each HU(i) has the form
HU(i) = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in
Q(i+1:n,i), and tau in Q(i,i).
Each FU(i) has the form
FU(i) = I - nu * w * w'
where nu is a real scalar, and w is a real vector with
w(1:i-1) = 0 and w(i) = 1; w(i+1:n) is stored on exit in
A(i+1:n,i), if op(A) = 'N', and in A(i,i+1:n), otherwise. The
scalar nu is stored in TAUL(i).
Each GU(i) is a Givens rotation acting on rows i and n+i,
where the cosine is stored in CSL(2*i-1) and the sine in
CSL(2*i).
Each HV(i) has the form
HV(i) = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in
Q(i,i+2:n), and tau in Q(i,i+1).
Each FV(i) has the form
FV(i) = I - nu * w * w'
where nu is a real scalar, and w is a real vector with
w(1:i) = 0 and w(i+1) = 1; w(i+2:n) is stored on exit in
B(i,i+2:n), if op(B) = 'N', and in B(i+2:n,i), otherwise.
The scalar nu is stored in TAUR(i).
Each GV(i) is a Givens rotation acting on columns i+1 and n+i+1,
where the cosine is stored in CSR(2*i-1) and the sine in
CSR(2*i).
Numerical Aspects
The algorithm requires 80/3 N**3 + 20 N**2 + O(N) floating point operations and is numerically backward stable.References
[1] Benner, P., Mehrmann, V., and Xu, H.
A numerically stable, structure preserving method for
computing the eigenvalues of real Hamiltonian or symplectic
pencils. Numer. Math., Vol 78 (3), pp. 329-358, 1998.
Further Comments
NoneExample
Program Text
* MB04TS/MB04WR EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 200 )
INTEGER LDA, LDB, LDG, LDQ, LDRES, LDU1, LDU2, LDV1,
$ LDV2, LDWORK
PARAMETER ( LDA = NMAX, LDB = NMAX, LDG = NMAX, LDQ = NMAX,
$ LDRES = NMAX, LDU1 = NMAX, LDU2 = NMAX,
$ LDV1 = NMAX, LDV2 = NMAX, LDWORK = NMAX )
* .. Local Scalars ..
CHARACTER*1 TRANA, TRANB, TRANV1
INTEGER I, INFO, J, N
DOUBLE PRECISION TEMP
* .. Local Arrays ..
DOUBLE PRECISION A(LDA, NMAX), B(LDB, NMAX), CSL(2*NMAX),
$ CSR(2*NMAX), DWORK(LDWORK), G(LDG,NMAX),
$ Q(LDQ,NMAX), RES(LDRES,5*NMAX), TAUL(NMAX),
$ TAUR(NMAX), U1(LDU1,NMAX), U2(LDU2, NMAX),
$ V1(LDV1, NMAX), V2(LDV2,NMAX)
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLANGE, DLAPY2, MA02JD
EXTERNAL DLANGE, DLAPY2, LSAME, MA02JD
* .. External Subroutines ..
EXTERNAL DGEMM, DLACPY, DLASET, MB04TS, MB04WR
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, TRANA, TRANB
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
CALL DLACPY( 'All', N, N, A, LDA, RES, LDRES )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,N ), I = 1,N )
CALL DLACPY( 'All', N, N, B, LDB, RES(1,N+1), LDRES )
READ ( NIN, FMT = * ) ( ( G(I,J), J = 1,N ), I = 1,N )
CALL DLACPY( 'All', N, N, G, LDG, RES(1,2*N+1), LDRES )
READ ( NIN, FMT = * ) ( ( Q(I,J), J = 1,N ), I = 1,N )
CALL DLACPY( 'All', N, N, Q, LDQ, RES(1,3*N+1), LDRES )
CALL MB04TS( TRANA, TRANB, N, 1, A, LDA, B, LDB, G, LDG, Q,
$ LDQ, CSL, CSR, TAUL, TAUR, DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
CALL DLACPY( 'All', N, N, A, LDA, U1, LDU1 )
CALL DLACPY( 'All', N, N, Q, LDQ, U2, LDU2 )
CALL MB04WR( 'U', TRANA, N, 1, U1, LDU1, U2, LDU2, CSL,
$ TAUL, DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
CALL DLACPY( 'All', N, N, Q, LDQ, V2, LDV2 )
CALL DLACPY( 'All', N, N, B, LDB, V1, LDV1 )
CALL MB04WR( 'V', TRANB, N, 1, V1, LDV1, V2, LDV2,
$ CSR, TAUR, DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99996 )
IF ( LSAME( TRANA, 'N' ) ) THEN
DO 10 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( U1(I,J), J = 1,N ), ( U2(I,J), J = 1,N )
10 CONTINUE
DO 20 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( -U2(I,J), J = 1,N ), ( U1(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99991 ) MA02JD( .FALSE.,
$ .FALSE., N, U1, LDU1, U2, LDU2,
$ RES(1,4*N+1), LDRES )
ELSE
DO 30 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( U1(J,I), J = 1,N ), ( U2(I,J), J = 1,N )
30 CONTINUE
DO 40 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( -U2(I,J), J = 1,N ), ( U1(J,I), J = 1,N )
40 CONTINUE
WRITE ( NOUT, FMT = 99991 ) MA02JD( .TRUE.,
$ .FALSE., N, U1, LDU1, U2, LDU2,
$ RES(1,4*N+1), LDRES )
END IF
WRITE ( NOUT, FMT = 99995 )
CALL DLASET( 'All', N, N, ZERO, ZERO, Q, LDQ )
IF ( LSAME( TRANA, 'N' ) ) THEN
CALL DLASET( 'Lower', N-1, N-1, ZERO, ZERO,
$ A(2,1), LDA )
DO 50 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( A(I,J), J = 1,N ), ( G(I,J), J = 1,N )
50 CONTINUE
ELSE
CALL DLASET( 'Upper', N-1, N-1, ZERO, ZERO,
$ A(1,2), LDA )
DO 60 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( A(J,I), J = 1,N ), ( G(I,J), J = 1,N )
60 CONTINUE
END IF
IF ( LSAME( TRANB, 'N' ) ) THEN
IF ( N.GT.1 ) THEN
CALL DLASET( 'Upper', N-2, N-2, ZERO, ZERO,
$ B(1,3), LDB )
END IF
DO 70 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( Q(I,J), J = 1,N ), ( B(I,J), J = 1,N )
70 CONTINUE
ELSE
IF ( N.GT.1 ) THEN
CALL DLASET( 'Lower', N-2, N-2, ZERO, ZERO,
$ B(3,1), LDB )
END IF
DO 80 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( Q(I,J), J = 1,N ), ( B(J,I), J = 1,N )
80 CONTINUE
END IF
C
IF ( LSAME( TRANB, 'N' ) ) THEN
TRANV1 = 'T'
ELSE
TRANV1 = 'N'
END IF
CALL DGEMM( TRANA, TRANV1, N, N, N, ONE, RES, LDRES,
$ V1, LDV1, ZERO, RES(1,4*N+1), LDRES )
CALL DGEMM( 'No Transpose', 'Transpose', N, N, N,
$ -ONE, RES(1,2*N+1), LDRES, V2, LDV2, ONE,
$ RES(1,4*N+1), LDRES )
CALL DGEMM( TRANA, TRANA, N, N, N, -ONE, U1, LDU1,
$ A, LDA, ONE, RES(1,4*N+1), LDRES )
TEMP = DLANGE( 'Frobenius', N, N, RES(1,4*N+1),
$ LDRES, DWORK )
CALL DGEMM( TRANA, 'Transpose', N, N, N, ONE, RES,
$ LDRES, V2, LDV2, ZERO, RES(1,4*N+1),
$ LDRES )
CALL DGEMM( 'No Transpose', TRANV1, N, N, N, ONE,
$ RES(1,2*N+1), LDRES, V1, LDV1, ONE,
$ RES(1,4*N+1), LDRES )
CALL DGEMM( TRANA, 'No Transpose', N, N, N, -ONE,
$ U1, LDU1, G, LDG, ONE, RES(1,4*N+1),
$ LDRES )
CALL DGEMM( 'No Transpose', TRANB, N, N, N, -ONE,
$ U2, LDU2, B, LDB, ONE, RES(1,4*N+1),
$ LDRES )
TEMP = DLAPY2( TEMP, DLANGE( 'Frobenius', N, N,
$ RES(1,4*N+1), LDRES, DWORK ) )
CALL DGEMM( 'No Transpose', TRANV1, N, N, N, ONE,
$ RES(1,3*N+1), LDRES, V1, LDV1, ZERO,
$ RES(1,4*N+1), LDRES )
CALL DGEMM( TRANB, 'Transpose', N, N, N, -ONE,
$ RES(1,N+1), LDRES, V2, LDV2, ONE,
$ RES(1,4*N+1), LDRES )
CALL DGEMM( 'No Transpose', TRANA, N, N, N, ONE,
$ U2, LDU2, A, LDA, ONE, RES(1,4*N+1),
$ LDRES )
TEMP = DLAPY2( TEMP, DLANGE( 'Frobenius', N, N,
$ RES(1,4*N+1), LDRES, DWORK ) )
CALL DGEMM( 'No Transpose', 'Transpose', N, N, N, ONE,
$ RES(1,3*N+1), LDRES, V2, LDV2, ZERO,
$ RES(1,4*N+1), LDRES )
CALL DGEMM( TRANB, TRANV1, N, N, N, ONE, RES(1,N+1),
$ LDRES, V1, LDV1, ONE, RES(1,4*N+1),
$ LDRES )
CALL DGEMM( 'No Transpose', 'No Transpose', N, N, N,
$ ONE, U2, LDU2, G, LDG, ONE, RES(1,4*N+1),
$ LDRES )
CALL DGEMM( TRANA, TRANB, N, N, N, -ONE, U1, LDU1,
$ B, LDB, ONE, RES(1,4*N+1), LDRES )
TEMP = DLAPY2( TEMP, DLANGE( 'Frobenius', N, N,
$ RES(1,4*N+1), LDRES, DWORK ) )
WRITE ( NOUT, FMT = 99990 ) TEMP
C
WRITE ( NOUT, FMT = 99994 )
IF ( LSAME( TRANB, 'N' ) ) THEN
DO 90 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( V1(J,I), J = 1,N ), ( V2(J,I), J = 1,N )
90 CONTINUE
DO 100 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( -V2(J,I), J = 1,N ), ( V1(J,I), J = 1,N )
100 CONTINUE
WRITE ( NOUT, FMT = 99989 ) MA02JD( .TRUE.,
$ .TRUE., N, V1, LDV1, V2, LDV2,
$ RES(1,4*N+1), LDRES )
ELSE
DO 110 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( V1(I,J), J = 1,N ), ( V2(J,I), J = 1,N )
110 CONTINUE
DO 120 I = 1, N
WRITE (NOUT, FMT = 99993)
$ ( -V2(J,I), J = 1,N ), ( V1(I,J), J = 1,N )
120 CONTINUE
WRITE ( NOUT, FMT = 99989 ) MA02JD( .FALSE.,
$ .TRUE., N, V1, LDV1, V2, LDV2,
$ RES(1,4*N+1), LDRES )
END IF
END IF
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' MB04TS EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB04TS = ',I2)
99997 FORMAT (' INFO on exit from MB04WR = ',I2)
99996 FORMAT (' The orthogonal symplectic factor U is ')
99995 FORMAT (/' The factor R is ')
99994 FORMAT (/' The orthogonal symplectic factor V is ')
99993 FORMAT (20(1X,F9.4))
99992 FORMAT (/' N is out of range.',/' N = ',I5)
99991 FORMAT (/' Orthogonality of U: || U^T U - I ||_F = ',G7.2)
99990 FORMAT (/' Residual: || H*V - U*R ||_F = ',G7.2)
99989 FORMAT (/' Orthogonality of V: || V^T V - I ||_F = ',G7.2)
END
Program Data
MB04TB EXAMPLE PROGRAM DATA
5 N N
0.4643 0.3655 0.6853 0.5090 0.3718
0.3688 0.6460 0.4227 0.6798 0.5135
0.7458 0.5043 0.9419 0.9717 0.9990
0.7140 0.4941 0.7802 0.5272 0.1220
0.7418 0.0339 0.7441 0.0436 0.6564
-0.4643 -0.3688 -0.7458 -0.7140 -0.7418
-0.3655 -0.6460 -0.5043 -0.4941 -0.0339
-0.6853 -0.4227 -0.9419 -0.7802 -0.7441
-0.5090 -0.6798 -0.9717 -0.5272 -0.0436
-0.3718 -0.5135 -0.9990 -0.1220 -0.6564
0.7933 1.5765 1.0711 1.0794 0.8481
1.5765 0.1167 1.5685 0.8756 0.5037
1.0711 1.5685 0.9902 0.3858 0.2109
1.0794 0.8756 0.3858 1.8834 1.4338
0.8481 0.5037 0.2109 1.4338 0.1439
1.0786 1.5264 1.1721 1.5343 0.4756
1.5264 0.8644 0.6872 1.1379 0.6499
1.1721 0.6872 1.5194 1.1197 1.0158
1.5343 1.1379 1.1197 0.6612 0.2004
0.4756 0.6499 1.0158 0.2004 1.2188
Program Results
MB04TS EXAMPLE PROGRAM RESULTS
The orthogonal symplectic factor U is
-0.1513 0.0756 -0.0027 0.1694 -0.2999 0.3515 -0.4843 0.6545 -0.1995 -0.1627
-0.1202 0.2320 0.1662 -0.2835 -0.0508 0.4975 0.3319 -0.2686 -0.4186 -0.4649
-0.2431 0.2724 0.3439 0.3954 0.0236 0.3820 -0.2863 -0.4324 0.3706 0.1984
-0.2327 -0.1509 -0.3710 -0.1240 -0.0393 0.5000 0.3659 0.1429 0.0493 0.6015
-0.2418 -0.2928 -0.0836 -0.5549 0.4824 0.1550 -0.4441 -0.0396 0.2376 -0.1702
-0.3515 0.4843 -0.6545 0.1995 0.1627 -0.1513 0.0756 -0.0027 0.1694 -0.2999
-0.4975 -0.3319 0.2686 0.4186 0.4649 -0.1202 0.2320 0.1662 -0.2835 -0.0508
-0.3820 0.2863 0.4324 -0.3706 -0.1984 -0.2431 0.2724 0.3439 0.3954 0.0236
-0.5000 -0.3659 -0.1429 -0.0493 -0.6015 -0.2327 -0.1509 -0.3710 -0.1240 -0.0393
-0.1550 0.4441 0.0396 -0.2376 0.1702 -0.2418 -0.2928 -0.0836 -0.5549 0.4824
Orthogonality of U: || U^T U - I ||_F = .24E-14
The factor R is
-3.0684 4.6724 -0.2613 -0.1996 0.0208 -0.1071 -0.1355 -0.1400 0.4652 -0.5032
0.0000 -1.8037 -0.0301 -0.1137 0.1771 0.0277 0.3929 0.5424 0.5220 -0.4843
0.0000 0.0000 -0.7617 -0.1874 0.2557 0.1244 -0.0012 0.4091 0.5123 -0.3522
0.0000 0.0000 0.0000 -0.6931 -0.4293 -0.3718 0.1542 -0.3635 0.0336 -0.9832
0.0000 0.0000 0.0000 0.0000 0.6469 0.2074 0.0266 0.2028 0.1995 0.2517
0.0000 0.0000 0.0000 0.0000 0.0000 2.6325 -4.7377 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 -0.2702 0.9347 -1.1210 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 -0.3219 -0.5394 0.1748 -0.4788 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 -0.1431 -0.1021 0.4974 -0.3565 -0.6402
0.0000 0.0000 0.0000 0.0000 0.0000 -0.1622 -0.2368 0.6126 -0.7369 0.6915
Residual: || H*V - U*R ||_F = .87E-14
The orthogonal symplectic factor V is
1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 -0.4740 0.6013 -0.2299 -0.4282 0.0000 0.0061 -0.1732 0.3134 0.2220
0.0000 -0.5553 -0.2623 0.6622 -0.3042 0.0000 -0.0382 0.2453 -0.1662 0.0509
0.0000 -0.5563 0.0322 -0.1431 0.4461 0.0000 -0.0665 -0.4132 -0.3100 -0.4457
0.0000 -0.3872 -0.4022 -0.4194 0.3541 0.0000 -0.0406 0.3820 0.3006 0.3861
0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000
0.0000 -0.0061 0.1732 -0.3134 -0.2220 0.0000 -0.4740 0.6013 -0.2299 -0.4282
0.0000 0.0382 -0.2453 0.1662 -0.0509 0.0000 -0.5553 -0.2623 0.6622 -0.3042
0.0000 0.0665 0.4132 0.3100 0.4457 0.0000 -0.5563 0.0322 -0.1431 0.4461
0.0000 0.0406 -0.3820 -0.3006 -0.3861 0.0000 -0.3872 -0.4022 -0.4194 0.3541
Orthogonality of V: || V^T V - I ||_F = .14E-14
Purpose
To perform the Givens transformation, defined by C (cos) and S
(sin), and interchange the vectors involved, i.e.
|X(i)| | 0 1 | | C S | |X(i)|
| | := | | x | | x | |, i = 1,...N.
|Y(i)| | 1 0 | |-S C | |Y(i)|
REMARK. This routine is a modification of DROT from BLAS.
This routine is called only by the SLICOT routines MB04TX
and MB04VX.
Numerical Aspects
The algorithm is backward stable.Specification
SUBROUTINE MB04TU( N, X, INCX, Y, INCY, C, S )
C .. Scalar Arguments ..
INTEGER INCX, INCY, N
DOUBLE PRECISION C, S
C .. Array Arguments ..
DOUBLE PRECISION X(*), Y(*)
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute orthogonal transformations Q and Z such that the transformed pencil Q'(sE-A)Z has the E matrix in column echelon form, where E and A are M-by-N matrices.Specification
SUBROUTINE MB04UD( JOBQ, JOBZ, M, N, A, LDA, E, LDE, Q, LDQ,
$ Z, LDZ, RANKE, ISTAIR, TOL, DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBQ, JOBZ
INTEGER INFO, LDA, LDE, LDQ, LDZ, M, N, RANKE
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER ISTAIR(*)
DOUBLE PRECISION A(LDA,*), DWORK(*), E(LDE,*), Q(LDQ,*), Z(LDZ,*)
Arguments
Mode Parameters
JOBQ CHARACTER*1
Indicates whether the user wishes to accumulate in a
matrix Q the unitary row permutations, as follows:
= 'N': Do not form Q;
= 'I': Q is initialized to the unit matrix and the
unitary row permutation matrix Q is returned;
= 'U': The given matrix Q is updated by the unitary
row permutations used in the reduction.
JOBZ CHARACTER*1
Indicates whether the user wishes to accumulate in a
matrix Z the unitary column transformations, as follows:
= 'N': Do not form Z;
= 'I': Z is initialized to the unit matrix and the
unitary transformation matrix Z is returned;
= 'U': The given matrix Z is updated by the unitary
transformations used in the reduction.
Input/Output Parameters
M (input) INTEGER
The number of rows in the matrices A, E and the order of
the matrix Q. M >= 0.
N (input) INTEGER
The number of columns in the matrices A, E and the order
of the matrix Z. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading M-by-N part of this array must
contain the A matrix of the pencil sE-A.
On exit, the leading M-by-N part of this array contains
the unitary transformed matrix Q' * A * Z.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,M).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading M-by-N part of this array must
contain the E matrix of the pencil sE-A, to be reduced to
column echelon form.
On exit, the leading M-by-N part of this array contains
the unitary transformed matrix Q' * E * Z, which is in
column echelon form.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,M).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,*)
On entry, if JOBQ = 'U', then the leading M-by-M part of
this array must contain a given matrix Q (e.g. from a
previous call to another SLICOT routine), and on exit, the
leading M-by-M part of this array contains the product of
the input matrix Q and the row permutation matrix used to
transform the rows of matrix E.
On exit, if JOBQ = 'I', then the leading M-by-M part of
this array contains the matrix of accumulated unitary
row transformations performed.
If JOBQ = 'N', the array Q is not referenced and can be
supplied as a dummy array (i.e. set parameter LDQ = 1 and
declare this array to be Q(1,1) in the calling program).
LDQ INTEGER
The leading dimension of array Q. If JOBQ = 'U' or
JOBQ = 'I', LDQ >= MAX(1,M); if JOBQ = 'N', LDQ >= 1.
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,*)
On entry, if JOBZ = 'U', then the leading N-by-N part of
this array must contain a given matrix Z (e.g. from a
previous call to another SLICOT routine), and on exit, the
leading N-by-N part of this array contains the product of
the input matrix Z and the column transformation matrix
used to transform the columns of matrix E.
On exit, if JOBZ = 'I', then the leading N-by-N part of
this array contains the matrix of accumulated unitary
column transformations performed.
If JOBZ = 'N', the array Z is not referenced and can be
supplied as a dummy array (i.e. set parameter LDZ = 1 and
declare this array to be Z(1,1) in the calling program).
LDZ INTEGER
The leading dimension of array Z. If JOBZ = 'U' or
JOBZ = 'I', LDZ >= MAX(1,N); if JOBZ = 'N', LDZ >= 1.
RANKE (output) INTEGER
The computed rank of the unitary transformed matrix E.
ISTAIR (output) INTEGER array, dimension (M)
This array contains information on the column echelon form
of the unitary transformed matrix E. Specifically,
ISTAIR(i) = +j if the first non-zero element E(i,j)
is a corner point and -j otherwise, for i = 1,2,...,M.
Tolerances
TOL DOUBLE PRECISION
A tolerance below which matrix elements are considered
to be zero. If the user sets TOL to be less than (or
equal to) zero then the tolerance is taken as
EPS * MAX(ABS(E(I,J))), where EPS is the machine
precision (see LAPACK Library routine DLAMCH),
I = 1,2,...,M and J = 1,2,...,N.
Workspace
DWORK DOUBLE PRECISION array, dimension (MAX(M,N))Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Given an M-by-N matrix pencil sE-A with E not necessarily regular,
the routine computes a unitary transformed pencil Q'(sE-A)Z such
that the matrix Q' * E * Z is in column echelon form (trapezoidal
form). Further details can be found in [1].
[An M-by-N matrix E with rank(E) = r is said to be in column
echelon form if the following conditions are satisfied:
(a) the first (N - r) columns contain only zero elements; and
(b) if E(i(k),k) is the last nonzero element in column k for
k = N-r+1,...,N, i.e. E(i(k),k) <> 0 and E(j,k) = 0 for
j > i(k), then 1 <= i(N-r+1) < i(N-r+2) < ... < i(N) <= M.]
References
[1] Beelen, Th. and Van Dooren, P.
An improved algorithm for the computation of Kronecker's
canonical form of a singular pencil.
Linear Algebra and Applications, 105, pp. 9-65, 1988.
Numerical Aspects
It is shown in [1] that the algorithm is numerically backward stable. The operations count is proportional to (MAX(M,N))**3.Further Comments
NoneExample
Program Text
* MB04UD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, NMAX
PARAMETER ( MMAX = 20, NMAX = 20 )
INTEGER LDA, LDE, LDQ, LDZ
PARAMETER ( LDA = MMAX, LDE = MMAX, LDQ = MMAX,
$ LDZ = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( NMAX,MMAX ) )
* PARAMETER ( LDWORK = NMAX+MMAX )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, J, M, N, RANKE
CHARACTER*1 JOBQ, JOBZ
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), E(LDE,NMAX),
$ Q(LDQ,MMAX), Z(LDZ,NMAX)
INTEGER ISTAIR(MMAX)
* .. External Subroutines ..
EXTERNAL MB04UD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, N, TOL
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) M
ELSE IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,M )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,M )
JOBQ = 'N'
JOBZ = 'N'
* Reduce E to column echelon form and compute Q'*A*Z.
CALL MB04UD( JOBQ, JOBZ, M, N, A, LDA, E, LDE, Q, LDQ, Z, LDZ,
$ RANKE, ISTAIR, TOL, DWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99991 )
DO 10 I = 1, M
WRITE ( NOUT, FMT = 99996 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99997 )
DO 100 I = 1, M
WRITE ( NOUT, FMT = 99996 ) ( E(I,J), J = 1,N )
100 CONTINUE
WRITE ( NOUT, FMT = 99995 ) RANKE
WRITE ( NOUT, FMT = 99994 ) ( ISTAIR(I), I = 1,M )
END IF
END IF
STOP
*
99999 FORMAT (' MB04UD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB04UD = ',I2)
99997 FORMAT (' The transformed matrix E is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The computed rank of E = ',I2)
99994 FORMAT (/' ISTAIR is ',/20(1X,I5))
99993 FORMAT (/' M is out of range.',/' M = ',I5)
99992 FORMAT (/' N is out of range.',/' N = ',I5)
99991 FORMAT (' The transformed matrix A is ')
END
Program Data
MB04UD EXAMPLE PROGRAM DATA 4 4 0.0 2.0 0.0 2.0 -2.0 0.0 -2.0 0.0 2.0 2.0 0.0 -2.0 0.0 2.0 -2.0 0.0 2.0 1.0 0.0 1.0 -1.0 0.0 -1.0 0.0 1.0 1.0 0.0 -1.0 0.0 1.0 -1.0 0.0 1.0Program Results
MB04UD EXAMPLE PROGRAM RESULTS
The transformed matrix A is
0.5164 1.0328 1.1547 -2.3094
0.0000 -2.5820 0.0000 -1.1547
0.0000 0.0000 -3.4641 0.0000
0.0000 0.0000 0.0000 -3.4641
The transformed matrix E is
0.2582 0.5164 0.5774 -1.1547
0.0000 -1.2910 0.0000 -0.5774
0.0000 0.0000 -1.7321 0.0000
0.0000 0.0000 0.0000 -1.7321
The computed rank of E = 4
ISTAIR is
1 2 3 4
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04VD.html 0000664 0000000 0000000 00000046315 14560147231 0020357 0 ustar 00root root 0000000 0000000
Purpose
To compute orthogonal transformations Q and Z such that the
transformed pencil Q'(sE-A)Z is in upper block triangular form,
where E is an M-by-N matrix in column echelon form (see SLICOT
Library routine MB04UD) and A is an M-by-N matrix.
If MODE = 'B', then the matrices A and E are transformed into the
following generalized Schur form by unitary transformations Q1
and Z1 :
| sE(eps,inf)-A(eps,inf) | X |
Q1'(sE-A)Z1 = |------------------------|------------|. (1)
| O | sE(r)-A(r) |
The pencil sE(eps,inf)-A(eps,inf) is in staircase form, and it
contains all Kronecker column indices and infinite elementary
divisors of the pencil sE-A. The pencil sE(r)-A(r) contains all
Kronecker row indices and elementary divisors of sE-A.
Note: X is a pencil.
If MODE = 'T', then the submatrices having full row and column
rank in the pencil sE(eps,inf)-A(eps,inf) in (1) are
triangularized by applying unitary transformations Q2 and Z2 to
Q1'*(sE-A)*Z1.
If MODE = 'S', then the pencil sE(eps,inf)-A(eps,inf) in (1) is
separated into sE(eps)-A(eps) and sE(inf)-A(inf) by applying
unitary transformations Q3 and Z3 to Q2'*Q1'*(sE-A)*Z1*Z2.
This gives
| sE(eps)-A(eps) | X | X |
|----------------|----------------|------------|
| O | sE(inf)-A(inf) | X |
Q'(sE-A)Z =|=================================|============| (2)
| | |
| O | sE(r)-A(r) |
where Q = Q1*Q2*Q3 and Z = Z1*Z2*Z3.
Note: the pencil sE(r)-A(r) is not reduced further.
Specification
SUBROUTINE MB04VD( MODE, JOBQ, JOBZ, M, N, RANKE, A, LDA, E, LDE,
$ Q, LDQ, Z, LDZ, ISTAIR, NBLCKS, NBLCKI, IMUK,
$ INUK, IMUK0, MNEI, TOL, IWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBQ, JOBZ, MODE
INTEGER INFO, LDA, LDE, LDQ, LDZ, M, N, NBLCKI, NBLCKS,
$ RANKE
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IMUK(*), IMUK0(*), INUK(*), ISTAIR(*), IWORK(*),
$ MNEI(*)
DOUBLE PRECISION A(LDA,*), E(LDE,*), Q(LDQ,*), Z(LDZ,*)
Arguments
Mode Parameters
MODE CHARACTER*1
Specifies the desired structure of the transformed
pencil Q'(sE-A)Z to be computed as follows:
= 'B': Basic reduction given by (1);
= 'T': Further reduction of (1) to triangular form;
= 'S': Further separation of sE(eps,inf)-A(eps,inf)
in (1) into the two pencils in (2).
JOBQ CHARACTER*1
Indicates whether the user wishes to accumulate in a
matrix Q the orthogonal row transformations, as follows:
= 'N': Do not form Q;
= 'I': Q is initialized to the unit matrix and the
orthogonal transformation matrix Q is returned;
= 'U': The given matrix Q is updated by the orthogonal
row transformations used in the reduction.
JOBZ CHARACTER*1
Indicates whether the user wishes to accumulate in a
matrix Z the orthogonal column transformations, as
follows:
= 'N': Do not form Z;
= 'I': Z is initialized to the unit matrix and the
orthogonal transformation matrix Z is returned;
= 'U': The given matrix Z is updated by the orthogonal
transformations used in the reduction.
Input/Output Parameters
M (input) INTEGER
The number of rows in the matrices A, E and the order of
the matrix Q. M >= 0.
N (input) INTEGER
The number of columns in the matrices A, E and the order
of the matrix Z. N >= 0.
RANKE (input) INTEGER
The rank of the matrix E in column echelon form.
RANKE >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading M-by-N part of this array must
contain the matrix to be row compressed.
On exit, the leading M-by-N part of this array contains
the matrix that has been row compressed while keeping
matrix E in column echelon form.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,M).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading M-by-N part of this array must
contain the matrix in column echelon form to be
transformed equivalent to matrix A.
On exit, the leading M-by-N part of this array contains
the matrix that has been transformed equivalent to matrix
A.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,M).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,*)
On entry, if JOBQ = 'U', then the leading M-by-M part of
this array must contain a given matrix Q (e.g. from a
previous call to another SLICOT routine), and on exit, the
leading M-by-M part of this array contains the product of
the input matrix Q and the row transformation matrix used
to transform the rows of matrices A and E.
On exit, if JOBQ = 'I', then the leading M-by-M part of
this array contains the matrix of accumulated orthogonal
row transformations performed.
If JOBQ = 'N', the array Q is not referenced and can be
supplied as a dummy array (i.e. set parameter LDQ = 1 and
declare this array to be Q(1,1) in the calling program).
LDQ INTEGER
The leading dimension of array Q. If JOBQ = 'U' or
JOBQ = 'I', LDQ >= MAX(1,M); if JOBQ = 'N', LDQ >= 1.
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,*)
On entry, if JOBZ = 'U', then the leading N-by-N part of
this array must contain a given matrix Z (e.g. from a
previous call to another SLICOT routine), and on exit, the
leading N-by-N part of this array contains the product of
the input matrix Z and the column transformation matrix
used to transform the columns of matrices A and E.
On exit, if JOBZ = 'I', then the leading N-by-N part of
this array contains the matrix of accumulated orthogonal
column transformations performed.
If JOBZ = 'N', the array Z is not referenced and can be
supplied as a dummy array (i.e. set parameter LDZ = 1 and
declare this array to be Z(1,1) in the calling program).
LDZ INTEGER
The leading dimension of array Z. If JOBZ = 'U' or
JOBZ = 'I', LDZ >= MAX(1,N); if JOBZ = 'N', LDZ >= 1.
ISTAIR (input/output) INTEGER array, dimension (M)
On entry, this array must contain information on the
column echelon form of the unitary transformed matrix E.
Specifically, ISTAIR(i) must be set to +j if the first
non-zero element E(i,j) is a corner point and -j
otherwise, for i = 1,2,...,M.
On exit, this array contains no useful information.
NBLCKS (output) INTEGER
The number of submatrices having full row rank greater
than or equal to 0 detected in matrix A in the pencil
sE(x)-A(x),
where x = eps,inf if MODE = 'B' or 'T',
or x = eps if MODE = 'S'.
NBLCKI (output) INTEGER
If MODE = 'S', the number of diagonal submatrices in the
pencil sE(inf)-A(inf). If MODE = 'B' or 'T' then
NBLCKI = 0.
IMUK (output) INTEGER array, dimension (MAX(N,M+1))
The leading NBLCKS elements of this array contain the
column dimensions mu(1),...,mu(NBLCKS) of the submatrices
having full column rank in the pencil sE(x)-A(x),
where x = eps,inf if MODE = 'B' or 'T',
or x = eps if MODE = 'S'.
INUK (output) INTEGER array, dimension (MAX(N,M+1))
The leading NBLCKS elements of this array contain the
row dimensions nu(1),...,nu(NBLCKS) of the submatrices
having full row rank in the pencil sE(x)-A(x),
where x = eps,inf if MODE = 'B' or 'T',
or x = eps if MODE = 'S'.
IMUK0 (output) INTEGER array, dimension (limuk0),
where limuk0 = N if MODE = 'S' and 1, otherwise.
If MODE = 'S', then the leading NBLCKI elements of this
array contain the dimensions mu0(1),...,mu0(NBLCKI)
of the square diagonal submatrices in the pencil
sE(inf)-A(inf).
Otherwise, IMUK0 is not referenced and can be supplied
as a dummy array.
MNEI (output) INTEGER array, dimension (3)
If MODE = 'B' or 'T' then
MNEI(1) contains the row dimension of
sE(eps,inf)-A(eps,inf);
MNEI(2) contains the column dimension of
sE(eps,inf)-A(eps,inf);
MNEI(3) = 0.
If MODE = 'S', then
MNEI(1) contains the row dimension of sE(eps)-A(eps);
MNEI(2) contains the column dimension of sE(eps)-A(eps);
MNEI(3) contains the order of the regular pencil
sE(inf)-A(inf).
Tolerances
TOL DOUBLE PRECISION
A tolerance below which matrix elements are considered
to be zero. If the user sets TOL to be less than (or
equal to) zero then the tolerance is taken as
EPS * MAX( ABS(A(I,J)), ABS(E(I,J)) ), where EPS is the
machine precision (see LAPACK Library routine DLAMCH),
I = 1,2,...,M and J = 1,2,...,N.
Workspace
IWORK INTEGER array, dimension (N)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
> 0: if incorrect rank decisions were revealed during the
triangularization phase. This failure is not likely
to occur. The possible values are:
= 1: if incorrect dimensions of a full column rank
submatrix;
= 2: if incorrect dimensions of a full row rank
submatrix.
Method
Let sE - A be an arbitrary pencil. Prior to calling the routine, this pencil must be transformed into a pencil with E in column echelon form. This may be accomplished by calling the SLICOT Library routine MB04UD. Depending on the value of MODE, submatrices of A and E are then reduced to one of the forms described above. Further details can be found in [1].References
[1] Beelen, Th. and Van Dooren, P.
An improved algorithm for the computation of Kronecker's
canonical form of a singular pencil.
Linear Algebra and Applications, 105, pp. 9-65, 1988.
Numerical Aspects
It is shown in [1] that the algorithm is numerically backward stable. The operations count is proportional to (MAX(M,N))**3.Further Comments
The difference mu(k)-nu(k), for k = 1,2,...,NBLCKS, is the number of elementary Kronecker blocks of size k x (k+1). If MODE = 'B' or 'T' on entry, then the difference nu(k)-mu(k+1), for k = 1,2,...,NBLCKS, is the number of infinite elementary divisors of degree k (with mu(NBLCKS+1) = 0). If MODE = 'S' on entry, then the difference mu0(k)-mu0(k+1), for k = 1,2,...,NBLCKI, is the number of infinite elementary divisors of degree k (with mu0(NBLCKI+1) = 0). In the pencil sE(r)-A(r), the pencils sE(f)-A(f) and sE(eta)-A(eta) can be separated by pertransposing the pencil sE(r)-A(r) and calling the routine with MODE set to 'B'. The result has got to be pertransposed again. (For more details see [1]).Example
Program Text
* MB04VD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, NMAX
PARAMETER ( MMAX = 20, NMAX = 20 )
INTEGER LDA, LDE, LDQ, LDZ
PARAMETER ( LDA = MMAX, LDE = MMAX, LDQ = MMAX,
$ LDZ = NMAX )
INTEGER LINUK
PARAMETER ( LINUK = MAX( NMAX,MMAX+1 ) )
* PARAMETER ( LINUK = NMAX+MMAX+1 )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( NMAX,MMAX ) )
* PARAMETER ( LDWORK = NMAX+MMAX )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, J, M, N, NBLCKI, NBLCKS, RANKE
CHARACTER*1 JOBQ, JOBZ, MODE
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), E(LDE,NMAX),
$ Q(LDQ,MMAX), Z(LDZ,NMAX)
INTEGER IMUK(LINUK), IMUK0(NMAX), INUK(LINUK),
$ ISTAIR(MMAX), IWORK(LIWORK), MNEI(3)
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL MB04UD, MB04VD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, N, TOL, MODE
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99984 ) M
ELSE IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99983 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,M )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,M )
JOBQ = 'I'
JOBZ = 'I'
* Reduce E to column echelon form and compute Q'*A*Z.
CALL MB04UD( JOBQ, JOBZ, M, N, A, LDA, E, LDE, Q, LDQ, Z, LDZ,
$ RANKE, ISTAIR, TOL, DWORK, INFO )
JOBQ = 'U'
JOBZ = 'U'
*
IF ( INFO.EQ.0 ) THEN
* Compute a unitary transformed pencil Q'*(s*E-A)*Z.
CALL MB04VD( MODE, JOBQ, JOBZ, M, N, RANKE, A, LDA, E, LDE,
$ Q, LDQ, Z, LDZ, ISTAIR, NBLCKS, NBLCKI, IMUK,
$ INUK, IMUK0, MNEI, TOL, IWORK, INFO )
*
IF ( INFO.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99996 )
WRITE ( NOUT, FMT = 99995 )
DO 140 I = 1, M
WRITE ( NOUT, FMT = 99994 ) ( Q(I,J), J = 1,M )
140 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 160 I = 1, M
WRITE ( NOUT, FMT = 99994 ) ( E(I,J), J = 1,N )
160 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 180 I = 1, M
WRITE ( NOUT, FMT = 99994 ) ( A(I,J), J = 1,N )
180 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 200 I = 1, N
WRITE ( NOUT, FMT = 99994 ) ( Z(I,J), J = 1,N )
200 CONTINUE
WRITE ( NOUT, FMT = 99990 ) NBLCKS
IF ( .NOT. LSAME( MODE, 'S' ) ) THEN
WRITE ( NOUT, FMT = 99989 ) ( IMUK(I), I = 1,NBLCKS )
WRITE ( NOUT, FMT = 99988 ) ( INUK(I), I = 1,NBLCKS )
ELSE
WRITE ( NOUT, FMT = 99987 ) ( IMUK(I), I = 1,NBLCKS )
WRITE ( NOUT, FMT = 99986 ) ( INUK(I), I = 1,NBLCKS )
WRITE ( NOUT, FMT = 99982 ) ( IMUK0(I), I = 1,NBLCKI )
WRITE ( NOUT, FMT = 99985 ) ( MNEI(I), I = 1,3 )
END IF
ELSE
WRITE ( NOUT, FMT = 99998 ) INFO
END IF
ELSE
WRITE ( NOUT, FMT = 99997 ) INFO
END IF
END IF
STOP
*
99999 FORMAT (' MB04VD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB04VD = ',I2)
99997 FORMAT (' INFO on exit from MB04UD = ',I2)
99996 FORMAT (' The unitary transformed pencil is Q''*(s*E-A)*Z, where',
$ /)
99995 FORMAT (' Matrix Q',/)
99994 FORMAT (20(1X,F8.4))
99993 FORMAT (/' Matrix E',/)
99992 FORMAT (/' Matrix A',/)
99991 FORMAT (/' Matrix Z',/)
99990 FORMAT (/' The number of submatrices having full row rank detect',
$ 'ed in matrix A = ',I3)
99989 FORMAT (/' The column dimensions of the submatrices having full ',
$ 'column rank in the pencil',/' sE(eps,inf) - A(eps,inf) a',
$ 're',/20(1X,I5))
99988 FORMAT (/' The row dimensions of the submatrices having full row',
$ ' rank in the pencil',/' sE(eps,inf) - A(eps,inf) are',
$ /20(1X,I5))
99987 FORMAT (/' The column dimensions of the submatrices having full ',
$ 'column rank in the pencil',/' sE(eps) - A(eps) are',
$ /20(1X,I5))
99986 FORMAT (/' The row dimensions of the submatrices having full row',
$ ' rank in the pencil',/' sE(eps) - A(eps) are',/20(1X,I5))
99985 FORMAT (/' MNEI is ',/20(1X,I5))
99984 FORMAT (/' M is out of range.',/' M = ',I5)
99983 FORMAT (/' N is out of range.',/' N = ',I5)
99982 FORMAT (/' The orders of the diagonal submatrices in the pencil ',
$ 'sE(inf) - A(inf) are',/20(1X,I5))
END
Program Data
MB04VD EXAMPLE PROGRAM DATA 2 4 0.0 S 1.0 0.0 -1.0 0.0 1.0 1.0 0.0 -1.0 0.0 -1.0 0.0 0.0 0.0 -1.0 0.0 0.0Program Results
MB04VD EXAMPLE PROGRAM RESULTS
The unitary transformed pencil is Q'*(s*E-A)*Z, where
Matrix Q
0.7071 -0.7071
0.7071 0.7071
Matrix E
0.0000 0.0000 -1.1547 0.8165
0.0000 0.0000 0.0000 0.0000
Matrix A
0.0000 1.7321 0.5774 -0.4082
0.0000 0.0000 0.0000 -1.2247
Matrix Z
0.5774 0.8165 0.0000 0.0000
0.0000 0.0000 0.8165 -0.5774
0.5774 -0.4082 -0.4082 -0.5774
0.5774 -0.4082 0.4082 0.5774
The number of submatrices having full row rank detected in matrix A = 2
The column dimensions of the submatrices having full column rank in the pencil
sE(eps) - A(eps) are
2 1
The row dimensions of the submatrices having full row rank in the pencil
sE(eps) - A(eps) are
1 0
The orders of the diagonal submatrices in the pencil sE(inf) - A(inf) are
1
MNEI is
1 3 1
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04WD.html 0000664 0000000 0000000 00000014371 14560147231 0020355 0 ustar 00root root 0000000 0000000
Purpose
To generate a matrix Q with orthogonal columns (spanning an
isotropic subspace), which is defined as the first n columns
of a product of symplectic reflectors and Givens rotations,
Q = diag( H(1),H(1) ) G(1) diag( F(1),F(1) )
diag( H(2),H(2) ) G(2) diag( F(2),F(2) )
....
diag( H(k),H(k) ) G(k) diag( F(k),F(k) ).
The matrix Q is returned in terms of its first 2*M rows
[ op( Q1 ) op( Q2 ) ]
Q = [ ].
[ -op( Q2 ) op( Q1 ) ]
Blocked version of the SLICOT Library routine MB04WU.
Specification
SUBROUTINE MB04WD( TRANQ1, TRANQ2, M, N, K, Q1, LDQ1, Q2, LDQ2,
$ CS, TAU, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TRANQ1, TRANQ2
INTEGER INFO, K, LDQ1, LDQ2, LDWORK, M, N
C .. Array Arguments ..
DOUBLE PRECISION CS(*), DWORK(*), Q1(LDQ1,*), Q2(LDQ2,*), TAU(*)
Arguments
Mode Parameters
TRANQ1 CHARACTER*1
Specifies the form of op( Q1 ) as follows:
= 'N': op( Q1 ) = Q1;
= 'T': op( Q1 ) = Q1';
= 'C': op( Q1 ) = Q1'.
TRANQ2 CHARACTER*1
Specifies the form of op( Q2 ) as follows:
= 'N': op( Q2 ) = Q2;
= 'T': op( Q2 ) = Q2';
= 'C': op( Q2 ) = Q2'.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrices Q1 and Q2. M >= 0.
N (input) INTEGER
The number of columns of the matrices Q1 and Q2.
M >= N >= 0.
K (input) INTEGER
The number of symplectic Givens rotations whose product
partly defines the matrix Q. N >= K >= 0.
Q1 (input/output) DOUBLE PRECISION array, dimension
(LDQ1,N) if TRANQ1 = 'N',
(LDQ1,M) if TRANQ1 = 'T' or TRANQ1 = 'C'
On entry with TRANQ1 = 'N', the leading M-by-K part of
this array must contain in its i-th column the vector
which defines the elementary reflector F(i).
On entry with TRANQ1 = 'T' or TRANQ1 = 'C', the leading
K-by-M part of this array must contain in its i-th row
the vector which defines the elementary reflector F(i).
On exit with TRANQ1 = 'N', the leading M-by-N part of this
array contains the matrix Q1.
On exit with TRANQ1 = 'T' or TRANQ1 = 'C', the leading
N-by-M part of this array contains the matrix Q1'.
LDQ1 INTEGER
The leading dimension of the array Q1.
LDQ1 >= MAX(1,M), if TRANQ1 = 'N';
LDQ1 >= MAX(1,N), if TRANQ1 = 'T' or TRANQ1 = 'C'.
Q2 (input/output) DOUBLE PRECISION array, dimension
(LDQ2,N) if TRANQ2 = 'N',
(LDQ2,M) if TRANQ2 = 'T' or TRANQ2 = 'C'
On entry with TRANQ2 = 'N', the leading M-by-K part of
this array must contain in its i-th column the vector
which defines the elementary reflector H(i) and, on the
diagonal, the scalar factor of H(i).
On entry with TRANQ2 = 'T' or TRANQ2 = 'C', the leading
K-by-M part of this array must contain in its i-th row the
vector which defines the elementary reflector H(i) and, on
the diagonal, the scalar factor of H(i).
On exit with TRANQ2 = 'N', the leading M-by-N part of this
array contains the matrix Q2.
On exit with TRANQ2 = 'T' or TRANQ2 = 'C', the leading
N-by-M part of this array contains the matrix Q2'.
LDQ2 INTEGER
The leading dimension of the array Q2.
LDQ2 >= MAX(1,M), if TRANQ2 = 'N';
LDQ2 >= MAX(1,N), if TRANQ2 = 'T' or TRANQ2 = 'C'.
CS (input) DOUBLE PRECISION array, dimension (2*K)
On entry, the first 2*K elements of this array must
contain the cosines and sines of the symplectic Givens
rotations G(i).
TAU (input) DOUBLE PRECISION array, dimension (K)
On entry, the first K elements of this array must
contain the scalar factors of the elementary reflectors
F(i).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK, MAX(M+N,8*N*NB + 15*NB*NB), where NB is
the optimal block size determined by the function UE01MD.
On exit, if INFO = -13, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,M+N).
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
References
[1] Kressner, D.
Block algorithms for orthogonal symplectic factorizations.
BIT, 43 (4), pp. 775-790, 2003.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To generate an orthogonal symplectic matrix U, which is defined as
a product of symplectic reflectors and Givens rotations
U = diag( H(1),H(1) ) G(1) diag( F(1),F(1) )
diag( H(2),H(2) ) G(2) diag( F(2),F(2) )
....
diag( H(n-1),H(n-1) ) G(n-1) diag( F(n-1),F(n-1) ).
as returned by MB04PU. The matrix U is returned in terms of its
first N rows
[ U1 U2 ]
U = [ ].
[ -U2 U1 ]
Specification
SUBROUTINE MB04WP( N, ILO, U1, LDU1, U2, LDU2, CS, TAU, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER ILO, INFO, LDU1, LDU2, LDWORK, N
C .. Array Arguments ..
DOUBLE PRECISION CS(*), DWORK(*), U1(LDU1,*), U2(LDU2,*), TAU(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrices U1 and U2. N >= 0.
ILO (input) INTEGER
ILO must have the same value as in the previous call of
MB04PU. U is equal to the unit matrix except in the
submatrix
U([ilo+1:n n+ilo+1:2*n], [ilo+1:n n+ilo+1:2*n]).
1 <= ILO <= N, if N > 0; ILO = 1, if N = 0.
U1 (input/output) DOUBLE PRECISION array, dimension (LDU1,N)
On entry, the leading N-by-N part of this array must
contain in its i-th column the vector which defines the
elementary reflector F(i).
On exit, the leading N-by-N part of this array contains
the matrix U1.
LDU1 INTEGER
The leading dimension of the array U1. LDU1 >= MAX(1,N).
U2 (input/output) DOUBLE PRECISION array, dimension (LDU2,N)
On entry, the leading N-by-N part of this array must
contain in its i-th column the vector which defines the
elementary reflector H(i) and, on the subdiagonal, the
scalar factor of H(i).
On exit, the leading N-by-N part of this array contains
the matrix U2.
LDU2 INTEGER
The leading dimension of the array U2. LDU2 >= MAX(1,N).
CS (input) DOUBLE PRECISION array, dimension (2N-2)
On entry, the first 2N-2 elements of this array must
contain the cosines and sines of the symplectic Givens
rotations G(i).
TAU (input) DOUBLE PRECISION array, dimension (N-1)
On entry, the first N-1 elements of this array must
contain the scalar factors of the elementary reflectors
F(i).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -10, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,2*(N-ILO)).
For optimum performance LDWORK should be larger. (See
SLICOT Library routine MB04WD).
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Numerical Aspects
The algorithm requires O(N**3) floating point operations and is strongly backward stable.References
[1] C. F. VAN LOAN:
A symplectic method for approximating all the eigenvalues of
a Hamiltonian matrix.
Linear Algebra and its Applications, 61, pp. 233-251, 1984.
[2] D. KRESSNER:
Block algorithms for orthogonal symplectic factorizations.
BIT, 43 (4), pp. 775-790, 2003.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To generate orthogonal symplectic matrices U or V, defined as
products of symplectic reflectors and Givens rotations
U = diag( HU(1),HU(1) ) GU(1) diag( FU(1),FU(1) )
diag( HU(2),HU(2) ) GU(2) diag( FU(2),FU(2) )
....
diag( HU(n),HU(n) ) GU(n) diag( FU(n),FU(n) ),
V = diag( HV(1),HV(1) ) GV(1) diag( FV(1),FV(1) )
diag( HV(2),HV(2) ) GV(2) diag( FV(2),FV(2) )
....
diag( HV(n-1),HV(n-1) ) GV(n-1) diag( FV(n-1),FV(n-1) ),
as returned by the SLICOT Library routines MB04TS or MB04TB. The
matrices U and V are returned in terms of their first N/2 rows:
[ U1 U2 ] [ V1 V2 ]
U = [ ], V = [ ].
[ -U2 U1 ] [ -V2 V1 ]
Specification
SUBROUTINE MB04WR( JOB, TRANS, N, ILO, Q1, LDQ1, Q2, LDQ2, CS,
$ TAU, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOB, TRANS
INTEGER ILO, INFO, LDQ1, LDQ2, LDWORK, N
C .. Array Arguments ..
DOUBLE PRECISION CS(*), DWORK(*), Q1(LDQ1,*), Q2(LDQ2,*), TAU(*)
Arguments
Input/Output Parameters
JOB CHARACTER*1
Specifies whether the matrix U or the matrix V is
required:
= 'U': generate U;
= 'V': generate V.
TRANS CHARACTER*1
If JOB = 'U' then TRANS must have the same value as
the argument TRANA in the previous call of MB04TS or
MB04TB.
If JOB = 'V' then TRANS must have the same value as
the argument TRANB in the previous call of MB04TS or
MB04TB.
N (input) INTEGER
The order of the matrices Q1 and Q2. N >= 0.
ILO (input) INTEGER
ILO must have the same value as in the previous call of
MB04TS or MB04TB. U and V are equal to the unit matrix
except in the submatrices
U([ilo:n n+ilo:2*n], [ilo:n n+ilo:2*n]) and
V([ilo+1:n n+ilo+1:2*n], [ilo+1:n n+ilo+1:2*n]),
respectively.
1 <= ILO <= N, if N > 0; ILO = 1, if N = 0.
Q1 (input/output) DOUBLE PRECISION array, dimension (LDQ1,N)
On entry, if JOB = 'U' and TRANS = 'N' then the
leading N-by-N part of this array must contain in its i-th
column the vector which defines the elementary reflector
FU(i).
If JOB = 'U' and TRANS = 'T' or TRANS = 'C' then the
leading N-by-N part of this array must contain in its i-th
row the vector which defines the elementary reflector
FU(i).
If JOB = 'V' and TRANS = 'N' then the leading N-by-N
part of this array must contain in its i-th row the vector
which defines the elementary reflector FV(i).
If JOB = 'V' and TRANS = 'T' or TRANS = 'C' then the
leading N-by-N part of this array must contain in its i-th
column the vector which defines the elementary reflector
FV(i).
On exit, if JOB = 'U' and TRANS = 'N' then the leading
N-by-N part of this array contains the matrix U1.
If JOB = 'U' and TRANS = 'T' or TRANS = 'C' then the
leading N-by-N part of this array contains the matrix
U1**T.
If JOB = 'V' and TRANS = 'N' then the leading N-by-N
part of this array contains the matrix V1**T.
If JOB = 'V' and TRANS = 'T' or TRANS = 'C' then the
leading N-by-N part of this array contains the matrix V1.
LDQ1 INTEGER
The leading dimension of the array Q1. LDQ1 >= MAX(1,N).
Q2 (input/output) DOUBLE PRECISION array, dimension (LDQ2,N)
On entry, if JOB = 'U' then the leading N-by-N part of
this array must contain in its i-th column the vector
which defines the elementary reflector HU(i).
If JOB = 'V' then the leading N-by-N part of this array
must contain in its i-th row the vector which defines the
elementary reflector HV(i).
On exit, if JOB = 'U' then the leading N-by-N part of
this array contains the matrix U2.
If JOB = 'V' then the leading N-by-N part of this array
contains the matrix V2**T.
LDQ2 INTEGER
The leading dimension of the array Q2. LDQ2 >= MAX(1,N).
CS (input) DOUBLE PRECISION array, dimension (2N)
On entry, if JOB = 'U' then the first 2N elements of
this array must contain the cosines and sines of the
symplectic Givens rotations GU(i).
If JOB = 'V' then the first 2N-2 elements of this array
must contain the cosines and sines of the symplectic
Givens rotations GV(i).
TAU (input) DOUBLE PRECISION array, dimension (N)
On entry, if JOB = 'U' then the first N elements of
this array must contain the scalar factors of the
elementary reflectors FU(i).
If JOB = 'V' then the first N-1 elements of this array
must contain the scalar factors of the elementary
reflectors FV(i).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -12, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,2*(N-ILO+1)).
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
References
[1] Benner, P., Mehrmann, V., and Xu, H.
A numerically stable, structure preserving method for
computing the eigenvalues of real Hamiltonian or symplectic
pencils. Numer. Math., Vol 78 (3), pp. 329-358, 1998.
[2] Kressner, D.
Block algorithms for orthogonal symplectic factorizations.
BIT, 43 (4), pp. 775-790, 2003.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To generate a matrix Q with orthogonal columns (spanning an
isotropic subspace), which is defined as the first n columns
of a product of symplectic reflectors and Givens rotations,
Q = diag( H(1),H(1) ) G(1) diag( F(1),F(1) )
diag( H(2),H(2) ) G(2) diag( F(2),F(2) )
....
diag( H(k),H(k) ) G(k) diag( F(k),F(k) ).
The matrix Q is returned in terms of its first 2*M rows
[ op( Q1 ) op( Q2 ) ]
Q = [ ].
[ -op( Q2 ) op( Q1 ) ]
Specification
SUBROUTINE MB04WU( TRANQ1, TRANQ2, M, N, K, Q1, LDQ1, Q2, LDQ2,
$ CS, TAU, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TRANQ1, TRANQ2
INTEGER INFO, K, LDQ1, LDQ2, LDWORK, M, N
C .. Array Arguments ..
DOUBLE PRECISION CS(*), DWORK(*), Q1(LDQ1,*), Q2(LDQ2,*), TAU(*)
Arguments
Mode Parameters
TRANQ1 CHARACTER*1
Specifies the form of op( Q1 ) as follows:
= 'N': op( Q1 ) = Q1;
= 'T': op( Q1 ) = Q1';
= 'C': op( Q1 ) = Q1'.
TRANQ2 CHARACTER*1
Specifies the form of op( Q2 ) as follows:
= 'N': op( Q2 ) = Q2;
= 'T': op( Q2 ) = Q2';
= 'C': op( Q2 ) = Q2'.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrices Q1 and Q2. M >= 0.
N (input) INTEGER
The number of columns of the matrices Q1 and Q2.
M >= N >= 0.
K (input) INTEGER
The number of symplectic Givens rotations whose product
partly defines the matrix Q. N >= K >= 0.
Q1 (input/output) DOUBLE PRECISION array, dimension
(LDQ1,N) if TRANQ1 = 'N',
(LDQ1,M) if TRANQ1 = 'T' or TRANQ1 = 'C'
On entry with TRANQ1 = 'N', the leading M-by-K part of
this array must contain in its i-th column the vector
which defines the elementary reflector F(i).
On entry with TRANQ1 = 'T' or TRANQ1 = 'C', the leading
K-by-M part of this array must contain in its i-th row
the vector which defines the elementary reflector F(i).
On exit with TRANQ1 = 'N', the leading M-by-N part of this
array contains the matrix Q1.
On exit with TRANQ1 = 'T' or TRANQ1 = 'C', the leading
N-by-M part of this array contains the matrix Q1'.
LDQ1 INTEGER
The leading dimension of the array Q1.
LDQ1 >= MAX(1,M), if TRANQ1 = 'N';
LDQ1 >= MAX(1,N), if TRANQ1 = 'T' or TRANQ1 = 'C'.
Q2 (input/output) DOUBLE PRECISION array, dimension
(LDQ2,N) if TRANQ2 = 'N',
(LDQ2,M) if TRANQ2 = 'T' or TRANQ2 = 'C'
On entry with TRANQ2 = 'N', the leading M-by-K part of
this array must contain in its i-th column the vector
which defines the elementary reflector H(i) and, on the
diagonal, the scalar factor of H(i).
On entry with TRANQ2 = 'T' or TRANQ2 = 'C', the leading
K-by-M part of this array must contain in its i-th row the
vector which defines the elementary reflector H(i) and, on
the diagonal, the scalar factor of H(i).
On exit with TRANQ2 = 'N', the leading M-by-N part of this
array contains the matrix Q2.
On exit with TRANQ2 = 'T' or TRANQ2 = 'C', the leading
N-by-M part of this array contains the matrix Q2'.
LDQ2 INTEGER
The leading dimension of the array Q2.
LDQ2 >= MAX(1,M), if TRANQ2 = 'N';
LDQ2 >= MAX(1,N), if TRANQ2 = 'T' or TRANQ2 = 'C'.
CS (input) DOUBLE PRECISION array, dimension (2*K)
On entry, the first 2*K elements of this array must
contain the cosines and sines of the symplectic Givens
rotations G(i).
TAU (input) DOUBLE PRECISION array, dimension (K)
On entry, the first K elements of this array must
contain the scalar factors of the elementary reflectors
F(i).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -13, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,M+N).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
References
[1] Bunse-Gerstner, A.
Matrix factorizations for symplectic QR-like methods.
Linear Algebra Appl., 83, pp. 49-77, 1986.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute a basis for the left and/or right singular subspace of an M-by-N matrix A corresponding to its smallest singular values.Specification
SUBROUTINE MB04XD( JOBU, JOBV, M, N, RANK, THETA, A, LDA, U, LDU,
$ V, LDV, Q, INUL, TOL, RELTOL, DWORK, LDWORK,
$ IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER JOBU, JOBV
INTEGER INFO, IWARN, LDA, LDU, LDV, LDWORK, M, N, RANK
DOUBLE PRECISION RELTOL, THETA, TOL
C .. Array Arguments ..
LOGICAL INUL(*)
DOUBLE PRECISION A(LDA,*), DWORK(*), Q(*), U(LDU,*), V(LDV,*)
Arguments
Mode Parameters
JOBU CHARACTER*1
Specifies whether to compute the left singular subspace
as follows:
= 'N': Do not compute the left singular subspace;
= 'A': Return the (M - RANK) base vectors of the desired
left singular subspace in U;
= 'S': Return the first (min(M,N) - RANK) base vectors
of the desired left singular subspace in U.
JOBV CHARACTER*1
Specifies whether to compute the right singular subspace
as follows:
= 'N': Do not compute the right singular subspace;
= 'A': Return the (N - RANK) base vectors of the desired
right singular subspace in V;
= 'S': Return the first (min(M,N) - RANK) base vectors
of the desired right singular subspace in V.
Input/Output Parameters
M (input) INTEGER
The number of rows in matrix A. M >= 0.
N (input) INTEGER
The number of columns in matrix A. N >= 0.
RANK (input/output) INTEGER
On entry, if RANK < 0, then the rank of matrix A is
computed by the routine as the number of singular values
greater than THETA.
Otherwise, RANK must specify the rank of matrix A.
RANK <= min(M,N).
On exit, if RANK < 0 on entry, then RANK contains the
computed rank of matrix A. That is, the number of singular
values of A greater than THETA.
Otherwise, the user-supplied value of RANK may be changed
by the routine on exit if the RANK-th and the (RANK+1)-th
singular values of A are considered to be equal.
See also the description of parameter TOL below.
THETA (input/output) DOUBLE PRECISION
On entry, if RANK < 0, then THETA must specify an upper
bound on the smallest singular values of A corresponding
to the singular subspace to be computed. THETA >= 0.0.
Otherwise, THETA must specify an initial estimate (t say)
for computing an upper bound on the (min(M,N) - RANK)
smallest singular values of A. If THETA < 0.0, then t is
computed by the routine.
On exit, if RANK >= 0 on entry, then THETA contains the
computed upper bound such that precisely RANK singular
values of A are greater than THETA + TOL.
Otherwise, THETA is unchanged.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading M-by-N part of this array must contain the
matrix A from which the basis of a desired singular
subspace is to be computed.
NOTE that this array is destroyed.
LDA INTEGER
The leading dimension of array A. LDA >= max(1,M).
U (output) DOUBLE PRECISION array, dimension (LDU,*)
If JOBU = 'A', then the leading M-by-M part of this array
contains the (M - RANK) M-dimensional base vectors of the
desired left singular subspace of A corresponding to its
singular values less than or equal to THETA. These vectors
are stored in the i-th column(s) of U for which
INUL(i) = .TRUE., where i = 1,2,...,M.
If JOBU = 'S', then the leading M-by-min(M,N) part of this
array contains the first (min(M,N) - RANK) M-dimensional
base vectors of the desired left singular subspace of A
corresponding to its singular values less than or equal to
THETA. These vectors are stored in the i-th column(s) of U
for which INUL(i) = .TRUE., where i = 1,2,..., min(M,N).
Otherwise, U is not referenced (since JOBU = 'N') and can
be supplied as a dummy array (i.e. set parameter LDU = 1
and declare this array to be U(1,1) in the calling
program).
LDU INTEGER
The leading dimension of array U.
LDU >= max(1,M) if JOBU = 'A' or JOBU = 'S',
LDU >= 1 if JOBU = 'N'.
V (output) DOUBLE PRECISION array, dimension (LDV,*)
If JOBV = 'A', then the leading N-by-N part of this array
contains the (N - RANK) N-dimensional base vectors of the
desired right singular subspace of A corresponding to its
singular values less than or equal to THETA. These vectors
are stored in the i-th column(s) of V for which
INUL(i) = .TRUE., where i = 1,2,...,N.
If JOBV = 'S', then the leading N-by-min(M,N) part of this
array contains the first (min(M,N) - RANK) N-dimensional
base vectors of the desired right singular subspace of A
corresponding to its singular values less than or equal to
THETA. These vectors are stored in the i-th column(s) of V
for which INUL(i) = .TRUE., where i = 1,2,...,MIN( M,N).
Otherwise, V is not referenced (since JOBV = 'N') and can
be supplied as a dummy array (i.e. set parameter LDV = 1
and declare this array to be V(1,1) in the calling
program).
LDV INTEGER
The leading dimension of array V.
LDV >= max(1,N) if JOBV = 'A' or JOBV = 'S',
LDV >= 1 if JOBV = 'N'.
Q (output) DOUBLE PRECISION array, dimension (2*min(M,N)-1)
This array contains the partially diagonalized bidiagonal
matrix J computed from A, at the moment that the desired
singular subspace has been found. Specifically, the
leading p = min(M,N) entries of Q contain the diagonal
elements q(1),q(2),...,q(p) and the entries Q(p+1),
Q(p+2),...,Q(2*p-1) contain the superdiagonal elements
e(1),e(2),...,e(p-1) of J.
INUL (output) LOGICAL array, dimension (max(M,N))
If JOBU <> 'N' or JOBV <> 'N', then the indices of the
elements of this array with value .TRUE. indicate the
columns in U and/or V containing the base vectors of the
desired left and/or right singular subspace of A. They
also equal the indices of the diagonal elements of the
bidiagonal submatrices in the array Q, which correspond
to the computed singular subspaces.
Tolerances
TOL DOUBLE PRECISION
This parameter defines the multiplicity of singular values
by considering all singular values within an interval of
length TOL as coinciding. TOL is used in checking how many
singular values are less than or equal to THETA. Also in
computing an appropriate upper bound THETA by a bisection
method, TOL is used as a stopping criterion defining the
minimum (absolute) subinterval width. TOL is also taken
as an absolute tolerance for negligible elements in the
QR/QL iterations. If the user sets TOL to be less than or
equal to 0, then the tolerance is taken as specified in
SLICOT Library routine MB04YD document.
RELTOL DOUBLE PRECISION
This parameter specifies the minimum relative width of an
interval. When an interval is narrower than TOL, or than
RELTOL times the larger (in magnitude) endpoint, then it
is considered to be sufficiently small and bisection has
converged. If the user sets RELTOL to be less than
BASE * EPS, where BASE is machine radix and EPS is machine
precision (see LAPACK Library routine DLAMCH), then the
tolerance is taken as BASE * EPS.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK = max(1, LDW + max(2*P + max(M,N), LDY)), where
P = min(M,N);
LDW = max(2*N, N*(N+1)/2), if JOBU <> 'N' and M large
enough than N;
LDW = 0, otherwise;
LDY = 8*P - 5, if JOBU <> 'N' or JOBV <> 'N';
LDY = 6*P - 3, if JOBU = 'N' and JOBV = 'N'.
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: if the rank of matrix A (as specified by the user)
has been lowered because a singular value of
multiplicity greater than 1 was found.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the maximum number of QR/QL iteration steps
(30*MIN(M,N)) has been exceeded.
Method
The method used is the Partial Singular Value Decomposition (PSVD)
approach proposed by Van Huffel, Vandewalle and Haegemans, which
is an efficient technique (see [1]) for computing the singular
subspace of a matrix corresponding to its smallest singular
values. It differs from the classical SVD algorithm [3] at three
points, which results in high efficiency. Firstly, the Householder
transformations of the bidiagonalization need only to be applied
on the base vectors of the desired singular subspaces; secondly,
the bidiagonal matrix need only be partially diagonalized; and
thirdly, the convergence rate of the iterative diagonalization can
be improved by an appropriate choice between QL and QR iterations.
(Note, however, that LAPACK Library routine DGESVD, for computing
SVD, also uses either QL and QR iterations.) Depending on the gap,
the desired numerical accuracy and the dimension of the desired
singular subspace, the PSVD can be up to three times faster than
the classical SVD algorithm.
The PSVD algorithm [1-2] for an M-by-N matrix A proceeds as
follows:
Step 1: Bidiagonalization phase
-----------------------
(a) If M is large enough than N, transform A into upper
triangular form R.
(b) Transform A (or R) into bidiagonal form:
|q(1) e(1) 0 ... 0 |
(0) | 0 q(2) e(2) . |
J = | . . |
| . e(N-1)|
| 0 ... q(N) |
if M >= N, or
|q(1) 0 0 ... 0 0 |
(0) |e(1) q(2) 0 . . |
J = | . . . |
| . q(M-1) . |
| 0 ... e(M-1) q(M)|
if M < N, using Householder transformations.
In the second case, transform the matrix to the upper bidiagonal
form by applying Givens rotations.
(c) If U is requested, initialize U with the identity matrix.
If V is requested, initialize V with the identity matrix.
Step 2: Partial diagonalization phase
-----------------------------
If the upper bound THETA is not given, then compute THETA such
that precisely (min(M,N) - RANK) singular values of the bidiagonal
matrix are less than or equal to THETA, using a bisection method
[4]. Diagonalize the given bidiagonal matrix J partially, using
either QR iterations (if the upper left diagonal element of the
considered bidiagonal submatrix is larger than the lower right
diagonal element) or QL iterations, such that J is split into
unreduced bidiagonal submatrices whose singular values are either
all larger than THETA or all less than or equal to THETA.
Accumulate the Givens rotations in U and/or V (if desired).
Step 3: Back transformation phase
-------------------------
(a) Apply the Householder transformations of Step 1(b) onto the
columns of U and/or V associated with the bidiagonal
submatrices with all singular values less than or equal to
THETA (if U and/or V is desired).
(b) If M is large enough than N, and U is desired, then apply the
Householder transformations of Step 1(a) onto each computed
column of U in Step 3(a).
References
[1] Van Huffel, S., Vandewalle, J. and Haegemans, A.
An efficient and reliable algorithm for computing the singular
subspace of a matrix associated with its smallest singular
values.
J. Comput. and Appl. Math., 19, pp. 313-330, 1987.
[2] Van Huffel, S.
Analysis of the total least squares problem and its use in
parameter estimation.
Doctoral dissertation, Dept. of Electr. Eng., Katholieke
Universiteit Leuven, Belgium, June 1987.
[3] Chan, T.F.
An improved algorithm for computing the singular value
decomposition.
ACM TOMS, 8, pp. 72-83, 1982.
[4] Van Huffel, S. and Vandewalle, J.
The partial total least squares algorithm.
J. Comput. and Appl. Math., 21, pp. 333-341, 1988.
Numerical Aspects
Using the PSVD a large reduction in computation time can be gained in total least squares applications (cf [2 - 4]), in the computation of the null space of a matrix and in solving (non)homogeneous linear equations.Further Comments
NoneExample
Program Text
* MB04XD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, NMAX
PARAMETER ( MMAX = 20, NMAX = 20 )
INTEGER LDA, LDU, LDV
PARAMETER ( LDA = MMAX, LDU = MMAX, LDV = NMAX )
INTEGER MAXMN, MNMIN
PARAMETER ( MAXMN = MAX( MMAX, NMAX ),
$ MNMIN = MIN( MMAX, NMAX ) )
INTEGER LENGQ
PARAMETER ( LENGQ = 2*MNMIN-1 )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( 2*NMAX, NMAX*( NMAX+1 )/2 )
$ + MAX( 2*MNMIN + MAXMN, 8*MNMIN - 5 ) )
* .. Local Scalars ..
DOUBLE PRECISION RELTOL, THETA, THETA1, TOL
INTEGER I, INFO, IWARN, J, K, LOOP, M, MINMN, N, NCOLU,
$ NCOLV, RANK, RANK1
CHARACTER*1 JOBU, JOBV
LOGICAL LJOBUA, LJOBUS, LJOBVA, LJOBVS, WANTU, WANTV
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), Q(LENGQ),
$ U(LDU,MMAX), V(LDV,NMAX)
LOGICAL INUL(MAXMN)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL MB04XD
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, N, RANK, THETA, TOL, RELTOL, JOBU, JOBV
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99983 ) M
ELSE IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99982 ) N
ELSE IF ( RANK.GT.MNMIN ) THEN
WRITE ( NOUT, FMT = 99981 ) RANK
ELSE IF ( RANK.LT.0 .AND. THETA.LT.ZERO ) THEN
WRITE ( NOUT, FMT = 99980 ) THETA
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,M )
RANK1 = RANK
THETA1 = THETA
* Compute a basis for the left and right singular subspace of A.
CALL MB04XD( JOBU, JOBV, M, N, RANK, THETA, A, LDA, U, LDU, V,
$ LDV, Q, INUL, TOL, RELTOL, DWORK, LDWORK, IWARN,
$ INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( IWARN.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) IWARN
WRITE ( NOUT, FMT = 99996 ) RANK
ELSE
IF ( RANK1.LT.0 ) WRITE ( NOUT, FMT = 99996 ) RANK
END IF
IF ( THETA1.LT.ZERO ) WRITE ( NOUT, FMT = 99995 ) THETA
LJOBUA = LSAME( JOBU, 'A' )
LJOBUS = LSAME( JOBU, 'S' )
LJOBVA = LSAME( JOBV, 'A' )
LJOBVS = LSAME( JOBV, 'S' )
WANTU = LJOBUA.OR.LJOBUS
WANTV = LJOBVA.OR.LJOBVS
WRITE ( NOUT, FMT = 99994 )
MINMN = MIN( M, N )
LOOP = MINMN - 1
DO 20 I = 1, LOOP
K = I + MINMN
WRITE ( NOUT, FMT = 99993 ) I, I, Q(I), I, I + 1, Q(K)
20 CONTINUE
WRITE ( NOUT, FMT = 99992 ) MINMN, MINMN, Q(MINMN)
IF ( WANTU ) THEN
NCOLU = M
IF ( LJOBUS ) NCOLU = MINMN
WRITE ( NOUT, FMT = 99986 )
DO 40 I = 1, M
WRITE ( NOUT, FMT = 99985 ) ( U(I,J), J = 1,NCOLU )
40 CONTINUE
WRITE ( NOUT, FMT = 99991 ) NCOLU
WRITE ( NOUT, FMT = 99990 )
DO 60 I = 1, NCOLU
WRITE ( NOUT, FMT = 99989 ) I, INUL(I)
60 CONTINUE
END IF
IF ( WANTV ) THEN
NCOLV = N
IF ( LJOBVS ) NCOLV = MINMN
WRITE ( NOUT, FMT = 99984 )
DO 80 I = 1, N
WRITE ( NOUT, FMT = 99985 ) ( V(I,J), J = 1,NCOLV )
80 CONTINUE
WRITE ( NOUT, FMT = 99988 ) NCOLV
WRITE ( NOUT, FMT = 99987 )
DO 100 J = 1, NCOLV
WRITE ( NOUT, FMT = 99989 ) J, INUL(J)
100 CONTINUE
END IF
END IF
END IF
STOP
*
99999 FORMAT (' MB04XD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB04XD = ',I2)
99997 FORMAT (' IWARN on exit from MB04XD = ',I2,/)
99996 FORMAT (' The computed rank of matrix A = ',I3,/)
99995 FORMAT (' The computed value of THETA = ',F7.4,/)
99994 FORMAT (' The elements of the partially diagonalized bidiagonal ',
$ 'matrix are',/)
99993 FORMAT (2(' (',I1,',',I1,') = ',F7.4,2X))
99992 FORMAT (' (',I1,',',I1,') = ',F7.4,/)
99991 FORMAT (/' Left singular subspace corresponds to the i-th column',
$ '(s) of U for which ',/' INUL(i) = .TRUE., i = 1,...,',I1,
$ /)
99990 FORMAT (' i INUL(i)',/)
99989 FORMAT (I3,L8)
99988 FORMAT (/' Right singular subspace corresponds to the j-th colum',
$ 'n(s) of V for which ',/' INUL(j) = .TRUE., j = 1,...,',I1,
$ /)
99987 FORMAT (' j INUL(j)',/)
99986 FORMAT (' Matrix U',/)
99985 FORMAT (20(1X,F8.4))
99984 FORMAT (/' Matrix V',/)
99983 FORMAT (/' M is out of range.',/' M = ',I5)
99982 FORMAT (/' N is out of range.',/' N = ',I5)
99981 FORMAT (/' RANK is out of range.',/' RANK = ',I5)
99980 FORMAT (/' THETA must be at least zero.',/' THETA = ',F8.4)
END
Program Data
MB04XD EXAMPLE PROGRAM DATA 6 4 -1 0.001 0.0 0.0 A A 0.80010 0.39985 0.60005 0.89999 0.29996 0.69990 0.39997 0.82997 0.49994 0.60003 0.20012 0.79011 0.90013 0.20016 0.79995 0.85002 0.39998 0.80006 0.49985 0.99016 0.20002 0.90007 0.70009 1.02994Program Results
MB04XD EXAMPLE PROGRAM RESULTS The computed rank of matrix A = 3 The elements of the partially diagonalized bidiagonal matrix are (1,1) = 3.2280 (1,2) = -0.0287 (2,2) = 0.8714 (2,3) = 0.0168 (3,3) = 0.3698 (3,4) = 0.0000 (4,4) = 0.0001 Matrix U 0.8933 0.4328 -0.1209 0.2499 -0.5812 0.4913 -0.4493 0.8555 -0.2572 0.1617 -0.4608 -0.7379 -0.0079 0.2841 0.9588 -0.5352 0.1892 0.0525 0.0000 0.0000 0.0003 -0.1741 0.3389 -0.3397 0.0000 0.0000 0.0000 0.6482 0.5428 0.1284 0.0000 0.0000 0.0000 -0.4176 -0.0674 0.2819 Left singular subspace corresponds to the i-th column(s) of U for which INUL(i) = .TRUE., i = 1,...,6 i INUL(i) 1 F 2 F 3 F 4 T 5 T 6 T Matrix V -0.3967 -0.7096 0.4612 -0.3555 0.9150 -0.2557 0.2414 -0.5687 -0.0728 0.6526 0.5215 -0.2128 0.0000 0.0720 0.6761 0.7106 Right singular subspace corresponds to the j-th column(s) of V for which INUL(j) = .TRUE., j = 1,...,4 j INUL(j) 1 F 2 F 3 F 4 T
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04XY.html 0000664 0000000 0000000 00000012377 14560147231 0020407 0 ustar 00root root 0000000 0000000
Purpose
To apply the Householder transformations Pj stored in factored form into the columns of the array X, to the desired columns of the matrix U by premultiplication, and/or the Householder transformations Qj stored in factored form into the rows of the array X, to the desired columns of the matrix V by premultiplication. The Householder transformations Pj and Qj are stored as produced by LAPACK Library routine DGEBRD.Specification
SUBROUTINE MB04XY( JOBU, JOBV, M, N, X, LDX, TAUP, TAUQ, U,
$ LDU, V, LDV, INUL, INFO )
C .. Scalar Arguments ..
CHARACTER JOBU, JOBV
INTEGER INFO, LDU, LDV, LDX, M, N
C .. Array Arguments ..
LOGICAL INUL(*)
DOUBLE PRECISION TAUP(*), TAUQ(*), U(LDU,*), V(LDV,*),
$ X(LDX,*)
Arguments
Mode Parameters
JOBU CHARACTER*1
Specifies whether to transform the columns in U as
follows:
= 'N': Do not transform the columns in U;
= 'A': Transform the columns in U (U has M columns);
= 'S': Transform the columns in U (U has min(M,N)
columns).
JOBV CHARACTER*1
Specifies whether to transform the columns in V as
follows:
= 'N': Do not transform the columns in V;
= 'A': Transform the columns in V (V has N columns);
= 'S': Transform the columns in V (V has min(M,N)
columns).
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix X. M >= 0.
N (input) INTEGER
The number of columns of the matrix X. N >= 0.
X (input) DOUBLE PRECISION array, dimension (LDX,N)
The leading M-by-N part contains in the columns of its
lower triangle the Householder transformations Pj, and
in the rows of its upper triangle the Householder
transformations Qj in factored form.
X is modified by the routine but restored on exit.
LDX INTEGER
The leading dimension of the array X. LDX >= MAX(1,M).
TAUP (input) DOUBLE PRECISION array, dimension (MIN(M,N))
The scalar factors of the Householder transformations Pj.
TAUQ (input) DOUBLE PRECISION array, dimension (MIN(M,N))
The scalar factors of the Householder transformations Qj.
U (input/output) DOUBLE PRECISION array, dimension (LDU,*)
On entry, U contains the M-by-M (if JOBU = 'A') or
M-by-min(M,N) (if JOBU = 'S') matrix U.
On exit, the Householder transformations Pj have been
applied to each column i of U corresponding to a parameter
INUL(i) = .TRUE.
NOTE that U is not referenced if JOBU = 'N'.
LDU INTEGER
The leading dimension of the array U.
LDU >= MAX(1,M), if JOBU = 'A' or JOBU = 'S';
LDU >= 1, if JOBU = 'N'.
V (input/output) DOUBLE PRECISION array, dimension (LDV,*)
On entry, V contains the N-by-N (if JOBV = 'A') or
N-by-min(M,N) (if JOBV = 'S') matrix V.
On exit, the Householder transformations Qj have been
applied to each column i of V corresponding to a parameter
INUL(i) = .TRUE.
NOTE that V is not referenced if JOBV = 'N'.
LDV INTEGER
The leading dimension of the array V.
LDV >= MAX(1,M), if JOBV = 'A' or JOBV = 'S';
LDV >= 1, if JOBV = 'N'.
INUL (input) LOGICAL array, dimension (MAX(M,N))
INUL(i) = .TRUE. if the i-th column of U and/or V is to be
transformed, and INUL(i) = .FALSE., otherwise.
(1 <= i <= MAX(M,N)).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The Householder transformations Pj or Qj are applied to the columns of U or V indexed by I for which INUL(I) = .TRUE..Numerical Aspects
The algorithm is backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To partially diagonalize the bidiagonal matrix
|q(1) e(1) 0 ... 0 |
| 0 q(2) e(2) . |
J = | . . | (1)
| . e(MIN(M,N)-1)|
| 0 ... ... q(MIN(M,N)) |
using QR or QL iterations in such a way that J is split into
unreduced bidiagonal submatrices whose singular values are either
all larger than a given bound or are all smaller than (or equal
to) this bound. The left- and right-hand Givens rotations
performed on J (corresponding to each QR or QL iteration step) may
be optionally accumulated in the arrays U and V.
Specification
SUBROUTINE MB04YD( JOBU, JOBV, M, N, RANK, THETA, Q, E, U, LDU, V,
$ LDV, INUL, TOL, RELTOL, DWORK, LDWORK, IWARN,
$ INFO )
C .. Scalar Arguments ..
CHARACTER JOBU, JOBV
INTEGER INFO, IWARN, LDU, LDV, LDWORK, M, N, RANK
DOUBLE PRECISION RELTOL, THETA, TOL
C .. Array Arguments ..
LOGICAL INUL(*)
DOUBLE PRECISION DWORK(*), E(*), Q(*), U(LDU,*), V(LDV,*)
Arguments
Mode Parameters
JOBU CHARACTER*1
Indicates whether the user wishes to accumulate in a
matrix U the left-hand Givens rotations, as follows:
= 'N': Do not form U;
= 'I': U is initialized to the M-by-MIN(M,N) submatrix of
the unit matrix and the left-hand Givens rotations
are accumulated in U;
= 'U': The given matrix U is updated by the left-hand
Givens rotations used in the calculation.
JOBV CHARACTER*1
Indicates whether the user wishes to accumulate in a
matrix V the right-hand Givens rotations, as follows:
= 'N': Do not form V;
= 'I': V is initialized to the N-by-MIN(M,N) submatrix of
the unit matrix and the right-hand Givens
rotations are accumulated in V;
= 'U': The given matrix V is updated by the right-hand
Givens rotations used in the calculation.
Input/Output Parameters
M (input) INTEGER
The number of rows in matrix U. M >= 0.
N (input) INTEGER
The number of rows in matrix V. N >= 0.
RANK (input/output) INTEGER
On entry, if RANK < 0, then the rank of matrix J is
computed by the routine as the number of singular values
larger than THETA.
Otherwise, RANK must specify the rank of matrix J.
RANK <= MIN(M,N).
On exit, if RANK < 0 on entry, then RANK contains the
computed rank of J. That is, the number of singular
values of J larger than THETA.
Otherwise, the user-supplied value of RANK may be
changed by the routine on exit if the RANK-th and the
(RANK+1)-th singular values of J are considered to be
equal. See also the parameter TOL.
THETA (input/output) DOUBLE PRECISION
On entry, if RANK < 0, then THETA must specify an upper
bound on the smallest singular values of J. THETA >= 0.0.
Otherwise, THETA must specify an initial estimate (t say)
for computing an upper bound such that precisely RANK
singular values are greater than this bound.
If THETA < 0.0, then t is computed by the routine.
On exit, if RANK >= 0 on entry, then THETA contains the
computed upper bound such that precisely RANK singular
values of J are greater than THETA + TOL.
Otherwise, THETA is unchanged.
Q (input/output) DOUBLE PRECISION array, dimension
(MIN(M,N))
On entry, this array must contain the diagonal elements
q(1),q(2),...,q(MIN(M,N)) of the bidiagonal matrix J. That
is, Q(i) = J(i,i) for i = 1,2,...,MIN(M,N).
On exit, this array contains the leading diagonal of the
transformed bidiagonal matrix J.
E (input/output) DOUBLE PRECISION array, dimension
(MIN(M,N)-1)
On entry, this array must contain the superdiagonal
elements e(1),e(2),...,e(MIN(M,N)-1) of the bidiagonal
matrix J. That is, E(k) = J(k,k+1) for k = 1,2,...,
MIN(M,N)-1.
On exit, this array contains the superdiagonal of the
transformed bidiagonal matrix J.
U (input/output) DOUBLE PRECISION array, dimension (LDU,*)
On entry, if JOBU = 'U', the leading M-by-MIN(M,N) part
of this array must contain a left transformation matrix
applied to the original matrix of the problem, and
on exit, the leading M-by-MIN(M,N) part of this array
contains the product of the input matrix U and the
left-hand Givens rotations.
On exit, if JOBU = 'I', then the leading M-by-MIN(M,N)
part of this array contains the matrix of accumulated
left-hand Givens rotations used.
If JOBU = 'N', the array U is not referenced and can be
supplied as a dummy array (i.e. set parameter LDU = 1 and
declare this array to be U(1,1) in the calling program).
LDU INTEGER
The leading dimension of array U. If JOBU = 'U' or
JOBU = 'I', LDU >= MAX(1,M); if JOBU = 'N', LDU >= 1.
V (input/output) DOUBLE PRECISION array, dimension (LDV,*)
On entry, if JOBV = 'U', the leading N-by-MIN(M,N) part
of this array must contain a right transformation matrix
applied to the original matrix of the problem, and
on exit, the leading N-by-MIN(M,N) part of this array
contains the product of the input matrix V and the
right-hand Givens rotations.
On exit, if JOBV = 'I', then the leading N-by-MIN(M,N)
part of this array contains the matrix of accumulated
right-hand Givens rotations used.
If JOBV = 'N', the array V is not referenced and can be
supplied as a dummy array (i.e. set parameter LDV = 1 and
declare this array to be V(1,1) in the calling program).
LDV INTEGER
The leading dimension of array V. If JOBV = 'U' or
JOBV = 'I', LDV >= MAX(1,N); if JOBV = 'N', LDV >= 1.
INUL (input/output) LOGICAL array, dimension (MIN(M,N))
On entry, the leading MIN(M,N) elements of this array must
be set to .FALSE. unless the i-th columns of U (if JOBU =
'U') and V (if JOBV = 'U') already contain a computed base
vector of the desired singular subspace of the original
matrix, in which case INUL(i) must be set to .TRUE.
for 1 <= i <= MIN(M,N).
On exit, the indices of the elements of this array with
value .TRUE. indicate the indices of the diagonal entries
of J which belong to those bidiagonal submatrices whose
singular values are all less than or equal to THETA.
Tolerances
TOL DOUBLE PRECISION
This parameter defines the multiplicity of singular values
by considering all singular values within an interval of
length TOL as coinciding. TOL is used in checking how many
singular values are less than or equal to THETA. Also in
computing an appropriate upper bound THETA by a bisection
method, TOL is used as a stopping criterion defining the
minimum (absolute) subinterval width. TOL is also taken
as an absolute tolerance for negligible elements in the
QR/QL iterations. If the user sets TOL to be less than or
equal to 0, then the tolerance is taken as
EPS * MAX(ABS(Q(i)), ABS(E(k))), where EPS is the
machine precision (see LAPACK Library routine DLAMCH),
i = 1,2,...,MIN(M,N) and k = 1,2,...,MIN(M,N)-1.
RELTOL DOUBLE PRECISION
This parameter specifies the minimum relative width of an
interval. When an interval is narrower than TOL, or than
RELTOL times the larger (in magnitude) endpoint, then it
is considered to be sufficiently small and bisection has
converged. If the user sets RELTOL to be less than
BASE * EPS, where BASE is machine radix and EPS is machine
precision (see LAPACK Library routine DLAMCH), then the
tolerance is taken as BASE * EPS.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,6*MIN(M,N)-5), if JOBU = 'I' or 'U', or
JOBV = 'I' or 'U';
LDWORK >= MAX(1,4*MIN(M,N)-3), if JOBU = 'N' and
JOBV = 'N'.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: if the rank of the bidiagonal matrix J (as specified
by the user) has been lowered because a singular
value of multiplicity larger than 1 was found.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value; this includes values like RANK > MIN(M,N), or
THETA < 0.0 and RANK < 0;
= 1: if the maximum number of QR/QL iteration steps
(30*MIN(M,N)) has been exceeded.
Method
If the upper bound THETA is not specified by the user, then it is
computed by the routine (using a bisection method) such that
precisely (MIN(M,N) - RANK) singular values of J are less than or
equal to THETA + TOL.
The method used by the routine (see [1]) then proceeds as follows.
The unreduced bidiagonal submatrices of J(j), where J(j) is the
transformed bidiagonal matrix after the j-th iteration step, are
classified into the following three classes:
- C1 contains the bidiagonal submatrices with all singular values
> THETA,
- C2 contains the bidiagonal submatrices with all singular values
<= THETA and
- C3 contains the bidiagonal submatrices with singular values
> THETA and also singular values <= THETA.
If C3 is empty, then the partial diagonalization is complete, and
RANK is the sum of the dimensions of the bidiagonal submatrices of
C1.
Otherwise, QR or QL iterations are performed on each bidiagonal
submatrix of C3, until this bidiagonal submatrix has been split
into two bidiagonal submatrices. These two submatrices are then
classified and the iterations are restarted.
If the upper left diagonal element of the bidiagonal submatrix is
larger than its lower right diagonal element, then QR iterations
are performed, else QL iterations are used. The shift is taken as
the smallest diagonal element of the bidiagonal submatrix (in
magnitude) unless its value exceeds THETA, in which case it is
taken as zero.
References
[1] Van Huffel, S., Vandewalle, J. and Haegemans, A.
An efficient and reliable algorithm for computing the
singular subspace of a matrix associated with its smallest
singular values.
J. Comput. and Appl. Math., 19, pp. 313-330, 1987.
Numerical Aspects
The algorithm is backward stable. To avoid overflow, matrix J is scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value (and not much smaller than that, for maximal accuracy).Further Comments
NoneExample
Program Text
* MB04YD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, NMAX
PARAMETER ( MMAX = 20, NMAX = 20 )
INTEGER MNMIN
PARAMETER ( MNMIN = MIN( MMAX, NMAX ) )
INTEGER LDU, LDV
PARAMETER ( LDU = MMAX, LDV = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 6*MNMIN - 5 )
* .. Local Scalars ..
DOUBLE PRECISION RELTOL, THETA, TOL
INTEGER I, INFO, IWARN, J, M, MINMN, N, RANK, RANK1
CHARACTER*1 JOBU, JOBV
LOGICAL LJOBUU, LJOBVU
* .. Local Arrays ..
DOUBLE PRECISION DWORK(LDWORK), E(MNMIN-1), Q(MNMIN),
$ U(LDU,MNMIN), V(LDV,MNMIN)
LOGICAL INUL(MNMIN)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL MB04YD
* .. Intrinsic Functions ..
INTRINSIC MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, N, THETA, RANK, TOL, RELTOL, JOBU, JOBV
MINMN = MIN( M, N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) M
ELSE IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) N
ELSE IF ( RANK.GT.MINMN ) THEN
WRITE ( NOUT, FMT = 99986 ) RANK
ELSE IF ( RANK.LT.0 .AND. THETA.LT.ZERO ) THEN
WRITE ( NOUT, FMT = 99985 ) THETA
ELSE
READ ( NIN, FMT = * ) ( Q(I), I = 1,MINMN )
READ ( NIN, FMT = * ) ( E(I), I = 1,MINMN-1 )
RANK1 = RANK
LJOBUU = LSAME( JOBU, 'U' )
LJOBVU = LSAME( JOBV, 'U' )
IF ( LJOBUU ) READ ( NIN, FMT = * )
$ ( ( U(I,J), J = 1,MINMN ), I = 1,M )
IF ( LJOBVU ) READ ( NIN, FMT = * )
$ ( ( V(I,J), J = 1,MINMN ), I = 1,N )
* Initialise the array INUL.
DO 20 I = 1, MINMN
INUL(I) = .FALSE.
20 CONTINUE
IF ( LJOBUU.OR.LJOBVU ) READ ( NIN, FMT = * )
$ ( INUL(I), I = 1,MINMN )
* Compute the number of singular values of J > THETA.
CALL MB04YD( JOBU, JOBV, M, N, RANK, THETA, Q, E, U, LDU, V,
$ LDV, INUL, TOL, RELTOL, DWORK, LDWORK, IWARN,
$ INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( IWARN.NE.0 ) THEN
WRITE ( NOUT, FMT = 99993 ) IWARN
WRITE ( NOUT, FMT = 99984 ) RANK
END IF
WRITE ( NOUT, FMT = 99997 )
DO 160 I = 1, MINMN - 1
WRITE ( NOUT, FMT = 99996 ) I, I, Q(I), I, (I+1), E(I)
160 CONTINUE
WRITE ( NOUT, FMT = 99995 ) MINMN, MINMN, Q(MINMN)
IF ( RANK1.LT.0 ) WRITE ( NOUT, FMT = 99994 ) RANK, THETA
IF ( .NOT.LSAME( JOBV, 'N' ) ) THEN
WRITE ( NOUT, FMT = 99992 )
DO 180 I = 1, N
WRITE ( NOUT, FMT = 99991 ) ( V(I,J), J = 1,MINMN )
180 CONTINUE
END IF
IF ( ( .NOT.LSAME( JOBU, 'N' ) ) .AND.
$ ( .NOT.LSAME( JOBV, 'N' ) ) )
$ WRITE ( NOUT, FMT = 99990 )
IF ( .NOT.LSAME( JOBU, 'N' ) ) THEN
WRITE ( NOUT, FMT = 99989 )
DO 200 I = 1, M
WRITE ( NOUT, FMT = 99991 ) ( U(I,J), J = 1,MINMN )
200 CONTINUE
END IF
END IF
END IF
STOP
*
99999 FORMAT (' MB04YD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB04YD = ',I2)
99997 FORMAT (' The transformed bidiagonal matrix J is',/)
99996 FORMAT (2(' (',I1,',',I1,') = ',F7.4,2X))
99995 FORMAT (' (',I1,',',I1,') = ',F7.4)
99994 FORMAT (/' J has ',I2,' singular values >',F7.4,/)
99993 FORMAT (' IWARN on exit from MB04YD = ',I2,/)
99992 FORMAT (' The product of the right-hand Givens rotation matrices',
$ ' equals ')
99991 FORMAT (20(1X,F8.4))
99990 FORMAT (' ')
99989 FORMAT (' The product of the left-hand Givens rotation matrices ',
$ 'equals ')
99988 FORMAT (/' M is out of range.',/' M = ',I5)
99987 FORMAT (/' N is out of range.',/' N = ',I5)
99986 FORMAT (/' RANK is out of range.',/' RANK = ',I5)
99985 FORMAT (/' THETA must be at least zero.',/' THETA = ',F8.4)
99984 FORMAT (/' The computed rank of matrix J = ',I3,/)
END
Program Data
MB04YD EXAMPLE PROGRAM DATA 5 5 2.0 -1 0.0 0.0 N N 1.0 2.0 3.0 4.0 5.0 2.0 3.0 4.0 5.0Program Results
MB04YD EXAMPLE PROGRAM RESULTS The transformed bidiagonal matrix J is (1,1) = 0.4045 (1,2) = 0.0000 (2,2) = 1.9839 (2,3) = 0.0000 (3,3) = 3.4815 (3,4) = 0.0128 (4,4) = 5.3723 (4,5) = 0.0273 (5,5) = 7.9948 J has 3 singular values > 2.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB04YW.html 0000664 0000000 0000000 00000016404 14560147231 0020401 0 ustar 00root root 0000000 0000000
Purpose
To perform either one QR or QL iteration step onto the unreduced
bidiagonal submatrix Jk:
|D(l) E(l) 0 ... 0 |
| 0 D(l+1) E(l+1) . |
Jk = | . . |
| . . |
| . E(k-1)|
| 0 ... ... D(k) |
with k <= p and l >= 1, p = MIN(M,N), of the bidiagonal matrix J:
|D(1) E(1) 0 ... 0 |
| 0 D(2) E(2) . |
J = | . . |.
| . . |
| . E(p-1)|
| 0 ... ... D(p) |
Hereby, Jk is transformed to S' Jk T with S and T products of
Givens rotations. These Givens rotations S (respectively, T) are
postmultiplied into U (respectively, V), if UPDATU (respectively,
UPDATV) is .TRUE..
Specification
SUBROUTINE MB04YW( QRIT, UPDATU, UPDATV, M, N, L, K, SHIFT, D, E,
$ U, LDU, V, LDV, DWORK )
C .. Scalar Arguments ..
LOGICAL QRIT, UPDATU, UPDATV
INTEGER K, L, LDU, LDV, M, N
DOUBLE PRECISION SHIFT
C .. Array Arguments ..
DOUBLE PRECISION D( * ), DWORK( * ), E( * ), U( LDU, * ),
$ V( LDV, * )
Arguments
Mode Parameters
QRIT LOGICAL
Indicates whether a QR or QL iteration step is to be
taken (from larger end diagonal element towards smaller),
as follows:
= .TRUE. : QR iteration step (chase bulge from top to
bottom);
= .FALSE.: QL iteration step (chase bulge from bottom to
top).
UPDATU LOGICAL
Indicates whether the user wishes to accumulate in a
matrix U the left-hand Givens rotations S, as follows:
= .FALSE.: Do not form U;
= .TRUE. : The given matrix U is updated (postmultiplied)
by the left-hand Givens rotations S.
UPDATV LOGICAL
Indicates whether the user wishes to accumulate in a
matrix V the right-hand Givens rotations S, as follows:
= .FALSE.: Do not form V;
= .TRUE. : The given matrix V is updated (postmultiplied)
by the right-hand Givens rotations T.
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix U. M >= 0.
N (input) INTEGER
The number of rows of the matrix V. N >= 0.
L (input) INTEGER
The index of the first diagonal entry of the considered
unreduced bidiagonal submatrix Jk of J.
K (input) INTEGER
The index of the last diagonal entry of the considered
unreduced bidiagonal submatrix Jk of J.
SHIFT (input) DOUBLE PRECISION
Value of the shift used in the QR or QL iteration step.
D (input/output) DOUBLE PRECISION array, dimension (p)
where p = MIN(M,N)
On entry, D must contain the diagonal entries of the
bidiagonal matrix J.
On exit, D contains the diagonal entries of the
transformed bidiagonal matrix S' J T.
E (input/output) DOUBLE PRECISION array, dimension (p-1)
On entry, E must contain the superdiagonal entries of J.
On exit, E contains the superdiagonal entries of the
transformed matrix S' J T.
U (input/output) DOUBLE PRECISION array, dimension (LDU,p)
On entry, if UPDATU = .TRUE., U must contain the M-by-p
left transformation matrix.
On exit, if UPDATU = .TRUE., the Givens rotations S on the
left have been postmultiplied into U, i.e., U * S is
returned.
U is not referenced if UPDATU = .FALSE..
LDU INTEGER
The leading dimension of the array U.
LDU >= max(1,M) if UPDATU = .TRUE.;
LDU >= 1 if UPDATU = .FALSE..
V (input/output) DOUBLE PRECISION array, dimension (LDV,p)
On entry, if UPDATV = .TRUE., V must contain the N-by-p
right transformation matrix.
On exit, if UPDATV = .TRUE., the Givens rotations T on the
right have been postmultiplied into V, i.e., V * T is
returned.
V is not referenced if UPDATV = .FALSE..
LDV INTEGER
The leading dimension of the array V.
LDV >= max(1,N) if UPDATV = .TRUE.;
LDV >= 1 if UPDATV = .FALSE..
Workspace
DWORK DOUBLE PRECISION array, dimension (MAX(1,LDWORK))
LDWORK >= 4*MIN(M,N)-4, if UPDATU = UPDATV = .TRUE.;
LDWORK >= 2*MIN(M,N)-2, if
UPDATU = .TRUE. and UPDATV = .FALSE. or
UPDATV = .TRUE. and UPDATU = .FALSE.;
LDWORK >= 1, if UPDATU = UPDATV = .FALSE..
Method
QR iterations diagonalize the bidiagonal matrix by zeroing the super-diagonal elements of Jk from bottom to top. QL iterations diagonalize the bidiagonal matrix by zeroing the super-diagonal elements of Jk from top to bottom. The routine overwrites Jk with the bidiagonal matrix S' Jk T, where S and T are products of Givens rotations. T is essentially the orthogonal matrix that would be obtained by applying one implicit symmetric shift QR (QL) step onto the matrix Jk'Jk. This step factors the matrix (Jk'Jk - shift*I) into a product of an orthogonal matrix T and a upper (lower) triangular matrix. See [1,Sec.8.2-8.3] and [2] for more details.References
[1] Golub, G.H. and Van Loan, C.F.
Matrix Computations.
The Johns Hopkins University Press, Baltimore, Maryland, 1983.
[2] Bowdler, H., Martin, R.S. and Wilkinson, J.H.
The QR and QL algorithms for symmetric matrices.
Numer. Math., 11, pp. 293-306, 1968.
[3] Demmel, J. and Kahan, W.
Computing small singular values of bidiagonal matrices with
guaranteed high relative accuracy.
SIAM J. Sci. Statist. Comput., 11, pp. 873-912, 1990.
Numerical Aspects
The algorithm is backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To transform a Hamiltonian matrix
( A G )
H = ( T ) (1)
( Q -A )
into a square-reduced Hamiltonian matrix
( A' G' )
H' = ( T ) (2)
( Q' -A' )
T
by an orthogonal symplectic similarity transformation H' = U H U,
where
( U1 U2 )
U = ( ). (3)
( -U2 U1 )
T
The square-reduced Hamiltonian matrix satisfies Q'A' - A' Q' = 0,
and
2 T 2 ( A'' G'' )
H' := (U H U) = ( T ).
( 0 A'' )
In addition, A'' is upper Hessenberg and G'' is skew symmetric.
The square roots of the eigenvalues of A'' = A'*A' + G'*Q' are the
eigenvalues of H.
Specification
SUBROUTINE MB04ZD( COMPU, N, A, LDA, QG, LDQG, U, LDU, DWORK, INFO
$ )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDQG, LDU, N
CHARACTER COMPU
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), QG(LDQG,*), U(LDU,*)
Arguments
Mode Parameters
COMPU CHARACTER*1
Indicates whether the orthogonal symplectic similarity
transformation matrix U in (3) is returned or
accumulated into an orthogonal symplectic matrix, or if
the transformation matrix is not required, as follows:
= 'N': U is not required;
= 'I' or 'F': on entry, U need not be set;
on exit, U contains the orthogonal
symplectic matrix U from (3);
= 'V' or 'A': the orthogonal symplectic similarity
transformations are accumulated into U;
on input, U must contain an orthogonal
symplectic matrix S;
on exit, U contains S*U with U from (3).
See the description of U below for details.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, G, and Q. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On input, the leading N-by-N part of this array must
contain the upper left block A of the Hamiltonian matrix H
in (1).
On output, the leading N-by-N part of this array contains
the upper left block A' of the square-reduced Hamiltonian
matrix H' in (2).
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
QG (input/output) DOUBLE PRECISION array, dimension
(LDQG,N+1)
On input, the leading N-by-N lower triangular part of this
array must contain the lower triangle of the lower left
symmetric block Q of the Hamiltonian matrix H in (1), and
the N-by-N upper triangular part of the submatrix in the
columns 2 to N+1 of this array must contain the upper
triangle of the upper right symmetric block G of H in (1).
So, if i >= j, then Q(i,j) = Q(j,i) is stored in QG(i,j)
and G(i,j) = G(j,i) is stored in QG(j,i+1).
On output, the leading N-by-N lower triangular part of
this array contains the lower triangle of the lower left
symmetric block Q', and the N-by-N upper triangular part
of the submatrix in the columns 2 to N+1 of this array
contains the upper triangle of the upper right symmetric
block G' of the square-reduced Hamiltonian matrix H'
in (2).
LDQG INTEGER
The leading dimension of the array QG. LDQG >= MAX(1,N).
U (input/output) DOUBLE PRECISION array, dimension (LDU,2*N)
If COMPU = 'N', then this array is not referenced.
If COMPU = 'I' or 'F', then the input contents of this
array are not specified. On output, the leading
N-by-(2*N) part of this array contains the first N rows
of the orthogonal symplectic matrix U in (3).
If COMPU = 'V' or 'A', then, on input, the leading
N-by-(2*N) part of this array must contain the first N
rows of an orthogonal symplectic matrix S. On output, the
leading N-by-(2*N) part of this array contains the first N
rows of the product S*U where U is the orthogonal
symplectic matrix from (3).
The storage scheme implied by (3) is used for orthogonal
symplectic matrices, i.e., only the first N rows are
stored, as they contain all relevant information.
LDU INTEGER
The leading dimension of the array U.
LDU >= MAX(1,N), if COMPU <> 'N';
LDU >= 1, if COMPU = 'N'.
Workspace
DWORK DOUBLE PRECISION array, dimension (2*N)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, then the i-th argument had an illegal
value.
Method
The Hamiltonian matrix H is transformed into a square-reduced Hamiltonian matrix H' using the implicit version of Van Loan's method as proposed in [1,2,3].References
[1] Van Loan, C. F.
A Symplectic Method for Approximating All the Eigenvalues of
a Hamiltonian Matrix.
Linear Algebra and its Applications, 61, pp. 233-251, 1984.
[2] Byers, R.
Hamiltonian and Symplectic Algorithms for the Algebraic
Riccati Equation.
Ph. D. Thesis, Cornell University, Ithaca, NY, January 1983.
[3] Benner, P., Byers, R., and Barth, E.
Fortran 77 Subroutines for Computing the Eigenvalues of
Hamiltonian Matrices. I: The Square-Reduced Method.
ACM Trans. Math. Software, 26, 1, pp. 49-77, 2000.
Numerical Aspects
This algorithm requires approximately 20*N**3 flops for
transforming H into square-reduced form. If the transformations
are required, this adds another 8*N**3 flops. The method is
strongly backward stable in the sense that if H' and U are the
computed square-reduced Hamiltonian and computed orthogonal
symplectic similarity transformation, then there is an orthogonal
symplectic matrix T and a Hamiltonian matrix M such that
H T = T M
|| T - U || <= c1 * eps
|| H' - M || <= c2 * eps * || H ||
where c1, c2 are modest constants depending on the dimension N and
eps is the machine precision.
Eigenvalues computed by explicitly forming the upper Hessenberg
matrix A'' = A'A' + G'Q', with A', G', and Q' as in (2), and
applying the Hessenberg QR iteration to A'' are exactly
eigenvalues of a perturbed Hamiltonian matrix H + E, where
|| E || <= c3 * sqrt(eps) * || H ||,
and c3 is a modest constant depending on the dimension N and eps
is the machine precision. Moreover, if the norm of H and an
eigenvalue lambda are of roughly the same magnitude, the computed
eigenvalue is essentially as accurate as the computed eigenvalue
from traditional methods. See [1] or [2].
Further Comments
NoneExample
Program Text
* MB04ZD EXAMPLE PROGRAM TEXT.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA, LDQG, LDU
PARAMETER ( LDA = NMAX, LDQG = NMAX, LDU = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = ( NMAX+NMAX )*( NMAX+NMAX+1 ) )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* .. Local Scalars ..
INTEGER I, INFO, IJ, J, JI, N, POS, WPOS
CHARACTER*1 COMPU
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), QG(LDQG,NMAX+1),
$ U(LDU,NMAX)
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMM, DSYMV, MB04ZD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, COMPU
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99998 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( QG(J,I+1), I = J,N ), J = 1,N )
READ ( NIN, FMT = * ) ( ( QG(I,J), I = J,N ), J = 1,N )
* Square-reduce by symplectic orthogonal similarity.
CALL MB04ZD( COMPU, N, A, LDA, QG, LDQG, U, LDU, DWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
* Show the square-reduced Hamiltonian.
WRITE ( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99994 ) ( A(I,J), J = 1,N ),
$ ( QG(J,I+1), J = 1,I-1 ), ( QG(I,J+1), J = I,N )
10 CONTINUE
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99994 ) ( QG(I,J), J = 1,I-1 ),
$ ( QG(J,I), J = I,N ), ( -A(J,I), J = 1,N )
20 CONTINUE
* Show the square of H.
WRITE ( NOUT, FMT = 99995 )
WPOS = ( NMAX+NMAX )*( NMAX+NMAX )
* T
* Compute N11 = A*A + G*Q and set N22 = N11 .
CALL DGEMM( 'N', 'N', N, N, N, ONE, A, LDA, A, LDA, ZERO,
$ DWORK, N+N )
DO 30 I = 1, N
CALL DCOPY( N-I+1, QG(I,I), 1, DWORK(WPOS+I), 1 )
CALL DCOPY( I-1, QG(I,1), LDQG, DWORK(WPOS+1), 1 )
CALL DSYMV( 'U', N, ONE, QG(1,2), LDQG, DWORK(WPOS+1), 1,
$ ONE, DWORK((I-1)*(N+N)+1), 1 )
POS = N*( N+N ) + N + I
CALL DCOPY( N, DWORK((I-1)*(N+N)+1), 1, DWORK(POS), N+N )
30 CONTINUE
DO 40 I = 1, N
CALL DSYMV( 'U', N, -ONE, QG(1,2), LDQG, A(I,1), LDA,
$ ZERO, DWORK((N+I-1)*(N+N)+1), 1 )
CALL DSYMV( 'L', N, ONE, QG, LDQG, A(1,I), 1, ZERO,
$ DWORK((I-1)*(N+N)+N+1), 1 )
40 CONTINUE
DO 60 J = 1, N
DO 50 I = J, N
IJ = ( N+J-1 )*( N+N ) + I
JI = ( N+I-1 )*( N+N ) + J
DWORK(IJ) = DWORK(IJ) - DWORK(JI)
DWORK(JI) = -DWORK(IJ)
IJ = N + I + ( J-1 )*( N+N )
JI = N + J + ( I-1 )*( N+N )
DWORK(IJ) = DWORK(IJ) - DWORK(JI)
DWORK(JI) = -DWORK(IJ)
50 CONTINUE
60 CONTINUE
DO 70 I = 1, N+N
WRITE ( NOUT, FMT = 99994 )
$ ( DWORK(I+(J-1)*(N+N) ), J = 1,N+N )
70 CONTINUE
ENDIF
END IF
STOP
*
99999 FORMAT (' MB04ZD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (/' N is out of range.',/' N = ',I5)
99997 FORMAT (' INFO on exit from MB04ZD = ',I2)
99996 FORMAT (/' The square-reduced Hamiltonian is ')
99995 FORMAT (/' The square of the square-reduced Hamiltonian is ')
99994 FORMAT (1X,8(F10.4))
END
Program Data
MB04ZD EXAMPLE PROGRAM DATA 3 N 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 1.0 1.0 1.0 2.0 2.0 3.0 7.0 6.0 5.0 8.0 4.0 9.0Program Results
MB04ZD EXAMPLE PROGRAM RESULTS
The square-reduced Hamiltonian is
1.0000 3.3485 0.3436 1.0000 1.9126 -0.1072
6.7566 11.0750 -0.3014 1.9126 8.4479 -1.0790
2.3478 1.6899 -2.3868 -0.1072 -1.0790 -2.9871
7.0000 8.6275 -0.6352 -1.0000 -6.7566 -2.3478
8.6275 16.2238 -0.1403 -3.3485 -11.0750 -1.6899
-0.6352 -0.1403 1.2371 -0.3436 0.3014 2.3868
The square of the square-reduced Hamiltonian is
48.0000 80.6858 -2.5217 0.0000 1.8590 -10.5824
167.8362 298.4815 -4.0310 -1.8590 0.0000 -33.1160
0.0000 4.5325 2.5185 10.5824 33.1160 0.0000
0.0000 0.0000 0.0000 48.0000 167.8362 0.0000
0.0000 0.0000 0.0000 80.6858 298.4815 4.5325
0.0000 0.0000 0.0000 -2.5217 -4.0310 2.5185
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB05MD.html 0000664 0000000 0000000 00000025366 14560147231 0020352 0 ustar 00root root 0000000 0000000
Purpose
To compute exp(A*delta) where A is a real N-by-N non-defective matrix with real or complex eigenvalues and delta is a scalar value. The routine also returns the eigenvalues and eigenvectors of A as well as (if all eigenvalues are real) the matrix product exp(Lambda*delta) times the inverse of the eigenvector matrix of A, where Lambda is the diagonal matrix of eigenvalues. Optionally, the routine computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors.Specification
SUBROUTINE MB05MD( BALANC, N, DELTA, A, LDA, V, LDV, Y, LDY, VALR,
$ VALI, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER BALANC
INTEGER INFO, LDA, LDV, LDWORK, LDY, N
DOUBLE PRECISION DELTA
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), DWORK(*), V(LDV,*), VALI(*), VALR(*),
$ Y(LDY,*)
Arguments
Mode Parameters
BALANC CHARACTER*1
Indicates how the input matrix should be diagonally scaled
to improve the conditioning of its eigenvalues as follows:
= 'N': Do not diagonally scale;
= 'S': Diagonally scale the matrix, i.e. replace A by
D*A*D**(-1), where D is a diagonal matrix chosen
to make the rows and columns of A more equal in
norm. Do not permute.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
DELTA (input) DOUBLE PRECISION
The scalar value delta of the problem.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A of the problem.
On exit, the leading N-by-N part of this array contains
the solution matrix exp(A*delta).
LDA INTEGER
The leading dimension of array A. LDA >= max(1,N).
V (output) DOUBLE PRECISION array, dimension (LDV,N)
The leading N-by-N part of this array contains the
eigenvector matrix for A.
If the k-th eigenvalue is real the k-th column of the
eigenvector matrix holds the eigenvector corresponding
to the k-th eigenvalue.
Otherwise, the k-th and (k+1)-th eigenvalues form a
complex conjugate pair and the k-th and (k+1)-th columns
of the eigenvector matrix hold the real and imaginary
parts of the eigenvectors corresponding to these
eigenvalues as follows.
If p and q denote the k-th and (k+1)-th columns of the
eigenvector matrix, respectively, then the eigenvector
corresponding to the complex eigenvalue with positive
(negative) imaginary value is given by
2
p + q*j (p - q*j), where j = -1.
LDV INTEGER
The leading dimension of array V. LDV >= max(1,N).
Y (output) DOUBLE PRECISION array, dimension (LDY,N)
The leading N-by-N part of this array contains an
intermediate result for computing the matrix exponential.
Specifically, exp(A*delta) is obtained as the product V*Y,
where V is the matrix stored in the leading N-by-N part of
the array V. If all eigenvalues of A are real, then the
leading N-by-N part of this array contains the matrix
product exp(Lambda*delta) times the inverse of the (right)
eigenvector matrix of A, where Lambda is the diagonal
matrix of eigenvalues.
LDY INTEGER
The leading dimension of array Y. LDY >= max(1,N).
VALR (output) DOUBLE PRECISION array, dimension (N)
VALI (output) DOUBLE PRECISION array, dimension (N)
These arrays contain the real and imaginary parts,
respectively, of the eigenvalues of the matrix A. The
eigenvalues are unordered except that complex conjugate
pairs of values appear consecutively with the eigenvalue
having positive imaginary part first.
Workspace
IWORK INTEGER array, dimension (N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK, and if N > 0, DWORK(2) returns the reciprocal
condition number of the triangular matrix used to obtain
the inverse of the eigenvector matrix.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= max(1,4*N).
For good performance, LDWORK must generally be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= i: if INFO = i, the QR algorithm failed to compute all
the eigenvalues; no eigenvectors have been computed;
elements i+1:N of VALR and VALI contain eigenvalues
which have converged;
= N+1: if the inverse of the eigenvector matrix could not
be formed due to an attempt to divide by zero, i.e.,
the eigenvector matrix is singular;
= N+2: if the matrix A is defective, possibly due to
rounding errors.
Method
This routine is an implementation of "Method 15" of the set of methods described in reference [1], which uses an eigenvalue/ eigenvector decomposition technique. A modification of LAPACK Library routine DGEEV is used for obtaining the right eigenvector matrix. A condition estimate is then employed to determine if the matrix A is near defective and hence the exponential solution is inaccurate. In this case the routine returns with the Error Indicator (INFO) set to N+2, and SLICOT Library routines MB05ND or MB05OD are the preferred alternative routines to be used.References
[1] Moler, C.B. and Van Loan, C.F.
Nineteen dubious ways to compute the exponential of a matrix.
SIAM Review, 20, pp. 801-836, 1978.
[2] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
Numerical Aspects
3 The algorithm requires 0(N ) operations.Further Comments
NoneExample
Program Text
* MB05MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA, LDV, LDY
PARAMETER ( LDA = NMAX, LDV = NMAX, LDY = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 4*NMAX )
* .. Local Scalars ..
DOUBLE PRECISION DELTA
INTEGER I, INFO, J, N
CHARACTER*1 BALANC
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), V(LDV,NMAX),
$ VALI(NMAX), VALR(NMAX), Y(LDY,NMAX)
INTEGER IWORK(NMAX)
* .. External Subroutines ..
EXTERNAL MB05MD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
BALANC = 'N'
READ ( NIN, FMT = * ) N, DELTA
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
* Find the exponential of the real non-defective matrix A*DELTA.
CALL MB05MD( BALANC, N, DELTA, A, LDA, V, LDV, Y, LDY, VALR,
$ VALI, IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( A(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 ) ( VALR(I), VALI(I), I = 1,N )
WRITE ( NOUT, FMT = 99994 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( V(I,J), J = 1,N )
40 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( Y(I,J), J = 1,N )
60 CONTINUE
END IF
END IF
STOP
*
99999 FORMAT (' MB05MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB05MD = ',I2)
99997 FORMAT (' The solution matrix exp(A*DELTA) is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The eigenvalues of A are ',/20(2F5.1,'*j '))
99994 FORMAT (/' The eigenvector matrix for A is ')
99993 FORMAT (/' The inverse eigenvector matrix for A (premultiplied by'
$ ,' exp(Lambda*DELTA)) is ')
99992 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
MB05MD EXAMPLE PROGRAM DATA 4 1.0 0.5 0.0 2.3 -2.6 0.0 0.5 -1.4 -0.7 2.3 -1.4 0.5 0.0 -2.6 -0.7 0.0 0.5Program Results
MB05MD EXAMPLE PROGRAM RESULTS The solution matrix exp(A*DELTA) is 26.8551 -3.2824 18.7409 -19.4430 -3.2824 4.3474 -5.1848 0.2700 18.7409 -5.1848 15.6012 -11.7228 -19.4430 0.2700 -11.7228 15.6012 The eigenvalues of A are -3.0 0.0*j 4.0 0.0*j -1.0 0.0*j 2.0 0.0*j The eigenvector matrix for A is -0.7000 0.7000 0.1000 -0.1000 0.1000 -0.1000 0.7000 -0.7000 0.5000 0.5000 0.5000 0.5000 -0.5000 -0.5000 0.5000 0.5000 The inverse eigenvector matrix for A (premultiplied by exp(Lambda*DELTA)) is -0.0349 0.0050 0.0249 -0.0249 38.2187 -5.4598 27.2991 -27.2991 0.0368 0.2575 0.1839 0.1839 -0.7389 -5.1723 3.6945 3.6945
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB05MY.html 0000664 0000000 0000000 00000014024 14560147231 0020364 0 ustar 00root root 0000000 0000000
Purpose
To compute, for an N-by-N real nonsymmetric matrix A, the
orthogonal matrix Q reducing it to real Schur form T, the
eigenvalues, and the right eigenvectors of T.
The right eigenvector r(j) of T satisfies
T * r(j) = lambda(j) * r(j)
where lambda(j) is its eigenvalue.
The matrix of right eigenvectors R is upper triangular, by
construction.
Specification
SUBROUTINE MB05MY( BALANC, N, A, LDA, WR, WI, R, LDR, Q, LDQ,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER BALANC
INTEGER INFO, LDA, LDQ, LDR, LDWORK, N
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), DWORK( * ), Q( LDQ, * ),
$ R( LDR, * ), WI( * ), WR( * )
Arguments
Mode Parameters
BALANC CHARACTER*1
Indicates how the input matrix should be diagonally scaled
to improve the conditioning of its eigenvalues as follows:
= 'N': Do not diagonally scale;
= 'S': Diagonally scale the matrix, i.e. replace A by
D*A*D**(-1), where D is a diagonal matrix chosen
to make the rows and columns of A more equal in
norm. Do not permute.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the given matrix A.
On exit, the leading N-by-N upper quasi-triangular part of
this array contains the real Schur canonical form of A.
LDA INTEGER
The leading dimension of array A. LDA >= max(1,N).
WR (output) DOUBLE PRECISION array, dimension (N)
WI (output) DOUBLE PRECISION array, dimension (N)
WR and WI contain the real and imaginary parts,
respectively, of the computed eigenvalues. Complex
conjugate pairs of eigenvalues appear consecutively
with the eigenvalue having the positive imaginary part
first.
R (output) DOUBLE PRECISION array, dimension (LDR,N)
The leading N-by-N upper triangular part of this array
contains the matrix of right eigenvectors R, in the same
order as their eigenvalues. The real and imaginary parts
of a complex eigenvector corresponding to an eigenvalue
with positive imaginary part are stored in consecutive
columns. (The corresponding conjugate eigenvector is not
stored.) The eigenvectors are not backward transformed
for balancing (when BALANC = 'S').
LDR INTEGER
The leading dimension of array R. LDR >= max(1,N).
Q (output) DOUBLE PRECISION array, dimension (LDQ,N)
The leading N-by-N part of this array contains the
orthogonal matrix Q which has reduced A to real Schur
form.
LDQ INTEGER
The leading dimension of array Q. LDQ >= MAX(1,N).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
If BALANC = 'S' and LDWORK > 0, DWORK(2),...,DWORK(N+1)
return the scaling factors used for balancing.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= max(1,4*N).
For good performance, LDWORK must generally be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, the QR algorithm failed to compute all
the eigenvalues, and no eigenvectors have been
computed; elements i+1:N of WR and WI contain
eigenvalues which have converged.
Method
This routine uses the QR algorithm to obtain the real Schur form T of matrix A. Then, the right eigenvectors of T are computed, but they are not backtransformed into the eigenvectors of A. MB05MY is a modification of the LAPACK driver routine DGEEV.References
[1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
Numerical Aspects
3 The algorithm requires 0(N ) operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute (a) F(delta) = exp(A*delta) and (b) H(delta) = Int[F(s) ds] from s = 0 to s = delta, where A is a real N-by-N matrix and delta is a scalar value.Specification
SUBROUTINE MB05ND( N, DELTA, A, LDA, EX, LDEX, EXINT, LDEXIN,
$ TOL, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDEX, LDEXIN, LDWORK, N
DOUBLE PRECISION DELTA, TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), DWORK(*), EX(LDEX,*), EXINT(LDEXIN,*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
DELTA (input) DOUBLE PRECISION
The scalar value delta of the problem.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
matrix A of the problem. (Array A need not be set if
DELTA = 0.)
LDA INTEGER
The leading dimension of array A. LDA >= max(1,N).
EX (output) DOUBLE PRECISION array, dimension (LDEX,N)
The leading N-by-N part of this array contains an
approximation to F(delta).
LDEX INTEGER
The leading dimension of array EX. LDEX >= MAX(1,N).
EXINT (output) DOUBLE PRECISION array, dimension (LDEXIN,N)
The leading N-by-N part of this array contains an
approximation to H(delta).
LDEXIN INTEGER
The leading dimension of array EXINT. LDEXIN >= MAX(1,N).
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in determining the order of the
Pade approximation to H(t), where t is a scale factor
determined by the routine. A reasonable value for TOL may
be SQRT(EPS), where EPS is the machine precision (see
LAPACK Library routine DLAMCH).
Workspace
IWORK INTEGER array, dimension (N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,N*(N+1)).
For optimum performance LDWORK should be larger (2*N*N).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, the (i,i) element of the denominator of
the Pade approximation is zero, so the denominator
is exactly singular;
= N+1: if DELTA = (delta * frobenius norm of matrix A) is
probably too large to permit meaningful computation.
That is, DELTA > SQRT(BIG), where BIG is a
representable number near the overflow threshold of
the machine (see LAPACK Library Routine DLAMCH).
Method
This routine uses a Pade approximation to H(t) for some small value of t (where 0 < t <= delta) and then calculates F(t) from H(t). Finally, the results are re-scaled to give F(delta) and H(delta). For a detailed description of the implementation of this algorithm see [1].References
[1] Benson, C.J.
The numerical evaluation of the matrix exponential and its
integral.
Report 82/03, Control Systems Research Group,
School of Electronic Engineering and Computer
Science, Kingston Polytechnic, January 1982.
[2] Ward, R.C.
Numerical computation of the matrix exponential with accuracy
estimate.
SIAM J. Numer. Anal., 14, pp. 600-610, 1977.
[3] Moler, C.B. and Van Loan, C.F.
Nineteen Dubious Ways to Compute the Exponential of a Matrix.
SIAM Rev., 20, pp. 801-836, 1978.
Numerical Aspects
3 The algorithm requires 0(N ) operations.Further Comments
NoneExample
Program Text
* MB05ND EXAMPLE PROGRAM TEXT.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA, LDEX, LDEXIN, LDWORK
PARAMETER ( LDA = NMAX, LDEX = NMAX, LDEXIN = NMAX,
$ LDWORK = NMAX*( NMAX+1 ) )
* .. Local Scalars ..
DOUBLE PRECISION DELTA, TOL
INTEGER I, INFO, J, N
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), EX(LDEX,NMAX),
$ EXINT(LDEXIN,NMAX)
INTEGER IWORK(NMAX)
* .. External Subroutines ..
EXTERNAL MB05ND
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, DELTA, TOL
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
* Find the matrix exponential of A*DELTA and its integral.
CALL MB05ND( N, DELTA, A, LDA, EX, LDEX, EXINT, LDEXIN, TOL,
$ IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( EX(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( EXINT(I,J), J = 1,N )
40 CONTINUE
END IF
END IF
STOP
*
99999 FORMAT (' MB05ND EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB05ND = ',I2)
99997 FORMAT (' The solution matrix exp(A*DELTA) is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' and its integral is ')
99994 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
MB05ND EXAMPLE PROGRAM DATA 5 0.1 0.0001 5.0 4.0 3.0 2.0 1.0 1.0 6.0 0.0 4.0 3.0 2.0 0.0 7.0 6.0 5.0 1.0 3.0 1.0 8.0 7.0 2.0 5.0 7.0 1.0 9.0Program Results
MB05ND EXAMPLE PROGRAM RESULTS The solution matrix exp(A*DELTA) is 1.8391 0.9476 0.7920 0.8216 0.7811 0.3359 2.2262 0.4013 1.0078 1.0957 0.6335 0.6776 2.6933 1.6155 1.8502 0.4804 1.1561 0.9110 2.7461 2.0854 0.7105 1.4244 1.8835 1.0966 3.4134 and its integral is 0.1347 0.0352 0.0284 0.0272 0.0231 0.0114 0.1477 0.0104 0.0369 0.0368 0.0218 0.0178 0.1624 0.0580 0.0619 0.0152 0.0385 0.0267 0.1660 0.0732 0.0240 0.0503 0.0679 0.0317 0.1863
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB05OD.html 0000664 0000000 0000000 00000017635 14560147231 0020354 0 ustar 00root root 0000000 0000000
Purpose
To compute exp(A*delta) where A is a real N-by-N matrix and delta is a scalar value. The routine also returns the minimal number of accurate digits in the 1-norm of exp(A*delta) and the number of accurate digits in the 1-norm of exp(A*delta) at 95% confidence level.Specification
SUBROUTINE MB05OD( BALANC, N, NDIAG, DELTA, A, LDA, MDIG, IDIG,
$ IWORK, DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER BALANC
INTEGER IDIG, INFO, IWARN, LDA, LDWORK, MDIG, N,
$ NDIAG
DOUBLE PRECISION DELTA
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), DWORK(*)
Arguments
Mode Parameters
BALANC CHARACTER*1
Specifies whether or not a balancing transformation (done
by SLICOT Library routine MB04MD) is required, as follows:
= 'N', do not use balancing;
= 'S', use balancing (scaling).
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
NDIAG (input) INTEGER
The specified order of the diagonal Pade approximant.
In the absence of further information NDIAG should
be set to 9. NDIAG should not exceed 15. NDIAG >= 1.
DELTA (input) DOUBLE PRECISION
The scalar value delta of the problem.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On input, the leading N-by-N part of this array must
contain the matrix A of the problem. (This is not needed
if DELTA = 0.)
On exit, if INFO = 0, the leading N-by-N part of this
array contains the solution matrix exp(A*delta).
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
MDIG (output) INTEGER
The minimal number of accurate digits in the 1-norm of
exp(A*delta).
IDIG (output) INTEGER
The number of accurate digits in the 1-norm of
exp(A*delta) at 95% confidence level.
Workspace
IWORK INTEGER array, dimension (N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= N*(2*N+NDIAG+1)+NDIAG, if N > 1.
LDWORK >= 1, if N <= 1.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: if MDIG = 0 and IDIG > 0, warning for possible
inaccuracy (the exponential has been computed);
= 2: if MDIG = 0 and IDIG = 0, warning for severe
inaccuracy (the exponential has been computed);
= 3: if balancing has been requested, but it failed to
reduce the matrix norm and was not actually used.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the norm of matrix A*delta (after a possible
balancing) is too large to obtain an accurate
result;
= 2: if the coefficient matrix (the denominator of the
Pade approximant) is exactly singular; try a
different value of NDIAG;
= 3: if the solution exponential would overflow, possibly
due to a too large value DELTA; the calculations
stopped prematurely. This error is not likely to
appear.
Method
The exponential of the matrix A is evaluated from a diagonal Pade
approximant. This routine is a modification of the subroutine
PADE, described in reference [1]. The routine implements an
algorithm which exploits the identity
(exp[(2**-m)*A]) ** (2**m) = exp(A),
where m is an integer determined by the algorithm, to improve the
accuracy for matrices with large norms.
References
[1] Ward, R.C.
Numerical computation of the matrix exponential with accuracy
estimate.
SIAM J. Numer. Anal., 14, pp. 600-610, 1977.
Numerical Aspects
3 The algorithm requires 0(N ) operations.Further Comments
NoneExample
Program Text
* MB05OD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA
PARAMETER ( LDA = NMAX )
INTEGER NDIAG
PARAMETER ( NDIAG = 9 )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX*( 2*NMAX+NDIAG+1 )+NDIAG )
* .. Local Scalars ..
DOUBLE PRECISION DELTA
INTEGER I, IDIG, INFO, IWARN, J, MDIG, N
CHARACTER*1 BALANC
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK)
INTEGER IWORK(NMAX)
* .. External Subroutines ..
EXTERNAL MB05OD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, DELTA, BALANC
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
* Find the exponential of the real defective matrix A*DELTA.
CALL MB05OD( BALANC, N, NDIAG, DELTA, A, LDA, MDIG, IDIG,
$ IWORK, DWORK, LDWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( IWARN.NE.0 )
$ WRITE ( NOUT, FMT = 99993 ) IWARN
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( A(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 ) MDIG, IDIG
END IF
END IF
STOP
*
99999 FORMAT (' MB05OD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB05OD = ',I2)
99997 FORMAT (' The solution matrix E = exp(A*DELTA) is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' Minimal number of accurate digits in the norm of E =',
$ I4,/' Number of accurate digits in the norm of E',/' ',
$ ' at 95 per cent confidence interval =',I4)
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (' IWARN on exit from MB05OD = ',I2)
END
Program Data
MB05OD EXAMPLE PROGRAM DATA 3 1.0 S 2.0 1.0 1.0 0.0 3.0 2.0 1.0 0.0 4.0Program Results
MB05OD EXAMPLE PROGRAM RESULTS
The solution matrix E = exp(A*DELTA) is
22.5984 17.2073 53.8144
24.4047 27.6033 83.2241
29.4097 12.2024 81.4177
Minimal number of accurate digits in the norm of E = 12
Number of accurate digits in the norm of E
at 95 per cent confidence interval = 15
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB05OY.html 0000664 0000000 0000000 00000006741 14560147231 0020375 0 ustar 00root root 0000000 0000000
Purpose
To restore a matrix after it has been transformed by applying balancing transformations (permutations and scalings), as determined by LAPACK Library routine DGEBAL.Specification
SUBROUTINE MB05OY( JOB, N, LOW, IGH, A, LDA, SCALE, INFO )
C .. Scalar Arguments ..
CHARACTER JOB
INTEGER IGH, INFO, LDA, LOW, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), SCALE(*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the type of backward transformation required,
as follows:
= 'N', do nothing, return immediately;
= 'P', do backward transformation for permutation only;
= 'S', do backward transformation for scaling only;
= 'B', do backward transformations for both permutation
and scaling.
JOB must be the same as the argument JOB supplied
to DGEBAL.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
LOW (input) INTEGER
IGH (input) INTEGER
The integers LOW and IGH determined by DGEBAL.
1 <= LOW <= IGH <= N, if N > 0; LOW=1 and IGH=0, if N=0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix to be back-transformed.
On exit, the leading N-by-N part of this array contains
the transformed matrix.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
SCALE (input) DOUBLE PRECISION array, dimension (N)
Details of the permutation and scaling factors, as
returned by DGEBAL.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Let P be a permutation matrix, and D a diagonal matrix of scaling
factors, both of order N. The routine computes
-1
A <-- P D A D P'.
where the permutation and scaling factors are encoded in the
array SCALE.
References
None.Numerical Aspects
2 The algorithm requires O(N ) operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To move the eigenvalues with strictly negative real parts of an
N-by-N complex skew-Hamiltonian/Hamiltonian pencil aS - bH in
structured Schur form to the leading principal subpencil, while
keeping the triangular form. On entry, we have
( A D ) ( B F )
S = ( ), H = ( ),
( 0 A' ) ( 0 -B' )
where A and B are upper triangular.
S and H are transformed by a unitary matrix Q such that
( Aout Dout )
Sout = J Q' J' S Q = ( ), and
( 0 Aout' )
(1)
( Bout Fout ) ( 0 I )
Hout = J Q' J' H Q = ( ), with J = ( ),
( 0 -Bout' ) ( -I 0 )
where Aout and Bout remain in upper triangular form. The notation
M' denotes the conjugate transpose of the matrix M.
Optionally, if COMPQ = 'I' or COMPQ = 'U', the unitary matrix Q
that fulfills (1) is computed.
Specification
SUBROUTINE MB3JZP( COMPQ, N, A, LDA, D, LDD, B, LDB, F, LDF, Q,
$ LDQ, NEIG, TOL, DWORK, ZWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ
INTEGER INFO, LDA, LDB, LDD, LDF, LDQ, N, NEIG
DOUBLE PRECISION TOL
C .. Array Arguments ..
COMPLEX*16 A( LDA, * ), B( LDB, * ), D( LDD, * ),
$ F( LDF, * ), Q( LDQ, * ), ZWORK( * )
DOUBLE PRECISION DWORK( * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
Specifies whether or not the unitary transformations
should be accumulated in the array Q, as follows:
= 'N': Q is not computed;
= 'I': the array Q is initialized internally to the unit
matrix, and the unitary matrix Q is returned;
= 'U': the array Q contains a unitary matrix Q0 on
entry, and the matrix Q0*Q is returned, where Q
is the product of the unitary transformations
that are applied to the pencil aS - bH to reorder
the eigenvalues.
Input/Output Parameters
N (input) INTEGER
The order of the pencil aS - bH. N >= 0, even.
A (input/output) COMPLEX*16 array, dimension (LDA, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular matrix A.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Aout.
The strictly lower triangular part of this array is not
referenced.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1, N/2).
D (input/output) COMPLEX*16 array, dimension (LDD, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular part of the skew-Hermitian
matrix D.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Dout.
The strictly lower triangular part of this array is not
referenced.
LDD INTEGER
The leading dimension of the array D. LDD >= MAX(1, N/2).
B (input/output) COMPLEX*16 array, dimension (LDB, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular matrix B.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Bout.
The strictly lower triangular part of this array is not
referenced.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N/2).
F (input/output) COMPLEX*16 array, dimension (LDF, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular part of the Hermitian matrix
F.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Fout.
The strictly lower triangular part of this array is not
referenced.
LDF INTEGER
The leading dimension of the array F. LDF >= MAX(1, N/2).
Q (input/output) COMPLEX*16 array, dimension (LDQ, N)
On entry, if COMPQ = 'U', then the leading N-by-N part of
this array must contain a given matrix Q0, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q0 and the transformation matrix Q
used to transform the matrices S and H.
On exit, if COMPQ = 'I', then the leading N-by-N part of
this array contains the unitary transformation matrix Q.
If COMPQ = 'N' this array is not referenced.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1, N), if COMPQ = 'I' or COMPQ = 'U'.
NEIG (output) INTEGER
The number of eigenvalues in aS - bH with strictly
negative real part.
Tolerances
TOL DOUBLE PRECISION
The tolerance used to decide the sign of the eigenvalues.
If the user sets TOL > 0, then the given value of TOL is
used. If the user sets TOL <= 0, then an implicitly
computed, default tolerance, defined by MIN(N,10)*EPS, is
used instead, where EPS is the machine precision (see
LAPACK Library routine DLAMCH). A larger value might be
needed for pencils with multiple eigenvalues.
Workspace
DWORK DOUBLE PRECISION array, dimension (N/2) ZWORK COMPLEX*16 array, dimension (N/2)Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value.
Method
The algorithm reorders the eigenvalues like the following scheme:
Step 1: Reorder the eigenvalues in the subpencil aA - bB.
I. Reorder the eigenvalues with negative real parts to the
top.
II. Reorder the eigenvalues with positive real parts to the
bottom.
Step 2: Reorder the remaining eigenvalues with negative real parts.
I. Exchange the eigenvalues between the last diagonal block
in aA - bB and the last diagonal block in aS - bH.
II. Move the eigenvalues in the N/2-th place to the (MM+1)-th
place, where MM denotes the current number of eigenvalues
with negative real parts in aA - bB.
The algorithm uses a sequence of unitary transformations as
described on page 43 in [1]. To achieve those transformations the
elementary SLICOT Library subroutines MB03DZ and MB03HZ are called
for the corresponding matrix structures.
References
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical Computation of Deflating Subspaces of Embedded
Hamiltonian Pencils.
Tech. Rep. SFB393/99-15, Technical University Chemnitz,
Germany, June 1999.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) complex floating point operations.Further Comments
For large values of N, the routine applies the transformations on panels of columns. The user may specify in INFO the desired number of columns. If on entry INFO <= 0, then the routine estimates a suitable value of this number.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the eigenvalues of a complex N-by-N skew-Hamiltonian/
Hamiltonian pencil aS - bH, with
( A D ) ( B F )
S = ( ) and H = ( ). (1)
( E A' ) ( G -B' )
The structured Schur form of the embedded real skew-Hamiltonian/
skew-Hamiltonian pencil aB_S - bB_T, defined as
( Re(A) -Im(A) | Re(D) -Im(D) )
( | )
( Im(A) Re(A) | Im(D) Re(D) )
( | )
B_S = (-----------------+-----------------) , and
( | )
( Re(E) -Im(E) | Re(A') Im(A') )
( | )
( Im(E) Re(E) | -Im(A') Re(A') )
(2)
( -Im(B) -Re(B) | -Im(F) -Re(F) )
( | )
( Re(B) -Im(B) | Re(F) -Im(F) )
( | )
B_T = (-----------------+-----------------) , T = i*H,
( | )
( -Im(G) -Re(G) | -Im(B') Re(B') )
( | )
( Re(G) -Im(G) | -Re(B') -Im(B') )
is determined and used to compute the eigenvalues. The notation M'
denotes the conjugate transpose of the matrix M. Optionally,
if COMPQ = 'C', an orthonormal basis of the right deflating
subspace of the pencil aS - bH, corresponding to the eigenvalues
with strictly negative real part, is computed. Namely, after
transforming aB_S - bB_H by unitary matrices, we have
( BA BD ) ( BB BF )
B_Sout = ( ) and B_Hout = ( ), (3)
( 0 BA' ) ( 0 -BB' )
and the eigenvalues with strictly negative real part of the
complex pencil aB_Sout - bB_Hout are moved to the top. The
embedding doubles the multiplicities of the eigenvalues of the
pencil aS - bH.
Specification
SUBROUTINE MB3LZP( COMPQ, ORTH, N, A, LDA, DE, LDDE, B, LDB, FG,
$ LDFG, NEIG, Q, LDQ, ALPHAR, ALPHAI, BETA,
$ IWORK, DWORK, LDWORK, ZWORK, LZWORK, BWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, ORTH
INTEGER INFO, LDA, LDB, LDDE, LDFG, LDQ, LDWORK,
$ LZWORK, N, NEIG
C .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION ALPHAI( * ), ALPHAR( * ), BETA( * ), DWORK( * )
COMPLEX*16 A( LDA, * ), B( LDB, * ), DE( LDDE, * ),
$ FG( LDFG, * ), Q( LDQ, * ), ZWORK( * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
Specifies whether to compute the deflating subspace
corresponding to the eigenvalues of aS - bH with strictly
negative real part.
= 'N': do not compute the deflating subspace; compute the
eigenvalues only;
= 'C': compute the deflating subspace and store it in the
leading subarray of Q.
ORTH CHARACTER*1
If COMPQ = 'C', specifies the technique for computing an
orthonormal basis of the deflating subspace, as follows:
= 'P': QR factorization with column pivoting;
= 'S': singular value decomposition.
If COMPQ = 'N', the ORTH value is not used.
Input/Output Parameters
N (input) INTEGER
The order of the pencil aS - bH. N >= 0, even.
A (input/output) COMPLEX*16 array, dimension (LDA, N)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix A.
On exit, if COMPQ = 'C', the leading N-by-N part of this
array contains the upper triangular matrix BA in (3) (see
also METHOD). The strictly lower triangular part is not
zeroed; it is preserved in the leading N/2-by-N/2 part.
If COMPQ = 'N', this array is unchanged on exit.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1, N).
DE (input/output) COMPLEX*16 array, dimension (LDDE, N)
On entry, the leading N/2-by-N/2 lower triangular part of
this array must contain the lower triangular part of the
skew-Hermitian matrix E, and the N/2-by-N/2 upper
triangular part of the submatrix in the columns 2 to N/2+1
of this array must contain the upper triangular part of
the skew-Hermitian matrix D.
On exit, if COMPQ = 'C', the leading N-by-N part of this
array contains the skew-Hermitian matrix BD in (3) (see
also METHOD). The strictly lower triangular part of the
input matrix is preserved.
If COMPQ = 'N', this array is unchanged on exit.
LDDE INTEGER
The leading dimension of the array DE. LDDE >= MAX(1, N).
B (input/output) COMPLEX*16 array, dimension (LDB, N)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix B.
On exit, if COMPQ = 'C', the leading N-by-N part of this
array contains the upper triangular matrix BB in (3) (see
also METHOD). The strictly lower triangular part is not
zeroed; the elements below the first subdiagonal of the
input matrix are preserved.
If COMPQ = 'N', this array is unchanged on exit.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N).
FG (input/output) COMPLEX*16 array, dimension (LDFG, N)
On entry, the leading N/2-by-N/2 lower triangular part of
this array must contain the lower triangular part of the
Hermitian matrix G, and the N/2-by-N/2 upper triangular
part of the submatrix in the columns 2 to N/2+1 of this
array must contain the upper triangular part of the
Hermitian matrix F.
On exit, if COMPQ = 'C', the leading N-by-N part of this
array contains the Hermitian matrix BF in (3) (see also
METHOD). The strictly lower triangular part of the input
matrix is preserved. The diagonal elements might have tiny
imaginary parts.
If COMPQ = 'N', this array is unchanged on exit.
LDFG INTEGER
The leading dimension of the array FG. LDFG >= MAX(1, N).
NEIG (output) INTEGER
If COMPQ = 'C', the number of eigenvalues in aS - bH with
strictly negative real part.
Q (output) COMPLEX*16 array, dimension (LDQ, 2*N)
On exit, if COMPQ = 'C', the leading N-by-NEIG part of
this array contains an orthonormal basis of the right
deflating subspace corresponding to the eigenvalues of the
pencil aS - bH with strictly negative real part.
The remaining entries are meaningless.
If COMPQ = 'N', this array is not referenced.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1, 2*N), if COMPQ = 'C'.
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
The real parts of each scalar alpha defining an eigenvalue
of the pencil aS - bH.
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
The imaginary parts of each scalar alpha defining an
eigenvalue of the pencil aS - bH.
If ALPHAI(j) is zero, then the j-th eigenvalue is real.
BETA (output) DOUBLE PRECISION array, dimension (N)
The scalars beta that define the eigenvalues of the pencil
aS - bH.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent the j-th eigenvalue of the pencil
aS - bH, in the form lambda = alpha/beta. Since lambda may
overflow, the ratios should not, in general, be computed.
Workspace
IWORK INTEGER array, dimension (N+1)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
On exit, if INFO = -20, DWORK(1) returns the minimum value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= MAX( 4*N*N + 2*N + MAX(3,N) ), if COMPQ = 'N';
LDWORK >= MAX( 1, 11*N*N + 2*N ), if COMPQ = 'C'.
For good performance LDWORK should be generally larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) returns the optimal LZWORK.
On exit, if INFO = -22, ZWORK(1) returns the minimum value
of LZWORK.
LZWORK INTEGER
The dimension of the array ZWORK.
LZWORK >= 1, if COMPQ = 'N';
LZWORK >= 8*N + 4, if COMPQ = 'C'.
For good performance LZWORK should be generally larger.
If LZWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
ZWORK array, returns this value as the first entry of
the ZWORK array, and no error message related to LZWORK
is issued by XERBLA.
BWORK LOGICAL array, dimension (LBWORK)
LBWORK >= 0, if COMPQ = 'N';
LBWORK >= N - 1, if COMPQ = 'C'.
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: QZ iteration failed in the SLICOT Library routine
MB04FP (QZ iteration did not converge or computation
of the shifts failed);
= 2: QZ iteration failed in the LAPACK routine ZHGEQZ when
trying to triangularize the 2-by-2 blocks;
= 3: the singular value decomposition failed in the LAPACK
routine ZGESVD (for ORTH = 'S');
= 4: warning: the pencil is numerically singular.
Method
First, T = i*H is set. Then, the embeddings, B_S and B_T, of the
matrices S and T, are determined and, subsequently, the SLICOT
Library routine MB04FP is applied to compute the structured Schur
form, i.e., the factorizations
~ ( S11 S12 )
B_S = J Q' J' B_S Q = ( ) and
( 0 S11' )
~ ( T11 T12 ) ( 0 I )
B_T = J Q' J' B_T Q = ( ), with J = ( ),
( 0 T11' ) ( -I 0 )
where Q is real orthogonal, S11 is upper triangular, and T11 is
upper quasi-triangular.
Second, the SLICOT Library routine MB3JZP is applied, to compute a
~
unitary matrix Q, such that
~ ~
~ ~ ~ ( S11 S12 )
J Q' J' B_S Q = ( ~ ) =: B_Sout,
( 0 S11' )
~ ~ ~ ( H11 H12 )
J Q' J'(-i*B_T) Q = ( ) =: B_Hout,
( 0 -H11' )
~ ~ ~
with S11, H11 upper triangular, and such that Spec_-(B_S, -i*B_T)
is contained in the spectrum of the 2*NEIG-by-2*NEIG leading
~
principal subpencil aS11 - bH11.
Finally, the right deflating subspace is computed.
See also page 22 in [1] for more details.
References
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical Computation of Deflating Subspaces of Embedded
Hamiltonian Pencils.
Tech. Rep. SFB393/99-15, Technical University Chemnitz,
Germany, June 1999.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) complex floating point operations.Further Comments
This routine does not perform any scaling of the matrices. Scaling might sometimes be useful, and it should be done externally. For large values of N, the routine applies the transformations on panels of columns. The user may specify in INFO the desired number of columns. If on entry INFO <= 0, then the routine estimates a suitable value of this number.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB3OYZ.html 0000664 0000000 0000000 00000016544 14560147231 0020447 0 ustar 00root root 0000000 0000000
Purpose
To compute a rank-revealing QR factorization of a complex general
M-by-N matrix A, which may be rank-deficient, and estimate its
effective rank using incremental condition estimation.
The routine uses a truncated QR factorization with column pivoting
[ R11 R12 ]
A * P = Q * R, where R = [ ],
[ 0 R22 ]
with R11 defined as the largest leading upper triangular submatrix
whose estimated condition number is less than 1/RCOND. The order
of R11, RANK, is the effective rank of A. Condition estimation is
performed during the QR factorization process. Matrix R22 is full
(but of small norm), or empty.
MB3OYZ does not perform any scaling of the matrix A.
Specification
SUBROUTINE MB3OYZ( M, N, A, LDA, RCOND, SVLMAX, RANK, SVAL, JPVT,
$ TAU, DWORK, ZWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, M, N, RANK
DOUBLE PRECISION RCOND, SVLMAX
C .. Array Arguments ..
INTEGER JPVT( * )
COMPLEX*16 A( LDA, * ), TAU( * ), ZWORK( * )
DOUBLE PRECISION DWORK( * ), SVAL( 3 )
Arguments
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
A (input/output) COMPLEX*16 array, dimension ( LDA, N )
On entry, the leading M-by-N part of this array must
contain the given matrix A.
On exit, the leading RANK-by-RANK upper triangular part
of A contains the triangular factor R11, and the elements
below the diagonal in the first RANK columns, with the
array TAU, represent the unitary matrix Q as a product
of RANK elementary reflectors.
The remaining N-RANK columns contain the result of the
QR factorization process used.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
RCOND (input) DOUBLE PRECISION
RCOND is used to determine the effective rank of A, which
is defined as the order of the largest leading triangular
submatrix R11 in the QR factorization with pivoting of A,
whose estimated condition number is less than 1/RCOND.
0 <= RCOND <= 1.
NOTE that when SVLMAX > 0, the estimated rank could be
less than that defined above (see SVLMAX).
SVLMAX (input) DOUBLE PRECISION
If A is a submatrix of another matrix B, and the rank
decision should be related to that matrix, then SVLMAX
should be an estimate of the largest singular value of B
(for instance, the Frobenius norm of B). If this is not
the case, the input value SVLMAX = 0 should work.
SVLMAX >= 0.
RANK (output) INTEGER
The effective (estimated) rank of A, i.e., the order of
the submatrix R11.
SVAL (output) DOUBLE PRECISION array, dimension ( 3 )
The estimates of some of the singular values of the
triangular factor R:
SVAL(1): largest singular value of R(1:RANK,1:RANK);
SVAL(2): smallest singular value of R(1:RANK,1:RANK);
SVAL(3): smallest singular value of R(1:RANK+1,1:RANK+1),
if RANK < MIN( M, N ), or of R(1:RANK,1:RANK),
otherwise.
If the triangular factorization is a rank-revealing one
(which will be the case if the leading columns were well-
conditioned), then SVAL(1) will also be an estimate for
the largest singular value of A, and SVAL(2) and SVAL(3)
will be estimates for the RANK-th and (RANK+1)-st singular
values of A, respectively.
By examining these values, one can confirm that the rank
is well defined with respect to the chosen value of RCOND.
The ratio SVAL(1)/SVAL(2) is an estimate of the condition
number of R(1:RANK,1:RANK).
JPVT (output) INTEGER array, dimension ( N )
If JPVT(i) = k, then the i-th column of A*P was the k-th
column of A.
TAU (output) COMPLEX*16 array, dimension ( MIN( M, N ) )
The leading RANK elements of TAU contain the scalar
factors of the elementary reflectors.
Workspace
DWORK DOUBLE PRECISION array, dimension ( 2*N ) ZWORK COMPLEX*16 array, dimension ( 3*N-1 )Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine computes a truncated QR factorization with column
pivoting of A, A * P = Q * R, with R defined above, and,
during this process, finds the largest leading submatrix whose
estimated condition number is less than 1/RCOND, taking the
possible positive value of SVLMAX into account. This is performed
using the LAPACK incremental condition estimation scheme and a
slightly modified rank decision test. The factorization process
stops when RANK has been determined.
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = rank <= min(m,n).
Each H(i) has the form
H = I - tau * v * v'
where tau is a complex scalar, and v is a complex vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
A(i+1:m,i), and tau in TAU(i).
The matrix P is represented in jpvt as follows: If
jpvt(j) = i
then the jth column of P is the ith canonical unit vector.
References
[1] Bischof, C.H. and P. Tang.
Generalizing Incremental Condition Estimation.
LAPACK Working Notes 32, Mathematics and Computer Science
Division, Argonne National Laboratory, UT, CS-91-132,
May 1991.
[2] Bischof, C.H. and P. Tang.
Robust Incremental Condition Estimation.
LAPACK Working Notes 33, Mathematics and Computer Science
Division, Argonne National Laboratory, UT, CS-91-133,
May 1991.
Numerical Aspects
The algorithm is backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute a rank-revealing RQ factorization of a complex general
M-by-N matrix A, which may be rank-deficient, and estimate its
effective rank using incremental condition estimation.
The routine uses a truncated RQ factorization with row pivoting:
[ R11 R12 ]
P * A = R * Q, where R = [ ],
[ 0 R22 ]
with R22 defined as the largest trailing upper triangular
submatrix whose estimated condition number is less than 1/RCOND.
The order of R22, RANK, is the effective rank of A. Condition
estimation is performed during the RQ factorization process.
Matrix R11 is full (but of small norm), or empty.
MB3PYZ does not perform any scaling of the matrix A.
Specification
SUBROUTINE MB3PYZ( M, N, A, LDA, RCOND, SVLMAX, RANK, SVAL, JPVT,
$ TAU, DWORK, ZWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, M, N, RANK
DOUBLE PRECISION RCOND, SVLMAX
C .. Array Arguments ..
INTEGER JPVT( * )
COMPLEX*16 A( LDA, * ), TAU( * ), ZWORK( * )
DOUBLE PRECISION DWORK( * ), SVAL( 3 )
Arguments
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
A (input/output) COMPLEX*16 array, dimension ( LDA, N )
On entry, the leading M-by-N part of this array must
contain the given matrix A.
On exit, the upper triangle of the subarray
A(M-RANK+1:M,N-RANK+1:N) contains the RANK-by-RANK upper
triangular matrix R22; the remaining elements in the last
RANK rows, with the array TAU, represent the unitary
matrix Q as a product of RANK elementary reflectors
(see METHOD). The first M-RANK rows contain the result
of the RQ factorization process used.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
RCOND (input) DOUBLE PRECISION
RCOND is used to determine the effective rank of A, which
is defined as the order of the largest trailing triangular
submatrix R22 in the RQ factorization with pivoting of A,
whose estimated condition number is less than 1/RCOND.
0 <= RCOND <= 1.
NOTE that when SVLMAX > 0, the estimated rank could be
less than that defined above (see SVLMAX).
SVLMAX (input) DOUBLE PRECISION
If A is a submatrix of another matrix B, and the rank
decision should be related to that matrix, then SVLMAX
should be an estimate of the largest singular value of B
(for instance, the Frobenius norm of B). If this is not
the case, the input value SVLMAX = 0 should work.
SVLMAX >= 0.
RANK (output) INTEGER
The effective (estimated) rank of A, i.e., the order of
the submatrix R22.
SVAL (output) DOUBLE PRECISION array, dimension ( 3 )
The estimates of some of the singular values of the
triangular factor R:
SVAL(1): largest singular value of
R(M-RANK+1:M,N-RANK+1:N);
SVAL(2): smallest singular value of
R(M-RANK+1:M,N-RANK+1:N);
SVAL(3): smallest singular value of R(M-RANK:M,N-RANK:N),
if RANK < MIN( M, N ), or of
R(M-RANK+1:M,N-RANK+1:N), otherwise.
If the triangular factorization is a rank-revealing one
(which will be the case if the trailing rows were well-
conditioned), then SVAL(1) will also be an estimate for
the largest singular value of A, and SVAL(2) and SVAL(3)
will be estimates for the RANK-th and (RANK+1)-st singular
values of A, respectively.
By examining these values, one can confirm that the rank
is well defined with respect to the chosen value of RCOND.
The ratio SVAL(1)/SVAL(2) is an estimate of the condition
number of R(M-RANK+1:M,N-RANK+1:N).
JPVT (output) INTEGER array, dimension ( M )
If JPVT(i) = k, then the i-th row of P*A was the k-th row
of A.
TAU (output) COMPLEX*16 array, dimension ( MIN( M, N ) )
The trailing RANK elements of TAU contain the scalar
factors of the elementary reflectors.
Workspace
DWORK DOUBLE PRECISION array, dimension ( 2*M ) ZWORK COMPLEX*16 array, dimension ( 3*M-1 )Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine computes a truncated RQ factorization with row
pivoting of A, P * A = R * Q, with R defined above, and,
during this process, finds the largest trailing submatrix whose
estimated condition number is less than 1/RCOND, taking the
possible positive value of SVLMAX into account. This is performed
using an adaptation of the LAPACK incremental condition estimation
scheme and a slightly modified rank decision test. The
factorization process stops when RANK has been determined.
The matrix Q is represented as a product of elementary reflectors
Q = H(k-rank+1)' H(k-rank+2)' . . . H(k)', where k = min(m,n).
Each H(i) has the form
H = I - tau * v * v'
where tau is a complex scalar, and v is a complex vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; conjg(v(1:n-k+i-1)) is stored
on exit in A(m-k+i,1:n-k+i-1), and tau in TAU(i).
The matrix P is represented in jpvt as follows: If
jpvt(j) = i
then the jth row of P is the ith canonical unit vector.
References
[1] Bischof, C.H. and P. Tang.
Generalizing Incremental Condition Estimation.
LAPACK Working Notes 32, Mathematics and Computer Science
Division, Argonne National Laboratory, UT, CS-91-132,
May 1991.
[2] Bischof, C.H. and P. Tang.
Robust Incremental Condition Estimation.
LAPACK Working Notes 33, Mathematics and Computer Science
Division, Argonne National Laboratory, UT, CS-91-133,
May 1991.
Numerical Aspects
The algorithm is backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To apply from the left the inverse of a balancing transformation,
computed by the SLICOT Library routine MB4DPZ, to the complex
matrix
[ V1 ]
[ ],
[ sgn*V2 ]
where sgn is either +1 or -1.
Specification
SUBROUTINE MB4DBZ( JOB, SGN, N, ILO, LSCALE, RSCALE, M, V1, LDV1,
$ V2, LDV2, INFO )
C .. Scalar Arguments ..
CHARACTER JOB, SGN
INTEGER ILO, INFO, LDV1, LDV2, M, N
C .. Array Arguments ..
DOUBLE PRECISION LSCALE(*), RSCALE(*)
COMPLEX*16 V1(LDV1,*), V2(LDV2,*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the type of inverse transformation required:
= 'N': do nothing, return immediately;
= 'P': do inverse transformation for permutation only;
= 'S': do inverse transformation for scaling only;
= 'B': do inverse transformations for both permutation
and scaling.
JOB must be the same as the argument JOB supplied to
MB4DPZ.
SGN CHARACTER*1
Specifies the sign to use for V2:
= 'P': sgn = +1;
= 'N': sgn = -1.
Input/Output Parameters
N (input) INTEGER
The number of rows of the matrices V1 and V2. N >= 0.
ILO (input) INTEGER
The integer ILO determined by MB4DPZ.
1 <= ILO <= N+1.
LSCALE (input) DOUBLE PRECISION array, dimension (N)
Details of the permutation and scaling factors applied
from the left, as returned by MB4DPZ.
RSCALE (input) DOUBLE PRECISION array, dimension (N)
Details of the permutation and scaling factors applied
from the right, as returned by MB4DPZ.
M (input) INTEGER
The number of columns of the matrices V1 and V2. M >= 0.
V1 (input/output) COMPLEX*16 array, dimension (LDV1,M)
On entry, the leading N-by-M part of this array must
contain the matrix V1.
On exit, the leading N-by-M part of this array is
overwritten by the updated matrix V1 of the transformed
matrix.
LDV1 INTEGER
The leading dimension of the array V1. LDV1 >= max(1,N).
V2 (input/output) COMPLEX*16 array, dimension (LDV2,M)
On entry, the leading N-by-M part of this array must
contain the matrix V2.
On exit, the leading N-by-M part of this array is
overwritten by the updated matrix V2 of the transformed
matrix.
LDV2 INTEGER
The leading dimension of the array V2. LDV2 >= max(1,N).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
References
[1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
[2] Benner, P.
Symplectic balancing of Hamiltonian matrices.
SIAM J. Sci. Comput., 22 (5), pp. 1885-1904, 2001.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To balance a pair of N-by-N complex matrices (A,B). This involves, first, permuting A and B by equivalence transformations to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal of A and B; and second, applying a diagonal equivalence transformation to rows and columns ILO to IHI to make the rows and columns as close in 1-norm as possible. Both steps are optional. Balancing may reduce the 1-norms of the matrices, and improve the accuracy of the computed eigenvalues and/or eigenvectors in the generalized eigenvalue problem A*x = lambda*B*x. This routine may optionally improve the conditioning of the scaling transformation compared to the LAPACK routine ZGGBAL.Specification
SUBROUTINE MB4DLZ( JOB, N, THRESH, A, LDA, B, LDB, ILO, IHI,
$ LSCALE, RSCALE, DWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER JOB
INTEGER IHI, ILO, INFO, IWARN, LDA, LDB, N
DOUBLE PRECISION THRESH
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), LSCALE(*), RSCALE(*)
COMPLEX*16 A(LDA,*), B(LDB,*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the operations to be performed on A and B:
= 'N': none: simply set ILO = 1, LSCALE(I) = 1.0 and
RSCALE(I) = 1.0 for I = 1,...,N.
= 'P': permute only;
= 'S': scale only;
= 'B': both permute and scale.
Input/Output Parameters
N (input) INTEGER
The order of matrices A and B. N >= 0.
THRESH (input) DOUBLE PRECISION
If JOB = 'S' or JOB = 'B', and THRESH >= 0, threshold
value for magnitude of the elements to be considered in
the scaling process: elements with magnitude less than or
equal to THRESH*MXNORM are ignored for scaling, where
MXNORM is the maximum of the 1-norms of the original
submatrices A(s,s) and B(s,s), with s = ILO:IHI.
If THRESH < 0, the subroutine finds the scaling factors
for which some conditions, detailed below, are fulfilled.
A sequence of increasing strictly positive threshold
values is used.
If THRESH = -1, the condition is that
max( norm(A(s,s),1)/norm(B(s,s),1),
norm(B(s,s),1)/norm(S(s,s),1) ) (1)
has the smallest value, for the threshold values used,
where A(s,s) and B(s,s) are the scaled submatrices.
If THRESH = -2, the norm ratio reduction (1) is tried, but
the subroutine may return IWARN = 1 and reset the scaling
factors to 1, if this seems suitable. See the description
of the argument IWARN and FURTHER COMMENTS.
If THRESH = -3, the condition is that
norm(A(s,s),1)*norm(B(s,s),1) (2)
has the smallest value for the scaled submatrices.
If THRESH = -4, the norm reduction in (2) is tried, but
the subroutine may return IWARN = 1 and reset the scaling
factors to 1, as for THRESH = -2 above.
If THRESH = -VALUE, with VALUE >= 10, the condition
numbers of the left and right scaling transformations will
be bounded by VALUE, i.e., the ratios between the largest
and smallest entries in LSCALE(s) and RSCALE(s), will be
at most VALUE. VALUE should be a power of 10.
If JOB = 'N' or JOB = 'P', the value of THRESH is
irrelevant.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A.
On exit, the leading N-by-N part of this array contains
the balanced matrix A.
In particular, the strictly lower triangular part of the
first ILO-1 columns and the last N-IHI rows of A is zero.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) COMPLEX*16 array, dimension (LDB, N)
On entry, the leading N-by-N part of this array must
contain the matrix B.
On exit, the leading N-by-N part of this array contains
the balanced matrix B.
In particular, the strictly lower triangular part of the
first ILO-1 columns and the last N-IHI rows of B is zero.
If JOB = 'N', the arrays A and B are not referenced.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N).
ILO (output) INTEGER
IHI (output) INTEGER
ILO and IHI are set to integers such that on exit
A(i,j) = 0 and B(i,j) = 0 if i > j and
j = 1,...,ILO-1 or i = IHI+1,...,N.
If JOB = 'N' or 'S', ILO = 1 and IHI = N.
LSCALE (output) DOUBLE PRECISION array, dimension (N)
Details of the permutations and scaling factors applied
to the left side of A and B. If P(j) is the index of the
row interchanged with row j, and D(j) is the scaling
factor applied to row j, then
LSCALE(j) = P(j) for j = 1,...,ILO-1
= D(j) for j = ILO,...,IHI
= P(j) for j = IHI+1,...,N.
The order in which the interchanges are made is N to
IHI+1, then 1 to ILO-1.
RSCALE (output) DOUBLE PRECISION array, dimension (N)
Details of the permutations and scaling factors applied
to the right side of A and B. If P(j) is the index of the
column interchanged with column j, and D(j) is the scaling
factor applied to column j, then
RSCALE(j) = P(j) for j = 1,...,ILO-1
= D(j) for j = ILO,...,IHI
= P(j) for j = IHI+1,...,N.
The order in which the interchanges are made is N to
IHI+1, then 1 to ILO-1.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK) where
LDWORK = 0, if JOB = 'N' or JOB = 'P', or N = 0;
LDWORK = 6*N, if (JOB = 'S' or JOB = 'B') and THRESH >= 0;
LDWORK = 8*N, if (JOB = 'S' or JOB = 'B') and THRESH < 0.
On exit, if JOB = 'S' or JOB = 'B', DWORK(1) and DWORK(2)
contain the initial 1-norms of A(s,s) and B(s,s), and
DWORK(3) and DWORK(4) contain their final 1-norms,
respectively. Moreover, DWORK(5) contains the THRESH value
used (irrelevant if IWARN = 1 or ILO = IHI).
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: scaling has been requested, for THRESH = -2 or
THRESH = -4, but it most probably would not improve
the accuracy of the computed solution for a related
eigenproblem (since maximum norm increased
significantly compared to the original pencil
matrices and (very) high and/or small scaling
factors occurred). The returned scaling factors have
been reset to 1, but information about permutations,
if requested, has been preserved.
Error Indicator
INFO INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Balancing consists of applying an equivalence transformation
to isolate eigenvalues and/or to make the 1-norms of the rows
and columns ILO,...,IHI of A and B nearly equal. If THRESH < 0,
a search is performed to find those scaling factors giving the
smallest norm ratio or product defined above (see the description
of the parameter THRESH).
Assuming JOB = 'S', let Dl and Dr be diagonal matrices containing
the vectors LSCALE and RSCALE, respectively. The returned matrices
are obtained using the equivalence transformation
Dl*A*Dr and Dl*B*Dr.
For THRESH = 0, the routine returns essentially the same results
as the LAPACK subroutine ZGGBAL [1]. Setting THRESH < 0, usually
gives better results than ZGGBAL for badly scaled matrix pencils.
References
[1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
Numerical Aspects
No rounding errors appear if JOB = 'P'.Further Comments
If THRESH = -2, the increase of the maximum norm of the scaled submatrices, compared to the maximum norm of the initial submatrices, is bounded by MXGAIN = 100. If THRESH = -2, or THRESH = -4, the maximum condition number of the scaling transformations is bounded by MXCOND = 1/SQRT(EPS), where EPS is the machine precision (see LAPACK Library routine DLAMCH).Example
Program Text
* MB4DLZ EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 10 )
INTEGER LDA, LDB
PARAMETER ( LDA = NMAX, LDB = NMAX )
* .. Local Scalars ..
CHARACTER*1 JOB
INTEGER I, ILO, INFO, IWARN, J, N
DOUBLE PRECISION THRESH
* .. Local Arrays ..
COMPLEX*16 A(LDA, NMAX ), B( LDB, NMAX )
DOUBLE PRECISION DWORK(8*NMAX), LSCALE(NMAX), RSCALE(NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL MB4DLZ
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, JOB, THRESH
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99985 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,N ), I = 1,N )
CALL MB4DLZ( JOB, N, THRESH, A, LDA, B, LDB, ILO, IHI, LSCALE,
$ RSCALE, DWORK, IWARN, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,N )
40 CONTINUE
WRITE ( NOUT, FMT = 99994 ) ILO
WRITE ( NOUT, FMT = 99993 ) IHI
WRITE ( NOUT, FMT = 99991 )
WRITE ( NOUT, FMT = 99992 ) ( LSCALE(I), I = 1,N )
WRITE ( NOUT, FMT = 99990 )
WRITE ( NOUT, FMT = 99992 ) ( RSCALE(I), I = 1,N )
IF ( LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'B' ) ) THEN
IF ( .NOT.( THRESH.EQ.-2 .OR. THRESH.EQ.-4 ) ) THEN
WRITE ( NOUT, FMT = 99989 )
WRITE ( NOUT, FMT = 99992 ) ( DWORK(I), I = 1,2 )
WRITE ( NOUT, FMT = 99988 )
WRITE ( NOUT, FMT = 99992 ) ( DWORK(I), I = 3,4 )
WRITE ( NOUT, FMT = 99987 )
WRITE ( NOUT, FMT = 99992 ) ( DWORK(5) )
ELSE
WRITE ( NOUT, FMT = 99986 ) IWARN
END IF
END IF
END IF
END IF
*
99999 FORMAT (' MB4DLZ EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB4DLZ = ',I2)
99997 FORMAT (' The balanced matrix A is ')
99996 FORMAT (/' The balanced matrix B is ')
99995 FORMAT (20( 1X, G11.4, SP, F9.3, S, 'i ') )
99994 FORMAT (/' ILO = ',I4)
99993 FORMAT (/' IHI = ',I4)
99992 FORMAT (20(1X,G11.4))
99991 FORMAT (/' The permutations and left scaling factors are ')
99990 FORMAT (/' The permutations and right scaling factors are ')
99989 FORMAT (/' The initial 1-norms of the (sub)matrices are ')
99988 FORMAT (/' The final 1-norms of the (sub)matrices are ')
99987 FORMAT (/' The threshold value finally used is ')
99986 FORMAT (/' IWARN on exit from MB4DLZ = ',I2)
99985 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
MB4DLZ EXAMPLE PROGRAM DATA
4 B -3
(1,0.5) 0 -1e-12 0
0 (-2,-1) 0 0
1 (-1,-0.5) (-1,0.5) 0
(-1,0.5) -1 0 (2,-1)
(1,0.5) 0 0 0
0 (1,0.5) 0 0
0 0 (1,-0.5) 0
0 0 0 (1,-0.5)
Program Results
MB4DLZ EXAMPLE PROGRAM RESULTS The balanced matrix A is 2.000 -1.000i -1.000 +0.500i 0.000 +0.000i -1.000 +0.000i 0.000 +0.000i 1.000 +0.500i -0.1000E-11 +0.000i 0.000 +0.000i 0.000 +0.000i 1.000 +0.000i -1.000 +0.500i -1.000 -0.500i 0.000 +0.000i 0.000 +0.000i 0.000 +0.000i -2.000 -1.000i The balanced matrix B is 1.000 -0.500i 0.000 +0.000i 0.000 +0.000i 0.000 +0.000i 0.000 +0.000i 1.000 +0.500i 0.000 +0.000i 0.000 +0.000i 0.000 +0.000i 0.000 +0.000i 1.000 -0.500i 0.000 +0.000i 0.000 +0.000i 0.000 +0.000i 0.000 +0.000i 1.000 +0.500i ILO = 2 IHI = 3 The permutations and left scaling factors are 2.000 1.000 1.000 2.000 The permutations and right scaling factors are 2.000 1.000 1.000 2.000 The initial 1-norms of the (sub)matrices are 2.118 1.118 The final 1-norms of the (sub)matrices are 2.118 1.118 The threshold value finally used is 0.2229E-12
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MB4DPZ.html 0000664 0000000 0000000 00000043475 14560147231 0020427 0 ustar 00root root 0000000 0000000
Purpose
To balance the 2*N-by-2*N complex skew-Hamiltonian/Hamiltonian
pencil aS - bH, with
( A D ) ( C V )
S = ( ) and H = ( ), A, C N-by-N, (1)
( E A' ) ( W -C' )
where D and E are skew-Hermitian, V and W are Hermitian matrices,
and ' denotes conjugate transpose. This involves, first, permuting
aS - bH by a symplectic equivalence transformation to isolate
eigenvalues in the first 1:ILO-1 elements on the diagonal of A
and C; and second, applying a diagonal equivalence transformation
to make the pairs of rows and columns ILO:N and N+ILO:2*N as close
in 1-norm as possible. Both steps are optional. Balancing may
reduce the 1-norms of the matrices S and H.
Specification
SUBROUTINE MB4DPZ( JOB, N, THRESH, A, LDA, DE, LDDE, C, LDC, VW,
$ LDVW, ILO, LSCALE, RSCALE, DWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER JOB
INTEGER ILO, INFO, IWARN, LDA, LDC, LDDE, LDVW, N
DOUBLE PRECISION THRESH
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), LSCALE(*), RSCALE(*)
COMPLEX*16 A(LDA,*), C(LDC,*), DE(LDDE,*), VW(LDVW,*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the operations to be performed on S and H:
= 'N': none: simply set ILO = 1, LSCALE(I) = 1.0 and
RSCALE(I) = 1.0 for i = 1,...,N.
= 'P': permute only;
= 'S': scale only;
= 'B': both permute and scale.
Input/Output Parameters
N (input) INTEGER
The order of matrices A, D, E, C, V, and W. N >= 0.
THRESH (input) DOUBLE PRECISION
If JOB = 'S' or JOB = 'B', and THRESH >= 0, threshold
value for magnitude of the elements to be considered in
the scaling process: elements with magnitude less than or
equal to THRESH*MXNORM are ignored for scaling, where
MXNORM is the maximum of the 1-norms of the original
submatrices S(s,s) and H(s,s), with s = [ILO:N,N+ILO:2*N].
If THRESH < 0, the subroutine finds the scaling factors
for which some conditions, detailed below, are fulfilled.
A sequence of increasing strictly positive threshold
values is used.
If THRESH = -1, the condition is that
max( norm(H(s,s),1)/norm(S(s,s),1),
norm(S(s,s),1)/norm(H(s,s),1) ) (1)
has the smallest value, for the threshold values used,
where S(s,s) and H(s,s) are the scaled submatrices.
If THRESH = -2, the norm ratio reduction (1) is tried, but
the subroutine may return IWARN = 1 and reset the scaling
factors to 1, if this seems suitable. See the description
of the argument IWARN and FURTHER COMMENTS.
If THRESH = -3, the condition is that
norm(H(s,s),1)*norm(S(s,s),1) (2)
has the smallest value for the scaled submatrices.
If THRESH = -4, the norm reduction in (2) is tried, but
the subroutine may return IWARN = 1 and reset the scaling
factors to 1, as for THRESH = -2 above.
If THRESH = -VALUE, with VALUE >= 10, the condition
numbers of the left and right scaling transformations will
be bounded by VALUE, i.e., the ratios between the largest
and smallest entries in [LSCALE(ILO:N); RSCALE(ILO:N)]
will be at most VALUE. VALUE should be a power of 10.
If JOB = 'N' or JOB = 'P', the value of THRESH is
irrelevant.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A.
On exit, the leading N-by-N part of this array contains
the matrix A of the balanced skew-Hamiltonian matrix S.
In particular, the strictly lower triangular part of the
first ILO-1 columns of A is zero.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
DE (input/output) COMPLEX*16 array, dimension (LDDE, N+1)
On entry, the leading N-by-N lower triangular part of
this array must contain the lower triangular part of the
skew-Hermitian matrix E, and the N-by-N upper triangular
part of the submatrix in the columns 2 to N+1 of this
array must contain the upper triangular part of the
skew-Hermitian matrix D. The real parts of the entries on
the diagonal and the first superdiagonal of this array
should be zero.
On exit, the leading N-by-N lower triangular part of this
array contains the lower triangular part of the balanced
matrix E, and the N-by-N upper triangular part of the
submatrix in the columns 2 to N+1 of this array contains
the upper triangular part of the balanced matrix D.
In particular, the lower triangular part of the first
ILO-1 columns of DE is zero.
LDDE INTEGER
The leading dimension of the array DE. LDDE >= MAX(1, N).
C (input/output) COMPLEX*16 array, dimension (LDC, N)
On entry, the leading N-by-N part of this array must
contain the matrix C.
On exit, the leading N-by-N part of this array contains
the matrix C of the balanced Hamiltonian matrix H.
In particular, the strictly lower triangular part of the
first ILO-1 columns of C is zero.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1, N).
VW (input/output) COMPLEX*16 array, dimension (LDVW, N+1)
On entry, the leading N-by-N lower triangular part of
this array must contain the lower triangular part of the
Hermitian matrix W, and the N-by-N upper triangular
part of the submatrix in the columns 2 to N+1 of this
array must contain the upper triangular part of the
Hermitian matrix V. The imaginary parts of the entries on
the diagonal and the first superdiagonal of this array
should be zero.
On exit, the leading N-by-N lower triangular part of this
array contains the lower triangular part of the balanced
matrix W, and the N-by-N upper triangular part of the
submatrix in the columns 2 to N+1 of this array contains
the upper triangular part of the balanced matrix V. In
particular, the lower triangular part of the first ILO-1
columns of VW is zero.
LDVW INTEGER
The leading dimension of the array VW. LDVW >= MAX(1, N).
ILO (output) INTEGER
ILO-1 is the number of deflated eigenvalues in the
balanced skew-Hamiltonian/Hamiltonian matrix pencil.
ILO is set to 1 if JOB = 'N' or JOB = 'S'.
LSCALE (output) DOUBLE PRECISION array, dimension (N)
Details of the permutations of S and H and scaling applied
to A, D, C, and V from the left. For j = 1,...,ILO-1 let
P(j) = LSCALE(j). If P(j) <= N, then rows and columns P(j)
and P(j)+N are interchanged with rows and columns j and
j+N, respectively. If P(j) > N, then row and column P(j)-N
are interchanged with row and column j+N by a generalized
symplectic permutation. For j = ILO,...,N the j-th element
of LSCALE contains the factor of the scaling applied to
row j of the matrices A, D, C, and V.
RSCALE (output) DOUBLE PRECISION array, dimension (N)
Details of the permutations of S and H and scaling applied
to A, E, C, and W from the right. For j = 1,...,ILO-1 let
P(j) = RSCALE(j). If P(j) <= N, then rows and columns P(j)
and P(j)+N are interchanged with rows and columns j and
j+N, respectively. If P(j) > N, then row and column P(j)-N
are interchanged with row and column j+N by a generalized
symplectic permutation. For j = ILO,...,N the j-th element
of RSCALE contains the factor of the scaling applied to
column j of the matrices A, E, C, and W.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK) where
LDWORK = 0, if JOB = 'N' or JOB = 'P', or N = 0;
LDWORK = 6*N, if (JOB = 'S' or JOB = 'B') and THRESH >= 0;
LDWORK = 8*N, if (JOB = 'S' or JOB = 'B') and THRESH < 0.
On exit, if JOB = 'S' or JOB = 'B', DWORK(1) and DWORK(2)
contain the initial 1-norms of S(s,s) and H(s,s), and
DWORK(3) and DWORK(4) contain their final 1-norms,
respectively. Moreover, DWORK(5) contains the THRESH value
used (irrelevant if IWARN = 1 or ILO = N).
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: scaling has been requested, for THRESH = -2 or
THRESH = -4, but it most probably would not improve
the accuracy of the computed solution for a related
eigenproblem (since maximum norm increased
significantly compared to the original pencil
matrices and (very) high and/or small scaling
factors occurred). The returned scaling factors have
been reset to 1, but information about permutations,
if requested, has been preserved.
Error Indicator
INFO INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Balancing consists of applying a (symplectic) equivalence
transformation to isolate eigenvalues and/or to make the 1-norms
of each pair of rows and columns indexed by s of S and H nearly
equal. If THRESH < 0, a search is performed to find those scaling
factors giving the smallest norm ratio or product defined above
(see the description of the parameter THRESH).
Assuming JOB = 'S', let Dl and Dr be diagonal matrices containing
the vectors LSCALE and RSCALE, respectively. The returned matrices
are obtained using the equivalence transformation
( Dl 0 ) ( A D ) ( Dr 0 ) ( Dl 0 ) ( C V ) ( Dr 0 )
( ) ( ) ( ), ( ) ( ) ( ).
( 0 Dr ) ( E A' ) ( 0 Dl ) ( 0 Dr ) ( W -C' ) ( 0 Dl )
For THRESH = 0, the routine returns essentially the same results
as the LAPACK subroutine ZGGBAL [1]. Setting THRESH < 0, usually
gives better results than ZGGBAL for badly scaled matrix pencils.
References
[1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
[2] Benner, P.
Symplectic balancing of Hamiltonian matrices.
SIAM J. Sci. Comput., 22 (5), pp. 1885-1904, 2001.
Numerical Aspects
The transformations used preserve the skew-Hamiltonian/Hamiltonian structure and do not introduce significant rounding errors. No rounding errors appear if JOB = 'P'. If T is the global transformation matrix applied to the right, then J'*T*J is the global transformation matrix applied to the left, where J = [ 0 I; -I 0 ], with blocks of order N.Further Comments
If THRESH = -2, the increase of the maximum norm of the scaled submatrices, compared to the maximum norm of the initial submatrices, is bounded by MXGAIN = 100. If THRESH = -2, or THRESH = -4, the maximum condition number of the scaling transformations is bounded by MXCOND = 1/SQRT(EPS), where EPS is the machine precision (see LAPACK Library routine DLAMCH).Example
Program Text
* MB4DPZ EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 10 )
INTEGER LDA, LDC, LDDE, LDVW
PARAMETER ( LDA = NMAX, LDC = NMAX, LDDE = NMAX,
$ LDVW = NMAX )
* .. Local Scalars ..
CHARACTER*1 JOB
INTEGER I, ILO, INFO, IWARN, J, N
DOUBLE PRECISION THRESH
* .. Local Arrays ..
COMPLEX*16 A(LDA, NMAX ), C( LDC, NMAX ), DE(LDDE, NMAX),
$ VW(LDVW, NMAX)
DOUBLE PRECISION DWORK(8*NMAX), LSCALE(NMAX), RSCALE(NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL MB4DPZ
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, JOB, THRESH
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99985 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( DE(I,J), J = 1,N+1 ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( VW(I,J), J = 1,N+1 ), I = 1,N )
CALL MB4DPZ( JOB, N, THRESH, A, LDA, DE, LDDE, C, LDC, VW,
$ LDVW, ILO, LSCALE, RSCALE, DWORK, IWARN, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99993 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99993 ) ( DE(I,J), J = 1,N+1 )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 30 I = 1, N
WRITE ( NOUT, FMT = 99993 ) ( C(I,J), J = 1,N )
30 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99993 ) ( VW(I,J), J = 1,N+1 )
40 CONTINUE
WRITE ( NOUT, FMT = 99992 ) ILO
WRITE ( NOUT, FMT = 99991 )
WRITE ( NOUT, FMT = 99984 ) ( LSCALE(I), I = 1,N )
WRITE ( NOUT, FMT = 99990 )
WRITE ( NOUT, FMT = 99984 ) ( RSCALE(I), I = 1,N )
IF ( LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'B' ) ) THEN
IF ( .NOT.( THRESH.EQ.-2 .OR. THRESH.EQ.-4 ) ) THEN
WRITE ( NOUT, FMT = 99989 )
WRITE ( NOUT, FMT = 99984 ) ( DWORK(I), I = 1,2 )
WRITE ( NOUT, FMT = 99988 )
WRITE ( NOUT, FMT = 99984 ) ( DWORK(I), I = 3,4 )
WRITE ( NOUT, FMT = 99987 )
WRITE ( NOUT, FMT = 99984 ) ( DWORK(5) )
ELSE
WRITE ( NOUT, FMT = 99986 ) IWARN
END IF
END IF
END IF
END IF
*
99999 FORMAT (' MB4DPZ EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB4DPZ = ',I2)
99997 FORMAT (' The balanced matrix A is ')
99996 FORMAT (/' The balanced matrix DE is ')
99995 FORMAT (' The balanced matrix C is ')
99994 FORMAT (/' The balanced matrix VW is ')
99993 FORMAT (20( 1X, G11.4, SP, F9.3, S, 'i ') )
99992 FORMAT (/' ILO = ',I4)
99991 FORMAT (/' The permutations and left scaling factors are ')
99990 FORMAT (/' The permutations and right scaling factors are ')
99989 FORMAT (/' The initial 1-norms of the (sub)matrices are ')
99988 FORMAT (/' The final 1-norms of the (sub)matrices are ')
99987 FORMAT (/' The threshold value finally used is ')
99986 FORMAT (/' IWARN on exit from MB4DPZ = ',I2)
99985 FORMAT (/' N is out of range.',/' N = ',I5)
99984 FORMAT (20(1X,G11.4))
END
Program Data
MB4DPZ EXAMPLE PROGRAM DATA
2 B -3
(1,0.5) 0
0 (1,0.5)
0 0 0
0 0 0
(1,0.5) 0
0 (-2,-1)
1 -1.e-12 0
(-1,0.5) -1 0
Program Results
MB4DPZ EXAMPLE PROGRAM RESULTS The balanced matrix A is 1.000 -0.500i 0.000 0.000i 0.000 0.000i 1.000 +0.500i The balanced matrix DE is 0.000 0.000i 0.000 +0.000i 0.000 +0.000i 0.000 +0.000i 0.000 +0.000i 0.000 +0.000i The balanced matrix C is 2.000 -1.000i 1.000 -0.500i 0.000 0.000i 1.000 +0.500i The balanced matrix VW is 0.000 0.000i 1.000 +0.000i 0.000 0.000i 0.000 0.000i 1.000 +0.000i -0.1000E-11 +0.000i ILO = 2 The permutations and left scaling factors are 4.000 1.000 The permutations and right scaling factors are 4.000 1.000 The initial 1-norms of the (sub)matrices are 1.118 2.118 The final 1-norms of the (sub)matrices are 1.118 2.118 The threshold value finally used is -3.000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MC01MD.html 0000664 0000000 0000000 00000013370 14560147231 0020337 0 ustar 00root root 0000000 0000000
Purpose
To calculate, for a given real polynomial P(x) and a real scalar
alpha, the leading K coefficients of the shifted polynomial
K-1
P(x) = q(1) + q(2) * (x-alpha) + ... + q(K) * (x-alpha) + ...
using Horner's algorithm.
Specification
SUBROUTINE MC01MD( DP, ALPHA, K, P, Q, INFO )
C .. Scalar Arguments ..
INTEGER DP, INFO, K
DOUBLE PRECISION ALPHA
C .. Array Arguments ..
DOUBLE PRECISION P(*), Q(*)
Arguments
Input/Output Parameters
DP (input) INTEGER
The degree of the polynomial P(x). DP >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar value alpha of the problem.
K (input) INTEGER
The number of coefficients of the shifted polynomial to be
computed. 1 <= K <= DP+1.
P (input) DOUBLE PRECISION array, dimension (DP+1)
This array must contain the coefficients of P(x) in
increasing powers of x.
Q (output) DOUBLE PRECISION array, dimension (DP+1)
The leading K elements of this array contain the first
K coefficients of the shifted polynomial in increasing
powers of (x - alpha), and the next (DP-K+1) elements
are used as internal workspace.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Given the real polynomial
2 DP
P(x) = p(1) + p(2) * x + p(3) * x + ... + p(DP+1) * x ,
the routine computes the leading K coefficients of the shifted
polynomial
K-1
P(x) = q(1) + q(2) * (x - alpha) + ... + q(K) * (x - alpha)
as follows.
Applying Horner's algorithm (see [1]) to P(x), i.e. dividing P(x)
by (x-alpha), yields
P(x) = q(1) + (x-alpha) * D(x),
where q(1) is the value of the constant term of the shifted
polynomial and D(x) is the quotient polynomial of degree (DP-1)
given by
2 DP-1
D(x) = d(2) + d(3) * x + d(4) * x + ... + d(DP+1) * x .
Applying Horner's algorithm to D(x) and subsequent quotient
polynomials yields q(2) and q(3), q(4), ..., q(K) respectively.
It follows immediately that q(1) = P(alpha), and in general
(i-1)
q(i) = P (alpha) / (i - 1)! for i = 1, 2, ..., K.
References
[1] STOER, J. and BULIRSCH, R.
Introduction to Numerical Analysis.
Springer-Verlag. 1980.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
* MC01MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER DPMAX
PARAMETER ( DPMAX = 20 )
* .. Local Scalars ..
DOUBLE PRECISION ALPHA
INTEGER DP, I, INFO, K
* .. Local Arrays ..
DOUBLE PRECISION P(DPMAX+1), Q(DPMAX+1)
* .. External Subroutines ..
EXTERNAL MC01MD
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) DP, ALPHA, K
IF ( DP.LE.-1 .OR. DP.GT.DPMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) DP
ELSE
READ ( NIN, FMT = * ) ( P(I), I = 1,DP+1 )
* Compute the leading K coefficients of the shifted polynomial.
CALL MC01MD( DP, ALPHA, K, P, Q, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) ALPHA
DO 20 I = 1, K
WRITE ( NOUT, FMT = 99996 ) I - 1, Q(I)
20 CONTINUE
END IF
END IF
*
STOP
*
99999 FORMAT (' MC01MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MC01MD = ',I2)
99997 FORMAT (' ALPHA = ',F8.4,//' The coefficients of the shifted pol',
$ 'ynomial are ',//' power of (x-ALPHA) coefficient ')
99996 FORMAT (5X,I5,15X,F9.4)
99995 FORMAT (/' DP is out of range.',/' DP = ',I5)
END
Program Data
MC01MD EXAMPLE PROGRAM DATA 5 2.0 6 6.0 5.0 4.0 3.0 2.0 1.0Program Results
MC01MD EXAMPLE PROGRAM RESULTS
ALPHA = 2.0000
The coefficients of the shifted polynomial are
power of (x-ALPHA) coefficient
0 120.0000
1 201.0000
2 150.0000
3 59.0000
4 12.0000
5 1.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MC01ND.html 0000664 0000000 0000000 00000010661 14560147231 0020340 0 ustar 00root root 0000000 0000000
Purpose
To compute the value of the real polynomial P(x) at a given complex point x = x0 using Horner's algorithm.Specification
SUBROUTINE MC01ND( DP, XR, XI, P, VR, VI, INFO )
C .. Scalar Arguments ..
INTEGER DP, INFO
DOUBLE PRECISION VI, VR, XI, XR
C .. Array Arguments ..
DOUBLE PRECISION P(*)
Arguments
Input/Output Parameters
DP (input) INTEGER
The degree of the polynomial P(x). DP >= 0.
XR (input) DOUBLE PRECISION
XI (input) DOUBLE PRECISION
The real and imaginary parts, respectively, of x0.
P (input) DOUBLE PRECISION array, dimension (DP+1)
This array must contain the coefficients of the polynomial
P(x) in increasing powers of x.
VR (output) DOUBLE PRECISION
VI (output) DOUBLE PRECISION
The real and imaginary parts, respectively, of P(x0).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Given the real polynomial
2 DP
P(x) = p(1) + p(2) * x + p(3) * x + ... + p(DP+1) * x ,
the routine computes the value of P(x0) using the recursion
q(DP+1) = p(DP+1),
q(i) = x0*q(i+1) + p(i) for i = DP, DP-1, ..., 1,
which is known as Horner's algorithm (see [1]). Then q(1) = P(x0).
References
[1] STOER, J and BULIRSCH, R.
Introduction to Numerical Analysis.
Springer-Verlag. 1980.
Numerical Aspects
The algorithm requires DP operations for real arguments and 4*DP for complex arguments.Further Comments
NoneExample
Program Text
* MC01ND EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER DPMAX
PARAMETER ( DPMAX = 20 )
* .. Local Scalars ..
DOUBLE PRECISION VI, VR, XI, XR
INTEGER DP, I, INFO
* .. Local Arrays ..
DOUBLE PRECISION P(DPMAX+1)
* .. External Subroutines ..
EXTERNAL MC01ND
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) DP, XR, XI
IF ( DP.LE.-1 .OR. DP.GT.DPMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) DP
ELSE
READ ( NIN, FMT = * ) ( P(I), I = 1,DP+1 )
* Evaluate the polynomial at the given (complex) point.
CALL MC01ND( DP, XR, XI, P, VR, VI, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) XR, XI, VR
WRITE ( NOUT, FMT = 99996 ) XR, XI, VI
END IF
END IF
*
STOP
*
99999 FORMAT (' MC01ND EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MC01ND = ',I2)
99997 FORMAT (' Real part of P(',F6.2,SP,F6.2,'*j ) = ',SS,F8.4)
99996 FORMAT (/' Imaginary part of P(',F6.2,SP,F6.2,'*j ) = ',SS,F8.4)
99995 FORMAT (/' DP is out of range.',/' DP = ',I5)
END
Program Data
MC01ND EXAMPLE PROGRAM DATA 4 -1.56 0.29 5.0 3.0 -1.0 2.0 1.0Program Results
MC01ND EXAMPLE PROGRAM RESULTS Real part of P( -1.56 +0.29*j ) = -4.1337 Imaginary part of P( -1.56 +0.29*j ) = 1.7088
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MC01OD.html 0000664 0000000 0000000 00000011236 14560147231 0020340 0 ustar 00root root 0000000 0000000
Purpose
To compute the coefficients of a complex polynomial P(x) from its zeros.Specification
SUBROUTINE MC01OD( K, REZ, IMZ, REP, IMP, DWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, K
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), IMP(*), IMZ(*), REP(*), REZ(*)
Arguments
Input/Output Parameters
K (input) INTEGER
The number of zeros (and hence the degree) of P(x).
K >= 0.
REZ (input) DOUBLE PRECISION array, dimension (K)
IMZ (input) DOUBLE PRECISION array, dimension (K)
The real and imaginary parts of the i-th zero of P(x)
must be stored in REZ(i) and IMZ(i), respectively, where
i = 1, 2, ..., K. The zeros may be supplied in any order.
REP (output) DOUBLE PRECISION array, dimension (K+1)
IMP (output) DOUBLE PRECISION array, dimension (K+1)
These arrays contain the real and imaginary parts,
respectively, of the coefficients of P(x) in increasing
powers of x. If K = 0, then REP(1) is set to one and
IMP(1) is set to zero.
Workspace
DWORK DOUBLE PRECISION array, dimension (2*K+2)
If K = 0, this array is not referenced.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine computes the coefficients of the complex K-th degree
polynomial P(x) as
P(x) = (x - r(1)) * (x - r(2)) * ... * (x - r(K))
where r(i) = (REZ(i),IMZ(i)), using real arithmetic.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
* MC01OD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER KMAX
PARAMETER ( KMAX = 10 )
* .. Local Scalars ..
INTEGER I, INFO, K
* .. Local Arrays ..
DOUBLE PRECISION DWORK(2*KMAX+2), IMP(KMAX+1), IMZ(KMAX),
$ REP(KMAX+1), REZ(KMAX)
* .. External Subroutines ..
EXTERNAL MC01OD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) K
IF ( K.LT.0 .OR. K.GT.KMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) K
ELSE
READ ( NIN, FMT = * ) ( REZ(I), IMZ(I), I = 1,K )
* Compute the coefficients of P(x) from the given zeros.
CALL MC01OD( K, REZ, IMZ, REP, IMP, DWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
WRITE ( NOUT, FMT = 99996 )
$ ( I, REP(I+1), IMP(I+1), I = 0,K )
END IF
END IF
STOP
*
99999 FORMAT (' MC01OD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MC01OD = ',I2)
99997 FORMAT (' The coefficients of the polynomial P(x) are ',//' powe',
$ 'r of x real part imag part ')
99996 FORMAT (2X,I5,8X,F9.4,5X,F9.4)
99995 FORMAT (/' K is out of range.',/' K = ',I5)
END
Program Data
MC01OD EXAMPLE PROGRAM DATA 5 1.1 0.9 0.6 -0.7 -2.0 0.3 -0.8 2.5 -0.3 -0.4Program Results
MC01OD EXAMPLE PROGRAM RESULTS
The coefficients of the polynomial P(x) are
power of x real part imag part
0 2.7494 -2.1300
1 -1.7590 -5.4205
2 0.0290 2.8290
3 -1.6500 -1.7300
4 1.4000 -2.6000
5 1.0000 0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MC01PD.html 0000664 0000000 0000000 00000011207 14560147231 0020337 0 ustar 00root root 0000000 0000000
Purpose
To compute the coefficients of a real polynomial P(x) from its zeros.Specification
SUBROUTINE MC01PD( K, REZ, IMZ, P, DWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, K
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), IMZ(*), P(*), REZ(*)
Arguments
Input/Output Parameters
K (input) INTEGER
The number of zeros (and hence the degree) of P(x).
K >= 0.
REZ (input) DOUBLE PRECISION array, dimension (K)
IMZ (input) DOUBLE PRECISION array, dimension (K)
The real and imaginary parts of the i-th zero of P(x)
must be stored in REZ(i) and IMZ(i), respectively, where
i = 1, 2, ..., K. The zeros may be supplied in any order,
except that complex conjugate zeros must appear
consecutively.
P (output) DOUBLE PRECISION array, dimension (K+1)
This array contains the coefficients of P(x) in increasing
powers of x. If K = 0, then P(1) is set to one.
Workspace
DWORK DOUBLE PRECISION array, dimension (K+1)
If K = 0, this array is not referenced.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, (REZ(i),IMZ(i)) is a complex zero but
(REZ(i-1),IMZ(i-1)) is not its conjugate.
Method
The routine computes the coefficients of the real K-th degree
polynomial P(x) as
P(x) = (x - r(1)) * (x - r(2)) * ... * (x - r(K))
where r(i) = (REZ(i),IMZ(i)).
Note that REZ(i) = REZ(j) and IMZ(i) = -IMZ(j) if r(i) and r(j)
form a complex conjugate pair (where i <> j), and that IMZ(i) = 0
if r(i) is real.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
* MC01PD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER KMAX
PARAMETER ( KMAX = 10 )
* .. Local Scalars ..
INTEGER I, INFO, K
* .. Local Arrays ..
DOUBLE PRECISION DWORK(KMAX+1), IMZ(KMAX), P(KMAX+1), REZ(KMAX)
* .. External Subroutines ..
EXTERNAL MC01PD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) K
IF ( K.LT.0 .OR. K.GT.KMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) K
ELSE
READ ( NIN, FMT = * ) ( REZ(I), IMZ(I), I = 1,K )
* Compute the coefficients of P(x) from the given zeros.
CALL MC01PD( K, REZ, IMZ, P, DWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
WRITE ( NOUT, FMT = 99996 ) ( I, P(I+1), I = 0,K )
END IF
END IF
STOP
*
99999 FORMAT (' MC01PD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MC01PD = ',I2)
99997 FORMAT (' The coefficients of the polynomial P(x) are ',//' powe',
$ 'r of x coefficient ')
99996 FORMAT (2X,I5,9X,F9.4)
99995 FORMAT (' K is out of range.',/' K = ',I5)
END
Program Data
MC01PD EXAMPLE PROGRAM DATA 5 0.0 1.0 0.0 -1.0 2.0 0.0 1.0 3.0 1.0 -3.0Program Results
MC01PD EXAMPLE PROGRAM RESULTS
The coefficients of the polynomial P(x) are
power of x coefficient
0 -20.0000
1 14.0000
2 -24.0000
3 15.0000
4 -4.0000
5 1.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MC01PY.html 0000664 0000000 0000000 00000005762 14560147231 0020375 0 ustar 00root root 0000000 0000000
Purpose
To compute the coefficients of a real polynomial P(x) from its zeros. The coefficients are stored in decreasing order of the powers of x.Specification
SUBROUTINE MC01PY( K, REZ, IMZ, P, DWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, K
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), IMZ(*), P(*), REZ(*)
Arguments
Input/Output Parameters
K (input) INTEGER
The number of zeros (and hence the degree) of P(x).
K >= 0.
REZ (input) DOUBLE PRECISION array, dimension (K)
IMZ (input) DOUBLE PRECISION array, dimension (K)
The real and imaginary parts of the i-th zero of P(x)
must be stored in REZ(i) and IMZ(i), respectively, where
i = 1, 2, ..., K. The zeros may be supplied in any order,
except that complex conjugate zeros must appear
consecutively.
P (output) DOUBLE PRECISION array, dimension (K+1)
This array contains the coefficients of P(x) in decreasing
powers of x.
Workspace
DWORK DOUBLE PRECISION array, dimension (K)
If K = 0, this array is not referenced.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, (REZ(i),IMZ(i)) is a complex zero but
(REZ(i-1),IMZ(i-1)) is not its conjugate.
Method
The routine computes the coefficients of the real K-th degree
polynomial P(x) as
P(x) = (x - r(1)) * (x - r(2)) * ... * (x - r(K))
where r(i) = (REZ(i),IMZ(i)).
Note that REZ(i) = REZ(j) and IMZ(i) = -IMZ(j) if r(i) and r(j)
form a complex conjugate pair (where i <> j), and that IMZ(i) = 0
if r(i) is real.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute, for two given real polynomials A(x) and B(x), the
quotient polynomial Q(x) and the remainder polynomial R(x) of
A(x) divided by B(x).
The polynomials Q(x) and R(x) satisfy the relationship
A(x) = B(x) * Q(x) + R(x),
where the degree of R(x) is less than the degree of B(x).
Specification
SUBROUTINE MC01QD( DA, DB, A, B, RQ, IWARN, INFO )
C .. Scalar Arguments ..
INTEGER DA, DB, INFO, IWARN
C .. Array Arguments ..
DOUBLE PRECISION A(*), B(*), RQ(*)
Arguments
Input/Output Parameters
DA (input) INTEGER
The degree of the numerator polynomial A(x). DA >= -1.
DB (input/output) INTEGER
On entry, the degree of the denominator polynomial B(x).
DB >= 0.
On exit, if B(DB+1) = 0.0 on entry, then DB contains the
index of the highest power of x for which B(DB+1) <> 0.0.
A (input) DOUBLE PRECISION array, dimension (DA+1)
This array must contain the coefficients of the
numerator polynomial A(x) in increasing powers of x
unless DA = -1 on entry, in which case A(x) is taken
to be the zero polynomial.
B (input) DOUBLE PRECISION array, dimension (DB+1)
This array must contain the coefficients of the
denominator polynomial B(x) in increasing powers of x.
RQ (output) DOUBLE PRECISION array, dimension (DA+1)
If DA < DB on exit, then this array contains the
coefficients of the remainder polynomial R(x) in
increasing powers of x; Q(x) is the zero polynomial.
Otherwise, the leading DB elements of this array contain
the coefficients of R(x) in increasing powers of x, and
the next (DA-DB+1) elements contain the coefficients of
Q(x) in increasing powers of x.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= k: if the degree of the denominator polynomial B(x) has
been reduced to (DB - k) because B(DB+1-j) = 0.0 on
entry for j = 0, 1, ..., k-1 and B(DB+1-k) <> 0.0.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if on entry, DB >= 0 and B(i) = 0.0, where
i = 1, 2, ..., DB+1.
Method
Given real polynomials
DA
A(x) = a(1) + a(2) * x + ... + a(DA+1) * x
and
DB
B(x) = b(1) + b(2) * x + ... + b(DB+1) * x
where b(DB+1) is non-zero, the routine computes the coeffcients of
the quotient polynomial
DA-DB
Q(x) = q(1) + q(2) * x + ... + q(DA-DB+1) * x
and the remainder polynomial
DB-1
R(x) = r(1) + r(2) * x + ... + r(DB) * x
such that A(x) = B(x) * Q(x) + R(x).
The algorithm used is synthetic division of polynomials (see [1]),
which involves the following steps:
(a) compute q(k+1) = a(DB+k+1) / b(DB+1)
and
(b) set a(j) = a(j) - q(k+1) * b(j-k) for j = k+1, ..., DB+k.
Steps (a) and (b) are performed for k = DA-DB, DA-DB-1, ..., 0 and
the algorithm terminates with r(i) = a(i) for i = 1, 2, ..., DB.
References
[1] Knuth, D.E.
The Art of Computer Programming, (Vol. 2, Seminumerical
Algorithms).
Addison-Wesley, Reading, Massachusetts (2nd Edition), 1981.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
* MC01QD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER DAMAX, DBMAX
PARAMETER ( DAMAX = 10, DBMAX = 10 )
* .. Local Scalars ..
INTEGER DA, DB, DBB, DQ, DR, I, IMAX, INFO, IWARN
* .. Local Arrays ..
DOUBLE PRECISION A(DAMAX+1), B(DBMAX+1), RQ(DAMAX+1)
* .. External Subroutines ..
EXTERNAL MC01QD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) DA
IF ( DA.LE.-2 .OR. DA.GT.DAMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) DA
ELSE
READ ( NIN, FMT = * ) ( A(I), I = 1,DA+1 )
READ ( NIN, FMT = * ) DB
DBB = DB
IF ( DB.LE.-1 .OR. DB.GT.DBMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) DB
ELSE
READ ( NIN, FMT = * ) ( B(I), I = 1,DB+1 )
* Compute Q(x) and R(x) from the given A(x) and B(x).
CALL MC01QD( DA, DB, A, B, RQ, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( IWARN.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) IWARN
WRITE ( NOUT, FMT = 99996 ) DBB, DB
END IF
WRITE ( NOUT, FMT = 99995 )
DQ = DA - DB
DR = DB - 1
IMAX = DQ
IF ( DR.GT.IMAX ) IMAX = DR
DO 20 I = 0, IMAX
IF ( I.LE.DQ .AND. I.LE.DR ) THEN
WRITE ( NOUT, FMT = 99994 ) I, RQ(DB+I+1), RQ(I+1)
ELSE IF ( I.LE.DQ ) THEN
WRITE ( NOUT, FMT = 99993 ) I, RQ(DB+I+1)
ELSE
WRITE ( NOUT, FMT = 99992 ) I, RQ(I+1)
END IF
20 CONTINUE
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' MC01QD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MC01QD = ',I2)
99997 FORMAT (' IWARN on exit from MC01QD = ',I2,/)
99996 FORMAT (' The degree of the denominator polynomial B(x) has been',
$ ' reduced from ',I2,' to ',I2,/)
99995 FORMAT (' The coefficients of the polynomials Q(x) and R(x) are ',
$ //' Q(x) R(x) ',/' power of',
$ ' x coefficient coefficient ')
99994 FORMAT (2X,I5,9X,F9.4,7X,F9.4)
99993 FORMAT (2X,I5,9X,F9.4)
99992 FORMAT (2X,I5,25X,F9.4)
99991 FORMAT (/' DA is out of range.',/' DA = ',I5)
99990 FORMAT (/' DB is out of range.',/' DB = ',I5)
END
Program Data
MC01QD EXAMPLE PROGRAM DATA 4 2.0 2.0 -1.0 2.0 1.0 2 1.0 -1.0 1.0Program Results
MC01QD EXAMPLE PROGRAM RESULTS
The coefficients of the polynomials Q(x) and R(x) are
Q(x) R(x)
power of x coefficient coefficient
0 1.0000 1.0000
1 3.0000 0.0000
2 1.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MC01RD.html 0000664 0000000 0000000 00000016740 14560147231 0020350 0 ustar 00root root 0000000 0000000
Purpose
To compute the coefficients of the polynomial
P(x) = P1(x) * P2(x) + alpha * P3(x),
where P1(x), P2(x) and P3(x) are given real polynomials and alpha
is a real scalar.
Each of the polynomials P1(x), P2(x) and P3(x) may be the zero
polynomial.
Specification
SUBROUTINE MC01RD( DP1, DP2, DP3, ALPHA, P1, P2, P3, INFO )
C .. Scalar Arguments ..
INTEGER DP1, DP2, DP3, INFO
DOUBLE PRECISION ALPHA
C .. Array Arguments ..
DOUBLE PRECISION P1(*), P2(*), P3(*)
Arguments
Input/Output Parameters
DP1 (input) INTEGER
The degree of the polynomial P1(x). DP1 >= -1.
DP2 (input) INTEGER
The degree of the polynomial P2(x). DP2 >= -1.
DP3 (input/output) INTEGER
On entry, the degree of the polynomial P3(x). DP3 >= -1.
On exit, the degree of the polynomial P(x).
ALPHA (input) DOUBLE PRECISION
The scalar value alpha of the problem.
P1 (input) DOUBLE PRECISION array, dimension (lenp1)
where lenp1 = DP1 + 1 if DP1 >= 0 and 1 otherwise.
If DP1 >= 0, then this array must contain the
coefficients of P1(x) in increasing powers of x.
If DP1 = -1, then P1(x) is taken to be the zero
polynomial, P1 is not referenced and can be supplied
as a dummy array.
P2 (input) DOUBLE PRECISION array, dimension (lenp2)
where lenp2 = DP2 + 1 if DP2 >= 0 and 1 otherwise.
If DP2 >= 0, then this array must contain the
coefficients of P2(x) in increasing powers of x.
If DP2 = -1, then P2(x) is taken to be the zero
polynomial, P2 is not referenced and can be supplied
as a dummy array.
P3 (input/output) DOUBLE PRECISION array, dimension (lenp3)
where lenp3 = MAX(DP1+DP2,DP3,0) + 1.
On entry, if DP3 >= 0, then this array must contain the
coefficients of P3(x) in increasing powers of x.
On entry, if DP3 = -1, then P3(x) is taken to be the zero
polynomial.
On exit, the leading (DP3+1) elements of this array
contain the coefficients of P(x) in increasing powers of x
unless DP3 = -1 on exit, in which case the coefficients of
P(x) (the zero polynomial) are not stored in the array.
This is the case, for instance, when ALPHA = 0.0 and
P1(x) or P2(x) is the zero polynomial.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Given real polynomials
DP1 i DP2 i
P1(x) = SUM a(i+1) * x , P2(x) = SUM b(i+1) * x and
i=0 i=0
DP3 i
P3(x) = SUM c(i+1) * x ,
i=0
the routine computes the coefficents of P(x) = P1(x) * P2(x) +
DP3 i
alpha * P3(x) = SUM d(i+1) * x as follows.
i=0
Let e(i) = c(i) for 1 <= i <= DP3+1 and e(i) = 0 for i > DP3+1.
Then if DP1 >= DP2,
i
d(i) = SUM a(k) * b(i-k+1) + f(i), for i = 1, ..., DP2+1,
k=1
i
d(i) = SUM a(k) * b(i-k+1) + f(i), for i = DP2+2, ..., DP1+1
k=i-DP2
and
DP1+1
d(i) = SUM a(k) * b(i-k+1) + f(i) for i = DP1+2,...,DP1+DP2+1,
k=i-DP2
where f(i) = alpha * e(i).
Similar formulas hold for the case DP1 < DP2.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
* MC01RD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER DP1MAX, DP2MAX, DP3MAX
PARAMETER ( DP1MAX = 10, DP2MAX = 10, DP3MAX = 10 )
INTEGER LENP3
PARAMETER ( LENP3 = MAX(DP1MAX+DP2MAX,DP3MAX)+1 )
* .. Local Scalars ..
DOUBLE PRECISION ALPHA
INTEGER DP1, DP2, DP3, I, INFO
* .. Local Arrays ..
DOUBLE PRECISION P1(DP1MAX+1), P2(DP2MAX+1), P3(LENP3)
* $ P3(DP1MAX+DP2MAX+DP3MAX+1)
* .. External Subroutines ..
EXTERNAL MC01RD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) DP1
IF ( DP1.LE.-2 .OR. DP1.GT.DP1MAX ) THEN
WRITE ( NOUT, FMT = 99994 ) DP1
ELSE
READ ( NIN, FMT = * ) ( P1(I), I = 1,DP1+1 )
READ ( NIN, FMT = * ) DP2
IF ( DP2.LE.-2 .OR. DP2.GT.DP2MAX ) THEN
WRITE ( NOUT, FMT = 99993 ) DP2
ELSE
READ ( NIN, FMT = * ) ( P2(I), I = 1,DP2+1 )
READ ( NIN, FMT = * ) DP3
IF ( DP3.LE.-2 .OR. DP3.GT.DP3MAX ) THEN
WRITE ( NOUT, FMT = 99992 ) DP3
ELSE
READ ( NIN, FMT = * ) ( P3(I), I = 1,DP3+1 )
END IF
READ ( NIN, FMT = * ) ALPHA
* Compute the coefficients of the polynomial P(x).
CALL MC01RD( DP1, DP2, DP3, ALPHA, P1, P2, P3, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) DP3
IF ( DP3.GE.0 ) THEN
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 0, DP3
WRITE ( NOUT, FMT = 99995 ) I, P3(I+1)
20 CONTINUE
END IF
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' MC01RD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MC01RD = ',I2)
99997 FORMAT (' Degree of the resulting polynomial P(x) = ',I2)
99996 FORMAT (/' The coefficients of P(x) are ',//' power of x coe',
$ 'fficient ')
99995 FORMAT (2X,I5,9X,F9.4)
99994 FORMAT (/' DP1 is out of range.',/' DP1 = ',I5)
99993 FORMAT (/' DP2 is out of range.',/' DP2 = ',I5)
99992 FORMAT (/' DP3 is out of range.',/' DP3 = ',I5)
END
Program Data
MC01RD EXAMPLE PROGRAM DATA 1 1.00 2.50 2 1.00 0.10 -0.40 1 1.15 1.50 -2.20Program Results
MC01RD EXAMPLE PROGRAM RESULTS
Degree of the resulting polynomial P(x) = 3
The coefficients of P(x) are
power of x coefficient
0 -1.5300
1 -0.7000
2 -0.1500
3 -1.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MC01SD.html 0000664 0000000 0000000 00000015721 14560147231 0020347 0 ustar 00root root 0000000 0000000
Purpose
To scale the coefficients of the real polynomial P(x) such that the coefficients of the scaled polynomial Q(x) = sP(tx) have minimal variation, where s and t are real scalars.Specification
SUBROUTINE MC01SD( DP, P, S, T, MANT, E, IWORK, INFO )
C .. Scalar Arguments ..
INTEGER DP, INFO, S, T
C .. Array Arguments ..
INTEGER E(*), IWORK(*)
DOUBLE PRECISION MANT(*), P(*)
Arguments
Input/Output Parameters
DP (input) INTEGER
The degree of the polynomial P(x). DP >= 0.
P (input/output) DOUBLE PRECISION array, dimension (DP+1)
On entry, this array must contain the coefficients of P(x)
in increasing powers of x.
On exit, this array contains the coefficients of the
scaled polynomial Q(x) in increasing powers of x.
S (output) INTEGER
The exponent of the floating-point representation of the
scaling factor s = BASE**S, where BASE is the base of the
machine representation of floating-point numbers (see
LAPACK Library Routine DLAMCH).
T (output) INTEGER
The exponent of the floating-point representation of the
scaling factor t = BASE**T.
MANT (output) DOUBLE PRECISION array, dimension (DP+1)
This array contains the mantissas of the standard
floating-point representation of the coefficients of the
scaled polynomial Q(x) in increasing powers of x.
E (output) INTEGER array, dimension (DP+1)
This array contains the exponents of the standard
floating-point representation of the coefficients of the
scaled polynomial Q(x) in increasing powers of x.
Workspace
IWORK INTEGER array, dimension (DP+1)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if on entry, P(x) is the zero polynomial.
Method
Define the variation of the coefficients of the real polynomial
2 DP
P(x) = p(0) + p(1) * x + p(2) * x + ... + p(DP) x
whose non-zero coefficients can be represented as
e(i)
p(i) = m(i) * BASE (where 1 <= ABS(m(i)) < BASE)
by
V = max(e(i)) - min(e(i)),
where max and min are taken over the indices i for which p(i) is
non-zero.
DP i i
For the scaled polynomial P(cx) = SUM p(i) * c * x with
i=0
j
c = (BASE) , the variation V(j) is given by
V(j) = max(e(i) + j * i) - min(e(i) + j * i).
Using the fact that V(j) is a convex function of j, the routine
determines scaling factors s = (BASE)**S and t = (BASE)**T such
that the coefficients of the scaled polynomial Q(x) = sP(tx)
satisfy the following conditions:
(a) 1 <= q(0) < BASE and
(b) the variation of the coefficients of Q(x) is minimal.
Further details can be found in [1].
References
[1] Dunaway, D.K.
Calculation of Zeros of a Real Polynomial through
Factorization using Euclid's Algorithm.
SIAM J. Numer. Anal., 11, pp. 1087-1104, 1974.
Numerical Aspects
Since the scaling is performed on the exponents of the floating- point representation of the coefficients of P(x), no rounding errors occur during the computation of the coefficients of Q(x).Further Comments
The scaling factors s and t are BASE dependent.Example
Program Text
* MC01SD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER DPMAX
PARAMETER ( DPMAX = 10 )
* .. Local Scalars ..
INTEGER BETA, DP, I, INFO, S, T
* .. Local Arrays ..
DOUBLE PRECISION MANT(DPMAX+1), P(DPMAX+1)
INTEGER E(DPMAX+1), IWORK(DPMAX+1)
C .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* .. External Subroutines ..
EXTERNAL MC01SD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) DP
IF ( DP.LE.-1 .OR. DP.GT.DPMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) DP
ELSE
READ ( NIN, FMT = * ) ( P(I), I = 1,DP+1 )
* Compute the coefficients of the scaled polynomial Q(x).
CALL MC01SD( DP, P, S, T, MANT, E, IWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
BETA = DLAMCH( 'Base' )
WRITE ( NOUT, FMT = 99995 ) BETA, S, T
WRITE ( NOUT,FMT = 99997 )
DO 20 I = 0, DP
WRITE ( NOUT, FMT = 99996 ) I, P(I+1)
20 CONTINUE
END IF
END IF
*
STOP
*
99999 FORMAT (' MC01SD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MC01SD = ',I2)
99997 FORMAT (/' The coefficients of the scaled polynomial Q(x) = s*P(',
$ 'tx) are ',//' power of x coefficient ')
99996 FORMAT (2X,I5,9X,F9.4)
99995 FORMAT (' The base of the machine (BETA) = ',I2,//' The scaling ',
$ 'factors are s = BETA**(',I3,') and t = BETA**(',I3,')')
99994 FORMAT (/' DP is out of range.',/' DP =',I5)
END
Program Data
MC01SD EXAMPLE PROGRAM DATA 5 10.0 -40.5 159.5 0.0 2560.0 -10236.5Program Results
MC01SD EXAMPLE PROGRAM RESULTS
The base of the machine (BETA) = 2
The scaling factors are s = BETA**( -3) and t = BETA**( -2)
The coefficients of the scaled polynomial Q(x) = s*P(tx) are
power of x coefficient
0 1.2500
1 -1.2656
2 1.2461
3 0.0000
4 1.2500
5 -1.2496
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MC01TD.html 0000664 0000000 0000000 00000020121 14560147231 0020336 0 ustar 00root root 0000000 0000000
Purpose
To determine whether or not a given polynomial P(x) with real coefficients is stable, either in the continuous-time or discrete- time case. A polynomial is said to be stable in the continuous-time case if all its zeros lie in the left half-plane, and stable in the discrete-time case if all its zeros lie inside the unit circle.Specification
SUBROUTINE MC01TD( DICO, DP, P, STABLE, NZ, DWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO
LOGICAL STABLE
INTEGER DP, INFO, IWARN, NZ
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), P(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Indicates whether the stability test to be applied to
P(x) is in the continuous-time or discrete-time case as
follows:
= 'C': Continuous-time case;
= 'D': Discrete-time case.
Input/Output Parameters
DP (input/output) INTEGER
On entry, the degree of the polynomial P(x). DP >= 0.
On exit, if P(DP+1) = 0.0 on entry, then DP contains the
index of the highest power of x for which P(DP+1) <> 0.0.
P (input) DOUBLE PRECISION array, dimension (DP+1)
This array must contain the coefficients of P(x) in
increasing powers of x.
STABLE (output) LOGICAL
Contains the value .TRUE. if P(x) is stable and the value
.FALSE. otherwise (see also NUMERICAL ASPECTS).
NZ (output) INTEGER
If INFO = 0, contains the number of unstable zeros - that
is, the number of zeros of P(x) in the right half-plane if
DICO = 'C' or the number of zeros of P(x) outside the unit
circle if DICO = 'D' (see also NUMERICAL ASPECTS).
Workspace
DWORK DOUBLE PRECISION array, dimension (2*DP+2)
The leading (DP+1) elements of DWORK contain the Routh
coefficients, if DICO = 'C', or the constant terms of
the Schur-Cohn transforms, if DICO = 'D'.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= k: if the degree of the polynomial P(x) has been
reduced to (DB - k) because P(DB+1-j) = 0.0 on entry
for j = 0, 1,..., k-1 and P(DB+1-k) <> 0.0.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if on entry, P(x) is the zero polynomial;
= 2: if the polynomial P(x) is most probably unstable,
although it may be stable with one or more zeros
very close to either the imaginary axis if
DICO = 'C' or the unit circle if DICO = 'D'.
The number of unstable zeros (NZ) is not determined.
Method
The stability of the real polynomial
2 DP
P(x) = p(0) + p(1) * x + p(2) * x + ... + p(DP) x
is determined as follows.
In the continuous-time case (DICO = 'C') the Routh algorithm
(see [1]) is used. The routine computes the Routh coefficients and
if they are non-zero then the number of sign changes in the
sequence of the coefficients is equal to the number of zeros with
positive imaginary part.
In the discrete-time case (DICO = 'D') the Schur-Cohn
algorithm (see [2] and [3]) is applied to the reciprocal
polynomial
2 DP
Q(x) = p(DP) + p(DP-1) * x + p(DP-2) * x + ... + p(0) x .
The routine computes the constant terms of the Schur transforms
and if all of them are non-zero then the number of zeros of P(x)
with modulus greater than unity is obtained from the sequence of
constant terms.
References
[1] Gantmacher, F.R.
Applications of the Theory of Matrices.
Interscience Publishers, New York, 1959.
[2] Kucera, V.
Discrete Linear Control. The Algorithmic Approach.
John Wiley & Sons, Chichester, 1979.
[3] Henrici, P.
Applied and Computational Complex Analysis (Vol. 1).
John Wiley & Sons, New York, 1974.
Numerical Aspects
The algorithm used by the routine is numerically stable. Note that if some of the Routh coefficients (DICO = 'C') or some of the constant terms of the Schur-Cohn transforms (DICO = 'D') are small relative to EPS (the machine precision), then the number of unstable zeros (and hence the value of STABLE) may be incorrect.Further Comments
NoneExample
Program Text
* MC01TD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER DPMAX
PARAMETER ( DPMAX = 10 )
* .. Local Scalars ..
INTEGER DP, DPP, I, INFO, IWARN, NZ
LOGICAL STABLE
CHARACTER*1 DICO
* .. Local Arrays ..
DOUBLE PRECISION DWORK(2*DPMAX+2), P(DPMAX+1)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL MC01TD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = * )
READ ( NIN, FMT = * ) DP, DICO
IF ( DP.LE.-1 .OR. DP.GT.DPMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) DP
ELSE
DPP = DP
READ ( NIN, FMT = * ) ( P(I), I = 1,DP+1 )
* Determine whether or not the given polynomial P(x) is stable.
CALL MC01TD( DICO, DP, P, STABLE, NZ, DWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( IWARN.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) IWARN
WRITE ( NOUT, FMT = 99996 ) DPP, DP
END IF
IF ( STABLE ) THEN
WRITE ( NOUT, FMT = 99995 )
ELSE
WRITE ( NOUT, FMT = 99994 )
IF ( LSAME( DICO, 'D' ) ) THEN
WRITE ( NOUT, FMT = 99992 ) NZ
ELSE
WRITE ( NOUT, FMT = 99991 ) NZ
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' MC01TD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MC01TD = ',I2)
99997 FORMAT (' IWARN on exit from MC01TD = ',I2,/)
99996 FORMAT (' The degree of the polynomial P(x) has been reduced fro',
$ 'm ',I2,' to ',I2,/)
99995 FORMAT (' The polynomial P(x) is stable ')
99994 FORMAT (' The polynomial P(x) is unstable ')
99993 FORMAT (/' DP is out of range. ',/' DP = ',I5)
99992 FORMAT (/' The number of zeros of P(x) outside the unit ',
$ 'circle = ',I2)
99991 FORMAT (/' The number of zeros of P(x) in the right ',
$ 'half-plane = ',I2)
END
Program Data
MC01TD EXAMPLE PROGRAM DATA 4 C 2.0 0.0 1.0 -1.0 1.0Program Results
MC01TD EXAMPLE PROGRAM RESULTS The polynomial P(x) is unstable The number of zeros of P(x) in the right half-plane = 2
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MC01VD.html 0000664 0000000 0000000 00000011345 14560147231 0020350 0 ustar 00root root 0000000 0000000
Purpose
To compute the roots of a quadratic equation with real coefficients.Specification
SUBROUTINE MC01VD( A, B, C, Z1RE, Z1IM, Z2RE, Z2IM, INFO )
C .. Scalar Arguments ..
INTEGER INFO
DOUBLE PRECISION A, B, C, Z1IM, Z1RE, Z2IM, Z2RE
Arguments
Input/Output Parameters
A (input) DOUBLE PRECISION
The value of the coefficient of the quadratic term.
B (input) DOUBLE PRECISION
The value of the coefficient of the linear term.
C (input) DOUBLE PRECISION
The value of the coefficient of the constant term.
Z1RE (output) DOUBLE PRECISION
Z1IM (output) DOUBLE PRECISION
The real and imaginary parts, respectively, of the largest
root in magnitude.
Z2RE (output) DOUBLE PRECISION
Z2IM (output) DOUBLE PRECISION
The real and imaginary parts, respectively, of the
smallest root in magnitude.
Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: if on entry, either A = B = 0.0 or A = 0.0 and the
root -C/B overflows; in this case Z1RE, Z1IM, Z2RE
and Z2IM are unassigned;
= 2: if on entry, A = 0.0; in this case Z1RE contains
BIG and Z1IM contains zero, where BIG is a
representable number near the overflow threshold
of the machine (see LAPACK Library Routine DLAMCH);
= 3: if on entry, either C = 0.0 and the root -B/A
overflows or A, B and C are non-zero and the largest
real root in magnitude cannot be computed without
overflow; in this case Z1RE contains BIG and Z1IM
contains zero;
= 4: if the roots cannot be computed without overflow; in
this case Z1RE, Z1IM, Z2RE and Z2IM are unassigned.
Method
The routine computes the roots (r1 and r2) of the real quadratic
equation
2
a * x + b * x + c = 0
as
- b - SIGN(b) * SQRT(b * b - 4 * a * c) c
r1 = --------------------------------------- and r2 = ------
2 * a a * r1
unless a = 0, in which case
-c
r1 = --.
b
Precautions are taken to avoid overflow and underflow wherever
possible.
Numerical Aspects
The algorithm is numerically stable.Further Comments
NoneExample
Program Text
* MC01VD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
* .. Local Scalars ..
DOUBLE PRECISION A, B, C, Z1IM, Z1RE, Z2IM, Z2RE
INTEGER INFO
* .. External Subroutines ..
EXTERNAL MC01VD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) A, B, C
* Solve the quadratic equation A*x**2 + B*x + C = 0.
CALL MC01VD( A, B, C, Z1RE, Z1IM, Z2RE, Z2IM, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
WRITE ( NOUT, FMT = 99996 ) Z1RE, Z1IM, Z2RE, Z2IM
END IF
*
STOP
*
99999 FORMAT (' MC01VD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MC01VD = ',I2)
99997 FORMAT (' The roots of the quadratic equation are ')
99996 FORMAT (/' x = ',F8.4,2X,SP,F8.4,'*j',SS,/' x = ',F8.4,2X,SP,F8.4,
$ '*j')
END
Program Data
MC01VD EXAMPLE PROGRAM DATA 0.5 -1.0 2.0Program Results
MC01VD EXAMPLE PROGRAM RESULTS The roots of the quadratic equation are x = 1.0000 +1.7321*j x = 1.0000 -1.7321*j
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MC01WD.html 0000664 0000000 0000000 00000013234 14560147231 0020350 0 ustar 00root root 0000000 0000000
Purpose
To compute, for a given real polynomial P(x) and a quadratic
polynomial B(x), the quotient polynomial Q(x) and the linear
remainder polynomial R(x) such that
P(x) = B(x) * Q(x) + R(x),
2
where B(x) = u1 + u2 * x + x , R(x) = q(1) + q(2) * (u2 + x)
and u1, u2, q(1) and q(2) are real scalars.
Specification
SUBROUTINE MC01WD( DP, P, U1, U2, Q, INFO )
C .. Scalar Arguments ..
INTEGER DP, INFO
DOUBLE PRECISION U1, U2
C .. Array Arguments ..
DOUBLE PRECISION P(*), Q(*)
Arguments
Input/Output Parameters
DP (input) INTEGER
The degree of the polynomial P(x). DP >= 0.
P (input) DOUBLE PRECISION array, dimension (DP+1)
This array must contain the coefficients of P(x) in
increasing powers of x.
U1 (input) DOUBLE PRECISION
The value of the constant term of the quadratic
polynomial B(x).
U2 (input) DOUBLE PRECISION
The value of the coefficient of x of the quadratic
polynomial B(x).
Q (output) DOUBLE PRECISION array, dimension (DP+1)
If DP >= 1 on entry, then elements Q(1) and Q(2) contain
the coefficients q(1) and q(2), respectively, of the
remainder polynomial R(x), and the next (DP-1) elements
of this array contain the coefficients of the quotient
polynomial Q(x) in increasing powers of x.
If DP = 0 on entry, then element Q(1) contains the
coefficient q(1) of the remainder polynomial R(x) = q(1);
Q(x) is the zero polynomial.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Given the real polynomials
DP i 2
P(x) = SUM p(i+1) * x and B(x) = u1 + u2 * x + x
i=0
the routine uses the recurrence relationships
q(DP+1) = p(DP+1),
q(DP) = p(DP) - u2 * q(DP+1) and
q(i) = p(i) - u2 * q(i+1) - u1 * q(i+2) for i = DP-1, ..., 1
to determine the coefficients of the quotient polynomial
DP-2 i
Q(x) = SUM q(i+3) * x
i=0
and the remainder polynomial
R(x) = q(1) + q(2) * (u2 + x).
Numerical Aspects
None.Further Comments
NoneExample
Program Text
* MC01WD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER DPMAX
PARAMETER ( DPMAX = 10 )
* .. Local Scalars ..
DOUBLE PRECISION U1, U2
INTEGER DP, I, INFO
* .. Local Arrays ..
DOUBLE PRECISION P(DPMAX+1), Q(DPMAX+1)
* .. External Subroutines ..
EXTERNAL MC01WD
* .. Executable Statements ..
*
WRITE ( NOUT,FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) DP
IF ( DP.LE.-1 .OR. DP.GT.DPMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) DP
ELSE
READ ( NIN, FMT = * ) ( P(I), I = 1,DP+1 )
READ ( NIN, FMT = * ) U1, U2
* Compute Q(x) and R(x) from P(x) = (x**2+U2*x+U1) * Q(x) + R(x).
CALL MC01WD( DP, P, U1, U2, Q, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 0, DP - 2
WRITE ( NOUT, FMT = 99996 ) I, Q(I+3)
20 CONTINUE
WRITE ( NOUT, FMT = 99995 ) Q(1) + Q(2)*U2, Q(2)
END IF
END IF
*
STOP
*
99999 FORMAT (' MC01WD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MC01WD = ',I2)
99997 FORMAT (' The coefficients of the quotient polynomial Q(x) are ',
$ //' power of x coefficient ')
99996 FORMAT (2X,I5,9X,F9.4)
99995 FORMAT (/' The coefficients of the remainder polynomial R(x) are '
$ ,//' power of x coefficient ',/' 0 ',F9.4,
$ /' 1 ',F9.4)
99994 FORMAT (/' DP is out of range.',/' DP = ',I5)
END
Program Data
MC01WD EXAMPLE PROGRAM DATA 6 0.62 1.10 1.64 1.88 2.12 1.70 1.00 0.60 0.80Program Results
MC01WD EXAMPLE PROGRAM RESULTS
The coefficients of the quotient polynomial Q(x) are
power of x coefficient
0 0.6000
1 0.7000
2 0.8000
3 0.9000
4 1.0000
The coefficients of the remainder polynomial R(x) are
power of x coefficient
0 0.2600
1 0.2000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MC01XD.html 0000664 0000000 0000000 00000014111 14560147231 0020344 0 ustar 00root root 0000000 0000000
Purpose
To compute the roots of the polynomial
P(t) = ALPHA + BETA*t + GAMMA*t^2 + DELTA*t^3 .
Specification
SUBROUTINE MC01XD( ALPHA, BETA, GAMMA, DELTA, EVR, EVI, EVQ,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDWORK
DOUBLE PRECISION ALPHA, BETA, DELTA, GAMMA
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), EVI(3), EVQ(3), EVR(3)
Arguments
Input/Output Parameters
ALPHA (input) DOUBLE PRECISION
BETA (input) DOUBLE PRECISION
GAMMA (input) DOUBLE PRECISION
DELTA (input) DOUBLE PRECISION
The coefficients of the polynomial P.
EVR (output) DOUBLE PRECISION array, DIMENSION at least 3
EVI (output) DOUBLE PRECISION array, DIMENSION at least 3
EVQ (output) DOUBLE PRECISION array, DIMENSION at least 3
On output, the kth root of P will be equal to
(EVR(K) + i*EVI(K))/EVQ(K) if EVQ(K) .NE. ZERO. Note that
the quotient may over- or underflow. If P has a degree d
less than 3, then 3-d computed roots will be infinite.
EVQ(K) >= 0.
Workspace
DWORK DOUBLE PRECISION array, DIMENSION (LDWORK)
On exit, if LDWORK = -1 on input, then DWORK(1) returns
the optimal value of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK. LDWORK >= 42.
If LDWORK = -1, an optimal workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = j, 1 <= j <= 6, an error occurred during
the call to one of the LAPACK Library routines DGGEV
or DGEEV (1 <= j <= 3). If INFO < 3, the values
returned in EVR(K), EVI(K), and EVQ(K) should be
correct for K = INFO+1,...,3.
Method
A matrix pencil is built, whose eigenvalues are the roots of the given polynomial, and they are computed using the QZ algorithm. However, when the ratio between the largest and smallest (in magnitude) polynomial coefficients is relatively big, and either ALPHA or DELTA has the largest magnitude, then a standard eigenproblem is solved using the QR algorithm, and EVQ(I) are set to 1, for I = 1,2,3.Numerical Aspects
The algorithm is numerically stable.Further Comments
NoneExample
Program Text
* MC01XD EXAMPLE PROGRAM TEXT.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NEV
PARAMETER ( NEV = 3 )
INTEGER LDWORK
PARAMETER ( LDWORK = 42 )
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
* .. Local Scalars ..
INTEGER I, INFO, J
DOUBLE PRECISION ALPHA, BETA, DELTA, GAMMA
* .. Local Arrays ..
DOUBLE PRECISION DWORK(LDWORK), EVI(NEV), EVQ(NEV), EVR(NEV),
$ RT(2)
* .. External Subroutines ..
EXTERNAL MC01XD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) ALPHA, BETA, GAMMA, DELTA
* Compute the roots of the polynomial.
CALL MC01XD( ALPHA, BETA, GAMMA, DELTA, EVR, EVI, EVQ,
$ DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99995 )
WRITE ( NOUT, FMT = 99996 ) ( EVR(J), J = 1, 3 )
*
WRITE ( NOUT, FMT = 99994 )
WRITE ( NOUT, FMT = 99996 ) ( EVI(J), J = 1, 3 )
*
WRITE ( NOUT, FMT = 99993 )
WRITE ( NOUT, FMT = 99996 ) ( EVQ(J), J = 1, 3 )
*
WRITE ( NOUT, FMT = 99992 )
DO 20 I = 1, 3
IF ( EVQ(I).NE.ZERO ) THEN
RT(1) = EVR(I)/EVQ(I)
RT(2) = EVI(I)/EVQ(I)
WRITE ( NOUT, FMT = 99996 ) ( RT(J), J = 1, 2 )
END IF
20 CONTINUE
*
END IF
STOP
*
99999 FORMAT (' MC01XD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MC01XD = ',I2)
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (' Real part of the numerators of the roots')
99994 FORMAT (' Imaginary part of the numerators of the roots')
99993 FORMAT (' Denominators of the roots')
99992 FORMAT (' Roots of the polynomial',/1X)
END
Program Data
MC01XD EXAMPLE PROGRAM DATA -3098110792.0746 4649783048.0746 -2327508384.0000 388574551.8120Program Results
MC01XD EXAMPLE PROGRAM RESULTS Real part of the numerators of the roots 1.0185 1.1048 0.5911 Imaginary part of the numerators of the roots 0.0000 0.0455 -0.0244 Denominators of the roots 0.5110 0.5528 0.2958 Roots of the polynomial 1.9931 0.0000 1.9984 0.0823 1.9984 -0.0823
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MC03MD.html 0000664 0000000 0000000 00000027562 14560147231 0020351 0 ustar 00root root 0000000 0000000
Purpose
To compute the coefficients of the real polynomial matrix
P(x) = P1(x) * P2(x) + alpha * P3(x),
where P1(x), P2(x) and P3(x) are given real polynomial matrices
and alpha is a real scalar.
Each of the polynomial matrices P1(x), P2(x) and P3(x) may be the
zero matrix.
Specification
SUBROUTINE MC03MD( RP1, CP1, CP2, DP1, DP2, DP3, ALPHA, P1,
$ LDP11, LDP12, P2, LDP21, LDP22, P3, LDP31,
$ LDP32, DWORK, INFO )
C .. Scalar Arguments ..
INTEGER CP1, CP2, DP1, DP2, DP3, INFO, LDP11, LDP12,
$ LDP21, LDP22, LDP31, LDP32, RP1
DOUBLE PRECISION ALPHA
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), P1(LDP11,LDP12,*), P2(LDP21,LDP22,*),
$ P3(LDP31,LDP32,*)
Arguments
Input/Output Parameters
RP1 (input) INTEGER
The number of rows of the matrices P1(x) and P3(x).
RP1 >= 0.
CP1 (input) INTEGER
The number of columns of matrix P1(x) and the number of
rows of matrix P2(x). CP1 >= 0.
CP2 (input) INTEGER
The number of columns of the matrices P2(x) and P3(x).
CP2 >= 0.
DP1 (input) INTEGER
The degree of the polynomial matrix P1(x). DP1 >= -1.
DP2 (input) INTEGER
The degree of the polynomial matrix P2(x). DP2 >= -1.
DP3 (input/output) INTEGER
On entry, the degree of the polynomial matrix P3(x).
DP3 >= -1.
On exit, the degree of the polynomial matrix P(x).
ALPHA (input) DOUBLE PRECISION
The scalar value alpha of the problem.
P1 (input) DOUBLE PRECISION array, dimension (LDP11,LDP12,*)
If DP1 >= 0, then the leading RP1-by-CP1-by-(DP1+1) part
of this array must contain the coefficients of the
polynomial matrix P1(x). Specifically, P1(i,j,k) must
contain the coefficient of x**(k-1) of the polynomial
which is the (i,j)-th element of P1(x), where i = 1,2,...,
RP1, j = 1,2,...,CP1 and k = 1,2,...,DP1+1.
If DP1 = -1, then P1(x) is taken to be the zero polynomial
matrix, P1 is not referenced and can be supplied as a
dummy array (i.e. set the parameters LDP11 = LDP12 = 1 and
declare this array to be P1(1,1,1) in the calling
program).
LDP11 INTEGER
The leading dimension of array P1.
LDP11 >= MAX(1,RP1) if DP1 >= 0,
LDP11 >= 1 if DP1 = -1.
LDP12 INTEGER
The second dimension of array P1.
LDP12 >= MAX(1,CP1) if DP1 >= 0,
LDP12 >= 1 if DP1 = -1.
P2 (input) DOUBLE PRECISION array, dimension (LDP21,LDP22,*)
If DP2 >= 0, then the leading CP1-by-CP2-by-(DP2+1) part
of this array must contain the coefficients of the
polynomial matrix P2(x). Specifically, P2(i,j,k) must
contain the coefficient of x**(k-1) of the polynomial
which is the (i,j)-th element of P2(x), where i = 1,2,...,
CP1, j = 1,2,...,CP2 and k = 1,2,...,DP2+1.
If DP2 = -1, then P2(x) is taken to be the zero polynomial
matrix, P2 is not referenced and can be supplied as a
dummy array (i.e. set the parameters LDP21 = LDP22 = 1 and
declare this array to be P2(1,1,1) in the calling
program).
LDP21 INTEGER
The leading dimension of array P2.
LDP21 >= MAX(1,CP1) if DP2 >= 0,
LDP21 >= 1 if DP2 = -1.
LDP22 INTEGER
The second dimension of array P2.
LDP22 >= MAX(1,CP2) if DP2 >= 0,
LDP22 >= 1 if DP2 = -1.
P3 (input/output) DOUBLE PRECISION array, dimension
(LDP31,LDP32,n), where n = MAX(DP1+DP2,DP3,0)+1.
On entry, if DP3 >= 0, then the leading
RP1-by-CP2-by-(DP3+1) part of this array must contain the
coefficients of the polynomial matrix P3(x). Specifically,
P3(i,j,k) must contain the coefficient of x**(k-1) of the
polynomial which is the (i,j)-th element of P3(x), where
i = 1,2,...,RP1, j = 1,2,...,CP2 and k = 1,2,...,DP3+1.
If DP3 = -1, then P3(x) is taken to be the zero polynomial
matrix.
On exit, if DP3 >= 0 on exit (ALPHA <> 0.0 and DP3 <> -1,
on entry, or DP1 <> -1 and DP2 <> -1), then the leading
RP1-by-CP2-by-(DP3+1) part of this array contains the
coefficients of P(x). Specifically, P3(i,j,k) contains the
coefficient of x**(k-1) of the polynomial which is the
(i,j)-th element of P(x), where i = 1,2,...,RP1, j = 1,2,
...,CP2 and k = 1,2,...,DP3+1.
If DP3 = -1 on exit, then the coefficients of P(x) (the
zero polynomial matrix) are not stored in the array.
LDP31 INTEGER
The leading dimension of array P3. LDP31 >= MAX(1,RP1).
LDP32 INTEGER
The second dimension of array P3. LDP32 >= MAX(1,CP2).
Workspace
DWORK DOUBLE PRECISION array, dimension (CP1)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Given real polynomial matrices
DP1 i
P1(x) = SUM (A(i+1) * x ),
i=0
DP2 i
P2(x) = SUM (B(i+1) * x ),
i=0
DP3 i
P3(x) = SUM (C(i+1) * x )
i=0
and a real scalar alpha, the routine computes the coefficients
d ,d ,..., of the polynomial matrix
1 2
P(x) = P1(x) * P2(x) + alpha * P3(x)
from the formula
s
d = SUM (A(k+1) * B(i-k+1)) + alpha * C(i+1),
i+1 k=r
where i = 0,1,...,DP1+DP2 and r and s depend on the value of i
(e.g. if i <= DP1 and i <= DP2, then r = 0 and s = i).
Numerical Aspects
None.Further Comments
Other elementary operations involving polynomial matrices can easily be obtained by calling the appropriate BLAS routine(s).Example
Program Text
* MC03MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER CP1MAX, CP2MAX, DP1MAX, DP2MAX, DP3MAX, RP1MAX
PARAMETER ( CP1MAX = 10, CP2MAX = 10, DP1MAX = 10,
$ DP2MAX = 10, DP3MAX = 20, RP1MAX = 10 )
INTEGER LDP11, LDP12, LDP21, LDP22, LDP31, LDP32
PARAMETER ( LDP11 = RP1MAX, LDP12 = CP1MAX,
$ LDP21 = CP1MAX, LDP22 = CP2MAX,
$ LDP31 = RP1MAX, LDP32 = CP2MAX )
* .. Local Scalars ..
DOUBLE PRECISION ALPHA
INTEGER CP1, CP2, DP1, DP2, DP3, I, INFO, J, K, RP1
* .. Local Arrays ..
DOUBLE PRECISION DWORK(CP1MAX),
$ P1(LDP11,LDP12,DP1MAX+1),
$ P2(LDP21,LDP22,DP2MAX+1),
$ P3(LDP31,LDP32,DP3MAX+1)
* .. External Subroutines ..
EXTERNAL MC03MD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) RP1, CP1, CP2
IF ( RP1.LT.0 .OR. RP1.GT.RP1MAX ) THEN
WRITE ( NOUT, FMT = 99995 ) RP1
ELSE IF ( CP1.LT.0 .OR. CP1.GT.CP1MAX ) THEN
WRITE ( NOUT, FMT = 99994 ) CP1
ELSE IF ( CP2.LT.0 .OR. CP2.GT.CP2MAX ) THEN
WRITE ( NOUT, FMT = 99993 ) CP2
ELSE
READ ( NIN, FMT = * ) DP1
IF ( DP1.LE.-2 .OR. DP1.GT.DP1MAX ) THEN
WRITE ( NOUT, FMT = 99992 ) DP1
ELSE
DO 40 K = 1, DP1 + 1
DO 20 J = 1, CP1
READ ( NIN, FMT = * ) ( P1(I,J,K), I = 1,RP1 )
20 CONTINUE
40 CONTINUE
READ ( NIN, FMT = * ) DP2
IF ( DP2.LE.-2 .OR. DP2.GT.DP2MAX ) THEN
WRITE ( NOUT, FMT = 99991 ) DP2
ELSE
DO 80 K = 1, DP2 + 1
DO 60 J = 1, CP2
READ ( NIN, FMT = * ) ( P2(I,J,K), I = 1,CP1 )
60 CONTINUE
80 CONTINUE
READ ( NIN, FMT = * ) DP3
IF ( DP3.LE.-2 .OR. DP3.GT.DP3MAX ) THEN
WRITE ( NOUT, FMT = 99990 ) DP3
ELSE
DO 120 K = 1, DP3 + 1
DO 100 J = 1, CP2
READ ( NIN, FMT = * ) ( P3(I,J,K), I = 1,RP1 )
100 CONTINUE
120 CONTINUE
READ ( NIN, FMT = * ) ALPHA
* Compute the coefficients of the polynomial matrix P(x)
CALL MC03MD( RP1, CP1, CP2, DP1, DP2, DP3, ALPHA, P1,
$ LDP11, LDP12, P2, LDP21, LDP22, P3,
$ LDP31, LDP32, DWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) DP3,
$ ( I-1, I = 1,DP3+1 )
DO 160 I = 1, RP1
DO 140 J = 1, CP2
WRITE ( NOUT, FMT = 99996 ) I, J,
$ ( P3(I,J,K), K = 1,DP3+1 )
140 CONTINUE
160 CONTINUE
END IF
END IF
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' MC03MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MC03MD = ',I2)
99997 FORMAT (' The polynomial matrix P(x) (of degree ',I2,') is ',
$ //' power of x ',20I8)
99996 FORMAT (/' element (',I2,',',I2,') is ',20(1X,F7.2))
99995 FORMAT (/' RP1 is out of range.',/' RP1 = ',I5)
99994 FORMAT (/' CP1 is out of range.',/' CP1 = ',I5)
99993 FORMAT (/' CP2 is out of range.',/' CP2 = ',I5)
99992 FORMAT (/' DP1 is out of range.',/' DP1 = ',I5)
99991 FORMAT (/' DP2 is out of range.',/' DP2 = ',I5)
99990 FORMAT (/' DP3 is out of range.',/' DP3 = ',I5)
END
Program Data
MC03MD EXAMPLE PROGRAM DATA 3 2 2 2 1.0 0.0 3.0 2.0 -1.0 2.0 -2.0 4.0 9.0 3.0 7.0 -2.0 6.0 2.0 -3.0 1.0 2.0 4.0 1 6.0 1.0 1.0 7.0 -9.0 -6.0 7.0 8.0 1 1.0 1.0 0.0 0.0 1.0 1.0 -1.0 1.0 1.0 -1.0 -1.0 1.0 1.0Program Results
MC03MD EXAMPLE PROGRAM RESULTS The polynomial matrix P(x) (of degree 3) is power of x 0 1 2 3 element ( 1, 1) is 9.00 -31.00 37.00 -60.00 element ( 1, 2) is 15.00 41.00 23.00 50.00 element ( 2, 1) is 0.00 38.00 -64.00 -30.00 element ( 2, 2) is -6.00 44.00 100.00 30.00 element ( 3, 1) is 20.00 14.00 -83.00 3.00 element ( 3, 2) is 18.00 33.00 72.00 11.00
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MC03ND.html 0000664 0000000 0000000 00000035461 14560147231 0020347 0 ustar 00root root 0000000 0000000
Purpose
To compute the coefficients of a minimal polynomial basis
DK
K(s) = K(0) + K(1) * s + ... + K(DK) * s
for the right nullspace of the MP-by-NP polynomial matrix of
degree DP, given by
DP
P(s) = P(0) + P(1) * s + ... + P(DP) * s ,
which corresponds to solving the polynomial matrix equation
P(s) * K(s) = 0.
Specification
SUBROUTINE MC03ND( MP, NP, DP, P, LDP1, LDP2, DK, GAM, NULLSP,
$ LDNULL, KER, LDKER1, LDKER2, TOL, IWORK, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER DK, DP, INFO, LDKER1, LDKER2, LDNULL, LDP1,
$ LDP2, LDWORK, MP, NP
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER GAM(*), IWORK(*)
DOUBLE PRECISION DWORK(*), KER(LDKER1,LDKER2,*),
$ NULLSP(LDNULL,*), P(LDP1,LDP2,*)
Arguments
Input/Output Parameters
MP (input) INTEGER
The number of rows of the polynomial matrix P(s).
MP >= 0.
NP (input) INTEGER
The number of columns of the polynomial matrix P(s).
NP >= 0.
DP (input) INTEGER
The degree of the polynomial matrix P(s). DP >= 1.
P (input) DOUBLE PRECISION array, dimension (LDP1,LDP2,DP+1)
The leading MP-by-NP-by-(DP+1) part of this array must
contain the coefficients of the polynomial matrix P(s).
Specifically, P(i,j,k) must contain the (i,j)-th element
of P(k-1), which is the cofficient of s**(k-1) of P(s),
where i = 1,2,...,MP, j = 1,2,...,NP and k = 1,2,...,DP+1.
LDP1 INTEGER
The leading dimension of array P. LDP1 >= MAX(1,MP).
LDP2 INTEGER
The second dimension of array P. LDP2 >= MAX(1,NP).
DK (output) INTEGER
The degree of the minimal polynomial basis K(s) for the
right nullspace of P(s) unless DK = -1, in which case
there is no right nullspace.
GAM (output) INTEGER array, dimension (DP*MP+1)
The leading (DK+1) elements of this array contain
information about the ordering of the right nullspace
vectors stored in array NULLSP.
NULLSP (output) DOUBLE PRECISION array, dimension
(LDNULL,(DP*MP+1)*NP)
The leading NP-by-SUM(i*GAM(i)) part of this array
contains the right nullspace vectors of P(s) in condensed
form (as defined in METHOD), where i = 1,2,...,DK+1.
LDNULL INTEGER
The leading dimension of array NULLSP.
LDNULL >= MAX(1,NP).
KER (output) DOUBLE PRECISION array, dimension
(LDKER1,LDKER2,DP*MP+1)
The leading NP-by-nk-by-(DK+1) part of this array contains
the coefficients of the minimal polynomial basis K(s),
where nk = SUM(GAM(i)) and i = 1,2,...,DK+1. Specifically,
KER(i,j,m) contains the (i,j)-th element of K(m-1), which
is the coefficient of s**(m-1) of K(s), where i = 1,2,...,
NP, j = 1,2,...,nk and m = 1,2,...,DK+1.
LDKER1 INTEGER
The leading dimension of array KER. LDKER1 >= MAX(1,NP).
LDKER2 INTEGER
The second dimension of array KER. LDKER2 >= MAX(1,NP).
Tolerances
TOL DOUBLE PRECISION
A tolerance below which matrix elements are considered
to be zero. If the user sets TOL to be less than
10 * EPS * MAX( ||A|| , ||E|| ), then the tolerance is
F F
taken as 10 * EPS * MAX( ||A|| , ||E|| ), where EPS is the
F F
machine precision (see LAPACK Library Routine DLAMCH) and
A and E are matrices (as defined in METHOD).
Workspace
IWORK INTEGER array, dimension (m+2*MAX(n,m+1)+n),
where m = DP*MP and n = (DP-1)*MP + NP.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK The length of the array DWORK.
LDWORK >= m*n*n + 2*m*n + 2*n*n.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
> 0: if incorrect rank decisions were taken during the
computations. This failure is not likely to occur.
The possible values are:
k, 1 <= k <= DK+1, the k-th diagonal submatrix had
not a full row rank;
DK+2, if incorrect dimensions of a full column
rank submatrix;
DK+3, if incorrect dimensions of a full row rank
submatrix.
Method
The computation of the right nullspace of the MP-by-NP polynomial
matrix P(s) of degree DP given by
DP-1 DP
P(s) = P(0) + P(1) * s + ... + P(DP-1) * s + P(DP) * s
is performed via the pencil s*E - A, associated with P(s), where
| I | | 0 -P(DP) |
| . | | I . . |
A = | . | and E = | . . . |. (1)
| . | | . 0 . |
| I | | I 0 -P(2) |
| P(0) | | I -P(1) |
The pencil s*E - A is transformed by unitary matrices Q and Z such
that
| sE(eps)-A(eps) | X | X |
|----------------|----------------|------------|
| 0 | sE(inf)-A(inf) | X |
Q'(s*E-A)Z = |=================================|============|.
| | |
| 0 | sE(r)-A(r) |
Since s*E(inf)-A(inf) and s*E(r)-A(r) have full column rank, the
minimal polynomial basis for the right nullspace of Q'(s*E-A)Z
(and consequently the basis for the right nullspace of s*E - A) is
completely determined by s*E(eps)-A(eps).
Let Veps(s) be a minimal polynomial basis for the right nullspace
of s*E(eps)-A(eps). Then
| Veps(s) |
V(s) = Z * |---------|
| 0 |
is a minimal polynomial basis for the right nullspace of s*E - A.
From the structure of s*E - A it can be shown that if V(s) is
partitioned as
| Vo(s) | (DP-1)*MP
V(s) = |------ |
| Ve(s) | NP
then the columns of Ve(s) form a minimal polynomial basis for the
right nullspace of P(s).
The vectors of Ve(s) are computed and stored in array NULLSP in
the following condensed form:
|| || | || | | || | |
|| U1,0 || U2,0 | U2,1 || U3,0 | U3,1 | U3,2 || U4,0 | ... |,
|| || | || | | || | |
where Ui,j is an NP-by-GAM(i) matrix which contains the i-th block
of columns of K(j), the j-th coefficient of the polynomial matrix
representation for the right nullspace
DK
K(s) = K(0) + K(1) * s + . . . + K(DK) * s .
The coefficients K(0), K(1), ..., K(DK) are NP-by-nk matrices
given by
K(0) = | U1,0 | U2,0 | U3,0 | . . . | U(DK+1,0) |
K(1) = | 0 | U2,1 | U3,1 | . . . | U(DK+1,1) |
K(2) = | 0 | 0 | U3,2 | . . . | U(DK+1,2) |
. . . . . . . . . .
K(DK) = | 0 | 0 | 0 | . . . | 0 | U(DK+1,DK)|.
Note that the degree of K(s) satisfies the inequality DK <=
DP * MIN(MP,NP) and that the dimension of K(s) satisfies the
inequality (NP-MP) <= nk <= NP.
References
[1] Beelen, Th.G.J.
New Algorithms for Computing the Kronecker structure of a
Pencil with Applications to Systems and Control Theory.
Ph.D.Thesis, Eindhoven University of Technology, 1987.
[2] Van Den Hurk, G.J.H.H.
New Algorithms for Solving Polynomial Matrix Problems.
Master's Thesis, Eindhoven University of Technology, 1987.
Numerical Aspects
The algorithm used by the routine involves the construction of a special block echelon form with pivots considered to be non-zero when they are larger than TOL. These pivots are then inverted in order to construct the columns of the kernel of the polynomial matrix. If TOL is chosen to be too small then these inversions may be sensitive whereas increasing TOL will make the inversions more robust but will affect the block echelon form (and hence the column degrees of the polynomial kernel). Furthermore, if the elements of the computed polynomial kernel are large relative to the polynomial matrix, then the user should consider trying several values of TOL.Further Comments
It also possible to compute a minimal polynomial basis for the right nullspace of a pencil, since a pencil is a polynomial matrix of degree 1. Thus for the pencil (s*E - A), the required input is P(1) = E and P(0) = -A. The routine can also be used to compute a minimal polynomial basis for the left nullspace of a polynomial matrix by simply transposing P(s).Example
Program Text
* MC03ND EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER DPMAX, MPMAX, NPMAX
* PARAMETER ( DPMAX = 5, MPMAX = 5, NPMAX = 5 )
PARAMETER ( DPMAX = 2, MPMAX = 5, NPMAX = 4 )
INTEGER LDP1, LDP2, LDNULL, LDKER1, LDKER2
PARAMETER ( LDP1 = MPMAX, LDP2 = NPMAX, LDNULL = NPMAX,
$ LDKER1 = NPMAX, LDKER2 = NPMAX )
INTEGER M, N
PARAMETER ( M = DPMAX*MPMAX, N = ( DPMAX-1 )*MPMAX+NPMAX )
INTEGER LIWORK, LDWORK
* PARAMETER ( LIWORK = 3*( N+M )+2,
PARAMETER ( LIWORK = M+2*MAX( N,M+1 )+N,
$ LDWORK = M*N**2+2*M*N+2*N**2 )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER DK, DP, I, INFO, J, K, M1, MP, NK, NP
* .. Local Arrays ..
DOUBLE PRECISION DWORK(LDWORK), KER(LDKER1,LDKER2,M+1),
$ NULLSP(LDNULL,(M+1)*NPMAX), P(LDP1,LDP2,DPMAX+1)
INTEGER GAM(M+1), IWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL MC03ND
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) MP, NP, DP, TOL
IF ( MP.LT.0 .OR. MP.GT.MPMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) MP
ELSE IF ( NP.LT.0 .OR. NP.GT.NPMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) NP
ELSE IF ( DP.LE.0 .OR. DP.GT.DPMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) DP
ELSE
DO 40 K = 1, DP + 1
DO 20 I = 1, MP
READ ( NIN, FMT = * ) ( P(I,J,K), J = 1,NP )
20 CONTINUE
40 CONTINUE
* Compute a minimal polynomial basis K(s) of the given P(s).
CALL MC03ND( MP, NP, DP, P, LDP1, LDP2, DK, GAM, NULLSP,
$ LDNULL, KER, LDKER1, LDKER2, TOL, IWORK, DWORK,
$ LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE IF ( DK.LT.0 ) THEN
WRITE ( NOUT, FMT = 99997 )
ELSE
NK = 0
M1 = 0
DO 60 I = 1, DK + 1
NK = NK + GAM(I)
M1 = M1 + GAM(I)*I
60 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 80 I = 1, NP
WRITE ( NOUT, FMT = 99995 ) ( NULLSP(I,J), J = 1,M1 )
80 CONTINUE
WRITE ( NOUT, FMT = 99994 ) DK, ( I-1, I = 1,DK+1 )
DO 120 I = 1, NP
DO 100 J = 1, NK
WRITE ( NOUT, FMT = 99993 )
$ I, J, ( KER(I,J,K), K = 1,DK+1 )
100 CONTINUE
120 CONTINUE
END IF
END IF
STOP
*
99999 FORMAT (' MC03ND EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MC03ND = ',I2)
99997 FORMAT (' The polynomial matrix P(s) has no right nullspace')
99996 FORMAT (' The right nullspace vectors of P(s) are ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' The minimal polynomial basis K(s) (of degree ',I2,') ',
$ 'for the right nullspace is ',//' power of s ',
$ 20I8)
99993 FORMAT (/' element (',I2,',',I2,') is ',20(1X,F7.2))
99992 FORMAT (/' DP is out of range.',/' DP = ',I5)
99991 FORMAT (/' NP is out of range.',/' NP = ',I5)
99990 FORMAT (/' MP is out of range.',/' MP = ',I5)
END
Program Data
MC03ND EXAMPLE PROGRAM DATA 5 4 2 0.0 2.0 2.0 0.0 3.0 0.0 4.0 0.0 6.0 8.0 8.0 0.0 12.0 0.0 0.0 0.0 0.0 2.0 2.0 0.0 3.0 1.0 0.0 1.0 0.0 0.0 0.0 2.0 0.0 4.0 0.0 4.0 0.0 2.0 2.0 0.0 3.0 3.0 2.0 1.0 3.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 1.0 0.0 1.0 0.0 1.0 0.0Program Results
MC03ND EXAMPLE PROGRAM RESULTS The right nullspace vectors of P(s) are 0.0000 0.0000 0.0000 -0.8321 0.0000 0.1538 0.0000 -1.0000 0.0000 0.5547 0.0000 0.2308 The minimal polynomial basis K(s) (of degree 1) for the right nullspace is power of s 0 1 element ( 1, 1) is 0.00 0.00 element ( 1, 2) is 0.00 0.00 element ( 2, 1) is -0.83 0.00 element ( 2, 2) is 0.00 0.15 element ( 3, 1) is 0.00 0.00 element ( 3, 2) is -1.00 0.00 element ( 4, 1) is 0.55 0.00 element ( 4, 2) is 0.00 0.23
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MC03NX.html 0000664 0000000 0000000 00000006753 14560147231 0020375 0 ustar 00root root 0000000 0000000
Purpose
Given an MP-by-NP polynomial matrix of degree dp
dp-1 dp
P(s) = P(0) + ... + P(dp-1) * s + P(dp) * s (1)
the routine composes the related pencil s*E-A where
| I | | O -P(dp) |
| . | | I . . |
A = | . | and E = | . . . |. (2)
| . | | . O . |
| I | | I O -P(2) |
| P(0) | | I -P(1) |
==================================================================
REMARK: This routine is intended to be called only from the SLICOT
routine MC03ND.
==================================================================
Specification
SUBROUTINE MC03NX( MP, NP, DP, P, LDP1, LDP2, A, LDA, E, LDE )
C .. Scalar Arguments ..
INTEGER DP, LDA, LDE, LDP1, LDP2, MP, NP
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), E(LDE,*), P(LDP1,LDP2,*)
Arguments
Input/Output Parameters
MP (input) INTEGER
The number of rows of the polynomial matrix P(s).
MP >= 0.
NP (input) INTEGER
The number of columns of the polynomial matrix P(s).
NP >= 0.
DP (input) INTEGER
The degree of the polynomial matrix P(s). DP >= 1.
P (input) DOUBLE PRECISION array, dimension (LDP1,LDP2,DP+1)
The leading MP-by-NP-by-(DP+1) part of this array must
contain the coefficients of the polynomial matrix P(s)
in (1) in increasing powers of s.
LDP1 INTEGER
The leading dimension of array P. LDP1 >= MAX(1,MP).
LDP2 INTEGER
The second dimension of array P. LDP2 >= MAX(1,NP).
A (output) DOUBLE PRECISION array, dimension
(LDA,(DP-1)*MP+NP)
The leading DP*MP-by-((DP-1)*MP+NP) part of this array
contains the matrix A as described in (2).
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,DP*MP).
E (output) DOUBLE PRECISION array, dimension
(LDE,(DP-1)*MP+NP)
The leading DP*MP-by-((DP-1)*MP+NP) part of this array
contains the matrix E as described in (2).
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,DP*MP).
Numerical Aspects
None.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To minimize the sum of the squares of m nonlinear functions, e, in n variables, x, by a modification of the Levenberg-Marquardt algorithm, using either a Cholesky-based or a conjugate gradients solver. The user must provide a subroutine FCN which calculates the functions and the Jacobian J (possibly by finite differences), and another subroutine JPJ, which computes either J'*J + par*I (if ALG = 'D'), or (J'*J + par*I)*x (if ALG = 'I'), where par is the Levenberg factor, exploiting the possible structure of the Jacobian matrix. Template implementations of these routines are included in the SLICOT Library.Specification
SUBROUTINE MD03AD( XINIT, ALG, STOR, UPLO, FCN, JPJ, M, N, ITMAX,
$ NPRINT, IPAR, LIPAR, DPAR1, LDPAR1, DPAR2,
$ LDPAR2, X, NFEV, NJEV, TOL, CGTOL, DWORK,
$ LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER ALG, STOR, UPLO, XINIT
INTEGER INFO, ITMAX, IWARN, LDPAR1, LDPAR2, LDWORK,
$ LIPAR, M, N, NFEV, NJEV, NPRINT
DOUBLE PRECISION CGTOL, TOL
C .. Array Arguments ..
DOUBLE PRECISION DPAR1(LDPAR1,*), DPAR2(LDPAR2,*), DWORK(*), X(*)
INTEGER IPAR(*)
Arguments
Mode Parameters
XINIT CHARACTER*1
Specifies how the variables x are initialized, as follows:
= 'R' : the array X is initialized to random values; the
entries DWORK(1:4) are used to initialize the
random number generator: the first three values
are converted to integers between 0 and 4095, and
the last one is converted to an odd integer
between 1 and 4095;
= 'G' : the given entries of X are used as initial values
of variables.
ALG CHARACTER*1
Specifies the algorithm used for solving the linear
systems involving a Jacobian matrix J, as follows:
= 'D' : a direct algorithm, which computes the Cholesky
factor of the matrix J'*J + par*I is used;
= 'I' : an iterative Conjugate Gradients algorithm, which
only needs the matrix J, is used.
In both cases, matrix J is stored in a compressed form.
STOR CHARACTER*1
If ALG = 'D', specifies the storage scheme for the
symmetric matrix J'*J, as follows:
= 'F' : full storage is used;
= 'P' : packed storage is used.
The option STOR = 'F' usually ensures a faster execution.
This parameter is not relevant if ALG = 'I'.
UPLO CHARACTER*1
If ALG = 'D', specifies which part of the matrix J'*J
is stored, as follows:
= 'U' : the upper triagular part is stored;
= 'L' : the lower triagular part is stored.
The option UPLO = 'U' usually ensures a faster execution.
This parameter is not relevant if ALG = 'I'.
Function Parameters
FCN EXTERNAL
Subroutine which evaluates the functions and the Jacobian.
FCN must be declared in an external statement in the user
calling program, and must have the following interface:
SUBROUTINE FCN( IFLAG, M, N, IPAR, LIPAR, DPAR1, LDPAR1,
$ DPAR2, LDPAR2, X, NFEVL, E, J, LDJ, JTE,
$ DWORK, LDWORK, INFO )
where
IFLAG (input/output) INTEGER
On entry, this parameter must contain a value
defining the computations to be performed:
= 0 : Optionally, print the current iterate X,
function values E, and Jacobian matrix J,
or other results defined in terms of these
values. See the argument NPRINT of MD03AD.
Do not alter E and J.
= 1 : Calculate the functions at X and return
this vector in E. Do not alter J.
= 2 : Calculate the Jacobian at X and return
this matrix in J. Also return J'*e in JTE
and NFEVL (see below). Do not alter E.
= 3 : Do not compute neither the functions nor
the Jacobian, but return in LDJ and
IPAR/DPAR1,DPAR2 (some of) the integer/real
parameters needed.
On exit, the value of this parameter should not be
changed by FCN unless the user wants to terminate
execution of MD03AD, in which case IFLAG must be
set to a negative integer.
M (input) INTEGER
The number of functions. M >= 0.
N (input) INTEGER
The number of variables. M >= N >= 0.
IPAR (input/output) INTEGER array, dimension (LIPAR)
The integer parameters describing the structure of
the Jacobian matrix or needed for problem solving.
IPAR is an input parameter, except for IFLAG = 3
on entry, when it is also an output parameter.
On exit, if IFLAG = 3, IPAR(1) contains the length
of the array J, for storing the Jacobian matrix,
and the entries IPAR(2:5) contain the workspace
required by FCN for IFLAG = 1, FCN for IFLAG = 2,
JPJ for ALG = 'D', and JPJ for ALG = 'I',
respectively.
LIPAR (input) INTEGER
The length of the array IPAR. LIPAR >= 5.
DPAR1 (input/output) DOUBLE PRECISION array, dimension
(LDPAR1,*) or (LDPAR1)
A first set of real parameters needed for
describing or solving the problem.
DPAR1 can also be used as an additional array for
intermediate results when computing the functions
or the Jacobian. For control problems, DPAR1 could
store the input trajectory of a system.
LDPAR1 (input) INTEGER
The leading dimension or the length of the array
DPAR1, as convenient. LDPAR1 >= 0. (LDPAR1 >= 1,
if leading dimension.)
DPAR2 (input/output) DOUBLE PRECISION array, dimension
(LDPAR2,*) or (LDPAR2)
A second set of real parameters needed for
describing or solving the problem.
DPAR2 can also be used as an additional array for
intermediate results when computing the functions
or the Jacobian. For control problems, DPAR2 could
store the output trajectory of a system.
LDPAR2 (input) INTEGER
The leading dimension or the length of the array
DPAR2, as convenient. LDPAR2 >= 0. (LDPAR2 >= 1,
if leading dimension.)
X (input) DOUBLE PRECISION array, dimension (N)
This array must contain the value of the
variables x where the functions or the Jacobian
must be evaluated.
NFEVL (input/output) INTEGER
The number of function evaluations needed to
compute the Jacobian by a finite difference
approximation.
NFEVL is an input parameter if IFLAG = 0, or an
output parameter if IFLAG = 2. If the Jacobian is
computed analytically, NFEVL should be set to a
non-positive value.
E (input/output) DOUBLE PRECISION array,
dimension (M)
This array contains the value of the (error)
functions e evaluated at X.
E is an input parameter if IFLAG = 0 or 2, or an
output parameter if IFLAG = 1.
J (input/output) DOUBLE PRECISION array, dimension
(LDJ,NC), where NC is the number of columns
needed.
This array contains a possibly compressed
representation of the Jacobian matrix evaluated
at X. If full Jacobian is stored, then NC = N.
J is an input parameter if IFLAG = 0, or an output
parameter if IFLAG = 2.
LDJ (input/output) INTEGER
The leading dimension of array J. LDJ >= 1.
LDJ is essentially used inside the routines FCN
and JPJ.
LDJ is an input parameter, except for IFLAG = 3
on entry, when it is an output parameter.
It is assumed in MD03AD that LDJ is not larger
than needed.
JTE (output) DOUBLE PRECISION array, dimension (N)
If IFLAG = 2, the matrix-vector product J'*e.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
The workspace array for subroutine FCN.
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
LDWORK (input) INTEGER
The size of the array DWORK (as large as needed
in the subroutine FCN). LDWORK >= 1.
INFO INTEGER
Error indicator, set to a negative value if an
input (scalar) argument is erroneous, and to
positive values for other possible errors in the
subroutine FCN. The LAPACK Library routine XERBLA
should be used in conjunction with negative INFO.
INFO must be zero if the subroutine finished
successfully.
Parameters marked with "(input)" must not be changed.
JPJ EXTERNAL
Subroutine which computes J'*J + par*I, if ALG = 'D', and
J'*J*x + par*x, if ALG = 'I', where J is the Jacobian as
described above.
JPJ must have the following interface:
SUBROUTINE JPJ( STOR, UPLO, N, IPAR, LIPAR, DPAR, LDPAR,
$ J, LDJ, JTJ, LDJTJ, DWORK, LDWORK, INFO )
if ALG = 'D', and
SUBROUTINE JPJ( N, IPAR, LIPAR, DPAR, LDPAR, J, LDJ, X,
$ INCX, DWORK, LDWORK, INFO )
if ALG = 'I', where
STOR (input) CHARACTER*1
Specifies the storage scheme for the symmetric
matrix J'*J, as follows:
= 'F' : full storage is used;
= 'P' : packed storage is used.
UPLO (input) CHARACTER*1
Specifies which part of the matrix J'*J is stored,
as follows:
= 'U' : the upper triagular part is stored;
= 'L' : the lower triagular part is stored.
N (input) INTEGER
The number of columns of the matrix J. N >= 0.
IPAR (input) INTEGER array, dimension (LIPAR)
The integer parameters describing the structure of
the Jacobian matrix.
LIPAR (input) INTEGER
The length of the array IPAR. LIPAR >= 0.
DPAR (input) DOUBLE PRECISION array, dimension (LDPAR)
DPAR(1) must contain an initial estimate of the
Levenberg-Marquardt parameter, par. DPAR(1) >= 0.
LDPAR (input) INTEGER
The length of the array DPAR. LDPAR >= 1.
J (input) DOUBLE PRECISION array, dimension
(LDJ, NC), where NC is the number of columns.
The leading NR-by-NC part of this array must
contain the (compressed) representation of the
Jacobian matrix J, where NR is the number of rows
of J (function of IPAR entries).
LDJ (input) INTEGER
The leading dimension of array J.
LDJ >= MAX(1,NR).
JTJ (output) DOUBLE PRECISION array,
dimension (LDJTJ,N), if STOR = 'F',
dimension (N*(N+1)/2), if STOR = 'P'.
The leading N-by-N (if STOR = 'F'), or N*(N+1)/2
(if STOR = 'P') part of this array contains the
upper or lower triangle of the matrix J'*J+par*I,
depending on UPLO = 'U', or UPLO = 'L',
respectively, stored either as a two-dimensional,
or one-dimensional array, depending on STOR.
LDJTJ (input) INTEGER
The leading dimension of the array JTJ.
LDJTJ >= MAX(1,N), if STOR = 'F'.
LDJTJ >= 1, if STOR = 'P'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
The workspace array for subroutine JPJ.
LDWORK (input) INTEGER
The size of the array DWORK (as large as needed
in the subroutine JPJ).
INFO INTEGER
Error indicator, set to a negative value if an
input (scalar) argument is erroneous, and to
positive values for other possible errors in the
subroutine JPJ. The LAPACK Library routine XERBLA
should be used in conjunction with negative INFO
values. INFO must be zero if the subroutine
finished successfully.
If ALG = 'I', the parameters in common with those for
ALG = 'D', have the same meaning, and the additional
parameters are:
X (input/output) DOUBLE PRECISION array, dimension
(1+(N-1)*INCX)
On entry, this incremented array must contain the
vector x.
On exit, this incremented array contains the value
of the matrix-vector product (J'*J + par)*x.
INCX (input) INTEGER
The increment for the elements of X. INCX > 0.
Parameters marked with "(input)" must not be changed.
Input/Output Parameters
M (input) INTEGER
The number of functions. M >= 0.
N (input) INTEGER
The number of variables. M >= N >= 0.
ITMAX (input) INTEGER
The maximum number of iterations. ITMAX >= 0.
NPRINT (input) INTEGER
This parameter enables controlled printing of iterates if
it is positive. In this case, FCN is called with IFLAG = 0
at the beginning of the first iteration and every NPRINT
iterations thereafter and immediately prior to return,
with X, E, and J available for printing. If NPRINT is not
positive, no special calls of FCN with IFLAG = 0 are made.
IPAR (input) INTEGER array, dimension (LIPAR)
The integer parameters needed, for instance, for
describing the structure of the Jacobian matrix, which
are handed over to the routines FCN and JPJ.
The first five entries of this array are modified
internally by a call to FCN (with IFLAG = 3), but are
restored on exit.
LIPAR (input) INTEGER
The length of the array IPAR. LIPAR >= 5.
DPAR1 (input/output) DOUBLE PRECISION array, dimension
(LDPAR1,*) or (LDPAR1)
A first set of real parameters needed for describing or
solving the problem. This argument is not used by MD03AD
routine, but it is passed to the routine FCN.
LDPAR1 (input) INTEGER
The leading dimension or the length of the array DPAR1, as
convenient. LDPAR1 >= 0. (LDPAR1 >= 1, if leading
dimension.)
DPAR2 (input/output) DOUBLE PRECISION array, dimension
(LDPAR2,*) or (LDPAR2)
A second set of real parameters needed for describing or
solving the problem. This argument is not used by MD03AD
routine, but it is passed to the routine FCN.
LDPAR2 (input) INTEGER
The leading dimension or the length of the array DPAR2, as
convenient. LDPAR2 >= 0. (LDPAR2 >= 1, if leading
dimension.)
X (input/output) DOUBLE PRECISION array, dimension (N)
On entry, if XINIT = 'G', this array must contain the
vector of initial variables x to be optimized.
If XINIT = 'R', this array need not be set before entry,
and random values will be used to initialize x.
On exit, if INFO = 0, this array contains the vector of
values that (approximately) minimize the sum of squares of
error functions. The values returned in IWARN and
DWORK(1:5) give details on the iterative process.
NFEV (output) INTEGER
The number of calls to FCN with IFLAG = 1. If FCN is
properly implemented, this includes the function
evaluations needed for finite difference approximation
of the Jacobian.
NJEV (output) INTEGER
The number of calls to FCN with IFLAG = 2.
Tolerances
TOL DOUBLE PRECISION
If TOL >= 0, the tolerance which measures the relative
error desired in the sum of squares. Termination occurs
when the actual relative reduction in the sum of squares
is at most TOL. If the user sets TOL < 0, then SQRT(EPS)
is used instead TOL, where EPS is the machine precision
(see LAPACK Library routine DLAMCH).
CGTOL DOUBLE PRECISION
If ALG = 'I' and CGTOL > 0, the tolerance which measures
the relative residual of the solutions computed by the
conjugate gradients (CG) algorithm. Termination of a
CG process occurs when the relative residual is at
most CGTOL. If the user sets CGTOL <= 0, then SQRT(EPS)
is used instead CGTOL.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK, DWORK(2) returns the residual error norm (the
sum of squares), DWORK(3) returns the number of iterations
performed, DWORK(4) returns the total number of conjugate
gradients iterations performed (zero, if ALG = 'D'), and
DWORK(5) returns the final Levenberg factor.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= max( 5, M + 2*N + size(J) +
max( DW( FCN|IFLAG = 1 ) + N,
DW( FCN|IFLAG = 2 ),
DW( sol ) ) ),
where size(J) is the size of the Jacobian (provided by FCN
in IPAR(1), for IFLAG = 3), DW( f ) is the workspace
needed by the routine f, where f is FCN or JPJ (provided
by FCN in IPAR(2:5), for IFLAG = 3), and DW( sol ) is the
workspace needed for solving linear systems,
DW( sol ) = N*N + DW( JPJ ), if ALG = 'D', STOR = 'F';
DW( sol ) = N*(N+1)/2 + DW( JPJ ),
if ALG = 'D', STOR = 'P';
DW( sol ) = 3*N + DW( JPJ ), if ALG = 'I'.
Warning Indicator
IWARN INTEGER
< 0: the user set IFLAG = IWARN in the subroutine FCN;
= 0: no warning;
= 1: if the iterative process did not converge in ITMAX
iterations with tolerance TOL;
= 2: if ALG = 'I', and in one or more iterations of the
Levenberg-Marquardt algorithm, the conjugate
gradient algorithm did not finish after 3*N
iterations, with the accuracy required in the
call;
= 3: the cosine of the angle between e and any column of
the Jacobian is at most FACTOR*EPS in absolute
value, where FACTOR = 100 is defined in a PARAMETER
statement;
= 4: TOL is too small: no further reduction in the sum
of squares is possible.
In all these cases, DWORK(1:5) are set as described
above.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: user-defined routine FCN returned with INFO <> 0
for IFLAG = 1;
= 2: user-defined routine FCN returned with INFO <> 0
for IFLAG = 2;
= 3: SLICOT Library routine MB02XD, if ALG = 'D', or
SLICOT Library routine MB02WD, if ALG = 'I' (or
user-defined routine JPJ), returned with INFO <> 0.
Method
If XINIT = 'R', the initial value for X is set to a vector of
pseudo-random values uniformly distributed in [-1,1].
The Levenberg-Marquardt algorithm (described in [1]) is used for
optimizing the parameters. This algorithm needs the Jacobian
matrix J, which is provided by the subroutine FCN. The algorithm
tries to update x by the formula
x = x - p,
using the solution of the system of linear equations
(J'*J + PAR*I)*p = J'*e,
where I is the identity matrix, and e the error function vector.
The Levenberg factor PAR is decreased after each successfull step
and increased in the other case.
If ALG = 'D', a direct method, which evaluates the matrix product
J'*J + par*I and then factors it using Cholesky algorithm,
implemented in the SLICOT Libray routine MB02XD, is used for
solving the linear system above.
If ALG = 'I', the Conjugate Gradients method, described in [2],
and implemented in the SLICOT Libray routine MB02WD, is used for
solving the linear system above. The main advantage of this method
is that in most cases the solution of the system can be computed
in less time than the time needed to compute the matrix J'*J
This is, however, problem dependent.
References
[1] Kelley, C.T.
Iterative Methods for Optimization.
Society for Industrial and Applied Mathematics (SIAM),
Philadelphia (Pa.), 1999.
[2] Golub, G.H. and van Loan, C.F.
Matrix Computations. Third Edition.
M. D. Johns Hopkins University Press, Baltimore, pp. 520-528,
1996.
[3] More, J.J.
The Levenberg-Marquardt algorithm: implementation and theory.
In Watson, G.A. (Ed.), Numerical Analysis, Lecture Notes in
Mathematics, vol. 630, Springer-Verlag, Berlin, Heidelberg
and New York, pp. 105-116, 1978.
Numerical Aspects
The Levenberg-Marquardt algorithm described in [3] is scaling invariant and globally convergent to (maybe local) minima. According to [1], the convergence rate near a local minimum is quadratic, if the Jacobian is computed analytically, and linear, if the Jacobian is computed numerically. Whether or not the direct algorithm is faster than the iterative Conjugate Gradients algorithm for solving the linear systems involved depends on several factors, including the conditioning of the Jacobian matrix, and the ratio between its dimensions.Further Comments
NoneExample
Program Text
* MD03AD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, NMAX
PARAMETER ( MMAX = 20, NMAX = 20 )
INTEGER LDWORK
PARAMETER ( LDWORK = MMAX + 2*NMAX + MMAX*NMAX +
$ MAX( NMAX*NMAX, 3*NMAX + MMAX ) )
* .. The lengths of DPAR1, DPAR2, IPAR are set to 1, 1, and 5 ..
INTEGER LDPAR1, LDPAR2, LIPAR
PARAMETER ( LDPAR1 = 1, LDPAR2 = 1, LIPAR = 5 )
* .. Local Scalars ..
CHARACTER*1 ALG, STOR, UPLO, XINIT
INTEGER I, INFO, ITMAX, IWARN, M, N, NFEV, NJEV, NPRINT
DOUBLE PRECISION CGTOL, TOL
* .. Array Arguments ..
INTEGER IPAR(LIPAR)
DOUBLE PRECISION DPAR1(LDPAR1), DPAR2(LDPAR2), DWORK(LDWORK),
$ X(NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL MD03AD, MD03AF, NF01BV, NF01BX
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, N, ITMAX, NPRINT, TOL, CGTOL, XINIT,
$ ALG, STOR, UPLO
IF( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) M
ELSE
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N
ELSE
IF ( LSAME( XINIT, 'G' ) )
$ READ ( NIN, FMT = * ) ( X(I), I = 1,N )
* Solve a standard nonlinear least squares problem.
IPAR(1) = M
IF ( LSAME( ALG, 'D' ) ) THEN
CALL MD03AD( XINIT, ALG, STOR, UPLO, MD03AF, NF01BV, M,
$ N, ITMAX, NPRINT, IPAR, LIPAR, DPAR1,
$ LDPAR1, DPAR2, LDPAR2, X, NFEV, NJEV, TOL,
$ CGTOL, DWORK, LDWORK, IWARN, INFO )
ELSE
CALL MD03AD( XINIT, ALG, STOR, UPLO, MD03AF, NF01BX, M,
$ N, ITMAX, NPRINT, IPAR, LIPAR, DPAR1,
$ LDPAR1, DPAR2, LDPAR2, X, NFEV, NJEV, TOL,
$ CGTOL, DWORK, LDWORK, IWARN, INFO )
END IF
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF( IWARN.NE.0 ) WRITE ( NOUT, FMT = 99991 ) IWARN
WRITE ( NOUT, FMT = 99997 ) DWORK(2)
WRITE ( NOUT, FMT = 99996 ) NFEV, NJEV
WRITE ( NOUT, FMT = 99994 )
WRITE ( NOUT, FMT = 99995 ) ( X(I), I = 1, N )
END IF
END IF
END IF
STOP
*
99999 FORMAT (' MD03AD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MD03AD = ',I2)
99997 FORMAT (/' Final 2-norm of the residuals = ',D15.7)
99996 FORMAT (/' The number of function and Jacobian evaluations = ',
$ 2I7)
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' Final approximate solution is ' )
99993 FORMAT (/' M is out of range.',/' M = ',I5)
99992 FORMAT (/' N is out of range.',/' N = ',I5)
99991 FORMAT (' IWARN on exit from MD03AD = ',I2)
END
C
SUBROUTINE MD03AF( IFLAG, M, N, IPAR, LIPAR, DPAR1, LDPAR1, DPAR2,
$ LDPAR2, X, NFEVL, E, J, LDJ, JTE, DWORK,
$ LDWORK, INFO )
C
C This is the FCN routine for solving a standard nonlinear least
C squares problem using SLICOT Library routine MD03AD. See the
C argument FCN in the routine MD03AD for the description of
C parameters.
C
C The example programmed in this routine is adapted from that
C accompanying the MINPACK routine LMDER.
C
C ******************************************************************
C
C .. Parameters ..
C .. NOUT is the unit number for printing intermediate results ..
INTEGER NOUT
PARAMETER ( NOUT = 6 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
C .. Scalar Arguments ..
INTEGER IFLAG, INFO, LDJ, LDPAR1, LDPAR2, LDWORK, LIPAR,
$ M, N, NFEVL
C .. Array Arguments ..
INTEGER IPAR(*)
DOUBLE PRECISION DPAR1(*), DPAR2(*), DWORK(*), E(*), J(LDJ,*),
$ JTE(*), X(*)
C .. Local Scalars ..
INTEGER I
DOUBLE PRECISION ERR, TMP1, TMP2, TMP3, TMP4
C .. External Functions ..
DOUBLE PRECISION DNRM2
EXTERNAL DNRM2
C .. External Subroutines ..
EXTERNAL DGEMV
C .. DATA Statements ..
DOUBLE PRECISION Y(15)
DATA Y(1), Y(2), Y(3), Y(4), Y(5), Y(6), Y(7), Y(8),
$ Y(9), Y(10), Y(11), Y(12), Y(13), Y(14), Y(15)
$ / 1.4D-1, 1.8D-1, 2.2D-1, 2.5D-1, 2.9D-1,
$ 3.2D-1, 3.5D-1, 3.9D-1, 3.7D-1, 5.8D-1,
$ 7.3D-1, 9.6D-1, 1.34D0, 2.1D0, 4.39D0 /
C
C .. Executable Statements ..
C
INFO = 0
IF ( IFLAG.EQ.1 ) THEN
C
C Compute the error function values, e.
C
DO 10 I = 1, 15
TMP1 = I
TMP2 = 16 - I
IF ( I.GT.8 ) THEN
TMP3 = TMP2
ELSE
TMP3 = TMP1
END IF
E(I) = Y(I) - ( X(1) + TMP1/( X(2)*TMP2 + X(3)*TMP3 ) )
10 CONTINUE
C
ELSE IF ( IFLAG.EQ.2 ) THEN
C
C Compute the Jacobian.
C
DO 30 I = 1, 15
TMP1 = I
TMP2 = 16 - I
IF ( I.GT.8 ) THEN
TMP3 = TMP2
ELSE
TMP3 = TMP1
END IF
TMP4 = ( X(2)*TMP2 + X(3)*TMP3 )**2
J(I,1) = -ONE
J(I,2) = TMP1*TMP2/TMP4
J(I,3) = TMP1*TMP3/TMP4
30 CONTINUE
C
C Compute the product J'*e (the error e was computed in array E).
C
CALL DGEMV( 'Transpose', M, N, ONE, J, LDJ, E, 1, ZERO, JTE,
$ 1 )
C
NFEVL = 0
C
ELSE IF ( IFLAG.EQ.3 ) THEN
C
C Set the parameter LDJ, the length of the array J, and the sizes
C of the workspace for MD03AF (IFLAG = 1 or 2), NF01BV and
C NF01BX.
C
LDJ = M
IPAR(1) = M*N
IPAR(2) = 0
IPAR(3) = 0
IPAR(4) = M
ELSE IF ( IFLAG.EQ.0 ) THEN
C
C Special call for printing intermediate results.
C
ERR = DNRM2( M, E, 1 )
WRITE( NOUT, '('' Norm of current error = '', D15.6)') ERR
C
END IF
C
DWORK(1) = ZERO
RETURN
C
C *** Last line of MD03AF ***
END
Program Data
MD03AD EXAMPLE PROGRAM DATA 15 3 100 0 -1. -1. G D F U 1.0 1.0 1.0Program Results
MD03AD EXAMPLE PROGRAM RESULTS Final 2-norm of the residuals = 0.9063596D-01 The number of function and Jacobian evaluations = 13 12 Final approximate solution is 0.0824 1.1330 2.3437
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MD03BD.html 0000664 0000000 0000000 00000123415 14560147231 0020331 0 ustar 00root root 0000000 0000000
Purpose
To minimize the sum of the squares of m nonlinear functions, e, in n variables, x, by a modification of the Levenberg-Marquardt algorithm. The user must provide a subroutine FCN which calculates the functions and the Jacobian (possibly by finite differences). In addition, specialized subroutines QRFACT, for QR factorization with pivoting of the Jacobian, and LMPARM, for the computation of Levenberg-Marquardt parameter, exploiting the possible structure of the Jacobian matrix, should be provided. Template implementations of these routines are included in SLICOT Library.Specification
SUBROUTINE MD03BD( XINIT, SCALE, COND, FCN, QRFACT, LMPARM, M, N,
$ ITMAX, FACTOR, NPRINT, IPAR, LIPAR, DPAR1,
$ LDPAR1, DPAR2, LDPAR2, X, DIAG, NFEV, NJEV,
$ FTOL, XTOL, GTOL, TOL, IWORK, DWORK, LDWORK,
$ IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER COND, SCALE, XINIT
INTEGER INFO, ITMAX, IWARN, LDPAR1, LDPAR2, LDWORK,
$ LIPAR, M, N, NFEV, NJEV, NPRINT
DOUBLE PRECISION FACTOR, FTOL, GTOL, TOL, XTOL
C .. Array Arguments ..
INTEGER IPAR(*), IWORK(*)
DOUBLE PRECISION DIAG(*), DPAR1(*), DPAR2(*), DWORK(*), X(*)
Arguments
Mode Parameters
XINIT CHARACTER*1
Specifies how the variables x are initialized, as follows:
= 'R' : the array X is initialized to random values; the
entries DWORK(1:4) are used to initialize the
random number generator: the first three values
are converted to integers between 0 and 4095, and
the last one is converted to an odd integer
between 1 and 4095;
= 'G' : the given entries of X are used as initial values
of variables.
SCALE CHARACTER*1
Specifies how the variables will be scaled, as follows:
= 'I' : use internal scaling;
= 'S' : use specified scaling factors, given in DIAG.
COND CHARACTER*1
Specifies whether the condition of the linear systems
involved should be estimated, as follows:
= 'E' : use incremental condition estimation to find the
numerical rank;
= 'N' : do not use condition estimation, but check the
diagonal entries of matrices for zero values.
Function Parameters
FCN EXTERNAL
Subroutine which evaluates the functions and the Jacobian.
FCN must be declared in an external statement in the user
calling program, and must have the following interface:
SUBROUTINE FCN( IFLAG, M, N, IPAR, LIPAR, DPAR1, LDPAR1,
$ DPAR2, LDPAR2, X, NFEVL, E, J, LDJ, DWORK,
$ LDWORK, INFO )
where
IFLAG (input/output) INTEGER
On entry, this parameter must contain a value
defining the computations to be performed:
= 0 : Optionally, print the current iterate X,
function values E, and Jacobian matrix J,
or other results defined in terms of these
values. See the argument NPRINT of MD03BD.
Do not alter E and J.
= 1 : Calculate the functions at X and return
this vector in E. Do not alter J.
= 2 : Calculate the Jacobian at X and return
this matrix in J. Also return NFEVL
(see below). Do not alter E.
= 3 : Do not compute neither the functions nor
the Jacobian, but return in LDJ and
IPAR/DPAR1,DPAR2 (some of) the integer/real
parameters needed.
On exit, the value of this parameter should not be
changed by FCN unless the user wants to terminate
execution of MD03BD, in which case IFLAG must be
set to a negative integer.
M (input) INTEGER
The number of functions. M >= 0.
N (input) INTEGER
The number of variables. M >= N >= 0.
IPAR (input/output) INTEGER array, dimension (LIPAR)
The integer parameters describing the structure of
the Jacobian matrix or needed for problem solving.
IPAR is an input parameter, except for IFLAG = 3
on entry, when it is also an output parameter.
On exit, if IFLAG = 3, IPAR(1) contains the length
of the array J, for storing the Jacobian matrix,
and the entries IPAR(2:5) contain the workspace
required by FCN for IFLAG = 1, FCN for IFLAG = 2,
QRFACT, and LMPARM, respectively.
LIPAR (input) INTEGER
The length of the array IPAR. LIPAR >= 5.
DPAR1 (input/output) DOUBLE PRECISION array, dimension
(LDPAR1,*) or (LDPAR1)
A first set of real parameters needed for
describing or solving the problem.
DPAR1 can also be used as an additional array for
intermediate results when computing the functions
or the Jacobian. For control problems, DPAR1 could
store the input trajectory of a system.
LDPAR1 (input) INTEGER
The leading dimension or the length of the array
DPAR1, as convenient. LDPAR1 >= 0. (LDPAR1 >= 1,
if leading dimension.)
DPAR2 (input/output) DOUBLE PRECISION array, dimension
(LDPAR2,*) or (LDPAR2)
A second set of real parameters needed for
describing or solving the problem.
DPAR2 can also be used as an additional array for
intermediate results when computing the functions
or the Jacobian. For control problems, DPAR2 could
store the output trajectory of a system.
LDPAR2 (input) INTEGER
The leading dimension or the length of the array
DPAR2, as convenient. LDPAR2 >= 0. (LDPAR2 >= 1,
if leading dimension.)
X (input) DOUBLE PRECISION array, dimension (N)
This array must contain the value of the
variables x where the functions or the Jacobian
must be evaluated.
NFEVL (input/output) INTEGER
The number of function evaluations needed to
compute the Jacobian by a finite difference
approximation.
NFEVL is an input parameter if IFLAG = 0, or an
output parameter if IFLAG = 2. If the Jacobian is
computed analytically, NFEVL should be set to a
non-positive value.
E (input/output) DOUBLE PRECISION array,
dimension (M)
This array contains the value of the (error)
functions e evaluated at X.
E is an input parameter if IFLAG = 0 or 2, or an
output parameter if IFLAG = 1.
J (input/output) DOUBLE PRECISION array, dimension
(LDJ,NC), where NC is the number of columns
needed.
This array contains a possibly compressed
representation of the Jacobian matrix evaluated
at X. If full Jacobian is stored, then NC = N.
J is an input parameter if IFLAG = 0, or an output
parameter if IFLAG = 2.
LDJ (input/output) INTEGER
The leading dimension of array J. LDJ >= 1.
LDJ is essentially used inside the routines FCN,
QRFACT and LMPARM.
LDJ is an input parameter, except for IFLAG = 3
on entry, when it is an output parameter.
It is assumed in MD03BD that LDJ is not larger
than needed.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
The workspace array for subroutine FCN.
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
LDWORK (input) INTEGER
The size of the array DWORK (as large as needed
in the subroutine FCN). LDWORK >= 1.
INFO INTEGER
Error indicator, set to a negative value if an
input (scalar) argument is erroneous, and to
positive values for other possible errors in the
subroutine FCN. The LAPACK Library routine XERBLA
should be used in conjunction with negative INFO.
INFO must be zero if the subroutine finished
successfully.
Parameters marked with "(input)" must not be changed.
QRFACT EXTERNAL
Subroutine which computes the QR factorization with
(block) column pivoting of the Jacobian matrix, J*P = Q*R.
QRFACT must be declared in an external statement in the
calling program, and must have the following interface:
SUBROUTINE QRFACT( N, IPAR, LIPAR, FNORM, J, LDJ, E,
$ JNORMS, GNORM, IPVT, DWORK, LDWORK,
$ INFO )
where
N (input) INTEGER
The number of columns of the Jacobian matrix J.
N >= 0.
IPAR (input) INTEGER array, dimension (LIPAR)
The integer parameters describing the structure of
the Jacobian matrix.
LIPAR (input) INTEGER
The length of the array IPAR. LIPAR >= 0.
FNORM (input) DOUBLE PRECISION
The Euclidean norm of the vector e. FNORM >= 0.
J (input/output) DOUBLE PRECISION array, dimension
(LDJ, NC), where NC is the number of columns.
On entry, the leading NR-by-NC part of this array
must contain the (compressed) representation
of the Jacobian matrix J, where NR is the number
of rows of J (function of IPAR entries).
On exit, the leading N-by-NC part of this array
contains a (compressed) representation of the
upper triangular factor R of the Jacobian matrix.
For efficiency of the later calculations, the
matrix R is delivered with the leading dimension
MAX(1,N), possibly much smaller than the value
of LDJ on entry.
LDJ (input/output) INTEGER
The leading dimension of array J.
On entry, LDJ >= MAX(1,NR).
On exit, LDJ >= MAX(1,N).
E (input/output) DOUBLE PRECISION array, dimension
(NR)
On entry, this array contains the error vector e.
On exit, this array contains the updated vector
Z*Q'*e, where Z is a block row permutation matrix
(possibly identity) used in the QR factorization
of J. (See, for example, the SLICOT Library
routine NF01BS, Section METHOD.)
JNORMS (output) DOUBLE PRECISION array, dimension (N)
This array contains the Euclidean norms of the
columns of the Jacobian matrix (in the original
order).
GNORM (output) DOUBLE PRECISION
If FNORM > 0, the 1-norm of the scaled vector
J'*e/FNORM, with each element i further divided
by JNORMS(i) (if JNORMS(i) is nonzero).
If FNORM = 0, the returned value of GNORM is 0.
IPVT (output) INTEGER array, dimension (N)
This array defines the permutation matrix P such
that J*P = Q*R. Column j of P is column IPVT(j) of
the identity matrix.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
The workspace array for subroutine QRFACT.
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
LDWORK (input) INTEGER
The size of the array DWORK (as large as needed
in the subroutine QRFACT). LDWORK >= 1.
INFO INTEGER
Error indicator, set to a negative value if an
input (scalar) argument is erroneous, and to
positive values for other possible errors in the
subroutine QRFACT. The LAPACK Library routine
XERBLA should be used in conjunction with negative
INFO. INFO must be zero if the subroutine finished
successfully.
Parameters marked with "(input)" must not be changed.
LMPARM EXTERNAL
Subroutine which determines a value for the Levenberg-
Marquardt parameter PAR such that if x solves the system
J*x = b , sqrt(PAR)*D*x = 0 ,
in the least squares sense, where J is an m-by-n matrix,
D is an n-by-n nonsingular diagonal matrix, and b is an
m-vector, and if DELTA is a positive number, DXNORM is
the Euclidean norm of D*x, then either PAR is zero and
( DXNORM - DELTA ) .LE. 0.1*DELTA ,
or PAR is positive and
ABS( DXNORM - DELTA ) .LE. 0.1*DELTA .
It is assumed that a block QR factorization, with column
pivoting, of J is available, that is, J*P = Q*R, where P
is a permutation matrix, Q has orthogonal columns, and
R is an upper triangular matrix (possibly stored in a
compressed form), with diagonal elements of nonincreasing
magnitude for each block. On output, LMPARM also provides
a (compressed) representation of an upper triangular
matrix S, such that
P'*(J'*J + PAR*D*D)*P = S'*S .
LMPARM must be declared in an external statement in the
calling program, and must have the following interface:
SUBROUTINE LMPARM( COND, N, IPAR, LIPAR, R, LDR, IPVT,
$ DIAG, QTB, DELTA, PAR, RANKS, X, RX,
$ TOL, DWORK, LDWORK, INFO )
where
COND CHARACTER*1
Specifies whether the condition of the linear
systems involved should be estimated, as follows:
= 'E' : use incremental condition estimation
to find the numerical rank;
= 'N' : do not use condition estimation, but
check the diagonal entries for zero
values;
= 'U' : use the ranks already stored in RANKS
(for R).
N (input) INTEGER
The order of the matrix R. N >= 0.
IPAR (input) INTEGER array, dimension (LIPAR)
The integer parameters describing the structure of
the Jacobian matrix.
LIPAR (input) INTEGER
The length of the array IPAR. LIPAR >= 0.
R (input/output) DOUBLE PRECISION array, dimension
(LDR, NC), where NC is the number of columns.
On entry, the leading N-by-NC part of this array
must contain the (compressed) representation (Rc)
of the upper triangular matrix R.
On exit, the full upper triangular part of R
(in representation Rc), is unaltered, and the
remaining part contains (part of) the (compressed)
representation of the transpose of the upper
triangular matrix S.
LDR (input) INTEGER
The leading dimension of array R.
LDR >= MAX(1,N).
IPVT (input) INTEGER array, dimension (N)
This array must define the permutation matrix P
such that J*P = Q*R. Column j of P is column
IPVT(j) of the identity matrix.
DIAG (input) DOUBLE PRECISION array, dimension (N)
This array must contain the diagonal elements of
the matrix D. DIAG(I) <> 0, I = 1,...,N.
QTB (input) DOUBLE PRECISION array, dimension (N)
This array must contain the first n elements of
the vector Q'*b.
DELTA (input) DOUBLE PRECISION
An upper bound on the Euclidean norm of D*x.
DELTA > 0.
PAR (input/output) DOUBLE PRECISION
On entry, PAR must contain an initial estimate of
the Levenberg-Marquardt parameter. PAR >= 0.
On exit, it contains the final estimate of this
parameter.
RANKS (input or output) INTEGER array, dimension (r),
where r is the number of diagonal blocks R_k in R,
corresponding to the block column structure of J.
On entry, if COND = 'U' and N > 0, this array must
contain the numerical ranks of the submatrices
R_k, k = 1:r. The number r is defined in terms of
the entries of IPAR.
On exit, if N > 0, this array contains the
numerical ranks of the submatrices S_k, k = 1:r.
X (output) DOUBLE PRECISION array, dimension (N)
This array contains the least squares solution of
the system J*x = b, sqrt(PAR)*D*x = 0.
RX (output) DOUBLE PRECISION array, dimension (N)
This array contains the matrix-vector product
-R*P'*x.
TOL (input) DOUBLE PRECISION
If COND = 'E', the tolerance to be used for
finding the ranks of the submatrices R_k and S_k.
If the user sets TOL > 0, then the given value of
TOL is used as a lower bound for the reciprocal
condition number; a (sub)matrix whose estimated
condition number is less than 1/TOL is considered
to be of full rank. If the user sets TOL <= 0,
then an implicitly computed, default tolerance,
defined by TOLDEF = N*EPS, is used instead,
where EPS is the machine precision (see LAPACK
Library routine DLAMCH).
This parameter is not relevant if COND = 'U'
or 'N'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
The workspace array for subroutine LMPARM.
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
LDWORK (input) INTEGER
The size of the array DWORK (as large as needed
in the subroutine LMPARM). LDWORK >= 1.
INFO INTEGER
Error indicator, set to a negative value if an
input (scalar) argument is erroneous, and to
positive values for other possible errors in the
subroutine LMPARM. The LAPACK Library routine
XERBLA should be used in conjunction with negative
INFO. INFO must be zero if the subroutine finished
successfully.
Parameters marked with "(input)" must not be changed.
Input/Output Parameters
M (input) INTEGER
The number of functions. M >= 0.
N (input) INTEGER
The number of variables. M >= N >= 0.
ITMAX (input) INTEGER
The maximum number of iterations. ITMAX >= 0.
FACTOR (input) DOUBLE PRECISION
The value used in determining the initial step bound. This
bound is set to the product of FACTOR and the Euclidean
norm of DIAG*X if nonzero, or else to FACTOR itself.
In most cases FACTOR should lie in the interval (.1,100).
A generally recommended value is 100. FACTOR > 0.
NPRINT (input) INTEGER
This parameter enables controlled printing of iterates if
it is positive. In this case, FCN is called with IFLAG = 0
at the beginning of the first iteration and every NPRINT
iterations thereafter and immediately prior to return,
with X, E, and J available for printing. Note that when
called immediately prior to return, J normally contains
the result returned by QRFACT and LMPARM (the compressed
R and S factors). If NPRINT is not positive, no special
calls of FCN with IFLAG = 0 are made.
IPAR (input) INTEGER array, dimension (LIPAR)
The integer parameters needed, for instance, for
describing the structure of the Jacobian matrix, which
are handed over to the routines FCN, QRFACT and LMPARM.
The first five entries of this array are modified
internally by a call to FCN (with IFLAG = 3), but are
restored on exit.
LIPAR (input) INTEGER
The length of the array IPAR. LIPAR >= 5.
DPAR1 (input/output) DOUBLE PRECISION array, dimension
(LDPAR1,*) or (LDPAR1)
A first set of real parameters needed for describing or
solving the problem. This argument is not used by MD03BD
routine, but it is passed to the routine FCN.
LDPAR1 (input) INTEGER
The leading dimension or the length of the array DPAR1, as
convenient. LDPAR1 >= 0. (LDPAR1 >= 1, if leading
dimension.)
DPAR2 (input/output) DOUBLE PRECISION array, dimension
(LDPAR2,*) or (LDPAR2)
A second set of real parameters needed for describing or
solving the problem. This argument is not used by MD03BD
routine, but it is passed to the routine FCN.
LDPAR2 (input) INTEGER
The leading dimension or the length of the array DPAR2, as
convenient. LDPAR2 >= 0. (LDPAR2 >= 1, if leading
dimension.)
X (input/output) DOUBLE PRECISION array, dimension (N)
On entry, if XINIT = 'G', this array must contain the
vector of initial variables x to be optimized.
If XINIT = 'R', this array need not be set before entry,
and random values will be used to initialize x.
On exit, if INFO = 0, this array contains the vector of
values that (approximately) minimize the sum of squares of
error functions. The values returned in IWARN and
DWORK(1:4) give details on the iterative process.
DIAG (input/output) DOUBLE PRECISION array, dimension (N)
On entry, if SCALE = 'S', this array must contain some
positive entries that serve as multiplicative scale
factors for the variables x. DIAG(I) > 0, I = 1,...,N.
If SCALE = 'I', DIAG is internally set.
On exit, this array contains the scale factors used
(or finally used, if SCALE = 'I').
NFEV (output) INTEGER
The number of calls to FCN with IFLAG = 1. If FCN is
properly implemented, this includes the function
evaluations needed for finite difference approximation
of the Jacobian.
NJEV (output) INTEGER
The number of calls to FCN with IFLAG = 2.
Tolerances
FTOL DOUBLE PRECISION
If FTOL >= 0, the tolerance which measures the relative
error desired in the sum of squares. Termination occurs
when both the actual and predicted relative reductions in
the sum of squares are at most FTOL. If the user sets
FTOL < 0, then SQRT(EPS) is used instead FTOL, where
EPS is the machine precision (see LAPACK Library routine
DLAMCH).
XTOL DOUBLE PRECISION
If XTOL >= 0, the tolerance which measures the relative
error desired in the approximate solution. Termination
occurs when the relative error between two consecutive
iterates is at most XTOL. If the user sets XTOL < 0,
then SQRT(EPS) is used instead XTOL.
GTOL DOUBLE PRECISION
If GTOL >= 0, the tolerance which measures the
orthogonality desired between the function vector e and
the columns of the Jacobian J. Termination occurs when
the cosine of the angle between e and any column of the
Jacobian J is at most GTOL in absolute value. If the user
sets GTOL < 0, then EPS is used instead GTOL.
TOL DOUBLE PRECISION
If COND = 'E', the tolerance to be used for finding the
ranks of the matrices of linear systems to be solved. If
the user sets TOL > 0, then the given value of TOL is used
as a lower bound for the reciprocal condition number; a
(sub)matrix whose estimated condition number is less than
1/TOL is considered to be of full rank. If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
defined by TOLDEF = N*EPS, is used instead.
This parameter is not relevant if COND = 'N'.
Workspace
IWORK INTEGER array, dimension (N+r), where r is the number
of diagonal blocks R_k in R (see description of LMPARM).
On output, if INFO = 0, the first N entries of this array
define a permutation matrix P such that J*P = Q*R, where
J is the final calculated Jacobian, Q is an orthogonal
matrix (not stored), and R is upper triangular with
diagonal elements of nonincreasing magnitude (possibly
for each block column of J). Column j of P is column
IWORK(j) of the identity matrix. If INFO = 0, the entries
N+1:N+r of this array contain the ranks of the final
submatrices S_k (see description of LMPARM).
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK, DWORK(2) returns the residual error norm (the
sum of squares), DWORK(3) returns the number of iterations
performed, and DWORK(4) returns the final Levenberg
factor. If INFO = 0, N > 0, and IWARN >= 0, the elements
DWORK(5) to DWORK(4+M) contain the final matrix-vector
product Z*Q'*e, and the elements DWORK(5+M) to
DWORK(4+M+N*NC) contain the (compressed) representation of
final upper triangular matrices R and S (if IWARN <> 4).
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= max( 4, M + max( size(J) +
max( DW( FCN|IFLAG = 1 ),
DW( FCN|IFLAG = 2 ),
DW( QRFACT ) + N ),
N*NC + N +
max( M + DW( FCN|IFLAG = 1 ),
N + DW( LMPARM ) ) ) ),
where size(J) is the size of the Jacobian (provided by FCN
in IPAR(1), for IFLAG = 3), and DW( f ) is the workspace
needed by the routine f, where f is FCN, QRFACT, or LMPARM
(provided by FCN in IPAR(2:5), for IFLAG = 3).
Warning Indicator
IWARN INTEGER
< 0: the user set IFLAG = IWARN in the subroutine FCN;
= 1: both actual and predicted relative reductions in
the sum of squares are at most FTOL;
= 2: relative error between two consecutive iterates is
at most XTOL;
= 3: conditions for IWARN = 1 and IWARN = 2 both hold;
= 4: the cosine of the angle between e and any column of
the Jacobian is at most GTOL in absolute value;
= 5: the number of iterations has reached ITMAX without
satisfying any convergence condition;
= 6: FTOL is too small: no further reduction in the sum
of squares is possible;
= 7: XTOL is too small: no further improvement in the
approximate solution x is possible;
= 8: GTOL is too small: e is orthogonal to the columns of
the Jacobian to machine precision.
In all these cases, DWORK(1:4) are set as described above.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: user-defined routine FCN returned with INFO <> 0
for IFLAG = 1;
= 2: user-defined routine FCN returned with INFO <> 0
for IFLAG = 2;
= 3: user-defined routine QRFACT returned with INFO <> 0;
= 4: user-defined routine LMPARM returned with INFO <> 0.
Method
If XINIT = 'R', the initial value for x is set to a vector of
pseudo-random values uniformly distributed in (-1,1).
The Levenberg-Marquardt algorithm (described in [1,3]) is used for
optimizing the variables x. This algorithm needs the Jacobian
matrix J, which is provided by the subroutine FCN. A trust region
method is used. The algorithm tries to update x by the formula
x = x - p,
using an approximate solution of the system of linear equations
(J'*J + PAR*D*D)*p = J'*e,
with e the error function vector, and D a diagonal nonsingular
matrix, where either PAR = 0 and
( norm( D*x ) - DELTA ) <= 0.1*DELTA ,
or PAR > 0 and
ABS( norm( D*x ) - DELTA ) <= 0.1*DELTA .
DELTA is the radius of the trust region. If the Gauss-Newton
direction is not acceptable, then an iterative algorithm obtains
improved lower and upper bounds for the Levenberg-Marquardt
parameter PAR. Only a few iterations are generally needed for
convergence of the algorithm. The trust region radius DELTA
and the Levenberg factor PAR are updated based on the ratio
between the actual and predicted reduction in the sum of squares.
References
[1] More, J.J., Garbow, B.S, and Hillstrom, K.E.
User's Guide for MINPACK-1.
Applied Math. Division, Argonne National Laboratory, Argonne,
Illinois, Report ANL-80-74, 1980.
[2] Golub, G.H. and van Loan, C.F.
Matrix Computations. Third Edition.
M. D. Johns Hopkins University Press, Baltimore, pp. 520-528,
1996.
[3] More, J.J.
The Levenberg-Marquardt algorithm: implementation and theory.
In Watson, G.A. (Ed.), Numerical Analysis, Lecture Notes in
Mathematics, vol. 630, Springer-Verlag, Berlin, Heidelberg
and New York, pp. 105-116, 1978.
Numerical Aspects
The Levenberg-Marquardt algorithm described in [3] is scaling invariant and globally convergent to (maybe local) minima. The convergence rate near a local minimum is quadratic, if the Jacobian is computed analytically, and linear, if the Jacobian is computed numerically.Further Comments
This routine is a more general version of the subroutines LMDER and LMDER1 from the MINPACK package [1], which enables to exploit the structure of the problem, and optionally use condition estimation. Unstructured problems could be solved as well. Template SLICOT Library implementations for FCN, QRFACT and LMPARM routines are: MD03BF, MD03BA, and MD03BB, respectively, for standard problems; NF01BF, NF01BS, and NF01BP, respectively, for optimizing the parameters of Wiener systems (structured problems).Example
Program Text
* MD03BD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, NMAX
PARAMETER ( MMAX = 20, NMAX = 20 )
INTEGER LDWORK
PARAMETER ( LDWORK = MMAX +
$ MAX( MMAX*NMAX + 5*NMAX + 1,
$ NMAX*NMAX + NMAX +
$ MAX( MMAX, 5*NMAX ) ) )
* .. Local Scalars ..
CHARACTER*1 COND, SCALE, XINIT
INTEGER I, INFO, ITMAX, IWARN, LDPAR1, LDPAR2, LIPAR, M,
$ N, NFEV, NJEV, NPRINT
DOUBLE PRECISION FACTOR, FTOL, GTOL, TOL, XTOL
* .. Array Arguments ..
INTEGER IPAR(5), IWORK(NMAX+1)
DOUBLE PRECISION DIAG(NMAX), DPAR1(1), DPAR2(1), DWORK(LDWORK),
$ X(NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL MD03BA, MD03BB, MD03BD, MD03BF
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, N, ITMAX, LIPAR, LDPAR1, LDPAR2, FACTOR,
$ NPRINT, FTOL, XTOL, GTOL, TOL, XINIT, SCALE,
$ COND
IF( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) M
ELSE
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N
ELSE
IF ( LSAME( SCALE, 'S' ) )
$ READ ( NIN, FMT = * ) ( DIAG(I), I = 1,N )
IF ( LSAME( XINIT, 'G' ) )
$ READ ( NIN, FMT = * ) ( X(I), I = 1,N )
* Solve a standard nonlinear least squares problem.
IPAR(1) = M
CALL MD03BD( XINIT, SCALE, COND, MD03BF, MD03BA, MD03BB,
$ M, N, ITMAX, FACTOR, NPRINT, IPAR, LIPAR,
$ DPAR1, LDPAR1, DPAR2, LDPAR2, X, DIAG, NFEV,
$ NJEV, FTOL, XTOL, GTOL, TOL, IWORK, DWORK,
$ LDWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF( IWARN.NE.0) WRITE ( NOUT, FMT = 99991 ) IWARN
WRITE ( NOUT, FMT = 99997 ) DWORK(2)
WRITE ( NOUT, FMT = 99996 ) NFEV, NJEV
WRITE ( NOUT, FMT = 99994 )
WRITE ( NOUT, FMT = 99995 ) ( X(I), I = 1, N )
END IF
END IF
END IF
STOP
*
99999 FORMAT (' MD03BD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MD03BD = ',I2)
99997 FORMAT (/' Final 2-norm of the residuals = ',D15.7)
99996 FORMAT (/' The number of function and Jacobian evaluations = ',
$ 2I7)
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' Final approximate solution is ' )
99993 FORMAT (/' M is out of range.',/' M = ',I5)
99992 FORMAT (/' N is out of range.',/' N = ',I5)
99991 FORMAT (' IWARN on exit from MD03BD = ',I2)
END
C
SUBROUTINE MD03BF( IFLAG, M, N, IPAR, LIPAR, DPAR1, LDPAR1, DPAR2,
$ LDPAR2, X, NFEVL, E, J, LDJ, DWORK, LDWORK,
$ INFO )
C
C This is the FCN routine for solving a standard nonlinear least
C squares problem using SLICOT Library routine MD03BD. See the
C argument FCN in the routine MD03BD for the description of
C parameters.
C
C The example programmed in this routine is adapted from that
C accompanying the MINPACK routine LMDER.
C
C ******************************************************************
C
C .. Parameters ..
C .. NOUT is the unit number for printing intermediate results ..
INTEGER NOUT
PARAMETER ( NOUT = 6 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
C .. Scalar Arguments ..
INTEGER IFLAG, INFO, LDJ, LDPAR1, LDPAR2, LDWORK, LIPAR,
$ M, N, NFEVL
C .. Array Arguments ..
INTEGER IPAR(*)
DOUBLE PRECISION DPAR1(*), DPAR2(*), DWORK(*), E(*), J(LDJ,*),
$ X(*)
C .. Local Scalars ..
INTEGER I
DOUBLE PRECISION ERR, TMP1, TMP2, TMP3, TMP4
C .. External Functions ..
DOUBLE PRECISION DNRM2
EXTERNAL DNRM2
C .. External Subroutines ..
EXTERNAL MD03BA, MD03BB
C .. DATA Statements ..
DOUBLE PRECISION Y(15)
DATA Y(1), Y(2), Y(3), Y(4), Y(5), Y(6), Y(7), Y(8),
$ Y(9), Y(10), Y(11), Y(12), Y(13), Y(14), Y(15)
$ / 1.4D-1, 1.8D-1, 2.2D-1, 2.5D-1, 2.9D-1,
$ 3.2D-1, 3.5D-1, 3.9D-1, 3.7D-1, 5.8D-1,
$ 7.3D-1, 9.6D-1, 1.34D0, 2.1D0, 4.39D0 /
C
C .. Executable Statements ..
C
INFO = 0
IF ( IFLAG.EQ.1 ) THEN
C
C Compute the error function values.
C
DO 10 I = 1, 15
TMP1 = I
TMP2 = 16 - I
IF ( I.GT.8 ) THEN
TMP3 = TMP2
ELSE
TMP3 = TMP1
END IF
E(I) = Y(I) - ( X(1) + TMP1/( X(2)*TMP2 + X(3)*TMP3 ) )
10 CONTINUE
C
ELSE IF ( IFLAG.EQ.2 ) THEN
C
C Compute the Jacobian.
C
DO 30 I = 1, 15
TMP1 = I
TMP2 = 16 - I
IF ( I.GT.8 ) THEN
TMP3 = TMP2
ELSE
TMP3 = TMP1
END IF
TMP4 = ( X(2)*TMP2 + X(3)*TMP3 )**2
J(I,1) = -ONE
J(I,2) = TMP1*TMP2/TMP4
J(I,3) = TMP1*TMP3/TMP4
30 CONTINUE
C
NFEVL = 0
C
ELSE IF ( IFLAG.EQ.3 ) THEN
C
C Set the parameter LDJ, the length of the array J, and the sizes
C of the workspace for MD03BF (IFLAG = 1 or 2), MD03BA and MD03BB.
C
LDJ = M
IPAR(1) = M*N
IPAR(2) = 0
IPAR(3) = 0
IPAR(4) = 4*N + 1
IPAR(5) = 4*N
ELSE IF ( IFLAG.EQ.0 ) THEN
C
C Special call for printing intermediate results.
C
ERR = DNRM2( M, E, 1 )
WRITE( NOUT, '('' Norm of current error = '', D15.6)') ERR
C
END IF
C
RETURN
C
C *** Last line of MD03BF ***
END
C
SUBROUTINE MD03BA( N, IPAR, LIPAR, FNORM, J, LDJ, E, JNORMS,
$ GNORM, IPVT, DWORK, LDWORK, INFO )
C
C This is the QRFACT routine for solving a standard nonlinear least
C squares problem using SLICOT Library routine MD03BD. See the
C argument QRFACT in the routine MD03BD for the description of
C parameters.
C
C For efficiency, the arguments are not checked. This is done in
C the routine MD03BX (except for LIPAR).
C
C ******************************************************************
C
C .. Scalar Arguments ..
INTEGER INFO, LDJ, LDWORK, LIPAR, N
DOUBLE PRECISION FNORM, GNORM
C .. Array Arguments ..
INTEGER IPAR(*), IPVT(*)
DOUBLE PRECISION DWORK(*), E(*), J(LDJ,*), JNORMS(*)
C .. External Subroutines ..
EXTERNAL MD03BX
C ..
C .. Executable Statements ..
C
CALL MD03BX( IPAR(1), N, FNORM, J, LDJ, E, JNORMS, GNORM, IPVT,
$ DWORK, LDWORK, INFO )
RETURN
C
C *** Last line of MD03BA ***
END
C
SUBROUTINE MD03BB( COND, N, IPAR, LIPAR, R, LDR, IPVT, DIAG, QTB,
$ DELTA, PAR, RANKS, X, RX, TOL, DWORK, LDWORK,
$ INFO )
C
C This is the LMPARM routine for solving a standard nonlinear least
C squares problem using SLICOT Library routine MD03BD. See the
C argument LMPARM in the routine MD03BD for the description of
C parameters.
C
C For efficiency, the arguments are not checked. This is done in
C the routine MD03BY (except for LIPAR).
C
C ******************************************************************
C
C .. Scalar Arguments ..
CHARACTER COND
INTEGER INFO, LDR, LDWORK, LIPAR, N
DOUBLE PRECISION DELTA, PAR, TOL
C .. Array Arguments ..
INTEGER IPAR(*), IPVT(*), RANKS(*)
DOUBLE PRECISION DIAG(*), DWORK(*), QTB(*), R(LDR,*), RX(*), X(*)
C .. External Subroutines ..
EXTERNAL MD03BY
C ..
C .. Executable Statements ..
C
CALL MD03BY( COND, N, R, LDR, IPVT, DIAG, QTB, DELTA, PAR,
$ RANKS(1), X, RX, TOL, DWORK, LDWORK, INFO )
RETURN
C
C *** Last line of MD03BB ***
END
Program Data
MD03BD EXAMPLE PROGRAM DATA 15 3 100 5 0 0 1.D2 0 -1. -1. -1. -1. G I E 1.0 1.0 1.0Program Results
MD03BD EXAMPLE PROGRAM RESULTS IWARN on exit from MD03BD = 1 Final 2-norm of the residuals = 0.9063596D-01 The number of function and Jacobian evaluations = 6 5 Final approximate solution is 0.0824 1.1330 2.3437
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/MD03BX.html 0000664 0000000 0000000 00000010717 14560147231 0020355 0 ustar 00root root 0000000 0000000
Purpose
To compute the QR factorization with column pivoting of an m-by-n matrix J (m >= n), that is, J*P = Q*R, where Q is a matrix with orthogonal columns, P a permutation matrix, and R an upper trapezoidal matrix with diagonal elements of nonincreasing magnitude, and to apply the transformation Q' on the error vector e (in-situ). The 1-norm of the scaled gradient is also returned. The matrix J could be the Jacobian of a nonlinear least squares problem.Specification
SUBROUTINE MD03BX( M, N, FNORM, J, LDJ, E, JNORMS, GNORM, IPVT,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDJ, LDWORK, M, N
DOUBLE PRECISION FNORM, GNORM
C .. Array Arguments ..
INTEGER IPVT(*)
DOUBLE PRECISION DWORK(*), E(*), J(*), JNORMS(*)
Arguments
Input/Output Parameters
M (input) INTEGER
The number of rows of the Jacobian matrix J. M >= 0.
N (input) INTEGER
The number of columns of the Jacobian matrix J.
M >= N >= 0.
FNORM (input) DOUBLE PRECISION
The Euclidean norm of the vector e. FNORM >= 0.
J (input/output) DOUBLE PRECISION array, dimension (LDJ, N)
On entry, the leading M-by-N part of this array must
contain the Jacobian matrix J.
On exit, the leading N-by-N upper triangular part of this
array contains the upper triangular factor R of the
Jacobian matrix. Note that for efficiency of the later
calculations, the matrix R is delivered with the leading
dimension MAX(1,N), possibly much smaller than the value
of LDJ on entry.
LDJ (input/output) INTEGER
The leading dimension of array J.
On entry, LDJ >= MAX(1,M).
On exit, LDJ >= MAX(1,N).
E (input/output) DOUBLE PRECISION array, dimension (M)
On entry, this array must contain the error vector e.
On exit, this array contains the updated vector Q'*e.
JNORMS (output) DOUBLE PRECISION array, dimension (N)
This array contains the Euclidean norms of the columns of
the Jacobian matrix, considered in the initial order.
GNORM (output) DOUBLE PRECISION
If FNORM > 0, the 1-norm of the scaled vector
J'*Q'*e/FNORM, with each element i further divided by
JNORMS(i) (if JNORMS(i) is nonzero).
If FNORM = 0, the returned value of GNORM is 0.
IPVT (output) INTEGER array, dimension (N)
This array defines the permutation matrix P such that
J*P = Q*R. Column j of P is column IPVT(j) of the identity
matrix.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 1, if N = 0 or M = 1;
LDWORK >= 4*N+1, if N > 1.
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The algorithm uses QR factorization with column pivoting of the matrix J, J*P = Q*R, and applies the orthogonal matrix Q' to the vector e.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To determine a value for the parameter PAR such that if x solves
the system
A*x = b , sqrt(PAR)*D*x = 0 ,
in the least squares sense, where A is an m-by-n matrix, D is an
n-by-n nonsingular diagonal matrix, and b is an m-vector, and if
DELTA is a positive number, DXNORM is the Euclidean norm of D*x,
then either PAR is zero and
( DXNORM - DELTA ) .LE. 0.1*DELTA ,
or PAR is positive and
ABS( DXNORM - DELTA ) .LE. 0.1*DELTA .
It is assumed that a QR factorization, with column pivoting, of A
is available, that is, A*P = Q*R, where P is a permutation matrix,
Q has orthogonal columns, and R is an upper triangular matrix
with diagonal elements of nonincreasing magnitude.
The routine needs the full upper triangle of R, the permutation
matrix P, and the first n components of Q'*b (' denotes the
transpose). On output, MD03BY also provides an upper triangular
matrix S such that
P'*(A'*A + PAR*D*D)*P = S'*S .
Matrix S is used in the solution process.
Specification
SUBROUTINE MD03BY( COND, N, R, LDR, IPVT, DIAG, QTB, DELTA, PAR,
$ RANK, X, RX, TOL, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COND
INTEGER INFO, LDR, LDWORK, N, RANK
DOUBLE PRECISION DELTA, PAR, TOL
C .. Array Arguments ..
INTEGER IPVT(*)
DOUBLE PRECISION DIAG(*), DWORK(*), QTB(*), R(LDR,*), RX(*), X(*)
Arguments
Mode Parameters
COND CHARACTER*1
Specifies whether the condition of the matrices R and S
should be estimated, as follows:
= 'E' : use incremental condition estimation for R and S;
= 'N' : do not use condition estimation, but check the
diagonal entries of R and S for zero values;
= 'U' : use the rank already stored in RANK (for R).
Input/Output Parameters
N (input) INTEGER
The order of the matrix R. N >= 0.
R (input/output) DOUBLE PRECISION array, dimension (LDR, N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix R.
On exit, the full upper triangle is unaltered, and the
strict lower triangle contains the strict upper triangle
(transposed) of the upper triangular matrix S.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
IPVT (input) INTEGER array, dimension (N)
This array must define the permutation matrix P such that
A*P = Q*R. Column j of P is column IPVT(j) of the identity
matrix.
DIAG (input) DOUBLE PRECISION array, dimension (N)
This array must contain the diagonal elements of the
matrix D. DIAG(I) <> 0, I = 1,...,N.
QTB (input) DOUBLE PRECISION array, dimension (N)
This array must contain the first n elements of the
vector Q'*b.
DELTA (input) DOUBLE PRECISION
An upper bound on the Euclidean norm of D*x. DELTA > 0.
PAR (input/output) DOUBLE PRECISION
On entry, PAR must contain an initial estimate of the
Levenberg-Marquardt parameter. PAR >= 0.
On exit, it contains the final estimate of this parameter.
RANK (input or output) INTEGER
On entry, if COND = 'U', this parameter must contain the
(numerical) rank of the matrix R.
On exit, this parameter contains the numerical rank of
the matrix S.
X (output) DOUBLE PRECISION array, dimension (N)
This array contains the least squares solution of the
system A*x = b, sqrt(PAR)*D*x = 0.
RX (output) DOUBLE PRECISION array, dimension (N)
This array contains the matrix-vector product -R*P'*x.
Tolerances
TOL DOUBLE PRECISION
If COND = 'E', the tolerance to be used for finding the
rank of the matrices R and S. If the user sets TOL > 0,
then the given value of TOL is used as a lower bound for
the reciprocal condition number; a (sub)matrix whose
estimated condition number is less than 1/TOL is
considered to be of full rank. If the user sets TOL <= 0,
then an implicitly computed, default tolerance, defined by
TOLDEF = N*EPS, is used instead, where EPS is the machine
precision (see LAPACK Library routine DLAMCH).
This parameter is not relevant if COND = 'U' or 'N'.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, the first N elements of this array contain the
diagonal elements of the upper triangular matrix S.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 4*N, if COND = 'E';
LDWORK >= 2*N, if COND <> 'E'.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The algorithm computes the Gauss-Newton direction. A least squares solution is found if the Jacobian is rank deficient. If the Gauss- Newton direction is not acceptable, then an iterative algorithm obtains improved lower and upper bounds for the parameter PAR. Only a few iterations are generally needed for convergence of the algorithm. If, however, the limit of ITMAX = 10 iterations is reached, then the output PAR will contain the best value obtained so far. If the Gauss-Newton step is acceptable, it is stored in x, and PAR is set to zero, hence S = R.References
[1] More, J.J., Garbow, B.S, and Hillstrom, K.E.
User's Guide for MINPACK-1.
Applied Math. Division, Argonne National Laboratory, Argonne,
Illinois, Report ANL-80-74, 1980.
Numerical Aspects
2 The algorithm requires 0(N ) operations and is backward stable.Further Comments
This routine is a LAPACK-based modification of LMPAR from the MINPACK package [1], and with optional condition estimation. The option COND = 'U' is useful when dealing with several right-hand side vectors, but RANK should be reset. If COND = 'E', but the matrix S is guaranteed to be nonsingular and well conditioned relative to TOL, i.e., rank(R) = N, and min(DIAG) > 0, then its condition is not estimated.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To calculate the output y of the Wiener system
x(t+1) = A*x(t) + B*u(t)
z(t) = C*x(t) + D*u(t),
y(t) = f(z(t),wb(1:L)),
where t = 1, 2, ..., NSMP, and f is a nonlinear function,
evaluated by the SLICOT Library routine NF01AY. The parameter
vector X is partitioned as X = ( wb(1), ..., wb(L), theta ),
where wb(i), i = 1:L, correspond to the nonlinear part, theta
corresponds to the linear part, and the notation is fully
described below.
Specification
SUBROUTINE NF01AD( NSMP, M, L, IPAR, LIPAR, X, LX, U, LDU, Y, LDY,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, L, LDU, LDWORK, LDY, LX, LIPAR, M, NSMP
C .. Array Arguments ..
INTEGER IPAR(*)
DOUBLE PRECISION DWORK(*), U(LDU,*), X(*), Y(LDY,*)
Arguments
Input/Output Parameters
NSMP (input) INTEGER
The number of training samples. NSMP >= 0.
M (input) INTEGER
The length of each input sample. M >= 0.
L (input) INTEGER
The length of each output sample. L >= 0.
IPAR (input) INTEGER array, dimension (LIPAR)
The integer parameters needed.
IPAR(1) must contain the order of the linear part,
referred to as N below. N >= 0.
IPAR(2) must contain the number of neurons for the
nonlinear part, referred to as NN below.
NN >= 0.
LIPAR (input) INTEGER
The length of IPAR. LIPAR >= 2.
X (input) DOUBLE PRECISION array, dimension (LX)
The parameter vector, partitioned as
X = (wb(1), ..., wb(L), theta), where the vectors
wb(i), of length NN*(L+2)+1, are parameters for the
static nonlinearity, which is simulated by the
SLICOT Library routine NF01AY. See the documentation of
NF01AY for further details. The vector theta, of length
N*(M + L + 1) + L*M, represents the matrices A, B, C,
D and x(1), and it can be retrieved from these matrices
by SLICOT Library routine TB01VD and retranslated by
TB01VY.
LX (input) INTEGER
The length of the array X.
LX >= ( NN*(L+2)+1 )*L + N*(M + L + 1) + L*M.
U (input) DOUBLE PRECISION array, dimension (LDU, M)
The leading NSMP-by-M part of this array must contain the
set of input samples,
U = ( U(1,1),...,U(1,M); ...; U(NSMP,1),...,U(NSMP,M) ).
LDU INTEGER
The leading dimension of the array U. LDU >= MAX(1,NSMP).
Y (output) DOUBLE PRECISION array, dimension (LDY, L)
The leading NSMP-by-L part of this array contains the
simulated output.
LDY INTEGER
The leading dimension of the array Y. LDY >= MAX(1,NSMP).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= NSMP*L + MAX( 2*NN, (N + L)*(N + M) + 2*N +
MAX( N*(N + L), N + M + L ) )
if M > 0;
LDWORK >= NSMP*L + MAX( 2*NN, (N + L)*N + 2*N +
MAX( N*(N + L), L ) ), if M = 0.
A larger value of LDWORK could improve the efficiency.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
BLAS routines are used for the matrix-vector multiplications and the routine NF01AY is called for the calculation of the nonlinear function.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To calculate the output of a set of neural networks with the
structure
- tanh(w1'*z+b1) -
/ : \
z --- : --- sum(ws(i)*...)+ b(n+1) --- y,
\ : /
- tanh(wn'*z+bn) -
given the input z and the parameter vectors wi, ws, and b,
where z, w1, ..., wn are vectors of length NZ, ws is a vector
of length n, b(1), ..., b(n+1) are scalars, and n is called the
number of neurons in the hidden layer, or just number of neurons.
Such a network is used for each L output variables.
Specification
SUBROUTINE NF01AY( NSMP, NZ, L, IPAR, LIPAR, WB, LWB, Z, LDZ,
$ Y, LDY, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, L, LDWORK, LDY, LDZ, LIPAR, LWB, NSMP, NZ
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), WB(*), Y(LDY,*), Z(LDZ,*)
INTEGER IPAR(*)
Arguments
Input/Output Parameters
NSMP (input) INTEGER
The number of training samples. NSMP >= 0.
NZ (input) INTEGER
The length of each input sample. NZ >= 0.
L (input) INTEGER
The length of each output sample. L >= 0.
IPAR (input) INTEGER array, dimension (LIPAR)
The integer parameters needed.
IPAR(1) must contain the number of neurons, n, per output
variable, denoted NN in the sequel. NN >= 0.
LIPAR (input) INTEGER
The length of the vector IPAR. LIPAR >= 1.
WB (input) DOUBLE PRECISION array, dimension (LWB)
The leading (NN*(NZ+2)+1)*L part of this array must
contain the weights and biases of the network. This vector
is partitioned into L vectors of length NN*(NZ+2)+1,
WB = [ wb(1), ..., wb(L) ]. Each wb(k), k = 1, ..., L,
corresponds to one output variable, and has the structure
wb(k) = [ w1(1), ..., w1(NZ), ..., wn(1), ..., wn(NZ),
ws(1), ..., ws(n), b(1), ..., b(n+1) ],
where wi(j) are the weights of the hidden layer,
ws(i) are the weights of the linear output layer, and
b(i) are the biases, as in the scheme above.
LWB (input) INTEGER
The length of the array WB.
LWB >= ( NN*(NZ + 2) + 1 )*L.
Z (input) DOUBLE PRECISION array, dimension (LDZ, NZ)
The leading NSMP-by-NZ part of this array must contain the
set of input samples,
Z = ( Z(1,1),...,Z(1,NZ); ...; Z(NSMP,1),...,Z(NSMP,NZ) ).
LDZ INTEGER
The leading dimension of the array Z. LDZ >= MAX(1,NSMP).
Y (output) DOUBLE PRECISION array, dimension (LDY, L)
The leading NSMP-by-L part of this array contains the set
of output samples,
Y = ( Y(1,1),...,Y(1,L); ...; Y(NSMP,1),...,Y(NSMP,L) ).
LDY INTEGER
The leading dimension of the array Y. LDY >= MAX(1,NSMP).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK. LDWORK >= 2*NN.
For better performance, LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
BLAS routines are used to compute the matrix-vector products.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To calculate the Jacobian dy/dX of the Wiener system
x(t+1) = A*x(t) + B*u(t)
z(t) = C*x(t) + D*u(t),
y(t,i) = sum( ws(k, i)*f(w(k, i)*z(t) + b(k,i)) ) + b(k+1,i),
where t = 1, 2, ..., NSMP,
i = 1, 2, ..., L,
k = 1, 2, ..., NN.
NN is arbitrary eligible and has to be provided in IPAR(2), and
X = ( wb(1), ..., wb(L), theta ) is described below.
Denoting y(j) = y(1:NSMP,j), the Jacobian J has the block form
dy(1)/dwb(1) 0 ..... 0 dy(1)/dtheta
0 dy(2)/dwb(2) ..... 0 dy(2)/dtheta
..... ..... ..... ..... .....
0 ..... 0 dy(L)/dwb(L) dy(L)/dtheta
but it will be returned without the zero blocks, in the form
dy(1)/dwb(1) dy(1)/dtheta
...
dy(L)/dwb(L) dy(L)/dtheta.
dy(i)/dwb(i) depends on f and is calculated by the routine NF01BY;
dy(i)/dtheta is computed by a forward-difference approximation.
Specification
SUBROUTINE NF01BD( CJTE, NSMP, M, L, IPAR, LIPAR, X, LX, U, LDU,
$ E, J, LDJ, JTE, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER CJTE
INTEGER INFO, L, LDJ, LDU, LDWORK, LX, LIPAR, M, NSMP
C .. Array Arguments ..
INTEGER IPAR(*)
DOUBLE PRECISION DWORK(*), E(*), J(LDJ, *), JTE(*), U(LDU,*),
$ X(*)
Arguments
Mode Parameters
CJTE CHARACTER*1
Specifies whether the matrix-vector product J'*e should be
computed or not, as follows:
= 'C' : compute J'*e;
= 'N' : do not compute J'*e.
Input/Output Parameters
NSMP (input) INTEGER
The number of training samples. NSMP >= 0.
M (input) INTEGER
The length of each input sample. M >= 0.
L (input) INTEGER
The length of each output sample. L >= 0.
IPAR (input/output) INTEGER array, dimension (LIPAR)
On entry, the first entries of this array must contain
the integer parameters needed; specifically,
IPAR(1) must contain the order of the linear part, N;
actually, N = abs(IPAR(1)), since setting
IPAR(1) < 0 has a special meaning (see below);
IPAR(2) must contain the number of neurons for the
nonlinear part, NN, NN >= 0.
On exit, if IPAR(1) < 0 on entry, then no computations are
performed, except the needed tests on input parameters,
but the following values are returned:
IPAR(1) contains the length of the array J, LJ;
LDJ contains the leading dimension of array J.
Otherwise, IPAR(1) and LDJ are unchanged on exit.
LIPAR (input) INTEGER
The length of the array IPAR. LIPAR >= 2.
X (input) DOUBLE PRECISION array, dimension (LX)
The leading LPAR entries of this array must contain the
set of system parameters, where
LPAR = (NN*(L + 2) + 1)*L + N*(M + L + 1) + L*M.
X has the form (wb(1), ..., wb(L), theta), where the
vectors wb(i) have the structure
(w(1,1), ..., w(1,L), ..., w(NN,1), ..., w(NN,L),
ws(1), ..., ws(NN), b(1), ..., b(NN+1) ),
and the vector theta represents the matrices A, B, C, D
and x(1), and it can be retrieved from these matrices
by SLICOT Library routine TB01VD and retranslated by
TB01VY.
LX (input) INTEGER
The length of X.
LX >= (NN*(L + 2) + 1)*L + N*(M + L + 1) + L*M.
U (input) DOUBLE PRECISION array, dimension (LDU, M)
The leading NSMP-by-M part of this array must contain the
set of input samples,
U = ( U(1,1),...,U(1,M); ...; U(NSMP,1),...,U(NSMP,M) ).
LDU INTEGER
The leading dimension of array U. LDU >= MAX(1,NSMP).
E (input) DOUBLE PRECISION array, dimension (NSMP*L)
If CJTE = 'C', this array must contain a vector e, which
will be premultiplied with J', e = vec( Y - y ), where
Y is set of output samples, and vec denotes the
concatenation of the columns of a matrix.
If CJTE = 'N', this array is not referenced.
J (output) DOUBLE PRECISION array, dimension (LDJ, *)
The leading NSMP*L-by-NCOLJ part of this array contains
the Jacobian of the error function stored in a compressed
form, as described above, where
NCOLJ = NN*(L + 2) + 1 + N*(M + L + 1) + L*M.
LDJ INTEGER
The leading dimension of array J. LDJ >= MAX(1,NSMP*L).
Note that LDJ is an input parameter, except for
IPAR(1) < 0 on entry, when it is an output parameter.
JTE (output) DOUBLE PRECISION array, dimension (LPAR)
If CJTE = 'C', this array contains the matrix-vector
product J'*e.
If CJTE = 'N', this array is not referenced.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 2*NSMP*L + MAX( 2*NN, (N + L)*(N + M) + 2*N +
MAX( N*(N + L), N + M + L ) )
if M > 0;
LDWORK >= 2*NSMP*L + MAX( 2*NN, (N + L)*N + 2*N +
MAX( N*(N + L), L ) ), if M = 0.
A larger value of LDWORK could improve the efficiency.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
BLAS routines are used for the matrix-vector multiplications, and the SLICOT Library routine TB01VY is called for the conversion of the output normal form parameters to an LTI-system; the routine NF01AD is then used for the simulation of the system with given parameters, and the routine NF01BY is called for the (analytically performed) calculation of the parts referring to the parameters of the static nonlinearity.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To determine a value for the Levenberg-Marquardt parameter PAR
such that if x solves the system
J*x = b , sqrt(PAR)*D*x = 0 ,
in the least squares sense, where J is an m-by-n matrix, D is an
n-by-n nonsingular diagonal matrix, and b is an m-vector, and if
DELTA is a positive number, DXNORM is the Euclidean norm of D*x,
then either PAR is zero and
( DXNORM - DELTA ) .LE. 0.1*DELTA ,
or PAR is positive and
ABS( DXNORM - DELTA ) .LE. 0.1*DELTA .
The matrix J is the current Jacobian matrix of a nonlinear least
squares problem, provided in a compressed form by SLICOT Library
routine NF01BD. It is assumed that a block QR factorization, with
column pivoting, of J is available, that is, J*P = Q*R, where P is
a permutation matrix, Q has orthogonal columns, and R is an upper
triangular matrix with diagonal elements of nonincreasing
magnitude for each block, as returned by SLICOT Library
routine NF01BS. The routine NF01BP needs the upper triangle of R
in compressed form, the permutation matrix P, and the first
n components of Q'*b (' denotes the transpose). On output,
NF01BP also provides a compressed representation of an upper
triangular matrix S, such that
P'*(J'*J + PAR*D*D)*P = S'*S .
Matrix S is used in the solution process. The matrix R has the
following structure
/ R_1 0 .. 0 | L_1 \
| 0 R_2 .. 0 | L_2 |
| : : .. : | : | ,
| 0 0 .. R_l | L_l |
\ 0 0 .. 0 | R_l+1 /
where the submatrices R_k, k = 1:l, have the same order BSN,
and R_k, k = 1:l+1, are square and upper triangular. This matrix
is stored in the compressed form
/ R_1 | L_1 \
| R_2 | L_2 |
Rc = | : | : | ,
| R_l | L_l |
\ X | R_l+1 /
where the submatrix X is irrelevant. The matrix S has the same
structure as R, and its diagonal blocks are denoted by S_k,
k = 1:l+1.
If l <= 1, then the full upper triangle of the matrix R is stored.
Specification
SUBROUTINE NF01BP( COND, N, IPAR, LIPAR, R, LDR, IPVT, DIAG, QTB,
$ DELTA, PAR, RANKS, X, RX, TOL, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER COND
INTEGER INFO, LDR, LDWORK, LIPAR, N
DOUBLE PRECISION DELTA, PAR, TOL
C .. Array Arguments ..
INTEGER IPAR(*), IPVT(*), RANKS(*)
DOUBLE PRECISION DIAG(*), DWORK(*), QTB(*), R(LDR,*), RX(*), X(*)
Arguments
Mode Parameters
COND CHARACTER*1
Specifies whether the condition of the diagonal blocks R_k
and S_k of the matrices R and S should be estimated,
as follows:
= 'E' : use incremental condition estimation for each
diagonal block of R_k and S_k to find its
numerical rank;
= 'N' : do not use condition estimation, but check the
diagonal entries of R_k and S_k for zero values;
= 'U' : use the ranks already stored in RANKS (for R).
Input/Output Parameters
N (input) INTEGER
The order of the matrix R. N = BN*BSN + ST >= 0.
(See parameter description below.)
IPAR (input) INTEGER array, dimension (LIPAR)
The integer parameters describing the structure of the
matrix R, as follows:
IPAR(1) must contain ST, the number of columns of the
submatrices L_k and the order of R_l+1. ST >= 0.
IPAR(2) must contain BN, the number of blocks, l, in the
block diagonal part of R. BN >= 0.
IPAR(3) must contain BSM, the number of rows of the blocks
R_k, k = 1:l. BSM >= 0.
IPAR(4) must contain BSN, the number of columns of the
blocks R_k, k = 1:l. BSN >= 0.
BSM is not used by this routine, but assumed equal to BSN.
LIPAR (input) INTEGER
The length of the array IPAR. LIPAR >= 4.
R (input/output) DOUBLE PRECISION array, dimension (LDR, NC)
where NC = N if BN <= 1, and NC = BSN+ST, if BN > 1.
On entry, the leading N-by-NC part of this array must
contain the (compressed) representation (Rc) of the upper
triangular matrix R. If BN > 1, the submatrix X in Rc is
not referenced. The zero strict lower triangles of R_k,
k = 1:l+1, need not be set. If BN <= 1 or BSN = 0, then
the full upper triangle of R must be stored.
On exit, the full upper triangles of R_k, k = 1:l+1, and
L_k, k = 1:l, are unaltered, and the strict lower
triangles of R_k, k = 1:l+1, contain the corresponding
strict upper triangles (transposed) of the upper
triangular matrix S.
If BN <= 1 or BSN = 0, then the transpose of the strict
upper triangle of S is stored in the strict lower triangle
of R.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
IPVT (input) INTEGER array, dimension (N)
This array must define the permutation matrix P such that
J*P = Q*R. Column j of P is column IPVT(j) of the identity
matrix.
DIAG (input) DOUBLE PRECISION array, dimension (N)
This array must contain the diagonal elements of the
matrix D. DIAG(I) <> 0, I = 1,...,N.
QTB (input) DOUBLE PRECISION array, dimension (N)
This array must contain the first n elements of the
vector Q'*b.
DELTA (input) DOUBLE PRECISION
An upper bound on the Euclidean norm of D*x. DELTA > 0.
PAR (input/output) DOUBLE PRECISION
On entry, PAR must contain an initial estimate of the
Levenberg-Marquardt parameter. PAR >= 0.
On exit, it contains the final estimate of this parameter.
RANKS (input or output) INTEGER array, dimension (r), where
r = BN + 1, if ST > 0, BSN > 0, and BN > 1;
r = BN, if ST = 0 and BSN > 0;
r = 1, if ST > 0 and ( BSN = 0 or BN <= 1 );
r = 0, if ST = 0 and BSN = 0.
On entry, if COND = 'U' and N > 0, this array must contain
the numerical ranks of the submatrices R_k, k = 1:l(+1).
On exit, if N > 0, this array contains the numerical ranks
of the submatrices S_k, k = 1:l(+1).
X (output) DOUBLE PRECISION array, dimension (N)
This array contains the least squares solution of the
system J*x = b, sqrt(PAR)*D*x = 0.
RX (output) DOUBLE PRECISION array, dimension (N)
This array contains the matrix-vector product -R*P'*x.
Tolerances
TOL DOUBLE PRECISION
If COND = 'E', the tolerance to be used for finding the
ranks of the submatrices R_k and S_k. If the user sets
TOL > 0, then the given value of TOL is used as a lower
bound for the reciprocal condition number; a (sub)matrix
whose estimated condition number is less than 1/TOL is
considered to be of full rank. If the user sets TOL <= 0,
then an implicitly computed, default tolerance, defined by
TOLDEF = N*EPS, is used instead, where EPS is the machine
precision (see LAPACK Library routine DLAMCH).
This parameter is not relevant if COND = 'U' or 'N'.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, the first N elements of this array contain the
diagonal elements of the upper triangular matrix S.
If BN > 1 and BSN > 0, the elements N+1 : N+ST*(N-ST)
contain the submatrix (S(1:N-ST,N-ST+1:N))' of the
matrix S.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 2*N, if BN <= 1 or BSN = 0 and
COND <> 'E';
LDWORK >= 4*N, if BN <= 1 or BSN = 0 and
COND = 'E';
LDWORK >= ST*(N-ST) + 2*N, if BN > 1 and BSN > 0 and
COND <> 'E';
LDWORK >= ST*(N-ST) + 2*N + 2*MAX(BSN,ST),
if BN > 1 and BSN > 0 and
COND = 'E'.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The algorithm computes the Gauss-Newton direction. An approximate basic least squares solution is found if the Jacobian is rank deficient. The computations exploit the special structure and storage scheme of the matrix R. If one or more of the submatrices R_k or S_k, k = 1:l+1, is singular, then the computed result is not the basic least squares solution for the whole problem, but a concatenation of (least squares) solutions of the individual subproblems involving R_k or S_k, k = 1:l+1 (with adapted right hand sides). If the Gauss-Newton direction is not acceptable, then an iterative algorithm obtains improved lower and upper bounds for the Levenberg-Marquardt parameter PAR. Only a few iterations are generally needed for convergence of the algorithm. If, however, the limit of ITMAX = 10 iterations is reached, then the output PAR will contain the best value obtained so far. If the Gauss-Newton step is acceptable, it is stored in x, and PAR is set to zero, hence S = R.References
[1] More, J.J., Garbow, B.S, and Hillstrom, K.E.
User's Guide for MINPACK-1.
Applied Math. Division, Argonne National Laboratory, Argonne,
Illinois, Report ANL-80-74, 1980.
Numerical Aspects
The algorithm requires 0(N*(BSN+ST)) operations and is backward stable, if R is nonsingular.Further Comments
This routine is a structure-exploiting, LAPACK-based modification of LMPAR from the MINPACK package [1], and with optional condition estimation. The option COND = 'U' is useful when dealing with several right-hand side vectors, but RANKS array should be reset. If COND = 'E', but the matrix S is guaranteed to be nonsingular and well conditioned relative to TOL, i.e., rank(R) = N, and min(DIAG) > 0, then its condition is not estimated.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To determine a vector x which solves the system of linear
equations
J*x = b , D*x = 0 ,
in the least squares sense, where J is an m-by-n matrix,
D is an n-by-n diagonal matrix, and b is an m-vector. The matrix J
is the current Jacobian of a nonlinear least squares problem,
provided in a compressed form by SLICOT Library routine NF01BD.
It is assumed that a block QR factorization, with column pivoting,
of J is available, that is, J*P = Q*R, where P is a permutation
matrix, Q has orthogonal columns, and R is an upper triangular
matrix with diagonal elements of nonincreasing magnitude for each
block, as returned by SLICOT Library routine NF01BS. The routine
NF01BQ needs the upper triangle of R in compressed form, the
permutation matrix P, and the first n components of Q'*b
(' denotes the transpose). The system J*x = b, D*x = 0, is then
equivalent to
R*z = Q'*b , P'*D*P*z = 0 , (1)
where x = P*z. If this system does not have full rank, then an
approximate least squares solution is obtained (see METHOD).
On output, NF01BQ also provides an upper triangular matrix S
such that
P'*(J'*J + D*D)*P = S'*S .
The system (1) is equivalent to S*z = c , where c contains the
first n components of the vector obtained by applying to
[ (Q'*b)' 0 ]' the transformations which triangularized
[ R' P'*D*P ]', getting S.
The matrix R has the following structure
/ R_1 0 .. 0 | L_1 \
| 0 R_2 .. 0 | L_2 |
| : : .. : | : | ,
| 0 0 .. R_l | L_l |
\ 0 0 .. 0 | R_l+1 /
where the submatrices R_k, k = 1:l, have the same order BSN,
and R_k, k = 1:l+1, are square and upper triangular. This matrix
is stored in the compressed form
/ R_1 | L_1 \
| R_2 | L_2 |
Rc = | : | : | ,
| R_l | L_l |
\ X | R_l+1 /
where the submatrix X is irrelevant. The matrix S has the same
structure as R, and its diagonal blocks are denoted by S_k,
k = 1:l+1.
If l <= 1, then the full upper triangle of the matrix R is stored.
Specification
SUBROUTINE NF01BQ( COND, N, IPAR, LIPAR, R, LDR, IPVT, DIAG, QTB,
$ RANKS, X, TOL, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COND
INTEGER INFO, LDR, LDWORK, LIPAR, N
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IPAR(*), IPVT(*), RANKS(*)
DOUBLE PRECISION DIAG(*), DWORK(*), QTB(*), R(LDR,*), X(*)
Arguments
Mode Parameters
COND CHARACTER*1
Specifies whether the condition of the matrices S_k should
be estimated, as follows:
= 'E' : use incremental condition estimation and store
the numerical rank of S_k in the array entry
RANKS(k), for k = 1:l+1;
= 'N' : do not use condition estimation, but check the
diagonal entries of S_k for zero values;
= 'U' : use the ranks already stored in RANKS(1:l+1).
Input/Output Parameters
N (input) INTEGER
The order of the matrix R. N = BN*BSN + ST >= 0.
(See parameter description below.)
IPAR (input) INTEGER array, dimension (LIPAR)
The integer parameters describing the structure of the
matrix R, as follows:
IPAR(1) must contain ST, the number of columns of the
submatrices L_k and the order of R_l+1. ST >= 0.
IPAR(2) must contain BN, the number of blocks, l, in the
block diagonal part of R. BN >= 0.
IPAR(3) must contain BSM, the number of rows of the blocks
R_k, k = 1:l. BSM >= 0.
IPAR(4) must contain BSN, the number of columns of the
blocks R_k, k = 1:l. BSN >= 0.
BSM is not used by this routine, but assumed equal to BSN.
LIPAR (input) INTEGER
The length of the array IPAR. LIPAR >= 4.
R (input/output) DOUBLE PRECISION array, dimension (LDR, NC)
where NC = N if BN <= 1, and NC = BSN+ST, if BN > 1.
On entry, the leading N-by-NC part of this array must
contain the (compressed) representation (Rc) of the upper
triangular matrix R. If BN > 1, the submatrix X in Rc is
not referenced. The zero strict lower triangles of R_k,
k = 1:l+1, need not be set. If BN <= 1 or BSN = 0, then
the full upper triangle of R must be stored.
On exit, the full upper triangles of R_k, k = 1:l+1, and
L_k, k = 1:l, are unaltered, and the strict lower
triangles of R_k, k = 1:l+1, contain the corresponding
strict upper triangles (transposed) of the upper
triangular matrix S.
If BN <= 1 or BSN = 0, then the transpose of the strict
upper triangle of S is stored in the strict lower triangle
of R.
LDR INTEGER
The leading dimension of the array R. LDR >= MAX(1,N).
IPVT (input) INTEGER array, dimension (N)
This array must define the permutation matrix P such that
J*P = Q*R. Column j of P is column IPVT(j) of the identity
matrix.
DIAG (input) DOUBLE PRECISION array, dimension (N)
This array must contain the diagonal elements of the
matrix D.
QTB (input) DOUBLE PRECISION array, dimension (N)
This array must contain the first n elements of the
vector Q'*b.
RANKS (input or output) INTEGER array, dimension (r), where
r = BN + 1, if ST > 0, BSN > 0, and BN > 1;
r = BN, if ST = 0 and BSN > 0;
r = 1, if ST > 0 and ( BSN = 0 or BN <= 1 );
r = 0, if ST = 0 and BSN = 0.
On entry, if COND = 'U' and N > 0, this array must contain
the numerical ranks of the submatrices S_k, k = 1:l(+1).
On exit, if COND = 'E' or 'N' and N > 0, this array
contains the numerical ranks of the submatrices S_k,
k = 1:l(+1), estimated according to the value of COND.
X (output) DOUBLE PRECISION array, dimension (N)
This array contains the least squares solution of the
system J*x = b, D*x = 0.
Tolerances
TOL DOUBLE PRECISION
If COND = 'E', the tolerance to be used for finding the
ranks of the submatrices S_k. If the user sets TOL > 0,
then the given value of TOL is used as a lower bound for
the reciprocal condition number; a (sub)matrix whose
estimated condition number is less than 1/TOL is
considered to be of full rank. If the user sets TOL <= 0,
then an implicitly computed, default tolerance, defined by
TOLDEF = N*EPS, is used instead, where EPS is the machine
precision (see LAPACK Library routine DLAMCH).
This parameter is not relevant if COND = 'U' or 'N'.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, the first N elements of this array contain the
diagonal elements of the upper triangular matrix S, and
the next N elements contain the solution z.
If BN > 1 and BSN > 0, the elements 2*N+1 : 2*N+ST*(N-ST)
contain the submatrix (S(1:N-ST,N-ST+1:N))' of the
matrix S.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 2*N, if BN <= 1 or BSN = 0 and
COND <> 'E';
LDWORK >= 4*N, if BN <= 1 or BSN = 0 and
COND = 'E';
LDWORK >= ST*(N-ST) + 2*N, if BN > 1 and BSN > 0 and
COND <> 'E';
LDWORK >= ST*(N-ST) + 2*N + 2*MAX(BSN,ST),
if BN > 1 and BSN > 0 and
COND = 'E'.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Standard plane rotations are used to annihilate the elements of the diagonal matrix D, updating the upper triangular matrix R and the first n elements of the vector Q'*b. A basic least squares solution is computed. The computations exploit the special structure and storage scheme of the matrix R. If one or more of the submatrices S_k, k = 1:l+1, is singular, then the computed result is not the basic least squares solution for the whole problem, but a concatenation of (least squares) solutions of the individual subproblems involving R_k, k = 1:l+1 (with adapted right hand sides).References
[1] More, J.J., Garbow, B.S, and Hillstrom, K.E.
User's Guide for MINPACK-1.
Applied Math. Division, Argonne National Laboratory, Argonne,
Illinois, Report ANL-80-74, 1980.
Numerical Aspects
The algorithm requires 0(N*(BSN+ST)) operations and is backward stable, if R is nonsingular.Further Comments
This routine is a structure-exploiting, LAPACK-based modification of QRSOLV from the MINPACK package [1], and with optional condition estimation. The option COND = 'U' is useful when dealing with several right-hand side vectors.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve one of the systems of linear equations
R*x = b , or R'*x = b ,
in the least squares sense, where R is an n-by-n block upper
triangular matrix, with the structure
/ R_1 0 .. 0 | L_1 \
| 0 R_2 .. 0 | L_2 |
| : : .. : | : | ,
| 0 0 .. R_l | L_l |
\ 0 0 .. 0 | R_l+1 /
with the upper triangular submatrices R_k, k = 1:l+1, square, and
the first l of the same order, BSN. The diagonal elements of each
block R_k have nonincreasing magnitude. The matrix R is stored in
the compressed form, as returned by SLICOT Library routine NF01BS,
/ R_1 | L_1 \
| R_2 | L_2 |
Rc = | : | : | ,
| R_l | L_l |
\ X | R_l+1 /
where the submatrix X is irrelevant. If the matrix R does not have
full rank, then a least squares solution is obtained. If l <= 1,
then R is an upper triangular matrix and its full upper triangle
is stored.
Optionally, the transpose of the matrix R can be stored in the
strict lower triangles of the submatrices R_k, k = 1:l+1, and in
the arrays SDIAG and S, as described at the parameter UPLO below.
Specification
SUBROUTINE NF01BR( COND, UPLO, TRANS, N, IPAR, LIPAR, R, LDR,
$ SDIAG, S, LDS, B, RANKS, TOL, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER COND, TRANS, UPLO
INTEGER INFO, LDR, LDS, LDWORK, LIPAR, N
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IPAR(*), RANKS(*)
DOUBLE PRECISION B(*), DWORK(*), R(LDR,*), S(LDS,*), SDIAG(*)
Arguments
Mode Parameters
COND CHARACTER*1
Specifies whether the condition of submatrices R_k should
be estimated, as follows:
= 'E' : use incremental condition estimation and store
the numerical rank of R_k in the array entry
RANKS(k), for k = 1:l+1;
= 'N' : do not use condition estimation, but check the
diagonal entries of R_k for zero values;
= 'U' : use the ranks already stored in RANKS(1:l+1).
UPLO CHARACTER*1
Specifies the storage scheme for the matrix R, as follows:
= 'U' : the upper triangular part is stored as in Rc;
= 'L' : the lower triangular part is stored, namely,
- the transpose of the strict upper triangle of
R_k is stored in the strict lower triangle of
R_k, for k = 1:l+1;
- the diagonal elements of R_k, k = 1:l+1, are
stored in the array SDIAG;
- the transpose of the last block column in R
(without R_l+1) is stored in the array S.
TRANS CHARACTER*1
Specifies the form of the system of equations, as follows:
= 'N': R*x = b (No transpose);
= 'T': R'*x = b (Transpose);
= 'C': R'*x = b (Transpose).
Input/Output Parameters
N (input) INTEGER
The order of the matrix R. N = BN*BSN + ST >= 0.
(See parameter description below.)
IPAR (input) INTEGER array, dimension (LIPAR)
The integer parameters describing the structure of the
matrix R, as follows:
IPAR(1) must contain ST, the number of columns of the
submatrices L_k and the order of R_l+1. ST >= 0.
IPAR(2) must contain BN, the number of blocks, l, in the
block diagonal part of R. BN >= 0.
IPAR(3) must contain BSM, the number of rows of the blocks
R_k, k = 1:l. BSM >= 0.
IPAR(4) must contain BSN, the number of columns of the
blocks R_k, k = 1:l. BSN >= 0.
BSM is not used by this routine, but assumed equal to BSN.
LIPAR (input) INTEGER
The length of the array IPAR. LIPAR >= 4.
R (input) DOUBLE PRECISION array, dimension (LDR, NC)
where NC = N if BN <= 1, and NC = BSN+ST, if BN > 1.
If UPLO = 'U', the leading N-by-NC part of this array must
contain the (compressed) representation (Rc) of the upper
triangular matrix R. The submatrix X in Rc and the strict
lower triangular parts of the diagonal blocks R_k,
k = 1:l+1, are not referenced. If BN <= 1 or BSN = 0, then
the full upper triangle of R must be stored.
If UPLO = 'L', BN > 1 and BSN > 0, the leading
(N-ST)-by-BSN part of this array must contain the
transposes of the strict upper triangles of R_k, k = 1:l,
stored in the strict lower triangles of R_k, and the
strict lower triangle of R_l+1 must contain the transpose
of the strict upper triangle of R_l+1. The submatrix X
in Rc is not referenced. The diagonal elements of R_k,
and, if COND = 'E', the upper triangular parts of R_k,
k = 1:l+1, are modified internally, but are restored
on exit.
If UPLO = 'L' and BN <= 1 or BSN = 0, the leading N-by-N
strict lower triangular part of this array must contain
the transpose of the strict upper triangular part of R.
The diagonal elements and, if COND = 'E', the upper
triangular elements are modified internally, but are
restored on exit.
LDR INTEGER
The leading dimension of the array R. LDR >= MAX(1,N).
SDIAG (input) DOUBLE PRECISION array, dimension (N)
If UPLO = 'L', this array must contain the diagonal
entries of R_k, k = 1:l+1. This array is modified
internally, but is restored on exit.
This parameter is not referenced if UPLO = 'U'.
S (input) DOUBLE PRECISION array, dimension (LDS,N-ST)
If UPLO = 'L', BN > 1, and BSN > 0, the leading
ST-by-(N-ST) part of this array must contain the transpose
of the rectangular part of the last block column in R,
that is [ L_1' L_2' ... L_l' ] . If COND = 'E', S is
modified internally, but is restored on exit.
This parameter is not referenced if UPLO = 'U', or
BN <= 1, or BSN = 0.
LDS INTEGER
The leading dimension of the array S.
LDS >= 1, if UPLO = 'U', or BN <= 1, or BSN = 0;
LDS >= MAX(1,ST), if UPLO = 'L', BN > 1, and BSN > 0.
B (input/output) DOUBLE PRECISION array, dimension (N)
On entry, this array must contain the right hand side
vector b.
On exit, this array contains the (least squares) solution
of the system R*x = b or R'*x = b.
RANKS (input or output) INTEGER array, dimension (r), where
r = BN + 1, if ST > 0, BSN > 0, and BN > 1;
r = BN, if ST = 0 and BSN > 0;
r = 1, if ST > 0 and ( BSN = 0 or BN <= 1 );
r = 0, if ST = 0 and BSN = 0.
On entry, if COND = 'U' and N > 0, this array must contain
the numerical ranks of the submatrices R_k, k = 1:l(+1).
On exit, if COND = 'E' or 'N' and N > 0, this array
contains the numerical ranks of the submatrices R_k,
k = 1:l(+1), estimated according to the value of COND.
Tolerances
TOL DOUBLE PRECISION
If COND = 'E', the tolerance to be used for finding the
ranks of the submatrices R_k. If the user sets TOL > 0,
then the given value of TOL is used as a lower bound for
the reciprocal condition number; a (sub)matrix whose
estimated condition number is less than 1/TOL is
considered to be of full rank. If the user sets TOL <= 0,
then an implicitly computed, default tolerance, defined by
TOLDEF = N*EPS, is used instead, where EPS is the machine
precision (see LAPACK Library routine DLAMCH).
This parameter is not relevant if COND = 'U' or 'N'.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK.
Denote Full = ( BN <= 1 or BSN = 0 );
Comp = ( BN > 1 and BSN > 0 ).
LDWORK >= 2*N, if Full and COND = 'E';
LDWORK >= 2*MAX(BSN,ST), if Comp and COND = 'E';
LDWORK >= 0, in the remaining cases.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Block back or forward substitution is used (depending on TRANS and UPLO), exploiting the special structure and storage scheme of the matrix R. If a submatrix R_k, k = 1:l+1, is singular, a local basic least squares solution is computed. Therefore, the returned result is not the basic least squares solution for the whole problem, but a concatenation of (least squares) solutions of the individual subproblems involving R_k, k = 1:l+1 (with adapted right hand sides).Numerical Aspects
2 2 The algorithm requires 0(BN*BSN + ST + N*ST) operations and is backward stable, if R is nonsingular.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the QR factorization of the Jacobian matrix J, as
received in compressed form from SLICOT Library routine NF01BD,
/ dy(1)/dwb(1) | dy(1)/ dtheta \
Jc = | : | : | ,
\ dy(L)/dwb(L) | dy(L)/ dtheta /
and to apply the transformation Q on the error vector e (in-situ).
The factorization is J*P = Q*R, where Q is a matrix with
orthogonal columns, P a permutation matrix, and R an upper
trapezoidal matrix with diagonal elements of nonincreasing
magnitude for each block column (see below). The 1-norm of the
scaled gradient is also returned.
Actually, the Jacobian J has the block form
dy(1)/dwb(1) 0 ..... 0 dy(1)/dtheta
0 dy(2)/dwb(2) ..... 0 dy(2)/dtheta
..... ..... ..... ..... .....
0 ..... 0 dy(L)/dwb(L) dy(L)/dtheta
but the zero blocks are omitted. The diagonal blocks have the
same size and correspond to the nonlinear part. The last block
column corresponds to the linear part. It is assumed that the
Jacobian matrix has at least as many rows as columns. The linear
or nonlinear parts can be empty. If L <= 1, the Jacobian is
represented as a full matrix.
Specification
SUBROUTINE NF01BS( N, IPAR, LIPAR, FNORM, J, LDJ, E, JNORMS,
$ GNORM, IPVT, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDJ, LDWORK, LIPAR, N
DOUBLE PRECISION FNORM, GNORM
C .. Array Arguments ..
INTEGER IPAR(*), IPVT(*)
DOUBLE PRECISION DWORK(*), E(*), J(*), JNORMS(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The number of columns of the Jacobian matrix J.
N = BN*BSN + ST >= 0. (See parameter description below.)
IPAR (input) INTEGER array, dimension (LIPAR)
The integer parameters describing the structure of the
matrix J, as follows:
IPAR(1) must contain ST, the number of parameters
corresponding to the linear part. ST >= 0.
IPAR(2) must contain BN, the number of blocks, BN = L,
for the parameters corresponding to the nonlinear
part. BN >= 0.
IPAR(3) must contain BSM, the number of rows of the blocks
J_k = dy(k)/dwb(k), k = 1:BN, if BN > 0, or the
number of rows of the matrix J, if BN <= 1.
BN*BSM >= N, if BN > 0;
BSM >= N, if BN = 0.
IPAR(4) must contain BSN, the number of columns of the
blocks J_k, k = 1:BN. BSN >= 0.
LIPAR (input) INTEGER
The length of the array IPAR. LIPAR >= 4.
FNORM (input) DOUBLE PRECISION
The Euclidean norm of the vector e. FNORM >= 0.
J (input/output) DOUBLE PRECISION array, dimension (LDJ, NC)
where NC = N if BN <= 1, and NC = BSN+ST, if BN > 1.
On entry, the leading NR-by-NC part of this array must
contain the (compressed) representation (Jc) of the
Jacobian matrix J, where NR = BSM if BN <= 1, and
NR = BN*BSM, if BN > 1.
On exit, the leading N-by-NC part of this array contains
a (compressed) representation of the upper triangular
factor R of the Jacobian matrix. The matrix R has the same
structure as the Jacobian matrix J, but with an additional
diagonal block. Note that for efficiency of the later
calculations, the matrix R is delivered with the leading
dimension MAX(1,N), possibly much smaller than the value
of LDJ on entry.
LDJ (input/output) INTEGER
The leading dimension of array J.
On entry, LDJ >= MAX(1,NR).
On exit, LDJ >= MAX(1,N).
E (input/output) DOUBLE PRECISION array, dimension (NR)
On entry, this array contains the vector e,
e = vec( Y - y ), where Y is set of output samples, and
vec denotes the concatenation of the columns of a matrix.
On exit, this array contains the updated vector Z*Q'*e,
where Z is the block row permutation matrix used in the
QR factorization of J (see METHOD).
JNORMS (output) DOUBLE PRECISION array, dimension (N)
This array contains the Euclidean norms of the columns
of the Jacobian matrix, considered in the initial order.
GNORM (output) DOUBLE PRECISION
If FNORM > 0, the 1-norm of the scaled vector J'*e/FNORM,
with each element i further divided by JNORMS(i) (if
JNORMS(i) is nonzero).
If FNORM = 0, the returned value of GNORM is 0.
IPVT (output) INTEGER array, dimension (N)
This array defines the permutation matrix P such that
J*P = Q*R. Column j of P is column IPVT(j) of the identity
matrix.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 1, if N = 0 or BN <= 1 and BSM = N = 1;
otherwise,
LDWORK >= 4*N+1, if BN <= 1 or BSN = 0;
LDWORK >= JWORK, if BN > 1 and BSN > 0, where JWORK is
given by the following procedure:
JWORK = BSN + MAX(3*BSN+1,ST);
JWORK = MAX(JWORK,4*ST+1), if BSM > BSN;
JWORK = MAX(JWORK,(BSM-BSN)*(BN-1)),
if BSN < BSM < 2*BSN.
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
A QR factorization with column pivoting of the matrix J is
computed, J*P = Q*R.
If l = L > 1, the R factor of the QR factorization has the same
structure as the Jacobian, but with an additional diagonal block.
Denote
/ J_1 0 .. 0 | L_1 \
| 0 J_2 .. 0 | L_2 |
J = | : : .. : | : | .
| : : .. : | : |
\ 0 0 .. J_l | L_l /
The algorithm consists in two phases. In the first phase, the
algorithm uses QR factorizations with column pivoting for each
block J_k, k = 1:l, and applies the orthogonal matrix Q'_k to the
corresponding part of the last block column and of e. After all
block rows have been processed, the block rows are interchanged
so that the zeroed submatrices in the first l block columns are
moved to the bottom part. The same block row permutation Z is
also applied to the vector e. At the end of the first phase,
the structure of the processed matrix J is
/ R_1 0 .. 0 | L^1_1 \
| 0 R_2 .. 0 | L^1_2 |
| : : .. : | : | .
| : : .. : | : |
| 0 0 .. R_l | L^1_l |
| 0 0 .. 0 | L^2_1 |
| : : .. : | : |
\ 0 0 .. 0 | L^2_l /
In the second phase, the submatrix L^2_1:l is triangularized
using an additional QR factorization with pivoting. (The columns
of L^1_1:l are also permuted accordingly.) Therefore, the column
pivoting is restricted to each such local block column.
If l <= 1, the matrix J is triangularized in one phase, by one
QR factorization with pivoting. In this case, the column
pivoting is global.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the matrix J'*J + c*I, for the Jacobian J as received
from SLICOT Library routine NF01BD:
/ dy(1)/dwb(1) | dy(1)/dtheta \
Jc = | : | : | .
\ dy(L)/dwb(L) | dy(L)/dtheta /
This is a compressed representation of the actual structure
/ J_1 0 .. 0 | L_1 \
| 0 J_2 .. 0 | L_2 |
J = | : : .. : | : | .
| : : .. : | : |
\ 0 0 .. J_L | L_L /
Specification
SUBROUTINE NF01BU( STOR, UPLO, N, IPAR, LIPAR, DPAR, LDPAR, J,
$ LDJ, JTJ, LDJTJ, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER STOR, UPLO
INTEGER INFO, LDJ, LDJTJ, LDPAR, LDWORK, LIPAR, N
C .. Array Arguments ..
DOUBLE PRECISION DPAR(*), DWORK(*), J(LDJ,*), JTJ(*)
INTEGER IPAR(*)
Arguments
Mode Parameters
STOR CHARACTER*1
Specifies the storage scheme for the symmetric
matrix J'*J + c*I, as follows:
= 'F' : full storage is used;
= 'P' : packed storage is used.
UPLO CHARACTER*1
Specifies which part of the matrix J'*J + c*I is stored,
as follows:
= 'U' : the upper triagular part is stored;
= 'L' : the lower triagular part is stored.
Input/Output Parameters
N (input) INTEGER
The order of the matrix J'*J + c*I.
N = BN*BSN + ST >= 0. (See parameter description below.)
IPAR (input) INTEGER array, dimension (LIPAR)
The integer parameters describing the structure of the
matrix J, as follows:
IPAR(1) must contain ST, the number of parameters
corresponding to the linear part. ST >= 0.
IPAR(2) must contain BN, the number of blocks, BN = L,
for the parameters corresponding to the nonlinear
part. BN >= 0.
IPAR(3) must contain BSM, the number of rows of the blocks
J_k = dy(k)/dwb(k), k = 1:BN, if BN > 0, or the
number of rows of the matrix J, if BN <= 1.
IPAR(4) must contain BSN, the number of columns of the
blocks J_k, k = 1:BN. BSN >= 0.
LIPAR (input) INTEGER
The length of the array IPAR. LIPAR >= 4.
DPAR (input) DOUBLE PRECISION array, dimension (LDPAR)
The real parameters needed for solving the problem.
The entry DPAR(1) must contain the real scalar c.
LDPAR (input) INTEGER
The length of the array DPAR. LDPAR >= 1.
J (input) DOUBLE PRECISION array, dimension (LDJ, NC)
where NC = N if BN <= 1, and NC = BSN+ST, if BN > 1.
The leading NR-by-NC part of this array must contain
the (compressed) representation (Jc) of the Jacobian
matrix J, where NR = BSM if BN <= 1, and NR = BN*BSM,
if BN > 1.
LDJ (input) INTEGER
The leading dimension of array J. LDJ >= MAX(1,NR).
JTJ (output) DOUBLE PRECISION array,
dimension (LDJTJ,N), if STOR = 'F',
dimension (N*(N+1)/2), if STOR = 'P'.
The leading N-by-N (if STOR = 'F'), or N*(N+1)/2 (if
STOR = 'P') part of this array contains the upper or
lower triangle of the matrix J'*J + c*I, depending on
UPLO = 'U', or UPLO = 'L', respectively, stored either as
a two-dimensional, or one-dimensional array, depending
on STOR.
LDJTJ INTEGER
The leading dimension of the array JTJ.
LDJTJ >= MAX(1,N), if STOR = 'F'.
LDJTJ >= 1, if STOR = 'P'.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
Currently, this array is not used.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= 0.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The matrix product is computed columnn-wise, exploiting the symmetry. BLAS 3 routines DGEMM and DSYRK are used if STOR = 'F', and BLAS 2 routine DGEMV is used if STOR = 'P'.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the matrix J'*J + c*I, for the Jacobian J as received from SLICOT Library routine NF01BY, for one output variable. NOTE: this routine must have the same arguments as SLICOT Library routine NF01BU.Specification
SUBROUTINE NF01BV( STOR, UPLO, N, IPAR, LIPAR, DPAR, LDPAR, J,
$ LDJ, JTJ, LDJTJ, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER STOR, UPLO
INTEGER INFO, LDJ, LDJTJ, LDPAR, LDWORK, LIPAR, N
C .. Array Arguments ..
INTEGER IPAR(*)
DOUBLE PRECISION DPAR(*), DWORK(*), J(LDJ,*), JTJ(*)
Arguments
Mode Parameters
STOR CHARACTER*1
Specifies the storage scheme for the symmetric
matrix J'*J + c*I, as follows:
= 'F' : full storage is used;
= 'P' : packed storage is used.
UPLO CHARACTER*1
Specifies which part of the matrix J'*J + c*I is stored,
as follows:
= 'U' : the upper triagular part is stored;
= 'L' : the lower triagular part is stored.
Input/Output Parameters
N (input) INTEGER
The number of columns of the Jacobian matrix J. N >= 0.
IPAR (input) INTEGER array, dimension (LIPAR)
The integer parameters describing the structure of the
matrix J, as follows:
IPAR(1) must contain the number of rows M of the Jacobian
matrix J. M >= 0.
IPAR is provided for compatibility with SLICOT Library
routine MD03AD.
LIPAR (input) INTEGER
The length of the array IPAR. LIPAR >= 1.
DPAR (input) DOUBLE PRECISION array, dimension (LDPAR)
The real parameters needed for solving the problem.
The entry DPAR(1) must contain the real scalar c.
LDPAR (input) INTEGER
The length of the array DPAR. LDPAR >= 1.
J (input) DOUBLE PRECISION array, dimension (LDJ,N)
The leading M-by-N part of this array must contain the
Jacobian matrix J.
LDJ INTEGER
The leading dimension of the array J. LDJ >= MAX(1,M).
JTJ (output) DOUBLE PRECISION array,
dimension (LDJTJ,N), if STOR = 'F',
dimension (N*(N+1)/2), if STOR = 'P'.
The leading N-by-N (if STOR = 'F'), or N*(N+1)/2 (if
STOR = 'P') part of this array contains the upper or
lower triangle of the matrix J'*J + c*I, depending on
UPLO = 'U', or UPLO = 'L', respectively, stored either as
a two-dimensional, or one-dimensional array, depending
on STOR.
LDJTJ INTEGER
The leading dimension of the array JTJ.
LDJTJ >= MAX(1,N), if STOR = 'F'.
LDJTJ >= 1, if STOR = 'P'.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
Currently, this array is not used.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= 0.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The matrix product is computed columnn-wise, exploiting the symmetry. BLAS 3 routine DSYRK is used if STOR = 'F', and BLAS 2 routine DGEMV is used if STOR = 'P'.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the matrix-vector product x <-- (J'*J + c*I)*x, for the
Jacobian J as received from SLICOT Library routine NF01BD:
/ dy(1)/dwb(1) | dy(1)/dtheta \
Jc = | : | : | .
\ dy(L)/dwb(L) | dy(L)/dtheta /
This is a compressed representation of the actual structure
/ J_1 0 .. 0 | L_1 \
| 0 J_2 .. 0 | L_2 |
J = | : : .. : | : | .
| : : .. : | : |
\ 0 0 .. J_L | L_L /
Specification
SUBROUTINE NF01BW( N, IPAR, LIPAR, DPAR, LDPAR, J, LDJ, X, INCX,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INCX, INFO, LDJ, LDPAR, LDWORK, LIPAR, N
C .. Array Arguments ..
DOUBLE PRECISION DPAR(*), DWORK(*), J(LDJ,*), X(*)
INTEGER IPAR(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The dimension of the vector x.
N = BN*BSN + ST >= 0. (See parameter description below.)
IPAR (input) INTEGER array, dimension (LIPAR)
The integer parameters describing the structure of the
matrix J, as follows:
IPAR(1) must contain ST, the number of parameters
corresponding to the linear part. ST >= 0.
IPAR(2) must contain BN, the number of blocks, BN = L,
for the parameters corresponding to the nonlinear
part. BN >= 0.
IPAR(3) must contain BSM, the number of rows of the blocks
J_k = dy(k)/dwb(k), k = 1:BN, if BN > 0, or the
number of rows of the matrix J, if BN <= 1.
IPAR(4) must contain BSN, the number of columns of the
blocks J_k, k = 1:BN. BSN >= 0.
LIPAR (input) INTEGER
The length of the array IPAR. LIPAR >= 4.
DPAR (input) DOUBLE PRECISION array, dimension (LDPAR)
The real parameters needed for solving the problem.
The entry DPAR(1) must contain the real scalar c.
LDPAR (input) INTEGER
The length of the array DPAR. LDPAR >= 1.
J (input) DOUBLE PRECISION array, dimension (LDJ, NC)
where NC = N if BN <= 1, and NC = BSN+ST, if BN > 1.
The leading NR-by-NC part of this array must contain
the (compressed) representation (Jc) of the Jacobian
matrix J, where NR = BSM if BN <= 1, and NR = BN*BSM,
if BN > 1.
LDJ (input) INTEGER
The leading dimension of array J. LDJ >= MAX(1,NR).
X (input/output) DOUBLE PRECISION array, dimension
(1+(N-1)*INCX)
On entry, this incremented array must contain the
vector x.
On exit, this incremented array contains the value of the
matrix-vector product (J'*J + c*I)*x.
INCX (input) INTEGER
The increment for the elements of X. INCX >= 1.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK. LDWORK >= NR.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The associativity of matrix multiplications is used; the result is obtained as: x_out = J'*( J*x ) + c*x.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute (J'*J + c*I)*x, where J is an m-by-n real matrix, c is a real scalar, I is the n-by-n identity matrix, and x is a real n-vector. NOTE: this routine must have the same arguments as SLICOT Library routine NF01BW.Specification
SUBROUTINE NF01BX( N, IPAR, LIPAR, DPAR, LDPAR, J, LDJ, X, INCX,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INCX, INFO, LDJ, LDPAR, LDWORK, LIPAR, N
C .. Array Arguments ..
INTEGER IPAR(*)
DOUBLE PRECISION DPAR(*), DWORK(*), J(LDJ,*), X(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The number of columns of the Jacobian matrix J. N >= 0.
IPAR (input) INTEGER array, dimension (LIPAR)
The integer parameters describing the structure of the
matrix J, as follows:
IPAR(1) must contain the number of rows M of the Jacobian
matrix J. M >= 0.
IPAR is provided for compatibility with SLICOT Library
routine MD03AD.
LIPAR (input) INTEGER
The length of the array IPAR. LIPAR >= 1.
DPAR (input) DOUBLE PRECISION array, dimension (LDPAR)
The real parameters needed for solving the problem.
The entry DPAR(1) must contain the real scalar c.
LDPAR (input) INTEGER
The length of the array DPAR. LDPAR >= 1.
J (input) DOUBLE PRECISION array, dimension (LDJ,N)
The leading M-by-N part of this array must contain the
Jacobian matrix J.
LDJ INTEGER
The leading dimension of the array J. LDJ >= MAX(1,M).
X (input/output) DOUBLE PRECISION array, dimension
(1+(N-1)*abs(INCX))
On entry, this incremented array must contain the
vector x.
On exit, this incremented array contains the value of the
matrix-vector product (J'*J + c*I)*x.
INCX (input) INTEGER
The increment for the elements of X. INCX <> 0.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK. LDWORK >= M.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The associativity of matrix multiplications is used; the result is obtained as: x_out = J'*( J*x ) + c*x.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the Jacobian of the error function for a neural network
of the structure
- tanh(w1*z+b1) -
/ : \
z --- : --- sum(ws(i)*...)+ b(n+1) --- y,
\ : /
- tanh(wn*z+bn) -
for the single-output case. The Jacobian has the form
d e(1) / d WB(1) ... d e(1) / d WB(NWB)
J = : : ,
d e(NSMP) / d WB(1) ... d e(NSMP) / d WB(NWB)
where e(z) is the error function, WB is the set of weights and
biases of the network (for the considered output), and NWB is
the number of elements of this set, NWB = IPAR(1)*(NZ+2)+1
(see below).
In the multi-output case, this routine should be called for each
output.
NOTE: this routine must have the same arguments as SLICOT Library
routine NF01BD.
Specification
SUBROUTINE NF01BY( CJTE, NSMP, NZ, L, IPAR, LIPAR, WB, LWB, Z,
$ LDZ, E, J, LDJ, JTE, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER CJTE
INTEGER INFO, L, LDJ, LDWORK, LDZ, LIPAR, LWB, NSMP, NZ
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), E(*), J(LDJ,*), JTE(*), WB(*),
$ Z(LDZ,*)
INTEGER IPAR(*)
Arguments
Mode Parameters
CJTE CHARACTER*1
Specifies whether the matrix-vector product J'*e should be
computed or not, as follows:
= 'C' : compute J'*e;
= 'N' : do not compute J'*e.
Input/Output Parameters
NSMP (input) INTEGER
The number of training samples. NSMP >= 0.
NZ (input) INTEGER
The length of each input sample. NZ >= 0.
L (input) INTEGER
The length of each output sample.
Currently, L must be 1.
IPAR (input/output) INTEGER array, dimension (LIPAR)
The integer parameters needed.
On entry, the first element of this array must contain
a value related to the number of neurons, n; specifically,
n = abs(IPAR(1)), since setting IPAR(1) < 0 has a special
meaning (see below).
On exit, if IPAR(1) < 0 on entry, then no computations are
performed, except the needed tests on input parameters,
but the following values are returned:
IPAR(1) contains the length of the array J, LJ;
LDJ contains the leading dimension of array J.
Otherwise, IPAR(1) and LDJ are unchanged on exit.
LIPAR (input) INTEGER
The length of the vector IPAR. LIPAR >= 1.
WB (input) DOUBLE PRECISION array, dimension (LWB)
The leading NWB = IPAR(1)*(NZ+2)+1 part of this array
must contain the weights and biases of the network,
WB = ( w(1,1), ..., w(1,NZ), ..., w(n,1), ..., w(n,NZ),
ws(1), ..., ws(n), b(1), ..., b(n+1) ),
where w(i,j) are the weights of the hidden layer,
ws(i) are the weights of the linear output layer and
b(i) are the biases.
LWB (input) INTEGER
The length of array WB. LWB >= NWB.
Z (input) DOUBLE PRECISION array, dimension (LDZ, NZ)
The leading NSMP-by-NZ part of this array must contain the
set of input samples,
Z = ( Z(1,1),...,Z(1,NZ); ...; Z(NSMP,1),...,Z(NSMP,NZ) ).
LDZ INTEGER
The leading dimension of array Z. LDZ >= MAX(1,NSMP).
E (input) DOUBLE PRECISION array, dimension (NSMP)
If CJTE = 'C', this array must contain the error vector e.
If CJTE = 'N', this array is not referenced.
J (output) DOUBLE PRECISION array, dimension (LDJ, NWB)
The leading NSMP-by-NWB part of this array contains the
Jacobian of the error function.
LDJ INTEGER
The leading dimension of array J. LDJ >= MAX(1,NSMP).
Note that LDJ is an input parameter, except for
IPAR(1) < 0 on entry, when it is an output parameter.
JTE (output) DOUBLE PRECISION array, dimension (NWB)
If CJTE = 'C', this array contains the matrix-vector
product J'*e.
If CJTE = 'N', this array is not referenced.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
This argument is included for combatibility with SLICOT
Library routine NF01BD.
LDWORK INTEGER
Normally, the length of the array DWORK. LDWORK >= 0.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The Jacobian is computed analytically.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
Interface for using a common entry point, DSblock compatible for defining Differential Algebraic Equations using several packages. The equations follow the form: dx/dt = f(x(t), (t), p, t) y(t) = g(x(t), (t), p, t) The user must define only the subroutines ODEDER and ODEOUT and the Jacobians (JACFX, JACFU, JACFP) if used, and the interface adapts the structure to fit all the codesSpecification
SUBROUTINE ODESolver(ISOLVER, CODEDER_, CODEOUT_, CJACFX_, $ CJACFU_, CJACFP_, $ NX, NY, NU, TINI, TOUT, X, U, Y, $ IPAR, DPAR, RTOL, ATOL, $ IWORK, LIWORK, DWORK, LDWORK, $ IWARN, INFO) .. Scalar Arguments .. DOUBLE PRECISION RTOL, TINI, TOUT INTEGER ISOLVER, IWARN, INFO, NX, NY, NU, $ LDWORK, LIWORK CHARACTER*9 CODEDER_, CODEOUT_, CJACFX_, CJACFU_, CJACFP_ .. Array Arguments .. DOUBLE PRECISION ATOL(*), DWORK(LDWORK), DPAR(*), $ X(NX), Y(NY), U(NU) INTEGER IWORK(LIWORK), IPAR(*)Arguments
Mode Parameters
ISOLVER (input) INTEGER Indicates the nonlinear solver package to be used: = 1: LSODE, = 2: LSODA, = 3: LSODES, = 4: RADAU5, = 5: DASSL, = 6: DASPK, = 7: DGELDA.Input/Output Parameters
ODEDER (input) EXTERNAL Evaluates the right hand side f of the ODE. ODEOUT (input) EXTERNAL Evaluates the output signals function g. JACFX (input) EXTERNAL Evaluates the jacobian matrix with respect to X. JACFU (input) EXTERNAL Evaluates the jacobian matrix with respect to U. JACFP (input) EXTERNAL Evaluates the jacobian matrix with respect to P. NX (input) INTEGER Dimension of the state vector. NY (input) INTEGER Dimension of the output vector. NU (input) INTEGER Dimension of the input vector.
TINI (input) DOUBLE PRECISION Initial value of time. TOUT (input) DOUBLE PRECISION Final value of time. X (input/output) DOUBLE PRECISION array, dimension (NX) On entry, array containing the initial state variables. On exit, it has the last value of the state variables. U (input) DOUBLE PRECISION array, dimension (NU) Array containing the input initial values. Y (input/output) DOUBLE PRECISION array, dimension (NY) On entry, array containing the initial values of Y. On exit, it has the results of the system. IPAR (input/output) INTEGER array, dimension (230) INPUT: 1..15 General 16..25 ODEPACK 26..35 RADAU5 36..50 DASSL/PK 51..60 GELDA 61..100 Reserved OUTPUT: 101..110 General 111..125 ODEPACK 126..135 RADAU5 136..145 DASSL/PK 146..155 GELDA 156..200 Reserved Any Mode: 201.. User Available Common integer parameters for SOLVERS: IPAR(1), Tolerance mode 0 : both RTOL and ATOL are scalars 1 : RTOL is scalar and ATOL is vector 2 : both RTOL and ATOL are vectors IPAR(2), Compute Output Values, must be 1 IPAR(3), mf, Method flag 0 : No jacobian used (non-stiff method). 1 : User supplied full jacobian (stiff). 2 : User supplied banded jacobian (stiff). 3 : User supplied sparse jacobian (stiff). 10 : internally generated full jacobian (stiff). 11 : internally generated banded jacobian (stiff). 12 : internally generated sparse jacobian (stiff). IPAR(5), Maximum number of steps allowed during one call to the solver. IPAR(6), ml, lower half-bandwithds of the banded jacobian, excluding tne main diagonal. IPAR(7), mu, upper half-bandwithds of the banded jacobian, excluding tne main diagonal. IPAR(8), Flag to generate extra printing at method switches: 0 means no extra printing 1 for minimal printing 2 for full printing IPAR(101) = Number of steps taken for the problem. IPAR(102) = Number of f evaluations. IPAR(103) = Number of jacobian evaluations. Common parameters for ODEPACK, DASSL and DASPK solvers: IPAR(111) = The method order last used(successfully). IPAR(112) = The order to be attempted on the next step. Common parameters for ODEPACK solver: IPAR(16), Status Flag IPAR(17), Optional inputs IPAR(18), Maximum number of messages printed, default value is 10. IPAR(113) = Index of the component of largest in the weighted local error vector ( e(i)/ewt(i) ). IPAR(114) = Length of rwork actually required. IPAR(115) = Length of iwork actually required. - LSODE and LSODES IPAR(19), Maximum order to be allowed. 12 if meth = 1 5 if meth = 2 If exceds the default value, it will be reduced to the default value. - LSODES IPAR(118), Number of nonzero elements in the jacobian matrix, including the diagonal (miter = 1 or 2). IPAR(119), Number of groups of column indices, used in difference quotient jacobian aproximations if miter = 2. IPAR(120), Number of sparse LU decompositions. IPAR(121), Base address in rwork of the history array. IPAR(122), Base address of the structure descriptor array ian. IPAR(123), Base address of the structure descriptor array jan. IPAR(124), Number of nonzero elements in the strict lower triangle of the LU factorization. IPAR(125), Number of nonzero elements in the strict upper triangle of the LU factorization. - LSODA IPAR(22), Maximum order to be allowed for the nonstiff method, default value is 12. If exceds the default value, it will be reduced to the default value. IPAR(23), Maximum order to be allowed for the stiff method, default value is 5. If exceds the default value, it will be reduced to the default value. IPAR(116), Method indicator for the last successful step 1 adams (nonstiff) 2 bdf (stiff) IPAR(117), Current method indicator 1 adams (nonstiff) 2 bdf (stiff) Parameters for RADAU5 solver: IPAR(26) Transforms the Jacobian matrix to Hessenberg form. IPAR(27) Maximum number of Newton iterations. IPAR(28) Starting values for Newton's method if 0 then is taken the extrapolated collocation solution if not equal 0 zero values are used. IPAR(29) Dimension of the index 1 variables. IPAR(30) Dimension of the index 2 variables. IPAR(31) Dimension of the index 3 variables. IPAR(32) Switch for step size strategy 0,1 Mod. Predictive controller(Gustafsson) 2 Classical step size control IPAR(33) Value of M1. IPAR(34) Value of M2. IPAR(126), Number of accepted steps. IPAR(127), Number of rejected steps. IPAR(128), Number of LU-Decompositions of both matrices IPAR(129), Number of forward-backward substitutions, of both systems. Common parameters for DASSL and DASPK solvers: IPAR(36), this parameter enables the code to initialize itself. Must set to 0 to indicate the start of every new problem. 0: Yes. (On each new problem) 1: No. (Allows 500 new steps) IPAR(37), code solve the problem without invoking any special non negativity contraints: 0: Yes 1: To have constraint checking only in the initial condition calculation. 2: To enforce nonnegativity in X during the integration. 3: To enforce both options 1 and 2. IPAR(38), Solver try to compute the initial T, X and XPRIME: 0: The initial T, X and XPRIME are consistent. 1: Given X_d calculate X_a and X'_d 2: Given X' calculate X. ( X_d differential variables in X X_a algebrac variables in X ) IPAR(136), Total number of error test failures so far. IPAR(137), Total number of convergence test failures. -Parameters for DASPK IPAR(39), DASPK use: 0: direct methods (compatible with DASSL) 1: Krylov method 2: Krylov method + Jac IPAR(40), DASPK uses scalars MAXLm KMP, NRMAX and EPLI when uses Krylov method. 0: uses default values. 1: uses user values. IPAR(41), Proceed to the integration after the initial condition calculation is done. Used when INFOV(11)>0 0: Yes 1: No IPAR(42), Errors are controled localy on all the variables. 0: Yes 1: No IPAR(43), Use default values for initial condition heuristic controls. 0: Yes 1: No and provide MXNIT, MXNJ, MXNH, LSOFF, STPTOL, EPINIT. IPAR(138), number of convergence failures of the linear iteration IPAR(139), length of IWORK actually required. IPAR(140), length of RWORK actually required. IPAR(141), total number of nonlinear iterations. IPAR(142), total number of linear (Krylov) iterations IPAR(143), number of PSOL calls. DPAR (input/output) DOUBLE PRECISION array, dimension (202) INPUT: 1..15 General 16..25 ODEPACK 26..35 RADAU5 36..50 DASSL/PK 51..60 GELDA 61..100 Reserved OUTPUT: 101..110 General 111..125 ODEPACK 126..135 RADAU5 136..145 DASSL/PK 146..155 GELDA 156..200 Reserved Any Mode: 201.. User Available Common real parameters for SOLVERS: DPAR(1), Initial step size guess. Optional in: ODEPACK, DASSL, .. DPAR(2), Maximum absolute step size allowed. Common parameters for ODEPACK and DASSL: DPAR(111), Step size in t last used (successfully). DPAR(113), Current value of the independent variable which the solver has actually reached Common parameters for ODEPACK solvers: DPAR(16), Critical value of t which the solver is not overshoot. DPAR(17), Minimum absolute step size allowed. DPAR(112), Step size to be attempted on the next step. DPAR(18), Tolerance scale factor, greater than 1.0. - LSODA DPAR(115) Value of t at the time of the last method switch, if any. - LSODA DPAR(115) Value of t at the time of the last method switch, if any. - LSODES DPAR(19), The element threshhold for sparsity determination when moss = 1 or 2. Parameters for RADAU5 solver: DPAR(26), The rounding unit, default 1E-16. DPAR(27), The safety factor in step size prediction, default 0.9D0. DPAR(28), Decides whether the jacobian should be recomputed, default 0.001D0. DPAR(29), Stopping criterion for Newton's method, default MIN(0.03D0, RTOL(1)**0.5D0) DPAR(30), DPAR(31): This saves, together with a large DPAR(28), LU-decompositions and computing time for large systems. DPAR(32), DPAR(33), Parameters for step size selection. Parameters for DASSL and DASPK solvers: DPAR(36), Stopping point (Tstop)Tolerances
RTOL DOUBLE PREISION Relative Tolerance. ATOL DOUBLE PREISION Absolute Tolerance.Workspace
IWORK INTEGER array, dimension (LIWORK) LIWORK INTEGER Size of IWORK, depending on solver: - LSODE 20 for mf = 10, 20 + neq for mf = 21, 22, 24, or 25. if mf = 24, 25, input in iwork(1),iwork(2) the lower and upper half-bandwidths ml,mu. -LSODA 20+NX -LSODES 30 -DASSL 20+NEQ DWORK DOUBLE PREISION array, dimension (LDWORK) LDWORK INTEGER Size of DWORK, depending on solver: - LSODE 20+16*NX , IPAR(3) = 10, 22+ 9*NX+NX**2 , IPAR(3) = 21 or 22, 22+10*NX+(2*IPAR(4)+IPAR(9))*NX, IPAR(3) = 24 or 25. - LSODA 22+NX*max(16,NX+9) - LSODES 20+16*NX , mf=10 20+(2+1./lenrat)*nnz + (11+9./lenrat)*NX, mf=121,222 - DASSL >= 40 LRW .GE. 40+(MAXORD+4)*NEQ+NEQ**2, IPAR(3) = 1 or 10 >= 40+(MAXORD+4)*NEQ+(2*ML+MU+1)*NEQ, IPAR(3) = 2 >= 40+(MAXORD+4)*NEQ+(2*ML+MU+1)*NEQ +2*(NEQ/(ML+MU+1)+1), IPAR(3) = 11Warning Indicator
IWARN INTEGER = 0: no warning; = 1: LSODE/LSODA/LSODES/RADAU5 do not use the input vector as argument; = 2: Only the 1st element of RTOL is used; = 3: Method (IPAR(3)) not allowed with LSODE/LSODA/LSODES/RADAU5/DASSL/DASPK; = 4: Only the 1st element of ATOL is used; = 5: Option not allowed for IPAR(37); = 6: Option not allowed for IPAR(38).Error Indicator
INFO INTEGER = 0: Successful exit; < 0: If INFO = -i, the i-th argument had an illegal value; = 1: Wrong tolerance mode; = 2: Sparse storage (IPAR(4)=1) incompatible with LSODE/LSODA/RADAU5; = 3: Dense storage (IPAR(4)=0) incompatible with LSODES = 100+ERROR: ODEDER returned ERROR = 200+ERROR: RADAU5 returned -ERROR = 300+ERROR: DDASSL returned -ERROR = 400+ERROR: DDASPK returned -ERROR = 500+ERROR: DGELDA returned -ERROR
Since the package integrates 9 different solvers, it is possible to solve differential equations by means of Backward Differential Formulas, Runge-Kutta, using direct or iterative methods (including preconditioning) for the linear system associated, differential equations with time-varying coefficients or of order higher than one. The interface facilitates the user the work of changing the integrator and testing the results, thus leading a more robust and efficient integrated package.References
[1] A.C. Hindmarsh, Brief Description of ODEPACK: A Systematized Collection
of ODE Solvers, http://www.netlib.org/odepack/doc
[2] L.R. Petzold DASSL Library Documentation, http://www.netlib.org/ode/
[3] P.N. Brown, A.C. Hindmarsh, L.R. Petzold, DASPK Package 1995 Revision
[4] R.S. Maier, Using DASPK on the TMC CM5. Experiences with Two Programming
Models, Minesota Supercomputer Center, Technical Report.
[5] E. Hairer, G. Wanner, Solving Ordinary Dirential Equations II. Stiánd
Dirential- Algebraic Problems., Springer Seried in Computational
Mathermatics 14, Springer-Verlag 1991, Second Edition 1996.
[6] P. Kunkel, V. Mehrmann, W. Rath und J. Weickert, `GELDA: A Software
Package for the Solution of General Linear Dirential Algebraic
equations', SIAM Journal Scienti^Lc Computing, Vol. 18, 1997, pp.
115 - 138.
[7] M. Otter, DSblock: A neutral description of dynamic systems.
Version 3.3, http://www.netlib.org/odepack/doc
[8] M. Otter, H. Elmqvist, The DSblock model interface for exchanging model
components, Proceedings of EUROSIM 95, ed. F.Brenenecker, Vienna, Sep.
11-15, 1995
[9] M. Otter, The DSblock model interface, version 4.0, Incomplete Draft,
http://dv.op.dlr.de/~otter7dsblock/dsblock4.0a.html
[10] Ch. Lubich, U. Novak, U. Pohle, Ch. Engstler, MEXX - Numerical
Software for the Integration of Constrained Mechanical Multibody
Systems, http://www.netlib.org/odepack/doc
[11] Working Group on Software (WGS), SLICOT Implementation and Documentation
Standards (version 1.0), WGS-Report 90-1, Eindhoven University of
Technology, May 1990.
[12] P. Kunkel and V. Mehrmann, Canonical forms for linear differential-
algebraic equations with variable coeÆcients., J. Comput. Appl.
Math., 56:225{259, 1994.
[13] Working Group on Software (WGS), SLICOT Implementation and Documentation
Standards, WGS-Report 96-1, Eindhoven University of Technology, updated:
Feb. 1998, ../../REPORTS/rep96-1.ps.Z.
[14] A. Varga, Standarization of Interface for Nonlinear Systems Software
in SLICOT, Deutsches Zentrum ur Luft un Raumfahrt, DLR. SLICOT-Working
Note 1998-4, 1998, Available at
../../REPORTS/SLWN1998-4.ps.Z.
[15] D. Kirk, Optimal Control Theory: An Introduction, Prentice-Hall.
Englewood Cli, NJ, 1970.
[16] F.L. Lewis and V.L. Syrmos, Optimal Control, Addison-Wesley.
New York, 1995.
[17] W.M.Lioen, J.J.B de Swart, Test Set for Initial Value Problem Solvers,
Technical Report NM-R9615, CWI, Amsterdam, 1996.
http://www.cwi.nl/cwi/projects/IVPTestset/.
[18] V.Hernandez, I.Blanquer, E.Arias, and P.Ruiz,
Definition and Implementation of a SLICOT Standard Interface and the
associated MATLAB Gateway for the Solution of Nonlinear Control Systems
by using ODE and DAE Packages}, Universidad Politecnica de Valencia,
DSIC. SLICOT Working Note 2000-3: July 2000. Available at
../../REPORTS/SLWN2000-3.ps.Z.
[19] J.J.B. de Swart, W.M. Lioen, W.A. van der Veen, SIDE, November 25,
1998. Available at http://www.cwi.nl/cwi/projects/PSIDE/.
[20] Kim, H.Young, F.L.Lewis, D.M.Dawson, Intelligent optimal control of
robotic manipulators using neural networks.
[21] J.C.Fernandez, E.Arias, V.Hernandez, L.Penalver, High Performance
Algorithm for Tracking Trajectories of Robot Manipulators,
Preprints of the Proceedings of the 6th IFAC International Workshop on
Algorithms and Architectures for Real-Time Control (AARTC-2000),
pages 127-134.
Numerical Aspects
The numerical aspects of the routine lie on the features of the different packages integrated. Several packages are more robust than others, and other packages simply cannot deal with problems that others do. For a detailed description of the numerical aspects of each method is recommended to check the references above.Further Comments
Several packages (LSODES, LSOIBT) deal only with sparse matrices. The interface checks the suitability of the methods to the parameters and show a warning message if problems could arise.Example
Program Text
* ODESOLVER EXAMPLE PROGRAM TEXT FOR LSODEX
PROBLEM
*
* .. Parameters ..
INTEGER
NIN, NOUT
PARAMETER
( NIN = 5, NOUT = 6 )
INTEGER LSODE_, LSODA_, LSODES_,
RADAU5_, DASSL_, DASPK_, DGELDA_
PARAMETER (LSODE_ = 1, LSODA_
= 2, LSODES_ = 3)
PARAMETER (RADAU5_ = 4, DASSL_ =
5, DASPK_ = 6)
PARAMETER (DGELDA_ = 7)
*
EXTERNAL IARGC_
INTEGER IARGC_
INTEGER NUMARGS
CHARACTER*80 NAME
CHARACTER*80 SOLVER
*
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
*
NUMARGS = IARGC_()
*
CALL GETARG_(0, NAME)
IF (NUMARGS .NE. 1) THEN
WRITE (*,*) 'Syntax
Error: ',NAME(1:8),' <solver>'
WRITE (*,*) 'Solvers
: LSODE, LSODA, LSODES, RADAU5, DASSL, DASP
&K'
ELSE
*
CALL GETARG_(1, SOLVER)
*
WRITE (*,*) 'Problem:
LSODEX Solver: ',SOLVER(1:7)
*
IF (SOLVER(1:5) .EQ.
'LSODE') THEN
CALL TEST(LSODE_)
ELSEIF (SOLVER(1:5)
.EQ. 'LSODA') THEN
CALL TEST(LSODA_)
ELSEIF (SOLVER(1:6)
.EQ. 'LSODES') THEN
CALL TEST(LSODES_)
ELSEIF (SOLVER(1:6)
.EQ. 'RADAU5') THEN
CALL TEST(RADAU5_)
ELSEIF (SOLVER(1:5)
.EQ. 'DASSL') THEN
CALL TEST(DASSL_)
ELSEIF (SOLVER(1:5)
.EQ. 'DASPK') THEN
CALL TEST(DASPK_)
ELSE
WRITE (*,*)
'Error: Solver: ', SOLVER,' unknown'
ENDIF
ENDIF
*
99999 FORMAT (' ODESOLVER EXAMPLE PROGRAM RESULTS FOR LSODEX PROBLEM'
.
,/1X)
END
*
*
*
*
SUBROUTINE TEST( ISOLVER )
*
*
* PURPOSE
*
* Testing subroutine ODESolver
*
* ARGUMENTS
*
* Input/Output Parameters
*
* ISOLVER (input) INTEGER
*
Indicates the nonlinear solver package to be used:
*
= 1: LSODE,
*
= 2: LSODA,
*
= 3: LSODES,
*
= 4: RADAU5,
*
= 5: DASSL,
*
= 6: DASPK,
*
= 7: DGELDA.
*
* METHOD
*
* REFERENCES
*
* CONTRIBUTORS
*
* REVISIONS
*
* -
*
* KEYWORDS
*
*
* ******************************************************************
* .. Parameters ..
INTEGER LSODE_, LSODA_, LSODES_,
RADAU5_, DASSL_, DASPK_, DGELDA_
PARAMETER (LSODE_ = 1, LSODA_
= 2, LSODES_ = 3)
PARAMETER (RADAU5_ = 4, DASSL_ =
5, DASPK_ = 6)
PARAMETER (DGELDA_ = 7)
INTEGER
NIN, NOUT
PARAMETER
( NIN = 5, NOUT = 6 )
INTEGER
MD, ND, LPAR, LWORK
PARAMETER
( MD = 400, ND = 100, LPAR = 250,
$
LWORK = 650000 )
* .. Scalar Arguments ..
INTEGER ISOLVER
* .. Local Scalars ..
INTEGER
NEQN, NDISC, MLJAC, MUJAC, MLMAS, MUMAS
INTEGER
IWARN, INFO
DOUBLE PRECISION ATOL(MD), RTOL,
NORM
LOGICAL
NUMJAC, NUMMAS, CONSIS
* .. Local Arrays ..
CHARACTER FULLNM*40, PROBLM*8, TYPE*3
CHARACTER*9 ODEDER, ODEOUT, JACFX,
JACFU, JACFP
INTEGER
IND(MD), IPAR(LPAR), IWORK(LWORK)
DOUBLE PRECISION T(0:ND), RPAR(LPAR),
DWORK(LWORK)
DOUBLE PRECISION X(MD), XPRIME(MD),
U(MD), Y(MD)
* .. External Functions ..
DOUBLE PRECISION DNRM2
EXTERNAL
DNRM2
* .. External Subroutines ..
EXTERNAL
PLSODEX,ILSODEX,SLSODEX
EXTERNAL
DAXPY
* .. Executable Statements ..
*
DO 20 I=1,NEQN
U(I)=0D0
Y(I)=0D0
20 CONTINUE
DO 40 I=1,LPAR
IPAR(I)=0
RPAR(I)=0D0
40 CONTINUE
DO 60 I=1,LWORK
IWORK(I)=0
DWORK(I)=0D0
60 CONTINUE
IPAR(2)=1
* Get the problem dependent parameters.
RPAR(1)=1D-3
IPAR(1)=0
ATOL(1)=1D-6
ATOL(2)=1D-10
ATOL(3)=1D-6
RTOL=1D-4
CALL PLSODEX(FULLNM,PROBLM,TYPE,NEQN,NDISC,T,NUMJAC,MLJAC,
$
MUJAC,NUMMAS,MLMAS,MUMAS,IND)
CALL ILSODEX(NEQN,T(0),X,XPRIME,CONSIS)
CALL SLSODEX(NEQN,T(1),XPRIME)
IF ( TYPE.NE.'ODE' ) THEN
WRITE ( NOUT,
FMT = 99998 )
ELSE
WRITE ( NOUT,
FMT = 99997 ) FULLNM, PROBLM, TYPE, ISOLVER
IF ( NUMJAC )
THEN
IPAR(3)=0
ELSE
IPAR(3)=1
END IF
IPAR(6)=MLJAC
IPAR(7)=MUJAC
ODEDER=''
ODEOUT=''
JACFX=''
JACFU=''
JACFP=''
CALL ODESolver(
ISOLVER, ODEDER, ODEOUT, JACFX, JACFU, JACFP,
$
NEQN, NEQN, NEQN, T(0), T(1), X, U, Y,
$
IPAR, RPAR, RTOL, ATOL,
$
IWORK, LWORK, DWORK, LWORK, IWARN, INFO )
IF ( INFO.NE.0
) THEN
WRITE ( NOUT, FMT = 99996 ) INFO
ELSE
IF ( IWARN.NE.0 ) THEN
WRITE ( NOUT, FMT = 99995 ) IWARN
ENDIF
IF ( NEQN .LE. 10 ) THEN
WRITE ( NOUT, FMT = 99994 )
DO 80 I=1,NEQN
WRITE ( NOUT, FMT = 99993 ) X(I), XPRIME(I)
80
CONTINUE
END IF
NORM=DNRM2(NEQN,XPRIME,1)
IF ( NORM.EQ.0D0 ) THEN
NORM=1D0
END IF
CALL DAXPY(NEQN,-1D0,X,1,XPRIME,1)
NORM=DNRM2(NEQN,XPRIME,1)/NORM
WRITE ( NOUT, FMT = 99992 ) NORM
END IF
END IF
*
99998 FORMAT (' ERROR: This test is only intended for ODE problems')
99997 FORMAT (' ',A,' (',A,' , ',A,') with SOLVER ',I2)
99996 FORMAT (' INFO on exit from ODESolver = ',I3)
99995 FORMAT (' IWARN on exit from ODESolver = ',I3)
99994 FORMAT (' Solution: (calculated) (reference)')
99993 FORMAT (F,F)
99992 FORMAT (' Relative error comparing with the reference solution:'
$
,E,/1X)
* *** Last line of TEST ***
END
SUBROUTINE ODEDER_( NX, NU, T, X,
U, RPAR, IPAR, F, INFO )
*
*
* PURPOSE
*
* Interface routine between ODESolver and
the problem function FEVAL
*
* ARGUMENTS
*
* Input/Output Parameters
*
* NX
(input) INTEGER
*
Dimension of the state vector.
*
* NU
(input) INTEGER
*
Dimension of the input vector.
*
* T
(input) INTEGER
*
The time point where the function is evaluated.
*
* X
(input) DOUBLE PRECISION array, dimension (NX)
*
Array containing the state variables.
*
* U
(input) DOUBLE PRECISION array, dimension (NU)
*
Array containing the input values.
*
* RPAR (input/output)
DOUBLE PRECISION array
*
Array for communication between the driver and FEVAL.
*
* IPAR (input/output)
INTEGER array
*
Array for communication between the driver and FEVAL.
*
* F
(output) DOUBLE PRECISION array, dimension (NX)
*
The resulting function value f(T,X).
*
* Error Indicator
*
* INFO INTEGER
*
Return values of error from FEVAL or 100 in case
*
a bad problem was choosen.
*
* METHOD
*
* REFERENCES
*
* CONTRIBUTORS
*
* REVISIONS
*
* -
*
* KEYWORDS
*
*
* ******************************************************************
*
* .. Scalar Arguments ..
INTEGER
NX, NU, INFO
DOUBLE PRECISION T
* .. Array Arguments ..
INTEGER
IPAR(*)
DOUBLE PRECISION X(NX), U(NU), RPAR(*),
F(NX)
* .. External Subroutines ..
EXTERNAL
FLSODEX
* .. Executable Statements ..
CALL FLSODEX(NX,T,X,X,F,INFO,RPAR,IPAR)
* *** Last line of ODEDER_ ***
END
SUBROUTINE JACFX_( NX, DUMMY,
LDFX, T, X, DUMMY2, RPAR, IPAR, FX,
$
INFO )
*
*
* PURPOSE
*
* Interface routine between ODESolver and
the problem function JEVAL
*
* ARGUMENTS
*
* Input/Output Parameters
*
* NX
(input) INTEGER
*
Dimension of the state vector.
*
* DUMMY (input) INTEGER
*
* LDFX (input)
INTEGER
*
The leading dimension of the array FX.
*
* T
(input) INTEGER
*
The time point where the derivative is evaluated.
*
* X
(input) DOUBLE PRECISION array, dimension (NX)
*
Array containing the state variables.
*
* DUMMY2 (input) DOUBLE PRECISION
*
* RPAR (input/output)
DOUBLE PRECISION array
*
Array for communication between the driver and FEVAL.
*
* IPAR (input/output)
INTEGER array
*
Array for communication between the driver and FEVAL.
*
* FX
(output) DOUBLE PRECISION array, dimension (LDFX,NX)
*
The array with the resulting Jacobian matrix.
*
* Error Indicator
*
* INFO INTEGER
*
Return values of error from JEVAL or 100 in case
*
a bad problem was choosen.
*
* METHOD
*
* REFERENCES
*
* CONTRIBUTORS
*
* REVISIONS
*
* -
*
* KEYWORDS
*
*
* ******************************************************************
*
* .. Scalar Arguments ..
INTEGER
NX, DUMMY, LDFX, INFO
DOUBLE PRECISION T
* .. Array Arguments ..
INTEGER
IPAR(*)
DOUBLE PRECISION X(NX), DUMMY2(*),
RPAR(*), FX(NX)
* .. External Subroutines ..
EXTERNAL
JLSODEX
* .. Executable Statements ..
CALL JLSODEX(LDFX,NX,T,X,X,FX,INFO,RPAR,IPAR)
* *** Last line of JACFX_ ***
END
Program Data
No data requiredProgram Results
ODESOLVER EXAMPLE PROGRAM RESULTS Problem: LSODEX Solver: LSODE IWARN on exit from ODESolver = 1 Solution: (calculated) 8.287534436182735E-08 3.329129749822125E-13 1.118553835127275E-07
Return to index
SLICOT-SLICOT-Reference-a037f7e/doc/SB01BD.html 0000664 0000000 0000000 00000037574 14560147231 0020345 0 ustar 00root root 0000000 0000000
Purpose
To determine the state feedback matrix F for a given system (A,B) such that the closed-loop state matrix A+B*F has specified eigenvalues.Specification
SUBROUTINE SB01BD( DICO, N, M, NP, ALPHA, A, LDA, B, LDB, WR, WI,
$ NFP, NAP, NUP, F, LDF, Z, LDZ, TOL, DWORK,
$ LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO
INTEGER INFO, IWARN, LDA, LDB, LDF, LDWORK, LDZ, M, N,
$ NAP, NFP, NP, NUP
DOUBLE PRECISION ALPHA, TOL
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), F(LDF,*),
$ WI(*), WR(*), Z(LDZ,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
Input/Output Parameters
N (input) INTEGER
The dimension of the state vector, i.e. the order of the
matrix A, and also the number of rows of the matrix B and
the number of columns of the matrix F. N >= 0.
M (input) INTEGER
The dimension of input vector, i.e. the number of columns
of the matrix B and the number of rows of the matrix F.
M >= 0.
NP (input) INTEGER
The number of given eigenvalues. At most N eigenvalues
can be assigned. 0 <= NP.
ALPHA (input) DOUBLE PRECISION
Specifies the maximum admissible value, either for real
parts, if DICO = 'C', or for moduli, if DICO = 'D',
of the eigenvalues of A which will not be modified by
the eigenvalue assignment algorithm.
ALPHA >= 0 if DICO = 'D'.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A.
On exit, the leading N-by-N part of this array contains
the matrix Z'*(A+B*F)*Z in a real Schur form.
The leading NFP-by-NFP diagonal block of A corresponds
to the fixed (unmodified) eigenvalues having real parts
less than ALPHA, if DICO = 'C', or moduli less than ALPHA,
if DICO = 'D'. The trailing NUP-by-NUP diagonal block of A
corresponds to the uncontrollable eigenvalues detected by
the eigenvalue assignment algorithm. The elements under
the first subdiagonal are set to zero.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
input/state matrix.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
WR,WI (input/output) DOUBLE PRECISION array, dimension (NP)
On entry, these arrays must contain the real and imaginary
parts, respectively, of the desired eigenvalues of the
closed-loop system state-matrix A+B*F. The eigenvalues
can be unordered, except that complex conjugate pairs
must appear consecutively in these arrays.
On exit, if INFO = 0, the leading NAP elements of these
arrays contain the real and imaginary parts, respectively,
of the assigned eigenvalues. The trailing NP-NAP elements
contain the unassigned eigenvalues.
NFP (output) INTEGER
The number of eigenvalues of A having real parts less than
ALPHA, if DICO = 'C', or moduli less than ALPHA, if
DICO = 'D'. These eigenvalues are not modified by the
eigenvalue assignment algorithm.
NAP (output) INTEGER
The number of assigned eigenvalues. If INFO = 0 on exit,
then NAP = N-NFP-NUP.
NUP (output) INTEGER
The number of uncontrollable eigenvalues detected by the
eigenvalue assignment algorithm (see METHOD).
F (output) DOUBLE PRECISION array, dimension (LDF,N)
The leading M-by-N part of this array contains the state
feedback F, which assigns NAP closed-loop eigenvalues and
keeps unaltered N-NAP open-loop eigenvalues.
LDF INTEGER
The leading dimension of array F. LDF >= MAX(1,M).
Z (output) DOUBLE PRECISION array, dimension (LDZ,N)
The leading N-by-N part of this array contains the
orthogonal matrix Z which reduces the closed-loop
system state matrix A + B*F to upper real Schur form.
LDZ INTEGER
The leading dimension of array Z. LDZ >= MAX(1,N).
Tolerances
TOL DOUBLE PRECISION
The absolute tolerance level below which the elements of A
or B are considered zero (used for controllability tests).
If the user sets TOL <= 0, then the default tolerance
TOL = N * EPS * max(NORM(A),NORM(B)) is used, where EPS is
the machine precision (see LAPACK Library routine DLAMCH)
and NORM(A) denotes the 1-norm of A.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The dimension of working array DWORK.
LDWORK >= MAX( 1,5*M,5*N,2*N+4*M ).
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= K: K violations of the numerical stability condition
NORM(F) <= 100*NORM(A)/NORM(B) occured during the
assignment of eigenvalues.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction of A to a real Schur form failed;
= 2: a failure was detected during the ordering of the
real Schur form of A, or in the iterative process
for reordering the eigenvalues of Z'*(A + B*F)*Z
along the diagonal.
= 3: the number of eigenvalues to be assigned is less
than the number of possibly assignable eigenvalues;
NAP eigenvalues have been properly assigned,
but some assignable eigenvalues remain unmodified.
= 4: an attempt is made to place a complex conjugate
pair on the location of a real eigenvalue. This
situation can only appear when N-NFP is odd,
NP > N-NFP-NUP is even, and for the last real
eigenvalue to be modified there exists no available
real eigenvalue to be assigned. However, NAP
eigenvalues have been already properly assigned.
Method
SB01BD is based on the factorization algorithm of [1].
Given the matrices A and B of dimensions N-by-N and N-by-M,
respectively, this subroutine constructs an M-by-N matrix F such
that A + BF has eigenvalues as follows.
Let NFP eigenvalues of A have real parts less than ALPHA, if
DICO = 'C', or moduli less then ALPHA, if DICO = 'D'. Then:
1) If the pair (A,B) is controllable, then A + B*F has
NAP = MIN(NP,N-NFP) eigenvalues assigned from those specified
by WR + j*WI and N-NAP unmodified eigenvalues;
2) If the pair (A,B) is uncontrollable, then the number of
assigned eigenvalues NAP satifies generally the condition
NAP <= MIN(NP,N-NFP).
At the beginning of the algorithm, F = 0 and the matrix A is
reduced to an ordered real Schur form by separating its spectrum
in two parts. The leading NFP-by-NFP part of the Schur form of
A corresponds to the eigenvalues which will not be modified.
These eigenvalues have real parts less than ALPHA, if
DICO = 'C', or moduli less than ALPHA, if DICO = 'D'.
The performed orthogonal transformations are accumulated in Z.
After this preliminary reduction, the algorithm proceeds
recursively.
Let F be the feedback matrix at the beginning of a typical step i.
At each step of the algorithm one real eigenvalue or two complex
conjugate eigenvalues are placed by a feedback Fi of rank 1 or
rank 2, respectively. Since the feedback Fi affects only the
last 1 or 2 columns of Z'*(A+B*F)*Z, the matrix Z'*(A+B*F+B*Fi)*Z
therefore remains in real Schur form. The assigned eigenvalue(s)
is (are) then moved to another diagonal position of the real
Schur form using reordering techniques and a new block is
transfered in the last diagonal position. The feedback matrix F
is updated as F <-- F + Fi. The eigenvalue(s) to be assigned at
each step is (are) chosen such that the norm of each Fi is
minimized.
If uncontrollable eigenvalues are encountered in the last diagonal
position of the real Schur matrix Z'*(A+B*F)*Z, the algorithm
deflates them at the bottom of the real Schur form and redefines
accordingly the position of the "last" block.
Note: Not all uncontrollable eigenvalues of the pair (A,B) are
necessarily detected by the eigenvalue assignment algorithm.
Undetected uncontrollable eigenvalues may exist if NFP > 0 and/or
NP < N-NFP.
References
[1] Varga A.
A Schur method for pole assignment.
IEEE Trans. Autom. Control, Vol. AC-26, pp. 517-519, 1981.
Numerical Aspects
3 The algorithm requires no more than 14N floating point operations. Although no proof of numerical stability is known, the algorithm has always been observed to yield reliable numerical results.Further Comments
NoneExample
Program Text
* SB01BD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX
PARAMETER ( NMAX = 20, MMAX = 20 )
INTEGER LDA, LDB, LDF, LDZ
PARAMETER ( LDA = NMAX, LDB = NMAX, LDF = MMAX,
$ LDZ = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( 5*MMAX,5*NMAX,2*NMAX+4*MMAX ) )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* .. Local Scalars ..
DOUBLE PRECISION ALPHA, ANORM, NRM, TOL
INTEGER I, INFO, IWARN, J, M, N, NAP, NFP, NP, NUP
CHARACTER*1 DICO
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), AIN(LDA,NMAX), B(LDB,MMAX),
$ DWORK(LDWORK), F(LDF,NMAX), WI(NMAX), WR(NMAX),
$ Z(LDZ,NMAX), ZTA(LDZ,NMAX)
C .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL DLAMCH, DLANGE, LSAME
* .. External Subroutines ..
EXTERNAL DGEMM, DLACPY, MB03QX, SB01BD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, NP, ALPHA, TOL, DICO
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF( NP.LT.0 .OR. NP.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) NP
ELSE
DO 10 I = 1, NP
READ ( NIN, FMT = * ) WR(I), WI(I)
10 CONTINUE
* Perform "eigenvalue assignment" to compute F.
CALL DLACPY( 'G', N, N, A, LDA, AIN, LDA )
CALL SB01BD( DICO, N, M, NP, ALPHA, A, LDA, B, LDB,
$ WR, WI, NFP, NAP, NUP, F, LDF, Z, LDZ,
$ TOL, DWORK, LDWORK, IWARN, INFO )
*
IF ( INFO.NE.0 .AND. INFO.LT.3 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
IF ( INFO .NE. 0 ) WRITE ( NOUT, FMT = 99997 ) INFO
IF ( IWARN .NE. 0 ) WRITE ( NOUT, FMT = 99991 ) IWARN
WRITE ( NOUT, FMT = 99990 ) NAP
WRITE ( NOUT, FMT = 99989 ) NFP
WRITE ( NOUT, FMT = 99988 ) NUP
WRITE ( NOUT, FMT = 99996 )
DO 60 I = 1, M
WRITE ( NOUT, FMT = 99995 ) ( F(I,J), J = 1,N )
60 CONTINUE
CALL MB03QX( N, A, LDA, WR, WI, INFO )
WRITE ( NOUT, FMT = 99998 ) ( WR(I), WI(I), I = 1,N )
* Compute NORM (Z*Aout*Z'-(A+B*F)) / (eps*NORM(A))
ANORM = DLANGE( 'F', N, N, AIN, LDA, DWORK )
CALL DGEMM( 'N', 'N', N, N, M, ONE, B, LDB, F, LDF,
$ ONE, AIN, LDA )
CALL DGEMM( 'N', 'N', N, N, N, ONE, Z, LDZ, A, LDA,
$ ZERO, ZTA, LDZ )
CALL DGEMM( 'N', 'T', N, N, N, ONE, ZTA, LDZ, Z, LDZ,
$ -ONE, AIN, LDA )
NRM = DLANGE( 'F', N, N, AIN, LDA, DWORK ) /
$ ( DLAMCH( 'E' )*ANORM )
WRITE ( NOUT, FMT = 99987 ) NRM
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB01BD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (/,' The eigenvalues of closed-loop matrix A+B*F',/
$ ( ' ( ',F8.4,',',F8.4,' )' ) )
99997 FORMAT (' INFO on exit from SB01BD = ',I2)
99996 FORMAT (/,' The state feedback matrix F is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' M is out of range.',/' M = ',I5)
99992 FORMAT (/' NP is out of range.',/' NP = ',I5)
99991 FORMAT (/' IWARN on exit from SB01BD = ', I2)
99990 FORMAT ( ' Number of assigned eigenvalues: NAP = ', I2 )
99989 FORMAT ( ' Number of fixed eigenvalues: NFP = ', I2)
99988 FORMAT ( ' Number of uncontrollable poles: NUP = ', I2)
99987 FORMAT (/,' NORM(A+B*F - Z*Aout*Z'') / (eps*NORM(A)) =',1PD12.5)
END
Program Data
SB01BD EXAMPLE PROGRAM DATA 4 2 2 -.4 1.E-8 C -6.8000 0.0000 -207.0000 0.0000 1.0000 0.0000 0.0000 0.0000 43.2000 0.0000 0.0000 -4.2000 0.0000 0.0000 1.0000 0.0000 5.6400 0.0000 0.0000 0.0000 0.0000 1.1800 0.0000 0.0000 -0.5000 0.1500 -0.5000 -0.1500 -2.0000 0.0000 -0.4000 0.0000Program Results
SB01BD EXAMPLE PROGRAM RESULTS Number of assigned eigenvalues: NAP = 2 Number of fixed eigenvalues: NFP = 2 Number of uncontrollable poles: NUP = 0 The state feedback matrix F is -0.0876 -4.2138 0.0837 -18.1412 -0.0233 18.2483 -0.4259 -4.8120 The eigenvalues of closed-loop matrix A+B*F ( -3.3984, 94.5253 ) ( -3.3984,-94.5253 ) ( -0.5000, 0.1500 ) ( -0.5000, -0.1500 ) NORM(A+B*F - Z*Aout*Z') / (eps*NORM(A)) = 1.03505D+01
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB01BX.html 0000664 0000000 0000000 00000006510 14560147231 0020353 0 ustar 00root root 0000000 0000000
Purpose
To choose a real eigenvalue or a pair of complex conjugate eigenvalues at "minimal" distance to a given real or complex value.Specification
SUBROUTINE SB01BX( REIG, N, XR, XI, WR, WI, S, P )
C .. Scalar Arguments ..
LOGICAL REIG
INTEGER N
DOUBLE PRECISION P, S, XI ,XR
C .. Array Arguments ..
DOUBLE PRECISION WI(*), WR(*)
Arguments
Mode Parameters
REIG LOGICAL
Specifies the type of eigenvalues as follows:
= .TRUE., a real eigenvalue is to be selected;
= .FALSE., a pair of complex eigenvalues is to be
selected.
Input/Output Parameters
N (input) INTEGER
The number of eigenvalues contained in the arrays WR
and WI. N >= 1.
XR,XI (input) DOUBLE PRECISION
If REIG = .TRUE., XR must contain the real value and XI
is assumed zero and therefore not referenced.
If REIG = .FALSE., XR must contain the real part and XI
the imaginary part, respectively, of the complex value.
WR,WI (input/output) DOUBLE PRECISION array, dimension (N)
On entry, if REIG = .TRUE., WR must contain the real
eigenvalues from which an eigenvalue at minimal distance
to XR is to be selected. In this case, WI is considered
zero and therefore not referenced.
On entry, if REIG = .FALSE., WR and WI must contain the
real and imaginary parts, respectively, of the eigenvalues
from which a pair of complex conjugate eigenvalues at
minimal "distance" to XR + jXI is to be selected.
The eigenvalues of each pair of complex conjugate
eigenvalues must appear consecutively.
On exit, the elements of these arrays are reordered such
that the selected eigenvalue(s) is (are) found in the
last element(s) of these arrays.
S,P (output) DOUBLE PRECISION
If REIG = .TRUE., S (and also P) contains the value of
the selected real eigenvalue.
If REIG = .FALSE., S and P contain the sum and product,
respectively, of the selected complex conjugate pair of
eigenvalues.
Further Comments
For efficiency reasons, |x| + |y| is used for a complex number x + jy, instead of its modulus.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve an N-by-N pole placement problem for the simple cases N = 1 or N = 2: given the N-by-N matrix A and N-by-M matrix B, construct an M-by-N matrix F such that A + B*F has prescribed eigenvalues. These eigenvalues are specified by their sum S and product P (if N = 2). The resulting F has minimum Frobenius norm.Specification
SUBROUTINE SB01BY( N, M, S, P, A, B, F, TOL, DWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, M, N
DOUBLE PRECISION P, S, TOL
C .. Array Arguments ..
DOUBLE PRECISION A(N,*), B(N,*), DWORK(*), F(M,*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix A and also the number of rows of
the matrix B and the number of columns of the matrix F.
N is either 1, if a single real eigenvalue is prescribed
or 2, if a complex conjugate pair or a set of two real
eigenvalues are prescribed.
M (input) INTEGER
The number of columns of the matrix B and also the number
of rows of the matrix F. M >= 1.
S (input) DOUBLE PRECISION
The sum of the prescribed eigenvalues if N = 2 or the
value of prescribed eigenvalue if N = 1.
P (input) DOUBLE PRECISION
The product of the prescribed eigenvalues if N = 2.
Not referenced if N = 1.
A (input/output) DOUBLE PRECISION array, dimension (N,N)
On entry, this array must contain the N-by-N state
dynamics matrix whose eigenvalues have to be moved to
prescribed locations.
On exit, this array contains no useful information.
B (input/output) DOUBLE PRECISION array, dimension (N,M)
On entry, this array must contain the N-by-M input/state
matrix B.
On exit, this array contains no useful information.
F (output) DOUBLE PRECISION array, dimension (M,N)
The state feedback matrix F which assigns one pole or two
poles of the closed-loop matrix A + B*F.
If N = 2 and the pair (A,B) is not controllable
(INFO = 1), then F(1,1) and F(1,2) contain the elements of
an orthogonal rotation which can be used to remove the
uncontrollable part of the pair (A,B).
Tolerances
TOL DOUBLE PRECISION
The absolute tolerance level below which the elements of A
and B are considered zero (used for controllability test).
Workspace
DWORK DOUBLE PRECISION array, dimension (M)Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: if uncontrollability of the pair (A,B) is detected.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute for a controllable matrix pair ( A, B ) a matrix G such that the matrix A - B*G has the desired eigenstructure, specified by desired eigenvalues and free eigenvector elements. The pair ( A, B ) should be given in orthogonal canonical form as returned by the SLICOT Library routine AB01ND.Specification
SUBROUTINE SB01DD( N, M, INDCON, A, LDA, B, LDB, NBLK, WR, WI,
$ Z, LDZ, Y, COUNT, G, LDG, TOL, IWORK, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER COUNT, INDCON, INFO, LDA, LDB, LDG, LDWORK,
$ LDZ, M, N
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK( * ), NBLK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DWORK( * ),
$ G( LDG, * ), WI( * ), WR( * ), Y( * ),
$ Z( LDZ, * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix A and the number of rows of the
matrix B. N >= 0.
M (input) INTEGER
The number of columns of the matrix B. M >= 0.
INDCON (input) INTEGER
The controllability index of the pair ( A, B ).
0 <= INDCON <= N.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the N-by-N matrix A in orthogonal canonical form,
as returned by SLICOT Library routine AB01ND.
On exit, the leading N-by-N part of this array contains
the real Schur form of the matrix A - B*G.
The elements below the real Schur form of A are set to
zero.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the N-by-M matrix B in orthogonal canonical form,
as returned by SLICOT Library routine AB01ND.
On exit, the leading N-by-M part of this array contains
the transformed matrix B.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
NBLK (input) INTEGER array, dimension (N)
The leading INDCON elements of this array must contain the
orders of the diagonal blocks in the orthogonal canonical
form of A, as returned by SLICOT Library routine AB01ND.
The values of these elements must satisfy the following
conditions:
NBLK(1) >= NBLK(2) >= ... >= NBLK(INDCON),
NBLK(1) + NBLK(2) + ... + NBLK(INDCON) = N.
WR (input) DOUBLE PRECISION array, dimension (N)
WI (input) DOUBLE PRECISION array, dimension (N)
These arrays must contain the real and imaginary parts,
respectively, of the desired poles of the closed-loop
system, i.e., the eigenvalues of A - B*G. The poles can be
unordered, except that complex conjugate pairs of poles
must appear consecutively.
The elements of WI for complex eigenvalues are modified
internally, but restored on exit.
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
On entry, the leading N-by-N part of this array must
contain the orthogonal matrix Z generated by SLICOT
Library routine AB01ND in the reduction of ( A, B ) to
orthogonal canonical form.
On exit, the leading N-by-N part of this array contains
the orthogonal transformation matrix which reduces A - B*G
to real Schur form.
LDZ INTEGER
The leading dimension of the array Z. LDZ >= max(1,N).
Y (input) DOUBLE PRECISION array, dimension (M*N)
Y contains elements which are used as free parameters
in the eigenstructure design. The values of these
parameters are often set by an external optimization
procedure.
COUNT (output) INTEGER
The actual number of elements in Y used as free
eigenvector and feedback matrix elements in the
eigenstructure design.
G (output) DOUBLE PRECISION array, dimension (LDG,N)
The leading M-by-N part of this array contains the
feedback matrix which assigns the desired eigenstructure
of A - B*G.
LDG INTEGER
The leading dimension of the array G. LDG >= max(1,M).
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in rank determination when
transforming (A, B). If the user sets TOL > 0, then
the given value of TOL is used as a lower bound for the
reciprocal condition number (see the description of the
argument RCOND in the SLICOT routine MB03OD); a
(sub)matrix whose estimated condition number is less than
1/TOL is considered to be of full rank. If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
defined by TOLDEF = N*N*EPS, is used instead, where
EPS is the machine precision (see LAPACK Library routine
DLAMCH).
Workspace
IWORK INTEGER array, dimension (M)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(M*N,M*M+2*N+4*M+1).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the pair ( A, B ) is not controllable or the free
parameters are not set appropriately.
Method
The routine implements the method proposed in [1], [2].References
[1] Petkov, P.Hr., Konstantinov, M.M., Gu, D.W. and
Postlethwaite, I.
Optimal pole assignment design of linear multi-input systems.
Report 96-11, Department of Engineering, Leicester University,
1996.
[2] Petkov, P.Hr., Christov, N.D. and Konstantinov, M.M.
A computational algorithm for pole assignment of linear multi
input systems. IEEE Trans. Automatic Control, vol. AC-31,
pp. 1044-1047, 1986.
Numerical Aspects
The method implemented is backward stable.Further Comments
The eigenvalues of the real Schur form matrix As, returned in the
array A, are very close to the desired eigenvalues WR+WI*i.
However, the eigenvalues of the closed-loop matrix A - B*G,
computed by the QR algorithm using the matrices A and B, given on
entry, may be far from WR+WI*i, although the relative error
norm( Z'*(A - B*G)*Z - As )/norm( As )
is close to machine accuracy. This may happen when the eigenvalue
problem for the matrix A - B*G is ill-conditioned.
Example
Program Text
* SB01DD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX
PARAMETER ( NMAX = 20, MMAX = 20 )
INTEGER LDA, LDB, LDG, LDZ
PARAMETER ( LDA = NMAX, LDB = NMAX, LDG = MMAX,
$ LDZ = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( 3*NMAX, MMAX*NMAX,
$ MMAX*MMAX + 2*NMAX + 4*MMAX + 1 )
$ )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER COUNT, I, INDCON, INFO1, INFO2, J, M, N, NCONT
CHARACTER*1 JOBZ
* .. Local Arrays ..
INTEGER IWORK(MMAX), NBLK(NMAX)
DOUBLE PRECISION A(LDA,NMAX), B(NMAX,MMAX), DWORK(LDWORK),
$ G(LDG,NMAX), WI(NMAX), WR(NMAX), Y(MMAX*NMAX),
$ Z(LDZ,NMAX)
* .. External Subroutines ..
EXTERNAL AB01ND, SB01DD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, TOL, JOBZ
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) M
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
READ ( NIN, FMT = * ) ( WR(I), I = 1,N )
READ ( NIN, FMT = * ) ( WI(I), I = 1,N )
READ ( NIN, FMT = * ) ( Y(I), I = 1,M*N )
* First reduce the given system to canonical form.
CALL AB01ND( JOBZ, N, M, A, LDA, B, LDB, NCONT, INDCON,
$ NBLK, Z, LDZ, DWORK, TOL, IWORK, DWORK(N+1),
$ LDWORK-N, INFO1 )
*
IF ( INFO1.EQ.0 ) THEN
* Find the state feedback matrix G.
CALL SB01DD( N, M, INDCON, A, LDA, B, LDB, NBLK, WR, WI,
$ Z, LDZ, Y, COUNT, G, LDG, TOL, IWORK, DWORK,
$ LDWORK, INFO2 )
*
IF ( INFO2.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO2
ELSE
WRITE ( NOUT, FMT = 99996 )
DO 10 I = 1, M
WRITE ( NOUT, FMT = 99995 ) ( G(I,J), J = 1,N )
10 CONTINUE
END IF
ELSE
WRITE ( NOUT, FMT = 99998 ) INFO1
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB01DD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB01ND =',I2)
99997 FORMAT (' INFO on exit from SB01DD =',I2)
99996 FORMAT (' The state feedback matrix G is')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' M is out of range.',/' M = ',I5)
END
Program Data
SB01DD EXAMPLE PROGRAM DATA 4 2 0.0 I -1.0 0.0 2.0 -3.0 1.0 -4.0 3.0 -1.0 0.0 2.0 4.0 -5.0 0.0 0.0 -1.0 -2.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 -1.0 -1.0 -1.0 -1.0 0.0 0.0 0.0 0.0 1.0 2.0 2.0 1.0 -1.0 -2.0 3.0 1.0Program Results
SB01DD EXAMPLE PROGRAM RESULTS The state feedback matrix G is -5.2339 3.1725 -15.7885 21.7043 -1.6022 0.8504 -5.1914 6.2339
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB01FY.html 0000664 0000000 0000000 00000007764 14560147231 0020374 0 ustar 00root root 0000000 0000000
Purpose
To compute the inner denominator of a right-coprime factorization of a system of order N, where N is either 1 or 2. Specifically, given the N-by-N unstable system state matrix A and the N-by-M system input matrix B, an M-by-N state-feedback matrix F and an M-by-M matrix V are constructed, such that the system (A + B*F, B*V, F, V) is inner.Specification
SUBROUTINE SB01FY( DISCR, N, M, A, LDA, B, LDB, F, LDF, V, LDV,
$ INFO )
C .. Scalar Arguments ..
LOGICAL DISCR
INTEGER INFO, LDA, LDB, LDF, LDV, M, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), F(LDF,*), V(LDV,*)
Arguments
Mode Parameters
DISCR LOGICAL
Specifies the type of system as follows:
= .FALSE.: continuous-time system;
= .TRUE. : discrete-time system.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A and also the number of rows of
the matrix B and the number of columns of the matrix F.
N is either 1 or 2.
M (input) INTEGER
The number of columns of the matrices B and V, and also
the number of rows of the matrix F. M >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A whose eigenvalues must have positive
real parts if DISCR = .FALSE. or moduli greater than unity
if DISCR = .TRUE..
LDA INTEGER
The leading dimension of array A. LDA >= N.
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
system input matrix B.
LDB INTEGER
The leading dimension of array B. LDB >= N.
F (output) DOUBLE PRECISION array, dimension (LDF,N)
The leading M-by-N part of this array contains the state-
feedback matrix F which assigns one eigenvalue (if N = 1)
or two eigenvalues (if N = 2) of the matrix A + B*F in
symmetric positions with respect to the imaginary axis
(if DISCR = .FALSE.) or the unit circle (if
DISCR = .TRUE.).
LDF INTEGER
The leading dimension of array F. LDF >= MAX(1,M).
V (output) DOUBLE PRECISION array, dimension (LDV,M)
The leading M-by-M upper triangular part of this array
contains the input/output matrix V of the resulting inner
system in upper triangular form.
If DISCR = .FALSE., the resulting V is an identity matrix.
LDV INTEGER
The leading dimension of array V. LDF >= MAX(1,M).
Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: if uncontrollability of the pair (A,B) is detected;
= 2: if A is stable or at the stability limit;
= 3: if N = 2 and A has a pair of real eigenvalues.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To determine the one-dimensional state feedback matrix G of the
linear time-invariant single-input system
dX/dt = A * X + B * U,
where A is an NCONT-by-NCONT matrix and B is an NCONT element
vector such that the closed-loop system
dX/dt = (A - B * G) * X
has desired poles. The system must be preliminarily reduced
to orthogonal canonical form using the SLICOT Library routine
AB01MD.
Specification
SUBROUTINE SB01MD( NCONT, N, A, LDA, B, WR, WI, Z, LDZ, G, DWORK,
$ INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDZ, N, NCONT
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(*), DWORK(*), G(*), WI(*), WR(*),
$ Z(LDZ,*)
Arguments
Input/Output Parameters
NCONT (input) INTEGER
The order of the matrix A as produced by SLICOT Library
routine AB01MD. NCONT >= 0.
N (input) INTEGER
The order of the matrix Z. N >= NCONT.
A (input/output) DOUBLE PRECISION array, dimension
(LDA,NCONT)
On entry, the leading NCONT-by-NCONT part of this array
must contain the canonical form of the state dynamics
matrix A as produced by SLICOT Library routine AB01MD.
On exit, the leading NCONT-by-NCONT part of this array
contains the upper quasi-triangular form S of the closed-
loop system matrix (A - B * G), that is triangular except
for possible 2-by-2 diagonal blocks.
(To reconstruct the closed-loop system matrix see
FURTHER COMMENTS below.)
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,NCONT).
B (input/output) DOUBLE PRECISION array, dimension (NCONT)
On entry, this array must contain the canonical form of
the input/state vector B as produced by SLICOT Library
routine AB01MD.
On exit, this array contains the transformed vector Z * B
of the closed-loop system.
WR (input) DOUBLE PRECISION array, dimension (NCONT)
WI (input) DOUBLE PRECISION array, dimension (NCONT)
These arrays must contain the real and imaginary parts,
respectively, of the desired poles of the closed-loop
system. The poles can be unordered, except that complex
conjugate pairs of poles must appear consecutively.
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
On entry, the leading N-by-N part of this array must
contain the orthogonal transformation matrix as produced
by SLICOT Library routine AB01MD, which reduces the system
to canonical form.
On exit, the leading NCONT-by-NCONT part of this array
contains the orthogonal matrix Z which reduces the closed-
loop system matrix (A - B * G) to upper quasi-triangular
form.
LDZ INTEGER
The leading dimension of array Z. LDZ >= MAX(1,N).
G (output) DOUBLE PRECISION array, dimension (NCONT)
This array contains the one-dimensional state feedback
matrix G of the original system.
Workspace
DWORK DOUBLE PRECISION array, dimension (3*NCONT)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The method is based on the orthogonal reduction of the closed-loop system matrix (A - B * G) to upper quasi-triangular form S whose 1-by-1 and 2-by-2 diagonal blocks correspond to the desired poles. That is, S = Z'*(A - B * G)*Z, where Z is an orthogonal matrix.References
[1] Petkov, P. Hr.
A Computational Algorithm for Pole Assignment of Linear
Single Input Systems.
Internal Report 81/2, Control Systems Research Group, School
of Electronic Engineering and Computer Science, Kingston
Polytechnic, 1981.
Numerical Aspects
3 The algorithm requires 0(NCONT ) operations and is backward stable.Further Comments
If required, the closed-loop system matrix (A - B * G) can be formed from the matrix product Z * S * Z' (where S and Z are the matrices output in arrays A and Z respectively).Example
Program Text
* SB01MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA, LDZ
PARAMETER ( LDA = NMAX, LDZ = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 3*NMAX )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO1, INFO2, J, N, NCONT
CHARACTER*1 JOBZ
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(NMAX), DWORK(LDWORK), G(NMAX),
$ WI(NMAX), WR(NMAX), Z(LDZ,NMAX)
* .. External Subroutines ..
EXTERNAL AB01MD, SB01MD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, TOL, JOBZ
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( B(I), I = 1,N )
READ ( NIN, FMT = * ) ( WR(I), I = 1,N )
READ ( NIN, FMT = * ) ( WI(I), I = 1,N )
* First reduce the given system to canonical form.
CALL AB01MD( JOBZ, N, A, LDA, B, NCONT, Z, LDZ, DWORK, TOL,
$ DWORK(N+1), LDWORK-N, INFO1 )
*
IF ( INFO1.EQ.0 ) THEN
* Find the one-dimensional state feedback matrix G.
CALL SB01MD( NCONT, N, A, LDA, B, WR, WI, Z, LDZ, G, DWORK,
$ INFO2 )
*
IF ( INFO2.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO2
ELSE
WRITE ( NOUT, FMT = 99996 ) ( G(I), I = 1,NCONT )
END IF
ELSE
WRITE ( NOUT, FMT = 99998 ) INFO1
END IF
END IF
STOP
*
99999 FORMAT (' SB01MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB01MD =',I2)
99997 FORMAT (' INFO on exit from SB01MD =',I2)
99996 FORMAT (' The one-dimensional state feedback matrix G is',
$ /20(1X,F8.4))
99995 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
SB01MD EXAMPLE PROGRAM DATA 4 0.0 I -1.0 0.0 2.0 -3.0 1.0 -4.0 3.0 -1.0 0.0 2.0 4.0 -5.0 0.0 0.0 -1.0 -2.0 1.0 0.0 0.0 0.0 -1.0 -1.0 -1.0 -1.0 0.0 0.0 0.0 0.0Program Results
SB01MD EXAMPLE PROGRAM RESULTS The one-dimensional state feedback matrix G is 1.0000 29.0000 93.0000 -76.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB02MD.html 0000664 0000000 0000000 00000036662 14560147231 0020356 0 ustar 00root root 0000000 0000000
Purpose
To solve for X either the continuous-time algebraic Riccati
equation
-1
Q + A'*X + X*A - X*B*R B'*X = 0 (1)
or the discrete-time algebraic Riccati equation
-1
X = A'*X*A - A'*X*B*(R + B'*X*B) B'*X*A + Q (2)
where A, B, Q and R are N-by-N, N-by-M, N-by-N and M-by-M matrices
respectively, with Q symmetric and R symmetric nonsingular; X is
an N-by-N symmetric matrix.
-1
The matrix G = B*R B' must be provided on input, instead of B and
R, that is, for instance, the continuous-time equation
Q + A'*X + X*A - X*G*X = 0 (3)
is solved, where G is an N-by-N symmetric matrix. SLICOT Library
routine SB02MT should be used to compute G, given B and R. SB02MT
also enables to solve Riccati equations corresponding to optimal
problems with coupling terms.
The routine also returns the computed values of the closed-loop
spectrum of the optimal system, i.e., the stable eigenvalues
lambda(1),...,lambda(N) of the corresponding Hamiltonian or
symplectic matrix associated to the optimal problem.
Specification
SUBROUTINE SB02MD( DICO, HINV, UPLO, SCAL, SORT, N, A, LDA, G,
$ LDG, Q, LDQ, RCOND, WR, WI, S, LDS, U, LDU,
$ IWORK, DWORK, LDWORK, BWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, HINV, SCAL, SORT, UPLO
INTEGER INFO, LDA, LDG, LDQ, LDS, LDU, LDWORK, N
DOUBLE PRECISION RCOND
C .. Array Arguments ..
LOGICAL BWORK(*)
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), DWORK(*), G(LDG,*), Q(LDQ,*),
$ S(LDS,*), U(LDU,*), WR(*), WI(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of Riccati equation to be solved as
follows:
= 'C': Equation (3), continuous-time case;
= 'D': Equation (2), discrete-time case.
HINV CHARACTER*1
If DICO = 'D', specifies which symplectic matrix is to be
constructed, as follows:
= 'D': The matrix H in (5) (see METHOD) is constructed;
= 'I': The inverse of the matrix H in (5) is constructed.
HINV is not used if DICO = 'C'.
UPLO CHARACTER*1
Specifies which triangle of the matrices G and Q is
stored, as follows:
= 'U': Upper triangle is stored;
= 'L': Lower triangle is stored.
SCAL CHARACTER*1
Specifies whether or not a scaling strategy should be
used, as follows:
= 'G': General scaling should be used;
= 'N': No scaling should be used.
SORT CHARACTER*1
Specifies which eigenvalues should be obtained in the top
of the Schur form, as follows:
= 'S': Stable eigenvalues come first;
= 'U': Unstable eigenvalues come first.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, Q, G and X. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the coefficient matrix A of the equation.
On exit, if DICO = 'D', and INFO = 0 or INFO > 1, the
-1
leading N-by-N part of this array contains the matrix A .
Otherwise, the array A is unchanged on exit.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
G (input) DOUBLE PRECISION array, dimension (LDG,N)
The leading N-by-N upper triangular part (if UPLO = 'U')
or lower triangular part (if UPLO = 'L') of this array
must contain the upper triangular part or lower triangular
part, respectively, of the symmetric matrix G.
The strictly lower triangular part (if UPLO = 'U') or
strictly upper triangular part (if UPLO = 'L') is not
referenced.
LDG INTEGER
The leading dimension of array G. LDG >= MAX(1,N).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, the leading N-by-N upper triangular part (if
UPLO = 'U') or lower triangular part (if UPLO = 'L') of
this array must contain the upper triangular part or lower
triangular part, respectively, of the symmetric matrix Q.
The strictly lower triangular part (if UPLO = 'U') or
strictly upper triangular part (if UPLO = 'L') is not
used.
On exit, if INFO = 0, the leading N-by-N part of this
array contains the solution matrix X of the problem.
LDQ INTEGER
The leading dimension of array N. LDQ >= MAX(1,N).
RCOND (output) DOUBLE PRECISION
An estimate of the reciprocal of the condition number (in
the 1-norm) of the N-th order system of algebraic
equations from which the solution matrix X is obtained.
WR (output) DOUBLE PRECISION array, dimension (2*N)
WI (output) DOUBLE PRECISION array, dimension (2*N)
If INFO = 0 or INFO = 5, these arrays contain the real and
imaginary parts, respectively, of the eigenvalues of the
2N-by-2N matrix S, ordered as specified by SORT (except
for the case HINV = 'D', when the order is opposite to
that specified by SORT). The leading N elements of these
arrays contain the closed-loop spectrum of the system
-1
matrix A - B*R *B'*X, if DICO = 'C', or of the matrix
-1
A - B*(R + B'*X*B) B'*X*A, if DICO = 'D'. Specifically,
lambda(k) = WR(k) + j*WI(k), for k = 1,2,...,N.
S (output) DOUBLE PRECISION array, dimension (LDS,2*N)
If INFO = 0 or INFO = 5, the leading 2N-by-2N part of this
array contains the ordered real Schur form S of the
Hamiltonian or symplectic matrix H. That is,
(S S )
( 11 12)
S = ( ),
(0 S )
( 22)
where S , S and S are N-by-N matrices.
11 12 22
LDS INTEGER
The leading dimension of array S. LDS >= MAX(1,2*N).
U (output) DOUBLE PRECISION array, dimension (LDU,2*N)
If INFO = 0 or INFO = 5, the leading 2N-by-2N part of this
array contains the transformation matrix U which reduces
the Hamiltonian or symplectic matrix H to the ordered real
Schur form S. That is,
(U U )
( 11 12)
U = ( ),
(U U )
( 21 22)
where U , U , U and U are N-by-N matrices.
11 12 21 22
LDU INTEGER
The leading dimension of array U. LDU >= MAX(1,2*N).
Workspace
IWORK INTEGER array, dimension (2*N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK and DWORK(2) returns the scaling factor used
(set to 1 if SCAL = 'N'), also set if INFO = 5;
if DICO = 'D', DWORK(3) returns the reciprocal condition
number of the given matrix A.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(2,6*N) if DICO = 'C';
LDWORK >= MAX(3,6*N) if DICO = 'D'.
For optimum performance LDWORK should be larger.
BWORK LOGICAL array, dimension (2*N)
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if matrix A is (numerically) singular in discrete-
time case;
= 2: if the Hamiltonian or symplectic matrix H cannot be
reduced to real Schur form;
= 3: if the real Schur form of the Hamiltonian or
symplectic matrix H cannot be appropriately ordered;
= 4: if the Hamiltonian or symplectic matrix H has less
than N stable eigenvalues;
= 5: if the N-th order system of linear algebraic
equations, from which the solution matrix X would
be obtained, is singular to working precision.
Method
The method used is the Schur vector approach proposed by Laub.
It is assumed that [A,B] is a stabilizable pair (where for (3) B
is any matrix such that B*B' = G with rank(B) = rank(G)), and
[E,A] is a detectable pair, where E is any matrix such that
E*E' = Q with rank(E) = rank(Q). Under these assumptions, any of
the algebraic Riccati equations (1)-(3) is known to have a unique
non-negative definite solution. See [2].
Now consider the 2N-by-2N Hamiltonian or symplectic matrix
( A -G )
H = ( ), (4)
(-Q -A'),
for continuous-time equation, and
-1 -1
( A A *G )
H = ( -1 -1 ), (5)
(Q*A A' + Q*A *G)
-1
for discrete-time equation, respectively, where G = B*R *B'.
The assumptions guarantee that H in (4) has no pure imaginary
eigenvalues, and H in (5) has no eigenvalues on the unit circle.
If Y is an N-by-N matrix then there exists an orthogonal matrix U
such that U'*Y*U is an upper quasi-triangular matrix. Moreover, U
can be chosen so that the 2-by-2 and 1-by-1 diagonal blocks
(corresponding to the complex conjugate eigenvalues and real
eigenvalues respectively) appear in any desired order. This is the
ordered real Schur form. Thus, we can find an orthogonal
similarity transformation U which puts (4) or (5) in ordered real
Schur form
U'*H*U = S = (S(1,1) S(1,2))
( 0 S(2,2))
where S(i,j) is an N-by-N matrix and the eigenvalues of S(1,1)
have negative real parts in case of (4), or moduli greater than
one in case of (5). If U is conformably partitioned into four
N-by-N blocks
U = (U(1,1) U(1,2))
(U(2,1) U(2,2))
with respect to the assumptions we then have
(a) U(1,1) is invertible and X = U(2,1)*inv(U(1,1)) solves (1),
(2), or (3) with X = X' and non-negative definite;
(b) the eigenvalues of S(1,1) (if DICO = 'C') or S(2,2) (if
DICO = 'D') are equal to the eigenvalues of optimal system
(the 'closed-loop' spectrum).
[A,B] is stabilizable if there exists a matrix F such that (A-BF)
is stable. [E,A] is detectable if [A',E'] is stabilizable.
References
[1] Laub, A.J.
A Schur Method for Solving Algebraic Riccati equations.
IEEE Trans. Auto. Contr., AC-24, pp. 913-921, 1979.
[2] Wonham, W.M.
On a matrix Riccati equation of stochastic control.
SIAM J. Contr., 6, pp. 681-697, 1968.
[3] Sima, V.
Algorithms for Linear-Quadratic Optimization.
Pure and Applied Mathematics: A Series of Monographs and
Textbooks, vol. 200, Marcel Dekker, Inc., New York, 1996.
Numerical Aspects
3 The algorithm requires 0(N ) operations.Further Comments
To obtain a stabilizing solution of the algebraic Riccati
equation for DICO = 'D', set SORT = 'U', if HINV = 'D', or set
SORT = 'S', if HINV = 'I'.
The routine can also compute the anti-stabilizing solutions of
the algebraic Riccati equations, by specifying
SORT = 'U' if DICO = 'D' and HINV = 'I', or DICO = 'C', or
SORT = 'S' if DICO = 'D' and HINV = 'D'.
Usually, the combinations HINV = 'D' and SORT = 'U', or HINV = 'I'
and SORT = 'U', will be faster then the other combinations [3].
Example
Program Text
* SB02MD EXAMPLE PROGRAM TEXT.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA, LDG, LDQ, LDS, LDU
PARAMETER ( LDA = NMAX, LDG = NMAX, LDQ = NMAX,
$ LDS = 2*NMAX, LDU = 2*NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = 2*NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 6*NMAX )
* .. Local Scalars ..
DOUBLE PRECISION RCOND
INTEGER I, INFO, J, N
CHARACTER DICO, HINV, SCAL, SORT, UPLO
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), G(LDG,NMAX),
$ Q(LDQ,NMAX), S(LDS,2*NMAX), U(LDU,2*NMAX),
$ WI(2*NMAX), WR(2*NMAX)
INTEGER IWORK(LIWORK)
LOGICAL BWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL SB02MD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, DICO, HINV, UPLO, SCAL, SORT
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( Q(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( G(I,J), J = 1,N ), I = 1,N )
* Find the solution matrix X.
CALL SB02MD( DICO, HINV, UPLO, SCAL, SORT, N, A, LDA, G, LDG,
$ Q, LDQ, RCOND, WR, WI, S, LDS, U, LDU, IWORK,
$ DWORK, LDWORK, BWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) RCOND
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( Q(I,J), J = 1,N )
20 CONTINUE
END IF
END IF
STOP
*
99999 FORMAT (' SB02MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB02MD = ',I2)
99997 FORMAT (' RCOND = ',F4.2,//' The solution matrix X is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
SB02MD EXAMPLE PROGRAM DATA 2 C D U N S 0.0 1.0 0.0 0.0 1.0 0.0 0.0 2.0 0.0 0.0 0.0 1.0Program Results
SB02MD EXAMPLE PROGRAM RESULTS RCOND = 0.31 The solution matrix X is 2.0000 1.0000 1.0000 2.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB02MT.html 0000664 0000000 0000000 00000030637 14560147231 0020372 0 ustar 00root root 0000000 0000000
Purpose
To compute the following matrices
-1
G = B*R *B',
- -1
A = A - B*R *L',
- -1
Q = Q - L*R *L',
where A, B, Q, R, L, and G are N-by-N, N-by-M, N-by-N, M-by-M,
N-by-M, and N-by-N matrices, respectively, with Q, R and G
symmetric matrices.
When R is well-conditioned with respect to inversion, standard
algorithms for solving linear-quadratic optimization problems will
then also solve optimization problems with coupling weighting
matrix L. Moreover, a gain in efficiency is possible using matrix
G in the deflating subspace algorithms (see SLICOT Library routine
SB02OD) or in the Newton's algorithms (see SLICOT Library routine
SG02CD).
Specification
SUBROUTINE SB02MT( JOBG, JOBL, FACT, UPLO, N, M, A, LDA, B, LDB,
$ Q, LDQ, R, LDR, L, LDL, IPIV, OUFACT, G, LDG,
$ IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER FACT, JOBG, JOBL, UPLO
INTEGER INFO, LDA, LDB, LDG, LDL, LDQ, LDR, LDWORK, M,
$ N, OUFACT
C .. Array Arguments ..
INTEGER IPIV(*), IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), G(LDG,*),
$ L(LDL,*), Q(LDQ,*), R(LDR,*)
Arguments
Mode Parameters
JOBG CHARACTER*1
Specifies whether or not the matrix G is to be computed,
as follows:
= 'G': Compute G;
= 'N': Do not compute G.
JOBL CHARACTER*1
Specifies whether or not the matrix L is zero, as follows:
= 'Z': L is zero;
= 'N': L is nonzero.
FACT CHARACTER*1
Specifies how the matrix R is given (factored or not), as
follows:
= 'N': Array R contains the matrix R;
= 'C': Array R contains the Cholesky factor of R;
= 'U': Array R contains the factors of the symmetric
indefinite UdU' or LdL' factorization of R.
UPLO CHARACTER*1
Specifies which triangle of the matrices R, Q (if
JOBL = 'N'), and G (if JOBG = 'G') is stored, as follows:
= 'U': Upper triangle is stored;
= 'L': Lower triangle is stored.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, Q, and G, and the number of
rows of the matrices B and L. N >= 0.
M (input) INTEGER
The order of the matrix R, and the number of columns of
the matrices B and L. M >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, if JOBL = 'N', the leading N-by-N part of this
array must contain the matrix A.
On exit, if JOBL = 'N', and INFO = 0, the leading N-by-N
- -1
part of this array contains the matrix A = A - B*R L'.
If JOBL = 'Z', this array is not referenced.
LDA INTEGER
The leading dimension of array A.
LDA >= MAX(1,N) if JOBL = 'N';
LDA >= 1 if JOBL = 'Z'.
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the matrix B.
On exit, if OUFACT = 1, and INFO = 0, the leading N-by-M
-1
part of this array contains the matrix B*chol(R) .
On exit, B is unchanged if OUFACT <> 1 (hence also when
FACT = 'U').
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if JOBL = 'N', the leading N-by-N upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array must contain the upper
triangular part or lower triangular part, respectively, of
the symmetric matrix Q. The strictly lower triangular part
(if UPLO = 'U') or strictly upper triangular part (if
UPLO = 'L') is not referenced.
On exit, if JOBL = 'N' and INFO = 0, the leading N-by-N
upper triangular part (if UPLO = 'U') or lower triangular
part (if UPLO = 'L') of this array contains the upper
triangular part or lower triangular part, respectively, of
- -1
the symmetric matrix Q = Q - L*R *L'.
If JOBL = 'Z', this array is not referenced.
LDQ INTEGER
The leading dimension of array Q.
LDQ >= MAX(1,N) if JOBL = 'N';
LDQ >= 1 if JOBL = 'Z'.
R (input/output) DOUBLE PRECISION array, dimension (LDR,M)
On entry, if FACT = 'N', the leading M-by-M upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array must contain the upper
triangular part or lower triangular part, respectively,
of the symmetric input weighting matrix R.
On entry, if FACT = 'C', the leading M-by-M upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array must contain the Cholesky
factor of the positive definite input weighting matrix R
(as produced by LAPACK routine DPOTRF).
On entry, if FACT = 'U', the leading M-by-M upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array must contain the factors of
the UdU' or LdL' factorization, respectively, of the
symmetric indefinite input weighting matrix R (as produced
by LAPACK routine DSYTRF).
If FACT = 'N', the strictly lower triangular part (if UPLO
= 'U') or strictly upper triangular part (if UPLO = 'L')
of this array is used as workspace (filled in by
symmetry).
On exit, if OUFACT = 1, and INFO = 0 (or INFO = M+1),
the leading M-by-M upper triangular part (if UPLO = 'U')
or lower triangular part (if UPLO = 'L') of this array
contains the Cholesky factor of the given input weighting
matrix.
On exit, if OUFACT = 2, and INFO = 0 (or INFO = M+1),
the leading M-by-M upper triangular part (if UPLO = 'U')
or lower triangular part (if UPLO = 'L') of this array
contains the factors of the UdU' or LdL' factorization,
respectively, of the given input weighting matrix.
On exit R is unchanged if FACT = 'C' or 'U'.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,M).
L (input/output) DOUBLE PRECISION array, dimension (LDL,M)
On entry, if JOBL = 'N', the leading N-by-M part of this
array must contain the matrix L.
On exit, if JOBL = 'N', OUFACT = 1, and INFO = 0, the
leading N-by-M part of this array contains the matrix
-1
L*chol(R) .
On exit, L is unchanged if OUFACT <> 1 (hence also when
FACT = 'U').
L is not referenced if JOBL = 'Z'.
LDL INTEGER
The leading dimension of array L.
LDL >= MAX(1,N) if JOBL = 'N';
LDL >= 1 if JOBL = 'Z'.
IPIV (input/output) INTEGER array, dimension (M)
On entry, if FACT = 'U', this array must contain details
of the interchanges performed and the block structure of
the d factor in the UdU' or LdL' factorization of matrix R
(as produced by LAPACK routine DSYTRF).
On exit, if OUFACT = 2, this array contains details of
the interchanges performed and the block structure of the
d factor in the UdU' or LdL' factorization of matrix R,
as produced by LAPACK routine DSYTRF.
This array is not referenced if FACT = 'C'.
OUFACT (output) INTEGER
Information about the factorization finally used.
OUFACT = 0: no factorization of R has been used (M = 0);
OUFACT = 1: Cholesky factorization of R has been used;
OUFACT = 2: UdU' (if UPLO = 'U') or LdL' (if UPLO = 'L')
factorization of R has been used.
G (output) DOUBLE PRECISION array, dimension (LDG,N)
If JOBG = 'G', and INFO = 0, the leading N-by-N upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array contains the upper
triangular part (if UPLO = 'U') or lower triangular part
-1
(if UPLO = 'L'), respectively, of the matrix G = B*R B'.
If JOBG = 'N', this array is not referenced.
LDG INTEGER
The leading dimension of array G.
LDG >= MAX(1,N) if JOBG = 'G';
LDG >= 1 if JOBG = 'N'.
Workspace
IWORK INTEGER array, dimension (M)
If FACT = 'C' or FACT = 'U', this array is not referenced.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or LDWORK = -1, DWORK(1) returns the
optimal value of LDWORK; if FACT = 'N' and LDWORK is set
as specified below, DWORK(2) contains the reciprocal
condition number of the given matrix R.
On exit, if LDWORK = -2 on input or INFO = -23, then
DWORK(1) returns the minimal value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 1 if FACT = 'C' or (FACT = 'U' and
JOBG = 'N' and JOBL = 'Z');
LDWORK >= MAX(2,3*M) if FACT = 'N' and JOBG = 'N' and
JOBL = 'Z';
LDWORK >= MAX(2,3*M,N*M) if FACT = 'N' and (JOBG = 'G' or
JOBL = 'N');
LDWORK >= MAX(1,N*M) if FACT = 'U' and (JOBG = 'G' or
JOBL = 'N').
For optimum performance LDWORK should be larger than 3*M,
if FACT = 'N'.
If LDWORK = -1, an optimal workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
If LDWORK = -2, a minimal workspace query is assumed; the
routine only calculates the minimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= i: if the i-th element (1 <= i <= M) of the d factor is
exactly zero; the UdU' (or LdL') factorization has
been completed, but the block diagonal matrix d is
exactly singular;
= M+1: if the matrix R is numerically singular.
Method
- - The matrices G, and/or A and Q are evaluated using the given or computed symmetric factorization of R.Numerical Aspects
The routine should not be used when R is ill-conditioned.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB02MU.html 0000664 0000000 0000000 00000015453 14560147231 0020372 0 ustar 00root root 0000000 0000000
Purpose
To construct the 2n-by-2n Hamiltonian or symplectic matrix S
associated to the linear-quadratic optimization problem, used to
solve the continuous- or discrete-time algebraic Riccati equation,
respectively.
For a continuous-time problem, S is defined by
( A -G )
S = ( ), (1)
( -Q -A')
and for a discrete-time problem by
-1 -1
( A A *G )
S = ( -1 -1 ), (2)
( QA A' + Q*A *G )
or
-T -T
( A + G*A *Q -G*A )
S = ( -T -T ), (3)
( -A *Q A )
where A, G, and Q are N-by-N matrices, with G and Q symmetric.
Matrix A must be nonsingular in the discrete-time case.
Specification
SUBROUTINE SB02MU( DICO, HINV, UPLO, N, A, LDA, G, LDG, Q, LDQ, S,
$ LDS, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, HINV, UPLO
INTEGER INFO, LDA, LDG, LDQ, LDS, LDWORK, N
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), DWORK(*), G(LDG,*), Q(LDQ,*),
$ S(LDS,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the system as follows:
= 'C': Continuous-time system;
= 'D': Discrete-time system.
HINV CHARACTER*1
If DICO = 'D', specifies which of the matrices (2) or (3)
is constructed, as follows:
= 'D': The matrix S in (2) is constructed;
= 'I': The (inverse) matrix S in (3) is constructed.
HINV is not referenced if DICO = 'C'.
UPLO CHARACTER*1
Specifies which triangle of the matrices G and Q is
stored, as follows:
= 'U': Upper triangle is stored;
= 'L': Lower triangle is stored.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, G, and Q. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A.
On exit, if DICO = 'D', and INFO = 0, the leading N-by-N
-1
part of this array contains the matrix A .
Otherwise, the array A is unchanged on exit.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
G (input) DOUBLE PRECISION array, dimension (LDG,N)
The leading N-by-N upper triangular part (if UPLO = 'U')
or lower triangular part (if UPLO = 'L') of this array
must contain the upper triangular part or lower triangular
part, respectively, of the symmetric matrix G.
The strictly lower triangular part (if UPLO = 'U') or
strictly upper triangular part (if UPLO = 'L') is not
referenced.
LDG INTEGER
The leading dimension of array G. LDG >= MAX(1,N).
Q (input) DOUBLE PRECISION array, dimension (LDQ,N)
The leading N-by-N upper triangular part (if UPLO = 'U')
or lower triangular part (if UPLO = 'L') of this array
must contain the upper triangular part or lower triangular
part, respectively, of the symmetric matrix Q.
The strictly lower triangular part (if UPLO = 'U') or
strictly upper triangular part (if UPLO = 'L') is not
referenced.
LDQ INTEGER
The leading dimension of array Q. LDQ >= MAX(1,N).
S (output) DOUBLE PRECISION array, dimension (LDS,2*N)
If INFO = 0, the leading 2N-by-2N part of this array
contains the Hamiltonian or symplectic matrix of the
problem.
LDS INTEGER
The leading dimension of array S. LDS >= MAX(1,2*N).
Workspace
IWORK INTEGER array, dimension (2*N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK; if DICO = 'D', DWORK(2) returns the reciprocal
condition number of the given matrix A.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 1 if DICO = 'C';
LDWORK >= MAX(2,4*N) if DICO = 'D'.
For optimum performance LDWORK should be larger, if
DICO = 'D'.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= i: if the leading i-by-i (1 <= i <= N) upper triangular
submatrix of A is singular in discrete-time case;
= N+1: if matrix A is numerically singular in discrete-
time case.
Method
For a continuous-time problem, the 2n-by-2n Hamiltonian matrix (1) is constructed. For a discrete-time problem, the 2n-by-2n symplectic matrix (2) or (3) - the inverse of the matrix in (2) - is constructed.Numerical Aspects
The discrete-time case needs the inverse of the matrix A, hence
the routine should not be used when A is ill-conditioned.
3
The algorithm requires 0(n ) floating point operations in the
discrete-time case.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the following matrices
-1
G = B*R *B',
- -1
A = A +/- op(B*R *L'),
- -1
Q = Q +/- L*R *L',
where A, B, Q, R, L, and G are N-by-N, N-by-M, N-by-N, M-by-M,
N-by-M, and N-by-N matrices, respectively, with Q, R and G
symmetric matrices, and op(W) is one of
op(W) = W or op(W) = W'.
When R is well-conditioned with respect to inversion, standard
algorithms for solving linear-quadratic optimization problems will
then also solve optimization problems with coupling weighting
matrix L. Moreover, a gain in efficiency is possible using matrix
G in the deflating subspace algorithms (see SLICOT Library routine
SB02OD) or in the Newton's algorithms (see SLICOT Library routine
SG02CD).
Specification
SUBROUTINE SB02MX( JOBG, JOBL, FACT, UPLO, TRANS, FLAG, DEF, N, M,
$ A, LDA, B, LDB, Q, LDQ, R, LDR, L, LDL, IPIV,
$ OUFACT, G, LDG, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DEF, FACT, FLAG, JOBG, JOBL, TRANS, UPLO
INTEGER INFO, LDA, LDB, LDG, LDL, LDQ, LDR, LDWORK, M,
$ N, OUFACT
C .. Array Arguments ..
INTEGER IPIV(*), IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), G(LDG,*),
$ L(LDL,*), Q(LDQ,*), R(LDR,*)
Arguments
Mode Parameters
JOBG CHARACTER*1
Specifies whether or not the matrix G is to be computed,
as follows:
= 'G': Compute G;
= 'N': Do not compute G.
JOBL CHARACTER*1
Specifies whether or not the matrix L is zero, as follows:
= 'Z': L is zero;
= 'N': L is nonzero.
FACT CHARACTER*1
Specifies how the matrix R is given (factored or not), as
follows:
= 'N': Array R contains the matrix R;
= 'C': Array R contains the Cholesky factor of R;
= 'U': Array R contains the factors of the symmetric
indefinite UdU' or LdL' factorization of R.
UPLO CHARACTER*1
Specifies which triangle of the matrices R, Q (if
JOBL = 'N'), and G (if JOBG = 'G') is stored, as follows:
= 'U': Upper triangle is stored;
= 'L': Lower triangle is stored.
TRANS CHARACTER*1
Specifies the form of op(W) to be used in the matrix
multiplication, as follows:
= 'N': op(W) = W;
= 'T': op(W) = W';
= 'C': op(W) = W'.
FLAG CHARACTER*1
Specifies which sign is used, as follows:
= 'P': The plus sign is used;
= 'M': The minus sign is used.
DEF CHARACTER*1
If FACT = 'N', specifies whether or not it is assumed that
matrix R is positive definite, as follows:
= 'D': Matrix R is assumed positive definite;
= 'I': Matrix R is assumed indefinite.
Both values can be used to perform the computations,
irrespective to the R definiteness, but using the adequate
value will save some computational effort (see FURTHER
COMMENTS).
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, Q, and G, and the number of
rows of the matrices B and L. N >= 0.
M (input) INTEGER
The order of the matrix R, and the number of columns of
the matrices B and L. M >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, if JOBL = 'N', the leading N-by-N part of this
array must contain the matrix A.
On exit, if JOBL = 'N', and INFO = 0, the leading N-by-N
-
part of this array contains the matrix A.
If JOBL = 'Z', this array is not referenced.
LDA INTEGER
The leading dimension of array A.
LDA >= MAX(1,N) if JOBL = 'N';
LDA >= 1 if JOBL = 'Z'.
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the matrix B.
On exit, if OUFACT = 1, and INFO = 0, the leading N-by-M
-1
part of this array contains the matrix B*chol(R) .
On exit, B is unchanged if OUFACT <> 1 (hence also when
FACT = 'U').
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if JOBL = 'N', the leading N-by-N upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array must contain the upper
triangular part or lower triangular part, respectively, of
the symmetric matrix Q. The strictly lower triangular part
(if UPLO = 'U') or strictly upper triangular part (if
UPLO = 'L') is not referenced.
On exit, if JOBL = 'N' and INFO = 0, the leading N-by-N
upper triangular part (if UPLO = 'U') or lower triangular
part (if UPLO = 'L') of this array contains the upper
triangular part or lower triangular part, respectively, of
- -1
the symmetric matrix Q = Q +/- L*R *L'.
If JOBL = 'Z', this array is not referenced.
LDQ INTEGER
The leading dimension of array Q.
LDQ >= MAX(1,N) if JOBL = 'N';
LDQ >= 1 if JOBL = 'Z'.
R (input/output) DOUBLE PRECISION array, dimension (LDR,M)
On entry, if FACT = 'N', the leading M-by-M upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array must contain the upper
triangular part or lower triangular part, respectively,
of the symmetric input weighting matrix R.
On entry, if FACT = 'C', the leading M-by-M upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array must contain the Cholesky
factor of the positive definite input weighting matrix R
(as produced by LAPACK routine DPOTRF).
On entry, if FACT = 'U', the leading M-by-M upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array must contain the factors of
the UdU' or LdL' factorization, respectively, of the
symmetric indefinite input weighting matrix R (as produced
by LAPACK routine DSYTRF).
If FACT = 'N' and DEF = 'D', the strictly lower triangular
part (if UPLO = 'U') or strictly upper triangular part
(if UPLO = 'L') of this array is used as workspace (filled
in by symmetry). If FACT = 'N' and DEF = 'I', the strictly
lower triangular part (if UPLO = 'U') or strictly upper
triangular part (if UPLO = 'L') is unchanged.
On exit, if OUFACT = 1, and INFO = 0 (or INFO = M+1),
the leading M-by-M upper triangular part (if UPLO = 'U')
or lower triangular part (if UPLO = 'L') of this array
contains the Cholesky factor of the given input weighting
matrix.
On exit, if OUFACT = 2, and INFO = 0 (or INFO = M+1),
the leading M-by-M upper triangular part (if UPLO = 'U')
or lower triangular part (if UPLO = 'L') of this array
contains the factors of the UdU' or LdL' factorization,
respectively, of the given input weighting matrix.
On exit R is unchanged if FACT = 'C' or 'U'.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,M).
L (input/output) DOUBLE PRECISION array, dimension (LDL,M)
On entry, if JOBL = 'N', the leading N-by-M part of this
array must contain the matrix L.
On exit, if JOBL = 'N', OUFACT = 1, and INFO = 0, the
leading N-by-M part of this array contains the matrix
-1
L*chol(R) .
On exit, L is unchanged if OUFACT <> 1 (hence also when
FACT = 'U').
L is not referenced if JOBL = 'Z'.
LDL INTEGER
The leading dimension of array L.
LDL >= MAX(1,N) if JOBL = 'N';
LDL >= 1 if JOBL = 'Z'.
IPIV (input/output) INTEGER array, dimension (M)
On entry, if FACT = 'U', this array must contain details
of the interchanges performed and the block structure of
the d factor in the UdU' or LdL' factorization of matrix R
(as produced by LAPACK routine DSYTRF).
On exit, if OUFACT = 2, this array contains details of
the interchanges performed and the block structure of the
d factor in the UdU' or LdL' factorization of matrix R,
as produced by LAPACK routine DSYTRF.
This array is not referenced if FACT = 'C'.
OUFACT (output) INTEGER
Information about the factorization finally used.
OUFACT = 0: no factorization of R has been used (M = 0);
OUFACT = 1: Cholesky factorization of R has been used;
OUFACT = 2: UdU' (if UPLO = 'U') or LdL' (if UPLO = 'L')
factorization of R has been used.
G (output) DOUBLE PRECISION array, dimension (LDG,N)
If JOBG = 'G', and INFO = 0, the leading N-by-N upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array contains the upper
triangular part (if UPLO = 'U') or lower triangular part
-1
(if UPLO = 'L'), respectively, of the matrix G = B*R B'.
If JOBG = 'N', this array is not referenced.
LDG INTEGER
The leading dimension of array G.
LDG >= MAX(1,N) if JOBG = 'G';
LDG >= 1 if JOBG = 'N'.
Workspace
IWORK INTEGER array, dimension (M)
If FACT = 'C' or FACT = 'U', this array is not referenced.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or LDWORK = -1, DWORK(1) returns the
optimal value of LDWORK; if FACT = 'N' and LDWORK is set
as specified below, DWORK(2) contains the reciprocal
condition number of the given matrix R. DWORK(2) is set to
zero if M = 0.
On exit, if LDWORK = -2 on input or INFO = -26, then
DWORK(1) returns the minimal value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 1 if FACT = 'C' or (FACT = 'U' and
JOBG = 'N' and JOBL = 'Z');
LDWORK >= MAX(2,3*M) if FACT = 'N' and JOBG = 'N' and
JOBL = 'Z';
LDWORK >= MAX(2,3*M,N*M) if FACT = 'N' and (JOBG = 'G' or
JOBL = 'N');
LDWORK >= MAX(1,N*M) if FACT = 'U' and (JOBG = 'G' or
JOBL = 'N').
For optimum performance LDWORK should be larger than 3*M,
if FACT = 'N'.
If LDWORK = -1, an optimal workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
If LDWORK = -2, a minimal workspace query is assumed; the
routine only calculates the minimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= i: if the i-th element (1 <= i <= M) of the d factor is
exactly zero; the UdU' (or LdL') factorization has
been completed, but the block diagonal matrix d is
exactly singular;
= M+1: if the matrix R is numerically singular.
Method
- - The matrices G, and/or A and Q are evaluated using the given or computed symmetric factorization of R.Numerical Aspects
The routine should not be used when R is ill-conditioned.Further Comments
Using argument TRANS allows to avoid the transposition of matrix A needed to solve optimal filtering/estimation problems by the same routines solving optimal control problems. If DEF is set to 'D', but R is indefinite, the computational effort for factorization will be approximately double, since Cholesky factorization, tried first, will fail, and symmetric indefinite factorization will then be used. If DEF is set to 'I', but R is positive definite, the computational effort will be slightly higher than that when using Cholesky factorization. It is recommended to use DEF = 'D' also if the definiteness is not known, but M is (much) smaller than N.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB02ND.html 0000664 0000000 0000000 00000046512 14560147231 0020352 0 ustar 00root root 0000000 0000000
Purpose
To compute the optimal feedback matrix F for the problem of
optimal control given by
-1
F = (R + B'XB) (B'XA + L') (1)
in the discrete-time case and
-1
F = R (B'X + L') (2)
in the continuous-time case, where A, B and L are N-by-N, N-by-M
and N-by-M matrices respectively; R and X are M-by-M and N-by-N
symmetric matrices respectively.
Optionally, matrix R may be specified in a factored form, and L
may be zero.
Specification
SUBROUTINE SB02ND( DICO, FACT, UPLO, JOBL, N, M, P, A, LDA, B,
$ LDB, R, LDR, IPIV, L, LDL, X, LDX, RNORM, F,
$ LDF, OUFACT, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, FACT, JOBL, UPLO
INTEGER INFO, LDA, LDB, LDF, LDL, LDR, LDWORK, LDX, M,
$ N, P
DOUBLE PRECISION RNORM
C .. Array Arguments ..
INTEGER IPIV(*), IWORK(*), OUFACT(2)
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), F(LDF,*),
$ L(LDL,*), R(LDR,*), X(LDX,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the equation from which F is to be determined,
as follows:
= 'D': Equation (1), discrete-time case;
= 'C': Equation (2), continuous-time case.
FACT CHARACTER*1
Specifies how the matrix R is given (factored or not), as
follows:
= 'N': Array R contains the matrix R;
= 'D': Array R contains a P-by-M matrix D, where R = D'D;
= 'C': Array R contains the Cholesky factor of R;
= 'U': Array R contains the symmetric indefinite UdU' or
LdL' factorization of R. This option is not
available for DICO = 'D'.
UPLO CHARACTER*1
Specifies which triangle of the possibly factored matrix R
(or R + B'XB, on exit) is or should be stored, as follows:
= 'U': Upper triangle is stored;
= 'L': Lower triangle is stored.
JOBL CHARACTER*1
Specifies whether or not the matrix L is zero, as follows:
= 'Z': L is zero;
= 'N': L is nonzero.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A and X. N >= 0.
No computations are performed if MIN(N,M) = 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of rows of the matrix D.
P >= M for DICO = 'C';
P >= 0 for DICO = 'D'.
This parameter must be specified only for FACT = 'D'.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
If DICO = 'D', the leading N-by-N part of this array must
contain the state matrix A of the system.
If DICO = 'C', this array is not referenced.
LDA INTEGER
The leading dimension of array A.
LDA >= MAX(1,N) if DICO = 'D';
LDA >= 1 if DICO = 'C'.
B (input/worksp.) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
input matrix B of the system.
If DICO = 'D' and FACT = 'D' or 'C', the contents of this
array is destroyed. Specifically, if, on exit,
OUFACT(2) = 1, this array contains chol(X)*B, and if
OUFACT(2) = 2 and INFO < M+2, but INFO >= 0, its trailing
part (in the first N rows) contains the submatrix of
sqrt(V)*U'B corresponding to the non-negligible, positive
eigenvalues of X, where V and U are the matrices with the
eigenvalues and eigenvectors of X.
Otherwise, B is unchanged on exit.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
R (input/output) DOUBLE PRECISION array, dimension (LDR,M)
On entry, if FACT = 'N', the leading M-by-M upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array must contain the upper
triangular part or lower triangular part, respectively,
of the symmetric input weighting matrix R.
On entry, if FACT = 'D', the leading P-by-M part of this
array must contain the direct transmission matrix D of the
system.
On entry, if FACT = 'C', the leading M-by-M upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array must contain the Cholesky
factor of the positive definite input weighting matrix R
(as produced by LAPACK routine DPOTRF).
On entry, if DICO = 'C' and FACT = 'U', the leading M-by-M
upper triangular part (if UPLO = 'U') or lower triangular
part (if UPLO = 'L') of this array must contain the
factors of the UdU' or LdL' factorization, respectively,
of the symmetric indefinite input weighting matrix R (as
produced by LAPACK routine DSYTRF).
The strictly lower triangular part (if UPLO = 'U') or
strictly upper triangular part (if UPLO = 'L') of this
array is used as workspace (filled in by symmetry with the
other strictly triangular part of R, of R+B'XB, or of the
result, if DICO = 'C', DICO = 'D', or (DICO = 'D' and
(FACT = 'D' or FACT = 'C') and UPLO = 'L'), respectively.
On exit, if OUFACT(1) = 1, and INFO = 0 (or INFO = M+1),
the leading M-by-M upper triangular part (if UPLO = 'U')
or lower triangular part (if UPLO = 'L') of this array
contains the Cholesky factor of the given input weighting
matrix R (for DICO = 'C'), or that of the matrix R + B'XB
(for DICO = 'D').
On exit, if OUFACT(1) = 2, and INFO = 0 (or INFO = M+1),
the leading M-by-M upper triangular part (if UPLO = 'U')
or lower triangular part (if UPLO = 'L') of this array
contains the factors of the UdU' or LdL' factorization,
respectively, of the given input weighting matrix
(for DICO = 'C'), or that of the matrix R + B'XB
(for DICO = 'D' and FACT = 'N').
On exit R is unchanged if FACT = 'U' or N = 0.
LDR INTEGER.
The leading dimension of the array R.
LDR >= MAX(1,M) if FACT <> 'D';
LDR >= MAX(1,M,P) if FACT = 'D'.
IPIV (input/output) INTEGER array, dimension (M)
On entry, if FACT = 'U', this array must contain details
of the interchanges performed and the block structure of
the d factor in the UdU' or LdL' factorization of matrix R
(as produced by LAPACK routine DSYTRF).
On exit, if OUFACT(1) = 2, this array contains details of
the interchanges performed and the block structure of the
d factor in the UdU' or LdL' factorization of matrix R or
R + B'XB, as produced by LAPACK routine DSYTRF.
This array is not referenced if FACT = 'D', or FACT = 'C',
or N = 0.
L (input) DOUBLE PRECISION array, dimension (LDL,M)
If JOBL = 'N', the leading N-by-M part of this array must
contain the cross weighting matrix L.
If JOBL = 'Z', this array is not referenced.
LDL INTEGER
The leading dimension of array L.
LDL >= MAX(1,N) if JOBL = 'N';
LDL >= 1 if JOBL = 'Z'.
X (input/output) DOUBLE PRECISION array, dimension (LDX,N)
On entry, the leading N-by-N part of this array must
contain the solution matrix X of the algebraic Riccati
equation as produced by SLICOT Library routines SB02MD or
SB02OD. Matrix X is assumed non-negative definite if
DICO = 'D' and (FACT = 'D' or FACT = 'C').
The full matrix X must be given on input if LDWORK < N*M
or if DICO = 'D' and (FACT = 'D' or FACT = 'C').
On exit, if DICO = 'D', FACT = 'D' or FACT = 'C', and
OUFACT(2) = 1, the N-by-N upper triangular part
(if UPLO = 'U') or lower triangular part (if UPLO = 'L')
of this array contains the Cholesky factor of the given
matrix X, which is found to be positive definite.
On exit, if DICO = 'D', FACT = 'D' or 'C', OUFACT(2) = 2,
and INFO < M+2 (but INFO >= 0), the leading N-by-N part of
this array contains the matrix of orthonormal eigenvectors
of X.
On exit X is unchanged if DICO = 'C' or FACT = 'N'.
LDX INTEGER
The leading dimension of array X. LDX >= MAX(1,N).
RNORM (input) DOUBLE PRECISION
If FACT = 'U', this parameter must contain the 1-norm of
the original matrix R (before factoring it).
Otherwise, this parameter is not used.
F (output) DOUBLE PRECISION array, dimension (LDF,N)
The leading M-by-N part of this array contains the
optimal feedback matrix F.
This array is not referenced if DICO = 'C' and FACT = 'D'
and P < M.
LDF INTEGER
The leading dimension of array F. LDF >= MAX(1,M).
OUFACT (output) INTEGER array, dimension (2)
Information about the factorization finally used.
OUFACT(1) = 1: Cholesky factorization of R (or R + B'XB)
has been used;
OUFACT(1) = 2: UdU' (if UPLO = 'U') or LdL' (if UPLO =
'L') factorization of R (or R + B'XB)
has been used;
OUFACT(2) = 1: Cholesky factorization of X has been used;
OUFACT(2) = 2: Spectral factorization of X has been used.
The value of OUFACT(2) is not set for DICO = 'C' or for
DICO = 'D' and FACT = 'N'.
This array is not set if N = 0 or M = 0.
Workspace
IWORK INTEGER array, dimension (M)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or LDWORK = -1, DWORK(1) returns the
optimal value of LDWORK, and for LDWORK set as specified
below, DWORK(2) contains the reciprocal condition number
of the matrix R (for DICO = 'C') or of R + B'XB (for
DICO = 'D'); DWORK(2) is set to 1 if N = 0.
On exit, if LDWORK = -2 on input or INFO = -25, then
DWORK(1) returns the minimal value of LDWORK.
If on exit INFO = 0, and OUFACT(2) = 2, then DWORK(3),...,
DWORK(N+2) contain the eigenvalues of X, in ascending
order.
LDWORK INTEGER
Dimension of working array DWORK.
LDWORK >= max(2,2*M) if FACT = 'U';
LDWORK >= max(2,3*M) if FACT <> 'U', DICO = 'C';
LDWORK >= max(2,3*M,N) if FACT = 'N', DICO = 'D';
LDWORK >= max(N+3*M+2,4*N+1) if FACT <> 'N', DICO = 'D'.
For optimum performance LDWORK should be larger.
If LDWORK = -1, an optimal workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
If LDWORK = -2, a minimal workspace query is assumed; the
routine only calculates the minimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= i: if the i-th element of the d factor is exactly zero;
the UdU' (or LdL') factorization has been completed,
but the block diagonal matrix d is exactly singular;
= M+1: if the matrix R (if DICO = 'C'), or R + B'XB
(if DICO = 'D') is numerically singular (to working
precision);
= M+2: if one or more of the eigenvalues of X has not
converged;
= M+3: if the matrix X is indefinite and updating the
triangular factorization failed.
If INFO > M+1, call the routine again with an appropriate,
unfactored matrix R.
Method
The optimal feedback matrix F is obtained as the solution to the
system of linear equations
(R + B'XB) * F = B'XA + L'
in the discrete-time case and
R * F = B'X + L'
in the continuous-time case, with R replaced by D'D if FACT = 'D'.
If FACT = 'N', Cholesky factorization is tried first, but
if the coefficient matrix is not positive definite, then UdU' (or
LdL') factorization is used. If FACT <> 'N', the factored form
of R is taken into account. The discrete-time case then involves
updating of a triangular factorization of R (or D'D); Cholesky or
symmetric spectral factorization of X is employed to avoid
squaring of the condition number of the matrix. When D is given,
its QR factorization is determined, and the triangular factor is
used as described above.
Numerical Aspects
The algorithm consists of numerically stable steps.
3 2
For DICO = 'C', it requires O(m + mn ) floating point operations
2
if FACT = 'N' and O(mn ) floating point operations, otherwise.
For DICO = 'D', the operation counts are similar, but additional
3
O(n ) floating point operations may be needed in the worst case.
These estimates assume that M <= N.
Further Comments
NoneExample
Program Text
* SB02ND EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER NMAX2
PARAMETER ( NMAX2 = 2*NMAX )
INTEGER LDA, LDB, LDC, LDL, LDR, LDS, LDT, LDU, LDX, LDF
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX, LDL = NMAX,
$ LDR = MAX(MMAX,PMAX), LDS = NMAX2+MMAX,
$ LDT = NMAX2+MMAX, LDU = NMAX2, LDX = NMAX,
$ LDF = MMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MAX( NMAX2,MMAX ) )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( NMAX+3*MMAX+2, 14*NMAX+23,
$ 16*NMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL, RCOND, RNORM
INTEGER I, INFO1, INFO2, J, M, N, P
CHARACTER*1 DICO, FACT, JOBB, JOBL, SORT, UPLO
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), ALFAI(2*NMAX), ALFAR(2*NMAX),
$ B(LDB,MMAX), BETA(2*NMAX), C(LDC,NMAX),
$ DWORK(LDWORK), F(LDF,NMAX), L(LDL,MMAX),
$ R(LDR,MMAX), S(LDS,NMAX2+MMAX), T(LDT,NMAX2),
$ U(LDU,NMAX2), X(LDX,NMAX)
INTEGER IPIV(LIWORK), IWORK(LIWORK), OUFACT(2)
LOGICAL BWORK(NMAX2)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL SB02ND, SB02OD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL, DICO, FACT, JOBL, UPLO
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
IF ( LSAME( FACT, 'D' ) ) THEN
READ ( NIN, FMT = * ) ( ( R(I,J), J = 1,M ), I = 1,P )
ELSE
READ ( NIN, FMT = * ) ( ( R(I,J), J = 1,M ), I = 1,M )
END IF
* Find the solution matrix X.
JOBB = 'B'
SORT = 'S'
CALL SB02OD( DICO, JOBB, 'Both', UPLO, JOBL, SORT, N, M,
$ P, A, LDA, B, LDB, C, LDC, R, LDR, L, LDL,
$ RCOND, X, LDX, ALFAR, ALFAI, BETA, S, LDS,
$ T, LDT, U, LDU, TOL, IWORK, DWORK, LDWORK,
$ BWORK, INFO1 )
*
IF ( INFO1.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO1
ELSE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99994 ) ( X(I,J), J = 1,N )
20 CONTINUE
* Compute the optimal feedback matrix F.
CALL SB02ND( DICO, FACT, UPLO, JOBL, N, M, P, A, LDA,
$ B, LDB, R, LDR, IPIV, L, LDL, X, LDX,
$ RNORM, F, LDF, OUFACT, IWORK, DWORK,
$ LDWORK, INFO2 )
*
IF ( INFO2.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO2
ELSE
WRITE ( NOUT, FMT = 99995 )
DO 40 I = 1, M
WRITE ( NOUT, FMT = 99994 ) ( F(I,J), J = 1,N )
40 CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB02ND EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB02OD = ',I2)
99997 FORMAT (' INFO on exit from SB02ND = ',I2)
99996 FORMAT (' The solution matrix X is ')
99995 FORMAT (/' The optimal feedback matrix F is ')
99994 FORMAT (20(1X,F8.4))
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' M is out of range.',/' M = ',I5)
99991 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
SB02ND EXAMPLE PROGRAM DATA 2 1 3 0.0 D N Z U 2.0 -1.0 1.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0Program Results
SB02ND EXAMPLE PROGRAM RESULTS The solution matrix X is 1.0000 0.0000 0.0000 1.0000 The optimal feedback matrix F is 2.0000 -1.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB02OD.html 0000664 0000000 0000000 00000054403 14560147231 0020351 0 ustar 00root root 0000000 0000000
Purpose
To solve for X either the continuous-time algebraic Riccati
equation
-1
Q + A'X + XA - (L+XB)R (L+XB)' = 0 (1)
or the discrete-time algebraic Riccati equation
-1
X = A'XA - (L+A'XB)(R + B'XB) (L+A'XB)' + Q (2)
where A, B, Q, R, and L are N-by-N, N-by-M, N-by-N, M-by-M and
N-by-M matrices, respectively, such that Q = C'C, R = D'D and
L = C'D; X is an N-by-N symmetric matrix.
The routine also returns the computed values of the closed-loop
spectrum of the system, i.e., the stable eigenvalues lambda(1),
..., lambda(N) of the corresponding Hamiltonian or symplectic
pencil, in the continuous-time case or discrete-time case,
respectively.
-1
Optionally, matrix G = BR B' may be given instead of B and R.
Other options include the case with Q and/or R given in a
factored form, Q = C'C, R = D'D, and with L a zero matrix.
The routine uses the method of deflating subspaces, based on
reordering the eigenvalues in a generalized Schur matrix pair.
A standard eigenproblem is solved in the continuous-time case
if G is given.
Specification
SUBROUTINE SB02OD( DICO, JOBB, FACT, UPLO, JOBL, SORT, N, M, P, A,
$ LDA, B, LDB, Q, LDQ, R, LDR, L, LDL, RCOND, X,
$ LDX, ALFAR, ALFAI, BETA, S, LDS, T, LDT, U,
$ LDU, TOL, IWORK, DWORK, LDWORK, BWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, FACT, JOBB, JOBL, SORT, UPLO
INTEGER INFO, LDA, LDB, LDL, LDQ, LDR, LDS, LDT, LDU,
$ LDWORK, LDX, M, N, P
DOUBLE PRECISION RCOND, TOL
C .. Array Arguments ..
LOGICAL BWORK(*)
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), ALFAI(*), ALFAR(*), B(LDB,*), BETA(*),
$ DWORK(*), L(LDL,*), Q(LDQ,*), R(LDR,*),
$ S(LDS,*), T(LDT,*), U(LDU,*), X(LDX,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of Riccati equation to be solved as
follows:
= 'C': Equation (1), continuous-time case;
= 'D': Equation (2), discrete-time case.
JOBB CHARACTER*1
Specifies whether or not the matrix G is given, instead
of the matrices B and R, as follows:
= 'B': B and R are given;
= 'G': G is given.
FACT CHARACTER*1
Specifies whether or not the matrices Q and/or R (if
JOBB = 'B') are factored, as follows:
= 'N': Not factored, Q and R are given;
= 'C': C is given, and Q = C'C;
= 'D': D is given, and R = D'D;
= 'B': Both factors C and D are given, Q = C'C, R = D'D.
UPLO CHARACTER*1
If JOBB = 'G', or FACT = 'N', specifies which triangle of
the matrices G and Q (if FACT = 'N'), or Q and R (if
JOBB = 'B'), is stored, as follows:
= 'U': Upper triangle is stored;
= 'L': Lower triangle is stored.
JOBL CHARACTER*1
Specifies whether or not the matrix L is zero, as follows:
= 'Z': L is zero;
= 'N': L is nonzero.
JOBL is not used if JOBB = 'G' and JOBL = 'Z' is assumed.
SLICOT Library routine SB02MT should be called just before
SB02OD, for obtaining the results when JOBB = 'G' and
JOBL = 'N'.
SORT CHARACTER*1
Specifies which eigenvalues should be obtained in the top
of the generalized Schur form, as follows:
= 'S': Stable eigenvalues come first;
= 'U': Unstable eigenvalues come first.
Input/Output Parameters
N (input) INTEGER
The actual state dimension, i.e. the order of the matrices
A, Q, and X, and the number of rows of the matrices B
and L. N >= 0.
M (input) INTEGER
The number of system inputs. If JOBB = 'B', M is the
order of the matrix R, and the number of columns of the
matrix B. M >= 0.
M is not used if JOBB = 'G'.
P (input) INTEGER
The number of system outputs. If FACT = 'C' or 'D' or 'B',
P is the number of rows of the matrices C and/or D.
P >= 0.
Otherwise, P is not used.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
state matrix A of the system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,*)
If JOBB = 'B', the leading N-by-M part of this array must
contain the input matrix B of the system.
If JOBB = 'G', the leading N-by-N upper triangular part
(if UPLO = 'U') or lower triangular part (if UPLO = 'L')
of this array must contain the upper triangular part or
lower triangular part, respectively, of the matrix
-1
G = BR B'. The strictly lower triangular part (if
UPLO = 'U') or strictly upper triangular part (if
UPLO = 'L') is not referenced.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
Q (input) DOUBLE PRECISION array, dimension (LDQ,N)
If FACT = 'N' or 'D', the leading N-by-N upper triangular
part (if UPLO = 'U') or lower triangular part (if UPLO =
'L') of this array must contain the upper triangular part
or lower triangular part, respectively, of the symmetric
state weighting matrix Q. The strictly lower triangular
part (if UPLO = 'U') or strictly upper triangular part (if
UPLO = 'L') is not referenced.
If JOBB = 'B', the triangular part of this array defined
by UPLO is modified internally, but is restored on exit.
If FACT = 'C' or 'B', the leading P-by-N part of this
array must contain the output matrix C of the system.
If JOBB = 'B', this part is modified internally, but is
restored on exit.
LDQ INTEGER
The leading dimension of array Q.
LDQ >= MAX(1,N) if FACT = 'N' or 'D',
LDQ >= MAX(1,P) if FACT = 'C' or 'B'.
R (input) DOUBLE PRECISION array, dimension (LDR,M)
If FACT = 'N' or 'C', the leading M-by-M upper triangular
part (if UPLO = 'U') or lower triangular part (if UPLO =
'L') of this array must contain the upper triangular part
or lower triangular part, respectively, of the symmetric
input weighting matrix R. The strictly lower triangular
part (if UPLO = 'U') or strictly upper triangular part (if
UPLO = 'L') is not referenced.
The triangular part of this array defined by UPLO is
modified internally, but is restored on exit.
If FACT = 'D' or 'B', the leading P-by-M part of this
array must contain the direct transmission matrix D of the
system. This part is modified internally, but is restored
on exit.
If JOBB = 'G', this array is not referenced.
LDR INTEGER
The leading dimension of array R.
LDR >= MAX(1,M) if JOBB = 'B' and FACT = 'N' or 'C';
LDR >= MAX(1,P) if JOBB = 'B' and FACT = 'D' or 'B';
LDR >= 1 if JOBB = 'G'.
L (input) DOUBLE PRECISION array, dimension (LDL,M)
If JOBL = 'N' (and JOBB = 'B'), the leading N-by-M part of
this array must contain the cross weighting matrix L.
This part is modified internally, but is restored on exit.
If JOBL = 'Z' or JOBB = 'G', this array is not referenced.
LDL INTEGER
The leading dimension of array L.
LDL >= MAX(1,N) if JOBL = 'N' and JOBB = 'B';
LDL >= 1 if JOBL = 'Z' or JOBB = 'G'.
RCOND (output) DOUBLE PRECISION
An estimate of the reciprocal of the condition number (in
the 1-norm) of the N-th order system of algebraic
equations from which the solution matrix X is obtained.
X (output) DOUBLE PRECISION array, dimension (LDX,N)
The leading N-by-N part of this array contains the
solution matrix X of the problem.
LDX INTEGER
The leading dimension of array X. LDX >= MAX(1,N).
ALFAR (output) DOUBLE PRECISION array, dimension (2*N)
ALFAI (output) DOUBLE PRECISION array, dimension (2*N)
BETA (output) DOUBLE PRECISION array, dimension (2*N)
The generalized eigenvalues of the 2N-by-2N matrix pair,
ordered as specified by SORT (if INFO = 0). For instance,
if SORT = 'S', the leading N elements of these arrays
contain the closed-loop spectrum of the system matrix
A - BF, where F is the optimal feedback matrix computed
based on the solution matrix X. Specifically,
lambda(k) = [ALFAR(k)+j*ALFAI(k)]/BETA(k) for
k = 1,2,...,N.
If DICO = 'C' and JOBB = 'G', the elements of BETA are
set to 1.
S (output) DOUBLE PRECISION array, dimension (LDS,*)
The leading 2N-by-2N part of this array contains the
ordered real Schur form S of the first matrix in the
reduced matrix pencil associated to the optimal problem,
or of the corresponding Hamiltonian matrix, if DICO = 'C'
and JOBB = 'G'. That is,
(S S )
( 11 12)
S = ( ),
(0 S )
( 22)
where S , S and S are N-by-N matrices.
11 12 22
Array S must have 2*N+M columns if JOBB = 'B', and 2*N
columns, otherwise.
LDS INTEGER
The leading dimension of array S.
LDS >= MAX(1,2*N+M) if JOBB = 'B',
LDS >= MAX(1,2*N) if JOBB = 'G'.
T (output) DOUBLE PRECISION array, dimension (LDT,2*N)
If DICO = 'D' or JOBB = 'B', the leading 2N-by-2N part of
this array contains the ordered upper triangular form T of
the second matrix in the reduced matrix pencil associated
to the optimal problem. That is,
(T T )
( 11 12)
T = ( ),
(0 T )
( 22)
where T , T and T are N-by-N matrices.
11 12 22
If DICO = 'C' and JOBB = 'G' this array is not referenced.
LDT INTEGER
The leading dimension of array T.
LDT >= MAX(1,2*N+M) if JOBB = 'B',
LDT >= MAX(1,2*N) if JOBB = 'G' and DICO = 'D',
LDT >= 1 if JOBB = 'G' and DICO = 'C'.
U (output) DOUBLE PRECISION array, dimension (LDU,2*N)
The leading 2N-by-2N part of this array contains the right
transformation matrix U which reduces the 2N-by-2N matrix
pencil to the ordered generalized real Schur form (S,T),
or the Hamiltonian matrix to the ordered real Schur
form S, if DICO = 'C' and JOBB = 'G'. That is,
(U U )
( 11 12)
U = ( ),
(U U )
( 21 22)
where U , U , U and U are N-by-N matrices.
11 12 21 22
LDU INTEGER
The leading dimension of array U. LDU >= MAX(1,2*N).
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used to test for near singularity of
the original matrix pencil, specifically of the triangular
factor obtained during the reduction process. If the user
sets TOL > 0, then the given value of TOL is used as a
lower bound for the reciprocal condition number of that
matrix; a matrix whose estimated condition number is less
than 1/TOL is considered to be nonsingular. If the user
sets TOL <= 0, then a default tolerance, defined by
TOLDEF = EPS, is used instead, where EPS is the machine
precision (see LAPACK Library routine DLAMCH).
This parameter is not referenced if JOBB = 'G'.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK >= MAX(1,M,2*N) if JOBB = 'B',
LIWORK >= MAX(1,2*N) if JOBB = 'G'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK. If JOBB = 'B' and N > 0, DWORK(2) returns the
reciprocal of the condition number of the M-by-M lower
triangular matrix obtained after compressing the matrix
pencil of order 2N+M to obtain a pencil of order 2N.
If INFO = 0 or INFO = 6, DWORK(3) returns the scaling
factor used internally, which should multiply the
submatrix Y2 to recover X from the first N columns of U
(see METHOD).
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(3,6*N), if JOBB = 'G',
DICO = 'C';
LDWORK >= MAX(7*(2*N+1)+16,16*N), if JOBB = 'G',
DICO = 'D';
LDWORK >= MAX(7*(2*N+1)+16,16*N,2*N+M,3*M), if JOBB = 'B'.
For optimum performance LDWORK should be larger.
BWORK LOGICAL array, dimension (2*N)
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the computed extended matrix pencil is singular,
possibly due to rounding errors;
= 2: if the QZ (or QR) algorithm failed;
= 3: if reordering of the (generalized) eigenvalues
failed;
= 4: if after reordering, roundoff changed values of
some complex eigenvalues so that leading eigenvalues
in the (generalized) Schur form no longer satisfy
the stability condition; this could also be caused
due to scaling;
= 5: if the computed dimension of the solution does not
equal N;
= 6: if a singular matrix was encountered during the
computation of the solution matrix X.
Method
The routine uses a variant of the method of deflating subspaces
proposed by van Dooren [1]. See also [2], [3].
It is assumed that (A,B) is stabilizable and (C,A) is detectable.
Under these assumptions the algebraic Riccati equation is known to
have a unique non-negative definite solution.
The first step in the method of deflating subspaces is to form the
extended Hamiltonian matrices, dimension 2N + M given by
discrete-time continuous-time
|A 0 B| |I 0 0| |A 0 B| |I 0 0|
|Q -I L| - z |0 -A' 0|, |Q A' L| - s |0 -I 0|.
|L' 0 R| |0 -B' 0| |L' B' R| |0 0 0|
Next, these pencils are compressed to a form (see [1])
lambda x A - B .
f f
This generalized eigenvalue problem is then solved using the QZ
algorithm and the stable deflating subspace Ys is determined.
If [Y1'|Y2']' is a basis for Ys, then the required solution is
-1
X = Y2 x Y1 .
A standard eigenvalue problem is solved using the QR algorithm in
the continuous-time case when G is given (DICO = 'C', JOBB = 'G').
References
[1] Van Dooren, P.
A Generalized Eigenvalue Approach for Solving Riccati
Equations.
SIAM J. Sci. Stat. Comp., 2, pp. 121-135, 1981.
[2] Mehrmann, V.
The Autonomous Linear Quadratic Control Problem. Theory and
Numerical Solution.
Lect. Notes in Control and Information Sciences, vol. 163,
Springer-Verlag, Berlin, 1991.
[3] Sima, V.
Algorithms for Linear-Quadratic Optimization.
Pure and Applied Mathematics: A Series of Monographs and
Textbooks, vol. 200, Marcel Dekker, Inc., New York, 1996.
Numerical Aspects
This routine is particularly suited for systems where the matrix R is ill-conditioned. Internal scaling is used.Further Comments
To obtain a stabilizing solution of the algebraic Riccati equations set SORT = 'S'. The routine can also compute the anti-stabilizing solutions of the algebraic Riccati equations, by specifying SORT = 'U'.Example
Program Text
* SB02OD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER NMAX2M, NMAX2
PARAMETER ( NMAX2M = 2*NMAX+MMAX, NMAX2 = 2*NMAX )
INTEGER LDA, LDB, LDL, LDQ, LDR, LDS, LDT, LDU, LDX
PARAMETER ( LDA = NMAX, LDB = NMAX, LDL = NMAX,
$ LDQ = MAX(NMAX,PMAX), LDR = MAX(MMAX,PMAX),
$ LDS = NMAX2M, LDT = NMAX2M, LDU = NMAX2,
$ LDX = NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MAX(MMAX,NMAX2) )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX(14*NMAX+23,16*NMAX) )
INTEGER LBWORK
PARAMETER ( LBWORK = NMAX2 )
* .. Local Scalars ..
DOUBLE PRECISION RCOND, TOL
INTEGER I, INFO, J, M, N, P
CHARACTER*1 DICO, FACT, JOBB, JOBL, SORT, UPLO
LOGICAL LJOBB
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), ALFAI(NMAX2), ALFAR(NMAX2),
$ B(LDB,MMAX), BETA(NMAX2), DWORK(LDWORK),
$ L(LDL,MMAX), Q(LDQ,NMAX), R(LDR,MMAX),
$ S(LDS,NMAX2M), T(LDT,NMAX2), U(LDU,NMAX2),
$ X(LDX,NMAX)
INTEGER IWORK(LIWORK)
LOGICAL BWORK(LBWORK)
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL SB02OD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL, DICO, JOBB, FACT, UPLO, JOBL,
$ SORT
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) M
ELSE
LJOBB = LSAME( JOBB, 'B' )
IF ( LJOBB ) THEN
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,N ), I = 1,N )
END IF
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) P
ELSE
IF ( LSAME( FACT, 'N' ) .OR. LSAME( FACT, 'D' ) ) THEN
READ ( NIN, FMT = * )
$ ( ( Q(I,J), J = 1,N ), I = 1,N )
ELSE
READ ( NIN, FMT = * )
$ ( ( Q(I,J), J = 1,N ), I = 1,P )
END IF
IF ( LJOBB ) THEN
IF ( LSAME( FACT, 'N' ) .OR. LSAME( FACT, 'C' ) ) THEN
READ ( NIN, FMT = * )
$ ( ( R(I,J), J = 1,M ), I = 1,M )
ELSE
READ ( NIN, FMT = * )
$ ( ( R(I,J), J = 1,M ), I = 1,P )
END IF
IF ( LSAME( JOBL, 'N' ) )
$ READ ( NIN, FMT = * )
$ ( ( L(I,J), J = 1,M ), I = 1,N )
END IF
* Find the solution matrix X.
CALL SB02OD( DICO, JOBB, FACT, UPLO, JOBL, SORT, N, M, P,
$ A, LDA, B, LDB, Q, LDQ, R, LDR, L, LDL,
$ RCOND, X, LDX, ALFAR, ALFAI, BETA, S, LDS,
$ T, LDT, U, LDU, TOL, IWORK, DWORK, LDWORK,
$ BWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( X(I,J), J = 1,N )
20 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB02OD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB02OD = ',I2)
99997 FORMAT (' The solution matrix X is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' N is out of range.',/' N = ',I5)
99994 FORMAT (/' M is out of range.',/' M = ',I5)
99993 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
SB02OD EXAMPLE PROGRAM DATA 2 1 3 0.0 C B B U Z S 0.0 1.0 0.0 0.0 0.0 1.0 1.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 1.0Program Results
SB02OD EXAMPLE PROGRAM RESULTS The solution matrix X is 1.7321 1.0000 1.0000 1.7321
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB02OY.html 0000664 0000000 0000000 00000032600 14560147231 0020371 0 ustar 00root root 0000000 0000000
Purpose
To construct the extended matrix pairs for the computation of the
solution of the algebraic matrix Riccati equations arising in the
problems of optimal control, both discrete and continuous-time,
and of spectral factorization, both discrete and continuous-time.
These matrix pairs, of dimension 2N + M, are given by
discrete-time continuous-time
|A 0 B| |E 0 0| |A 0 B| |E 0 0|
|Q -E' L| - z |0 -A' 0|, |Q A' L| - s |0 -E' 0|. (1)
|L' 0 R| |0 -B' 0| |L' B' R| |0 0 0|
After construction, these pencils are compressed to a form
(see [1])
lambda x A - B ,
f f
where A and B are 2N-by-2N matrices.
f f
-1
Optionally, matrix G = BR B' may be given instead of B and R;
then, for L = 0, 2N-by-2N matrix pairs are directly constructed as
discrete-time continuous-time
|A 0 | |E G | |A -G | |E 0 |
| | - z | |, | | - s | |. (2)
|Q -E'| |0 -A'| |Q A'| |0 -E'|
Similar pairs are obtained for non-zero L, if SLICOT Library
routine SB02MT is called before SB02OY.
Other options include the case with E identity matrix, L a zero
matrix, or Q and/or R given in a factored form, Q = C'C, R = D'D.
For spectral factorization problems, there are minor differences
(e.g., B is replaced by C').
The second matrix in (2) is not constructed in the continuous-time
case if E is specified as being an identity matrix.
Specification
SUBROUTINE SB02OY( TYPE, DICO, JOBB, FACT, UPLO, JOBL, JOBE, N, M,
$ P, A, LDA, B, LDB, Q, LDQ, R, LDR, L, LDL, E,
$ LDE, AF, LDAF, BF, LDBF, TOL, IWORK, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, FACT, JOBB, JOBE, JOBL, TYPE, UPLO
INTEGER INFO, LDA, LDAF, LDB, LDBF, LDE, LDL, LDQ, LDR,
$ LDWORK, M, N, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), AF(LDAF,*), B(LDB,*), BF(LDBF,*),
$ DWORK(*), E(LDE,*), L(LDL,*), Q(LDQ,*), R(LDR,*)
Arguments
Mode Parameters
TYPE CHARACTER*1
Specifies the type of problem to be addressed as follows:
= 'O': Optimal control problem;
= 'S': Spectral factorization problem.
DICO CHARACTER*1
Specifies the type of linear system considered as follows:
= 'C': Continuous-time system;
= 'D': Discrete-time system.
JOBB CHARACTER*1
Specifies whether or not the matrix G is given, instead
of the matrices B and R, as follows:
= 'B': B and R are given;
= 'G': G is given.
For JOBB = 'G', a 2N-by-2N matrix pair is directly
obtained assuming L = 0 (see the description of JOBL).
FACT CHARACTER*1
Specifies whether or not the matrices Q and/or R (if
JOBB = 'B') are factored, as follows:
= 'N': Not factored, Q and R are given;
= 'C': C is given, and Q = C'C;
= 'D': D is given, and R = D'D (if TYPE = 'O'), or
R = D + D' (if TYPE = 'S');
= 'B': Both factors C and D are given, Q = C'C, R = D'D
(or R = D + D').
UPLO CHARACTER*1
If JOBB = 'G', or FACT = 'N', specifies which triangle of
the matrices G and Q (if FACT = 'N'), or Q and R (if
JOBB = 'B'), is stored, as follows:
= 'U': Upper triangle is stored;
= 'L': Lower triangle is stored.
JOBL CHARACTER*1
Specifies whether or not the matrix L is zero, as follows:
= 'Z': L is zero;
= 'N': L is nonzero.
JOBL is not used if JOBB = 'G' and JOBL = 'Z' is assumed.
Using SLICOT Library routine SB02MT to compute the
corresponding A and Q in this case, before calling SB02OY,
enables to obtain 2N-by-2N matrix pairs directly.
JOBE CHARACTER*1
Specifies whether or not the matrix E is identity, as
follows:
= 'I': E is the identity matrix;
= 'N': E is a general matrix.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, Q, and E, and the number
of rows of the matrices B and L. N >= 0.
M (input) INTEGER
If JOBB = 'B', M is the order of the matrix R, and the
number of columns of the matrix B. M >= 0.
M is not used if JOBB = 'G'.
P (input) INTEGER
If FACT = 'C' or 'D' or 'B', or if TYPE = 'S', P is the
number of rows of the matrix C and/or D, respectively.
P >= 0, and if JOBB = 'B' and TYPE = 'S', then P = M.
Otherwise, P is not used.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
state matrix A of the system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,*)
If JOBB = 'B', the leading N-by-M part of this array must
contain the input matrix B of the system.
If JOBB = 'G', the leading N-by-N upper triangular part
(if UPLO = 'U') or lower triangular part (if UPLO = 'L')
of this array must contain the upper triangular part or
lower triangular part, respectively, of the matrix
-1
G = BR B'. The strictly lower triangular part (if
UPLO = 'U') or strictly upper triangular part (if
UPLO = 'L') is not referenced.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
Q (input) DOUBLE PRECISION array, dimension (LDQ,N)
If FACT = 'N' or 'D', the leading N-by-N upper triangular
part (if UPLO = 'U') or lower triangular part (if UPLO =
'L') of this array must contain the upper triangular part
or lower triangular part, respectively, of the symmetric
output weighting matrix Q. The strictly lower triangular
part (if UPLO = 'U') or strictly upper triangular part (if
UPLO = 'L') is not referenced.
If FACT = 'C' or 'B', the leading P-by-N part of this
array must contain the output matrix C of the system.
LDQ INTEGER
The leading dimension of array Q.
LDQ >= MAX(1,N) if FACT = 'N' or 'D',
LDQ >= MAX(1,P) if FACT = 'C' or 'B'.
R (input) DOUBLE PRECISION array, dimension (LDR,M)
If FACT = 'N' or 'C', the leading M-by-M upper triangular
part (if UPLO = 'U') or lower triangular part (if UPLO =
'L') of this array must contain the upper triangular part
or lower triangular part, respectively, of the symmetric
input weighting matrix R. The strictly lower triangular
part (if UPLO = 'U') or strictly upper triangular part (if
UPLO = 'L') is not referenced.
If FACT = 'D' or 'B', the leading P-by-M part of this
array must contain the direct transmission matrix D of the
system.
If JOBB = 'G', this array is not referenced.
LDR INTEGER
The leading dimension of array R.
LDR >= MAX(1,M) if JOBB = 'B' and FACT = 'N' or 'C';
LDR >= MAX(1,P) if JOBB = 'B' and FACT = 'D' or 'B';
LDR >= 1 if JOBB = 'G'.
L (input) DOUBLE PRECISION array, dimension (LDL,M)
If JOBL = 'N' (and JOBB = 'B'), the leading N-by-M part of
this array must contain the cross weighting matrix L.
If JOBL = 'Z' or JOBB = 'G', this array is not referenced.
LDL INTEGER
The leading dimension of array L.
LDL >= MAX(1,N) if JOBL = 'N';
LDL >= 1 if JOBL = 'Z' or JOBB = 'G'.
E (input) DOUBLE PRECISION array, dimension (LDE,N)
If JOBE = 'N', the leading N-by-N part of this array must
contain the matrix E of the descriptor system.
If JOBE = 'I', E is taken as identity and this array is
not referenced.
LDE INTEGER
The leading dimension of array E.
LDE >= MAX(1,N) if JOBE = 'N';
LDE >= 1 if JOBE = 'I'.
AF (output) DOUBLE PRECISION array, dimension (LDAF,*)
The leading 2N-by-2N part of this array contains the
matrix A in the matrix pencil.
f
Array AF must have 2*N+M columns if JOBB = 'B', and 2*N
columns, otherwise.
LDAF INTEGER
The leading dimension of array AF.
LDAF >= MAX(1,2*N+M) if JOBB = 'B',
LDAF >= MAX(1,2*N) if JOBB = 'G'.
BF (output) DOUBLE PRECISION array, dimension (LDBF,2*N)
If DICO = 'D' or JOBB = 'B' or JOBE = 'N', the leading
2N-by-2N part of this array contains the matrix B in the
f
matrix pencil.
The last M zero columns are never constructed.
If DICO = 'C' and JOBB = 'G' and JOBE = 'I', this array
is not referenced.
LDBF INTEGER
The leading dimension of array BF.
LDBF >= MAX(1,2*N+M) if JOBB = 'B',
LDBF >= MAX(1,2*N) if JOBB = 'G' and ( DICO = 'D' or
JOBE = 'N' ),
LDBF >= 1 if JOBB = 'G' and ( DICO = 'C' and
JOBE = 'I' ).
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used to test for near singularity of
the original matrix pencil, specifically of the triangular
factor obtained during the reduction process. If the user
sets TOL > 0, then the given value of TOL is used as a
lower bound for the reciprocal condition number of that
matrix; a matrix whose estimated condition number is less
than 1/TOL is considered to be nonsingular. If the user
sets TOL <= 0, then a default tolerance, defined by
TOLDEF = EPS, is used instead, where EPS is the machine
precision (see LAPACK Library routine DLAMCH).
This parameter is not referenced if JOBB = 'G'.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK >= M if JOBB = 'B',
LIWORK >= 1 if JOBB = 'G'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK. If JOBB = 'B', DWORK(2) returns the reciprocal
of the condition number of the M-by-M lower triangular
matrix obtained after compression.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 1 if JOBB = 'G',
LDWORK >= MAX(1,2*N + M,3*M) if JOBB = 'B'.
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the computed extended matrix pencil is singular,
possibly due to rounding errors.
Method
The extended matrix pairs are constructed, taking various options into account. If JOBB = 'B', the problem order is reduced from 2N+M to 2N (see [1]).References
[1] Van Dooren, P.
A Generalized Eigenvalue Approach for Solving Riccati
Equations.
SIAM J. Sci. Stat. Comp., 2, pp. 121-135, 1981.
[2] Mehrmann, V.
The Autonomous Linear Quadratic Control Problem. Theory and
Numerical Solution.
Lect. Notes in Control and Information Sciences, vol. 163,
Springer-Verlag, Berlin, 1991.
[3] Sima, V.
Algorithms for Linear-Quadratic Optimization.
Pure and Applied Mathematics: A Series of Monographs and
Textbooks, vol. 200, Marcel Dekker, Inc., New York, 1996.
Numerical Aspects
The algorithm is backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve the real continuous-time matrix algebraic Riccati
equation
op(A)'*X + X*op(A) + Q - X*G*X = 0,
where op(A) = A or A' = A**T and G, Q are symmetric (G = G**T,
Q = Q**T). The matrices A, G and Q are N-by-N and the solution X
is an N-by-N symmetric matrix.
An error bound on the solution and a condition estimate are also
optionally provided.
It is assumed that the matrices A, G and Q are such that the
corresponding Hamiltonian matrix has N eigenvalues with negative
real parts.
Specification
SUBROUTINE SB02PD( JOB, TRANA, UPLO, N, A, LDA, G, LDG, Q, LDQ, X,
$ LDX, RCOND, FERR, WR, WI, IWORK, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER JOB, TRANA, UPLO
INTEGER INFO, LDA, LDG, LDQ, LDWORK, LDX, N
DOUBLE PRECISION FERR, RCOND
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), DWORK( * ), G( LDG, * ),
$ Q( LDQ, * ), WI( * ), WR( * ), X( LDX, * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'X': Compute the solution only;
= 'A': Compute all: the solution, reciprocal condition
number, and the error bound.
TRANA CHARACTER*1
Specifies the option op(A):
= 'N': op(A) = A (No transpose);
= 'T': op(A) = A**T (Transpose);
= 'C': op(A) = A**T (Conjugate transpose = Transpose).
UPLO CHARACTER*1
Specifies which triangle of the matrices G and Q is
stored, as follows:
= 'U': Upper triangles of G and Q are stored;
= 'L': Lower triangles of G and Q are stored.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, G, Q, and X. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
coefficient matrix A of the equation.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
G (input) DOUBLE PRECISION array, dimension (LDG,N)
If UPLO = 'U', the leading N-by-N upper triangular part of
this array must contain the upper triangular part of the
matrix G.
If UPLO = 'L', the leading N-by-N lower triangular part of
this array must contain the lower triangular part of the
matrix G.
LDG INTEGER
The leading dimension of the array G. LDG >= max(1,N).
Q (input) DOUBLE PRECISION array, dimension (LDQ,N)
If UPLO = 'U', the leading N-by-N upper triangular part of
this array must contain the upper triangular part of the
matrix Q.
If UPLO = 'L', the leading N-by-N lower triangular part of
this array must contain the lower triangular part of the
matrix Q.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
X (output) DOUBLE PRECISION array, dimension (LDX,N)
If INFO = 0, INFO = 2, or INFO = 4, the leading N-by-N
part of this array contains the symmetric solution matrix
X of the algebraic Riccati equation.
LDX INTEGER
The leading dimension of the array X. LDX >= max(1,N).
RCOND (output) DOUBLE PRECISION
If JOB = 'A', the estimate of the reciprocal condition
number of the Riccati equation.
FERR (output) DOUBLE PRECISION
If JOB = 'A', the estimated forward error bound for the
solution X. If XTRUE is the true solution, FERR bounds the
magnitude of the largest entry in (X - XTRUE) divided by
the magnitude of the largest entry in X.
WR (output) DOUBLE PRECISION array, dimension (N)
WI (output) DOUBLE PRECISION array, dimension (N)
If JOB = 'A' and TRANA = 'N', WR and WI contain the real
and imaginary parts, respectively, of the eigenvalues of
the matrix A - G*X, i.e., the closed-loop system poles.
If JOB = 'A' and TRANA = 'T' or 'C', WR and WI contain the
real and imaginary parts, respectively, of the eigenvalues
of the matrix A - X*G, i.e., the closed-loop system poles.
If JOB = 'X', these arrays are not referenced.
Workspace
IWORK INTEGER array, dimension (LIWORK), where
LIWORK >= 2*N, if JOB = 'X';
LIWORK >= max(2*N,N*N), if JOB = 'A'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or INFO = 2, DWORK(1) contains the
optimal value of LDWORK. If JOB = 'A', then DWORK(2:N*N+1)
and DWORK(N*N+2:2*N*N+1) contain a real Schur form of the
closed-loop system matrix, Ac = A - G*X (if TRANA = 'N')
or Ac = A - X*G (if TRANA = 'T' or 'C'), and the
orthogonal matrix which reduced Ac to real Schur form,
respectively.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= 4*N*N + 8*N + 1, if JOB = 'X';
LDWORK >= max( 4*N*N + 8*N + 1, 6*N*N ), if JOB = 'A'.
For good performance, LDWORK should be larger, e.g.,
LDWORK >= 4*N*N + 6*N +( 2*N+1 )*NB, if JOB = 'X',
where NB is the optimal blocksize.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the Hamiltonian matrix has eigenvalues on the
imaginary axis, so the solution and error bounds
could not be computed;
= 2: the iteration for the matrix sign function failed to
converge after 50 iterations, but an approximate
solution and error bounds (if JOB = 'A') have been
computed;
= 3: the system of linear equations for the solution is
singular to working precision, so the solution and
error bounds could not be computed;
= 4: the matrix A-G*X (or A-X*G) cannot be reduced to
Schur canonical form and condition number estimate
and forward error estimate have not been computed.
Method
The Riccati equation is solved by the matrix sign function approach [1], [2], implementing a scaling which enhances the numerical stability [4].References
[1] Bai, Z., Demmel, J., Dongarra, J., Petitet, A., Robinson, H.,
and Stanley, K.
The spectral decomposition of nonsymmetric matrices on
distributed memory parallel computers.
SIAM J. Sci. Comput., vol. 18, pp. 1446-1461, 1997.
[2] Byers, R., He, C., and Mehrmann, V.
The matrix sign function method and the computation of
invariant subspaces.
SIAM J. Matrix Anal. Appl., vol. 18, pp. 615-632, 1997.
[3] Higham, N.J.
Perturbation theory and backward error for AX-XB=C.
BIT, vol. 33, pp. 124-136, 1993.
[4] Petkov, P.Hr., Konstantinov, M.M., and Mehrmann, V.,
DGRSVX and DMSRIC: Fortran 77 subroutines for solving
continuous-time matrix algebraic Riccati equations with
condition and accuracy estimates.
Preprint SFB393/98-16, Fak. f. Mathematik, Technical
University Chemnitz, May 1998.
Numerical Aspects
The solution accuracy can be controlled by the output parameter FERR.Further Comments
The condition number of the Riccati equation is estimated as
cond = ( norm(Theta)*norm(A) + norm(inv(Omega))*norm(Q) +
norm(Pi)*norm(G) ) / norm(X),
where Omega, Theta and Pi are linear operators defined by
Omega(W) = op(Ac)'*W + W*op(Ac),
Theta(W) = inv(Omega(op(W)'*X + X*op(W))),
Pi(W) = inv(Omega(X*W*X)),
and the matrix Ac (the closed-loop system matrix) is given by
Ac = A - G*X, if TRANA = 'N', or
Ac = A - X*G, if TRANA = 'T' or 'C'.
The program estimates the quantities
sep(op(Ac),-op(Ac)') = 1 / norm(inv(Omega)),
norm(Theta) and norm(Pi) using 1-norm condition estimator.
The forward error bound is estimated using a practical error bound
similar to the one proposed in [3].
Example
Program Text
* SB02PD EXAMPLE PROGRAM TEXT.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA, LDG, LDQ, LDX
PARAMETER ( LDA = NMAX, LDG = NMAX, LDQ = NMAX,
$ LDX = NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MAX( 2*NMAX, NMAX*NMAX ) )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( 4*NMAX*NMAX + 8*NMAX,
$ 6*NMAX*NMAX ) + 1 )
* .. Local Scalars ..
DOUBLE PRECISION FERR, RCOND
INTEGER I, INFO, J, N
CHARACTER JOB, TRANA, UPLO
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), G(LDG,NMAX),
$ Q(LDQ,NMAX), WI(NMAX), WR(NMAX),
$ X(LDX,NMAX)
INTEGER IWORK(LIWORK)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL SB02PD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, JOB, TRANA, UPLO
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( Q(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( G(I,J), J = 1,N ), I = 1,N )
* Find the solution matrix X.
CALL SB02PD( JOB, TRANA, UPLO, N, A, LDA, G, LDG, Q, LDQ, X,
$ LDX, RCOND, FERR, WR, WI, IWORK, DWORK, LDWORK,
$ INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
END IF
IF ( INFO.EQ.0 .OR. INFO.EQ.2 .OR. INFO.EQ.4 ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( X(I,J), J = 1,N )
20 CONTINUE
IF ( LSAME( JOB, 'A' ) .AND. INFO.NE.4 ) THEN
WRITE ( NOUT, FMT = 99994 ) RCOND
WRITE ( NOUT, FMT = 99993 ) FERR
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB02PD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB02PD = ',I2)
99997 FORMAT (' The solution matrix X is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' N is out of range.',/' N = ',I5)
99994 FORMAT (/' Estimated reciprocal condition number = ',F8.4)
99993 FORMAT (/' Estimated error bound = ',F20.16)
END
Program Data
SB02PD EXAMPLE PROGRAM DATA 2 A N U 0.0 1.0 0.0 0.0 1.0 0.0 0.0 2.0 0.0 0.0 0.0 1.0Program Results
SB02PD EXAMPLE PROGRAM RESULTS The solution matrix X is 2.0000 1.0000 1.0000 2.0000 Estimated reciprocal condition number = 0.1333 Estimated error bound = 0.0000000000000063
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB02QD.html 0000664 0000000 0000000 00000044340 14560147231 0020352 0 ustar 00root root 0000000 0000000
Purpose
To estimate the conditioning and compute an error bound on the
solution of the real continuous-time matrix algebraic Riccati
equation
op(A)'*X + X*op(A) + Q - X*G*X = 0, (1)
where op(A) = A or A' (A**T) and Q, G are symmetric (Q = Q**T,
G = G**T). The matrices A, Q and G are N-by-N and the solution X
is N-by-N.
Specification
SUBROUTINE SB02QD( JOB, FACT, TRANA, UPLO, LYAPUN, N, A, LDA, T,
$ LDT, U, LDU, G, LDG, Q, LDQ, X, LDX, SEP,
$ RCOND, FERR, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER FACT, JOB, LYAPUN, TRANA, UPLO
INTEGER INFO, LDA, LDG, LDQ, LDT, LDU, LDWORK, LDX, N
DOUBLE PRECISION FERR, RCOND, SEP
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), DWORK( * ), G( LDG, * ),
$ Q( LDQ, * ), T( LDT, * ), U( LDU, * ),
$ X( LDX, * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'C': Compute the reciprocal condition number only;
= 'E': Compute the error bound only;
= 'B': Compute both the reciprocal condition number and
the error bound.
FACT CHARACTER*1
Specifies whether or not the real Schur factorization of
the matrix Ac = A - G*X (if TRANA = 'N') or Ac = A - X*G
(if TRANA = 'T' or 'C') is supplied on entry, as follows:
= 'F': On entry, T and U (if LYAPUN = 'O') contain the
factors from the real Schur factorization of the
matrix Ac;
= 'N': The Schur factorization of Ac will be computed
and the factors will be stored in T and U (if
LYAPUN = 'O').
TRANA CHARACTER*1
Specifies the form of op(A) to be used, as follows:
= 'N': op(A) = A (No transpose);
= 'T': op(A) = A**T (Transpose);
= 'C': op(A) = A**T (Conjugate transpose = Transpose).
UPLO CHARACTER*1
Specifies which part of the symmetric matrices Q and G is
to be used, as follows:
= 'U': Upper triangular part;
= 'L': Lower triangular part.
LYAPUN CHARACTER*1
Specifies whether or not the original Lyapunov equations
should be solved in the iterative estimation process,
as follows:
= 'O': Solve the original Lyapunov equations, updating
the right-hand sides and solutions with the
matrix U, e.g., RHS <-- U'*RHS*U;
= 'R': Solve reduced Lyapunov equations only, without
updating the right-hand sides and solutions.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, X, Q, and G. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
If FACT = 'N' or LYAPUN = 'O', the leading N-by-N part of
this array must contain the matrix A.
If FACT = 'F' and LYAPUN = 'R', A is not referenced.
LDA INTEGER
The leading dimension of the array A.
LDA >= max(1,N), if FACT = 'N' or LYAPUN = 'O';
LDA >= 1, if FACT = 'F' and LYAPUN = 'R'.
T (input or output) DOUBLE PRECISION array, dimension
(LDT,N)
If FACT = 'F', then T is an input argument and on entry,
the leading N-by-N upper Hessenberg part of this array
must contain the upper quasi-triangular matrix T in Schur
canonical form from a Schur factorization of Ac (see
argument FACT).
If FACT = 'N', then T is an output argument and on exit,
if INFO = 0 or INFO = N+1, the leading N-by-N upper
Hessenberg part of this array contains the upper quasi-
triangular matrix T in Schur canonical form from a Schur
factorization of Ac (see argument FACT).
LDT INTEGER
The leading dimension of the array T. LDT >= max(1,N).
U (input or output) DOUBLE PRECISION array, dimension
(LDU,N)
If LYAPUN = 'O' and FACT = 'F', then U is an input
argument and on entry, the leading N-by-N part of this
array must contain the orthogonal matrix U from a real
Schur factorization of Ac (see argument FACT).
If LYAPUN = 'O' and FACT = 'N', then U is an output
argument and on exit, if INFO = 0 or INFO = N+1, it
contains the orthogonal N-by-N matrix from a real Schur
factorization of Ac (see argument FACT).
If LYAPUN = 'R', the array U is not referenced.
LDU INTEGER
The leading dimension of the array U.
LDU >= 1, if LYAPUN = 'R';
LDU >= MAX(1,N), if LYAPUN = 'O'.
G (input) DOUBLE PRECISION array, dimension (LDG,N)
If UPLO = 'U', the leading N-by-N upper triangular part of
this array must contain the upper triangular part of the
matrix G.
If UPLO = 'L', the leading N-by-N lower triangular part of
this array must contain the lower triangular part of the
matrix G. _
Matrix G should correspond to G in the "reduced" Riccati
equation (with matrix T, instead of A), if LYAPUN = 'R'.
See METHOD.
LDG INTEGER
The leading dimension of the array G. LDG >= max(1,N).
Q (input) DOUBLE PRECISION array, dimension (LDQ,N)
If UPLO = 'U', the leading N-by-N upper triangular part of
this array must contain the upper triangular part of the
matrix Q.
If UPLO = 'L', the leading N-by-N lower triangular part of
this array must contain the lower triangular part of the
matrix Q. _
Matrix Q should correspond to Q in the "reduced" Riccati
equation (with matrix T, instead of A), if LYAPUN = 'R'.
See METHOD.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
X (input) DOUBLE PRECISION array, dimension (LDX,N)
The leading N-by-N part of this array must contain the
symmetric solution matrix of the original Riccati
equation (with matrix A), if LYAPUN = 'O', or of the
"reduced" Riccati equation (with matrix T), if
LYAPUN = 'R'. See METHOD.
LDX INTEGER
The leading dimension of the array X. LDX >= max(1,N).
SEP (output) DOUBLE PRECISION
If JOB = 'C' or JOB = 'B', the estimated quantity
sep(op(Ac),-op(Ac)').
If N = 0, or X = 0, or JOB = 'E', SEP is not referenced.
RCOND (output) DOUBLE PRECISION
If JOB = 'C' or JOB = 'B', an estimate of the reciprocal
condition number of the continuous-time Riccati equation.
If N = 0 or X = 0, RCOND is set to 1 or 0, respectively.
If JOB = 'E', RCOND is not referenced.
FERR (output) DOUBLE PRECISION
If JOB = 'E' or JOB = 'B', an estimated forward error
bound for the solution X. If XTRUE is the true solution,
FERR bounds the magnitude of the largest entry in
(X - XTRUE) divided by the magnitude of the largest entry
in X.
If N = 0 or X = 0, FERR is set to 0.
If JOB = 'C', FERR is not referenced.
Workspace
IWORK INTEGER array, dimension (N*N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or INFO = N+1, DWORK(1) returns the
optimal value of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
Let LWA = N*N, if LYAPUN = 'O' and JOB = 'E' or 'B';
LWA = 0, otherwise.
If FACT = 'N', then
LDWORK = MAX(1, 5*N, 2*N*N), if JOB = 'C';
LDWORK = MAX(1, LWA + 5*N, 4*N*N ), if JOB = 'E', 'B'.
If FACT = 'F', then
LDWORK = MAX(1, 2*N*N), if JOB = 'C';
LDWORK = MAX(1, 4*N*N ), if JOB = 'E' or 'B'.
For good performance, LDWORK must generally be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, i <= N, the QR algorithm failed to
complete the reduction of the matrix Ac to Schur
canonical form (see LAPACK Library routine DGEES);
on exit, the matrix T(i+1:N,i+1:N) contains the
partially converged Schur form, and DWORK(i+1:N) and
DWORK(N+i+1:2*N) contain the real and imaginary
parts, respectively, of the converged eigenvalues;
this error is unlikely to appear;
= N+1: if the matrices T and -T' have common or very
close eigenvalues; perturbed values were used to
solve Lyapunov equations, but the matrix T, if given
(for FACT = 'F'), is unchanged.
Method
The condition number of the Riccati equation is estimated as
cond = ( norm(Theta)*norm(A) + norm(inv(Omega))*norm(Q) +
norm(Pi)*norm(G) ) / norm(X),
where Omega, Theta and Pi are linear operators defined by
Omega(W) = op(Ac)'*W + W*op(Ac),
Theta(W) = inv(Omega(op(W)'*X + X*op(W))),
Pi(W) = inv(Omega(X*W*X)),
and Ac = A - G*X (if TRANA = 'N') or Ac = A - X*G (if TRANA = 'T'
or 'C'). Note that the Riccati equation (1) is equivalent to
_ _ _ _ _ _
op(T)'*X + X*op(T) + Q + X*G*X = 0, (2)
_ _ _
where X = U'*X*U, Q = U'*Q*U, and G = U'*G*U, with U the
orthogonal matrix reducing Ac to a real Schur form, T = U'*Ac*U.
The routine estimates the quantities
sep(op(Ac),-op(Ac)') = 1 / norm(inv(Omega)),
norm(Theta) and norm(Pi) using 1-norm condition estimator.
The forward error bound is estimated using a practical error bound
similar to the one proposed in [2].
References
[1] Ghavimi, A.R. and Laub, A.J.
Backward error, sensitivity, and refinement of computed
solutions of algebraic Riccati equations.
Numerical Linear Algebra with Applications, vol. 2, pp. 29-49,
1995.
[2] Higham, N.J.
Perturbation theory and backward error for AX-XB=C.
BIT, vol. 33, pp. 124-136, 1993.
[3] Petkov, P.Hr., Konstantinov, M.M., and Mehrmann, V.
DGRSVX and DMSRIC: Fortran 77 subroutines for solving
continuous-time matrix algebraic Riccati equations with
condition and accuracy estimates.
Preprint SFB393/98-16, Fak. f. Mathematik, Tech. Univ.
Chemnitz, May 1998.
Numerical Aspects
3 The algorithm requires 0(N ) operations. The accuracy of the estimates obtained depends on the solution accuracy and on the properties of the 1-norm estimator.Further Comments
The option LYAPUN = 'R' may occasionally produce slightly worse or better estimates, and it is much faster than the option 'O'. When SEP is computed and it is zero, the routine returns immediately, with RCOND and FERR (if requested) set to 0 and 1, respectively. In this case, the equation is singular.Example
Program Text
* SB02QD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA, LDG, LDQ, LDT, LDU, LDX
PARAMETER ( LDA = NMAX, LDG = NMAX, LDQ = NMAX, LDT = NMAX,
$ LDU = NMAX, LDX = NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX*NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 8*NMAX*NMAX + 10*NMAX )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* .. Local Scalars ..
DOUBLE PRECISION FERR, RCND, RCOND, SEP
INTEGER I, INFO1, INFO2, INFO3, IS, IU, IW, J, N, N2,
$ SDIM
CHARACTER*1 FACT, JOB, JOBS, LYAPUN, TRANA, TRANAT, UPLO
* .. Local Arrays ..
LOGICAL BWORK(2*NMAX)
INTEGER IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), G(LDG,NMAX),
$ Q(LDQ,NMAX), T(LDT,NMAX), U(LDU,NMAX),
$ X(LDX,NMAX)
* .. External Functions ..
LOGICAL LSAME, SELECT
EXTERNAL LSAME, SELECT
* .. External Subroutines ..
EXTERNAL DGEES, DLACPY, DSYMM, MA02ED, MB01RU, SB02MD,
$ SB02QD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, JOB, FACT, TRANA, UPLO, LYAPUN
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( Q(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( G(I,J), J = 1,N ), I = 1,N )
CALL DLACPY( UPLO, N, N, Q, LDQ, X, LDX )
N2 = 2*N
IS = 2*N2 + 1
IU = IS + N2*N2
IW = IU + N2*N2
* Solve the continuous-time Riccati equation.
CALL SB02MD( 'continuous', 'direct', UPLO, 'no scaling',
$ 'stable', N, A, LDA, G, LDG, X, LDX, RCND,
$ DWORK(1), DWORK(N2+1), DWORK(IS), N2, DWORK(IU),
$ N2, IWORK, DWORK(IW), LDWORK-IW+1, BWORK, INFO1 )
*
IF ( INFO1.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99995 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99994 ) ( X(I,J), J = 1,N )
10 CONTINUE
IF ( LSAME( FACT, 'F' ) .OR. LSAME( LYAPUN, 'R' ) ) THEN
CALL DLACPY( 'Full', N, N, A, LDA, T, LDT )
IF ( LSAME( TRANA, 'N' ) ) THEN
* Compute Ac = A-G*X.
CALL DSYMM( 'Left', UPLO, N, N, -ONE, G, LDG, X, LDX,
$ ONE, T, LDT )
ELSE
* Compute Ac = A-X*G.
CALL DSYMM( 'Right', UPLO, N, N, -ONE, G, LDG, X, LDX,
$ ONE, T, LDT )
END IF
* Compute the Schur factorization of Ac.
JOBS = 'V'
CALL DGEES( JOBS, 'Not ordered', SELECT, N, T, LDT, SDIM,
$ DWORK(1), DWORK(N+1), U, LDU, DWORK(2*N+1),
$ LDWORK-2*N, BWORK, INFO3 )
IF( INFO3.NE.0 ) THEN
WRITE ( NOUT, FMT = 99996 ) INFO3
STOP
END IF
END IF
*
IF ( LSAME( LYAPUN, 'R' ) ) THEN
IF( LSAME( TRANA, 'N' ) ) THEN
TRANAT = 'T'
ELSE
TRANAT = 'N'
END IF
*
CALL MB01RU( UPLO, TRANAT, N, N, ZERO, ONE, X, LDX,
$ U, LDU, X, LDX, DWORK, N*N, INFO2 )
CALL MA02ED( UPLO, N, X, LDX )
CALL MB01RU( UPLO, TRANAT, N, N, ZERO, ONE, G, LDG,
$ U, LDU, G, LDG, DWORK, N*N, INFO2 )
CALL MB01RU( UPLO, TRANAT, N, N, ZERO, ONE, Q, LDQ,
$ U, LDU, Q, LDQ, DWORK, N*N, INFO2 )
END IF
* Estimate the condition and error bound on the solution.
CALL SB02QD( JOB, FACT, TRANA, UPLO, LYAPUN, N, A, LDA, T,
$ LDT, U, LDU, G, LDG, Q, LDQ, X, LDX, SEP,
$ RCOND, FERR, IWORK, DWORK, LDWORK, INFO2 )
*
IF ( INFO2.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO2
END IF
IF ( INFO2.EQ.0 .OR. INFO2.EQ.N+1 ) THEN
WRITE ( NOUT, FMT = 99992 ) SEP
WRITE ( NOUT, FMT = 99991 ) RCOND
WRITE ( NOUT, FMT = 99990 ) FERR
END IF
ELSE
WRITE ( NOUT, FMT = 99998 ) INFO1
END IF
END IF
STOP
*
99999 FORMAT (' SB02QD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB02MD =',I2)
99997 FORMAT (' INFO on exit from SB02QD =',I2)
99996 FORMAT (' INFO on exit from DGEES =',I2)
99995 FORMAT (' The solution matrix X is')
99994 FORMAT (20(1X,F8.4))
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' Estimated separation = ',F8.4)
99991 FORMAT (/' Estimated reciprocal condition number = ',F8.4)
99990 FORMAT (/' Estimated error bound = ',F8.4)
END
Program Data
SB02QD EXAMPLE PROGRAM DATA 2 B N N U O 0.0 1.0 0.0 0.0 1.0 0.0 0.0 2.0 0.0 0.0 0.0 1.0Program Results
SB02QD EXAMPLE PROGRAM RESULTS The solution matrix X is 2.0000 1.0000 1.0000 2.0000 Estimated separation = 0.4000 Estimated reciprocal condition number = 0.1333 Estimated error bound = 0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB02RD.html 0000664 0000000 0000000 00000071755 14560147231 0020365 0 ustar 00root root 0000000 0000000
Purpose
To solve for X either the continuous-time algebraic Riccati
equation
-1
Q + op(A)'*X + X*op(A) - X*op(B)*R op(B)'*X = 0, (1)
or the discrete-time algebraic Riccati equation
-1
X = op(A)'*X*op(A) - op(A)'*X*op(B)*(R + op(B)'*X*op(B)) *
op(B)'*X*op(A) + Q, (2)
where op(M) = M or M' (M**T), A, op(B), Q, and R are N-by-N,
N-by-M, N-by-N, and M-by-M matrices respectively, with Q symmetric
and R symmetric nonsingular; X is an N-by-N symmetric matrix.
-1
The matrix G = op(B)*R *op(B)' must be provided on input, instead
of B and R, that is, the continuous-time equation
Q + op(A)'*X + X*op(A) - X*G*X = 0, (3)
or the discrete-time equation
-1
Q + op(A)'*X*(I_n + G*X) *op(A) - X = 0, (4)
are solved, where G is an N-by-N symmetric matrix. SLICOT Library
routine SB02MT should be used to compute G, given B and R. SB02MT
also enables to solve Riccati equations corresponding to optimal
problems with coupling terms.
The routine also returns the computed values of the closed-loop
spectrum of the optimal system, i.e., the stable eigenvalues
lambda(1),...,lambda(N) of the corresponding Hamiltonian or
symplectic matrix associated to the optimal problem. It is assumed
that the matrices A, G, and Q are such that the associated
Hamiltonian or symplectic matrix has N stable eigenvalues, i.e.,
with negative real parts, in the continuous-time case, and with
moduli less than one, in the discrete-time case.
Optionally, estimates of the conditioning and error bound on the
solution of the Riccati equation (3) or (4) are returned.
Specification
SUBROUTINE SB02RD( JOB, DICO, HINV, TRANA, UPLO, SCAL, SORT, FACT,
$ LYAPUN, N, A, LDA, T, LDT, V, LDV, G, LDG, Q,
$ LDQ, X, LDX, SEP, RCOND, FERR, WR, WI, S, LDS,
$ IWORK, DWORK, LDWORK, BWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, FACT, HINV, JOB, LYAPUN, SCAL, SORT,
$ TRANA, UPLO
INTEGER INFO, LDA, LDG, LDQ, LDS, LDT, LDV, LDWORK, LDX,
$ N
DOUBLE PRECISION FERR, RCOND, SEP
C .. Array Arguments ..
LOGICAL BWORK(*)
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), DWORK(*), G(LDG,*), Q(LDQ,*),
$ S(LDS,*), T(LDT,*), V(LDV,*), WI(*), WR(*),
$ X(LDX,*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'X': Compute the solution only;
= 'C': Compute the reciprocal condition number only;
= 'E': Compute the error bound only;
= 'A': Compute all: the solution, reciprocal condition
number, and the error bound.
DICO CHARACTER*1
Specifies the type of Riccati equation to be solved or
analyzed, as follows:
= 'C': Equation (3), continuous-time case;
= 'D': Equation (4), discrete-time case.
HINV CHARACTER*1
If DICO = 'D' and JOB = 'X' or JOB = 'A', specifies which
symplectic matrix is to be constructed, as follows:
= 'D': The matrix H in (6) (see METHOD) is constructed;
= 'I': The inverse of the matrix H in (6) is constructed.
HINV is not used if DICO = 'C', or JOB = 'C' or 'E'.
TRANA CHARACTER*1
Specifies the form of op(A) to be used, as follows:
= 'N': op(A) = A (No transpose);
= 'T': op(A) = A**T (Transpose);
= 'C': op(A) = A**T (Conjugate transpose = Transpose).
UPLO CHARACTER*1
Specifies which triangle of the matrices G and Q is
stored, as follows:
= 'U': Upper triangle is stored;
= 'L': Lower triangle is stored.
SCAL CHARACTER*1
If JOB = 'X' or JOB = 'A', specifies whether or not a
scaling strategy should be used, as follows:
= 'G': General scaling should be used;
= 'N': No scaling should be used.
SCAL is not used if JOB = 'C' or 'E'.
SORT CHARACTER*1
If JOB = 'X' or JOB = 'A', specifies which eigenvalues
should be obtained in the top of the Schur form, as
follows:
= 'S': Stable eigenvalues come first;
= 'U': Unstable eigenvalues come first.
SORT is not used if JOB = 'C' or 'E'.
FACT CHARACTER*1
If JOB <> 'X', specifies whether or not a real Schur
factorization of the closed-loop system matrix Ac is
supplied on entry, as follows:
= 'F': On entry, T and V contain the factors from a real
Schur factorization of the matrix Ac;
= 'N': A Schur factorization of Ac will be computed
and the factors will be stored in T and V.
For a continuous-time system, the matrix Ac is given by
Ac = A - G*X, if TRANA = 'N', or
Ac = A - X*G, if TRANA = 'T' or 'C',
and for a discrete-time system, the matrix Ac is given by
Ac = inv(I_n + G*X)*A, if TRANA = 'N', or
Ac = A*inv(I_n + X*G), if TRANA = 'T' or 'C'.
FACT is not used if JOB = 'X'.
LYAPUN CHARACTER*1
If JOB <> 'X', specifies whether or not the original or
"reduced" Lyapunov equations should be solved for
estimating reciprocal condition number and/or the error
bound, as follows:
= 'O': Solve the original Lyapunov equations, updating
the right-hand sides and solutions with the
matrix V, e.g., X <-- V'*X*V;
= 'R': Solve reduced Lyapunov equations only, without
updating the right-hand sides and solutions.
This means that a real Schur form T of Ac appears
in the equations, instead of Ac.
LYAPUN is not used if JOB = 'X'.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, Q, G, and X. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
If JOB = 'X' or JOB = 'A' or FACT = 'N' or LYAPUN = 'O',
the leading N-by-N part of this array must contain the
coefficient matrix A of the equation.
If JOB = 'C' or 'E' and FACT = 'F' and LYAPUN = 'R', A is
not referenced.
LDA INTEGER
The leading dimension of the array A.
LDA >= MAX(1,N), if JOB = 'X' or JOB = 'A' or
FACT = 'N' or LYAPUN = 'O'.
LDA >= 1, otherwise.
T (input or output) DOUBLE PRECISION array, dimension
(LDT,N)
If JOB <> 'X' and FACT = 'F', then T is an input argument
and on entry, the leading N-by-N upper Hessenberg part of
this array must contain the upper quasi-triangular matrix
T in Schur canonical form from a Schur factorization of Ac
(see argument FACT).
If JOB <> 'X' and FACT = 'N', then T is an output argument
and on exit, if INFO = 0 or INFO = 7, the leading N-by-N
upper Hessenberg part of this array contains the upper
quasi-triangular matrix T in Schur canonical form from a
Schur factorization of Ac (see argument FACT).
If JOB = 'X', the array T is not referenced.
LDT INTEGER
The leading dimension of the array T.
LDT >= 1, if JOB = 'X';
LDT >= MAX(1,N), if JOB <> 'X'.
V (input or output) DOUBLE PRECISION array, dimension
(LDV,N)
If JOB <> 'X' and FACT = 'F', then V is an input argument
and on entry, the leading N-by-N part of this array must
contain the orthogonal matrix V from a real Schur
factorization of Ac (see argument FACT).
If JOB <> 'X' and FACT = 'N', then V is an output argument
and on exit, if INFO = 0 or INFO = 7, the leading N-by-N
part of this array contains the orthogonal N-by-N matrix
from a real Schur factorization of Ac (see argument FACT).
If JOB = 'X', the array V is not referenced.
LDV INTEGER
The leading dimension of the array V.
LDV >= 1, if JOB = 'X';
LDV >= MAX(1,N), if JOB <> 'X'.
G (input/output) DOUBLE PRECISION array, dimension (LDG,N)
On entry, the leading N-by-N upper triangular part (if
UPLO = 'U') or lower triangular part (if UPLO = 'L') of
this array must contain the upper triangular part or lower
triangular part, respectively, of the symmetric matrix G.
On exit, if JOB = 'X' and DICO = 'D', or JOB <> 'X' and
LYAPUN = 'R', the leading N-by-N part of this array
contains the symmetric matrix G fully stored.
If JOB <> 'X' and LYAPUN = 'R', this array is modified
internally, but restored on exit.
LDG INTEGER
The leading dimension of the array G. LDG >= MAX(1,N).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, the leading N-by-N upper triangular part (if
UPLO = 'U') or lower triangular part (if UPLO = 'L') of
this array must contain the upper triangular part or lower
triangular part, respectively, of the symmetric matrix Q.
On exit, if JOB = 'X' and DICO = 'D', or JOB <> 'X' and
LYAPUN = 'R', the leading N-by-N part of this array
contains the symmetric matrix Q fully stored.
If JOB <> 'X' and LYAPUN = 'R', this array is modified
internally, but restored on exit.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= MAX(1,N).
X (input or output) DOUBLE PRECISION array, dimension
(LDX,N)
If JOB = 'C' or JOB = 'E', then X is an input argument
and on entry, the leading N-by-N part of this array must
contain the symmetric solution matrix of the algebraic
Riccati equation. If LYAPUN = 'R', this array is modified
internally, but restored on exit; however, it could differ
from the input matrix at the round-off error level.
If JOB = 'X' or JOB = 'A', then X is an output argument
and on exit, if INFO = 0 or INFO >= 6, the leading N-by-N
part of this array contains the symmetric solution matrix
X of the algebraic Riccati equation.
LDX INTEGER
The leading dimension of the array X. LDX >= MAX(1,N).
SEP (output) DOUBLE PRECISION
If JOB = 'C' or JOB = 'A', and INFO = 0 or INFO = 7, the
estimated quantity
sep(op(Ac),-op(Ac)'), if DICO = 'C', or
sepd(op(Ac),op(Ac)'), if DICO = 'D'. (See METHOD.)
If JOB = 'C' or JOB = 'A' and X = 0, or JOB = 'E', SEP is
not referenced.
If JOB = 'X', and INFO = 0, INFO = 5 or INFO = 7,
SEP contains the scaling factor used, which should
multiply the (2,1) submatrix of U to recover X from the
first N columns of U (see METHOD). If SCAL = 'N', SEP is
set to 1.
RCOND (output) DOUBLE PRECISION
If JOB = 'C' or JOB = 'A', and INFO = 0 or INFO = 7, an
estimate of the reciprocal condition number of the
algebraic Riccati equation.
If N = 0 or X = 0, RCOND is set to 1 or 0, respectively.
If JOB = 'X', or JOB = 'E', RCOND is not referenced.
FERR (output) DOUBLE PRECISION
If JOB = 'E' or JOB = 'A', and INFO = 0 or INFO = 7, an
estimated forward error bound for the solution X. If XTRUE
is the true solution, FERR bounds the magnitude of the
largest entry in (X - XTRUE) divided by the magnitude of
the largest entry in X.
If N = 0 or X = 0, FERR is set to 0.
If JOB = 'X', or JOB = 'C', FERR is not referenced.
WR (output) DOUBLE PRECISION array, dimension (2*N)
WI (output) DOUBLE PRECISION array, dimension (2*N)
If JOB = 'X' or JOB = 'A', and INFO = 0 or INFO >= 5,
these arrays contain the real and imaginary parts,
respectively, of the eigenvalues of the 2N-by-2N matrix S,
ordered as specified by SORT (except for the case
HINV = 'D', when the order is opposite to that specified
by SORT). The leading N elements of these arrays contain
the closed-loop spectrum of the system matrix Ac (see
argument FACT). Specifically,
lambda(k) = WR(k) + j*WI(k), for k = 1,2,...,N.
If JOB = 'C' or JOB = 'E', these arrays are not
referenced.
S (output) DOUBLE PRECISION array, dimension (LDS,2*N)
If JOB = 'X' or JOB = 'A', and INFO = 0 or INFO >= 5, the
leading 2N-by-2N part of this array contains the ordered
real Schur form S of the (scaled, if SCAL = 'G')
Hamiltonian or symplectic matrix H. That is,
( S S )
( 11 12 )
S = ( ),
( 0 S )
( 22 )
where S , S and S are N-by-N matrices.
11 12 22
If JOB = 'C' or JOB = 'E', this array is not referenced.
LDS INTEGER
The leading dimension of the array S.
LDS >= MAX(1,2*N), if JOB = 'X' or JOB = 'A';
LDS >= 1, if JOB = 'C' or JOB = 'E'.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK >= 2*N, if JOB = 'X';
LIWORK >= N*N, if JOB = 'C' or JOB = 'E';
LIWORK >= MAX(2*N,N*N), if JOB = 'A'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, or INFO = 7, DWORK(1) returns the
optimal value of LDWORK. If INFO = 0, or INFO >= 5, and
JOB = 'X', or JOB = 'A', then DWORK(2) returns an estimate
RCONDU of the reciprocal of the condition number (in the
1-norm) of the N-th order system of algebraic equations
from which the solution matrix X is obtained, and DWORK(3)
returns the reciprocal pivot growth factor for the LU
factorization of the coefficient matrix of that system
(see SLICOT Library routine MB02PD); if DWORK(3) is much
less than 1, then the computed X and RCONDU could be
unreliable.
If DICO = 'D', and JOB = 'X', or JOB = 'A', then DWORK(4)
returns the reciprocal condition number RCONDA of the
given matrix A, and DWORK(5) returns the reciprocal pivot
growth factor for A or for its leading columns, if A is
singular (see SLICOT Library routine MB02PD); if DWORK(5)
is much less than 1, then the computed S and RCONDA could
be unreliable.
On exit, if INFO = 0, or INFO >= 4, and JOB = 'X', the
elements DWORK(6:5+4*N*N) contain the 2*N-by-2*N
transformation matrix U which reduced the Hamiltonian or
symplectic matrix H to the ordered real Schur form S.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 5+MAX(1,4*N*N+8*N), if JOB = 'X' or JOB = 'A';
This may also be used for JOB = 'C' or JOB = 'E', but
exact bounds are as follows:
LDWORK >= 5 + MAX(1,LWS,LWE) + LWN, where
LWS = 0, if FACT = 'F' or LYAPUN = 'R';
= 5*N, if FACT = 'N' and LYAPUN = 'O' and
DICO = 'C' and JOB = 'C';
= 5*N+N*N, if FACT = 'N' and LYAPUN = 'O' and
DICO = 'C' and JOB = 'E';
= 5*N+N*N, if FACT = 'N' and LYAPUN = 'O' and
DICO = 'D';
LWE = 2*N*N, if DICO = 'C' and JOB = 'C';
= 4*N*N, if DICO = 'C' and JOB = 'E';
= MAX(3,2*N*N) + N*N, if DICO = 'D' and JOB = 'C';
= MAX(3,2*N*N) + 2*N*N, if DICO = 'D' and JOB = 'E';
LWN = 0, if LYAPUN = 'O' or JOB = 'C';
= 2*N, if LYAPUN = 'R' and DICO = 'C' and JOB = 'E';
= 3*N, if LYAPUN = 'R' and DICO = 'D' and JOB = 'E'.
For optimum performance LDWORK should sometimes be larger.
BWORK LOGICAL array, dimension (LBWORK)
LBWORK >= 2*N, if JOB = 'X' or JOB = 'A';
LBWORK >= 1, if JOB = 'C' or JOB = 'E', and
FACT = 'N' and LYAPUN = 'R';
LBWORK >= 0, otherwise.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if matrix A is (numerically) singular in discrete-
time case;
= 2: if the Hamiltonian or symplectic matrix H cannot be
reduced to real Schur form;
= 3: if the real Schur form of the Hamiltonian or
symplectic matrix H cannot be appropriately ordered;
= 4: if the Hamiltonian or symplectic matrix H has less
than N stable eigenvalues;
= 5: if the N-th order system of linear algebraic
equations, from which the solution matrix X would
be obtained, is singular to working precision;
= 6: if the QR algorithm failed to complete the reduction
of the matrix Ac to Schur canonical form, T;
= 7: if T and -T' have some almost equal eigenvalues, if
DICO = 'C', or T has almost reciprocal eigenvalues,
if DICO = 'D'; perturbed values were used to solve
Lyapunov equations, but the matrix T, if given (for
FACT = 'F'), is unchanged. (This is a warning
indicator.)
Method
The method used is the Schur vector approach proposed by Laub [1],
but with an optional scaling, which enhances the numerical
stability [6]. It is assumed that [A,B] is a stabilizable pair
(where for (3) or (4), B is any matrix such that B*B' = G with
rank(B) = rank(G)), and [E,A] is a detectable pair, where E is any
matrix such that E*E' = Q with rank(E) = rank(Q). Under these
assumptions, any of the algebraic Riccati equations (1)-(4) is
known to have a unique non-negative definite solution. See [2].
Now consider the 2N-by-2N Hamiltonian or symplectic matrix
( op(A) -G )
H = ( ), (5)
( -Q -op(A)' ),
for continuous-time equation, and
-1 -1
( op(A) op(A) *G )
H = ( -1 -1 ), (6)
( Q*op(A) op(A)' + Q*op(A) *G )
for discrete-time equation, respectively, where
-1
G = op(B)*R *op(B)'.
The assumptions guarantee that H in (5) has no pure imaginary
eigenvalues, and H in (6) has no eigenvalues on the unit circle.
If Y is an N-by-N matrix then there exists an orthogonal matrix U
such that U'*Y*U is an upper quasi-triangular matrix. Moreover, U
can be chosen so that the 2-by-2 and 1-by-1 diagonal blocks
(corresponding to the complex conjugate eigenvalues and real
eigenvalues respectively) appear in any desired order. This is the
ordered real Schur form. Thus, we can find an orthogonal
similarity transformation U which puts (5) or (6) in ordered real
Schur form
U'*H*U = S = (S(1,1) S(1,2))
( 0 S(2,2))
where S(i,j) is an N-by-N matrix and the eigenvalues of S(1,1)
have negative real parts in case of (5), or moduli greater than
one in case of (6). If U is conformably partitioned into four
N-by-N blocks
U = (U(1,1) U(1,2))
(U(2,1) U(2,2))
with respect to the assumptions we then have
(a) U(1,1) is invertible and X = U(2,1)*inv(U(1,1)) solves (1),
(2), (3), or (4) with X = X' and non-negative definite;
(b) the eigenvalues of S(1,1) (if DICO = 'C') or S(2,2) (if
DICO = 'D') are equal to the eigenvalues of optimal system
(the 'closed-loop' spectrum).
[A,B] is stabilizable if there exists a matrix F such that (A-BF)
is stable. [E,A] is detectable if [A',E'] is stabilizable.
The condition number of a Riccati equation is estimated as
cond = ( norm(Theta)*norm(A) + norm(inv(Omega))*norm(Q) +
norm(Pi)*norm(G) ) / norm(X),
where Omega, Theta and Pi are linear operators defined by
Omega(W) = op(Ac)'*W + W*op(Ac),
Theta(W) = inv(Omega(op(W)'*X + X*op(W))),
Pi(W) = inv(Omega(X*W*X)),
in the continuous-time case, and
Omega(W) = op(Ac)'*W*op(Ac) - W,
Theta(W) = inv(Omega(op(W)'*X*op(Ac) + op(Ac)'X*op(W))),
Pi(W) = inv(Omega(op(Ac)'*X*W*X*op(Ac))),
in the discrete-time case, and Ac has been defined (see argument
FACT). Details are given in the comments of SLICOT Library
routines SB02QD and SB02SD.
The routine estimates the quantities
sep(op(Ac),-op(Ac)') = 1 / norm(inv(Omega)),
sepd(op(Ac),op(Ac)') = 1 / norm(inv(Omega)),
norm(Theta) and norm(Pi) using 1-norm condition estimator.
The forward error bound is estimated using a practical error bound
similar to the one proposed in [5].
References
[1] Laub, A.J.
A Schur Method for Solving Algebraic Riccati equations.
IEEE Trans. Auto. Contr., AC-24, pp. 913-921, 1979.
[2] Wonham, W.M.
On a matrix Riccati equation of stochastic control.
SIAM J. Contr., 6, pp. 681-697, 1968.
[3] Sima, V.
Algorithms for Linear-Quadratic Optimization.
Pure and Applied Mathematics: A Series of Monographs and
Textbooks, vol. 200, Marcel Dekker, Inc., New York, 1996.
[4] Ghavimi, A.R. and Laub, A.J.
Backward error, sensitivity, and refinement of computed
solutions of algebraic Riccati equations.
Numerical Linear Algebra with Applications, vol. 2, pp. 29-49,
1995.
[5] Higham, N.J.
Perturbation theory and backward error for AX-XB=C.
BIT, vol. 33, pp. 124-136, 1993.
[6] Petkov, P.Hr., Konstantinov, M.M., and Mehrmann, V.
DGRSVX and DMSRIC: Fortran 77 subroutines for solving
continuous-time matrix algebraic Riccati equations with
condition and accuracy estimates.
Preprint SFB393/98-16, Fak. f. Mathematik, Tech. Univ.
Chemnitz, May 1998.
Numerical Aspects
3 The algorithm requires 0(N ) operations. The solution accuracy can be controlled by the output parameter FERR.Further Comments
To obtain a stabilizing solution of the algebraic Riccati
equation for DICO = 'D', set SORT = 'U', if HINV = 'D', or set
SORT = 'S', if HINV = 'I'.
The routine can also compute the anti-stabilizing solutions of
the algebraic Riccati equations, by specifying
SORT = 'U' if DICO = 'D' and HINV = 'I', or DICO = 'C', or
SORT = 'S' if DICO = 'D' and HINV = 'D'.
Usually, the combinations HINV = 'D' and SORT = 'U', or HINV = 'I'
and SORT = 'U', for stabilizing and anti-stabilizing solutions,
respectively, will be faster then the other combinations [3].
The option LYAPUN = 'R' may produce slightly worse or better
estimates, and it is faster than the option 'O'.
This routine is a functionally extended and more accurate
version of the SLICOT Library routine SB02MD. Transposed problems
can be dealt with as well. Iterative refinement is used whenever
useful to solve linear algebraic systems. Condition numbers and
error bounds on the solutions are optionally provided.
Example
Program Text
* SB02RD EXAMPLE PROGRAM TEXT.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA, LDG, LDQ, LDS, LDT, LDV, LDX
PARAMETER ( LDA = NMAX, LDG = NMAX, LDQ = NMAX,
$ LDS = 2*NMAX, LDT = NMAX, LDV = NMAX,
$ LDX = NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MAX( 2*NMAX, NMAX*NMAX ) )
INTEGER LDWORK
PARAMETER ( LDWORK = 5 + 4*NMAX*NMAX + 8*NMAX )
* .. Local Scalars ..
DOUBLE PRECISION FERR, RCOND, SEP
INTEGER I, INFO, J, N
CHARACTER DICO, FACT, HINV, JOB, LYAPUN, SCAL, SORT, TRANA,
$ UPLO
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), G(LDG,NMAX),
$ Q(LDQ,NMAX), S(LDS,2*NMAX), T(LDT,NMAX),
$ V(LDV,NMAX), WI(2*NMAX), WR(2*NMAX), X(LDX,NMAX)
INTEGER IWORK(LIWORK)
LOGICAL BWORK(LIWORK)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL SB02RD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, JOB, DICO, HINV, TRANA, UPLO, SCAL, SORT,
$ FACT, LYAPUN
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) N
ELSE
IF ( LSAME( JOB, 'X' ) .OR. LSAME( JOB, 'A' ) .OR.
$ LSAME( FACT, 'N' ) .OR. LSAME( LYAPUN, 'O' ) )
$ READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( .NOT.LSAME( JOB, 'X' ) .AND. LSAME( FACT, 'F' ) ) THEN
READ ( NIN, FMT = * ) ( ( T(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( V(I,J), J = 1,N ), I = 1,N )
END IF
READ ( NIN, FMT = * ) ( ( Q(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( G(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( JOB, 'C' ) .OR. LSAME( JOB, 'E' ) )
$ READ ( NIN, FMT = * ) ( ( X(I,J), J = 1,N ), I = 1,N )
* Find the solution matrix X.
CALL SB02RD( JOB, DICO, HINV, TRANA, UPLO, SCAL, SORT, FACT,
$ LYAPUN, N, A, LDA, T, LDT, V, LDV, G, LDG, Q, LDQ,
$ X, LDX, SEP, RCOND, FERR, WR, WI, S, LDS, IWORK,
$ DWORK, LDWORK, BWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
END IF
IF ( INFO.EQ.0 .OR. INFO.EQ.7 ) THEN
IF ( LSAME( JOB, 'X' ) .OR. LSAME( JOB, 'A' ) ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( X(I,J), J = 1,N )
20 CONTINUE
END IF
IF ( LSAME( JOB, 'C' ) .OR. LSAME( JOB, 'A' ) ) THEN
WRITE ( NOUT, FMT = 99994 ) SEP
WRITE ( NOUT, FMT = 99993 ) RCOND
END IF
IF ( LSAME( JOB, 'E' ) .OR. LSAME( JOB, 'A' ) )
$ WRITE ( NOUT, FMT = 99992 ) FERR
END IF
END IF
STOP
*
99999 FORMAT (' SB02RD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB02RD = ',I2)
99997 FORMAT (' The solution matrix X is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' N is out of range.',/' N = ',I5)
99994 FORMAT (/' Estimated separation = ',F8.4)
99993 FORMAT (/' Estimated reciprocal condition number = ',F8.4)
99992 FORMAT (/' Estimated error bound = ',F8.4)
END
Program Data
SB02RD EXAMPLE PROGRAM DATA 2 A C D N U N S N O 0.0 1.0 0.0 0.0 1.0 0.0 0.0 2.0 0.0 0.0 0.0 1.0Program Results
SB02RD EXAMPLE PROGRAM RESULTS The solution matrix X is 2.0000 1.0000 1.0000 2.0000 Estimated separation = 0.4000 Estimated reciprocal condition number = 0.1333 Estimated error bound = 0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB02RU.html 0000664 0000000 0000000 00000017264 14560147231 0020401 0 ustar 00root root 0000000 0000000
Purpose
To construct the 2n-by-2n Hamiltonian or symplectic matrix S
associated to the linear-quadratic optimization problem, used to
solve the continuous- or discrete-time algebraic Riccati equation,
respectively.
For a continuous-time problem, S is defined by
( op(A) -G )
S = ( ), (1)
( -Q -op(A)' )
and for a discrete-time problem by
-1 -1
( op(A) op(A) *G )
S = ( -1 -1 ), (2)
( Q*op(A) op(A)' + Q*op(A) *G )
or
-T -T
( op(A) + G*op(A) *Q -G*op(A) )
S = ( -T -T ), (3)
( -op(A) *Q op(A) )
where op(A) = A or A' (A**T), A, G, and Q are n-by-n matrices,
with G and Q symmetric. Matrix A must be nonsingular in the
discrete-time case.
Specification
SUBROUTINE SB02RU( DICO, HINV, TRANA, UPLO, N, A, LDA, G, LDG, Q,
$ LDQ, S, LDS, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, HINV, TRANA, UPLO
INTEGER INFO, LDA, LDG, LDQ, LDS, LDWORK, N
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), DWORK(*), G(LDG,*), Q(LDQ,*),
$ S(LDS,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the system as follows:
= 'C': Continuous-time system;
= 'D': Discrete-time system.
HINV CHARACTER*1
If DICO = 'D', specifies which of the matrices (2) or (3)
is constructed, as follows:
= 'D': The matrix S in (2) is constructed;
= 'I': The (inverse) matrix S in (3) is constructed.
HINV is not referenced if DICO = 'C'.
TRANA CHARACTER*1
Specifies the form of op(A) to be used, as follows:
= 'N': op(A) = A (No transpose);
= 'T': op(A) = A**T (Transpose);
= 'C': op(A) = A**T (Conjugate transpose = Transpose).
UPLO CHARACTER*1
Specifies which triangle of the matrices G and Q is
stored, as follows:
= 'U': Upper triangle is stored;
= 'L': Lower triangle is stored.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, G, and Q. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
G (input/output) DOUBLE PRECISION array, dimension (LDG,N)
On entry, the leading N-by-N upper triangular part (if
UPLO = 'U') or lower triangular part (if UPLO = 'L') of
this array must contain the upper triangular part or lower
triangular part, respectively, of the symmetric matrix G.
On exit, if DICO = 'D', the leading N-by-N part of this
array contains the symmetric matrix G fully stored.
If DICO = 'C', this array is not modified on exit, and the
strictly lower triangular part (if UPLO = 'U') or strictly
upper triangular part (if UPLO = 'L') is not referenced.
LDG INTEGER
The leading dimension of the array G. LDG >= MAX(1,N).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, the leading N-by-N upper triangular part (if
UPLO = 'U') or lower triangular part (if UPLO = 'L') of
this array must contain the upper triangular part or lower
triangular part, respectively, of the symmetric matrix Q.
On exit, if DICO = 'D', the leading N-by-N part of this
array contains the symmetric matrix Q fully stored.
If DICO = 'C', this array is not modified on exit, and the
strictly lower triangular part (if UPLO = 'U') or strictly
upper triangular part (if UPLO = 'L') is not referenced.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= MAX(1,N).
S (output) DOUBLE PRECISION array, dimension (LDS,2*N)
If INFO = 0, the leading 2N-by-2N part of this array
contains the Hamiltonian or symplectic matrix of the
problem.
LDS INTEGER
The leading dimension of the array S. LDS >= MAX(1,2*N).
Workspace
IWORK INTEGER array, dimension (LIWORK), where
LIWORK >= 0, if DICO = 'C';
LIWORK >= 2*N, if DICO = 'D'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if DICO = 'D', DWORK(1) returns the reciprocal
condition number RCOND of the given matrix A, and
DWORK(2) returns the reciprocal pivot growth factor
norm(A)/norm(U) (see SLICOT Library routine MB02PD).
If DWORK(2) is much less than 1, then the computed S
and RCOND could be unreliable. If 0 < INFO <= N, then
DWORK(2) contains the reciprocal pivot growth factor for
the leading INFO columns of A.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 0, if DICO = 'C';
LDWORK >= MAX(2,6*N), if DICO = 'D'.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= i: if the leading i-by-i (1 <= i <= N) upper triangular
submatrix of A is singular in discrete-time case;
= N+1: if matrix A is numerically singular in discrete-
time case.
Method
For a continuous-time problem, the 2n-by-2n Hamiltonian matrix (1) is constructed. For a discrete-time problem, the 2n-by-2n symplectic matrix (2) or (3) - the inverse of the matrix in (2) - is constructed.Numerical Aspects
The discrete-time case needs the inverse of the matrix A, hence
the routine should not be used when A is ill-conditioned.
3
The algorithm requires 0(n ) floating point operations in the
discrete-time case.
Further Comments
This routine is a functionally extended and with improved accuracy version of the SLICOT Library routine SB02MU. Transposed problems can be dealt with as well. The LU factorization of op(A) (with no equilibration) and iterative refinement are used for solving the various linear algebraic systems involved.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To estimate the conditioning and compute an error bound on the
solution of the real discrete-time matrix algebraic Riccati
equation (see FURTHER COMMENTS)
-1
X = op(A)'*X*(I_n + G*X) *op(A) + Q, (1)
where op(A) = A or A' (A**T) and Q, G are symmetric (Q = Q**T,
G = G**T). The matrices A, Q and G are N-by-N and the solution X
is N-by-N.
Specification
SUBROUTINE SB02SD( JOB, FACT, TRANA, UPLO, LYAPUN, N, A, LDA, T,
$ LDT, U, LDU, G, LDG, Q, LDQ, X, LDX, SEPD,
$ RCOND, FERR, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER FACT, JOB, LYAPUN, TRANA, UPLO
INTEGER INFO, LDA, LDG, LDQ, LDT, LDU, LDWORK, LDX, N
DOUBLE PRECISION FERR, RCOND, SEPD
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), DWORK( * ), G( LDG, * ),
$ Q( LDQ, * ), T( LDT, * ), U( LDU, * ),
$ X( LDX, * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'C': Compute the reciprocal condition number only;
= 'E': Compute the error bound only;
= 'B': Compute both the reciprocal condition number and
the error bound.
FACT CHARACTER*1
Specifies whether or not the real Schur factorization of
the matrix Ac = inv(I_n + G*X)*A (if TRANA = 'N'), or
Ac = A*inv(I_n + X*G) (if TRANA = 'T' or 'C'), is supplied
on entry, as follows:
= 'F': On entry, T and U (if LYAPUN = 'O') contain the
factors from the real Schur factorization of the
matrix Ac;
= 'N': The Schur factorization of Ac will be computed
and the factors will be stored in T and U (if
LYAPUN = 'O').
TRANA CHARACTER*1
Specifies the form of op(A) to be used, as follows:
= 'N': op(A) = A (No transpose);
= 'T': op(A) = A**T (Transpose);
= 'C': op(A) = A**T (Conjugate transpose = Transpose).
UPLO CHARACTER*1
Specifies which part of the symmetric matrices Q and G is
to be used, as follows:
= 'U': Upper triangular part;
= 'L': Lower triangular part.
LYAPUN CHARACTER*1
Specifies whether or not the original Lyapunov equations
should be solved in the iterative estimation process,
as follows:
= 'O': Solve the original Lyapunov equations, updating
the right-hand sides and solutions with the
matrix U, e.g., RHS <-- U'*RHS*U;
= 'R': Solve reduced Lyapunov equations only, without
updating the right-hand sides and solutions.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, X, Q, and G. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
If FACT = 'N' or LYAPUN = 'O', the leading N-by-N part of
this array must contain the matrix A.
If FACT = 'F' and LYAPUN = 'R', A is not referenced.
LDA INTEGER
The leading dimension of the array A.
LDA >= max(1,N), if FACT = 'N' or LYAPUN = 'O';
LDA >= 1, if FACT = 'F' and LYAPUN = 'R'.
T (input or output) DOUBLE PRECISION array, dimension
(LDT,N)
If FACT = 'F', then T is an input argument and on entry,
the leading N-by-N upper Hessenberg part of this array
must contain the upper quasi-triangular matrix T in Schur
canonical form from a Schur factorization of Ac (see
argument FACT).
If FACT = 'N', then T is an output argument and on exit,
if INFO = 0 or INFO = N+1, the leading N-by-N upper
Hessenberg part of this array contains the upper quasi-
triangular matrix T in Schur canonical form from a Schur
factorization of Ac (see argument FACT).
LDT INTEGER
The leading dimension of the array T. LDT >= max(1,N).
U (input or output) DOUBLE PRECISION array, dimension
(LDU,N)
If LYAPUN = 'O' and FACT = 'F', then U is an input
argument and on entry, the leading N-by-N part of this
array must contain the orthogonal matrix U from a real
Schur factorization of Ac (see argument FACT).
If LYAPUN = 'O' and FACT = 'N', then U is an output
argument and on exit, if INFO = 0 or INFO = N+1, it
contains the orthogonal N-by-N matrix from a real Schur
factorization of Ac (see argument FACT).
If LYAPUN = 'R', the array U is not referenced.
LDU INTEGER
The leading dimension of the array U.
LDU >= 1, if LYAPUN = 'R';
LDU >= MAX(1,N), if LYAPUN = 'O'.
G (input) DOUBLE PRECISION array, dimension (LDG,N)
If UPLO = 'U', the leading N-by-N upper triangular part of
this array must contain the upper triangular part of the
matrix G.
If UPLO = 'L', the leading N-by-N lower triangular part of
this array must contain the lower triangular part of the
matrix G. _
Matrix G should correspond to G in the "reduced" Riccati
equation (with matrix T, instead of A), if LYAPUN = 'R'.
See METHOD.
LDG INTEGER
The leading dimension of the array G. LDG >= max(1,N).
Q (input) DOUBLE PRECISION array, dimension (LDQ,N)
If UPLO = 'U', the leading N-by-N upper triangular part of
this array must contain the upper triangular part of the
matrix Q.
If UPLO = 'L', the leading N-by-N lower triangular part of
this array must contain the lower triangular part of the
matrix Q. _
Matrix Q should correspond to Q in the "reduced" Riccati
equation (with matrix T, instead of A), if LYAPUN = 'R'.
See METHOD.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
X (input) DOUBLE PRECISION array, dimension (LDX,N)
The leading N-by-N part of this array must contain the
symmetric solution matrix of the original Riccati
equation (with matrix A), if LYAPUN = 'O', or of the
"reduced" Riccati equation (with matrix T), if
LYAPUN = 'R'. See METHOD.
LDX INTEGER
The leading dimension of the array X. LDX >= max(1,N).
SEPD (output) DOUBLE PRECISION
If JOB = 'C' or JOB = 'B', the estimated quantity
sepd(op(Ac),op(Ac)').
If N = 0, or X = 0, or JOB = 'E', SEPD is not referenced.
RCOND (output) DOUBLE PRECISION
If JOB = 'C' or JOB = 'B', an estimate of the reciprocal
condition number of the discrete-time Riccati equation.
If N = 0 or X = 0, RCOND is set to 1 or 0, respectively.
If JOB = 'E', RCOND is not referenced.
FERR (output) DOUBLE PRECISION
If JOB = 'E' or JOB = 'B', an estimated forward error
bound for the solution X. If XTRUE is the true solution,
FERR bounds the magnitude of the largest entry in
(X - XTRUE) divided by the magnitude of the largest entry
in X.
If N = 0 or X = 0, FERR is set to 0.
If JOB = 'C', FERR is not referenced.
Workspace
IWORK INTEGER array, dimension (N*N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or INFO = N+1, DWORK(1) returns the
optimal value of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
Let LWA = N*N, if LYAPUN = 'O';
LWA = 0, otherwise,
and LWN = N, if LYAPUN = 'R' and JOB = 'E' or 'B';
LWN = 0, otherwise.
If FACT = 'N', then
LDWORK = MAX(LWA + 5*N, MAX(3,2*N*N) + N*N),
if JOB = 'C';
LDWORK = MAX(LWA + 5*N, MAX(3,2*N*N) + 2*N*N + LWN),
if JOB = 'E' or 'B'.
If FACT = 'F', then
LDWORK = MAX(3,2*N*N) + N*N, if JOB = 'C';
LDWORK = MAX(3,2*N*N) + 2*N*N + LWN,
if JOB = 'E' or 'B'.
For good performance, LDWORK must generally be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, i <= N, the QR algorithm failed to
complete the reduction of the matrix Ac to Schur
canonical form (see LAPACK Library routine DGEES);
on exit, the matrix T(i+1:N,i+1:N) contains the
partially converged Schur form, and DWORK(i+1:N) and
DWORK(N+i+1:2*N) contain the real and imaginary
parts, respectively, of the converged eigenvalues;
this error is unlikely to appear;
= N+1: if T has almost reciprocal eigenvalues; perturbed
values were used to solve Lyapunov equations, but
the matrix T, if given (for FACT = 'F'), is
unchanged.
Method
The condition number of the Riccati equation is estimated as
cond = ( norm(Theta)*norm(A) + norm(inv(Omega))*norm(Q) +
norm(Pi)*norm(G) ) / norm(X),
where Omega, Theta and Pi are linear operators defined by
Omega(W) = op(Ac)'*W*op(Ac) - W,
Theta(W) = inv(Omega(op(W)'*X*op(Ac) + op(Ac)'X*op(W))),
Pi(W) = inv(Omega(op(Ac)'*X*W*X*op(Ac))),
and Ac = inv(I_n + G*X)*A (if TRANA = 'N'), or
Ac = A*inv(I_n + X*G) (if TRANA = 'T' or 'C').
Note that the Riccati equation (1) is equivalent to
X = op(Ac)'*X*op(Ac) + op(Ac)'*X*G*X*op(Ac) + Q, (2)
and to
_ _ _ _ _ _
X = op(T)'*X*op(T) + op(T)'*X*G*X*op(T) + Q, (3)
_ _ _
where X = U'*X*U, Q = U'*Q*U, and G = U'*G*U, with U the
orthogonal matrix reducing Ac to a real Schur form, T = U'*Ac*U.
The routine estimates the quantities
sepd(op(Ac),op(Ac)') = 1 / norm(inv(Omega)),
norm(Theta) and norm(Pi) using 1-norm condition estimator.
The forward error bound is estimated using a practical error bound
similar to the one proposed in [2].
References
[1] Ghavimi, A.R. and Laub, A.J.
Backward error, sensitivity, and refinement of computed
solutions of algebraic Riccati equations.
Numerical Linear Algebra with Applications, vol. 2, pp. 29-49,
1995.
[2] Higham, N.J.
Perturbation theory and backward error for AX-XB=C.
BIT, vol. 33, pp. 124-136, 1993.
[3] Petkov, P.Hr., Konstantinov, M.M., and Mehrmann, V.
DGRSVX and DMSRIC: Fortran 77 subroutines for solving
continuous-time matrix algebraic Riccati equations with
condition and accuracy estimates.
Preprint SFB393/98-16, Fak. f. Mathematik, Tech. Univ.
Chemnitz, May 1998.
Numerical Aspects
3 The algorithm requires 0(N ) operations. The accuracy of the estimates obtained depends on the solution accuracy and on the properties of the 1-norm estimator.Further Comments
The option LYAPUN = 'R' may occasionally produce slightly worse
or better estimates, and it is much faster than the option 'O'.
When SEPD is computed and it is zero, the routine returns
immediately, with RCOND and FERR (if requested) set to 0 and 1,
respectively. In this case, the equation is singular.
Let B be an N-by-M matrix (if TRANA = 'N') or an M-by-N matrix
(if TRANA = 'T' or 'C'), let R be an M-by-M symmetric positive
definite matrix (R = R**T), and denote G = op(B)*inv(R)*op(B)'.
Then, the Riccati equation (1) is equivalent to the standard
discrete-time matrix algebraic Riccati equation
X = op(A)'*X*op(A) - (4)
-1
op(A)'*X*op(B)*(R + op(B)'*X*op(B)) *op(B)'*X*op(A) + Q.
By symmetry, the equation (1) is also equivalent to
-1
X = op(A)'*(I_n + X*G) *X*op(A) + Q.
Example
Program Text
* SB02SD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA, LDG, LDQ, LDT, LDU, LDX
PARAMETER ( LDA = NMAX, LDG = NMAX, LDQ = NMAX, LDT = NMAX,
$ LDU = NMAX, LDX = NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX*NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 8*NMAX*NMAX + 10*NMAX )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* .. Local Scalars ..
DOUBLE PRECISION FERR, RCND, RCOND, SEPD
INTEGER I, INFO1, INFO2, INFO3, IS, IU, IW, J, N, N2,
$ SDIM
CHARACTER*1 FACT, JOB, JOBS, LYAPUN, TRANA, TRANAT, UPLO
* .. Local Arrays ..
LOGICAL BWORK(2*NMAX)
INTEGER IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), AS(LDA,NMAX), DWORK(LDWORK),
$ G(LDG,NMAX), Q(LDQ,NMAX), T(LDT,NMAX),
$ U(LDU,NMAX), X(LDX,NMAX)
* .. External Functions ..
LOGICAL LSAME, SELECT
EXTERNAL LSAME, SELECT
* .. External Subroutines ..
EXTERNAL DGEES, DGESV, DLACPY, DLASET, DSWAP, DSYMM,
$ MA02AD, MA02ED, MB01RU, SB02MD, SB02SD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, JOB, FACT, TRANA, UPLO, LYAPUN
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( Q(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( G(I,J), J = 1,N ), I = 1,N )
CALL DLACPY( 'Full', N, N, A, LDA, AS, LDA )
CALL DLACPY( UPLO, N, N, Q, LDQ, X, LDX )
N2 = 2*N
IS = 2*N2 + 1
IU = IS + N2*N2
IW = IU + N2*N2
* Solve the discrete-time Riccati equation.
CALL SB02MD( 'discrete', 'direct', UPLO, 'no scaling',
$ 'stable', N, AS, LDA, G, LDG, X, LDX, RCND,
$ DWORK(1), DWORK(N2+1), DWORK(IS), N2, DWORK(IU),
$ N2, IWORK, DWORK(IW), LDWORK-IW+1, BWORK, INFO1 )
*
IF ( INFO1.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99995 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99994 ) ( X(I,J), J = 1,N )
10 CONTINUE
IF ( LSAME( FACT, 'F' ) .OR. LSAME( LYAPUN, 'R' ) ) THEN
CALL DLASET( 'Full', N, N, ZERO, ONE, DWORK, N )
CALL DSYMM( 'Left', UPLO, N, N, ONE, G, LDG, X, LDX,
$ ONE, DWORK, N )
IF ( LSAME( TRANA, 'N' ) ) THEN
* Compute Ac = inv(I_n + G*X)*A.
CALL DLACPY( 'Full', N, N, A, LDA, T, LDT )
CALL DGESV( N, N, DWORK, N, IWORK, T, LDT, INFO3 )
ELSE
* Compute Ac = A*inv(I_n + X*G)
CALL MA02AD( 'Full', N, N, A, LDA, T, LDT )
CALL DGESV( N, N, DWORK, N, IWORK, T, LDT, INFO3 )
DO 20 J = 2, N
CALL DSWAP( J-1, T(1,J), 1, T(J,1), LDT )
20 CONTINUE
END IF
* Compute the Schur factorization of Ac.
JOBS = 'V'
CALL DGEES( JOBS, 'Not ordered', SELECT, N, T, LDT, SDIM,
$ DWORK(1), DWORK(N+1), U, LDU, DWORK(2*N+1),
$ LDWORK-2*N, BWORK, INFO3 )
IF( INFO3.NE.0 ) THEN
WRITE ( NOUT, FMT = 99996 ) INFO3
STOP
END IF
END IF
*
IF ( LSAME( LYAPUN, 'R' ) ) THEN
IF( LSAME( TRANA, 'N' ) ) THEN
TRANAT = 'T'
ELSE
TRANAT = 'N'
END IF
*
CALL MB01RU( UPLO, TRANAT, N, N, ZERO, ONE, X, LDX,
$ U, LDU, X, LDX, DWORK, N*N, INFO2 )
CALL MA02ED( UPLO, N, X, LDX )
CALL MB01RU( UPLO, TRANAT, N, N, ZERO, ONE, G, LDG,
$ U, LDU, G, LDG, DWORK, N*N, INFO2 )
CALL MB01RU( UPLO, TRANAT, N, N, ZERO, ONE, Q, LDQ,
$ U, LDU, Q, LDQ, DWORK, N*N, INFO2 )
END IF
* Estimate the condition and error bound on the solution.
CALL SB02SD( JOB, FACT, TRANA, UPLO, LYAPUN, N, A, LDA, T,
$ LDT, U, LDU, G, LDG, Q, LDQ, X, LDX, SEPD,
$ RCOND, FERR, IWORK, DWORK, LDWORK, INFO2 )
*
IF ( INFO2.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO2
END IF
IF ( INFO2.EQ.0 .OR. INFO2.EQ.N+1 ) THEN
WRITE ( NOUT, FMT = 99992 ) SEPD
WRITE ( NOUT, FMT = 99991 ) RCOND
WRITE ( NOUT, FMT = 99990 ) FERR
END IF
ELSE
WRITE ( NOUT, FMT = 99998 ) INFO1
END IF
END IF
STOP
*
99999 FORMAT (' SB02SD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB02MD =',I2)
99997 FORMAT (' INFO on exit from SB02SD =',I2)
99996 FORMAT (' INFO on exit from DGEES =',I2)
99995 FORMAT (' The solution matrix X is')
99994 FORMAT (20(1X,F8.4))
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' Estimated separation = ',F8.4)
99991 FORMAT (/' Estimated reciprocal condition number = ',F8.4)
99990 FORMAT (/' Estimated error bound = ',F8.4)
END
Program Data
SB02SD EXAMPLE PROGRAM DATA 2 B N N U O 2.0 -1.0 1.0 0.0 0.0 0.0 0.0 1.0 1.0 0.0 0.0 0.0Program Results
SB02SD EXAMPLE PROGRAM RESULTS The solution matrix X is -0.7691 1.2496 1.2496 -2.3306 Estimated separation = 0.4456 Estimated reciprocal condition number = 0.1445 Estimated error bound = 0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB03MD.html 0000664 0000000 0000000 00000040760 14560147231 0020351 0 ustar 00root root 0000000 0000000
Purpose
To solve for X either the real continuous-time Lyapunov equation
op(A)'*X + X*op(A) = scale*C (1)
or the real discrete-time Lyapunov equation
op(A)'*X*op(A) - X = scale*C (2)
and/or estimate an associated condition number, called separation,
where op(A) = A or A' (A**T) and C is symmetric (C = C').
(A' denotes the transpose of the matrix A.) A is N-by-N, the right
hand side C and the solution X are N-by-N, and scale is an output
scale factor, set less than or equal to 1 to avoid overflow in X.
Specification
SUBROUTINE SB03MD( DICO, JOB, FACT, TRANA, N, A, LDA, U, LDU, C,
$ LDC, SCALE, SEP, FERR, WR, WI, IWORK, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, FACT, JOB, TRANA
INTEGER INFO, LDA, LDC, LDU, LDWORK, N
DOUBLE PRECISION FERR, SCALE, SEP
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), DWORK( * ),
$ U( LDU, * ), WI( * ), WR( * )
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the equation from which X is to be determined
as follows:
= 'C': Equation (1), continuous-time case;
= 'D': Equation (2), discrete-time case.
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'X': Compute the solution only;
= 'S': Compute the separation only;
= 'B': Compute both the solution and the separation.
FACT CHARACTER*1
Specifies whether or not the real Schur factorization
of the matrix A is supplied on entry, as follows:
= 'F': On entry, A and U contain the factors from the
real Schur factorization of the matrix A;
= 'N': The Schur factorization of A will be computed
and the factors will be stored in A and U.
TRANA CHARACTER*1
Specifies the form of op(A) to be used, as follows:
= 'N': op(A) = A (No transpose);
= 'T': op(A) = A**T (Transpose);
= 'C': op(A) = A**T (Conjugate transpose = Transpose).
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, X, and C. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A. If FACT = 'F', then A contains
an upper quasi-triangular matrix in Schur canonical form;
the elements below the upper Hessenberg part of the
array A are not referenced.
On exit, if INFO = 0 or INFO = N+1, the leading N-by-N
upper Hessenberg part of this array contains the upper
quasi-triangular matrix in Schur canonical form from the
Schur factorization of A. The contents of array A is not
modified if FACT = 'F'.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
U (input or output) DOUBLE PRECISION array, dimension
(LDU,N)
If FACT = 'F', then U is an input argument and on entry
the leading N-by-N part of this array must contain the
orthogonal matrix U of the real Schur factorization of A.
If FACT = 'N', then U is an output argument and on exit,
if INFO = 0 or INFO = N+1, it contains the orthogonal
N-by-N matrix from the real Schur factorization of A.
LDU INTEGER
The leading dimension of array U. LDU >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry with JOB = 'X' or 'B', the leading N-by-N part of
this array must contain the symmetric matrix C.
On exit with JOB = 'X' or 'B', if INFO = 0 or INFO = N+1,
the leading N-by-N part of C has been overwritten by the
symmetric solution matrix X.
If JOB = 'S', C is not referenced.
LDC INTEGER
The leading dimension of array C.
LDC >= 1, if JOB = 'S';
LDC >= MAX(1,N), otherwise.
SCALE (output) DOUBLE PRECISION
The scale factor, scale, set less than or equal to 1 to
prevent the solution overflowing.
SEP (output) DOUBLE PRECISION
If JOB = 'S' or JOB = 'B', and INFO = 0 or INFO = N+1, SEP
contains the estimated separation of the matrices op(A)
and -op(A)', if DICO = 'C' or of op(A) and op(A)', if
DICO = 'D'.
If JOB = 'X', SEP is not referenced.
FERR (output) DOUBLE PRECISION
If JOB = 'B', and INFO = 0 or INFO = N+1, FERR contains an
estimated forward error bound for the solution X.
If XTRUE is the true solution, FERR bounds the relative
error in the computed solution, measured in the Frobenius
norm: norm(X - XTRUE)/norm(XTRUE).
If JOB = 'X' or JOB = 'S', FERR is not referenced.
WR (output) DOUBLE PRECISION array, dimension (N)
WI (output) DOUBLE PRECISION array, dimension (N)
If FACT = 'N', and INFO = 0 or INFO = N+1, WR and WI
contain the real and imaginary parts, respectively, of
the eigenvalues of A.
If FACT = 'F', WR and WI are not referenced.
Workspace
IWORK INTEGER array, dimension (N*N)
This array is not referenced if JOB = 'X'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or INFO = N+1, DWORK(1) returns the
optimal value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= 1, and
If JOB = 'X' then
If FACT = 'F', LDWORK >= N*N, for DICO = 'C';
LDWORK >= MAX(N*N, 2*N), for DICO = 'D';
If FACT = 'N', LDWORK >= MAX(N*N, 3*N).
If JOB = 'S' or JOB = 'B' then
If FACT = 'F', LDWORK >= 2*N*N, for DICO = 'C';
LDWORK >= 2*N*N + 2*N, for DICO = 'D'.
If FACT = 'N', LDWORK >= MAX(2*N*N, 3*N), DICO = 'C';
LDWORK >= 2*N*N + 2*N, for DICO = 'D'.
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message related to LDWORK is issued by
XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, the QR algorithm failed to compute all
the eigenvalues (see LAPACK Library routine DGEES);
elements i+1:n of WR and WI contain eigenvalues
which have converged, and A contains the partially
converged Schur form;
= N+1: if DICO = 'C', and the matrices A and -A' have
common or very close eigenvalues, or
if DICO = 'D', and matrix A has almost reciprocal
eigenvalues (that is, lambda(i) = 1/lambda(j) for
some i and j, where lambda(i) and lambda(j) are
eigenvalues of A and i <> j); perturbed values were
used to solve the equation (but the matrix A is
unchanged).
Method
The Schur factorization of a square matrix A is given by
A = U*S*U'
where U is orthogonal and S is block upper triangular with 1-by-1
and 2-by-2 blocks on its diagonal, these blocks corresponding to
the eigenvalues of A, the 2-by-2 blocks being complex conjugate
pairs. This factorization is obtained by numerically stable
methods: first A is reduced to upper Hessenberg form (if FACT =
'N') by means of Householder transformations and then the
QR Algorithm is applied to reduce the Hessenberg form to S, the
transformation matrices being accumulated at each step to give U.
If A has already been factorized prior to calling the routine
however, then the factors U and S may be supplied and the initial
factorization omitted.
_ _
If we now put C = U'CU and X = UXU' equations (1) and (2) (see
PURPOSE) become (for TRANS = 'N')
_ _ _
S'X + XS = C, (3)
and
_ _ _
S'XS - X = C, (4)
respectively. Partition S, C and X as
_ _ _ _
(s s') (c c') (x x')
( 11 ) _ ( 11 ) _ ( 11 )
S = ( ), C = ( ), X = ( )
( ) ( _ ) ( _ )
( 0 S ) ( c C ) ( x X )
1 1 1
_ _
where s , c and x are either scalars or 2-by-2 matrices and s,
11 11 11
_ _
c and x are either (N-1) element vectors or matrices with two
columns. Equations (3) and (4) can then be re-written as
_ _ _
s' x + x s = c (3.1)
11 11 11 11 11
_ _ _ _
S'x + xs = c - sx (3.2)
1 11 11
_ _
S'X + X S = C - (sx' + xs') (3.3)
1 1 1 1 1
and
_ _ _
s' x s - x = c (4.1)
11 11 11 11 11
_ _ _ _
S'xs - x = c - sx s (4.2)
1 11 11 11
_ _ _
S'X S - X = C - sx s' - [s(S'x)' + (S'x)s'] (4.3)
1 1 1 1 1 11 1 1
_
respectively. If DICO = 'C' ['D'], then once x has been
11
found from equation (3.1) [(4.1)], equation (3.2) [(4.2)] can be
_
solved by forward substitution for x and then equation (3.3)
[(4.3)] is of the same form as (3) [(4)] but of the order (N-1) or
(N-2) depending upon whether s is 1-by-1 or 2-by-2.
11
_ _
When s is 2-by-2 then x and c will be 1-by-2 matrices and s,
11 11 11
_ _
x and c are matrices with two columns. In this case, equation
(3.1) [(4.1)] defines the three equations in the unknown elements
_
of x and equation (3.2) [(4.2)] can then be solved by forward
11 _
substitution, a row of x being found at each step.
References
[1] Barraud, A.Y. T
A numerical algorithm to solve A XA - X = Q.
IEEE Trans. Auto. Contr., AC-22, pp. 883-885, 1977.
[2] Bartels, R.H. and Stewart, G.W. T
Solution of the matrix equation A X + XB = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
[3] Hammarling, S.J.
Numerical solution of the stable, non-negative definite
Lyapunov equation.
IMA J. Num. Anal., 2, pp. 303-325, 1982.
Numerical Aspects
3 The algorithm requires 0(N ) operations and is backward stable.Further Comments
If DICO = 'C', SEP is defined as the separation of op(A) and
-op(A)':
sep( op(A), -op(A)' ) = sigma_min( T )
and if DICO = 'D', SEP is defined as
sep( op(A), op(A)' ) = sigma_min( T )
where sigma_min(T) is the smallest singular value of the
N*N-by-N*N matrix
T = kprod( I(N), op(A)' ) + kprod( op(A)', I(N) ) (DICO = 'C'),
T = kprod( op(A)', op(A)' ) - I(N**2) (DICO = 'D').
I(x) is an x-by-x identity matrix, and kprod denotes the Kronecker
product. The program estimates sigma_min(T) by the reciprocal of
an estimate of the 1-norm of inverse(T). The true reciprocal
1-norm of inverse(T) cannot differ from sigma_min(T) by more
than a factor of N.
When SEP is small, small changes in A, C can cause large changes
in the solution of the equation. An approximate bound on the
maximum relative error in the computed solution is
EPS * norm(A) / SEP (DICO = 'C'),
EPS * norm(A)**2 / SEP (DICO = 'D'),
where EPS is the machine precision.
Example
Program Text
* SB03MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA, LDC, LDU
PARAMETER ( LDA = NMAX, LDC = NMAX, LDU = NMAX )
INTEGER LDWORK, LIWORK
PARAMETER ( LDWORK = 2*NMAX*NMAX + 3*NMAX,
$ LIWORK = NMAX*NMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, N
CHARACTER*1 DICO, FACT, JOB, TRANA
DOUBLE PRECISION FERR, SCALE, SEP
* .. Local Arrays ..
INTEGER IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), C(LDC,NMAX), DWORK(LDWORK),
$ U(LDU,NMAX), WI(NMAX), WR(NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL SB03MD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, DICO, FACT, JOB, TRANA
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( FACT, 'F' ) ) READ ( NIN, FMT = * )
$ ( ( U(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,N )
* Find the solution matrix X.
CALL SB03MD( DICO, JOB, FACT, TRANA, N, A, LDA, U, LDU, C, LDC,
$ SCALE, SEP, FERR, WR, WI, IWORK, DWORK, LDWORK,
$ INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( C(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99994 ) SCALE
IF ( .NOT.LSAME( JOB, 'X' ) )
$ WRITE ( NOUT, FMT = 99993 ) SEP
IF ( LSAME( JOB, 'B' ) )
$ WRITE ( NOUT, FMT = 99992 ) FERR
END IF
END IF
STOP
*
99999 FORMAT (' SB03MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB03MD = ',I2)
99997 FORMAT (' The solution matrix X is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' N is out of range.',/' N = ',I5)
99994 FORMAT (/' Scaling factor = ',F8.4)
99993 FORMAT (/' Estimated separation = ',F8.4)
99992 FORMAT (/' Estimated forward error bound = ',F8.4)
END
Program Data
SB03MD EXAMPLE PROGRAM DATA 3 D N X N 3.0 1.0 1.0 1.0 3.0 0.0 0.0 0.0 3.0 25.0 24.0 15.0 24.0 32.0 8.0 15.0 8.0 40.0Program Results
SB03MD EXAMPLE PROGRAM RESULTS The solution matrix X is 2.0000 1.0000 1.0000 1.0000 3.0000 0.0000 1.0000 0.0000 4.0000 Scaling factor = 1.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB03MU.html 0000664 0000000 0000000 00000011263 14560147231 0020366 0 ustar 00root root 0000000 0000000
Purpose
To solve for the N1-by-N2 matrix X, 1 <= N1,N2 <= 2, in
ISGN*op(TL)*X*op(TR) - X = SCALE*B,
where TL is N1-by-N1, TR is N2-by-N2, B is N1-by-N2, and ISGN = 1
or -1. op(T) = T or T', where T' denotes the transpose of T.
Specification
SUBROUTINE SB03MU( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
$ LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO )
C .. Scalar Arguments ..
LOGICAL LTRANL, LTRANR
INTEGER INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2
DOUBLE PRECISION SCALE, XNORM
C .. Array Arguments ..
DOUBLE PRECISION B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),
$ X( LDX, * )
Arguments
Mode Parameters
LTRANL LOGICAL
Specifies the form of op(TL) to be used, as follows:
= .FALSE.: op(TL) = TL,
= .TRUE. : op(TL) = TL'.
LTRANR LOGICAL
Specifies the form of op(TR) to be used, as follows:
= .FALSE.: op(TR) = TR,
= .TRUE. : op(TR) = TR'.
ISGN INTEGER
Specifies the sign of the equation as described before.
ISGN may only be 1 or -1.
Input/Output Parameters
N1 (input) INTEGER
The order of matrix TL. N1 may only be 0, 1 or 2.
N2 (input) INTEGER
The order of matrix TR. N2 may only be 0, 1 or 2.
TL (input) DOUBLE PRECISION array, dimension (LDTL,2)
The leading N1-by-N1 part of this array must contain the
matrix TL.
LDTL INTEGER
The leading dimension of array TL. LDTL >= MAX(1,N1).
TR (input) DOUBLE PRECISION array, dimension (LDTR,2)
The leading N2-by-N2 part of this array must contain the
matrix TR.
LDTR INTEGER
The leading dimension of array TR. LDTR >= MAX(1,N2).
B (input) DOUBLE PRECISION array, dimension (LDB,2)
The leading N1-by-N2 part of this array must contain the
right-hand side of the equation.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N1).
SCALE (output) DOUBLE PRECISION
The scale factor. SCALE is chosen less than or equal to 1
to prevent the solution overflowing.
X (output) DOUBLE PRECISION array, dimension (LDX,N2)
The leading N1-by-N2 part of this array contains the
solution of the equation.
Note that X may be identified with B in the calling
statement.
LDX INTEGER
The leading dimension of array X. LDX >= MAX(1,N1).
XNORM (output) DOUBLE PRECISION
The infinity-norm of the solution.
Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: if TL and TR have almost reciprocal eigenvalues, so
TL or TR is perturbed to get a nonsingular equation.
NOTE: In the interests of speed, this routine does not
check the inputs for errors.
Method
The equivalent linear algebraic system of equations is formed and solved using Gaussian elimination with complete pivoting.References
[1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
Numerical Aspects
The algorithm is stable and reliable, since Gaussian elimination with complete pivoting is used.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve for the 2-by-2 symmetric matrix X in
op(T)'*X*op(T) - X = SCALE*B,
where T is 2-by-2, B is symmetric 2-by-2, and op(T) = T or T',
where T' denotes the transpose of T.
Specification
SUBROUTINE SB03MV( LTRAN, LUPPER, T, LDT, B, LDB, SCALE, X, LDX,
$ XNORM, INFO )
C .. Scalar Arguments ..
LOGICAL LTRAN, LUPPER
INTEGER INFO, LDB, LDT, LDX
DOUBLE PRECISION SCALE, XNORM
C .. Array Arguments ..
DOUBLE PRECISION B( LDB, * ), T( LDT, * ), X( LDX, * )
Arguments
Mode Parameters
LTRAN LOGICAL
Specifies the form of op(T) to be used, as follows:
= .FALSE.: op(T) = T,
= .TRUE. : op(T) = T'.
LUPPER LOGICAL
Specifies which triangle of the matrix B is used, and
which triangle of the matrix X is computed, as follows:
= .TRUE. : The upper triangular part;
= .FALSE.: The lower triangular part.
Input/Output Parameters
T (input) DOUBLE PRECISION array, dimension (LDT,2)
The leading 2-by-2 part of this array must contain the
matrix T.
LDT INTEGER
The leading dimension of array T. LDT >= 2.
B (input) DOUBLE PRECISION array, dimension (LDB,2)
On entry with LUPPER = .TRUE., the leading 2-by-2 upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix B and the strictly
lower triangular part of B is not referenced.
On entry with LUPPER = .FALSE., the leading 2-by-2 lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix B and the strictly
upper triangular part of B is not referenced.
LDB INTEGER
The leading dimension of array B. LDB >= 2.
SCALE (output) DOUBLE PRECISION
The scale factor. SCALE is chosen less than or equal to 1
to prevent the solution overflowing.
X (output) DOUBLE PRECISION array, dimension (LDX,2)
On exit with LUPPER = .TRUE., the leading 2-by-2 upper
triangular part of this array contains the upper
triangular part of the symmetric solution matrix X and the
strictly lower triangular part of X is not referenced.
On exit with LUPPER = .FALSE., the leading 2-by-2 lower
triangular part of this array contains the lower
triangular part of the symmetric solution matrix X and the
strictly upper triangular part of X is not referenced.
Note that X may be identified with B in the calling
statement.
LDX INTEGER
The leading dimension of array X. LDX >= 2.
XNORM (output) DOUBLE PRECISION
The infinity-norm of the solution.
Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: if T has almost reciprocal eigenvalues, so T
is perturbed to get a nonsingular equation.
NOTE: In the interests of speed, this routine does not
check the inputs for errors.
Method
The equivalent linear algebraic system of equations is formed and solved using Gaussian elimination with complete pivoting.References
[1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
Numerical Aspects
The algorithm is stable and reliable, since Gaussian elimination with complete pivoting is used.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve for the 2-by-2 symmetric matrix X in
op(T)'*X + X*op(T) = SCALE*B,
where T is 2-by-2, B is symmetric 2-by-2, and op(T) = T or T',
where T' denotes the transpose of T.
Specification
SUBROUTINE SB03MW( LTRAN, LUPPER, T, LDT, B, LDB, SCALE, X, LDX,
$ XNORM, INFO )
C .. Scalar Arguments ..
LOGICAL LTRAN, LUPPER
INTEGER INFO, LDB, LDT, LDX
DOUBLE PRECISION SCALE, XNORM
C .. Array Arguments ..
DOUBLE PRECISION B( LDB, * ), T( LDT, * ), X( LDX, * )
Arguments
Mode Parameters
LTRAN LOGICAL
Specifies the form of op(T) to be used, as follows:
= .FALSE.: op(T) = T,
= .TRUE. : op(T) = T'.
LUPPER LOGICAL
Specifies which triangle of the matrix B is used, and
which triangle of the matrix X is computed, as follows:
= .TRUE. : The upper triangular part;
= .FALSE.: The lower triangular part.
Input/Output Parameters
T (input) DOUBLE PRECISION array, dimension (LDT,2)
The leading 2-by-2 part of this array must contain the
matrix T.
LDT INTEGER
The leading dimension of array T. LDT >= 2.
B (input) DOUBLE PRECISION array, dimension (LDB,2)
On entry with LUPPER = .TRUE., the leading 2-by-2 upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix B and the strictly
lower triangular part of B is not referenced.
On entry with LUPPER = .FALSE., the leading 2-by-2 lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix B and the strictly
upper triangular part of B is not referenced.
LDB INTEGER
The leading dimension of array B. LDB >= 2.
SCALE (output) DOUBLE PRECISION
The scale factor. SCALE is chosen less than or equal to 1
to prevent the solution overflowing.
X (output) DOUBLE PRECISION array, dimension (LDX,2)
On exit with LUPPER = .TRUE., the leading 2-by-2 upper
triangular part of this array contains the upper
triangular part of the symmetric solution matrix X and the
strictly lower triangular part of X is not referenced.
On exit with LUPPER = .FALSE., the leading 2-by-2 lower
triangular part of this array contains the lower
triangular part of the symmetric solution matrix X and the
strictly upper triangular part of X is not referenced.
Note that X may be identified with B in the calling
statement.
LDX INTEGER
The leading dimension of array X. LDX >= 2.
XNORM (output) DOUBLE PRECISION
The infinity-norm of the solution.
Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: if T and -T have too close eigenvalues, so T
is perturbed to get a nonsingular equation.
NOTE: In the interests of speed, this routine does not
check the inputs for errors.
Method
The equivalent linear algebraic system of equations is formed and solved using Gaussian elimination with complete pivoting.References
[1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
Numerical Aspects
The algorithm is stable and reliable, since Gaussian elimination with complete pivoting is used.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve the real discrete Lyapunov matrix equation
op(A)'*X*op(A) - X = scale*C
where op(A) = A or A' (A**T), A is upper quasi-triangular and C is
symmetric (C = C'). (A' denotes the transpose of the matrix A.)
A is N-by-N, the right hand side C and the solution X are N-by-N,
and scale is an output scale factor, set less than or equal to 1
to avoid overflow in X. The solution matrix X is overwritten
onto C.
A must be in Schur canonical form (as returned by LAPACK routines
DGEES or DHSEQR), that is, block upper triangular with 1-by-1 and
2-by-2 diagonal blocks; each 2-by-2 diagonal block has its
diagonal elements equal and its off-diagonal elements of opposite
sign.
Specification
SUBROUTINE SB03MX( TRANA, N, A, LDA, C, LDC, SCALE, DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TRANA
INTEGER INFO, LDA, LDC, N
DOUBLE PRECISION SCALE
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), DWORK( * )
Arguments
Mode Parameters
TRANA CHARACTER*1
Specifies the form of op(A) to be used, as follows:
= 'N': op(A) = A (No transpose);
= 'T': op(A) = A**T (Transpose);
= 'C': op(A) = A**T (Conjugate transpose = Transpose).
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, X, and C. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
upper quasi-triangular matrix A, in Schur canonical form.
The part of A below the first sub-diagonal is not
referenced.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading N-by-N part of this array must
contain the symmetric matrix C.
On exit, if INFO >= 0, the leading N-by-N part of this
array contains the symmetric solution matrix X.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,N).
SCALE (output) DOUBLE PRECISION
The scale factor, scale, set less than or equal to 1 to
prevent the solution overflowing.
Workspace
DWORK DOUBLE PRECISION array, dimension (2*N)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if A has almost reciprocal eigenvalues; perturbed
values were used to solve the equation (but the
matrix A is unchanged).
Method
A discrete-time version of the Bartels-Stewart algorithm is used. A set of equivalent linear algebraic systems of equations of order at most four are formed and solved using Gaussian elimination with complete pivoting.References
[1] Barraud, A.Y. T
A numerical algorithm to solve A XA - X = Q.
IEEE Trans. Auto. Contr., AC-22, pp. 883-885, 1977.
[2] Bartels, R.H. and Stewart, G.W. T
Solution of the matrix equation A X + XB = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
Numerical Aspects
3 The algorithm requires 0(N ) operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve the real Lyapunov matrix equation
op(A)'*X + X*op(A) = scale*C
where op(A) = A or A' (A**T), A is upper quasi-triangular and C is
symmetric (C = C'). (A' denotes the transpose of the matrix A.)
A is N-by-N, the right hand side C and the solution X are N-by-N,
and scale is an output scale factor, set less than or equal to 1
to avoid overflow in X. The solution matrix X is overwritten
onto C.
A must be in Schur canonical form (as returned by LAPACK routines
DGEES or DHSEQR), that is, block upper triangular with 1-by-1 and
2-by-2 diagonal blocks; each 2-by-2 diagonal block has its
diagonal elements equal and its off-diagonal elements of opposite
sign.
Specification
SUBROUTINE SB03MY( TRANA, N, A, LDA, C, LDC, SCALE, INFO )
C .. Scalar Arguments ..
CHARACTER TRANA
INTEGER INFO, LDA, LDC, N
DOUBLE PRECISION SCALE
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * )
Arguments
Mode Parameters
TRANA CHARACTER*1
Specifies the form of op(A) to be used, as follows:
= 'N': op(A) = A (No transpose);
= 'T': op(A) = A**T (Transpose);
= 'C': op(A) = A**T (Conjugate transpose = Transpose).
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, X, and C. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
upper quasi-triangular matrix A, in Schur canonical form.
The part of A below the first sub-diagonal is not
referenced.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading N-by-N part of this array must
contain the symmetric matrix C.
On exit, if INFO >= 0, the leading N-by-N part of this
array contains the symmetric solution matrix X.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,N).
SCALE (output) DOUBLE PRECISION
The scale factor, scale, set less than or equal to 1 to
prevent the solution overflowing.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if A and -A have common or very close eigenvalues;
perturbed values were used to solve the equation
(but the matrix A is unchanged).
Method
Bartels-Stewart algorithm is used. A set of equivalent linear algebraic systems of equations of order at most four are formed and solved using Gaussian elimination with complete pivoting.References
[1] Bartels, R.H. and Stewart, G.W. T
Solution of the matrix equation A X + XB = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
Numerical Aspects
3 The algorithm requires 0(N ) operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve for X = op(U)'*op(U) either the stable non-negative
definite continuous-time Lyapunov equation
2
op(A)'*X + X*op(A) = -scale *op(B)'*op(B), (1)
or the convergent non-negative definite discrete-time Lyapunov
equation
2
op(A)'*X*op(A) - X = -scale *op(B)'*op(B), (2)
where op(K) = K or K' (i.e., the transpose of the matrix K), A is
an N-by-N matrix, op(B) is an M-by-N matrix, U is an upper
triangular matrix containing the Cholesky factor of the solution
matrix X, X = op(U)'*op(U), and scale is an output scale factor,
set less than or equal to 1 to avoid overflow in X. If matrix B
has full rank then the solution matrix X will be positive definite
and hence the Cholesky factor U will be nonsingular, but if B is
rank deficient, then X may be only positive semi-definite and U
will be singular.
In the case of equation (1) the matrix A must be stable (that is,
all the eigenvalues of A must have negative real parts), and for
equation (2) the matrix A must be convergent (that is, all the
eigenvalues of A must lie inside the unit circle).
Specification
SUBROUTINE SB03OD( DICO, FACT, TRANS, N, M, A, LDA, Q, LDQ, B,
$ LDB, SCALE, WR, WI, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, FACT, TRANS
INTEGER INFO, LDA, LDB, LDQ, LDWORK, M, N
DOUBLE PRECISION SCALE
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), Q(LDQ,*), WI(*),
$ WR(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of Lyapunov equation to be solved, as
follows:
= 'C': Equation (1), continuous-time case;
= 'D': Equation (2), discrete-time case.
FACT CHARACTER*1
Specifies whether or not the real Schur factorization
of the matrix A is supplied on entry, as follows:
= 'F': On entry, A and Q contain the factors from the
real Schur factorization of the matrix A;
= 'N': The Schur factorization of A will be computed
and the factors will be stored in A and Q.
TRANS CHARACTER*1
Specifies the form of op(K) to be used, as follows:
= 'N': op(K) = K (No transpose);
= 'T': op(K) = K**T (Transpose).
Input/Output Parameters
N (input) INTEGER
The order of the matrix A and the number of columns of
the matrix op(B). N >= 0.
M (input) INTEGER
The number of rows of the matrix op(B). M >= 0.
If M = 0, A is unchanged on exit, and Q, WR and WI are not
set.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A. If FACT = 'F', then A contains
an upper quasi-triangular matrix S in Schur canonical
form; the elements below the upper Hessenberg part of the
array A are then not referenced.
On exit, the leading N-by-N upper Hessenberg part of this
array contains the upper quasi-triangular matrix S in
Schur canonical form from the Shur factorization of A.
The contents of the array A is not modified if FACT = 'F'.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
Q (input or output) DOUBLE PRECISION array, dimension
(LDQ,N)
On entry, if FACT = 'F', then the leading N-by-N part of
this array must contain the orthogonal matrix Q of the
Schur factorization of A.
Otherwise, Q need not be set on entry.
On exit, the leading N-by-N part of this array contains
the orthogonal matrix Q of the Schur factorization of A.
The contents of the array Q is not modified if FACT = 'F'.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
if TRANS = 'N', and dimension (LDB,max(M,N)), if
TRANS = 'T'.
On entry, if TRANS = 'N', the leading M-by-N part of this
array must contain the coefficient matrix B of the
equation.
On entry, if TRANS = 'T', the leading N-by-M part of this
array must contain the coefficient matrix B of the
equation.
On exit, the leading N-by-N part of this array contains
the upper triangular Cholesky factor U of the solution
matrix X of the problem, X = op(U)'*op(U).
If M = 0 and N > 0, then U is set to zero.
LDB INTEGER
The leading dimension of the array B.
LDB >= MAX(1,N,M), if TRANS = 'N';
LDB >= MAX(1,N), if TRANS = 'T'.
SCALE (output) DOUBLE PRECISION
The scale factor, scale, set less than or equal to 1 to
prevent the solution overflowing.
WR (output) DOUBLE PRECISION array, dimension (N)
WI (output) DOUBLE PRECISION array, dimension (N)
If INFO >= 0 and INFO <= 3, WR and WI contain the real and
imaginary parts, respectively, of the eigenvalues of A.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or INFO = 1, DWORK(1) returns the
optimal value of LDWORK.
On exit, if INFO = -16, DWORK(1) returns the minimum value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
If M > 0, LDWORK >= MAX(1,4*N);
If M = 0, LDWORK >= 1.
For optimum performance LDWORK should sometimes be larger.
If LDWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message related to LDWORK is issued by
XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the Lyapunov equation is (nearly) singular
(warning indicator);
if DICO = 'C' this means that while the matrix A
(or the factor S) has computed eigenvalues with
negative real parts, it is only just stable in the
sense that small perturbations in A can make one or
more of the eigenvalues have a non-negative real
part;
if DICO = 'D' this means that while the matrix A
(or the factor S) has computed eigenvalues inside
the unit circle, it is nevertheless only just
convergent, in the sense that small perturbations
in A can make one or more of the eigenvalues lie
outside the unit circle;
perturbed values were used to solve the equation;
= 2: if FACT = 'N' and DICO = 'C', but the matrix A is
not stable (that is, one or more of the eigenvalues
of A has a non-negative real part), or DICO = 'D',
but the matrix A is not convergent (that is, one or
more of the eigenvalues of A lies outside the unit
circle); however, A will still have been factored
and the eigenvalues of A returned in WR and WI.
= 3: if FACT = 'F' and DICO = 'C', but the Schur factor S
supplied in the array A is not stable (that is, one
or more of the eigenvalues of S has a non-negative
real part), or DICO = 'D', but the Schur factor S
supplied in the array A is not convergent (that is,
one or more of the eigenvalues of S lies outside the
unit circle); the eigenvalues of A are still
returned in WR and WI;
= 4: if FACT = 'F' and the Schur factor S supplied in
the array A has two or more consecutive non-zero
elements on the first subdiagonal, so that there is
a block larger than 2-by-2 on the diagonal;
= 5: if FACT = 'F' and the Schur factor S supplied in
the array A has a 2-by-2 diagonal block with real
eigenvalues instead of a complex conjugate pair;
= 6: if FACT = 'N' and the LAPACK Library routine DGEES
has failed to converge. This failure is not likely
to occur. The matrix B will be unaltered but A will
be destroyed.
Method
The method used by the routine is based on the Bartels and Stewart
method [1], except that it finds the upper triangular matrix U
directly without first finding X and without the need to form the
normal matrix op(B)'*op(B).
The Schur factorization of a square matrix A is given by
A = QSQ',
where Q is orthogonal and S is an N-by-N block upper triangular
matrix with 1-by-1 and 2-by-2 blocks on its diagonal (which
correspond to the eigenvalues of A). If A has already been
factored prior to calling the routine however, then the factors
Q and S may be supplied and the initial factorization omitted.
If TRANS = 'N' and 6*M > 7*N, the matrix B is factored as
(QR factorization)
_ _
B = P ( R ),
( 0 )
_ _
where P is an M-by-M orthogonal matrix and R is a square upper
_ _
triangular matrix. Then, the matrix B = RQ is factored as
_
B = PR.
If TRANS = 'N' and 6*M <= 7*N, the matrix BQ is factored as
BQ = P ( R ), M >= N, BQ = P ( R Z ), M < N.
( 0 )
If TRANS = 'T' and 6*M > 7*N, the matrix B is factored as
(RQ factorization)
_ _
B = ( 0 R ) P,
_ _
where P is an M-by-M orthogonal matrix and R is a square upper
_ _
triangular matrix. Then, the matrix B = Q' R is factored as
_
B = RP.
If TRANS = 'T' and 6*M <= 7*N, the matrix Q' B is factored as
( Z )
Q' B = ( 0 R ) P, M >= N, Q' B = ( ) P, M < N.
( R )
These factorizations are utilised to either transform the
continuous-time Lyapunov equation to the canonical form
2
op(S)'*op(V)'*op(V) + op(V)'*op(V)*op(S) = -scale *op(F)'*op(F),
or the discrete-time Lyapunov equation to the canonical form
2
op(S)'*op(V)'*op(V)*op(S) - op(V)'*op(V) = -scale *op(F)'*op(F),
where V and F are upper triangular, and
F = R, M >= N, F = ( R Z ), M < N, if TRANS = 'N';
( 0 0 )
F = R, M >= N, F = ( 0 Z ), M < N, if TRANS = 'T'.
( 0 R )
The transformed equation is then solved for V, from which U is
obtained via the QR factorization of V*Q', if TRANS = 'N', or
via the RQ factorization of Q*V, if TRANS = 'T'.
References
[1] Bartels, R.H. and Stewart, G.W.
Solution of the matrix equation A'X + XB = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
[2] Hammarling, S.J.
Numerical solution of the stable, non-negative definite
Lyapunov equation.
IMA J. Num. Anal., 2, pp. 303-325, 1982.
Numerical Aspects
3 The algorithm requires 0(N ) operations and is backward stable.Further Comments
The Lyapunov equation may be very ill-conditioned. In particular,
if A is only just stable (or convergent) then the Lyapunov
equation will be ill-conditioned. A symptom of ill-conditioning
is "large" elements in U relative to those of A and B, or a
"small" value for scale. A condition estimate can be computed
using SLICOT Library routine SB03MD.
SB03OD routine can be also used for solving "unstable" Lyapunov
equations, i.e., when matrix A has all eigenvalues with positive
real parts, if DICO = 'C', or with moduli greater than one,
if DICO = 'D'. Specifically, one may solve for X = op(U)'*op(U)
either the continuous-time Lyapunov equation
2
op(A)'*X + X*op(A) = scale *op(B)'*op(B), (3)
or the discrete-time Lyapunov equation
2
op(A)'*X*op(A) - X = scale *op(B)'*op(B), (4)
provided, for equation (3), the given matrix A is replaced by -A,
or, for equation (4), the given matrices A and B are replaced by
inv(A) and B*inv(A), if TRANS = 'N' (or inv(A)*B, if TRANS = 'T'),
respectively. Although the inversion generally can rise numerical
problems, in case of equation (4) it is expected that the matrix A
is enough well-conditioned, having only eigenvalues with moduli
greater than 1. However, if A is ill-conditioned, it could be
preferable to use the more general SLICOT Lyapunov solver SB03MD.
Example
Program Text
* SB03OD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX
PARAMETER ( NMAX = 20, MMAX = 20 )
INTEGER LDA, LDB, LDQ, LDX, LDWORK
PARAMETER ( LDA = NMAX, LDB = MAX( MMAX,NMAX ),
$ LDQ = NMAX, LDX = NMAX )
PARAMETER ( LDWORK = 4*NMAX+MIN(MMAX,NMAX) )
* .. Local Scalars ..
DOUBLE PRECISION SCALE, TEMP
INTEGER I, INFO, J, K, M, N
CHARACTER*1 DICO, FACT, TRANS
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,LDB), DWORK(LDWORK),
$ Q(LDQ,NMAX), WR(NMAX), WI(NMAX), X(LDX,NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL SB03OD
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, DICO, FACT, TRANS
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( FACT, 'F' ) ) READ ( NIN, FMT = * )
$ ( ( Q(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) M
ELSE
IF ( LSAME( TRANS, 'N' ) ) THEN
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,N ), I = 1,M )
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
END IF
* Find the Cholesky factor U.
CALL SB03OD( DICO, FACT, TRANS, N, M, A, LDA, Q, LDQ, B,
$ LDB, SCALE, WR, WI, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 J = 1, N
WRITE ( NOUT, FMT = 99996 ) ( B(I,J), I = 1,J )
20 CONTINUE
* Form the solution matrix X = op(U)'*op(U).
IF ( LSAME( TRANS, 'N' ) ) THEN
DO 80 I = 1, N
DO 60 J = I, N
TEMP = ZERO
DO 40 K = 1, I
TEMP = TEMP + B(K,I)*B(K,J)
40 CONTINUE
X(I,J) = TEMP
X(J,I) = TEMP
60 CONTINUE
80 CONTINUE
ELSE
DO 140 I = 1, N
DO 120 J = I, N
TEMP = ZERO
DO 100 K = J, N
TEMP = TEMP + B(I,K)*B(J,K)
100 CONTINUE
X(I,J) = TEMP
X(J,I) = TEMP
120 CONTINUE
140 CONTINUE
END IF
WRITE ( NOUT, FMT = 99995 )
DO 160 J = 1, N
WRITE ( NOUT, FMT = 99996 ) ( X(I,J), I = 1,N )
160 CONTINUE
WRITE ( NOUT, FMT = 99992 ) SCALE
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB03OD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB03OD = ',I2)
99997 FORMAT (' The transpose of the Cholesky factor U is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The solution matrix X = op(U)''*op(U) is ')
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' M is out of range.',/' M = ',I5)
99992 FORMAT (/' Scaling factor = ',F8.4)
END
Program Data
SB03OD EXAMPLE PROGRAM DATA 4 5 C N N -1.0 37.0 -12.0 -12.0 -1.0 -10.0 0.0 4.0 2.0 -4.0 7.0 -6.0 2.0 2.0 7.0 -9.0 1.0 2.5 1.0 3.5 0.0 1.0 0.0 1.0 -1.0 -2.5 -1.0 -1.5 1.0 2.5 4.0 -5.5 -1.0 -2.5 -4.0 3.5Program Results
SB03OD EXAMPLE PROGRAM RESULTS The transpose of the Cholesky factor U is 1.0000 3.0000 1.0000 2.0000 -1.0000 1.0000 -1.0000 1.0000 -2.0000 1.0000 The solution matrix X = op(U)'*op(U) is 1.0000 3.0000 2.0000 -1.0000 3.0000 10.0000 5.0000 -2.0000 2.0000 5.0000 6.0000 -5.0000 -1.0000 -2.0000 -5.0000 7.0000 Scaling factor = 1.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB03OR.html 0000664 0000000 0000000 00000011654 14560147231 0020371 0 ustar 00root root 0000000 0000000
Purpose
To compute the solution of the Sylvester equations
op(S)'*X + X*op(A) = scale*C, if DISCR = .FALSE. or
op(S)'*X*op(A) - X = scale*C, if DISCR = .TRUE.
where op(K) = K or K' (i.e., the transpose of the matrix K), S is
an N-by-N block upper triangular matrix with one-by-one and
two-by-two blocks on the diagonal, A is an M-by-M matrix (M = 1 or
M = 2), X and C are each N-by-M matrices, and scale is an output
scale factor, set less than or equal to 1 to avoid overflow in X.
The solution X is overwritten on C.
SB03OR is a service routine for the Lyapunov solver SB03OT.
Specification
SUBROUTINE SB03OR( DISCR, LTRANS, N, M, S, LDS, A, LDA, C, LDC,
$ SCALE, INFO )
C .. Scalar Arguments ..
LOGICAL DISCR, LTRANS
INTEGER INFO, LDA, LDS, LDC, M, N
DOUBLE PRECISION SCALE
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), S( LDS, * )
Arguments
Mode Parameters
DISCR LOGICAL
Specifies the equation to be solved:
= .FALSE.: op(S)'*X + X*op(A) = scale*C;
= .TRUE. : op(S)'*X*op(A) - X = scale*C.
LTRANS LOGICAL
Specifies the form of op(K) to be used, as follows:
= .FALSE.: op(K) = K (No transpose);
= .TRUE. : op(K) = K**T (Transpose).
Input/Output Parameters
N (input) INTEGER
The order of the matrix S and also the number of rows of
matrices X and C. N >= 0.
M (input) INTEGER
The order of the matrix A and also the number of columns
of matrices X and C. M = 1 or M = 2.
S (input) DOUBLE PRECISION array, dimension (LDS,N)
The leading N-by-N upper Hessenberg part of the array S
must contain the block upper triangular matrix. The
elements below the upper Hessenberg part of the array S
are not referenced. The array S must not contain
diagonal blocks larger than two-by-two and the two-by-two
blocks must only correspond to complex conjugate pairs of
eigenvalues, not to real eigenvalues.
LDS INTEGER
The leading dimension of array S. LDS >= MAX(1,N).
A (input) DOUBLE PRECISION array, dimension (LDS,M)
The leading M-by-M part of this array must contain a
given matrix, where M = 1 or M = 2.
LDA INTEGER
The leading dimension of array A. LDA >= M.
C (input/output) DOUBLE PRECISION array, dimension (LDC,M)
On entry, C must contain an N-by-M matrix, where M = 1 or
M = 2.
On exit, C contains the N-by-M matrix X, the solution of
the Sylvester equation.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,N).
SCALE (output) DOUBLE PRECISION
The scale factor, scale, set less than or equal to 1 to
prevent the solution overflowing.
Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: if DISCR = .FALSE., and S and -A have common
eigenvalues, or if DISCR = .TRUE., and S and A have
eigenvalues whose product is equal to unity;
a solution has been computed using slightly
perturbed values.
Method
The LAPACK scheme for solving Sylvester equations is adapted.References
[1] Hammarling, S.J.
Numerical solution of the stable, non-negative definite
Lyapunov equation.
IMA J. Num. Anal., 2, pp. 303-325, 1982.
Numerical Aspects
2 The algorithm requires 0(N M) operations and is backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
H
To solve for X = op(U) *op(U) either the stable non-negative
definite continuous-time Lyapunov equation
H 2 H
op(S) *X + X*op(S) = -scale *op(R) *op(R), (1)
or the convergent non-negative definite discrete-time Lyapunov
equation
H 2 H
op(S) *X*op(S) - X = -scale *op(R) *op(R), (2)
where op(K) = K or K**H (i.e., the conjugate transpose of the
matrix K), S and R are complex N-by-N upper triangular matrices,
and scale is an output scale factor, set less than or equal to 1
to avoid overflow in X. The diagonal elements of the matrix R must
be real non-negative.
In the case of equation (1) the matrix S must be stable (that is,
all the eigenvalues of S must have negative real parts), and for
equation (2) the matrix S must be convergent (that is, all the
eigenvalues of S must lie inside the unit circle).
Specification
SUBROUTINE SB03OS( DISCR, LTRANS, N, S, LDS, R, LDR, SCALE, DWORK,
$ ZWORK, INFO )
C .. Scalar Arguments ..
DOUBLE PRECISION SCALE
INTEGER INFO, LDR, LDS, N
LOGICAL DISCR, LTRANS
C .. Array Arguments ..
COMPLEX*16 R(LDR,*), S(LDS,*), ZWORK(*)
DOUBLE PRECISION DWORK(*)
Arguments
Mode Parameters
DISCR LOGICAL
Specifies the type of Lyapunov equation to be solved as
follows:
= .TRUE. : Equation (2), discrete-time case;
= .FALSE.: Equation (1), continuous-time case.
LTRANS LOGICAL
Specifies the form of op(K) to be used, as follows:
= .FALSE.: op(K) = K (No transpose);
= .TRUE. : op(K) = K**H (Conjugate transpose).
Input/Output Parameters
N (input) INTEGER
The order of the matrices S and R. N >= 0.
S (input) COMPLEX*16 array of dimension (LDS,N)
The leading N-by-N upper triangular part of this array
must contain the upper triangular matrix.
The elements below the upper triangular part of the array
S are not referenced.
LDS INTEGER
The leading dimension of array S. LDS >= MAX(1,N).
R (input/output) COMPLEX*16 array of dimension (LDR,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix R, with
real non-negative entries on its main diagonal.
On exit, the leading N-by-N upper triangular part of this
array contains the upper triangular matrix U, with real
non-negative entries on its main diagonal.
The strictly lower triangle of R is not referenced.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
SCALE (output) DOUBLE PRECISION
The scale factor, scale, set less than or equal to 1 to
prevent the solution overflowing.
Workspace
DWORK DOUBLE PRECISION array, dimension (N-1) ZWORK COMPLEX*16 array, dimension (2*N-2)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 3: if the matrix S is not stable (that is, one or more
of the eigenvalues of S has a non-negative real
part), if DISCR = .FALSE., or not convergent (that
is, one or more of the eigenvalues of S lies outside
the unit circle), if DISCR = .TRUE..
Method
The method used by the routine is based on a variant of the
Bartels and Stewart backward substitution method [1], that finds
the Cholesky factor op(U) directly without first finding X and
without the need to form the normal matrix op(R)'*op(R) [2].
The continuous-time Lyapunov equation in the canonical form
H H H 2 H
op(S) *op(U) *op(U) + op(U) *op(U)*op(S) = -scale *op(R) *op(R),
or the discrete-time Lyapunov equation in the canonical form
H H H 2 H
op(S) *op(U) *op(U)*op(S) - op(U) *op(U) = -scale *op(R) *op(R),
where U and R are upper triangular, is solved for U.
References
[1] Bartels, R.H. and Stewart, G.W.
Solution of the matrix equation A'X + XB = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
[2] Hammarling, S.J.
Numerical solution of the stable, non-negative definite
Lyapunov equation.
IMA J. Num. Anal., 2, pp. 303-325, 1982.
Numerical Aspects
3 The algorithm requires 0(N ) operations and is backward stable.Further Comments
The Lyapunov equation may be very ill-conditioned. In particular if S is only just stable (or convergent) then the Lyapunov equation will be ill-conditioned. "Large" elements in U relative to those of S and R, or a "small" value for scale, is a symptom of ill-conditioning. A condition estimate can be computed using SLICOT Library routine SB03MD.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve for X = op(U)'*op(U) either the stable non-negative
definite continuous-time Lyapunov equation
2
op(S)'*X + X*op(S) = -scale *op(R)'*op(R) (1)
or the convergent non-negative definite discrete-time Lyapunov
equation
2
op(S)'*X*op(S) - X = -scale *op(R)'*op(R) (2)
where op(K) = K or K' (i.e., the transpose of the matrix K), S is
an N-by-N block upper triangular matrix with one-by-one or
two-by-two blocks on the diagonal, R is an N-by-N upper triangular
matrix, and scale is an output scale factor, set less than or
equal to 1 to avoid overflow in X.
In the case of equation (1) the matrix S must be stable (that
is, all the eigenvalues of S must have negative real parts),
and for equation (2) the matrix S must be convergent (that is,
all the eigenvalues of S must lie inside the unit circle).
Specification
SUBROUTINE SB03OT( DISCR, LTRANS, N, S, LDS, R, LDR, SCALE, DWORK,
$ INFO )
C .. Scalar Arguments ..
LOGICAL DISCR, LTRANS
INTEGER INFO, LDR, LDS, N
DOUBLE PRECISION SCALE
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), R(LDR,*), S(LDS,*)
Arguments
Mode Parameters
DISCR LOGICAL
Specifies the type of Lyapunov equation to be solved as
follows:
= .TRUE. : Equation (2), discrete-time case;
= .FALSE.: Equation (1), continuous-time case.
LTRANS LOGICAL
Specifies the form of op(K) to be used, as follows:
= .FALSE.: op(K) = K (No transpose);
= .TRUE. : op(K) = K**T (Transpose).
Input/Output Parameters
N (input) INTEGER
The order of the matrices S and R. N >= 0.
S (input) DOUBLE PRECISION array of dimension (LDS,N)
The leading N-by-N upper Hessenberg part of this array
must contain the block upper triangular matrix.
The elements below the upper Hessenberg part of the array
S are not referenced. The 2-by-2 blocks must only
correspond to complex conjugate pairs of eigenvalues (not
to real eigenvalues).
LDS INTEGER
The leading dimension of array S. LDS >= MAX(1,N).
R (input/output) DOUBLE PRECISION array of dimension (LDR,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix R.
On exit, the leading N-by-N upper triangular part of this
array contains the upper triangular matrix U.
The strict lower triangle of R is not referenced.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
SCALE (output) DOUBLE PRECISION
The scale factor, scale, set less than or equal to 1 to
prevent the solution overflowing.
Workspace
DWORK DOUBLE PRECISION array, dimension (4*N)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the Lyapunov equation is (nearly) singular
(warning indicator);
if DISCR = .FALSE., this means that while the
matrix S has computed eigenvalues with negative real
parts, it is only just stable in the sense that
small perturbations in S can make one or more of the
eigenvalues have a non-negative real part;
if DISCR = .TRUE., this means that while the
matrix S has computed eigenvalues inside the unit
circle, it is nevertheless only just convergent, in
the sense that small perturbations in S can make one
or more of the eigenvalues lie outside the unit
circle;
perturbed values were used to solve the equation
(but the matrix S is unchanged);
= 2: if the matrix S is not stable (that is, one or more
of the eigenvalues of S has a non-negative real
part), if DISCR = .FALSE., or not convergent (that
is, one or more of the eigenvalues of S lies outside
the unit circle), if DISCR = .TRUE.;
= 3: if the matrix S has two or more consecutive non-zero
elements on the first sub-diagonal, so that there is
a block larger than 2-by-2 on the diagonal;
= 4: if the matrix S has a 2-by-2 diagonal block with
real eigenvalues instead of a complex conjugate
pair.
Method
The method used by the routine is based on a variant of the
Bartels and Stewart backward substitution method [1], that finds
the Cholesky factor op(U) directly without first finding X and
without the need to form the normal matrix op(R)'*op(R) [2].
The continuous-time Lyapunov equation in the canonical form
2
op(S)'*op(U)'*op(U) + op(U)'*op(U)*op(S) = -scale *op(R)'*op(R),
or the discrete-time Lyapunov equation in the canonical form
2
op(S)'*op(U)'*op(U)*op(S) - op(U)'*op(U) = -scale *op(R)'*op(R),
where U and R are upper triangular, is solved for U.
References
[1] Bartels, R.H. and Stewart, G.W.
Solution of the matrix equation A'X + XB = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
[2] Hammarling, S.J.
Numerical solution of the stable, non-negative definite
Lyapunov equation.
IMA J. Num. Anal., 2, pp. 303-325, 1982.
Numerical Aspects
3 The algorithm requires 0(N ) operations and is backward stable.Further Comments
The Lyapunov equation may be very ill-conditioned. In particular if S is only just stable (or convergent) then the Lyapunov equation will be ill-conditioned. "Large" elements in U relative to those of S and R, or a "small" value for scale, is a symptom of ill-conditioning. A condition estimate can be computed using SLICOT Library routine SB03MD.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve for X = op(U)'*op(U) either the stable non-negative
definite continuous-time Lyapunov equation
2
op(A)'*X + X*op(A) = -scale *op(B)'*op(B) (1)
or the convergent non-negative definite discrete-time Lyapunov
equation
2
op(A)'*X*op(A) - X = -scale *op(B)'*op(B) (2)
where op(K) = K or K' (i.e., the transpose of the matrix K), A is
an N-by-N matrix in real Schur form, op(B) is an M-by-N matrix,
U is an upper triangular matrix containing the Cholesky factor of
the solution matrix X, X = op(U)'*op(U), and scale is an output
scale factor, set less than or equal to 1 to avoid overflow in X.
If matrix B has full rank then the solution matrix X will be
positive-definite and hence the Cholesky factor U will be
nonsingular, but if B is rank deficient then X may only be
positive semi-definite and U will be singular.
In the case of equation (1) the matrix A must be stable (that
is, all the eigenvalues of A must have negative real parts),
and for equation (2) the matrix A must be convergent (that is,
all the eigenvalues of A must lie inside the unit circle).
Specification
SUBROUTINE SB03OU( DISCR, LTRANS, N, M, A, LDA, B, LDB, TAU, U,
$ LDU, SCALE, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
LOGICAL DISCR, LTRANS
INTEGER INFO, LDA, LDB, LDU, LDWORK, M, N
DOUBLE PRECISION SCALE
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), TAU(*), U(LDU,*)
Arguments
Mode Parameters
DISCR LOGICAL
Specifies the type of Lyapunov equation to be solved as
follows:
= .TRUE. : Equation (2), discrete-time case;
= .FALSE.: Equation (1), continuous-time case.
LTRANS LOGICAL
Specifies the form of op(K) to be used, as follows:
= .FALSE.: op(K) = K (No transpose);
= .TRUE. : op(K) = K**T (Transpose).
Input/Output Parameters
N (input) INTEGER
The order of the matrix A and the number of columns in
matrix op(B). N >= 0.
M (input) INTEGER
The number of rows in matrix op(B). M >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N upper Hessenberg part of this array
must contain a real Schur form matrix S. The elements
below the upper Hessenberg part of the array A are not
referenced. The 2-by-2 blocks must only correspond to
complex conjugate pairs of eigenvalues (not to real
eigenvalues).
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
if LTRANS = .FALSE., and dimension (LDB,M), if
LTRANS = .TRUE..
On entry, if LTRANS = .FALSE., the leading M-by-N part of
this array must contain the coefficient matrix B of the
equation.
On entry, if LTRANS = .TRUE., the leading N-by-M part of
this array must contain the coefficient matrix B of the
equation.
On exit, if LTRANS = .FALSE., the leading
MIN(M,N)-by-MIN(M,N) upper triangular part of this array
contains the upper triangular matrix R (as defined in
METHOD), and the M-by-MIN(M,N) strictly lower triangular
part together with the elements of the array TAU are
overwritten by details of the matrix P (also defined in
METHOD). When M < N, columns (M+1),...,N of the array B
are overwritten by the matrix Z (see METHOD).
On exit, if LTRANS = .TRUE., the leading
MIN(M,N)-by-MIN(M,N) upper triangular part of
B(1:N,M-N+1), if M >= N, or of B(N-M+1:N,1:M), if M < N,
contains the upper triangular matrix R (as defined in
METHOD), and the remaining elements (below the diagonal
of R) together with the elements of the array TAU are
overwritten by details of the matrix P (also defined in
METHOD). When M < N, rows 1,...,(N-M) of the array B
are overwritten by the matrix Z (see METHOD).
LDB INTEGER
The leading dimension of array B.
LDB >= MAX(1,M), if LTRANS = .FALSE.,
LDB >= MAX(1,N), if LTRANS = .TRUE..
TAU (output) DOUBLE PRECISION array of dimension (MIN(N,M))
This array contains the scalar factors of the elementary
reflectors defining the matrix P.
U (output) DOUBLE PRECISION array of dimension (LDU,N)
The leading N-by-N upper triangular part of this array
contains the Cholesky factor of the solution matrix X of
the problem, X = op(U)'*op(U).
The array U may be identified with B in the calling
statement, if B is properly dimensioned, and the
intermediate results returned in B are not needed.
LDU INTEGER
The leading dimension of array U. LDU >= MAX(1,N).
SCALE (output) DOUBLE PRECISION
The scale factor, scale, set less than or equal to 1 to
prevent the solution overflowing.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, or INFO = 1, DWORK(1) returns the
optimal value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,4*N).
For optimum performance LDWORK should sometimes be larger.
If LDWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message related to LDWORK is issued by
XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the Lyapunov equation is (nearly) singular
(warning indicator);
if DISCR = .FALSE., this means that while the matrix
A has computed eigenvalues with negative real parts,
it is only just stable in the sense that small
perturbations in A can make one or more of the
eigenvalues have a non-negative real part;
if DISCR = .TRUE., this means that while the matrix
A has computed eigenvalues inside the unit circle,
it is nevertheless only just convergent, in the
sense that small perturbations in A can make one or
more of the eigenvalues lie outside the unit circle;
perturbed values were used to solve the equation
(but the matrix A is unchanged);
= 2: if matrix A is not stable (that is, one or more of
the eigenvalues of A has a non-negative real part),
if DISCR = .FALSE., or not convergent (that is, one
or more of the eigenvalues of A lies outside the
unit circle), if DISCR = .TRUE.;
= 3: if matrix A has two or more consecutive non-zero
elements on the first sub-diagonal, so that there is
a block larger than 2-by-2 on the diagonal;
= 4: if matrix A has a 2-by-2 diagonal block with real
eigenvalues instead of a complex conjugate pair.
Method
The method used by the routine is based on the Bartels and
Stewart method [1], except that it finds the upper triangular
matrix U directly without first finding X and without the need
to form the normal matrix op(B)'*op(B) [2].
If LTRANS = .FALSE., the matrix B is factored as
B = P ( R ), M >= N, B = P ( R Z ), M < N,
( 0 )
(QR factorization), where P is an M-by-M orthogonal matrix and
R is a square upper triangular matrix.
If LTRANS = .TRUE., the matrix B is factored as
B = ( 0 R ) P, M >= N, B = ( Z ) P, M < N,
( R )
(RQ factorization), where P is an M-by-M orthogonal matrix and
R is a square upper triangular matrix.
These factorizations are used to solve the continuous-time
Lyapunov equation in the canonical form
2
op(A)'*op(U)'*op(U) + op(U)'*op(U)*op(A) = -scale *op(F)'*op(F),
or the discrete-time Lyapunov equation in the canonical form
2
op(A)'*op(U)'*op(U)*op(A) - op(U)'*op(U) = -scale *op(F)'*op(F),
where U and F are N-by-N upper triangular matrices, and
F = R, if M >= N, or
F = ( R ), if LTRANS = .FALSE., or
( 0 )
F = ( 0 Z ), if LTRANS = .TRUE., if M < N.
( 0 R )
The canonical equation is solved for U.
References
[1] Bartels, R.H. and Stewart, G.W.
Solution of the matrix equation A'X + XB = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
[2] Hammarling, S.J.
Numerical solution of the stable, non-negative definite
Lyapunov equation.
IMA J. Num. Anal., 2, pp. 303-325, 1982.
Numerical Aspects
3 The algorithm requires 0(N ) operations and is backward stable.Further Comments
The Lyapunov equation may be very ill-conditioned. In particular, if A is only just stable (or convergent) then the Lyapunov equation will be ill-conditioned. "Large" elements in U relative to those of A and B, or a "small" value for scale, are symptoms of ill-conditioning. A condition estimate can be computed using SLICOT Library routine SB03MD.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To construct a complex plane rotation such that, for a complex
number a and a real number b,
( conjg( c ) s )*( a ) = ( d ),
( -s c ) ( b ) ( 0 )
where d is always real and is overwritten on a, so that on
return the imaginary part of a is zero. b is unaltered.
This routine has A and C declared as REAL, because it is intended
for use within a real Lyapunov solver and the REAL declarations
mean that a standard Fortran DOUBLE PRECISION version may be
readily constructed. However A and C could safely be declared
COMPLEX in the calling program, although some systems may give a
type mismatch warning.
Specification
SUBROUTINE SB03OV( A, B, SMALL, C, S )
C .. Scalar Arguments ..
DOUBLE PRECISION B, S, SMALL
C .. Array Arguments ..
DOUBLE PRECISION A(2), C(2)
Arguments
Input/Output Parameters
A (input/output) DOUBLE PRECISION array, dimension (2)
On entry, A(1) and A(2) must contain the real and
imaginary part, respectively, of the complex number a.
On exit, A(1) contains the real part of d, and A(2) is
set to zero.
B (input) DOUBLE PRECISION
The real number b.
SMALL (input) DOUBLE PRECISION
A small real number. If the norm d of [ a; b ] is smaller
than SMALL, then the rotation is taken as a unit matrix,
and A(1) and A(2) are set to d and 0, respectively.
C (output) DOUBLE PRECISION array, dimension (2)
C(1) and C(2) contain the real and imaginary part,
respectively, of the complex number c, the cosines of
the plane rotation.
S (output) DOUBLE PRECISION
The real number s, the sines of the plane rotation.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve for the Cholesky factor U of X,
op(U)'*op(U) = X,
where U is a two-by-two upper triangular matrix, either the
continuous-time two-by-two Lyapunov equation
2
op(S)'*X + X*op(S) = -ISGN*scale *op(R)'*op(R),
when DISCR = .FALSE., or the discrete-time two-by-two Lyapunov
equation
2
op(S)'*X*op(S) - X = -ISGN*scale *op(R)'*op(R),
when DISCR = .TRUE., where op(K) = K or K' (i.e., the transpose of
the matrix K), S is a two-by-two matrix with complex conjugate
eigenvalues, R is a two-by-two upper triangular matrix,
ISGN = -1 or 1, and scale is an output scale factor, set less
than or equal to 1 to avoid overflow in X. The routine also
computes two matrices, B and A, so that
2
B*U = U*S and A*U = scale *R, if LTRANS = .FALSE., or
2
U*B = S*U and U*A = scale *R, if LTRANS = .TRUE.,
which are used by the general Lyapunov solver.
In the continuous-time case ISGN*S must be stable, so that its
eigenvalues must have strictly negative real parts.
In the discrete-time case S must be convergent if ISGN = 1, that
is, its eigenvalues must have moduli less than unity, or S must
be completely divergent if ISGN = -1, that is, its eigenvalues
must have moduli greater than unity.
Specification
SUBROUTINE SB03OY( DISCR, LTRANS, ISGN, S, LDS, R, LDR, A, LDA,
$ SCALE, INFO )
C .. Scalar Arguments ..
LOGICAL DISCR, LTRANS
INTEGER INFO, ISGN, LDA, LDR, LDS
DOUBLE PRECISION SCALE
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), R(LDR,*), S(LDS,*)
Arguments
Mode Parameters
DISCR LOGICAL
Specifies the equation to be solved: 2
= .FALSE.: op(S)'*X + X*op(S) = -ISGN*scale *op(R)'*op(R);
2
= .TRUE. : op(S)'*X*op(S) - X = -ISGN*scale *op(R)'*op(R).
LTRANS LOGICAL
Specifies the form of op(K) to be used, as follows:
= .FALSE.: op(K) = K (No transpose);
= .TRUE. : op(K) = K**T (Transpose).
ISGN INTEGER
Specifies the sign of the equation as described before.
ISGN may only be 1 or -1.
Input/Output Parameters
S (input/output) DOUBLE PRECISION array, dimension (LDS,2)
On entry, S must contain a 2-by-2 matrix.
On exit, S contains a 2-by-2 matrix B such that B*U = U*S,
if LTRANS = .FALSE., or U*B = S*U, if LTRANS = .TRUE..
Notice that if U is nonsingular then
B = U*S*inv( U ), if LTRANS = .FALSE.
B = inv( U )*S*U, if LTRANS = .TRUE..
LDS INTEGER
The leading dimension of array S. LDS >= 2.
R (input/output) DOUBLE PRECISION array, dimension (LDR,2)
On entry, R must contain a 2-by-2 upper triangular matrix.
The element R( 2, 1 ) is not referenced.
On exit, R contains U, the 2-by-2 upper triangular
Cholesky factor of the solution X, X = op(U)'*op(U).
LDR INTEGER
The leading dimension of array R. LDR >= 2.
A (output) DOUBLE PRECISION array, dimension (LDA,2)
A contains a 2-by-2 upper triangular matrix A satisfying
A*U/scale = scale*R, if LTRANS = .FALSE., or
U*A/scale = scale*R, if LTRANS = .TRUE..
Notice that if U is nonsingular then
A = scale*scale*R*inv( U ), if LTRANS = .FALSE.
A = scale*scale*inv( U )*R, if LTRANS = .TRUE..
LDA INTEGER
The leading dimension of array A. LDA >= 2.
SCALE (output) DOUBLE PRECISION
The scale factor, scale, set less than or equal to 1 to
prevent the solution overflowing.
Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: if the Lyapunov equation is (nearly) singular
(warning indicator);
if DISCR = .FALSE., this means that while the
matrix S has computed eigenvalues with negative real
parts, it is only just stable in the sense that
small perturbations in S can make one or more of the
eigenvalues have a non-negative real part;
if DISCR = .TRUE., this means that while the
matrix S has computed eigenvalues inside the unit
circle, it is nevertheless only just convergent, in
the sense that small perturbations in S can make one
or more of the eigenvalues lie outside the unit
circle;
perturbed values were used to solve the equation
(but the matrix S is unchanged);
= 2: if DISCR = .FALSE., and ISGN*S is not stable or
if DISCR = .TRUE., ISGN = 1 and S is not convergent
or if DISCR = .TRUE., ISGN = -1 and S is not
completely divergent;
= 4: if S has real eigenvalues.
NOTE: In the interests of speed, this routine does not check all
inputs for errors.
Method
The LAPACK scheme for solving 2-by-2 Sylvester equations is adapted for 2-by-2 Lyapunov equations, but directly computing the Cholesky factor of the solution.References
[1] Hammarling S. J.
Numerical solution of the stable, non-negative definite
Lyapunov equation.
IMA J. Num. Anal., 2, pp. 303-325, 1982.
Numerical Aspects
The algorithm is backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
H
To solve for X = op(U) *op(U) either the stable non-negative
definite continuous-time Lyapunov equation
H 2 H
op(A) *X + X*op(A) = -scale *op(B) *op(B), (1)
or the convergent non-negative definite discrete-time Lyapunov
equation
H 2 H
op(A) *X*op(A) - X = -scale *op(B) *op(B), (2)
where op(K) = K or K**H (i.e., the conjugate transpose of the
matrix K), A is an N-by-N matrix, op(B) is an M-by-N matrix, U is
an upper triangular matrix containing the Cholesky factor of the
solution matrix X, and scale is an output scale factor, set less
than or equal to 1 to avoid overflow in X. If matrix B has full
rank, then the solution matrix X will be positive definite and
hence the Cholesky factor U will be nonsingular, but if B is rank
deficient, then X may be only positive semi-definite and U will be
singular.
In the case of equation (1) the matrix A must be stable (that is,
all the eigenvalues of A must have negative real parts), and for
equation (2) the matrix A must be convergent (that is, all the
eigenvalues of A must lie inside the unit circle).
Specification
SUBROUTINE SB03OZ( DICO, FACT, TRANS, N, M, A, LDA, Q, LDQ, B,
$ LDB, SCALE, W, DWORK, ZWORK, LZWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, FACT, TRANS
INTEGER INFO, LDA, LDB, LDQ, LZWORK, M, N
DOUBLE PRECISION SCALE
C .. Array Arguments ..
COMPLEX*16 A(LDA,*), B(LDB,*), Q(LDQ,*), W(*), ZWORK(*)
DOUBLE PRECISION DWORK(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of Lyapunov equation to be solved, as
follows:
= 'C': Equation (1), continuous-time case;
= 'D': Equation (2), discrete-time case.
FACT CHARACTER*1
Specifies whether or not the Schur factorization of the
matrix A is supplied on entry, as follows:
= 'F': On entry, A and Q contain the factors from the
Schur factorization of the matrix A;
= 'N': The Schur factorization of A will be computed
and the factors will be stored in A and Q.
TRANS CHARACTER*1
Specifies the form of op(K) to be used, as follows:
= 'N': op(K) = K (No transpose);
= 'C': op(K) = K**H (Conjugate transpose).
Input/Output Parameters
N (input) INTEGER
The order of the matrix A and the number of columns of
the matrix op(B). N >= 0.
M (input) INTEGER
The number of rows of the matrix op(B). M >= 0.
If M = 0, A is unchanged on exit, and Q and W are not set.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A. If FACT = 'F', then A contains
an upper triangular matrix S in Schur form; the elements
below the diagonal of the array A are then not referenced.
On exit, the leading N-by-N upper triangular part of this
array contains the upper triangle of the matrix S.
The contents of the array A is not modified if FACT = 'F'.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
Q (input or output) COMPLEX*16 array, dimension (LDQ,N)
On entry, if FACT = 'F', then the leading N-by-N part of
this array must contain the unitary matrix Q of the Schur
factorization of A.
Otherwise, Q need not be set on entry.
On exit, the leading N-by-N part of this array contains
the unitary matrix Q of the Schur factorization of A.
The contents of the array Q is not modified if FACT = 'F'.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= MAX(1,N).
B (input/output) COMPLEX*16 array, dimension (LDB,N)
if TRANS = 'N', and dimension (LDB,max(M,N)), if
TRANS = 'C'.
On entry, if TRANS = 'N', the leading M-by-N part of this
array must contain the coefficient matrix B of the
equation.
On entry, if TRANS = 'C', the leading N-by-M part of this
array must contain the coefficient matrix B of the
equation.
On exit, the leading N-by-N part of this array contains
the upper triangular Cholesky factor U of the solution
matrix X of the problem, X = op(U)**H * op(U).
If M = 0 and N > 0, then U is set to zero.
LDB INTEGER
The leading dimension of the array B.
LDB >= MAX(1,N,M), if TRANS = 'N';
LDB >= MAX(1,N), if TRANS = 'C'.
SCALE (output) DOUBLE PRECISION
The scale factor, scale, set less than or equal to 1 to
prevent the solution overflowing.
W (output) COMPLEX*16 array, dimension (N)
If INFO >= 0 and INFO <= 3, W contains the eigenvalues of
the matrix A.
Workspace
DWORK DOUBLE PRECISION array, dimension (N)
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0 or INFO = 1, ZWORK(1) returns the
optimal value of LZWORK.
On exit, if INFO = -16, ZWORK(1) returns the minimum value
of LZWORK.
LZWORK INTEGER
The length of the array ZWORK.
If M > 0, LZWORK >= MAX(1,2*N+MAX(MIN(N,M)-2,0));
If M = 0, LZWORK >= 1.
For optimum performance LZWORK should sometimes be larger.
If LZWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the ZWORK
array, returns this value as the first entry of the ZWORK
array, and no error message related to LZWORK is issued by
XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the Lyapunov equation is (nearly) singular
(warning indicator);
if DICO = 'C' this means that while the matrix A
(or the factor S) has computed eigenvalues with
negative real parts, it is only just stable in the
sense that small perturbations in A can make one or
more of the eigenvalues have a non-negative real
part;
if DICO = 'D' this means that while the matrix A
(or the factor S) has computed eigenvalues inside
the unit circle, it is nevertheless only just
convergent, in the sense that small perturbations
in A can make one or more of the eigenvalues lie
outside the unit circle;
perturbed values were used to solve the equation;
= 2: if FACT = 'N' and DICO = 'C', but the matrix A is
not stable (that is, one or more of the eigenvalues
of A has a non-negative real part), or DICO = 'D',
but the matrix A is not convergent (that is, one or
more of the eigenvalues of A lies outside the unit
circle); however, A will still have been factored
and the eigenvalues of A returned in W;
= 3: if FACT = 'F' and DICO = 'C', but the Schur factor S
supplied in the array A is not stable (that is, one
or more of the eigenvalues of S has a non-negative
real part), or DICO = 'D', but the Schur factor S
supplied in the array A is not convergent (that is,
one or more of the eigenvalues of S lies outside the
unit circle); the eigenvalues of A are still
returned in W;
= 6: if FACT = 'N' and the LAPACK Library routine ZGEES
has failed to converge. This failure is not likely
to occur. The matrix B will be unaltered but A will
be destroyed.
Method
The method used by the routine is based on the Bartels and Stewart
method [1], except that it finds the upper triangular matrix U
directly without first finding X and without the need to form the
normal matrix op(B)**H * op(B).
The Schur factorization of a square matrix A is given by
H
A = QSQ ,
where Q is unitary and S is an N-by-N upper triangular matrix.
If A has already been factored prior to calling the routine, then
the factors Q and S may be supplied and the initial factorization
omitted.
If TRANS = 'N' and 6*M > 7*N, the matrix B is factored as
(QR factorization)
_ _
B = P ( R ),
( 0 )
_ _
where P is an M-by-M unitary matrix and R is a square upper
_ _
triangular matrix. Then, the matrix B = RQ is factored as
_
B = PR.
If TRANS = 'N' and 6*M <= 7*N, the matrix BQ is factored as
BQ = P ( R ), M >= N, BQ = P ( R Z ), M < N.
( 0 )
If TRANS = 'C' and 6*M > 7*N, the matrix B is factored as
(RQ factorization)
_ _
B = ( 0 R ) P,
_ _
where P is an M-by-M unitary matrix and R is a square upper
_ H _
triangular matrix. Then, the matrix B = Q R is factored as
_
B = RP.
H
If TRANS = 'C' and 6*M <= 7*N, the matrix Q B is factored as
H H ( Z )
Q B = ( 0 R ) P, M >= N, Q B = ( ) P, M < N.
( R )
These factorizations are utilised to either transform the
continuous-time Lyapunov equation to the canonical form
H H H 2 H
op(S) *op(V) *op(V) + op(V) *op(V)*op(S) = -scale *op(F) *op(F),
or the discrete-time Lyapunov equation to the canonical form
H H H 2 H
op(S) *op(V) *op(V)*op(S) - op(V) *op(V) = -scale *op(F) *op(F),
where V and F are upper triangular, and
F = R, M >= N, F = ( R Z ), M < N, if TRANS = 'N';
( 0 0 )
F = R, M >= N, F = ( 0 Z ), M < N, if TRANS = 'C'.
( 0 R )
The transformed equation is then solved for V, from which U is
obtained via the QR factorization of V*Q**H, if TRANS = 'N', or
via the RQ factorization of Q*V, if TRANS = 'C'.
References
[1] Bartels, R.H. and Stewart, G.W.
Solution of the matrix equation A'X + XB = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
[2] Hammarling, S.J.
Numerical solution of the stable, non-negative definite
Lyapunov equation.
IMA J. Num. Anal., 2, pp. 303-325, 1982.
Numerical Aspects
3 The algorithm requires 0(N ) operations and is backward stable.Further Comments
The Lyapunov equation may be very ill-conditioned. In particular,
if A is only just stable (or convergent) then the Lyapunov
equation will be ill-conditioned. A symptom of ill-conditioning
is "large" elements in U relative to those of A and B, or a
"small" value for scale.
SB03OZ routine can be also used for solving "unstable" Lyapunov
equations, i.e., when matrix A has all eigenvalues with positive
real parts, if DICO = 'C', or with moduli greater than one,
if DICO = 'D'. Specifically, one may solve for X = op(U)**H*op(U)
either the continuous-time Lyapunov equation
H 2 H
op(A) *X + X*op(A) = scale *op(B) *op(B), (3)
or the discrete-time Lyapunov equation
H 2 H
op(A) *X*op(A) - X = scale *op(B) *op(B), (4)
provided, for equation (3), the given matrix A is replaced by -A,
or, for equation (4), the given matrices A and B are replaced by
inv(A) and B*inv(A), if TRANS = 'N' (or inv(A)*B, if TRANS = 'C'),
respectively. Although the inversion generally can rise numerical
problems, in case of equation (4) it is expected that the matrix A
is enough well-conditioned, having only eigenvalues with moduli
greater than 1.
Example
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB03PD.html 0000664 0000000 0000000 00000021300 14560147231 0020341 0 ustar 00root root 0000000 0000000
Purpose
To solve the real discrete Lyapunov matrix equation
op(A)'*X*op(A) - X = scale*C
and/or estimate the quantity, called separation,
sepd(op(A),op(A)') = min norm(op(A)'*X*op(A) - X)/norm(X)
where op(A) = A or A' (A**T) and C is symmetric (C = C').
(A' denotes the transpose of the matrix A.) A is N-by-N, the right
hand side C and the solution X are N-by-N, and scale is an output
scale factor, set less than or equal to 1 to avoid overflow in X.
Specification
SUBROUTINE SB03PD( JOB, FACT, TRANA, N, A, LDA, U, LDU, C, LDC,
$ SCALE, SEPD, FERR, WR, WI, IWORK, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER FACT, JOB, TRANA
INTEGER INFO, LDA, LDC, LDU, LDWORK, N
DOUBLE PRECISION FERR, SCALE, SEPD
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), DWORK( * ),
$ U( LDU, * ), WI( * ), WR( * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'X': Compute the solution only;
= 'S': Compute the separation only;
= 'B': Compute both the solution and the separation.
FACT CHARACTER*1
Specifies whether or not the real Schur factorization
of the matrix A is supplied on entry, as follows:
= 'F': On entry, A and U contain the factors from the
real Schur factorization of the matrix A;
= 'N': The Schur factorization of A will be computed
and the factors will be stored in A and U.
TRANA CHARACTER*1
Specifies the form of op(A) to be used, as follows:
= 'N': op(A) = A (No transpose);
= 'T': op(A) = A**T (Transpose);
= 'C': op(A) = A**T (Conjugate transpose = Transpose).
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, X, and C. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A. If FACT = 'F', then A contains
an upper quasi-triangular matrix in Schur canonical form.
On exit, if INFO = 0 or INFO = N+1, the leading N-by-N
part of this array contains the upper quasi-triangular
matrix in Schur canonical form from the Shur factorization
of A. The contents of array A is not modified if
FACT = 'F'.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
U (input or output) DOUBLE PRECISION array, dimension
(LDU,N)
If FACT = 'F', then U is an input argument and on entry
it must contain the orthogonal matrix U from the real
Schur factorization of A.
If FACT = 'N', then U is an output argument and on exit,
if INFO = 0 or INFO = N+1, it contains the orthogonal
N-by-N matrix from the real Schur factorization of A.
LDU INTEGER
The leading dimension of array U. LDU >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry with JOB = 'X' or 'B', the leading N-by-N part of
this array must contain the symmetric matrix C.
On exit with JOB = 'X' or 'B', if INFO = 0 or INFO = N+1,
the leading N-by-N part of C has been overwritten by the
symmetric solution matrix X.
If JOB = 'S', C is not referenced.
LDC INTEGER
The leading dimension of array C.
LDC >= 1, if JOB = 'S';
LDC >= MAX(1,N), otherwise.
SCALE (output) DOUBLE PRECISION
The scale factor, scale, set less than or equal to 1 to
prevent the solution overflowing.
SEPD (output) DOUBLE PRECISION
If JOB = 'S' or JOB = 'B', and INFO = 0 or INFO = N+1,
SEPD contains the estimate in the 1-norm of
sepd(op(A),op(A)').
If JOB = 'X' or N = 0, SEPD is not referenced.
FERR (output) DOUBLE PRECISION
If JOB = 'B', and INFO = 0 or INFO = N+1, FERR contains
an estimated forward error bound for the solution X.
If XTRUE is the true solution, FERR bounds the relative
error in the computed solution, measured in the Frobenius
norm: norm(X - XTRUE)/norm(XTRUE).
If JOB = 'X' or JOB = 'S', FERR is not referenced.
WR (output) DOUBLE PRECISION array, dimension (N)
WI (output) DOUBLE PRECISION array, dimension (N)
If FACT = 'N', and INFO = 0 or INFO = N+1, WR and WI
contain the real and imaginary parts, respectively, of the
eigenvalues of A.
If FACT = 'F', WR and WI are not referenced.
Workspace
IWORK INTEGER array, dimension (N*N)
This array is not referenced if JOB = 'X'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or INFO = N+1, DWORK(1) returns the
optimal value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= 1 and
If JOB = 'X' then
If FACT = 'F', LDWORK >= MAX(N*N,2*N);
If FACT = 'N', LDWORK >= MAX(N*N,3*N).
If JOB = 'S' or JOB = 'B' then
LDWORK >= 2*N*N + 2*N.
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, the QR algorithm failed to compute all
the eigenvalues (see LAPACK Library routine DGEES);
elements i+1:n of WR and WI contain eigenvalues
which have converged, and A contains the partially
converged Schur form;
= N+1: if matrix A has almost reciprocal eigenvalues;
perturbed values were used to solve the equation
(but the matrix A is unchanged).
Method
After reducing matrix A to real Schur canonical form (if needed), a discrete-time version of the Bartels-Stewart algorithm is used. A set of equivalent linear algebraic systems of equations of order at most four are formed and solved using Gaussian elimination with complete pivoting.References
[1] Barraud, A.Y. T
A numerical algorithm to solve A XA - X = Q.
IEEE Trans. Auto. Contr., AC-22, pp. 883-885, 1977.
[2] Bartels, R.H. and Stewart, G.W. T
Solution of the matrix equation A X + XB = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
Numerical Aspects
3 The algorithm requires 0(N ) operations.Further Comments
SEPD is defined as
sepd( op(A), op(A)' ) = sigma_min( T )
where sigma_min(T) is the smallest singular value of the
N*N-by-N*N matrix
T = kprod( op(A)', op(A)' ) - I(N**2).
I(N**2) is an N*N-by-N*N identity matrix, and kprod denotes the
Kronecker product. The program estimates sigma_min(T) by the
reciprocal of an estimate of the 1-norm of inverse(T). The true
reciprocal 1-norm of inverse(T) cannot differ from sigma_min(T) by
more than a factor of N.
When SEPD is small, small changes in A, C can cause large changes
in the solution of the equation. An approximate bound on the
maximum relative error in the computed solution is
EPS * norm(A)**2 / SEPD
where EPS is the machine precision.
Example
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB03QD.html 0000664 0000000 0000000 00000037256 14560147231 0020363 0 ustar 00root root 0000000 0000000
Purpose
To estimate the conditioning and compute an error bound on the
solution of the real continuous-time Lyapunov matrix equation
op(A)'*X + X*op(A) = scale*C
where op(A) = A or A' (A**T) and C is symmetric (C = C**T). The
matrix A is N-by-N, the right hand side C and the solution X are
N-by-N symmetric matrices, and scale is a given scale factor.
Specification
SUBROUTINE SB03QD( JOB, FACT, TRANA, UPLO, LYAPUN, N, SCALE, A,
$ LDA, T, LDT, U, LDU, C, LDC, X, LDX, SEP,
$ RCOND, FERR, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER FACT, JOB, LYAPUN, TRANA, UPLO
INTEGER INFO, LDA, LDC, LDT, LDU, LDWORK, LDX, N
DOUBLE PRECISION FERR, RCOND, SCALE, SEP
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), DWORK( * ),
$ T( LDT, * ), U( LDU, * ), X( LDX, * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'C': Compute the reciprocal condition number only;
= 'E': Compute the error bound only;
= 'B': Compute both the reciprocal condition number and
the error bound.
FACT CHARACTER*1
Specifies whether or not the real Schur factorization
of the matrix A is supplied on entry, as follows:
= 'F': On entry, T and U (if LYAPUN = 'O') contain the
factors from the real Schur factorization of the
matrix A;
= 'N': The Schur factorization of A will be computed
and the factors will be stored in T and U (if
LYAPUN = 'O').
TRANA CHARACTER*1
Specifies the form of op(A) to be used, as follows:
= 'N': op(A) = A (No transpose);
= 'T': op(A) = A**T (Transpose);
= 'C': op(A) = A**T (Conjugate transpose = Transpose).
UPLO CHARACTER*1
Specifies which part of the symmetric matrix C is to be
used, as follows:
= 'U': Upper triangular part;
= 'L': Lower triangular part.
LYAPUN CHARACTER*1
Specifies whether or not the original Lyapunov equations
should be solved in the iterative estimation process,
as follows:
= 'O': Solve the original Lyapunov equations, updating
the right-hand sides and solutions with the
matrix U, e.g., X <-- U'*X*U;
= 'R': Solve reduced Lyapunov equations only, without
updating the right-hand sides and solutions.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, X and C. N >= 0.
SCALE (input) DOUBLE PRECISION
The scale factor, scale, set by a Lyapunov solver.
0 <= SCALE <= 1.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
If FACT = 'N' or LYAPUN = 'O', the leading N-by-N part of
this array must contain the original matrix A.
If FACT = 'F' and LYAPUN = 'R', A is not referenced.
LDA INTEGER
The leading dimension of the array A.
LDA >= MAX(1,N), if FACT = 'N' or LYAPUN = 'O';
LDA >= 1, if FACT = 'F' and LYAPUN = 'R'.
T (input/output) DOUBLE PRECISION array, dimension
(LDT,N)
If FACT = 'F', then on entry the leading N-by-N upper
Hessenberg part of this array must contain the upper
quasi-triangular matrix T in Schur canonical form from a
Schur factorization of A.
If FACT = 'N', then this array need not be set on input.
On exit, (if INFO = 0 or INFO = N+1, for FACT = 'N') the
leading N-by-N upper Hessenberg part of this array
contains the upper quasi-triangular matrix T in Schur
canonical form from a Schur factorization of A.
LDT INTEGER
The leading dimension of the array T. LDT >= MAX(1,N).
U (input or output) DOUBLE PRECISION array, dimension
(LDU,N)
If LYAPUN = 'O' and FACT = 'F', then U is an input
argument and on entry, the leading N-by-N part of this
array must contain the orthogonal matrix U from a real
Schur factorization of A.
If LYAPUN = 'O' and FACT = 'N', then U is an output
argument and on exit, if INFO = 0 or INFO = N+1, it
contains the orthogonal N-by-N matrix from a real Schur
factorization of A.
If LYAPUN = 'R', the array U is not referenced.
LDU INTEGER
The leading dimension of the array U.
LDU >= 1, if LYAPUN = 'R';
LDU >= MAX(1,N), if LYAPUN = 'O'.
C (input) DOUBLE PRECISION array, dimension (LDC,N)
If UPLO = 'U', the leading N-by-N upper triangular part of
this array must contain the upper triangular part of the
matrix C of the original Lyapunov equation (with
matrix A), if LYAPUN = 'O', or of the reduced Lyapunov
equation (with matrix T), if LYAPUN = 'R'.
If UPLO = 'L', the leading N-by-N lower triangular part of
this array must contain the lower triangular part of the
matrix C of the original Lyapunov equation (with
matrix A), if LYAPUN = 'O', or of the reduced Lyapunov
equation (with matrix T), if LYAPUN = 'R'.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,N).
X (input) DOUBLE PRECISION array, dimension (LDX,N)
The leading N-by-N part of this array must contain the
symmetric solution matrix X of the original Lyapunov
equation (with matrix A), if LYAPUN = 'O', or of the
reduced Lyapunov equation (with matrix T), if
LYAPUN = 'R'.
LDX INTEGER
The leading dimension of the array X. LDX >= MAX(1,N).
SEP (output) DOUBLE PRECISION
If JOB = 'C' or JOB = 'B', the estimated quantity
sep(op(A),-op(A)').
If N = 0, or X = 0, or JOB = 'E', SEP is not referenced.
RCOND (output) DOUBLE PRECISION
If JOB = 'C' or JOB = 'B', an estimate of the reciprocal
condition number of the continuous-time Lyapunov equation.
If N = 0 or X = 0, RCOND is set to 1 or 0, respectively.
If JOB = 'E', RCOND is not referenced.
FERR (output) DOUBLE PRECISION
If JOB = 'E' or JOB = 'B', an estimated forward error
bound for the solution X. If XTRUE is the true solution,
FERR bounds the magnitude of the largest entry in
(X - XTRUE) divided by the magnitude of the largest entry
in X.
If N = 0 or X = 0, FERR is set to 0.
If JOB = 'C', FERR is not referenced.
Workspace
IWORK INTEGER array, dimension (N*N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or INFO = N+1, DWORK(1) returns the
optimal value of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
If JOB = 'C', then
LDWORK >= MAX(1,2*N*N), if FACT = 'F';
LDWORK >= MAX(1,2*N*N,5*N), if FACT = 'N'.
If JOB = 'E', or JOB = 'B', and LYAPUN = 'O', then
LDWORK >= MAX(1,3*N*N), if FACT = 'F';
LDWORK >= MAX(1,3*N*N,5*N), if FACT = 'N'.
If JOB = 'E', or JOB = 'B', and LYAPUN = 'R', then
LDWORK >= MAX(1,3*N*N+N-1), if FACT = 'F';
LDWORK >= MAX(1,3*N*N+N-1,5*N), if FACT = 'N'.
For optimum performance LDWORK should sometimes be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, i <= N, the QR algorithm failed to
complete the reduction to Schur canonical form (see
LAPACK Library routine DGEES); on exit, the matrix
T(i+1:N,i+1:N) contains the partially converged
Schur form, and DWORK(i+1:N) and DWORK(N+i+1:2*N)
contain the real and imaginary parts, respectively,
of the converged eigenvalues; this error is unlikely
to appear;
= N+1: if the matrices T and -T' have common or very
close eigenvalues; perturbed values were used to
solve Lyapunov equations, but the matrix T, if given
(for FACT = 'F'), is unchanged.
Method
The condition number of the continuous-time Lyapunov equation is estimated as cond = (norm(Theta)*norm(A) + norm(inv(Omega))*norm(C))/norm(X), where Omega and Theta are linear operators defined by Omega(W) = op(A)'*W + W*op(A), Theta(W) = inv(Omega(op(W)'*X + X*op(W))). The routine estimates the quantities sep(op(A),-op(A)') = 1 / norm(inv(Omega)) and norm(Theta) using 1-norm condition estimators. The forward error bound is estimated using a practical error bound similar to the one proposed in [1].References
[1] Higham, N.J.
Perturbation theory and backward error for AX-XB=C.
BIT, vol. 33, pp. 124-136, 1993.
Numerical Aspects
3 The algorithm requires 0(N ) operations. The accuracy of the estimates obtained depends on the solution accuracy and on the properties of the 1-norm estimator.Further Comments
The option LYAPUN = 'R' may occasionally produce slightly worse or better estimates, and it is much faster than the option 'O'. When SEP is computed and it is zero, the routine returns immediately, with RCOND and FERR (if requested) set to 0 and 1, respectively. In this case, the equation is singular.Example
Program Text
* SB03QD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA, LDC, LDT, LDU, LDX
PARAMETER ( LDA = NMAX, LDC = NMAX, LDT = NMAX,
$ LDU = NMAX, LDX = NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX*NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( 1, 3*NMAX*NMAX + NMAX - 1,
$ 5*NMAX ) )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* .. Local Scalars ..
DOUBLE PRECISION FERR, RCOND, SCALE, SEP
INTEGER I, INFO1, INFO2, J, N
CHARACTER*1 DICO, FACT, JOB, LYAPUN, TRANA, TRANAT, UPLO
* .. Local Arrays ..
INTEGER IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), C(LDC,NMAX), DWORK(LDWORK),
$ T(LDT,NMAX), U(LDU,NMAX), X(LDX,NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL DLACPY, MA02ED, MB01RU, SB03MD, SB03QD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
DICO = 'C'
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, JOB, FACT, TRANA, UPLO, LYAPUN
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( FACT, 'F' ) ) READ ( NIN, FMT = * )
$ ( ( U(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,N )
CALL DLACPY( 'Full', N, N, A, LDA, T, LDT )
CALL DLACPY( 'Full', N, N, C, LDC, X, LDX )
* Solve the continuous-time Lyapunov matrix equation.
CALL SB03MD( DICO, 'X', FACT, TRANA, N, T, LDT, U, LDU, X, LDX,
$ SCALE, SEP, FERR, DWORK(1), DWORK(N+1), IWORK,
$ DWORK(2*N+1), LDWORK-2*N, INFO1 )
*
IF ( INFO1.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( X(I,J), J = 1,N )
10 CONTINUE
IF ( LSAME( LYAPUN, 'R' ) ) THEN
IF( LSAME( TRANA, 'N' ) ) THEN
TRANAT = 'T'
ELSE
TRANAT = 'N'
END IF
CALL MB01RU( UPLO, TRANAT, N, N, ZERO, ONE, X, LDX,
$ U, LDU, X, LDX, DWORK, N*N, INFO2 )
CALL MA02ED( UPLO, N, X, LDX )
CALL MB01RU( UPLO, TRANAT, N, N, ZERO, ONE, C, LDC,
$ U, LDU, C, LDC, DWORK, N*N, INFO2 )
END IF
* Estimate the condition and error bound on the solution.
CALL SB03QD( JOB, 'F', TRANA, UPLO, LYAPUN, N, SCALE, A,
$ LDA, T, LDT, U, LDU, C, LDC, X, LDX, SEP,
$ RCOND, FERR, IWORK, DWORK, LDWORK, INFO2 )
*
IF ( INFO2.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO2
ELSE
WRITE ( NOUT, FMT = 99993 ) SCALE
WRITE ( NOUT, FMT = 99992 ) SEP
WRITE ( NOUT, FMT = 99991 ) RCOND
WRITE ( NOUT, FMT = 99990 ) FERR
END IF
ELSE
WRITE ( NOUT, FMT = 99998 ) INFO1
END IF
END IF
STOP
*
99999 FORMAT (' SB03QD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB03MD =',I2)
99997 FORMAT (' INFO on exit from SB03QD =',I2)
99996 FORMAT (' The solution matrix X is')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' Scaling factor = ',F8.4)
99992 FORMAT (/' Estimated separation = ',F8.4)
99991 FORMAT (/' Estimated reciprocal condition number = ',F8.4)
99990 FORMAT (/' Estimated error bound = ',F8.4)
END
Program Data
SB03QD EXAMPLE PROGRAM DATA 3 B N N U O 3.0 1.0 1.0 1.0 3.0 0.0 0.0 0.0 3.0 25.0 24.0 15.0 24.0 32.0 8.0 15.0 8.0 40.0Program Results
SB03QD EXAMPLE PROGRAM RESULTS The solution matrix X is 3.2604 2.7187 1.8616 2.7187 4.4271 0.5699 1.8616 0.5699 6.0461 Scaling factor = 1.0000 Estimated separation = 4.9068 Estimated reciprocal condition number = 0.3611 Estimated error bound = 0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB03QX.html 0000664 0000000 0000000 00000015100 14560147231 0020367 0 ustar 00root root 0000000 0000000
Purpose
To estimate a forward error bound for the solution X of a real
continuous-time Lyapunov matrix equation,
op(A)'*X + X*op(A) = C,
where op(A) = A or A' (A**T) and C is symmetric (C = C**T). The
matrix A, the right hand side C, and the solution X are N-by-N.
An absolute residual matrix, which takes into account the rounding
errors in forming it, is given in the array R.
Specification
SUBROUTINE SB03QX( TRANA, UPLO, LYAPUN, N, XANORM, T, LDT, U, LDU,
$ R, LDR, FERR, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER LYAPUN, TRANA, UPLO
INTEGER INFO, LDR, LDT, LDU, LDWORK, N
DOUBLE PRECISION FERR, XANORM
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION DWORK( * ), R( LDR, * ), T( LDT, * ),
$ U( LDU, * )
Arguments
Mode Parameters
TRANA CHARACTER*1
Specifies the form of op(A) to be used, as follows:
= 'N': op(A) = A (No transpose);
= 'T': op(A) = A**T (Transpose);
= 'C': op(A) = A**T (Conjugate transpose = Transpose).
UPLO CHARACTER*1
Specifies which part of the symmetric matrix R is to be
used, as follows:
= 'U': Upper triangular part;
= 'L': Lower triangular part.
LYAPUN CHARACTER*1
Specifies whether or not the original Lyapunov equations
should be solved, as follows:
= 'O': Solve the original Lyapunov equations, updating
the right-hand sides and solutions with the
matrix U, e.g., X <-- U'*X*U;
= 'R': Solve reduced Lyapunov equations only, without
updating the right-hand sides and solutions.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A and R. N >= 0.
XANORM (input) DOUBLE PRECISION
The absolute (maximal) norm of the symmetric solution
matrix X of the Lyapunov equation. XANORM >= 0.
T (input) DOUBLE PRECISION array, dimension (LDT,N)
The leading N-by-N upper Hessenberg part of this array
must contain the upper quasi-triangular matrix T in Schur
canonical form from a Schur factorization of A.
LDT INTEGER
The leading dimension of array T. LDT >= MAX(1,N).
U (input) DOUBLE PRECISION array, dimension (LDU,N)
The leading N-by-N part of this array must contain the
orthogonal matrix U from a real Schur factorization of A.
If LYAPUN = 'R', the array U is not referenced.
LDU INTEGER
The leading dimension of array U.
LDU >= 1, if LYAPUN = 'R';
LDU >= MAX(1,N), if LYAPUN = 'O'.
R (input/output) DOUBLE PRECISION array, dimension (LDR,N)
On entry, if UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the absolute residual matrix R, with
bounds on rounding errors added.
On entry, if UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the absolute residual matrix R, with
bounds on rounding errors added.
On exit, the leading N-by-N part of this array contains
the symmetric absolute residual matrix R (with bounds on
rounding errors added), fully stored.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
FERR (output) DOUBLE PRECISION
An estimated forward error bound for the solution X.
If XTRUE is the true solution, FERR bounds the magnitude
of the largest entry in (X - XTRUE) divided by the
magnitude of the largest entry in X.
If N = 0 or XANORM = 0, FERR is set to 0, without any
calculations.
Workspace
IWORK INTEGER array, dimension (N*N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK. LDWORK >= 2*N*N.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= N+1: if the matrices T and -T' have common or very
close eigenvalues; perturbed values were used to
solve Lyapunov equations (but the matrix T is
unchanged).
Method
The forward error bound is estimated using a practical error bound similar to the one proposed in [1], based on the 1-norm estimator in [2].References
[1] Higham, N.J.
Perturbation theory and backward error for AX-XB=C.
BIT, vol. 33, pp. 124-136, 1993.
[2] Higham, N.J.
FORTRAN codes for estimating the one-norm of a real or
complex matrix, with applications to condition estimation.
ACM Trans. Math. Softw., 14, pp. 381-396, 1988.
Numerical Aspects
3 The algorithm requires 0(N ) operations.Further Comments
The option LYAPUN = 'R' may occasionally produce slightly worse or better estimates, and it is much faster than the option 'O'. The routine can be also used as a final step in estimating a forward error bound for the solution of a continuous-time algebraic matrix Riccati equation.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To estimate the separation between the matrices op(A) and -op(A)',
sep(op(A),-op(A)') = min norm(op(A)'*X + X*op(A))/norm(X)
= 1 / norm(inv(Omega))
and/or the 1-norm of Theta, where op(A) = A or A' (A**T), and
Omega and Theta are linear operators associated to the real
continuous-time Lyapunov matrix equation
op(A)'*X + X*op(A) = C,
defined by
Omega(W) = op(A)'*W + W*op(A),
Theta(W) = inv(Omega(op(W)'*X + X*op(W))).
The 1-norm condition estimators are used.
Specification
SUBROUTINE SB03QY( JOB, TRANA, LYAPUN, N, T, LDT, U, LDU, X, LDX,
$ SEP, THNORM, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOB, LYAPUN, TRANA
INTEGER INFO, LDT, LDU, LDWORK, LDX, N
DOUBLE PRECISION SEP, THNORM
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION DWORK( * ), T( LDT, * ), U( LDU, * ),
$ X( LDX, * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'S': Compute the separation only;
= 'T': Compute the norm of Theta only;
= 'B': Compute both the separation and the norm of Theta.
TRANA CHARACTER*1
Specifies the form of op(A) to be used, as follows:
= 'N': op(A) = A (No transpose);
= 'T': op(A) = A**T (Transpose);
= 'C': op(A) = A**T (Conjugate transpose = Transpose).
LYAPUN CHARACTER*1
Specifies whether or not the original Lyapunov equations
should be solved, as follows:
= 'O': Solve the original Lyapunov equations, updating
the right-hand sides and solutions with the
matrix U, e.g., X <-- U'*X*U;
= 'R': Solve reduced Lyapunov equations only, without
updating the right-hand sides and solutions.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A and X. N >= 0.
T (input) DOUBLE PRECISION array, dimension (LDT,N)
The leading N-by-N upper Hessenberg part of this array
must contain the upper quasi-triangular matrix T in Schur
canonical form from a Schur factorization of A.
LDT INTEGER
The leading dimension of array T. LDT >= MAX(1,N).
U (input) DOUBLE PRECISION array, dimension (LDU,N)
The leading N-by-N part of this array must contain the
orthogonal matrix U from a real Schur factorization of A.
If LYAPUN = 'R', the array U is not referenced.
LDU INTEGER
The leading dimension of array U.
LDU >= 1, if LYAPUN = 'R';
LDU >= MAX(1,N), if LYAPUN = 'O'.
X (input) DOUBLE PRECISION array, dimension (LDX,N)
The leading N-by-N part of this array must contain the
solution matrix X of the Lyapunov equation (reduced
Lyapunov equation if LYAPUN = 'R').
If JOB = 'S', the array X is not referenced.
LDX INTEGER
The leading dimension of array X.
LDX >= 1, if JOB = 'S';
LDX >= MAX(1,N), if JOB = 'T' or 'B'.
SEP (output) DOUBLE PRECISION
If JOB = 'S' or JOB = 'B', and INFO >= 0, SEP contains the
estimated separation of the matrices op(A) and -op(A)'.
If JOB = 'T' or N = 0, SEP is not referenced.
THNORM (output) DOUBLE PRECISION
If JOB = 'T' or JOB = 'B', and INFO >= 0, THNORM contains
the estimated 1-norm of operator Theta.
If JOB = 'S' or N = 0, THNORM is not referenced.
Workspace
IWORK INTEGER array, dimension (N*N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK. LDWORK >= 2*N*N.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= N+1: if the matrices T and -T' have common or very
close eigenvalues; perturbed values were used to
solve Lyapunov equations (but the matrix T is
unchanged).
Method
SEP is defined as the separation of op(A) and -op(A)':
sep( op(A), -op(A)' ) = sigma_min( K )
where sigma_min(K) is the smallest singular value of the
N*N-by-N*N matrix
K = kprod( I(N), op(A)' ) + kprod( op(A)', I(N) ).
I(N) is an N-by-N identity matrix, and kprod denotes the Kronecker
product. The routine estimates sigma_min(K) by the reciprocal of
an estimate of the 1-norm of inverse(K), computed as suggested in
[1]. This involves the solution of several continuous-time
Lyapunov equations, either direct or transposed. The true
reciprocal 1-norm of inverse(K) cannot differ from sigma_min(K) by
more than a factor of N.
The 1-norm of Theta is estimated similarly.
References
[1] Higham, N.J.
FORTRAN codes for estimating the one-norm of a real or
complex matrix, with applications to condition estimation.
ACM Trans. Math. Softw., 14, pp. 381-396, 1988.
Numerical Aspects
3 The algorithm requires 0(N ) operations.Further Comments
When SEP is zero, the routine returns immediately, with THNORM (if requested) not set. In this case, the equation is singular. The option LYAPUN = 'R' may occasionally produce slightly worse or better estimates, and it is much faster than the option 'O'.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve the real Lyapunov matrix equation
op(A)'*X + X*op(A) = scale*C
and/or estimate the separation between the matrices op(A) and
-op(A)', where op(A) = A or A' (A**T) and C is symmetric (C = C').
(A' denotes the transpose of the matrix A.) A is N-by-N, the right
hand side C and the solution X are N-by-N, and scale is an output
scale factor, set less than or equal to 1 to avoid overflow in X.
Specification
SUBROUTINE SB03RD( JOB, FACT, TRANA, N, A, LDA, U, LDU, C, LDC,
$ SCALE, SEP, FERR, WR, WI, IWORK, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER FACT, JOB, TRANA
INTEGER INFO, LDA, LDC, LDU, LDWORK, N
DOUBLE PRECISION FERR, SCALE, SEP
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), DWORK( * ),
$ U( LDU, * ), WI( * ), WR( * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'X': Compute the solution only;
= 'S': Compute the separation only;
= 'B': Compute both the solution and the separation.
FACT CHARACTER*1
Specifies whether or not the real Schur factorization
of the matrix A is supplied on entry, as follows:
= 'F': On entry, A and U contain the factors from the
real Schur factorization of the matrix A;
= 'N': The Schur factorization of A will be computed
and the factors will be stored in A and U.
TRANA CHARACTER*1
Specifies the form of op(A) to be used, as follows:
= 'N': op(A) = A (No transpose);
= 'T': op(A) = A**T (Transpose);
= 'C': op(A) = A**T (Conjugate transpose = Transpose).
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, X, and C. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A. If FACT = 'F', then A contains
an upper quasi-triangular matrix in Schur canonical form.
On exit, if INFO = 0 or INFO = N+1, the leading N-by-N
part of this array contains the upper quasi-triangular
matrix in Schur canonical form from the Shur factorization
of A. The contents of array A is not modified if
FACT = 'F'.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
U (input or output) DOUBLE PRECISION array, dimension
(LDU,N)
If FACT = 'F', then U is an input argument and on entry
it must contain the orthogonal matrix U from the real
Schur factorization of A.
If FACT = 'N', then U is an output argument and on exit,
if INFO = 0 or INFO = N+1, it contains the orthogonal
N-by-N matrix from the real Schur factorization of A.
LDU INTEGER
The leading dimension of array U. LDU >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry with JOB = 'X' or 'B', the leading N-by-N part of
this array must contain the symmetric matrix C.
On exit with JOB = 'X' or 'B', if INFO = 0 or INFO = N+1,
the leading N-by-N part of C has been overwritten by the
symmetric solution matrix X.
If JOB = 'S', C is not referenced.
LDC INTEGER
The leading dimension of array C.
LDC >= 1, if JOB = 'S';
LDC >= MAX(1,N), otherwise.
SCALE (output) DOUBLE PRECISION
The scale factor, scale, set less than or equal to 1 to
prevent the solution overflowing.
SEP (output) DOUBLE PRECISION
If JOB = 'S' or JOB = 'B', and INFO = 0 or INFO = N+1, SEP
contains the estimated separation of the matrices op(A)
and -op(A)'.
If JOB = 'X' or N = 0, SEP is not referenced.
FERR (output) DOUBLE PRECISION
If JOB = 'B', and INFO = 0 or INFO = N+1, FERR contains
an estimated forward error bound for the solution X.
If XTRUE is the true solution, FERR bounds the relative
error in the computed solution, measured in the Frobenius
norm: norm(X - XTRUE)/norm(XTRUE).
If JOB = 'X' or JOB = 'S', FERR is not referenced.
WR (output) DOUBLE PRECISION array, dimension (N)
WI (output) DOUBLE PRECISION array, dimension (N)
If FACT = 'N', and INFO = 0 or INFO = N+1, WR and WI
contain the real and imaginary parts, respectively, of the
eigenvalues of A.
If FACT = 'F', WR and WI are not referenced.
Workspace
IWORK INTEGER array, dimension (N*N)
This array is not referenced if JOB = 'X'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or INFO = N+1, DWORK(1) returns the
optimal value of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= 1 and
If JOB = 'X' then
If FACT = 'F', LDWORK >= N*N;
If FACT = 'N', LDWORK >= MAX(N*N,3*N).
If JOB = 'S' or JOB = 'B' then
If FACT = 'F', LDWORK >= 2*N*N;
If FACT = 'N', LDWORK >= MAX(2*N*N,3*N).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, the QR algorithm failed to compute all
the eigenvalues (see LAPACK Library routine DGEES);
elements i+1:n of WR and WI contain eigenvalues
which have converged, and A contains the partially
converged Schur form;
= N+1: if the matrices A and -A' have common or very
close eigenvalues; perturbed values were used to
solve the equation (but the matrix A is unchanged).
Method
After reducing matrix A to real Schur canonical form (if needed), the Bartels-Stewart algorithm is used. A set of equivalent linear algebraic systems of equations of order at most four are formed and solved using Gaussian elimination with complete pivoting.References
[1] Bartels, R.H. and Stewart, G.W. T
Solution of the matrix equation A X + XB = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
Numerical Aspects
3 The algorithm requires 0(N ) operations.Further Comments
SEP is defined as the separation of op(A) and -op(A)':
sep( op(A), -op(A)' ) = sigma_min( T )
where sigma_min(T) is the smallest singular value of the
N*N-by-N*N matrix
T = kprod( I(N), op(A)' ) + kprod( op(A), I(N) ).
I(N) is an N-by-N identity matrix, and kprod denotes the Kronecker
product. The program estimates sigma_min(T) by the reciprocal of
an estimate of the 1-norm of inverse(T). The true reciprocal
1-norm of inverse(T) cannot differ from sigma_min(T) by more
than a factor of N.
When SEP is small, small changes in A, C can cause large changes
in the solution of the equation. An approximate bound on the
maximum relative error in the computed solution is
EPS * norm(A) / SEP
where EPS is the machine precision.
Example
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB03SD.html 0000664 0000000 0000000 00000037304 14560147231 0020357 0 ustar 00root root 0000000 0000000
Purpose
To estimate the conditioning and compute an error bound on the
solution of the real discrete-time Lyapunov matrix equation
op(A)'*X*op(A) - X = scale*C
where op(A) = A or A' (A**T) and C is symmetric (C = C**T). The
matrix A is N-by-N, the right hand side C and the solution X are
N-by-N symmetric matrices, and scale is a given scale factor.
Specification
SUBROUTINE SB03SD( JOB, FACT, TRANA, UPLO, LYAPUN, N, SCALE, A,
$ LDA, T, LDT, U, LDU, C, LDC, X, LDX, SEPD,
$ RCOND, FERR, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER FACT, JOB, LYAPUN, TRANA, UPLO
INTEGER INFO, LDA, LDC, LDT, LDU, LDWORK, LDX, N
DOUBLE PRECISION FERR, RCOND, SCALE, SEPD
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), DWORK( * ),
$ T( LDT, * ), U( LDU, * ), X( LDX, * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'C': Compute the reciprocal condition number only;
= 'E': Compute the error bound only;
= 'B': Compute both the reciprocal condition number and
the error bound.
FACT CHARACTER*1
Specifies whether or not the real Schur factorization
of the matrix A is supplied on entry, as follows:
= 'F': On entry, T and U (if LYAPUN = 'O') contain the
factors from the real Schur factorization of the
matrix A;
= 'N': The Schur factorization of A will be computed
and the factors will be stored in T and U (if
LYAPUN = 'O').
TRANA CHARACTER*1
Specifies the form of op(A) to be used, as follows:
= 'N': op(A) = A (No transpose);
= 'T': op(A) = A**T (Transpose);
= 'C': op(A) = A**T (Conjugate transpose = Transpose).
UPLO CHARACTER*1
Specifies which part of the symmetric matrix C is to be
used, as follows:
= 'U': Upper triangular part;
= 'L': Lower triangular part.
LYAPUN CHARACTER*1
Specifies whether or not the original Lyapunov equations
should be solved in the iterative estimation process,
as follows:
= 'O': Solve the original Lyapunov equations, updating
the right-hand sides and solutions with the
matrix U, e.g., X <-- U'*X*U;
= 'R': Solve reduced Lyapunov equations only, without
updating the right-hand sides and solutions.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, X and C. N >= 0.
SCALE (input) DOUBLE PRECISION
The scale factor, scale, set by a Lyapunov solver.
0 <= SCALE <= 1.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
If FACT = 'N' or LYAPUN = 'O', the leading N-by-N part of
this array must contain the original matrix A.
If FACT = 'F' and LYAPUN = 'R', A is not referenced.
LDA INTEGER
The leading dimension of the array A.
LDA >= MAX(1,N), if FACT = 'N' or LYAPUN = 'O';
LDA >= 1, if FACT = 'F' and LYAPUN = 'R'.
T (input/output) DOUBLE PRECISION array, dimension
(LDT,N)
If FACT = 'F', then on entry the leading N-by-N upper
Hessenberg part of this array must contain the upper
quasi-triangular matrix T in Schur canonical form from a
Schur factorization of A.
If FACT = 'N', then this array need not be set on input.
On exit, (if INFO = 0 or INFO = N+1, for FACT = 'N') the
leading N-by-N upper Hessenberg part of this array
contains the upper quasi-triangular matrix T in Schur
canonical form from a Schur factorization of A.
LDT INTEGER
The leading dimension of the array T. LDT >= MAX(1,N).
U (input or output) DOUBLE PRECISION array, dimension
(LDU,N)
If LYAPUN = 'O' and FACT = 'F', then U is an input
argument and on entry, the leading N-by-N part of this
array must contain the orthogonal matrix U from a real
Schur factorization of A.
If LYAPUN = 'O' and FACT = 'N', then U is an output
argument and on exit, if INFO = 0 or INFO = N+1, it
contains the orthogonal N-by-N matrix from a real Schur
factorization of A.
If LYAPUN = 'R', the array U is not referenced.
LDU INTEGER
The leading dimension of the array U.
LDU >= 1, if LYAPUN = 'R';
LDU >= MAX(1,N), if LYAPUN = 'O'.
C (input) DOUBLE PRECISION array, dimension (LDC,N)
If UPLO = 'U', the leading N-by-N upper triangular part of
this array must contain the upper triangular part of the
matrix C of the original Lyapunov equation (with
matrix A), if LYAPUN = 'O', or of the reduced Lyapunov
equation (with matrix T), if LYAPUN = 'R'.
If UPLO = 'L', the leading N-by-N lower triangular part of
this array must contain the lower triangular part of the
matrix C of the original Lyapunov equation (with
matrix A), if LYAPUN = 'O', or of the reduced Lyapunov
equation (with matrix T), if LYAPUN = 'R'.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,N).
X (input) DOUBLE PRECISION array, dimension (LDX,N)
The leading N-by-N part of this array must contain the
symmetric solution matrix X of the original Lyapunov
equation (with matrix A), if LYAPUN = 'O', or of the
reduced Lyapunov equation (with matrix T), if
LYAPUN = 'R'.
The array X is modified internally, but restored on exit.
LDX INTEGER
The leading dimension of the array X. LDX >= MAX(1,N).
SEPD (output) DOUBLE PRECISION
If JOB = 'C' or JOB = 'B', the estimated quantity
sepd(op(A),op(A)').
If N = 0, or X = 0, or JOB = 'E', SEPD is not referenced.
RCOND (output) DOUBLE PRECISION
If JOB = 'C' or JOB = 'B', an estimate of the reciprocal
condition number of the discrete-time Lyapunov equation.
If N = 0 or X = 0, RCOND is set to 1 or 0, respectively.
If JOB = 'E', RCOND is not referenced.
FERR (output) DOUBLE PRECISION
If JOB = 'E' or JOB = 'B', an estimated forward error
bound for the solution X. If XTRUE is the true solution,
FERR bounds the magnitude of the largest entry in
(X - XTRUE) divided by the magnitude of the largest entry
in X.
If N = 0 or X = 0, FERR is set to 0.
If JOB = 'C', FERR is not referenced.
Workspace
IWORK INTEGER array, dimension (N*N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or INFO = N+1, DWORK(1) returns the
optimal value of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= 1, if N = 0; else,
LDWORK >= MAX(3,2*N*N) + N*N, if JOB = 'C',
FACT = 'F';
LDWORK >= MAX(MAX(3,2*N*N) + N*N, 5*N), if JOB = 'C',
FACT = 'N';
LDWORK >= MAX(3,2*N*N) + N*N + 2*N, if JOB = 'E', or
JOB = 'B'.
For optimum performance LDWORK should sometimes be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, i <= N, the QR algorithm failed to
complete the reduction to Schur canonical form (see
LAPACK Library routine DGEES); on exit, the matrix
T(i+1:N,i+1:N) contains the partially converged
Schur form, and DWORK(i+1:N) and DWORK(N+i+1:2*N)
contain the real and imaginary parts, respectively,
of the converged eigenvalues; this error is unlikely
to appear;
= N+1: if the matrix T has almost reciprocal eigenvalues;
perturbed values were used to solve Lyapunov
equations, but the matrix T, if given (for
FACT = 'F'), is unchanged.
Method
The condition number of the discrete-time Lyapunov equation is estimated as cond = (norm(Theta)*norm(A) + norm(inv(Omega))*norm(C))/norm(X), where Omega and Theta are linear operators defined by Omega(W) = op(A)'*W*op(A) - W, Theta(W) = inv(Omega(op(W)'*X*op(A) + op(A)'*X*op(W))). The routine estimates the quantities sepd(op(A),op(A)') = 1 / norm(inv(Omega)) and norm(Theta) using 1-norm condition estimators. The forward error bound is estimated using a practical error bound similar to the one proposed in [1].References
[1] Higham, N.J.
Perturbation theory and backward error for AX-XB=C.
BIT, vol. 33, pp. 124-136, 1993.
Numerical Aspects
3 The algorithm requires 0(N ) operations. The accuracy of the estimates obtained depends on the solution accuracy and on the properties of the 1-norm estimator.Further Comments
The option LYAPUN = 'R' may occasionally produce slightly worse or better estimates, and it is much faster than the option 'O'. When SEPD is computed and it is zero, the routine returns immediately, with RCOND and FERR (if requested) set to 0 and 1, respectively. In this case, the equation is singular.Example
Program Text
* SB03SD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA, LDC, LDT, LDU, LDX
PARAMETER ( LDA = NMAX, LDC = NMAX, LDT = NMAX,
$ LDU = NMAX, LDX = NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX*NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( 3, 2*NMAX*NMAX ) +
$ NMAX*NMAX + 2*NMAX )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* .. Local Scalars ..
DOUBLE PRECISION FERR, RCOND, SCALE, SEPD
INTEGER I, INFO1, INFO2, J, N
CHARACTER*1 DICO, FACT, JOB, LYAPUN, TRANA, TRANAT, UPLO
* .. Local Arrays ..
INTEGER IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), C(LDC,NMAX), DWORK(LDWORK),
$ T(LDT,NMAX), U(LDU,NMAX), X(LDX,NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL DLACPY, MA02ED, MB01RU, SB03MD, SB03SD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
DICO = 'D'
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, JOB, FACT, TRANA, UPLO, LYAPUN
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( FACT, 'F' ) ) READ ( NIN, FMT = * )
$ ( ( U(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,N )
CALL DLACPY( 'Full', N, N, A, LDA, T, LDT )
CALL DLACPY( 'Full', N, N, C, LDC, X, LDX )
* Solve the discrete-time Lyapunov matrix equation.
CALL SB03MD( DICO, 'X', FACT, TRANA, N, T, LDT, U, LDU, X, LDX,
$ SCALE, SEPD, FERR, DWORK(1), DWORK(N+1), IWORK,
$ DWORK(2*N+1), LDWORK-2*N, INFO1 )
*
IF ( INFO1.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( X(I,J), J = 1,N )
10 CONTINUE
IF ( LSAME( LYAPUN, 'R' ) ) THEN
IF( LSAME( TRANA, 'N' ) ) THEN
TRANAT = 'T'
ELSE
TRANAT = 'N'
END IF
CALL MB01RU( UPLO, TRANAT, N, N, ZERO, ONE, X, LDX,
$ U, LDU, X, LDX, DWORK, N*N, INFO2 )
CALL MA02ED( UPLO, N, X, LDX )
CALL MB01RU( UPLO, TRANAT, N, N, ZERO, ONE, C, LDC,
$ U, LDU, C, LDC, DWORK, N*N, INFO2 )
END IF
* Estimate the condition and error bound on the solution.
CALL SB03SD( JOB, 'F', TRANA, UPLO, LYAPUN, N, SCALE, A,
$ LDA, T, LDT, U, LDU, C, LDC, X, LDX, SEPD,
$ RCOND, FERR, IWORK, DWORK, LDWORK, INFO2 )
*
IF ( INFO2.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO2
ELSE
WRITE ( NOUT, FMT = 99993 ) SCALE
WRITE ( NOUT, FMT = 99992 ) SEPD
WRITE ( NOUT, FMT = 99991 ) RCOND
WRITE ( NOUT, FMT = 99990 ) FERR
END IF
ELSE
WRITE ( NOUT, FMT = 99998 ) INFO1
END IF
END IF
STOP
*
99999 FORMAT (' SB03SD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB03MD =',I2)
99997 FORMAT (' INFO on exit from SB03SD =',I2)
99996 FORMAT (' The solution matrix X is')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' Scaling factor = ',F8.4)
99992 FORMAT (/' Estimated separation = ',F8.4)
99991 FORMAT (/' Estimated reciprocal condition number = ',F8.4)
99990 FORMAT (/' Estimated error bound = ',F8.4)
END
Program Data
SB03SD EXAMPLE PROGRAM DATA 3 B N N U O 3.0 1.0 1.0 1.0 3.0 0.0 0.0 0.0 3.0 25.0 24.0 15.0 24.0 32.0 8.0 15.0 8.0 40.0Program Results
SB03SD EXAMPLE PROGRAM RESULTS The solution matrix X is 2.0000 1.0000 1.0000 1.0000 3.0000 0.0000 1.0000 0.0000 4.0000 Scaling factor = 1.0000 Estimated separation = 5.2302 Estimated reciprocal condition number = 0.1832 Estimated error bound = 0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB03SX.html 0000664 0000000 0000000 00000015135 14560147231 0020401 0 ustar 00root root 0000000 0000000
Purpose
To estimate a forward error bound for the solution X of a real
discrete-time Lyapunov matrix equation,
op(A)'*X*op(A) - X = C,
where op(A) = A or A' (A**T) and C is symmetric (C = C**T). The
matrix A, the right hand side C, and the solution X are N-by-N.
An absolute residual matrix, which takes into account the rounding
errors in forming it, is given in the array R.
Specification
SUBROUTINE SB03SX( TRANA, UPLO, LYAPUN, N, XANORM, T, LDT, U, LDU,
$ R, LDR, FERR, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER LYAPUN, TRANA, UPLO
INTEGER INFO, LDR, LDT, LDU, LDWORK, N
DOUBLE PRECISION FERR, XANORM
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION DWORK( * ), R( LDR, * ), T( LDT, * ),
$ U( LDU, * )
Arguments
Mode Parameters
TRANA CHARACTER*1
Specifies the form of op(A) to be used, as follows:
= 'N': op(A) = A (No transpose);
= 'T': op(A) = A**T (Transpose);
= 'C': op(A) = A**T (Conjugate transpose = Transpose).
UPLO CHARACTER*1
Specifies which part of the symmetric matrix R is to be
used, as follows:
= 'U': Upper triangular part;
= 'L': Lower triangular part.
LYAPUN CHARACTER*1
Specifies whether or not the original Lyapunov equations
should be solved, as follows:
= 'O': Solve the original Lyapunov equations, updating
the right-hand sides and solutions with the
matrix U, e.g., X <-- U'*X*U;
= 'R': Solve reduced Lyapunov equations only, without
updating the right-hand sides and solutions.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A and R. N >= 0.
XANORM (input) DOUBLE PRECISION
The absolute (maximal) norm of the symmetric solution
matrix X of the Lyapunov equation. XANORM >= 0.
T (input) DOUBLE PRECISION array, dimension (LDT,N)
The leading N-by-N upper Hessenberg part of this array
must contain the upper quasi-triangular matrix T in Schur
canonical form from a Schur factorization of A.
LDT INTEGER
The leading dimension of array T. LDT >= MAX(1,N).
U (input) DOUBLE PRECISION array, dimension (LDU,N)
The leading N-by-N part of this array must contain the
orthogonal matrix U from a real Schur factorization of A.
If LYAPUN = 'R', the array U is not referenced.
LDU INTEGER
The leading dimension of array U.
LDU >= 1, if LYAPUN = 'R';
LDU >= MAX(1,N), if LYAPUN = 'O'.
R (input/output) DOUBLE PRECISION array, dimension (LDR,N)
On entry, if UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the absolute residual matrix R, with
bounds on rounding errors added.
On entry, if UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the absolute residual matrix R, with
bounds on rounding errors added.
On exit, the leading N-by-N part of this array contains
the symmetric absolute residual matrix R (with bounds on
rounding errors added), fully stored.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
FERR (output) DOUBLE PRECISION
An estimated forward error bound for the solution X.
If XTRUE is the true solution, FERR bounds the magnitude
of the largest entry in (X - XTRUE) divided by the
magnitude of the largest entry in X.
If N = 0 or XANORM = 0, FERR is set to 0, without any
calculations.
Workspace
IWORK INTEGER array, dimension (N*N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 0, if N = 0;
LDWORK >= MAX(3,2*N*N), if N > 0.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= N+1: if T has almost reciprocal eigenvalues; perturbed
values were used to solve Lyapunov equations (but
the matrix T is unchanged).
Method
The forward error bound is estimated using a practical error bound similar to the one proposed in [1], based on the 1-norm estimator in [2].References
[1] Higham, N.J.
Perturbation theory and backward error for AX-XB=C.
BIT, vol. 33, pp. 124-136, 1993.
[2] Higham, N.J.
FORTRAN codes for estimating the one-norm of a real or
complex matrix, with applications to condition estimation.
ACM Trans. Math. Softw., 14, pp. 381-396, 1988.
Numerical Aspects
3 The algorithm requires 0(N ) operations.Further Comments
The option LYAPUN = 'R' may occasionally produce slightly worse or better estimates, and it is much faster than the option 'O'. The routine can be also used as a final step in estimating a forward error bound for the solution of a discrete-time algebraic matrix Riccati equation.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To estimate the "separation" between the matrices op(A) and
op(A)',
sepd(op(A),op(A)') = min norm(op(A)'*X*op(A) - X)/norm(X)
= 1 / norm(inv(Omega))
and/or the 1-norm of Theta, where op(A) = A or A' (A**T), and
Omega and Theta are linear operators associated to the real
discrete-time Lyapunov matrix equation
op(A)'*X*op(A) - X = C,
defined by
Omega(W) = op(A)'*W*op(A) - W,
Theta(W) = inv(Omega(op(W)'*X*op(A) + op(A)'*X*op(W))).
The 1-norm condition estimators are used.
Specification
SUBROUTINE SB03SY( JOB, TRANA, LYAPUN, N, T, LDT, U, LDU, XA,
$ LDXA, SEPD, THNORM, IWORK, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER JOB, LYAPUN, TRANA
INTEGER INFO, LDT, LDU, LDWORK, LDXA, N
DOUBLE PRECISION SEPD, THNORM
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION DWORK( * ), T( LDT, * ), U( LDU, * ),
$ XA( LDXA, * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'S': Compute the separation only;
= 'T': Compute the norm of Theta only;
= 'B': Compute both the separation and the norm of Theta.
TRANA CHARACTER*1
Specifies the form of op(A) to be used, as follows:
= 'N': op(A) = A (No transpose);
= 'T': op(A) = A**T (Transpose);
= 'C': op(A) = A**T (Conjugate transpose = Transpose).
LYAPUN CHARACTER*1
Specifies whether or not the original Lyapunov equations
should be solved, as follows:
= 'O': Solve the original Lyapunov equations, updating
the right-hand sides and solutions with the
matrix U, e.g., X <-- U'*X*U;
= 'R': Solve reduced Lyapunov equations only, without
updating the right-hand sides and solutions.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A and X. N >= 0.
T (input) DOUBLE PRECISION array, dimension (LDT,N)
The leading N-by-N upper Hessenberg part of this array
must contain the upper quasi-triangular matrix T in Schur
canonical form from a Schur factorization of A.
LDT INTEGER
The leading dimension of array T. LDT >= MAX(1,N).
U (input) DOUBLE PRECISION array, dimension (LDU,N)
The leading N-by-N part of this array must contain the
orthogonal matrix U from a real Schur factorization of A.
If LYAPUN = 'R', the array U is not referenced.
LDU INTEGER
The leading dimension of array U.
LDU >= 1, if LYAPUN = 'R';
LDU >= MAX(1,N), if LYAPUN = 'O'.
XA (input) DOUBLE PRECISION array, dimension (LDXA,N)
The leading N-by-N part of this array must contain the
matrix product X*op(A), if LYAPUN = 'O', or U'*X*U*op(T),
if LYAPUN = 'R', in the Lyapunov equation.
If JOB = 'S', the array XA is not referenced.
LDXA INTEGER
The leading dimension of array XA.
LDXA >= 1, if JOB = 'S';
LDXA >= MAX(1,N), if JOB = 'T' or 'B'.
SEPD (output) DOUBLE PRECISION
If JOB = 'S' or JOB = 'B', and INFO >= 0, SEPD contains
the estimated quantity sepd(op(A),op(A)').
If JOB = 'T' or N = 0, SEPD is not referenced.
THNORM (output) DOUBLE PRECISION
If JOB = 'T' or JOB = 'B', and INFO >= 0, THNORM contains
the estimated 1-norm of operator Theta.
If JOB = 'S' or N = 0, THNORM is not referenced.
Workspace
IWORK INTEGER array, dimension (N*N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 0, if N = 0;
LDWORK >= MAX(3,2*N*N), if N > 0.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= N+1: if T has (almost) reciprocal eigenvalues;
perturbed values were used to solve Lyapunov
equations (but the matrix T is unchanged).
Method
SEPD is defined as
sepd( op(A), op(A)' ) = sigma_min( K )
where sigma_min(K) is the smallest singular value of the
N*N-by-N*N matrix
K = kprod( op(A)', op(A)' ) - I(N**2).
I(N**2) is an N*N-by-N*N identity matrix, and kprod denotes the
Kronecker product. The routine estimates sigma_min(K) by the
reciprocal of an estimate of the 1-norm of inverse(K), computed as
suggested in [1]. This involves the solution of several discrete-
time Lyapunov equations, either direct or transposed. The true
reciprocal 1-norm of inverse(K) cannot differ from sigma_min(K) by
more than a factor of N.
The 1-norm of Theta is estimated similarly.
References
[1] Higham, N.J.
FORTRAN codes for estimating the one-norm of a real or
complex matrix, with applications to condition estimation.
ACM Trans. Math. Softw., 14, pp. 381-396, 1988.
Numerical Aspects
3 The algorithm requires 0(N ) operations.Further Comments
When SEPD is zero, the routine returns immediately, with THNORM (if requested) not set. In this case, the equation is singular. The option LYAPUN = 'R' may occasionally produce slightly worse or better estimates, and it is much faster than the option 'O'.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve the real continuous-time Lyapunov matrix equation
op(A)'*X + X*op(A) = scale*C,
estimate the conditioning, and compute an error bound on the
solution X, where op(A) = A or A' (A**T), the matrix A is N-by-N,
the right hand side C and the solution X are N-by-N symmetric
matrices (C = C', X = X'), and scale is an output scale factor,
set less than or equal to 1 to avoid overflow in X.
Specification
SUBROUTINE SB03TD( JOB, FACT, TRANA, UPLO, LYAPUN, N, SCALE, A,
$ LDA, T, LDT, U, LDU, C, LDC, X, LDX, SEP,
$ RCOND, FERR, WR, WI, IWORK, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER FACT, JOB, LYAPUN, TRANA, UPLO
INTEGER INFO, LDA, LDC, LDT, LDU, LDWORK, LDX, N
DOUBLE PRECISION FERR, RCOND, SCALE, SEP
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), DWORK( * ),
$ T( LDT, * ), U( LDU, * ), WI( * ), WR( * ),
$ X( LDX, * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'X': Compute the solution only;
= 'S': Compute the separation only;
= 'C': Compute the reciprocal condition number only;
= 'E': Compute the error bound only;
= 'A': Compute all: the solution, separation, reciprocal
condition number, and the error bound.
FACT CHARACTER*1
Specifies whether or not the real Schur factorization
of the matrix A is supplied on entry, as follows:
= 'F': On entry, T and U (if LYAPUN = 'O') contain the
factors from the real Schur factorization of the
matrix A;
= 'N': The Schur factorization of A will be computed
and the factors will be stored in T and U (if
LYAPUN = 'O').
TRANA CHARACTER*1
Specifies the form of op(A) to be used, as follows:
= 'N': op(A) = A (No transpose);
= 'T': op(A) = A**T (Transpose);
= 'C': op(A) = A**T (Conjugate transpose = Transpose).
UPLO CHARACTER*1
Specifies which part of the symmetric matrix C is to be
used, as follows:
= 'U': Upper triangular part;
= 'L': Lower triangular part.
LYAPUN CHARACTER*1
Specifies whether or not the original or "reduced"
Lyapunov equations should be solved, as follows:
= 'O': Solve the original Lyapunov equations, updating
the right-hand sides and solutions with the
matrix U, e.g., X <-- U'*X*U;
= 'R': Solve reduced Lyapunov equations only, without
updating the right-hand sides and solutions.
This means that a real Schur form T of A appears
in the equation, instead of A.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, X, and C. N >= 0.
SCALE (input or output) DOUBLE PRECISION
If JOB = 'C' or JOB = 'E', SCALE is an input argument:
the scale factor, set by a Lyapunov solver.
0 <= SCALE <= 1.
If JOB = 'X' or JOB = 'A', SCALE is an output argument:
the scale factor, scale, set less than or equal to 1 to
prevent the solution overflowing.
If JOB = 'S', this argument is not used.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
If FACT = 'N' or (LYAPUN = 'O' and JOB <> 'X'), the
leading N-by-N part of this array must contain the
original matrix A.
If FACT = 'F' and (LYAPUN = 'R' or JOB = 'X'), A is
not referenced.
LDA INTEGER
The leading dimension of the array A.
LDA >= MAX(1,N), if FACT = 'N' or LYAPUN = 'O' and
JOB <> 'X';
LDA >= 1, otherwise.
T (input/output) DOUBLE PRECISION array, dimension
(LDT,N)
If FACT = 'F', then on entry the leading N-by-N upper
Hessenberg part of this array must contain the upper
quasi-triangular matrix T in Schur canonical form from a
Schur factorization of A.
If FACT = 'N', then this array need not be set on input.
On exit, (if INFO = 0 or INFO = N+1, for FACT = 'N') the
leading N-by-N upper Hessenberg part of this array
contains the upper quasi-triangular matrix T in Schur
canonical form from a Schur factorization of A.
The contents of array T is not modified if FACT = 'F'.
LDT INTEGER
The leading dimension of the array T. LDT >= MAX(1,N).
U (input or output) DOUBLE PRECISION array, dimension
(LDU,N)
If LYAPUN = 'O' and FACT = 'F', then U is an input
argument and on entry, the leading N-by-N part of this
array must contain the orthogonal matrix U from a real
Schur factorization of A.
If LYAPUN = 'O' and FACT = 'N', then U is an output
argument and on exit, if INFO = 0 or INFO = N+1, it
contains the orthogonal N-by-N matrix from a real Schur
factorization of A.
If LYAPUN = 'R', the array U is not referenced.
LDU INTEGER
The leading dimension of the array U.
LDU >= 1, if LYAPUN = 'R';
LDU >= MAX(1,N), if LYAPUN = 'O'.
C (input) DOUBLE PRECISION array, dimension (LDC,N)
If JOB <> 'S' and UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the matrix C of the original Lyapunov
equation (with matrix A), if LYAPUN = 'O', or of the
reduced Lyapunov equation (with matrix T), if
LYAPUN = 'R'.
If JOB <> 'S' and UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the matrix C of the original Lyapunov
equation (with matrix A), if LYAPUN = 'O', or of the
reduced Lyapunov equation (with matrix T), if
LYAPUN = 'R'.
The remaining strictly triangular part of this array is
used as workspace.
If JOB = 'X', then this array may be identified with X
in the call of this routine.
If JOB = 'S', the array C is not referenced.
LDC INTEGER
The leading dimension of the array C.
LDC >= 1, if JOB = 'S';
LDC >= MAX(1,N), otherwise.
X (input or output) DOUBLE PRECISION array, dimension
(LDX,N)
If JOB = 'C' or 'E', then X is an input argument and on
entry, the leading N-by-N part of this array must contain
the symmetric solution matrix X of the original Lyapunov
equation (with matrix A), if LYAPUN = 'O', or of the
reduced Lyapunov equation (with matrix T), if
LYAPUN = 'R'.
If JOB = 'X' or 'A', then X is an output argument and on
exit, if INFO = 0 or INFO = N+1, the leading N-by-N part
of this array contains the symmetric solution matrix X of
of the original Lyapunov equation (with matrix A), if
LYAPUN = 'O', or of the reduced Lyapunov equation (with
matrix T), if LYAPUN = 'R'.
If JOB = 'S', the array X is not referenced.
LDX INTEGER
The leading dimension of the array X.
LDX >= 1, if JOB = 'S';
LDX >= MAX(1,N), otherwise.
SEP (output) DOUBLE PRECISION
If JOB = 'S' or JOB = 'C' or JOB = 'A', and INFO = 0 or
INFO = N+1, SEP contains the estimated separation of the
matrices op(A) and -op(A)', sep(op(A),-op(A)').
If N = 0, or X = 0, or JOB = 'X' or JOB = 'E', SEP is not
referenced.
RCOND (output) DOUBLE PRECISION
If JOB = 'C' or JOB = 'A', an estimate of the reciprocal
condition number of the continuous-time Lyapunov equation.
If N = 0 or X = 0, RCOND is set to 1 or 0, respectively.
If JOB = 'X' or JOB = 'S' or JOB = 'E', RCOND is not
referenced.
FERR (output) DOUBLE PRECISION
If JOB = 'E' or JOB = 'A', and INFO = 0 or INFO = N+1,
FERR contains an estimated forward error bound for the
solution X. If XTRUE is the true solution, FERR bounds the
relative error in the computed solution, measured in the
Frobenius norm: norm(X - XTRUE)/norm(XTRUE).
If N = 0 or X = 0, FERR is set to 0.
If JOB = 'X' or JOB = 'S' or JOB = 'C', FERR is not
referenced.
WR (output) DOUBLE PRECISION array, dimension (N)
WI (output) DOUBLE PRECISION array, dimension (N)
If FACT = 'N', and INFO = 0 or INFO = N+1, WR and WI
contain the real and imaginary parts, respectively, of the
eigenvalues of A.
If FACT = 'F', WR and WI are not referenced.
Workspace
IWORK INTEGER array, dimension (N*N)
This array is not referenced if JOB = 'X'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or INFO = N+1, DWORK(1) returns the
optimal value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
If JOB = 'X', then
LDWORK >= MAX(1,N*N), if FACT = 'F';
LDWORK >= MAX(1,MAX(N*N,3*N)), if FACT = 'N'.
If JOB = 'S' or JOB = 'C', then
LDWORK >= MAX(1,2*N*N), if FACT = 'F';
LDWORK >= MAX(1,2*N*N,3*N), if FACT = 'N'.
If JOB = 'E', or JOB = 'A', and LYAPUN = 'O', then
LDWORK >= MAX(1,3*N*N);
If JOB = 'E', or JOB = 'A', and LYAPUN = 'R', then
LDWORK >= MAX(1,3*N*N+N-1).
For optimum performance LDWORK should sometimes be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, i <= N, the QR algorithm failed to
complete the reduction to Schur canonical form (see
LAPACK Library routine DGEES); on exit, the matrix
T(i+1:N,i+1:N) contains the partially converged
Schur form, and the elements i+1:n of WR and WI
contain the real and imaginary parts, respectively,
of the converged eigenvalues; this error is unlikely
to appear;
= N+1: if the matrices T and -T' have common or very
close eigenvalues; perturbed values were used to
solve Lyapunov equations, but the matrix T, if given
(for FACT = 'F'), is unchanged.
Method
After reducing matrix A to real Schur canonical form (if needed), the Bartels-Stewart algorithm is used. A set of equivalent linear algebraic systems of equations of order at most four are formed and solved using Gaussian elimination with complete pivoting. The condition number of the continuous-time Lyapunov equation is estimated as cond = (norm(Theta)*norm(A) + norm(inv(Omega))*norm(C))/norm(X), where Omega and Theta are linear operators defined by Omega(W) = op(A)'*W + W*op(A), Theta(W) = inv(Omega(op(W)'*X + X*op(W))). The routine estimates the quantities sep(op(A),-op(A)') = 1 / norm(inv(Omega)) and norm(Theta) using 1-norm condition estimators. The forward error bound is estimated using a practical error bound similar to the one proposed in [2].References
[1] Bartels, R.H. and Stewart, G.W. T
Solution of the matrix equation A X + XB = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
[2] Higham, N.J.
Perturbation theory and backward error for AX-XB=C.
BIT, vol. 33, pp. 124-136, 1993.
Numerical Aspects
3 The algorithm requires 0(N ) operations. The accuracy of the estimates obtained depends on the solution accuracy and on the properties of the 1-norm estimator.Further Comments
The separation of op(A) and -op(A)' can also be defined as
sep( op(A), -op(A)' ) = sigma_min( T ),
where sigma_min(T) is the smallest singular value of the
N*N-by-N*N matrix
T = kprod( I(N), op(A)' ) + kprod( op(A)', I(N) ).
I(N) is an N-by-N identity matrix, and kprod denotes the Kronecker
product. The routine estimates sigma_min(T) by the reciprocal of
an estimate of the 1-norm of inverse(T). The true reciprocal
1-norm of inverse(T) cannot differ from sigma_min(T) by more
than a factor of N.
Example
Program Text
* SB03TD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA, LDC, LDT, LDU, LDX
PARAMETER ( LDA = NMAX, LDC = NMAX, LDT = NMAX,
$ LDU = NMAX, LDX = NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX*NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( 1, 3*NMAX*NMAX + NMAX - 1 ) )
* .. Local Scalars ..
DOUBLE PRECISION FERR, RCOND, SCALE, SEP
INTEGER I, INFO, J, N
CHARACTER*1 DICO, FACT, JOB, LYAPUN, TRANA, UPLO
* .. Local Arrays ..
INTEGER IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), C(LDC,NMAX), DWORK(LDWORK),
$ T(LDT,NMAX), U(LDU,NMAX), X(LDX,NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL SB03TD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
DICO = 'C'
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, JOB, FACT, TRANA, UPLO, LYAPUN
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
IF ( LSAME( JOB, 'C' ) .OR. LSAME( JOB, 'E' ) )
$ READ ( NIN, FMT = * ) SCALE
IF ( LSAME( FACT, 'N' ) .OR. ( LSAME( LYAPUN, 'O' ) .AND.
$ .NOT.LSAME( JOB, 'X') ) )
$ READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( FACT, 'F' ) ) THEN
READ ( NIN, FMT = * ) ( ( T(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( LYAPUN, 'O' ) )
$ READ ( NIN, FMT = * ) ( ( U(I,J), J = 1,N ), I = 1,N )
END IF
IF ( .NOT.LSAME( JOB, 'S' ) )
$ READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( JOB, 'C' ) .OR. LSAME( JOB, 'E' ) )
$ READ ( NIN, FMT = * ) ( ( X(I,J), J = 1,N ), I = 1,N )
* Solve the continuous-time Lyapunov matrix equation and/or
* estimate the condition and error bound on the solution.
CALL SB03TD( JOB, FACT, TRANA, UPLO, LYAPUN, N, SCALE, A, LDA,
$ T, LDT, U, LDU, C, LDC, X, LDX, SEP, RCOND, FERR,
$ DWORK(1), DWORK(N+1), IWORK, DWORK(2*N+1),
$ LDWORK-2*N, INFO )
*
IF ( INFO.EQ.0 ) THEN
IF ( LSAME( JOB, 'X' ) .OR. LSAME( JOB, 'A' ) ) THEN
WRITE ( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( X(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99993 ) SCALE
END IF
IF ( LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'C' )
$ .OR. LSAME( JOB, 'A' ) )
$ WRITE ( NOUT, FMT = 99992 ) SEP
IF ( LSAME( JOB, 'C' ) .OR. LSAME( JOB, 'A' ) )
$ WRITE ( NOUT, FMT = 99991 ) RCOND
IF ( LSAME( JOB, 'E' ) .OR. LSAME( JOB, 'A' ) )
$ WRITE ( NOUT, FMT = 99990 ) FERR
ELSE
WRITE ( NOUT, FMT = 99998 ) INFO
END IF
END IF
STOP
*
99999 FORMAT (' SB03TD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB03TD =',I2)
99996 FORMAT (' The solution matrix X is')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' Scaling factor = ',F8.4)
99992 FORMAT (/' Estimated separation = ',F8.4)
99991 FORMAT (/' Estimated reciprocal condition number = ',F8.4)
99990 FORMAT (/' Estimated error bound = ',F8.4)
END
Program Data
SB03TD EXAMPLE PROGRAM DATA 3 A N N U O 3.0 1.0 1.0 1.0 3.0 0.0 0.0 0.0 3.0 25.0 24.0 15.0 24.0 32.0 8.0 15.0 8.0 40.0Program Results
SB03TD EXAMPLE PROGRAM RESULTS The solution matrix X is 3.2604 2.7187 1.8616 2.7187 4.4271 0.5699 1.8616 0.5699 6.0461 Scaling factor = 1.0000 Estimated separation = 4.9068 Estimated reciprocal condition number = 0.3611 Estimated error bound = 0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB03UD.html 0000664 0000000 0000000 00000044206 14560147231 0020360 0 ustar 00root root 0000000 0000000
Purpose
To solve the real discrete-time Lyapunov matrix equation
op(A)'*X*op(A) - X = scale*C,
estimate the conditioning, and compute an error bound on the
solution X, where op(A) = A or A' (A**T), the matrix A is N-by-N,
the right hand side C and the solution X are N-by-N symmetric
matrices (C = C', X = X'), and scale is an output scale factor,
set less than or equal to 1 to avoid overflow in X.
Specification
SUBROUTINE SB03UD( JOB, FACT, TRANA, UPLO, LYAPUN, N, SCALE, A,
$ LDA, T, LDT, U, LDU, C, LDC, X, LDX, SEPD,
$ RCOND, FERR, WR, WI, IWORK, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER FACT, JOB, LYAPUN, TRANA, UPLO
INTEGER INFO, LDA, LDC, LDT, LDU, LDWORK, LDX, N
DOUBLE PRECISION FERR, RCOND, SCALE, SEPD
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), DWORK( * ),
$ T( LDT, * ), U( LDU, * ), WI( * ), WR( * ),
$ X( LDX, * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'X': Compute the solution only;
= 'S': Compute the separation only;
= 'C': Compute the reciprocal condition number only;
= 'E': Compute the error bound only;
= 'A': Compute all: the solution, separation, reciprocal
condition number, and the error bound.
FACT CHARACTER*1
Specifies whether or not the real Schur factorization
of the matrix A is supplied on entry, as follows:
= 'F': On entry, T and U (if LYAPUN = 'O') contain the
factors from the real Schur factorization of the
matrix A;
= 'N': The Schur factorization of A will be computed
and the factors will be stored in T and U (if
LYAPUN = 'O').
TRANA CHARACTER*1
Specifies the form of op(A) to be used, as follows:
= 'N': op(A) = A (No transpose);
= 'T': op(A) = A**T (Transpose);
= 'C': op(A) = A**T (Conjugate transpose = Transpose).
UPLO CHARACTER*1
Specifies which part of the symmetric matrix C is to be
used, as follows:
= 'U': Upper triangular part;
= 'L': Lower triangular part.
LYAPUN CHARACTER*1
Specifies whether or not the original or "reduced"
Lyapunov equations should be solved, as follows:
= 'O': Solve the original Lyapunov equations, updating
the right-hand sides and solutions with the
matrix U, e.g., X <-- U'*X*U;
= 'R': Solve reduced Lyapunov equations only, without
updating the right-hand sides and solutions.
This means that a real Schur form T of A appears
in the equation, instead of A.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, X, and C. N >= 0.
SCALE (input or output) DOUBLE PRECISION
If JOB = 'C' or JOB = 'E', SCALE is an input argument:
the scale factor, set by a Lyapunov solver.
0 <= SCALE <= 1.
If JOB = 'X' or JOB = 'A', SCALE is an output argument:
the scale factor, scale, set less than or equal to 1 to
prevent the solution overflowing.
If JOB = 'S', this argument is not used.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
If FACT = 'N' or (LYAPUN = 'O' and JOB <> 'X'), the
leading N-by-N part of this array must contain the
original matrix A.
If FACT = 'F' and (LYAPUN = 'R' or JOB = 'X'), A is
not referenced.
LDA INTEGER
The leading dimension of the array A.
LDA >= MAX(1,N), if FACT = 'N' or LYAPUN = 'O' and
JOB <> 'X';
LDA >= 1, otherwise.
T (input/output) DOUBLE PRECISION array, dimension
(LDT,N)
If FACT = 'F', then on entry the leading N-by-N upper
Hessenberg part of this array must contain the upper
quasi-triangular matrix T in Schur canonical form from a
Schur factorization of A.
If FACT = 'N', then this array need not be set on input.
On exit, (if INFO = 0 or INFO = N+1, for FACT = 'N') the
leading N-by-N upper Hessenberg part of this array
contains the upper quasi-triangular matrix T in Schur
canonical form from a Schur factorization of A.
The contents of array T is not modified if FACT = 'F'.
LDT INTEGER
The leading dimension of the array T. LDT >= MAX(1,N).
U (input or output) DOUBLE PRECISION array, dimension
(LDU,N)
If LYAPUN = 'O' and FACT = 'F', then U is an input
argument and on entry, the leading N-by-N part of this
array must contain the orthogonal matrix U from a real
Schur factorization of A.
If LYAPUN = 'O' and FACT = 'N', then U is an output
argument and on exit, if INFO = 0 or INFO = N+1, it
contains the orthogonal N-by-N matrix from a real Schur
factorization of A.
If LYAPUN = 'R', the array U is not referenced.
LDU INTEGER
The leading dimension of the array U.
LDU >= 1, if LYAPUN = 'R';
LDU >= MAX(1,N), if LYAPUN = 'O'.
C (input) DOUBLE PRECISION array, dimension (LDC,N)
If JOB <> 'S' and UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the matrix C of the original Lyapunov
equation (with matrix A), if LYAPUN = 'O', or of the
reduced Lyapunov equation (with matrix T), if
LYAPUN = 'R'.
If JOB <> 'S' and UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the matrix C of the original Lyapunov
equation (with matrix A), if LYAPUN = 'O', or of the
reduced Lyapunov equation (with matrix T), if
LYAPUN = 'R'.
The remaining strictly triangular part of this array is
used as workspace.
If JOB = 'X', then this array may be identified with X
in the call of this routine.
If JOB = 'S', the array C is not referenced.
LDC INTEGER
The leading dimension of the array C.
LDC >= 1, if JOB = 'S';
LDC >= MAX(1,N), otherwise.
X (input or output) DOUBLE PRECISION array, dimension
(LDX,N)
If JOB = 'C' or 'E', then X is an input argument and on
entry, the leading N-by-N part of this array must contain
the symmetric solution matrix X of the original Lyapunov
equation (with matrix A), if LYAPUN = 'O', or of the
reduced Lyapunov equation (with matrix T), if
LYAPUN = 'R'.
If JOB = 'X' or 'A', then X is an output argument and on
exit, if INFO = 0 or INFO = N+1, the leading N-by-N part
of this array contains the symmetric solution matrix X of
of the original Lyapunov equation (with matrix A), if
LYAPUN = 'O', or of the reduced Lyapunov equation (with
matrix T), if LYAPUN = 'R'.
If JOB = 'S', the array X is not referenced.
LDX INTEGER
The leading dimension of the array X.
LDX >= 1, if JOB = 'S';
LDX >= MAX(1,N), otherwise.
SEPD (output) DOUBLE PRECISION
If JOB = 'S' or JOB = 'C' or JOB = 'A', and INFO = 0 or
INFO = N+1, SEPD contains the estimated separation of the
matrices op(A) and op(A)', sepd(op(A),op(A)').
If N = 0, or X = 0, or JOB = 'X' or JOB = 'E', SEPD is not
referenced.
RCOND (output) DOUBLE PRECISION
If JOB = 'C' or JOB = 'A', an estimate of the reciprocal
condition number of the continuous-time Lyapunov equation.
If N = 0 or X = 0, RCOND is set to 1 or 0, respectively.
If JOB = 'X' or JOB = 'S' or JOB = 'E', RCOND is not
referenced.
FERR (output) DOUBLE PRECISION
If JOB = 'E' or JOB = 'A', and INFO = 0 or INFO = N+1,
FERR contains an estimated forward error bound for the
solution X. If XTRUE is the true solution, FERR bounds the
relative error in the computed solution, measured in the
Frobenius norm: norm(X - XTRUE)/norm(XTRUE).
If N = 0 or X = 0, FERR is set to 0.
If JOB = 'X' or JOB = 'S' or JOB = 'C', FERR is not
referenced.
WR (output) DOUBLE PRECISION array, dimension (N)
WI (output) DOUBLE PRECISION array, dimension (N)
If FACT = 'N', and INFO = 0 or INFO = N+1, WR and WI
contain the real and imaginary parts, respectively, of the
eigenvalues of A.
If FACT = 'F', WR and WI are not referenced.
Workspace
IWORK INTEGER array, dimension (N*N)
This array is not referenced if JOB = 'X'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or INFO = N+1, DWORK(1) returns the
optimal value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
If JOB = 'X', then
LDWORK >= MAX(1,N*N,2*N), if FACT = 'F';
LDWORK >= MAX(1,N*N,3*N), if FACT = 'N'.
If JOB = 'S', then
LDWORK >= MAX(3,2*N*N).
If JOB = 'C', then
LDWORK >= MAX(3,2*N*N) + N*N.
If JOB = 'E', or JOB = 'A', then
LDWORK >= MAX(3,2*N*N) + N*N + 2*N.
For optimum performance LDWORK should sometimes be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, i <= N, the QR algorithm failed to
complete the reduction to Schur canonical form (see
LAPACK Library routine DGEES); on exit, the matrix
T(i+1:N,i+1:N) contains the partially converged
Schur form, and the elements i+1:n of WR and WI
contain the real and imaginary parts, respectively,
of the converged eigenvalues; this error is unlikely
to appear;
= N+1: if the matrix T has almost reciprocal eigenvalues;
perturbed values were used to solve Lyapunov
equations, but the matrix T, if given (for
FACT = 'F'), is unchanged.
Method
After reducing matrix A to real Schur canonical form (if needed), a discrete-time version of the Bartels-Stewart algorithm is used. A set of equivalent linear algebraic systems of equations of order at most four are formed and solved using Gaussian elimination with complete pivoting. The condition number of the discrete-time Lyapunov equation is estimated as cond = (norm(Theta)*norm(A) + norm(inv(Omega))*norm(C))/norm(X), where Omega and Theta are linear operators defined by Omega(W) = op(A)'*W*op(A) - W, Theta(W) = inv(Omega(op(W)'*X*op(A) + op(A)'*X*op(W))). The routine estimates the quantities sepd(op(A),op(A)') = 1 / norm(inv(Omega)) and norm(Theta) using 1-norm condition estimators. The forward error bound is estimated using a practical error bound similar to the one proposed in [3].References
[1] Barraud, A.Y. T
A numerical algorithm to solve A XA - X = Q.
IEEE Trans. Auto. Contr., AC-22, pp. 883-885, 1977.
[2] Bartels, R.H. and Stewart, G.W. T
Solution of the matrix equation A X + XB = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
[3] Higham, N.J.
Perturbation theory and backward error for AX-XB=C.
BIT, vol. 33, pp. 124-136, 1993.
Numerical Aspects
3 The algorithm requires 0(N ) operations. The accuracy of the estimates obtained depends on the solution accuracy and on the properties of the 1-norm estimator.Further Comments
The "separation" sepd of op(A) and op(A)' can also be defined as
sepd( op(A), op(A)' ) = sigma_min( T ),
where sigma_min(T) is the smallest singular value of the
N*N-by-N*N matrix
T = kprod( op(A)', op(A)' ) - I(N**2).
I(N**2) is an N*N-by-N*N identity matrix, and kprod denotes the
Kronecker product. The routine estimates sigma_min(T) by the
reciprocal of an estimate of the 1-norm of inverse(T). The true
reciprocal 1-norm of inverse(T) cannot differ from sigma_min(T) by
more than a factor of N.
Example
Program Text
* SB03UD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA, LDC, LDT, LDU, LDX
PARAMETER ( LDA = NMAX, LDC = NMAX, LDT = NMAX,
$ LDU = NMAX, LDX = NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX*NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( 3, 2*NMAX*NMAX ) +
$ NMAX*NMAX + 2*NMAX )
* .. Local Scalars ..
DOUBLE PRECISION FERR, RCOND, SCALE, SEPD
INTEGER I, INFO, J, N
CHARACTER*1 DICO, FACT, JOB, LYAPUN, TRANA, UPLO
* .. Local Arrays ..
INTEGER IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), C(LDC,NMAX), DWORK(LDWORK),
$ T(LDT,NMAX), U(LDU,NMAX), X(LDX,NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL SB03UD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
DICO = 'D'
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, JOB, FACT, TRANA, UPLO, LYAPUN
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
IF ( LSAME( JOB, 'C' ) .OR. LSAME( JOB, 'E' ) )
$ READ ( NIN, FMT = * ) SCALE
IF ( LSAME( FACT, 'N' ) .OR. ( LSAME( LYAPUN, 'O' ) .AND.
$ .NOT.LSAME( JOB, 'X') ) )
$ READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( FACT, 'F' ) ) THEN
READ ( NIN, FMT = * ) ( ( T(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( LYAPUN, 'O' ) )
$ READ ( NIN, FMT = * ) ( ( U(I,J), J = 1,N ), I = 1,N )
END IF
IF ( .NOT.LSAME( JOB, 'S' ) )
$ READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( JOB, 'C' ) .OR. LSAME( JOB, 'E' ) )
$ READ ( NIN, FMT = * ) ( ( X(I,J), J = 1,N ), I = 1,N )
* Solve the discrete-time Lyapunov matrix equation and/or
* estimate the condition and error bound on the solution.
CALL SB03UD( JOB, FACT, TRANA, UPLO, LYAPUN, N, SCALE, A, LDA,
$ T, LDT, U, LDU, C, LDC, X, LDX, SEPD, RCOND, FERR,
$ DWORK(1), DWORK(N+1), IWORK, DWORK(2*N+1),
$ LDWORK-2*N, INFO )
*
IF ( INFO.EQ.0 ) THEN
IF ( LSAME( JOB, 'X' ) .OR. LSAME( JOB, 'A' ) ) THEN
WRITE ( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( X(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99993 ) SCALE
END IF
IF ( LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'C' )
$ .OR. LSAME( JOB, 'A' ) )
$ WRITE ( NOUT, FMT = 99992 ) SEPD
IF ( LSAME( JOB, 'C' ) .OR. LSAME( JOB, 'A' ) )
$ WRITE ( NOUT, FMT = 99991 ) RCOND
IF ( LSAME( JOB, 'E' ) .OR. LSAME( JOB, 'A' ) )
$ WRITE ( NOUT, FMT = 99990 ) FERR
ELSE
WRITE ( NOUT, FMT = 99998 ) INFO
END IF
END IF
STOP
*
99999 FORMAT (' SB03UD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB03UD =',I2)
99996 FORMAT (' The solution matrix X is')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' Scaling factor = ',F8.4)
99992 FORMAT (/' Estimated separation = ',F8.4)
99991 FORMAT (/' Estimated reciprocal condition number = ',F8.4)
99990 FORMAT (/' Estimated error bound = ',F8.4)
END
Program Data
SB03UD EXAMPLE PROGRAM DATA 3 A N N U O 3.0 1.0 1.0 1.0 3.0 0.0 0.0 0.0 3.0 25.0 24.0 15.0 24.0 32.0 8.0 15.0 8.0 40.0Program Results
SB03UD EXAMPLE PROGRAM RESULTS The solution matrix X is 2.0000 1.0000 1.0000 1.0000 3.0000 0.0000 1.0000 0.0000 4.0000 Scaling factor = 1.0000 Estimated separation = 5.2302 Estimated reciprocal condition number = 0.1832 Estimated error bound = 0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB04MD.html 0000664 0000000 0000000 00000021056 14560147231 0020347 0 ustar 00root root 0000000 0000000
Purpose
To solve for X the continuous-time Sylvester equation
AX + XB = C
where A, B, C and X are general N-by-N, M-by-M, N-by-M and
N-by-M matrices respectively.
Specification
SUBROUTINE SB04MD( N, M, A, LDA, B, LDB, C, LDC, Z, LDZ, IWORK,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDWORK, LDZ, M, N
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), Z(LDZ,*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
M (input) INTEGER
The order of the matrix B. M >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the coefficient matrix A of the equation.
On exit, the leading N-by-N upper Hessenberg part of this
array contains the matrix H, and the remainder of the
leading N-by-N part, together with the elements 2,3,...,N
of array DWORK, contain the orthogonal transformation
matrix U (stored in factored form).
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading M-by-M part of this array must
contain the coefficient matrix B of the equation.
On exit, the leading M-by-M part of this array contains
the quasi-triangular Schur factor S of the matrix B'.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,M).
C (input/output) DOUBLE PRECISION array, dimension (LDC,M)
On entry, the leading N-by-M part of this array must
contain the coefficient matrix C of the equation.
On exit, the leading N-by-M part of this array contains
the solution matrix X of the problem.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,N).
Z (output) DOUBLE PRECISION array, dimension (LDZ,M)
The leading M-by-M part of this array contains the
orthogonal matrix Z used to transform B' to real upper
Schur form.
LDZ INTEGER
The leading dimension of array Z. LDZ >= MAX(1,M).
Workspace
IWORK INTEGER array, dimension (4*N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK, and DWORK(2), DWORK(3),..., DWORK(N) contain
the scalar factors of the elementary reflectors used to
reduce A to upper Hessenberg form, as returned by LAPACK
Library routine DGEHRD.
LDWORK INTEGER
The length of the array DWORK.
LDWORK = MAX(1, 2*N*N + 8*N, 5*M, N + M).
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message related to LDWORK is issued by
XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, 1 <= i <= M, the QR algorithm failed to
compute all the eigenvalues (see LAPACK Library
routine DGEES);
> M: if a singular matrix was encountered whilst solving
for the (INFO-M)-th column of matrix X.
Method
The matrix A is transformed to upper Hessenberg form H = U'AU by
the orthogonal transformation matrix U; matrix B' is transformed
to real upper Schur form S = Z'B'Z using the orthogonal
transformation matrix Z. The matrix C is also multiplied by the
transformations, F = U'CZ, and the solution matrix Y of the
transformed system
HY + YS' = F
is computed by back substitution. Finally, the matrix Y is then
multiplied by the orthogonal transformation matrices, X = UYZ', in
order to obtain the solution matrix X to the original problem.
References
[1] Golub, G.H., Nash, S. and Van Loan, C.F.
A Hessenberg-Schur method for the problem AX + XB = C.
IEEE Trans. Auto. Contr., AC-24, pp. 909-913, 1979.
Numerical Aspects
3 3 2 2 The algorithm requires about (5/3) N + 10 M + 5 N M + 2.5 M N operations and is backward stable.Further Comments
NoneExample
Program Text
* SB04MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX
PARAMETER ( NMAX = 20, MMAX = 20 )
INTEGER LDA, LDB, LDC, LDZ
PARAMETER ( LDA = NMAX, LDB = MMAX, LDC = NMAX,
$ LDZ = MMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = 4*NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( 1, 2*NMAX*NMAX+8*NMAX, 5*MMAX,
$ NMAX+MMAX ) )
* .. Local Scalars ..
INTEGER I, INFO, J, M, N
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,MMAX),
$ DWORK(LDWORK), Z(LDZ,MMAX)
INTEGER IWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL SB04MD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,M )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,M ), I = 1,N )
* Find the solution matrix X.
CALL SB04MD( N, M, A, LDA, B, LDB, C, LDC, Z, LDZ, IWORK,
$ DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( C(I,J), J = 1,M )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 40 I = 1, M
WRITE ( NOUT, FMT = 99996 ) ( Z(I,J), J = 1,M )
40 CONTINUE
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB04MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB04MD = ',I2)
99997 FORMAT (' The solution matrix X is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The orthogonal matrix Z is ')
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' M is out of range.',/' M = ',I5)
END
Program Data
SB04MD EXAMPLE PROGRAM DATA 3 2 2.0 1.0 3.0 0.0 2.0 1.0 6.0 1.0 2.0 2.0 1.0 1.0 6.0 2.0 1.0 1.0 4.0 0.0 5.0Program Results
SB04MD EXAMPLE PROGRAM RESULTS The solution matrix X is -2.7685 0.5498 -1.0531 0.6865 4.5257 -0.4389 The orthogonal matrix Z is -0.9732 -0.2298 0.2298 -0.9732
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB04MR.html 0000664 0000000 0000000 00000006247 14560147231 0020372 0 ustar 00root root 0000000 0000000
Purpose
To solve a linear algebraic system of order M whose coefficient matrix has zeros below the second subdiagonal. The matrix is stored compactly, row-wise.Specification
SUBROUTINE SB04MR( M, D, IPR, INFO )
C .. Scalar Arguments ..
INTEGER INFO, M
C .. Array Arguments ..
INTEGER IPR(*)
DOUBLE PRECISION D(*)
Arguments
Input/Output Parameters
M (input) INTEGER
The order of the system. M >= 0.
Note that parameter M should have twice the value in the
original problem (see SLICOT Library routine SB04MU).
D (input/output) DOUBLE PRECISION array, dimension
(M*(M+1)/2+3*M)
On entry, the first M*(M+1)/2 + 2*M elements of this array
must contain the coefficient matrix, stored compactly,
row-wise, and the next M elements must contain the right
hand side of the linear system, as set by SLICOT Library
routine SB04MU.
On exit, the content of this array is updated, the last M
elements containing the solution with components
interchanged (see IPR).
IPR (output) INTEGER array, dimension (2*M)
The leading M elements contain information about the
row interchanges performed for solving the system.
Specifically, the i-th component of the solution is
specified by IPR(i).
Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: if a singular matrix was encountered.
Method
Gaussian elimination with partial pivoting is used. The rows of the matrix are not actually permuted, only their indices are interchanged in array IPR.References
[1] Golub, G.H., Nash, S. and Van Loan, C.F.
A Hessenberg-Schur method for the problem AX + XB = C.
IEEE Trans. Auto. Contr., AC-24, pp. 909-913, 1979.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To construct and solve a linear algebraic system of order 2*M whose coefficient matrix has zeros below the second subdiagonal. Such systems appear when solving continuous-time Sylvester equations using the Hessenberg-Schur method.Specification
SUBROUTINE SB04MU( N, M, IND, A, LDA, B, LDB, C, LDC, D, IPR,
$ INFO )
C .. Scalar Arguments ..
INTEGER INFO, IND, LDA, LDB, LDC, M, N
C .. Array Arguments ..
INTEGER IPR(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix B. N >= 0.
M (input) INTEGER
The order of the matrix A. M >= 0.
IND (input) INTEGER
IND and IND - 1 specify the indices of the columns in C
to be computed. IND > 1.
A (input) DOUBLE PRECISION array, dimension (LDA,M)
The leading M-by-M part of this array must contain an
upper Hessenberg matrix.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,M).
B (input) DOUBLE PRECISION array, dimension (LDB,N)
The leading N-by-N part of this array must contain a
matrix in real Schur form.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading M-by-N part of this array must
contain the coefficient matrix C of the equation.
On exit, the leading M-by-N part of this array contains
the matrix C with columns IND-1 and IND updated.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,M).
Workspace
D DOUBLE PRECISION array, dimension (2*M*M+7*M) IPR INTEGER array, dimension (4*M)Error Indicator
INFO INTEGER
= 0: successful exit;
> 0: if INFO = IND, a singular matrix was encountered.
Method
A special linear algebraic system of order 2*M, whose coefficient matrix has zeros below the second subdiagonal is constructed and solved. The coefficient matrix is stored compactly, row-wise.References
[1] Golub, G.H., Nash, S. and Van Loan, C.F.
A Hessenberg-Schur method for the problem AX + XB = C.
IEEE Trans. Auto. Contr., AC-24, pp. 909-913, 1979.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve a linear algebraic system of order M whose coefficient matrix is in upper Hessenberg form, stored compactly, row-wise.Specification
SUBROUTINE SB04MW( M, D, IPR, INFO )
C .. Scalar Arguments ..
INTEGER INFO, M
C .. Array Arguments ..
INTEGER IPR(*)
DOUBLE PRECISION D(*)
Arguments
Input/Output Parameters
M (input) INTEGER
The order of the system. M >= 0.
D (input/output) DOUBLE PRECISION array, dimension
(M*(M+1)/2+2*M)
On entry, the first M*(M+1)/2 + M elements of this array
must contain an upper Hessenberg matrix, stored compactly,
row-wise, and the next M elements must contain the right
hand side of the linear system, as set by SLICOT Library
routine SB04MY.
On exit, the content of this array is updated, the last M
elements containing the solution with components
interchanged (see IPR).
IPR (output) INTEGER array, dimension (2*M)
The leading M elements contain information about the
row interchanges performed for solving the system.
Specifically, the i-th component of the solution is
specified by IPR(i).
Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: if a singular matrix was encountered.
Method
Gaussian elimination with partial pivoting is used. The rows of the matrix are not actually permuted, only their indices are interchanged in array IPR.References
[1] Golub, G.H., Nash, S. and Van Loan, C.F.
A Hessenberg-Schur method for the problem AX + XB = C.
IEEE Trans. Auto. Contr., AC-24, pp. 909-913, 1979.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To construct and solve a linear algebraic system of order M whose coefficient matrix is in upper Hessenberg form. Such systems appear when solving Sylvester equations using the Hessenberg-Schur method.Specification
SUBROUTINE SB04MY( N, M, IND, A, LDA, B, LDB, C, LDC, D, IPR,
$ INFO )
C .. Scalar Arguments ..
INTEGER INFO, IND, LDA, LDB, LDC, M, N
C .. Array Arguments ..
INTEGER IPR(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix B. N >= 0.
M (input) INTEGER
The order of the matrix A. M >= 0.
IND (input) INTEGER
The index of the column in C to be computed. IND >= 1.
A (input) DOUBLE PRECISION array, dimension (LDA,M)
The leading M-by-M part of this array must contain an
upper Hessenberg matrix.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,M).
B (input) DOUBLE PRECISION array, dimension (LDB,N)
The leading N-by-N part of this array must contain a
matrix in real Schur form.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading M-by-N part of this array must
contain the coefficient matrix C of the equation.
On exit, the leading M-by-N part of this array contains
the matrix C with column IND updated.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,M).
Workspace
D DOUBLE PRECISION array, dimension (M*(M+1)/2+2*M) IPR INTEGER array, dimension (2*M)Error Indicator
INFO INTEGER
= 0: successful exit;
> 0: if INFO = IND, a singular matrix was encountered.
Method
A special linear algebraic system of order M, with coefficient matrix in upper Hessenberg form is constructed and solved. The coefficient matrix is stored compactly, row-wise.References
[1] Golub, G.H., Nash, S. and Van Loan, C.F.
A Hessenberg-Schur method for the problem AX + XB = C.
IEEE Trans. Auto. Contr., AC-24, pp. 909-913, 1979.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve for X the continuous-time Sylvester equation
AX + XB = C,
with at least one of the matrices A or B in Schur form and the
other in Hessenberg or Schur form (both either upper or lower);
A, B, C and X are N-by-N, M-by-M, N-by-M, and N-by-M matrices,
respectively.
Specification
SUBROUTINE SB04ND( ABSCHU, ULA, ULB, N, M, A, LDA, B, LDB, C,
$ LDC, TOL, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER ABSCHU, ULA, ULB
INTEGER INFO, LDA, LDB, LDC, LDWORK, M, N
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*)
Arguments
Mode Parameters
ABSCHU CHARACTER*1
Indicates whether A and/or B is/are in Schur or
Hessenberg form as follows:
= 'A': A is in Schur form, B is in Hessenberg form;
= 'B': B is in Schur form, A is in Hessenberg form;
= 'S': Both A and B are in Schur form.
ULA CHARACTER*1
Indicates whether A is in upper or lower Schur form or
upper or lower Hessenberg form as follows:
= 'U': A is in upper Hessenberg form if ABSCHU = 'B' and
upper Schur form otherwise;
= 'L': A is in lower Hessenberg form if ABSCHU = 'B' and
lower Schur form otherwise.
ULB CHARACTER*1
Indicates whether B is in upper or lower Schur form or
upper or lower Hessenberg form as follows:
= 'U': B is in upper Hessenberg form if ABSCHU = 'A' and
upper Schur form otherwise;
= 'L': B is in lower Hessenberg form if ABSCHU = 'A' and
lower Schur form otherwise.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
M (input) INTEGER
The order of the matrix B. M >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
coefficient matrix A of the equation.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading M-by-M part of this array must contain the
coefficient matrix B of the equation.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,M).
C (input/output) DOUBLE PRECISION array, dimension (LDC,M)
On entry, the leading N-by-M part of this array must
contain the coefficient matrix C of the equation.
On exit, if INFO = 0, the leading N-by-M part of this
array contains the solution matrix X of the problem.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,N).
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used to test for near singularity in
the Sylvester equation. If the user sets TOL > 0, then the
given value of TOL is used as a lower bound for the
reciprocal condition number; a matrix whose estimated
condition number is less than 1/TOL is considered to be
nonsingular. If the user sets TOL <= 0, then a default
tolerance, defined by TOLDEF = EPS, is used instead, where
EPS is the machine precision (see LAPACK Library routine
DLAMCH).
This parameter is not referenced if ABSCHU = 'S',
ULA = 'U', and ULB = 'U'.
Workspace
IWORK INTEGER array, dimension (2*MAX(M,N))
This parameter is not referenced if ABSCHU = 'S',
ULA = 'U', and ULB = 'U'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
This parameter is not referenced if ABSCHU = 'S',
ULA = 'U', and ULB = 'U'.
LDWORK INTEGER
The length of the array DWORK.
LDWORK = 0, if ABSCHU = 'S', ULA = 'U', and ULB = 'U';
LDWORK = 2*MAX(M,N)*(4 + 2*MAX(M,N)), otherwise.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if a (numerically) singular matrix T was encountered
during the computation of the solution matrix X.
That is, the estimated reciprocal condition number
of T is less than or equal to TOL.
Method
Matrices A and B are assumed to be in (upper or lower) Hessenberg or Schur form (with at least one of them in Schur form). The solution matrix X is then computed by rows or columns via the back substitution scheme proposed by Golub, Nash and Van Loan (see [1]), which involves the solution of triangular systems of equations that are constructed recursively and which may be nearly singular if A and -B have close eigenvalues. If near singularity is detected, then the routine returns with the Error Indicator (INFO) set to 1.References
[1] Golub, G.H., Nash, S. and Van Loan, C.F.
A Hessenberg-Schur method for the problem AX + XB = C.
IEEE Trans. Auto. Contr., AC-24, pp. 909-913, 1979.
Numerical Aspects
2 2
The algorithm requires approximately 5M N + 0.5MN operations in
2 2
the worst case and 2.5M N + 0.5MN operations in the best case
(where M is the order of the matrix in Hessenberg form and N is
the order of the matrix in Schur form) and is mixed stable (see
[1]).
Further Comments
NoneExample
Program Text
* SB04ND EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX
PARAMETER ( NMAX = 20, MMAX = 20 )
INTEGER LDA, LDB, LDC
PARAMETER ( LDA = NMAX, LDB = MMAX, LDC = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 2*( MAX( NMAX,MMAX ) )*
$ ( 4+2*( MAX( NMAX,MMAX ) ) ) )
INTEGER LIWORK
PARAMETER ( LIWORK = 2*MAX( NMAX,MMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, J, M, N
CHARACTER*1 ABSCHU, ULA, ULB
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,MMAX),
$ DWORK(LDWORK)
INTEGER IWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL SB04ND
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, TOL, ULA, ULB, ABSCHU
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,M )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,M ), I = 1,N )
* Find the solution matrix X.
CALL SB04ND( ABSCHU, ULA, ULB, N, M, A, LDA, B, LDB, C,
$ LDC, TOL, IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( C(I,J), J = 1,M )
20 CONTINUE
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB04ND EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB04ND = ',I2)
99997 FORMAT (' The solution matrix X is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' N is out of range.',/' N = ',I5)
99994 FORMAT (/' M is out of range.',/' M = ',I5)
END
Program Data
SB04ND EXAMPLE PROGRAM DATA 5 3 0.0 U U B 17.0 24.0 1.0 8.0 15.0 23.0 5.0 7.0 14.0 16.0 0.0 6.0 13.0 20.0 22.0 0.0 0.0 19.0 21.0 3.0 0.0 0.0 0.0 2.0 9.0 8.0 1.0 6.0 0.0 5.0 7.0 0.0 9.0 2.0 62.0 -12.0 26.0 59.0 -10.0 31.0 70.0 -6.0 9.0 35.0 31.0 -7.0 36.0 -15.0 7.0Program Results
SB04ND EXAMPLE PROGRAM RESULTS The solution matrix X is 0.0000 0.0000 1.0000 1.0000 0.0000 0.0000 0.0000 1.0000 0.0000 1.0000 1.0000 -1.0000 2.0000 -2.0000 1.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB04NV.html 0000664 0000000 0000000 00000006500 14560147231 0020367 0 ustar 00root root 0000000 0000000
Purpose
To construct the right-hand sides D for a system of equations in Hessenberg form solved via SB04NX (case with 2 right-hand sides).Specification
SUBROUTINE SB04NV( ABSCHR, UL, N, M, C, LDC, INDX, AB, LDAB, D )
C .. Scalar Arguments ..
CHARACTER ABSCHR, UL
INTEGER INDX, LDAB, LDC, M, N
C .. Array Arguments ..
DOUBLE PRECISION AB(LDAB,*), C(LDC,*), D(*)
Arguments
Mode Parameters
ABSCHR CHARACTER*1
Indicates whether AB contains A or B, as follows:
= 'A': AB contains A;
= 'B': AB contains B.
UL CHARACTER*1
Indicates whether AB is upper or lower Hessenberg matrix,
as follows:
= 'U': AB is upper Hessenberg;
= 'L': AB is lower Hessenberg.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
M (input) INTEGER
The order of the matrix B. M >= 0.
C (input) DOUBLE PRECISION array, dimension (LDC,M)
The leading N-by-M part of this array must contain both
the not yet modified part of the coefficient matrix C of
the Sylvester equation AX + XB = C, and both the currently
computed part of the solution of the Sylvester equation.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,N).
INDX (input) INTEGER
The position of the first column/row of C to be used in
the construction of the right-hand side D.
AB (input) DOUBLE PRECISION array, dimension (LDAB,*)
The leading N-by-N or M-by-M part of this array must
contain either A or B of the Sylvester equation
AX + XB = C.
LDAB INTEGER
The leading dimension of array AB.
LDAB >= MAX(1,N) or LDAB >= MAX(1,M) (depending on
ABSCHR = 'A' or ABSCHR = 'B', respectively).
D (output) DOUBLE PRECISION array, dimension (*)
The leading 2*N or 2*M part of this array (depending on
ABSCHR = 'B' or ABSCHR = 'A', respectively) contains the
right-hand side stored as a matrix with two rows.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To construct the right-hand side D for a system of equations in Hessenberg form solved via SB04NY (case with 1 right-hand side).Specification
SUBROUTINE SB04NW( ABSCHR, UL, N, M, C, LDC, INDX, AB, LDAB, D )
C .. Scalar Arguments ..
CHARACTER ABSCHR, UL
INTEGER INDX, LDAB, LDC, M, N
C .. Array Arguments ..
DOUBLE PRECISION AB(LDAB,*), C(LDC,*), D(*)
Arguments
Mode Parameters
ABSCHR CHARACTER*1
Indicates whether AB contains A or B, as follows:
= 'A': AB contains A;
= 'B': AB contains B.
UL CHARACTER*1
Indicates whether AB is upper or lower Hessenberg matrix,
as follows:
= 'U': AB is upper Hessenberg;
= 'L': AB is lower Hessenberg.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
M (input) INTEGER
The order of the matrix B. M >= 0.
C (input) DOUBLE PRECISION array, dimension (LDC,M)
The leading N-by-M part of this array must contain both
the not yet modified part of the coefficient matrix C of
the Sylvester equation AX + XB = C, and both the currently
computed part of the solution of the Sylvester equation.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,N).
INDX (input) INTEGER
The position of the column/row of C to be used in the
construction of the right-hand side D.
AB (input) DOUBLE PRECISION array, dimension (LDAB,*)
The leading N-by-N or M-by-M part of this array must
contain either A or B of the Sylvester equation
AX + XB = C.
LDAB INTEGER
The leading dimension of array AB.
LDAB >= MAX(1,N) or LDAB >= MAX(1,M) (depending on
ABSCHR = 'A' or ABSCHR = 'B', respectively).
D (output) DOUBLE PRECISION array, dimension (*)
The leading N or M part of this array (depending on
ABSCHR = 'B' or ABSCHR = 'A', respectively) contains the
right-hand side.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve a system of equations in Hessenberg form with two consecutive offdiagonals and two right-hand sides.Specification
SUBROUTINE SB04NX( RC, UL, M, A, LDA, LAMBD1, LAMBD2, LAMBD3,
$ LAMBD4, D, TOL, IWORK, DWORK, LDDWOR, INFO )
C .. Scalar Arguments ..
CHARACTER RC, UL
INTEGER INFO, LDA, LDDWOR, M
DOUBLE PRECISION LAMBD1, LAMBD2, LAMBD3, LAMBD4, TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), D(*), DWORK(LDDWOR,*)
Arguments
Mode Parameters
RC CHARACTER*1
Indicates processing by columns or rows, as follows:
= 'R': Row transformations are applied;
= 'C': Column transformations are applied.
UL CHARACTER*1
Indicates whether AB is upper or lower Hessenberg matrix,
as follows:
= 'U': AB is upper Hessenberg;
= 'L': AB is lower Hessenberg.
Input/Output Parameters
M (input) INTEGER
The order of the matrix A. M >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,M)
The leading M-by-M part of this array must contain a
matrix A in Hessenberg form.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,M).
LAMBD1, (input) DOUBLE PRECISION
LAMBD2, These variables must contain the 2-by-2 block to be added
LAMBD3, to the diagonal blocks of A.
LAMBD4
D (input/output) DOUBLE PRECISION array, dimension (2*M)
On entry, this array must contain the two right-hand
side vectors of the Hessenberg system, stored row-wise.
On exit, if INFO = 0, this array contains the two solution
vectors of the Hessenberg system, stored row-wise.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used to test for near singularity of
the triangular factor R of the Hessenberg matrix. A matrix
whose estimated condition number is less than 1/TOL is
considered to be nonsingular.
Workspace
IWORK INTEGER array, dimension (2*M)
DWORK DOUBLE PRECISION array, dimension (LDDWOR,2*M+3)
The leading 2*M-by-2*M part of this array is used for
computing the triangular factor of the QR decomposition
of the Hessenberg matrix. The remaining 6*M elements are
used as workspace for the computation of the reciprocal
condition estimate.
LDDWOR INTEGER
The leading dimension of array DWORK.
LDDWOR >= MAX(1,2*M).
Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: if the Hessenberg matrix is (numerically) singular.
That is, its estimated reciprocal condition number
is less than or equal to TOL.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve a system of equations in Hessenberg form with one offdiagonal and one right-hand side.Specification
SUBROUTINE SB04NY( RC, UL, M, A, LDA, LAMBDA, D, TOL, IWORK,
$ DWORK, LDDWOR, INFO )
C .. Scalar Arguments ..
CHARACTER RC, UL
INTEGER INFO, LDA, LDDWOR, M
DOUBLE PRECISION LAMBDA, TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), D(*), DWORK(LDDWOR,*)
Arguments
Mode Parameters
RC CHARACTER*1
Indicates processing by columns or rows, as follows:
= 'R': Row transformations are applied;
= 'C': Column transformations are applied.
UL CHARACTER*1
Indicates whether AB is upper or lower Hessenberg matrix,
as follows:
= 'U': AB is upper Hessenberg;
= 'L': AB is lower Hessenberg.
Input/Output Parameters
M (input) INTEGER
The order of the matrix A. M >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,M)
The leading M-by-M part of this array must contain a
matrix A in Hessenberg form.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,M).
LAMBDA (input) DOUBLE PRECISION
This variable must contain the value to be added to the
diagonal elements of A.
D (input/output) DOUBLE PRECISION array, dimension (M)
On entry, this array must contain the right-hand side
vector of the Hessenberg system.
On exit, if INFO = 0, this array contains the solution
vector of the Hessenberg system.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used to test for near singularity of
the triangular factor R of the Hessenberg matrix. A matrix
whose estimated condition number is less than 1/TOL is
considered to be nonsingular.
Workspace
IWORK INTEGER array, dimension (M)
DWORK DOUBLE PRECISION array, dimension (LDDWOR,M+3)
The leading M-by-M part of this array is used for
computing the triangular factor of the QR decomposition
of the Hessenberg matrix. The remaining 3*M elements are
used as workspace for the computation of the reciprocal
condition estimate.
LDDWOR INTEGER
The leading dimension of array DWORK. LDDWOR >= MAX(1,M).
Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: if the Hessenberg matrix is (numerically) singular.
That is, its estimated reciprocal condition number
is less than or equal to TOL.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve for R and L one of the generalized Sylvester equations
A * R - L * B = scale * C )
) (1)
D * R - L * E = scale * F )
or
A' * R + D' * L = scale * C )
) (2)
R * B' + L * E' = scale * (-F) )
where A and D are M-by-M matrices, B and E are N-by-N matrices and
C, F, R and L are M-by-N matrices.
The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an
output scaling factor chosen to avoid overflow.
The routine also optionally computes a Dif estimate, which
measures the separation of the spectrum of the matrix pair (A,D)
from the spectrum of the matrix pair (B,E), Dif[(A,D),(B,E)].
Specification
SUBROUTINE SB04OD( REDUCE, TRANS, JOBD, M, N, A, LDA, B, LDB, C,
$ LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, P,
$ LDP, Q, LDQ, U, LDU, V, LDV, IWORK, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBD, REDUCE, TRANS
INTEGER INFO, LDA, LDB, LDC, LDD, LDE, LDF, LDP, LDQ,
$ LDU, LDV, LDWORK, M, N
DOUBLE PRECISION DIF, SCALE
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), E(LDE,*), F(LDF,*), P(LDP,*),
$ Q(LDQ,*), U(LDU,*), V(LDV,*)
Arguments
Mode Parameters
REDUCE CHARACTER*1
Indicates whether the matrix pairs (A,D) and/or (B,E) are
to be reduced to generalized Schur form as follows:
= 'R': The matrix pairs (A,D) and (B,E) are to be reduced
to generalized (real) Schur canonical form;
= 'A': The matrix pair (A,D) only is to be reduced
to generalized (real) Schur canonical form,
and the matrix pair (B,E) already is in this form;
= 'B': The matrix pair (B,E) only is to be reduced
to generalized (real) Schur canonical form,
and the matrix pair (A,D) already is in this form;
= 'N': The matrix pairs (A,D) and (B,E) are already in
generalized (real) Schur canonical form, as
produced by LAPACK routine DGGES.
TRANS CHARACTER*1
Indicates which of the equations, (1) or (2), is to be
solved as follows:
= 'N': The generalized Sylvester equation (1) is to be
solved;
= 'T': The "transposed" generalized Sylvester equation
(2) is to be solved.
JOBD CHARACTER*1
Indicates whether the Dif estimator is to be computed as
follows:
= '1': Only the one-norm-based Dif estimate is computed
and stored in DIF;
= '2': Only the Frobenius norm-based Dif estimate is
computed and stored in DIF;
= 'D': The equation (1) is solved and the one-norm-based
Dif estimate is computed and stored in DIF;
= 'F': The equation (1) is solved and the Frobenius norm-
based Dif estimate is computed and stored in DIF;
= 'N': The Dif estimator is not required and hence DIF is
not referenced. (Solve either (1) or (2) only.)
JOBD is not referenced if TRANS = 'T'.
Input/Output Parameters
M (input) INTEGER
The order of the matrices A and D and the number of rows
of the matrices C, F, R and L. M >= 0.
N (input) INTEGER
The order of the matrices B and E and the number of
columns of the matrices C, F, R and L. N >= 0.
No computations are performed if N = 0 or M = 0, but SCALE
and DIF (if JOB <> 'N') are set to 1.
A (input/output) DOUBLE PRECISION array, dimension (LDA,M)
On entry, the leading M-by-M part of this array must
contain the coefficient matrix A of the equation; A must
be in upper quasi-triangular form if REDUCE = 'B' or 'N'.
On exit, the leading M-by-M part of this array contains
the upper quasi-triangular form of A.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,M).
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the leading N-by-N part of this array must
contain the coefficient matrix B of the equation; B must
be in upper quasi-triangular form if REDUCE = 'A' or 'N'.
On exit, the leading N-by-N part of this array contains
the upper quasi-triangular form of B.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading M-by-N part of this array must
contain the right-hand side matrix C of the first equation
in (1) or (2).
On exit, if JOBD = 'N', 'D' or 'F', the leading M-by-N
part of this array contains the solution matrix R of the
problem; if JOBD = '1' or '2' and TRANS = 'N', the leading
M-by-N part of this array contains the solution matrix R
achieved during the computation of the Dif estimate.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,M).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading M-by-M part of this array must
contain the coefficient matrix D of the equation; D must
be in upper triangular form if REDUCE = 'B' or 'N'.
On exit, the leading M-by-M part of this array contains
the upper triangular form of D.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,M).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain the coefficient matrix E of the equation; E must
be in upper triangular form if REDUCE = 'A' or 'N'.
On exit, the leading N-by-N part of this array contains
the upper triangular form of E.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,N).
F (input/output) DOUBLE PRECISION array, dimension (LDF,N)
On entry, the leading M-by-N part of this array must
contain the right-hand side matrix F of the second
equation in (1) or (2).
On exit, if JOBD = 'N', 'D' or 'F', the leading M-by-N
part of this array contains the solution matrix L of the
problem; if JOBD = '1' or '2' and TRANS = 'N', the leading
M-by-N part of this array contains the solution matrix L
achieved during the computation of the Dif estimate.
LDF INTEGER
The leading dimension of array F. LDF >= MAX(1,M).
SCALE (output) DOUBLE PRECISION
The scaling factor in (1) or (2). If 0 < SCALE < 1, C and
F hold the solutions R and L, respectively, to a slightly
perturbed system, but the computed generalized (real)
Schur canonical form matrices P'*A*Q, U'*B*V, P'*D*Q and
U'*E*V (see METHOD), or input matrices A, B, D, and E (if
already reduced to these forms), have not been changed.
If SCALE = 0, C and F hold the solutions R and L,
respectively, to the homogeneous system with C = F = 0.
Normally, SCALE = 1.
DIF (output) DOUBLE PRECISION
If TRANS = 'N' and JOBD <> 'N', then DIF contains the
value of the Dif estimator, which is an upper bound of
-1
Dif[(A,D),(B,E)] = sigma_min(Z) = 1/||Z ||, in either the
one-norm, or Frobenius norm, respectively (see METHOD).
Otherwise, DIF is not referenced.
P (output) DOUBLE PRECISION array, dimension (LDP,*)
If REDUCE = 'R' or 'A', then the leading M-by-M part of
this array contains the (left) transformation matrix used
to reduce (A,D) to generalized Schur form.
Otherwise, P is not referenced and can be supplied as a
dummy array (i.e. set parameter LDP = 1 and declare this
array to be P(1,1) in the calling program).
LDP INTEGER
The leading dimension of array P.
LDP >= MAX(1,M) if REDUCE = 'R' or 'A',
LDP >= 1 if REDUCE = 'B' or 'N'.
Q (output) DOUBLE PRECISION array, dimension (LDQ,*)
If REDUCE = 'R' or 'A', then the leading M-by-M part of
this array contains the (right) transformation matrix used
to reduce (A,D) to generalized Schur form.
Otherwise, Q is not referenced and can be supplied as a
dummy array (i.e. set parameter LDQ = 1 and declare this
array to be Q(1,1) in the calling program).
LDQ INTEGER
The leading dimension of array Q.
LDQ >= MAX(1,M) if REDUCE = 'R' or 'A',
LDQ >= 1 if REDUCE = 'B' or 'N'.
U (output) DOUBLE PRECISION array, dimension (LDU,*)
If REDUCE = 'R' or 'B', then the leading N-by-N part of
this array contains the (left) transformation matrix used
to reduce (B,E) to generalized Schur form.
Otherwise, U is not referenced and can be supplied as a
dummy array (i.e. set parameter LDU = 1 and declare this
array to be U(1,1) in the calling program).
LDU INTEGER
The leading dimension of array U.
LDU >= MAX(1,N) if REDUCE = 'R' or 'B',
LDU >= 1 if REDUCE = 'A' or 'N'.
V (output) DOUBLE PRECISION array, dimension (LDV,*)
If REDUCE = 'R' or 'B', then the leading N-by-N part of
this array contains the (right) transformation matrix used
to reduce (B,E) to generalized Schur form.
Otherwise, V is not referenced and can be supplied as a
dummy array (i.e. set parameter LDV = 1 and declare this
array to be V(1,1) in the calling program).
LDV INTEGER
The leading dimension of array V.
LDV >= MAX(1,N) if REDUCE = 'R' or 'B',
LDV >= 1 if REDUCE = 'A' or 'N'.
Workspace
IWORK INTEGER array, dimension (M+N+6)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
If TRANS = 'N' and JOBD = 'D' or 'F', then
LDWORK = MAX(1,11*MN,10*MN+23,2*M*N) if REDUCE = 'R';
LDWORK = MAX(1,11*M, 10*M+23, 2*M*N) if REDUCE = 'A';
LDWORK = MAX(1,11*N, 10*N+23, 2*M*N) if REDUCE = 'B';
LDWORK = MAX(1,2*M*N) if REDUCE = 'N',
where MN = max(M,N).
Otherwise, the term 2*M*N above should be omitted.
For optimum performance LDWORK should be larger.
If LDWORK = -1 a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if REDUCE <> 'N' and either (A,D) and/or (B,E)
cannot be reduced to generalized Schur form;
= 2: if REDUCE = 'N' and either A or B is not in
upper quasi-triangular form;
= 3: if a singular matrix was encountered during the
computation of the solution matrices R and L, that
is (A,D) and (B,E) have common or close eigenvalues.
Method
For the case TRANS = 'N', and REDUCE = 'R' or 'N', the algorithm
used by the routine consists of four steps (see [1] and [2]) as
follows:
(a) if REDUCE = 'R', then the matrix pairs (A,D) and (B,E) are
transformed to generalized Schur form, i.e. orthogonal
matrices P, Q, U and V are computed such that P' * A * Q
and U' * B * V are in upper quasi-triangular form and
P' * D * Q and U' * E * V are in upper triangular form;
(b) if REDUCE = 'R', then the matrices C and F are transformed
to give P' * C * V and P' * F * V respectively;
(c) if REDUCE = 'R', then the transformed system
P' * A * Q * R1 - L1 * U' * B * V = scale * P' * C * V
P' * D * Q * R1 - L1 * U' * E * V = scale * P' * F * V
is solved to give R1 and L1; otherwise, equation (1) is
solved to give R and L directly. The Dif estimator
is also computed if JOBD <> 'N'.
(d) if REDUCE = 'R', then the solution is transformed back
to give R = Q * R1 * V' and L = P * L1 * U'.
By using Kronecker products, equation (1) can also be written as
the system of linear equations Z * x = scale*y (see [1]), where
| I*A I*D |
Z = | |.
|-B'*I -E'*I |
-1
If JOBD <> 'N', then a lower bound on ||Z ||, in either the one-
norm or Frobenius norm, is computed, which in most cases is
a reliable estimate of the true value. Notice that since Z is a
matrix of order 2 * M * N, the exact value of Dif (i.e., in the
Frobenius norm case, the smallest singular value of Z) may be very
expensive to compute.
The case TRANS = 'N', and REDUCE = 'A' or 'B', is similar, but
only one of the matrix pairs should be reduced and the
calculations simplify.
For the case TRANS = 'T', and REDUCE = 'R' or 'N', the algorithm
is similar, but the steps (b), (c), and (d) are as follows:
(b) if REDUCE = 'R', then the matrices C and F are transformed
to give Q' * C * V and P' * F * U respectively;
(c) if REDUCE = 'R', then the transformed system
Q' * A' * P * R1 + Q' * D' * P * L1 = scale * Q' * C * V
R1 * V' * B' * U + L1 * V' * E' * U = -scale * P' * F * U
is solved to give R1 and L1; otherwise, equation (2) is
solved to give R and L directly.
(d) if REDUCE = 'R', then the solution is transformed back
to give R = P * R1 * V' and L = P * L1 * V'.
References
[1] Kagstrom, B. and Westin, L.
Generalized Schur Methods with Condition Estimators for
Solving the Generalized Sylvester Equation.
IEEE Trans. Auto. Contr., 34, pp. 745-751, 1989.
[2] Kagstrom, B. and Westin, L.
GSYLV - Fortran Routines for the Generalized Schur Method with
Dif Estimators for Solving the Generalized Sylvester
Equation.
Report UMINF-132.86, Institute of Information Processing,
Univ. of Umea, Sweden, July 1987.
[3] Golub, G.H., Nash, S. and Van Loan, C.F.
A Hessenberg-Schur Method for the Problem AX + XB = C.
IEEE Trans. Auto. Contr., AC-24, pp. 909-913, 1979.
[4] Kagstrom, B. and Van Dooren, P.
Additive Decomposition of a Transfer Function with respect to
a Specified Region.
In: "Signal Processing, Scattering and Operator Theory, and
Numerical Methods" (Eds. M.A. Kaashoek et al.).
Proceedings of MTNS-89, Vol. 3, pp. 469-477, Birkhauser Boston
Inc., 1990.
[5] Kagstrom, B. and Van Dooren, P.
A Generalized State-space Approach for the Additive
Decomposition of a Transfer Matrix.
Report UMINF-91.12, Institute of Information Processing, Univ.
of Umea, Sweden, April 1991.
Numerical Aspects
The algorithm is backward stable. A reliable estimate for the
condition number of Z in the Frobenius norm, is (see [1])
K(Z) = SQRT( ||A||**2 + ||B||**2 + ||C||**2 + ||D||**2 )/DIF.
If mu is an upper bound on the relative error of the elements of
the matrices A, B, C, D, E and F, then the relative error in the
actual solution is approximately mu * K(Z).
The relative error in the computed solution (due to rounding
errors) is approximately EPS * K(Z), where EPS is the machine
precision (see LAPACK Library routine DLAMCH).
Further Comments
For applications of the generalized Sylvester equation in control theory, see [4] and [5].Example
Program Text
* SB04OD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, NMAX
PARAMETER ( MMAX = 10, NMAX = 10 )
INTEGER LDA, LDB, LDC, LDD, LDE, LDF, LDP, LDQ, LDU, LDV
PARAMETER ( LDA = MMAX, LDB = NMAX, LDC = MMAX, LDD = MMAX,
$ LDE = NMAX, LDF = MMAX, LDP = MMAX, LDQ = MMAX,
$ LDU = NMAX, LDV = NMAX )
INTEGER LDWORK, LIWORK
PARAMETER ( LDWORK = MAX(11*MAX(MMAX,NMAX),
$ 10*MAX(MMAX,NMAX)+23,2*MMAX*NMAX),
$ LIWORK = MMAX+NMAX+6 )
* .. Local Scalars ..
DOUBLE PRECISION DIF, SCALE
INTEGER I, INFO, J, M, N
CHARACTER*1 JOBD, REDUCE, TRANS
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,MMAX), B(LDB,NMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK), E(LDE,NMAX),
$ F(LDF,NMAX), P(LDP,MMAX), Q(LDQ,MMAX),
$ U(LDU,NMAX), V(LDV,NMAX)
INTEGER IWORK(LIWORK)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL SB04OD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, N, REDUCE, TRANS, JOBD
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,M ), I = 1,M )
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,M )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,M )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( F(I,J), J = 1,N ), I = 1,M )
* Find the solution matrices L and R.
CALL SB04OD( REDUCE, TRANS, JOBD, M, N, A, LDA, B, LDB, C,
$ LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, P,
$ LDP, Q, LDQ, U, LDU, V, LDV, IWORK, DWORK,
$ LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, M
WRITE ( NOUT, FMT = 99991 ) ( F(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 40 I = 1, M
WRITE ( NOUT, FMT = 99991 ) ( C(I,J), J = 1,N )
40 CONTINUE
IF ( LSAME( REDUCE, 'R' ).OR.LSAME( REDUCE, 'A' ) ) THEN
WRITE ( NOUT, FMT = 99995 )
DO 60 I = 1, M
WRITE ( NOUT, FMT = 99991 ) ( P(I,J), J = 1,M )
60 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 80 I = 1, M
WRITE ( NOUT, FMT = 99991 ) ( Q(I,J), J = 1,M )
80 CONTINUE
END IF
IF ( LSAME( REDUCE, 'R' ).OR.LSAME( REDUCE, 'B' ) ) THEN
WRITE ( NOUT, FMT = 99993 )
DO 100 I = 1, N
WRITE ( NOUT, FMT = 99991 ) ( U(I,J), J = 1,N )
100 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 120 I = 1, N
WRITE ( NOUT, FMT = 99991 ) ( V(I,J), J = 1,N )
120 CONTINUE
END IF
IF ( SCALE.NE.ONE ) WRITE ( NOUT, FMT = 99987 ) SCALE
IF ( .NOT.LSAME( JOBD, 'N' ) )
$ WRITE ( NOUT, FMT = 99990 ) DIF
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' SB04OD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB04OD = ',I2)
99997 FORMAT (' The solution matrix L is ')
99996 FORMAT (/' The solution matrix R is ')
99995 FORMAT (/' The left transformation matrix P is ')
99994 FORMAT (/' The right transformation matrix Q is ')
99993 FORMAT (/' The left transformation matrix U is ')
99992 FORMAT (/' The right transformation matrix V is ')
99991 FORMAT (20(1X,F8.4))
99990 FORMAT (/' DIF = ',F8.4)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' SCALE = ',F8.4)
END
Program Data
SB04OD EXAMPLE PROGRAM DATA
3 2 R N D
1.6 -3.1 1.9
-3.8 4.2 2.4
0.5 2.2 -4.5
1.1 0.1
-1.3 -3.1
-2.0 28.9
-5.7 -11.8
12.9 -31.7
2.5 0.1 1.7
-2.5 0.0 0.9
0.1 5.1 -7.3
6.0 2.4
-3.6 2.5
0.5 23.8
-11.0 -10.4
39.5 -74.8
Program Results
SB04OD EXAMPLE PROGRAM RESULTS The solution matrix L is -0.7538 -1.6210 2.1778 1.7005 -3.5029 2.7961 The solution matrix R is 1.3064 2.7989 0.3698 -5.3376 -0.8767 6.7500 The left transformation matrix P is -0.3093 -0.9502 0.0383 0.9366 -0.2974 0.1851 -0.1645 0.0932 0.9820 The right transformation matrix Q is -0.6097 -0.7920 -0.0314 0.6310 -0.5090 0.5854 0.4796 -0.3371 -0.8102 The left transformation matrix U is -0.8121 0.5835 0.5835 0.8121 The right transformation matrix V is -0.9861 0.1660 0.1660 0.9861 DIF = 0.1147
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB04OW.html 0000664 0000000 0000000 00000014115 14560147231 0020372 0 ustar 00root root 0000000 0000000
Purpose
To solve a periodic Sylvester equation
A * R - L * B = scale * C (1)
D * L - R * E = scale * F,
using Level 1 and 2 BLAS, where R and L are unknown M-by-N
matrices, (A, D), (B, E) and (C, F) are given matrix pairs of
size M-by-M, N-by-N and M-by-N, respectively, with real entries.
(A, D) and (B, E) must be in periodic Schur form, i.e. A, B are
upper quasi triangular and D, E are upper triangular. The solution
(R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output scaling
factor chosen to avoid overflow.
This routine is largely based on the LAPACK routine DTGSY2
developed by Bo Kagstrom and Peter Poromaa.
Specification
SUBROUTINE SB04OW( M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE,
$ F, LDF, SCALE, IWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N
DOUBLE PRECISION SCALE
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ E(LDE,*), F(LDF,*)
Arguments
Input/Output Parameters
M (input) INTEGER
The order of A and D, and the row dimension of C, F, R
and L. M >= 0.
N (input) INTEGER
The order of B and E, and the column dimension of C, F, R
and L. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,M)
On entry, the leading M-by-M part of this array must
contain the upper quasi triangular matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,M).
B (input) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the leading N-by-N part of this array must
contain the upper quasi triangular matrix B.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading M-by-N part of this array must
contain the right-hand-side of the first matrix equation
in (1).
On exit, the leading M-by-N part of this array contains
the solution R.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,M).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading M-by-M part of this array must
contain the upper triangular matrix D.
LDD INTEGER
The leading dimension of the array D. LDD >= MAX(1,M).
E (input) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain the upper triangular matrix E.
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,N).
F (input/output) DOUBLE PRECISION array, dimension (LDF,N)
On entry, the leading M-by-N part of this array must
contain the right-hand-side of the second matrix equation
in (1).
On exit, the leading M-by-N part of this array contains
the solution L.
LDF INTEGER
The leading dimension of the array F. LDF >= MAX(1,M).
SCALE (output) DOUBLE PRECISION
On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the arrays
C and F will hold the solutions R and L, respectively, to
a slightly perturbed system but the input matrices A, B, D
and E have not been changed. If SCALE = 0, C and F will
hold solutions to the homogeneous system with C = F = 0.
Normally, SCALE = 1.
Workspace
IWORK INTEGER array, dimension (M+N+2)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: the matrix products A*D and B*E have common or very
close eigenvalues.
Method
In matrix notation solving equation (1) corresponds to solving
Z*x = scale*b, where Z is defined as
Z = [ kron(In, A) -kron(B', Im) ] (2)
[ -kron(E', Im) kron(In, D) ],
Ik is the identity matrix of size k and X' is the transpose of X.
kron(X, Y) is the Kronecker product between the matrices X and Y.
In the process of solving (1), we solve a number of such systems
where Dim(Im), Dim(In) = 1 or 2.
References
[1] Kagstrom, B.
A Direct Method for Reordering Eigenvalues in the Generalized
Real Schur Form of a Regular Matrix Pair (A,B). M.S. Moonen
et al (eds.), Linear Algebra for Large Scale and Real-Time
Applications, Kluwer Academic Publ., pp. 195-218, 1993.
[2] Sreedhar, J. and Van Dooren, P.
A Schur approach for solving some periodic matrix equations.
U. Helmke et al (eds.), Systems and Networks: Mathematical
Theory and Applications, Akademie Verlag, Berlin, vol. 77,
pp. 339-362, 1994.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve for X either the real continuous-time Sylvester equation
op(A)*X + ISGN*X*op(B) = scale*C, (1)
or the real discrete-time Sylvester equation
op(A)*X*op(B) + ISGN*X = scale*C, (2)
where op(M) = M or M**T, and ISGN = 1 or -1. A is M-by-M and
B is N-by-N; the right hand side C and the solution X are M-by-N;
and scale is an output scale factor, set less than or equal to 1
to avoid overflow in X. The solution matrix X is overwritten
onto C.
If A and/or B are not (upper) quasi-triangular, that is, block
upper triangular with 1-by-1 and 2-by-2 diagonal blocks, they are
reduced to Schur canonical form, that is, quasi-triangular with
each 2-by-2 diagonal block having its diagonal elements equal and
its off-diagonal elements of opposite sign.
Specification
SUBROUTINE SB04PD( DICO, FACTA, FACTB, TRANA, TRANB, ISGN, M, N,
$ A, LDA, U, LDU, B, LDB, V, LDV, C, LDC, SCALE,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, FACTA, FACTB, TRANA, TRANB
INTEGER INFO, ISGN, LDA, LDB, LDC, LDU, LDV, LDWORK, M,
$ N
DOUBLE PRECISION SCALE
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ DWORK( * ), U( LDU, * ), V( LDV, * )
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the equation from which X is to be determined
as follows:
= 'C': Equation (1), continuous-time case;
= 'D': Equation (2), discrete-time case.
FACTA CHARACTER*1
Specifies whether or not the real Schur factorization
of the matrix A is supplied on entry, as follows:
= 'F': On entry, A and U contain the factors from the
real Schur factorization of the matrix A;
= 'N': The Schur factorization of A will be computed
and the factors will be stored in A and U;
= 'S': The matrix A is quasi-triangular (or Schur).
FACTB CHARACTER*1
Specifies whether or not the real Schur factorization
of the matrix B is supplied on entry, as follows:
= 'F': On entry, B and V contain the factors from the
real Schur factorization of the matrix B;
= 'N': The Schur factorization of B will be computed
and the factors will be stored in B and V;
= 'S': The matrix B is quasi-triangular (or Schur).
TRANA CHARACTER*1
Specifies the form of op(A) to be used, as follows:
= 'N': op(A) = A (No transpose);
= 'T': op(A) = A**T (Transpose);
= 'C': op(A) = A**T (Conjugate transpose = Transpose).
TRANB CHARACTER*1
Specifies the form of op(B) to be used, as follows:
= 'N': op(B) = B (No transpose);
= 'T': op(B) = B**T (Transpose);
= 'C': op(B) = B**T (Conjugate transpose = Transpose).
ISGN INTEGER
Specifies the sign of the equation as described before.
ISGN may only be 1 or -1.
Input/Output Parameters
M (input) INTEGER
The order of the matrix A, and the number of rows in the
matrices X and C. M >= 0.
N (input) INTEGER
The order of the matrix B, and the number of columns in
the matrices X and C. N >= 0.
A (input or input/output) DOUBLE PRECISION array,
dimension (LDA,M)
On entry, the leading M-by-M part of this array must
contain the matrix A. If FACTA = 'S', then A contains
a quasi-triangular matrix, and if FACTA = 'F', then A
is in Schur canonical form; the elements below the upper
Hessenberg part of the array A are not referenced.
On exit, if FACTA = 'N', and INFO = 0 or INFO >= M+1, the
leading M-by-M upper Hessenberg part of this array
contains the upper quasi-triangular matrix in Schur
canonical form from the Schur factorization of A. The
contents of array A is not modified if FACTA = 'F' or 'S'.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,M).
U (input or output) DOUBLE PRECISION array, dimension
(LDU,M)
If FACTA = 'F', then U is an input argument and on entry
the leading M-by-M part of this array must contain the
orthogonal matrix U of the real Schur factorization of A.
If FACTA = 'N', then U is an output argument and on exit,
if INFO = 0 or INFO >= M+1, it contains the orthogonal
M-by-M matrix from the real Schur factorization of A.
If FACTA = 'S', the array U is not referenced.
LDU INTEGER
The leading dimension of array U.
LDU >= MAX(1,M), if FACTA = 'F' or 'N';
LDU >= 1, if FACTA = 'S'.
B (input or input/output) DOUBLE PRECISION array,
dimension (LDB,N)
On entry, the leading N-by-N part of this array must
contain the matrix B. If FACTB = 'S', then B contains
a quasi-triangular matrix, and if FACTB = 'F', then B
is in Schur canonical form; the elements below the upper
Hessenberg part of the array B are not referenced.
On exit, if FACTB = 'N', and INFO = 0 or INFO = M+N+1,
the leading N-by-N upper Hessenberg part of this array
contains the upper quasi-triangular matrix in Schur
canonical form from the Schur factorization of B. The
contents of array B is not modified if FACTB = 'F' or 'S'.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
V (input or output) DOUBLE PRECISION array, dimension
(LDV,N)
If FACTB = 'F', then V is an input argument and on entry
the leading N-by-N part of this array must contain the
orthogonal matrix V of the real Schur factorization of B.
If FACTB = 'N', then V is an output argument and on exit,
if INFO = 0 or INFO = M+N+1, it contains the orthogonal
N-by-N matrix from the real Schur factorization of B.
If FACTB = 'S', the array V is not referenced.
LDV INTEGER
The leading dimension of array V.
LDV >= MAX(1,N), if FACTB = 'F' or 'N';
LDV >= 1, if FACTB = 'S'.
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading M-by-N part of this array must
contain the right hand side matrix C.
On exit, if INFO = 0 or INFO = M+N+1, the leading M-by-N
part of this array contains the solution matrix X.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,M).
SCALE (output) DOUBLE PRECISION
The scale factor, scale, set less than or equal to 1 to
prevent the solution overflowing.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or M+N+1, then: DWORK(1) returns the
optimal value of LDWORK; if FACTA = 'N', DWORK(1+i) and
DWORK(1+M+i), i = 1,...,M, contain the real and imaginary
parts, respectively, of the eigenvalues of A; and, if
FACTB = 'N', DWORK(1+f+j) and DWORK(1+f+N+j), j = 1,...,N,
with f = 2*M if FACTA = 'N', and f = 0, otherwise, contain
the real and imaginary parts, respectively, of the
eigenvalues of B.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( 1, a+MAX( c, b+d, b+e ) ),
where a = 1+2*M, if FACTA = 'N',
a = 0, if FACTA <> 'N',
b = 2*N, if FACTB = 'N', FACTA = 'N',
b = 1+2*N, if FACTB = 'N', FACTA <> 'N',
b = 0, if FACTB <> 'N',
c = 3*M, if FACTA = 'N',
c = M, if FACTA = 'F',
c = 0, if FACTA = 'S',
d = 3*N, if FACTB = 'N',
d = N, if FACTB = 'F',
d = 0, if FACTB = 'S',
e = M, if DICO = 'C', FACTA <> 'S',
e = 0, if DICO = 'C', FACTA = 'S',
e = 2*M, if DICO = 'D'.
An upper bound is
LDWORK = 1+2*M+MAX( 3*M, 5*N, 2*N+2*M ).
For good performance, LDWORK should be larger, e.g.,
LDWORK = 1+2*M+MAX( 3*M, 5*N, 2*N+2*M, 2*N+M*N ).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= i: if INFO = i, i = 1,...,M, the QR algorithm failed
to compute all the eigenvalues of the matrix A
(see LAPACK Library routine DGEES); the elements
2+i:1+M and 2+i+M:1+2*M of DWORK contain the real
and imaginary parts, respectively, of the
eigenvalues of A which have converged, and the
array A contains the partially converged Schur form;
= M+j: if INFO = M+j, j = 1,...,N, the QR algorithm
failed to compute all the eigenvalues of the matrix
B (see LAPACK Library routine DGEES); the elements
2+f+j:1+f+N and 2+f+j+N:1+f+2*N of DWORK contain the
real and imaginary parts, respectively, of the
eigenvalues of B which have converged, and the
array B contains the partially converged Schur form;
as defined for the parameter DWORK,
f = 2*M, if FACTA = 'N',
f = 0, if FACTA <> 'N';
= M+N+1: if DICO = 'C', and the matrices A and -ISGN*B
have common or very close eigenvalues, or
if DICO = 'D', and the matrices A and -ISGN*B have
almost reciprocal eigenvalues (that is, if lambda(i)
and mu(j) are eigenvalues of A and -ISGN*B, then
lambda(i) = 1/mu(j) for some i and j);
perturbed values were used to solve the equation
(but the matrices A and B are unchanged).
Method
An extension and refinement of the algorithms in [1,2] is used.
If the matrices A and/or B are not quasi-triangular (see PURPOSE),
they are reduced to Schur canonical form
A = U*S*U', B = V*T*V',
where U, V are orthogonal, and S, T are block upper triangular
with 1-by-1 and 2-by-2 blocks on their diagonal. The right hand
side matrix C is updated accordingly,
C = U'*C*V;
then, the solution matrix X of the "reduced" Sylvester equation
(with A and B in (1) or (2) replaced by S and T, respectively),
is computed column-wise via a back substitution scheme. A set of
equivalent linear algebraic systems of equations of order at most
four are formed and solved using Gaussian elimination with
complete pivoting. Finally, the solution X of the original
equation is obtained from the updating formula
X = U*X*V'.
If A and/or B are already quasi-triangular (or in Schur form), the
initial factorizations and the corresponding updating steps are
omitted.
References
[1] Bartels, R.H. and Stewart, G.W. T
Solution of the matrix equation A X + XB = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
[2] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
Numerical Aspects
The algorithm is stable and reliable, since orthogonal transformations and Gaussian elimination with complete pivoting are used. If INFO = M+N+1, the Sylvester equation is numerically singular.Further Comments
NoneExample
Program Text
* SB04PD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, NMAX
PARAMETER ( MMAX = 20, NMAX = 20 )
INTEGER LDA, LDB, LDC, LDU, LDV
PARAMETER ( LDA = MMAX, LDB = NMAX, LDC = MMAX,
$ LDU = MMAX, LDV = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 1 + 2*MMAX + MAX( 3*MMAX, 5*NMAX,
$ 2*( NMAX + MMAX ) ) )
* .. Local Scalars ..
CHARACTER DICO, FACTA, FACTB, TRANA, TRANB
INTEGER I, INFO, ISGN, J, M, N
DOUBLE PRECISION SCALE
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,MMAX), B(LDB,NMAX), C(LDC,NMAX),
$ DWORK(LDWORK), U(LDU,MMAX), V(LDV,NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL SB04PD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, N, ISGN, DICO, FACTA, FACTB, TRANA, TRANB
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) M
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,M ), I = 1,M )
IF ( LSAME( FACTA, 'F' ) )
$ READ ( NIN, FMT = * ) ( ( U(I,J), J = 1,M ), I = 1,M )
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) N
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( FACTB, 'F' ) )
$ READ ( NIN, FMT = * ) ( ( V(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,M )
* Find the solution matrix X.
CALL SB04PD( DICO, FACTA, FACTB, TRANA, TRANB, ISGN, M, N,
$ A, LDA, U, LDU, B, LDB, V, LDV, C, LDC, SCALE,
$ DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 )
$ WRITE ( NOUT, FMT = 99998 ) INFO
IF ( INFO.EQ.0 .OR. INFO.EQ.M+N+1 ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, M
WRITE ( NOUT, FMT = 99996 ) ( C(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 ) SCALE
IF ( LSAME( FACTA, 'N' ) ) THEN
WRITE ( NOUT, FMT = 99994 )
DO 40 I = 1, M
WRITE ( NOUT, FMT = 99996 ) ( U(I,J), J = 1,M )
40 CONTINUE
END IF
IF ( LSAME( FACTB, 'N' ) ) THEN
WRITE ( NOUT, FMT = 99993 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( V(I,J), J = 1,N )
60 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB04PD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB04PD = ',I2)
99997 FORMAT (' The solution matrix X is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' Scaling factor = ',F8.4)
99994 FORMAT (/' The orthogonal matrix U is ')
99993 FORMAT (/' The orthogonal matrix V is ')
99992 FORMAT (/' M is out of range.',/' M = ',I5)
99991 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
SB04PD EXAMPLE PROGRAM DATA 3 2 1 D N N N N 2.0 1.0 3.0 0.0 2.0 1.0 6.0 1.0 2.0 2.0 1.0 1.0 6.0 2.0 1.0 1.0 4.0 0.0 5.0Program Results
SB04PD EXAMPLE PROGRAM RESULTS The solution matrix X is -0.3430 0.1995 -0.1856 0.4192 0.6922 -0.2952 Scaling factor = 1.0000 The orthogonal matrix U is 0.5396 -0.7797 0.3178 0.1954 -0.2512 -0.9480 -0.8190 -0.5736 -0.0168 The orthogonal matrix V is -0.9732 -0.2298 0.2298 -0.9732
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB04PX.html 0000664 0000000 0000000 00000011314 14560147231 0020372 0 ustar 00root root 0000000 0000000
Purpose
To solve for the N1-by-N2 matrix X, 1 <= N1,N2 <= 2, in
op(TL)*X*op(TR) + ISGN*X = SCALE*B,
where TL is N1-by-N1, TR is N2-by-N2, B is N1-by-N2, and ISGN = 1
or -1. op(T) = T or T', where T' denotes the transpose of T.
Specification
SUBROUTINE SB04PX( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
$ LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO )
C .. Scalar Arguments ..
LOGICAL LTRANL, LTRANR
INTEGER INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2
DOUBLE PRECISION SCALE, XNORM
C .. Array Arguments ..
DOUBLE PRECISION B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),
$ X( LDX, * )
Arguments
Mode Parameters
LTRANL LOGICAL
Specifies the form of op(TL) to be used, as follows:
= .FALSE.: op(TL) = TL,
= .TRUE. : op(TL) = TL'.
LTRANR LOGICAL
Specifies the form of op(TR) to be used, as follows:
= .FALSE.: op(TR) = TR,
= .TRUE. : op(TR) = TR'.
ISGN INTEGER
Specifies the sign of the equation as described before.
ISGN may only be 1 or -1.
Input/Output Parameters
N1 (input) INTEGER
The order of matrix TL. N1 may only be 0, 1 or 2.
N2 (input) INTEGER
The order of matrix TR. N2 may only be 0, 1 or 2.
TL (input) DOUBLE PRECISION array, dimension (LDTL,N1)
The leading N1-by-N1 part of this array must contain the
matrix TL.
LDTL INTEGER
The leading dimension of array TL. LDTL >= MAX(1,N1).
TR (input) DOUBLE PRECISION array, dimension (LDTR,N2)
The leading N2-by-N2 part of this array must contain the
matrix TR.
LDTR INTEGER
The leading dimension of array TR. LDTR >= MAX(1,N2).
B (input) DOUBLE PRECISION array, dimension (LDB,N2)
The leading N1-by-N2 part of this array must contain the
right-hand side of the equation.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N1).
SCALE (output) DOUBLE PRECISION
The scale factor. SCALE is chosen less than or equal to 1
to prevent the solution overflowing.
X (output) DOUBLE PRECISION array, dimension (LDX,N2)
The leading N1-by-N2 part of this array contains the
solution of the equation.
Note that X may be identified with B in the calling
statement.
LDX INTEGER
The leading dimension of array X. LDX >= MAX(1,N1).
XNORM (output) DOUBLE PRECISION
The infinity-norm of the solution.
Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: if TL and -ISGN*TR have almost reciprocal
eigenvalues, so TL or TR is perturbed to get a
nonsingular equation.
NOTE: In the interests of speed, this routine does not
check the inputs for errors.
Method
The equivalent linear algebraic system of equations is formed and solved using Gaussian elimination with complete pivoting.References
[1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
Numerical Aspects
The algorithm is stable and reliable, since Gaussian elimination with complete pivoting is used.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve for X the discrete-time Sylvester equation
op(A)*X*op(B) + ISGN*X = scale*C,
where op(A) = A or A**T, A and B are both upper quasi-triangular,
and ISGN = 1 or -1. A is M-by-M and B is N-by-N; the right hand
side C and the solution X are M-by-N; and scale is an output scale
factor, set less than or equal to 1 to avoid overflow in X. The
solution matrix X is overwritten onto C.
A and B must be in Schur canonical form (as returned by LAPACK
Library routine DHSEQR), that is, block upper triangular with
1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has
its diagonal elements equal and its off-diagonal elements of
opposite sign.
Specification
SUBROUTINE SB04PY( TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C,
$ LDC, SCALE, DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TRANA, TRANB
INTEGER INFO, ISGN, LDA, LDB, LDC, M, N
DOUBLE PRECISION SCALE
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ DWORK( * )
Arguments
Mode Parameters
TRANA CHARACTER*1
Specifies the form of op(A) to be used, as follows:
= 'N': op(A) = A (No transpose);
= 'T': op(A) = A**T (Transpose);
= 'C': op(A) = A**T (Conjugate transpose = Transpose).
TRANB CHARACTER*1
Specifies the form of op(B) to be used, as follows:
= 'N': op(B) = B (No transpose);
= 'T': op(B) = B**T (Transpose);
= 'C': op(B) = B**T (Conjugate transpose = Transpose).
ISGN INTEGER
Specifies the sign of the equation as described before.
ISGN may only be 1 or -1.
Input/Output Parameters
M (input) INTEGER
The order of the matrix A, and the number of rows in the
matrices X and C. M >= 0.
N (input) INTEGER
The order of the matrix B, and the number of columns in
the matrices X and C. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,M)
The leading M-by-M part of this array must contain the
upper quasi-triangular matrix A, in Schur canonical form.
The part of A below the first sub-diagonal is not
referenced.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,M).
B (input) DOUBLE PRECISION array, dimension (LDB,N)
The leading N-by-N part of this array must contain the
upper quasi-triangular matrix B, in Schur canonical form.
The part of B below the first sub-diagonal is not
referenced.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading M-by-N part of this array must
contain the right hand side matrix C.
On exit, if INFO >= 0, the leading M-by-N part of this
array contains the solution matrix X.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,M).
SCALE (output) DOUBLE PRECISION
The scale factor, scale, set less than or equal to 1 to
prevent the solution overflowing.
Workspace
DWORK DOUBLE PRECISION array, dimension (2*M)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: A and -ISGN*B have almost reciprocal eigenvalues;
perturbed values were used to solve the equation
(but the matrices A and B are unchanged).
Method
The solution matrix X is computed column-wise via a back substitution scheme, an extension and refinement of the algorithm in [1], similar to that used in [2] for continuous-time Sylvester equations. A set of equivalent linear algebraic systems of equations of order at most four are formed and solved using Gaussian elimination with complete pivoting.References
[1] Bartels, R.H. and Stewart, G.W. T
Solution of the matrix equation A X + XB = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
[2] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
Numerical Aspects
The algorithm is stable and reliable, since Gaussian elimination with complete pivoting is used.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve for X the discrete-time Sylvester equation
X + AXB = C,
where A, B, C and X are general N-by-N, M-by-M, N-by-M and
N-by-M matrices respectively. A Hessenberg-Schur method, which
reduces A to upper Hessenberg form, H = U'AU, and B' to real
Schur form, S = Z'B'Z (with U, Z orthogonal matrices), is used.
Specification
SUBROUTINE SB04QD( N, M, A, LDA, B, LDB, C, LDC, Z, LDZ, IWORK,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDWORK, LDZ, M, N
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), Z(LDZ,*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
M (input) INTEGER
The order of the matrix B. M >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the coefficient matrix A of the equation.
On exit, the leading N-by-N upper Hessenberg part of this
array contains the matrix H, and the remainder of the
leading N-by-N part, together with the elements 2,3,...,N
of array DWORK, contain the orthogonal transformation
matrix U (stored in factored form).
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading M-by-M part of this array must
contain the coefficient matrix B of the equation.
On exit, the leading M-by-M part of this array contains
the quasi-triangular Schur factor S of the matrix B'.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,M).
C (input/output) DOUBLE PRECISION array, dimension (LDC,M)
On entry, the leading N-by-M part of this array must
contain the coefficient matrix C of the equation.
On exit, the leading N-by-M part of this array contains
the solution matrix X of the problem.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,N).
Z (output) DOUBLE PRECISION array, dimension (LDZ,M)
The leading M-by-M part of this array contains the
orthogonal matrix Z used to transform B' to real upper
Schur form.
LDZ INTEGER
The leading dimension of array Z. LDZ >= MAX(1,M).
Workspace
IWORK INTEGER array, dimension (4*N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK, and DWORK(2), DWORK(3),..., DWORK(N) contain
the scalar factors of the elementary reflectors used to
reduce A to upper Hessenberg form, as returned by LAPACK
Library routine DGEHRD.
LDWORK INTEGER
The length of the array DWORK.
LDWORK = MAX(1, 2*N*N + 9*N, 5*M, N + M).
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message related to LDWORK is issued by
XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, 1 <= i <= M, the QR algorithm failed to
compute all the eigenvalues of B (see LAPACK Library
routine DGEES);
> M: if a singular matrix was encountered whilst solving
for the (INFO-M)-th column of matrix X.
Method
The matrix A is transformed to upper Hessenberg form H = U'AU by
the orthogonal transformation matrix U; matrix B' is transformed
to real upper Schur form S = Z'B'Z using the orthogonal
transformation matrix Z. The matrix C is also multiplied by the
transformations, F = U'CZ, and the solution matrix Y of the
transformed system
Y + HYS' = F
is computed by back substitution. Finally, the matrix Y is then
multiplied by the orthogonal transformation matrices, X = UYZ', in
order to obtain the solution matrix X to the original problem.
References
[1] Golub, G.H., Nash, S. and Van Loan, C.F.
A Hessenberg-Schur method for the problem AX + XB = C.
IEEE Trans. Auto. Contr., AC-24, pp. 909-913, 1979.
[2] Sima, V.
Algorithms for Linear-quadratic Optimization.
Marcel Dekker, Inc., New York, 1996.
Numerical Aspects
3 3 2 2 The algorithm requires about (5/3) N + 10 M + 5 N M + 2.5 M N operations and is backward stable.Further Comments
NoneExample
Program Text
* SB04QD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX
PARAMETER ( NMAX = 20, MMAX = 20 )
INTEGER LDA, LDB, LDC, LDZ
PARAMETER ( LDA = NMAX, LDB = MMAX, LDC = NMAX,
$ LDZ = MMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = 4*NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( 1, 2*NMAX*NMAX+9*NMAX, 5*MMAX,
$ NMAX+MMAX ) )
* .. Local Scalars ..
INTEGER I, INFO, J, M, N
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,MMAX),
$ DWORK(LDWORK), Z(LDZ,MMAX)
INTEGER IWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL SB04QD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,M )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,M ), I = 1,N )
* Find the solution matrix X.
CALL SB04QD( N, M, A, LDA, B, LDB, C, LDC, Z, LDZ, IWORK,
$ DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( C(I,J), J = 1,M )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 40 I = 1, M
WRITE ( NOUT, FMT = 99996 ) ( Z(I,J), J = 1,M )
40 CONTINUE
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB04QD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB04QD = ',I2)
99997 FORMAT (' The solution matrix X is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The orthogonal matrix Z is ')
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' M is out of range.',/' M = ',I5)
END
Program Data
SB04QD EXAMPLE PROGRAM DATA 3 3 1.0 2.0 3.0 6.0 7.0 8.0 9.0 2.0 3.0 7.0 2.0 3.0 2.0 1.0 2.0 3.0 4.0 1.0 271.0 135.0 147.0 923.0 494.0 482.0 578.0 383.0 287.0Program Results
SB04QD EXAMPLE PROGRAM RESULTS The solution matrix X is 2.0000 3.0000 6.0000 4.0000 7.0000 1.0000 5.0000 3.0000 2.0000 The orthogonal matrix Z is 0.8337 0.5204 -0.1845 0.3881 -0.7900 -0.4746 0.3928 -0.3241 0.8606
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB04QR.html 0000664 0000000 0000000 00000006532 14560147231 0020373 0 ustar 00root root 0000000 0000000
Purpose
To solve a linear algebraic system of order M whose coefficient matrix has zeros below the third subdiagonal and zero elements on the third subdiagonal with even column indices. The matrix is stored compactly, row-wise.Specification
SUBROUTINE SB04QR( M, D, IPR, INFO )
C .. Scalar Arguments ..
INTEGER INFO, M
C .. Array Arguments ..
INTEGER IPR(*)
DOUBLE PRECISION D(*)
Arguments
Input/Output Parameters
M (input) INTEGER
The order of the system. M >= 0, M even.
Note that parameter M should have twice the value in the
original problem (see SLICOT Library routine SB04QU).
D (input/output) DOUBLE PRECISION array, dimension
(M*M/2+4*M)
On entry, the first M*M/2 + 3*M elements of this array
must contain the coefficient matrix, stored compactly,
row-wise, and the next M elements must contain the right
hand side of the linear system, as set by SLICOT Library
routine SB04QU.
On exit, the content of this array is updated, the last M
elements containing the solution with components
interchanged (see IPR).
IPR (output) INTEGER array, dimension (2*M)
The leading M elements contain information about the
row interchanges performed for solving the system.
Specifically, the i-th component of the solution is
specified by IPR(i).
Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: if a singular matrix was encountered.
Method
Gaussian elimination with partial pivoting is used. The rows of the matrix are not actually permuted, only their indices are interchanged in array IPR.References
[1] Golub, G.H., Nash, S. and Van Loan, C.F.
A Hessenberg-Schur method for the problem AX + XB = C.
IEEE Trans. Auto. Contr., AC-24, pp. 909-913, 1979.
[2] Sima, V.
Algorithms for Linear-quadratic Optimization.
Marcel Dekker, Inc., New York, 1996.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To construct and solve a linear algebraic system of order 2*M whose coefficient matrix has zeros below the third subdiagonal, and zero elements on the third subdiagonal with even column indices. Such systems appear when solving discrete-time Sylvester equations using the Hessenberg-Schur method.Specification
SUBROUTINE SB04QU( N, M, IND, A, LDA, B, LDB, C, LDC, D, IPR,
$ INFO )
C .. Scalar Arguments ..
INTEGER INFO, IND, LDA, LDB, LDC, M, N
C .. Array Arguments ..
INTEGER IPR(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix B. N >= 0.
M (input) INTEGER
The order of the matrix A. M >= 0.
IND (input) INTEGER
IND and IND - 1 specify the indices of the columns in C
to be computed. IND > 1.
A (input) DOUBLE PRECISION array, dimension (LDA,M)
The leading M-by-M part of this array must contain an
upper Hessenberg matrix.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,M).
B (input) DOUBLE PRECISION array, dimension (LDB,N)
The leading N-by-N part of this array must contain a
matrix in real Schur form.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading M-by-N part of this array must
contain the coefficient matrix C of the equation.
On exit, the leading M-by-N part of this array contains
the matrix C with columns IND-1 and IND updated.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,M).
Workspace
D DOUBLE PRECISION array, dimension (2*M*M+8*M) IPR INTEGER array, dimension (4*M)Error Indicator
INFO INTEGER
= 0: successful exit;
> 0: if INFO = IND, a singular matrix was encountered.
Method
A special linear algebraic system of order 2*M, whose coefficient matrix has zeros below the third subdiagonal and zero elements on the third subdiagonal with even column indices, is constructed and solved. The coefficient matrix is stored compactly, row-wise.References
[1] Golub, G.H., Nash, S. and Van Loan, C.F.
A Hessenberg-Schur method for the problem AX + XB = C.
IEEE Trans. Auto. Contr., AC-24, pp. 909-913, 1979.
[2] Sima, V.
Algorithms for Linear-quadratic Optimization.
Marcel Dekker, Inc., New York, 1996.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To construct and solve a linear algebraic system of order M whose coefficient matrix is in upper Hessenberg form. Such systems appear when solving discrete-time Sylvester equations using the Hessenberg-Schur method.Specification
SUBROUTINE SB04QY( N, M, IND, A, LDA, B, LDB, C, LDC, D, IPR,
$ INFO )
C .. Scalar Arguments ..
INTEGER INFO, IND, LDA, LDB, LDC, M, N
C .. Array Arguments ..
INTEGER IPR(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix B. N >= 0.
M (input) INTEGER
The order of the matrix A. M >= 0.
IND (input) INTEGER
The index of the column in C to be computed. IND >= 1.
A (input) DOUBLE PRECISION array, dimension (LDA,M)
The leading M-by-M part of this array must contain an
upper Hessenberg matrix.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,M).
B (input) DOUBLE PRECISION array, dimension (LDB,N)
The leading N-by-N part of this array must contain a
matrix in real Schur form.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading M-by-N part of this array must
contain the coefficient matrix C of the equation.
On exit, the leading M-by-N part of this array contains
the matrix C with column IND updated.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,M).
Workspace
D DOUBLE PRECISION array, dimension (M*(M+1)/2+2*M) IPR INTEGER array, dimension (2*M)Error Indicator
INFO INTEGER
= 0: successful exit;
> 0: if INFO = IND, a singular matrix was encountered.
Method
A special linear algebraic system of order M, with coefficient matrix in upper Hessenberg form is constructed and solved. The coefficient matrix is stored compactly, row-wise.References
[1] Golub, G.H., Nash, S. and Van Loan, C.F.
A Hessenberg-Schur method for the problem AX + XB = C.
IEEE Trans. Auto. Contr., AC-24, pp. 909-913, 1979.
[2] Sima, V.
Algorithms for Linear-quadratic Optimization.
Marcel Dekker, Inc., New York, 1996.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve for X the discrete-time Sylvester equation
X + AXB = C,
with at least one of the matrices A or B in Schur form and the
other in Hessenberg or Schur form (both either upper or lower);
A, B, C and X are N-by-N, M-by-M, N-by-M, and N-by-M matrices,
respectively.
Specification
SUBROUTINE SB04RD( ABSCHU, ULA, ULB, N, M, A, LDA, B, LDB, C,
$ LDC, TOL, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER ABSCHU, ULA, ULB
INTEGER INFO, LDA, LDB, LDC, LDWORK, M, N
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*)
Arguments
Mode Parameters
ABSCHU CHARACTER*1
Indicates whether A and/or B is/are in Schur or
Hessenberg form as follows:
= 'A': A is in Schur form, B is in Hessenberg form;
= 'B': B is in Schur form, A is in Hessenberg form;
= 'S': Both A and B are in Schur form.
ULA CHARACTER*1
Indicates whether A is in upper or lower Schur form or
upper or lower Hessenberg form as follows:
= 'U': A is in upper Hessenberg form if ABSCHU = 'B' and
upper Schur form otherwise;
= 'L': A is in lower Hessenberg form if ABSCHU = 'B' and
lower Schur form otherwise.
ULB CHARACTER*1
Indicates whether B is in upper or lower Schur form or
upper or lower Hessenberg form as follows:
= 'U': B is in upper Hessenberg form if ABSCHU = 'A' and
upper Schur form otherwise;
= 'L': B is in lower Hessenberg form if ABSCHU = 'A' and
lower Schur form otherwise.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
M (input) INTEGER
The order of the matrix B. M >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
coefficient matrix A of the equation.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading M-by-M part of this array must contain the
coefficient matrix B of the equation.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,M).
C (input/output) DOUBLE PRECISION array, dimension (LDC,M)
On entry, the leading N-by-M part of this array must
contain the coefficient matrix C of the equation.
On exit, if INFO = 0, the leading N-by-M part of this
array contains the solution matrix X of the problem.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,N).
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used to test for near singularity in
the Sylvester equation. If the user sets TOL > 0, then the
given value of TOL is used as a lower bound for the
reciprocal condition number; a matrix whose estimated
condition number is less than 1/TOL is considered to be
nonsingular. If the user sets TOL <= 0, then a default
tolerance, defined by TOLDEF = EPS, is used instead, where
EPS is the machine precision (see LAPACK Library routine
DLAMCH).
This parameter is not referenced if ABSCHU = 'S',
ULA = 'U', and ULB = 'U'.
Workspace
IWORK INTEGER array, dimension (2*MAX(M,N))
This parameter is not referenced if ABSCHU = 'S',
ULA = 'U', and ULB = 'U'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK.
LDWORK = 2*N, if ABSCHU = 'S', ULA = 'U', and ULB = 'U';
LDWORK = 2*MAX(M,N)*(4 + 2*MAX(M,N)), otherwise.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if a (numerically) singular matrix T was encountered
during the computation of the solution matrix X.
That is, the estimated reciprocal condition number
of T is less than or equal to TOL.
Method
Matrices A and B are assumed to be in (upper or lower) Hessenberg or Schur form (with at least one of them in Schur form). The solution matrix X is then computed by rows or columns via the back substitution scheme proposed by Golub, Nash and Van Loan (see [1]), which involves the solution of triangular systems of equations that are constructed recursively and which may be nearly singular if A and -B have almost reciprocal eigenvalues. If near singularity is detected, then the routine returns with the Error Indicator (INFO) set to 1.References
[1] Golub, G.H., Nash, S. and Van Loan, C.F.
A Hessenberg-Schur method for the problem AX + XB = C.
IEEE Trans. Auto. Contr., AC-24, pp. 909-913, 1979.
[2] Sima, V.
Algorithms for Linear-quadratic Optimization.
Marcel Dekker, Inc., New York, 1996.
Numerical Aspects
2 2
The algorithm requires approximately 5M N + 0.5MN operations in
2 2
the worst case and 2.5M N + 0.5MN operations in the best case
(where M is the order of the matrix in Hessenberg form and N is
the order of the matrix in Schur form) and is mixed stable (see
[1]).
Further Comments
NoneExample
Program Text
* SB04RD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX
PARAMETER ( NMAX = 20, MMAX = 20 )
INTEGER LDA, LDB, LDC
PARAMETER ( LDA = NMAX, LDB = MMAX, LDC = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 2*( MAX( NMAX,MMAX ) )*
$ ( 4+2*( MAX( NMAX,MMAX ) ) ) )
INTEGER LIWORK
PARAMETER ( LIWORK = 2*MAX( NMAX,MMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, J, M, N
CHARACTER*1 ABSCHU, ULA, ULB
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,MMAX),
$ DWORK(LDWORK)
INTEGER IWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL SB04RD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, TOL, ULA, ULB, ABSCHU
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,M )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,M ), I = 1,N )
* Find the solution matrix X.
CALL SB04RD( ABSCHU, ULA, ULB, N, M, A, LDA, B, LDB, C,
$ LDC, TOL, IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( C(I,J), J = 1,M )
20 CONTINUE
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB04RD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB04RD = ',I2)
99997 FORMAT (' The solution matrix X is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' N is out of range.',/' N = ',I5)
99994 FORMAT (/' M is out of range.',/' M = ',I5)
END
Program Data
SB04RD EXAMPLE PROGRAM DATA 5 5 0.0 U U B 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 1.0 0.0 2.0 3.0 4.0 5.0 0.0 0.0 6.0 7.0 8.0 0.0 0.0 0.0 9.0 1.0 1.0 2.0 3.0 4.0 5.0 0.0 1.0 2.0 3.0 4.0 0.0 0.0 1.0 2.0 3.0 0.0 0.0 0.0 1.0 -5.0 0.0 0.0 0.0 4.0 1.0 2.0 4.0 10.0 40.0 7.0 6.0 20.0 40.0 74.0 38.0 0.0 2.0 8.0 36.0 2.0 0.0 0.0 6.0 52.0 -9.0 0.0 0.0 0.0 13.0 -43.0Program Results
SB04RD EXAMPLE PROGRAM RESULTS The solution matrix X is 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB04RV.html 0000664 0000000 0000000 00000010003 14560147231 0020364 0 ustar 00root root 0000000 0000000
Purpose
To construct the right-hand sides D for a system of equations in quasi-Hessenberg form solved via SB04RX (case with 2 right-hand sides).Specification
SUBROUTINE SB04RV( ABSCHR, UL, N, M, C, LDC, INDX, AB, LDAB, BA,
$ LDBA, D, DWORK )
C .. Scalar Arguments ..
CHARACTER ABSCHR, UL
INTEGER INDX, LDAB, LDBA, LDC, M, N
C .. Array Arguments ..
DOUBLE PRECISION AB(LDAB,*), BA(LDBA,*), C(LDC,*), D(*), DWORK(*)
Arguments
Mode Parameters
ABSCHR CHARACTER*1
Indicates whether AB contains A or B, as follows:
= 'A': AB contains A;
= 'B': AB contains B.
UL CHARACTER*1
Indicates whether AB is upper or lower Hessenberg matrix,
as follows:
= 'U': AB is upper Hessenberg;
= 'L': AB is lower Hessenberg.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
M (input) INTEGER
The order of the matrix B. M >= 0.
C (input) DOUBLE PRECISION array, dimension (LDC,M)
The leading N-by-M part of this array must contain both
the not yet modified part of the coefficient matrix C of
the Sylvester equation X + AXB = C, and both the currently
computed part of the solution of the Sylvester equation.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,N).
INDX (input) INTEGER
The position of the first column/row of C to be used in
the construction of the right-hand side D.
AB (input) DOUBLE PRECISION array, dimension (LDAB,*)
The leading N-by-N or M-by-M part of this array must
contain either A or B of the Sylvester equation
X + AXB = C.
LDAB INTEGER
The leading dimension of array AB.
LDAB >= MAX(1,N) or LDAB >= MAX(1,M) (depending on
ABSCHR = 'A' or ABSCHR = 'B', respectively).
BA (input) DOUBLE PRECISION array, dimension (LDBA,*)
The leading N-by-N or M-by-M part of this array must
contain either A or B of the Sylvester equation
X + AXB = C, the matrix not contained in AB.
LDBA INTEGER
The leading dimension of array BA.
LDBA >= MAX(1,N) or LDBA >= MAX(1,M) (depending on
ABSCHR = 'B' or ABSCHR = 'A', respectively).
D (output) DOUBLE PRECISION array, dimension (*)
The leading 2*N or 2*M part of this array (depending on
ABSCHR = 'B' or ABSCHR = 'A', respectively) contains the
right-hand side stored as a matrix with two rows.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
where LDWORK is equal to 2*N or 2*M (depending on
ABSCHR = 'B' or ABSCHR = 'A', respectively).
Numerical Aspects
None.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To construct the right-hand side D for a system of equations in Hessenberg form solved via SB04RY (case with 1 right-hand side).Specification
SUBROUTINE SB04RW( ABSCHR, UL, N, M, C, LDC, INDX, AB, LDAB, BA,
$ LDBA, D, DWORK )
C .. Scalar Arguments ..
CHARACTER ABSCHR, UL
INTEGER INDX, LDAB, LDBA, LDC, M, N
C .. Array Arguments ..
DOUBLE PRECISION AB(LDAB,*), BA(LDBA,*), C(LDC,*), D(*), DWORK(*)
Arguments
Mode Parameters
ABSCHR CHARACTER*1
Indicates whether AB contains A or B, as follows:
= 'A': AB contains A;
= 'B': AB contains B.
UL CHARACTER*1
Indicates whether AB is upper or lower Hessenberg matrix,
as follows:
= 'U': AB is upper Hessenberg;
= 'L': AB is lower Hessenberg.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
M (input) INTEGER
The order of the matrix B. M >= 0.
C (input) DOUBLE PRECISION array, dimension (LDC,M)
The leading N-by-M part of this array must contain both
the not yet modified part of the coefficient matrix C of
the Sylvester equation X + AXB = C, and both the currently
computed part of the solution of the Sylvester equation.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,N).
INDX (input) INTEGER
The position of the column/row of C to be used in the
construction of the right-hand side D.
AB (input) DOUBLE PRECISION array, dimension (LDAB,*)
The leading N-by-N or M-by-M part of this array must
contain either A or B of the Sylvester equation
X + AXB = C.
LDAB INTEGER
The leading dimension of array AB.
LDAB >= MAX(1,N) or LDAB >= MAX(1,M) (depending on
ABSCHR = 'A' or ABSCHR = 'B', respectively).
BA (input) DOUBLE PRECISION array, dimension (LDBA,*)
The leading N-by-N or M-by-M part of this array must
contain either A or B of the Sylvester equation
X + AXB = C, the matrix not contained in AB.
LDBA INTEGER
The leading dimension of array BA.
LDBA >= MAX(1,N) or LDBA >= MAX(1,M) (depending on
ABSCHR = 'B' or ABSCHR = 'A', respectively).
D (output) DOUBLE PRECISION array, dimension (*)
The leading N or M part of this array (depending on
ABSCHR = 'B' or ABSCHR = 'A', respectively) contains the
right-hand side.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
where LDWORK is equal to N or M (depending on ABSCHR = 'B'
or ABSCHR = 'A', respectively).
Numerical Aspects
None.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve a system of equations in quasi-Hessenberg form (Hessenberg form plus two consecutive offdiagonals) with two right-hand sides.Specification
SUBROUTINE SB04RX( RC, UL, M, A, LDA, LAMBD1, LAMBD2, LAMBD3,
$ LAMBD4, D, TOL, IWORK, DWORK, LDDWOR, INFO )
C .. Scalar Arguments ..
CHARACTER RC, UL
INTEGER INFO, LDA, LDDWOR, M
DOUBLE PRECISION LAMBD1, LAMBD2, LAMBD3, LAMBD4, TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), D(*), DWORK(LDDWOR,*)
Arguments
Mode Parameters
RC CHARACTER*1
Indicates processing by columns or rows, as follows:
= 'R': Row transformations are applied;
= 'C': Column transformations are applied.
UL CHARACTER*1
Indicates whether A is upper or lower Hessenberg matrix,
as follows:
= 'U': A is upper Hessenberg;
= 'L': A is lower Hessenberg.
Input/Output Parameters
M (input) INTEGER
The order of the matrix A. M >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,M)
The leading M-by-M part of this array must contain a
matrix A in Hessenberg form.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,M).
LAMBD1, (input) DOUBLE PRECISION
LAMBD2, These variables must contain the 2-by-2 block to be
LAMBD3, multiplied to the elements of A.
LAMBD4
D (input/output) DOUBLE PRECISION array, dimension (2*M)
On entry, this array must contain the two right-hand
side vectors of the quasi-Hessenberg system, stored
row-wise.
On exit, if INFO = 0, this array contains the two solution
vectors of the quasi-Hessenberg system, stored row-wise.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used to test for near singularity of
the triangular factor R of the quasi-Hessenberg matrix.
A matrix whose estimated condition number is less
than 1/TOL is considered to be nonsingular.
Workspace
IWORK INTEGER array, dimension (2*M)
DWORK DOUBLE PRECISION array, dimension (LDDWOR,2*M+3)
The leading 2*M-by-2*M part of this array is used for
computing the triangular factor of the QR decomposition
of the quasi-Hessenberg matrix. The remaining 6*M elements
are used as workspace for the computation of the
reciprocal condition estimate.
LDDWOR INTEGER
The leading dimension of array DWORK.
LDDWOR >= MAX(1,2*M).
Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: if the quasi-Hessenberg matrix is (numerically)
singular. That is, its estimated reciprocal
condition number is less than or equal to TOL.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve a system of equations in Hessenberg form with one right-hand side.Specification
SUBROUTINE SB04RY( RC, UL, M, A, LDA, LAMBDA, D, TOL, IWORK,
$ DWORK, LDDWOR, INFO )
C .. Scalar Arguments ..
CHARACTER RC, UL
INTEGER INFO, LDA, LDDWOR, M
DOUBLE PRECISION LAMBDA, TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), D(*), DWORK(LDDWOR,*)
Arguments
Mode Parameters
RC CHARACTER*1
Indicates processing by columns or rows, as follows:
= 'R': Row transformations are applied;
= 'C': Column transformations are applied.
UL CHARACTER*1
Indicates whether A is upper or lower Hessenberg matrix,
as follows:
= 'U': A is upper Hessenberg;
= 'L': A is lower Hessenberg.
Input/Output Parameters
M (input) INTEGER
The order of the matrix A. M >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,M)
The leading M-by-M part of this array must contain a
matrix A in Hessenberg form.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,M).
LAMBDA (input) DOUBLE PRECISION
This variable must contain the value to be multiplied with
the elements of A.
D (input/output) DOUBLE PRECISION array, dimension (M)
On entry, this array must contain the right-hand side
vector of the Hessenberg system.
On exit, if INFO = 0, this array contains the solution
vector of the Hessenberg system.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used to test for near singularity of
the triangular factor R of the Hessenberg matrix. A matrix
whose estimated condition number is less than 1/TOL is
considered to be nonsingular.
Workspace
IWORK INTEGER array, dimension (M)
DWORK DOUBLE PRECISION array, dimension (LDDWOR,M+3)
The leading M-by-M part of this array is used for
computing the triangular factor of the QR decomposition
of the Hessenberg matrix. The remaining 3*M elements are
used as workspace for the computation of the reciprocal
condition estimate.
LDDWOR INTEGER
The leading dimension of array DWORK. LDDWOR >= MAX(1,M).
Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: if the Hessenberg matrix is (numerically) singular.
That is, its estimated reciprocal condition number
is less than or equal to TOL.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To construct the minimum norm feedback matrix F to perform "deadbeat control" on a (A,B)-pair of a state-space model (which must be preliminarily reduced to upper "staircase" form using SLICOT Library routine AB01OD) such that the matrix R = A + BFU' is nilpotent. (The transformation matrix U reduces R to upper Schur form with zero blocks on its diagonal (of dimension KSTAIR(i)) and therefore contains bases for the i-th controllable subspaces, where i = 1,...,KMAX).Specification
SUBROUTINE SB06ND( N, M, KMAX, A, LDA, B, LDB, KSTAIR, U, LDU, F,
$ LDF, DWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, KMAX, LDA, LDB, LDF, LDU, M, N
C .. Array Arguments ..
INTEGER KSTAIR(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), F(LDF,*), U(LDU,*)
Arguments
Input/Output Parameters
N (input) INTEGER
The actual state dimension, i.e. the order of the
matrix A. N >= 0.
M (input) INTEGER
The actual input dimension. M >= 0.
KMAX (input) INTEGER
The number of "stairs" in the staircase form as produced
by SLICOT Library routine AB01OD. 0 <= KMAX <= N.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the transformed state-space matrix of the
(A,B)-pair with triangular stairs, as produced by SLICOT
Library routine AB01OD (with option STAGES = 'A').
On exit, the leading N-by-N part of this array contains
the matrix U'AU + U'BF.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the transformed triangular input matrix of the
(A,B)-pair as produced by SLICOT Library routine AB01OD
(with option STAGES = 'A').
On exit, the leading N-by-M part of this array contains
the matrix U'B.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
KSTAIR (input) INTEGER array, dimension (KMAX)
The leading KMAX elements of this array must contain the
dimensions of each "stair" as produced by SLICOT Library
routine AB01OD.
U (input/output) DOUBLE PRECISION array, dimension (LDU,N)
On entry, the leading N-by-N part of this array must
contain either a transformation matrix (e.g. from a
previous call to other SLICOT routine) or be initialised
as the identity matrix.
On exit, the leading N-by-N part of this array contains
the product of the input matrix U and the state-space
transformation matrix which reduces A + BFU' to real
Schur form.
LDU INTEGER
The leading dimension of array U. LDU >= MAX(1,N).
F (output) DOUBLE PRECISION array, dimension (LDF,N)
The leading M-by-N part of this array contains the
deadbeat feedback matrix F.
LDF INTEGER
The leading dimension of array F. LDF >= MAX(1,M).
Workspace
DWORK DOUBLE PRECISION array, dimension (2*N)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Starting from the (A,B)-pair in "staircase form" with "triangular"
stairs, dimensions KSTAIR(i+1) x KSTAIR(i), (described by the
vector KSTAIR):
| B | A * . . . * |
| 1| 11 . . |
| | A A . . |
| | 21 22 . . |
| | . . . |
[ B | A ] = | | . . * |
| | . . |
| 0 | 0 |
| | A A |
| | r,r-1 rr |
where the i-th diagonal block of A has dimension KSTAIR(i), for
i = 1,2,...,r, the feedback matrix F is constructed recursively in
r steps (where the number of "stairs" r is given by KMAX). In each
step a unitary state-space transformation U and a part of F are
updated in order to achieve the final form:
| 0 A * . . . * |
| 12 . . |
| . . |
| 0 A . . |
| 23 . . |
| . . |
[ U'AU + U'BF ] = | . . * | .
| . . |
| |
| A |
| r-1,r|
| |
| 0 |
References
[1] Van Dooren, P.
Deadbeat control: a special inverse eigenvalue problem.
BIT, 24, pp. 681-699, 1984.
Numerical Aspects
The algorithm requires O((N + M) * N**2) operations and is mixed numerical stable (see [1]).Further Comments
NoneExample
Program Text
* SB06ND EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX
PARAMETER ( NMAX = 20, MMAX = 20 )
INTEGER LDA, LDB, LDU, LDV, LDF
PARAMETER ( LDA = NMAX, LDB = NMAX, LDU = NMAX,
$ LDV = MMAX, LDF = MMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX + MAX(NMAX,3*MMAX) )
* PARAMETER ( LDWORK = 4*NMAX)
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, J, KMAX, M, N, NCONT
CHARACTER*1 JOBU, JOBV
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), DWORK(LDWORK),
$ F(LDF,NMAX), U(LDU,NMAX), V(LDV,MMAX)
INTEGER IWORK(LIWORK), KSTAIR(NMAX)
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL AB01OD, DLASET, SB06ND
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, TOL, JOBU, JOBV
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), I = 1,N ), J = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
* First put (A,B) into staircase form with triangular pivots
* and determine the stairsizes.
CALL AB01OD( 'A', JOBU, JOBV, N, M, A, LDA, B, LDB, U,
$ LDU, V, LDV, NCONT, KMAX, KSTAIR, TOL, IWORK,
$ DWORK, LDWORK, INFO )
*
IF ( INFO.EQ.0 ) THEN
IF( LSAME( JOBU, 'N' ) ) THEN
* Initialize U as the identity matrix.
CALL DLASET( 'Full', N, N, ZERO, ONE, U, LDU )
END IF
* Perform "deadbeat control" to give F.
CALL SB06ND( N, M, KMAX, A, LDA, B, LDB, KSTAIR, U, LDU,
$ F, LDF, DWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99996 )
DO 60 I = 1, M
WRITE ( NOUT, FMT = 99995 ) ( F(I,J), J = 1,N )
60 CONTINUE
END IF
ELSE
WRITE ( NOUT, FMT = 99998 ) INFO
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB06ND EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB01OD = ',I2)
99997 FORMAT (' INFO on exit from SB06ND = ',I2)
99996 FORMAT (' The deadbeat feedback matrix F is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' M is out of range.',/' M = ',I5)
END
Program Data
SB06ND EXAMPLE PROGRAM DATA 5 2 0.0 N N -17.0 24.0 41.0 68.0 15.0 23.0 -35.0 27.0 14.0 16.0 34.0 26.0 -13.0 20.0 22.0 10.0 12.0 19.0 -21.0 63.0 11.0 18.0 25.0 52.0 -29.0 -31.0 14.0 74.0 -69.0 -59.0 16.0 16.0 -25.0 -25.0 36.0Program Results
SB06ND EXAMPLE PROGRAM RESULTS The deadbeat feedback matrix F is -0.4819 -0.5782 -2.7595 -3.1093 0.0000 0.2121 -0.4462 0.7698 -1.5421 -0.5773
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB08CD.html 0000664 0000000 0000000 00000040565 14560147231 0020347 0 ustar 00root root 0000000 0000000
Purpose
To construct, for a given system G = (A,B,C,D), an output
injection matrix H, an orthogonal transformation matrix Z, and a
gain matrix V, such that the systems
Q = (Z'*(A+H*C)*Z, Z'*(B+H*D), V*C*Z, V*D)
and
R = (Z'*(A+H*C)*Z, Z'*H, V*C*Z, V)
provide a stable left coprime factorization of G in the form
-1
G = R * Q,
where G, Q and R are the corresponding transfer-function matrices
and the denominator R is co-inner, that is, R(s)*R'(-s) = I in
the continuous-time case, or R(z)*R'(1/z) = I in the discrete-time
case. The Z matrix is not explicitly computed.
Note: G must have no observable poles on the imaginary axis
for a continuous-time system, or on the unit circle for a
discrete-time system. If the given state-space representation
is not detectable, the undetectable part of the original
system is automatically deflated and the order of the systems
Q and R is accordingly reduced.
Specification
SUBROUTINE SB08CD( DICO, N, M, P, A, LDA, B, LDB, C, LDC, D, LDD,
$ NQ, NR, BR, LDBR, DR, LDDR, TOL, DWORK, LDWORK,
$ IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO
INTEGER INFO, IWARN, LDA, LDB, LDBR, LDC, LDD, LDDR,
$ LDWORK, M, N, NQ, NR, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), BR(LDBR,*), C(LDC,*),
$ D(LDD,*), DR(LDDR,*), DWORK(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
Input/Output Parameters
N (input) INTEGER
The dimension of the state vector, i.e. the order of the
matrix A, and also the number of rows of the matrices B
and BR, and the number of columns of the matrix C.
N >= 0.
M (input) INTEGER
The dimension of input vector, i.e. the number of columns
of the matrices B and D. M >= 0.
P (input) INTEGER
The dimension of output vector, i.e. the number of rows
of the matrices C, D and DR, and the number of columns
of the matrices BR and DR. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A. The matrix A must not
have observable eigenvalues on the imaginary axis, if
DICO = 'C', or on the unit circle, if DICO = 'D'.
On exit, the leading NQ-by-NQ part of this array contains
the leading NQ-by-NQ part of the matrix Z'*(A+H*C)*Z, the
state dynamics matrix of the numerator factor Q, in a
real Schur form. The leading NR-by-NR part of this matrix
represents the state dynamics matrix of a minimal
realization of the denominator factor R.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension
(LDB,MAX(M,P))
On entry, the leading N-by-M part of this array must
contain the input/state matrix.
On exit, the leading NQ-by-M part of this array contains
the leading NQ-by-M part of the matrix Z'*(B+H*D), the
input/state matrix of the numerator factor Q.
The remaining part of this array is needed as workspace.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, the leading P-by-NQ part of this array contains
the leading P-by-NQ part of the matrix V*C*Z, the
state/output matrix of the numerator factor Q.
The first NR columns of this array represent the
state/output matrix of a minimal realization of the
denominator factor R.
The remaining part of this array is needed as workspace.
LDC INTEGER
The leading dimension of array C.
LDC >= MAX(1,M,P), if N > 0.
LDC >= 1, if N = 0.
D (input/output) DOUBLE PRECISION array, dimension
(LDD,MAX(M,P))
On entry, the leading P-by-M part of this array must
contain the input/output matrix.
On exit, the leading P-by-M part of this array contains
the matrix V*D representing the input/output matrix
of the numerator factor Q.
The remaining part of this array is needed as workspace.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,M,P).
NQ (output) INTEGER
The order of the resulting factors Q and R.
Generally, NQ = N - NS, where NS is the number of
unobservable eigenvalues outside the stability region.
NR (output) INTEGER
The order of the minimal realization of the factor R.
Generally, NR is the number of observable eigenvalues
of A outside the stability region (the number of modified
eigenvalues).
BR (output) DOUBLE PRECISION array, dimension (LDBR,P)
The leading NQ-by-P part of this array contains the
leading NQ-by-P part of the output injection matrix
Z'*H, which reflects the eigenvalues of A lying outside
the stable region to values which are symmetric with
respect to the imaginary axis (if DICO = 'C') or the unit
circle (if DICO = 'D'). The first NR rows of this matrix
form the input/state matrix of a minimal realization of
the denominator factor R.
LDBR INTEGER
The leading dimension of array BR. LDBR >= MAX(1,N).
DR (output) DOUBLE PRECISION array, dimension (LDDR,P)
The leading P-by-P part of this array contains the lower
triangular matrix V representing the input/output matrix
of the denominator factor R.
LDDR INTEGER
The leading dimension of array DR. LDDR >= MAX(1,P).
Tolerances
TOL DOUBLE PRECISION
The absolute tolerance level below which the elements of
C are considered zero (used for observability tests).
If the user sets TOL <= 0, then an implicitly computed,
default tolerance, defined by TOLDEF = N*EPS*NORM(C),
is used instead, where EPS is the machine precision
(see LAPACK Library routine DLAMCH) and NORM(C) denotes
the infinity-norm of C.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The dimension of working array DWORK.
LDWORK >= MAX( 1, P*N + MAX( N*(N+5),P*(P+2),4*P,4*M ) ).
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= K: K violations of the numerical stability condition
NORM(H) <= 10*NORM(A)/NORM(C) occured during the
assignment of eigenvalues.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction of A to a real Schur form failed;
= 2: a failure was detected during the ordering of the
real Schur form of A, or in the iterative process
for reordering the eigenvalues of Z'*(A + H*C)*Z
along the diagonal;
= 3: if DICO = 'C' and the matrix A has an observable
eigenvalue on the imaginary axis, or DICO = 'D' and
A has an observable eigenvalue on the unit circle.
Method
The subroutine uses the right coprime factorization algorithm with inner denominator of [1] applied to G'.References
[1] Varga A.
A Schur method for computing coprime factorizations with
inner denominators and applications in model reduction.
Proc. ACC'93, San Francisco, CA, pp. 2130-2131, 1993.
Numerical Aspects
3 The algorithm requires no more than 14N floating point operations.Further Comments
NoneExample
Program Text
* SB08CD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER MPMAX
PARAMETER ( MPMAX = MAX( MMAX, PMAX ) )
INTEGER LDA, LDB, LDBR, LDC, LDD, LDDR
PARAMETER ( LDA = NMAX, LDB = NMAX, LDBR = NMAX,
$ LDC = MPMAX, LDD = MPMAX, LDDR = PMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX*PMAX + MAX( NMAX*( NMAX + 5 ),
$ PMAX*( PMAX + 2 ), 4*PMAX, 4*MMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, IWARN, J, M, N, NQ, NR, P
CHARACTER*1 DICO
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MPMAX), BR(LDBR,PMAX),
$ C(LDC,NMAX), D(LDD,MPMAX), DR(LDDR,PMAX),
$ DWORK(LDWORK)
* .. External Subroutines ..
EXTERNAL SB08CD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL, DICO
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1, N ), I = 1, N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1, M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1, N ), I = 1, P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1, M ), I = 1, P )
* Find a RCFID for (A,B,C,D).
CALL SB08CD( DICO, N, M, P, A, LDA, B, LDB, C, LDC,
$ D, LDD, NQ, NR, BR, LDBR, DR, LDDR, TOL,
$ DWORK, LDWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NQ
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1, NQ )
20 CONTINUE
IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NQ
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1, M )
40 CONTINUE
IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1, NQ )
60 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 70 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1, M )
70 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99986 )
DO 80 I = 1, NR
WRITE ( NOUT, FMT = 99995 )
$ ( A(I,J), J = 1, NR )
80 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99985 )
DO 90 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( BR(I,J), J = 1, P )
90 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99984 )
DO 100 I = 1, P
WRITE ( NOUT, FMT = 99995 )
$ ( C(I,J), J = 1, NR )
100 CONTINUE
WRITE ( NOUT, FMT = 99983 )
DO 110 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( DR(I,J), J = 1, P )
110 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB08CD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB08CD = ',I2)
99996 FORMAT (/' The numerator state dynamics matrix AQ is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The numerator input/state matrix BQ is ')
99992 FORMAT (/' The numerator state/output matrix CQ is ')
99991 FORMAT (/' The numerator input/output matrix DQ is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99986 FORMAT (/' The denominator state dynamics matrix AR is ')
99985 FORMAT (/' The denominator input/state matrix BR is ')
99984 FORMAT (/' The denominator state/output matrix CR is ')
99983 FORMAT (/' The denominator input/output matrix DR is ')
END
Program Data
SB08CD EXAMPLE PROGRAM DATA (Continuous system) 7 2 3 1.E-10 C -0.04165 0.0000 4.9200 0.4920 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 0.0000 0.0000 0.5450 0.0000 0.0000 0.0000 0.0545 0.0000 0.0000 0.0000 0.0000 0.0000 -0.49200 0.004165 0.0000 4.9200 0.0000 0.0000 0.0000 0.0000 0.5210 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 12.500 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000Program Results
SB08CD EXAMPLE PROGRAM RESULTS The numerator state dynamics matrix AQ is -0.1605 0.0523 0.9423 2.0193 0.4166 0.2518 1.6140 -0.4489 -0.1605 1.7955 3.8719 -0.2394 0.0491 -0.8740 0.0000 0.0000 -12.4245 3.5463 -0.0057 0.0254 -0.0053 0.0000 0.0000 0.0000 -3.5957 -0.0153 -0.0290 -0.0616 0.0000 0.0000 0.0000 0.0000 -13.1627 -1.9835 -3.6182 0.0000 0.0000 0.0000 0.0000 0.0000 -1.4178 5.6218 0.0000 0.0000 0.0000 0.0000 0.0000 -0.8374 -1.4178 The numerator input/state matrix BQ is -1.0157 0.2554 0.5523 0.4443 0.0056 -11.6989 0.0490 4.3728 11.7198 -0.0038 -2.8173 0.0308 3.1018 -0.0009 The numerator state/output matrix CQ is 0.1975 -0.1063 -0.0006 -0.0083 0.1279 0.8797 0.3994 0.8541 -0.4513 -0.0007 -0.0041 0.0305 -0.2562 0.0122 0.4668 0.8826 0.0248 -0.0506 0.0000 0.0022 -0.0017 The numerator input/output matrix DQ is 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 The denominator state dynamics matrix AR is -0.1605 0.0523 -0.4489 -0.1605 The denominator input/state matrix BR is -0.0158 -0.0692 -0.1688 0.0306 0.1281 -0.4984 The denominator state/output matrix CR is 0.1975 -0.1063 0.8541 -0.4513 0.4668 0.8826 The denominator input/output matrix DR is 1.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 1.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB08DD.html 0000664 0000000 0000000 00000037627 14560147231 0020355 0 ustar 00root root 0000000 0000000
Purpose
To construct, for a given system G = (A,B,C,D), a feedback matrix
F, an orthogonal transformation matrix Z, and a gain matrix V,
such that the systems
Q = (Z'*(A+B*F)*Z, Z'*B*V, (C+D*F)*Z, D*V)
and
R = (Z'*(A+B*F)*Z, Z'*B*V, F*Z, V)
provide a stable right coprime factorization of G in the form
-1
G = Q * R ,
where G, Q and R are the corresponding transfer-function matrices
and the denominator R is inner, that is, R'(-s)*R(s) = I in the
continuous-time case, or R'(1/z)*R(z) = I in the discrete-time
case. The Z matrix is not explicitly computed.
Note: G must have no controllable poles on the imaginary axis
for a continuous-time system, or on the unit circle for a
discrete-time system. If the given state-space representation
is not stabilizable, the unstabilizable part of the original
system is automatically deflated and the order of the systems
Q and R is accordingly reduced.
Specification
SUBROUTINE SB08DD( DICO, N, M, P, A, LDA, B, LDB, C, LDC, D, LDD,
$ NQ, NR, CR, LDCR, DR, LDDR, TOL, DWORK, LDWORK,
$ IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO
INTEGER INFO, IWARN, LDA, LDB, LDC, LDCR, LDD, LDDR,
$ LDWORK, M, N, NQ, NR, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), CR(LDCR,*),
$ D(LDD,*), DR(LDDR,*), DWORK(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
Input/Output Parameters
N (input) INTEGER
The dimension of the state vector, i.e. the order of the
matrix A, and also the number of rows of the matrix B and
the number of columns of the matrices C and CR. N >= 0.
M (input) INTEGER
The dimension of input vector, i.e. the number of columns
of the matrices B, D and DR and the number of rows of the
matrices CR and DR. M >= 0.
P (input) INTEGER
The dimension of output vector, i.e. the number of rows
of the matrices C and D. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A. The matrix A must not
have controllable eigenvalues on the imaginary axis, if
DICO = 'C', or on the unit circle, if DICO = 'D'.
On exit, the leading NQ-by-NQ part of this array contains
the leading NQ-by-NQ part of the matrix Z'*(A+B*F)*Z, the
state dynamics matrix of the numerator factor Q, in a
real Schur form. The trailing NR-by-NR part of this matrix
represents the state dynamics matrix of a minimal
realization of the denominator factor R.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input/state matrix.
On exit, the leading NQ-by-M part of this array contains
the leading NQ-by-M part of the matrix Z'*B*V, the
input/state matrix of the numerator factor Q. The last
NR rows of this matrix form the input/state matrix of
a minimal realization of the denominator factor R.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, the leading P-by-NQ part of this array contains
the leading P-by-NQ part of the matrix (C+D*F)*Z,
the state/output matrix of the numerator factor Q.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the input/output matrix.
On exit, the leading P-by-M part of this array contains
the matrix D*V representing the input/output matrix
of the numerator factor Q.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
NQ (output) INTEGER
The order of the resulting factors Q and R.
Generally, NQ = N - NS, where NS is the number of
uncontrollable eigenvalues outside the stability region.
NR (output) INTEGER
The order of the minimal realization of the factor R.
Generally, NR is the number of controllable eigenvalues
of A outside the stability region (the number of modified
eigenvalues).
CR (output) DOUBLE PRECISION array, dimension (LDCR,N)
The leading M-by-NQ part of this array contains the
leading M-by-NQ part of the feedback matrix F*Z, which
reflects the eigenvalues of A lying outside the stable
region to values which are symmetric with respect to the
imaginary axis (if DICO = 'C') or the unit circle (if
DICO = 'D'). The last NR columns of this matrix form the
state/output matrix of a minimal realization of the
denominator factor R.
LDCR INTEGER
The leading dimension of array CR. LDCR >= MAX(1,M).
DR (output) DOUBLE PRECISION array, dimension (LDDR,M)
The leading M-by-M part of this array contains the upper
triangular matrix V of order M representing the
input/output matrix of the denominator factor R.
LDDR INTEGER
The leading dimension of array DR. LDDR >= MAX(1,M).
Tolerances
TOL DOUBLE PRECISION
The absolute tolerance level below which the elements of
B are considered zero (used for controllability tests).
If the user sets TOL <= 0, then an implicitly computed,
default tolerance, defined by TOLDEF = N*EPS*NORM(B),
is used instead, where EPS is the machine precision
(see LAPACK Library routine DLAMCH) and NORM(B) denotes
the 1-norm of B.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The dimension of working array DWORK.
LDWORK >= MAX( 1, N*(N+5), M*(M+2), 4*M, 4*P ).
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= K: K violations of the numerical stability condition
NORM(F) <= 10*NORM(A)/NORM(B) occured during the
assignment of eigenvalues.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction of A to a real Schur form failed;
= 2: a failure was detected during the ordering of the
real Schur form of A, or in the iterative process
for reordering the eigenvalues of Z'*(A + B*F)*Z
along the diagonal;
= 3: if DICO = 'C' and the matrix A has a controllable
eigenvalue on the imaginary axis, or DICO = 'D'
and A has a controllable eigenvalue on the unit
circle.
Method
The subroutine is based on the factorization algorithm of [1].References
[1] Varga A.
A Schur method for computing coprime factorizations with inner
denominators and applications in model reduction.
Proc. ACC'93, San Francisco, CA, pp. 2130-2131, 1993.
Numerical Aspects
3 The algorithm requires no more than 14N floating point operations.Further Comments
NoneExample
Program Text
* SB08DD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDCR, LDD, LDDR
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDCR = MMAX, LDD = PMAX, LDDR = MMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( NMAX*( NMAX + 5 ),
$ MMAX*( MMAX + 2 ),
$ 4*NMAX, 4*PMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, IWARN, J, M, N, NQ, NR, P
CHARACTER*1 DICO
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ CR(LDCR,NMAX), D(LDD,MMAX), DR(LDDR,MMAX),
$ DWORK(LDWORK)
* .. External Subroutines ..
EXTERNAL SB08DD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL, DICO
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1, N ), I = 1, N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1, M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1, N ), I = 1, P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1, M ), I = 1, P )
* Find a RCFID for (A,B,C,D).
CALL SB08DD( DICO, N, M, P, A, LDA, B, LDB, C, LDC,
$ D, LDD, NQ, NR, CR, LDCR, DR, LDDR, TOL,
$ DWORK, LDWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NQ
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1, NQ )
20 CONTINUE
IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NQ
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1, M )
40 CONTINUE
IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1, NQ )
60 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 70 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1, M )
70 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99986 )
DO 80 I = NQ-NR+1, NQ
WRITE ( NOUT, FMT = 99995 )
$ ( A(I,J), J = NQ-NR+1, NQ )
80 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99985 )
DO 90 I = NQ-NR+1, NQ
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1, M )
90 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99984 )
DO 100 I = 1, M
WRITE ( NOUT, FMT = 99995 )
$ ( CR(I,J), J = NQ-NR+1, NQ )
100 CONTINUE
WRITE ( NOUT, FMT = 99983 )
DO 110 I = 1, M
WRITE ( NOUT, FMT = 99995 ) ( DR(I,J), J = 1, M )
110 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB08DD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB08DD = ',I2)
99996 FORMAT (/' The numerator state dynamics matrix AQ is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The numerator input/state matrix BQ is ')
99992 FORMAT (/' The numerator state/output matrix CQ is ')
99991 FORMAT (/' The numerator input/output matrix DQ is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99986 FORMAT (/' The denominator state dynamics matrix AR is ')
99985 FORMAT (/' The denominator input/state matrix BR is ')
99984 FORMAT (/' The denominator state/output matrix CR is ')
99983 FORMAT (/' The denominator input/output matrix DR is ')
END
Program Data
SB08DD EXAMPLE PROGRAM DATA (Continuous system) 7 2 3 1.E-10 C -0.04165 0.0000 4.9200 0.4920 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 0.0000 0.0000 0.5450 0.0000 0.0000 0.0000 0.0545 0.0000 0.0000 0.0000 0.0000 0.0000 -0.49200 0.004165 0.0000 4.9200 0.0000 0.0000 0.0000 0.0000 0.5210 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 12.500 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000Program Results
SB08DD EXAMPLE PROGRAM RESULTS The numerator state dynamics matrix AQ is -1.4178 -5.1682 3.2450 -0.2173 0.0564 -4.1066 -0.2336 0.9109 -1.4178 -2.1262 0.1231 0.0805 -0.4816 0.2196 0.0000 0.0000 -13.1627 0.0608 -0.0218 3.8320 0.3429 0.0000 0.0000 0.0000 -3.5957 -3.3373 0.0816 -4.1237 0.0000 0.0000 0.0000 0.0000 -12.4245 -0.3133 4.4255 0.0000 0.0000 0.0000 0.0000 0.0000 -0.1605 -0.0772 0.0000 0.0000 0.0000 0.0000 0.0000 0.3040 -0.1605 The numerator input/state matrix BQ is 5.0302 -0.0063 0.7078 -0.0409 -11.3663 0.0051 0.1760 0.5879 -0.0265 12.2119 1.1050 0.3215 0.0066 -2.5822 The numerator state/output matrix CQ is -0.8659 0.2787 -0.3432 0.0020 0.0000 0.2325 0.0265 0.0797 -0.3951 0.0976 -0.0292 0.0062 0.8985 0.1406 -0.0165 -0.0645 0.0097 0.8032 -0.1602 0.0874 -0.5630 The numerator input/output matrix DQ is 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 The denominator state dynamics matrix AR is -0.1605 -0.0772 0.3040 -0.1605 The denominator input/state matrix BR is 1.1050 0.3215 0.0066 -2.5822 The denominator state/output matrix CR is -0.2288 -0.0259 -0.0070 0.1497 The denominator input/output matrix DR is 1.0000 0.0000 0.0000 1.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB08ED.html 0000664 0000000 0000000 00000040500 14560147231 0020336 0 ustar 00root root 0000000 0000000
Purpose
To construct, for a given system G = (A,B,C,D), an output
injection matrix H and an orthogonal transformation matrix Z, such
that the systems
Q = (Z'*(A+H*C)*Z, Z'*(B+H*D), C*Z, D)
and
R = (Z'*(A+H*C)*Z, Z'*H, C*Z, I)
provide a stable left coprime factorization of G in the form
-1
G = R * Q,
where G, Q and R are the corresponding transfer-function matrices.
The resulting state dynamics matrix of the systems Q and R has
eigenvalues lying inside a given stability domain.
The Z matrix is not explicitly computed.
Note: If the given state-space representation is not detectable,
the undetectable part of the original system is automatically
deflated and the order of the systems Q and R is accordingly
reduced.
Specification
SUBROUTINE SB08ED( DICO, N, M, P, ALPHA, A, LDA, B, LDB, C, LDC,
$ D, LDD, NQ, NR, BR, LDBR, DR, LDDR, TOL, DWORK,
$ LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO
INTEGER INFO, IWARN, LDA, LDB, LDBR, LDC, LDD, LDDR,
$ LDWORK, M, N, NQ, NR, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), ALPHA(*), B(LDB,*), BR(LDBR,*),
$ C(LDC,*), D(LDD,*), DR(LDDR,*), DWORK(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
Input/Output Parameters
N (input) INTEGER
The dimension of the state vector, i.e. the order of the
matrix A, and also the number of rows of the matrices B
and BR, and the number of columns of the matrix C.
N >= 0.
M (input) INTEGER
The dimension of input vector, i.e. the number of columns
of the matrices B and D. M >= 0.
P (input) INTEGER
The dimension of output vector, i.e. the number of rows
of the matrices C, D and DR, and the number of columns of
the matrices BR and DR. P >= 0.
ALPHA (input) DOUBLE PRECISION array, dimension (2)
ALPHA(1) contains the desired stability degree to be
assigned for the eigenvalues of A+H*C, and ALPHA(2)
the stability margin. The eigenvalues outside the
ALPHA(2)-stability region will be assigned to have the
real parts equal to ALPHA(1) < 0 and unmodified
imaginary parts for a continuous-time system
(DICO = 'C'), or moduli equal to 0 <= ALPHA(2) < 1
for a discrete-time system (DICO = 'D').
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A.
On exit, the leading NQ-by-NQ part of this array contains
the leading NQ-by-NQ part of the matrix Z'*(A+H*C)*Z, the
state dynamics matrix of the numerator factor Q, in a
real Schur form. The leading NR-by-NR part of this matrix
represents the state dynamics matrix of a minimal
realization of the denominator factor R.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension
(LDB,MAX(M,P))
On entry, the leading N-by-M part of this array must
contain the input/state matrix of the system.
On exit, the leading NQ-by-M part of this array contains
the leading NQ-by-M part of the matrix Z'*(B+H*D), the
input/state matrix of the numerator factor Q.
The remaining part of this array is needed as workspace.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix of the system.
On exit, the leading P-by-NQ part of this array contains
the leading P-by-NQ part of the matrix C*Z, the
state/output matrix of the numerator factor Q.
The first NR columns of this array represent the
state/output matrix of a minimal realization of the
denominator factor R.
The remaining part of this array is needed as workspace.
LDC INTEGER
The leading dimension of array C.
LDC >= MAX(1,M,P), if N > 0.
LDC >= 1, if N = 0.
D (input) DOUBLE PRECISION array, dimension (LDD,MAX(M,P))
The leading P-by-M part of this array must contain the
input/output matrix. D represents also the input/output
matrix of the numerator factor Q.
This array is modified internally, but restored on exit.
The remaining part of this array is needed as workspace.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,M,P).
NQ (output) INTEGER
The order of the resulting factors Q and R.
Generally, NQ = N - NS, where NS is the number of
unobservable eigenvalues outside the stability region.
NR (output) INTEGER
The order of the minimal realization of the factor R.
Generally, NR is the number of observable eigenvalues
of A outside the stability region (the number of modified
eigenvalues).
BR (output) DOUBLE PRECISION array, dimension (LDBR,P)
The leading NQ-by-P part of this array contains the
leading NQ-by-P part of the output injection matrix
Z'*H, which moves the eigenvalues of A lying outside
the ALPHA-stable region to values on the ALPHA-stability
boundary. The first NR rows of this matrix form the
input/state matrix of a minimal realization of the
denominator factor R.
LDBR INTEGER
The leading dimension of array BR. LDBR >= MAX(1,N).
DR (output) DOUBLE PRECISION array, dimension (LDDR,P)
The leading P-by-P part of this array contains an
identity matrix representing the input/output matrix
of the denominator factor R.
LDDR INTEGER
The leading dimension of array DR. LDDR >= MAX(1,P).
Tolerances
TOL DOUBLE PRECISION
The absolute tolerance level below which the elements of
C are considered zero (used for observability tests).
If the user sets TOL <= 0, then an implicitly computed,
default tolerance, defined by TOLDEF = N*EPS*NORM(C),
is used instead, where EPS is the machine precision
(see LAPACK Library routine DLAMCH) and NORM(C) denotes
the infinity-norm of C.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The dimension of working array DWORK.
LDWORK >= MAX( 1, N*P + MAX( N*(N+5), 5*P, 4*M ) ).
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= K: K violations of the numerical stability condition
NORM(H) <= 10*NORM(A)/NORM(C) occured during the
assignment of eigenvalues.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction of A to a real Schur form failed;
= 2: a failure was detected during the ordering of the
real Schur form of A, or in the iterative process
for reordering the eigenvalues of Z'*(A + H*C)*Z
along the diagonal.
Method
The subroutine uses the right coprime factorization algorithm of [1] applied to G'.References
[1] Varga A.
Coprime factors model reduction method based on
square-root balancing-free techniques.
System Analysis, Modelling and Simulation,
vol. 11, pp. 303-311, 1993.
Numerical Aspects
3 The algorithm requires no more than 14N floating point operations.Further Comments
NoneExample
Program Text
* SB08ED EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER MPMAX
PARAMETER ( MPMAX = MAX( MMAX, PMAX ) )
INTEGER LDA, LDB, LDBR, LDC, LDD, LDDR
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = MPMAX,
$ LDD = MPMAX, LDBR = NMAX, LDDR = PMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX*PMAX + MAX( NMAX*( NMAX + 5 ),
$ 5*PMAX, 4*MMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, IWARN, J, M, N, NQ, NR, P
CHARACTER*1 DICO
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), ALPHA(2), B(LDB,MPMAX),
$ BR(LDBR,PMAX), C(LDC,NMAX), D(LDD,MPMAX),
$ DR(LDDR,PMAX), DWORK(LDWORK)
* .. External Subroutines ..
EXTERNAL SB08ED
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, ALPHA(1), TOL, DICO
ALPHA(2) = ALPHA(1)
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1, N ), I = 1, N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1, M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1, N ), I = 1, P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1, M ), I = 1, P )
* Find a LCF for (A,B,C,D).
CALL SB08ED( DICO, N, M, P, ALPHA, A, LDA, B, LDB, C,
$ LDC, D, LDD, NQ, NR, BR, LDBR, DR, LDDR,
$ TOL, DWORK, LDWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NQ
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1, NQ )
20 CONTINUE
IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NQ
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1, M )
40 CONTINUE
IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1, NQ )
60 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 70 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1, M )
70 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99986 )
DO 80 I = 1, NR
WRITE ( NOUT, FMT = 99995 )
$ ( A(I,J), J = 1, NR )
80 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99985 )
DO 90 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( BR(I,J), J = 1, P )
90 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99984 )
DO 100 I = 1, P
WRITE ( NOUT, FMT = 99995 )
$ ( C(I,J), J = 1, NR )
100 CONTINUE
WRITE ( NOUT, FMT = 99983 )
DO 110 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( DR(I,J), J = 1, P )
110 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB08ED EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB08ED = ',I2)
99996 FORMAT (/' The numerator state dynamics matrix AQ is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The numerator input/state matrix BQ is ')
99992 FORMAT (/' The numerator state/output matrix CQ is ')
99991 FORMAT (/' The numerator input/output matrix DQ is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99986 FORMAT (/' The denominator state dynamics matrix AR is ')
99985 FORMAT (/' The denominator input/state matrix BR is ')
99984 FORMAT (/' The denominator state/output matrix CR is ')
99983 FORMAT (/' The denominator input/output matrix DR is ')
END
Program Data
SB08ED EXAMPLE PROGRAM DATA (Continuous system) 7 2 3 -1.0 1.E-10 C -0.04165 0.0000 4.9200 0.4920 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 0.0000 0.0000 0.5450 0.0000 0.0000 0.0000 0.0545 0.0000 0.0000 0.0000 0.0000 0.0000 -0.49200 0.004165 0.0000 4.9200 0.0000 0.0000 0.0000 0.0000 0.5210 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 12.500 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000Program Results
SB08ED EXAMPLE PROGRAM RESULTS The numerator state dynamics matrix AQ is -1.0000 0.0526 -0.1408 -0.3060 0.4199 0.2408 1.7274 -0.4463 -1.0000 2.0067 4.3895 0.0062 0.1813 0.0895 0.0000 0.0000 -12.4245 3.5463 -0.0057 0.0254 -0.0053 0.0000 0.0000 0.0000 -3.5957 -0.0153 -0.0290 -0.0616 0.0000 0.0000 0.0000 0.0000 -13.1627 -1.9835 -3.6182 0.0000 0.0000 0.0000 0.0000 0.0000 -1.4178 5.6218 0.0000 0.0000 0.0000 0.0000 0.0000 -0.8374 -1.4178 The numerator input/state matrix BQ is -1.1544 -0.0159 -0.0631 0.5122 0.0056 -11.6989 0.0490 4.3728 11.7198 -0.0038 -2.8173 0.0308 3.1018 -0.0009 The numerator state/output matrix CQ is 0.2238 0.0132 -0.0006 -0.0083 0.1279 0.8797 0.3994 0.9639 0.0643 -0.0007 -0.0041 0.0305 -0.2562 0.0122 -0.0660 0.9962 0.0248 -0.0506 0.0000 0.0022 -0.0017 The numerator input/output matrix DQ is 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 The denominator state dynamics matrix AR is -1.0000 0.0526 -0.4463 -1.0000 The denominator input/state matrix BR is -0.2623 -1.1297 0.0764 -0.0155 -0.0752 -1.1676 The denominator state/output matrix CR is 0.2238 0.0132 0.9639 0.0643 -0.0660 0.9962 The denominator input/output matrix DR is 1.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 1.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB08FD.html 0000664 0000000 0000000 00000037326 14560147231 0020353 0 ustar 00root root 0000000 0000000
Purpose
To construct, for a given system G = (A,B,C,D), a feedback
matrix F and an orthogonal transformation matrix Z, such that
the systems
Q = (Z'*(A+B*F)*Z, Z'*B, (C+D*F)*Z, D)
and
R = (Z'*(A+B*F)*Z, Z'*B, F*Z, I)
provide a stable right coprime factorization of G in the form
-1
G = Q * R ,
where G, Q and R are the corresponding transfer-function matrices.
The resulting state dynamics matrix of the systems Q and R has
eigenvalues lying inside a given stability domain.
The Z matrix is not explicitly computed.
Note: If the given state-space representation is not stabilizable,
the unstabilizable part of the original system is automatically
deflated and the order of the systems Q and R is accordingly
reduced.
Specification
SUBROUTINE SB08FD( DICO, N, M, P, ALPHA, A, LDA, B, LDB, C, LDC,
$ D, LDD, NQ, NR, CR, LDCR, DR, LDDR, TOL, DWORK,
$ LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO
INTEGER INFO, IWARN, LDA, LDB, LDC, LDCR, LDD, LDDR,
$ LDWORK, M, N, NQ, NR, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), ALPHA(*), B(LDB,*), C(LDC,*),
$ CR(LDCR,*), D(LDD,*), DR(LDDR,*), DWORK(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
Input/Output Parameters
N (input) INTEGER
The dimension of the state vector, i.e. the order of the
matrix A, and also the number of rows of the matrix B and
the number of columns of the matrices C and CR. N >= 0.
M (input) INTEGER
The dimension of input vector, i.e. the number of columns
of the matrices B, D and DR and the number of rows of the
matrices CR and DR. M >= 0.
P (input) INTEGER
The dimension of output vector, i.e. the number of rows
of the matrices C and D. P >= 0.
ALPHA (input) DOUBLE PRECISION array, dimension (2)
ALPHA(1) contains the desired stability degree to be
assigned for the eigenvalues of A+B*F, and ALPHA(2)
the stability margin. The eigenvalues outside the
ALPHA(2)-stability region will be assigned to have the
real parts equal to ALPHA(1) < 0 and unmodified
imaginary parts for a continuous-time system
(DICO = 'C'), or moduli equal to 0 <= ALPHA(2) < 1
for a discrete-time system (DICO = 'D').
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A.
On exit, the leading NQ-by-NQ part of this array contains
the leading NQ-by-NQ part of the matrix Z'*(A+B*F)*Z, the
state dynamics matrix of the numerator factor Q, in a
real Schur form. The trailing NR-by-NR part of this matrix
represents the state dynamics matrix of a minimal
realization of the denominator factor R.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input/state matrix.
On exit, the leading NQ-by-M part of this array contains
the leading NQ-by-M part of the matrix Z'*B, the
input/state matrix of the numerator factor Q. The last
NR rows of this matrix form the input/state matrix of
a minimal realization of the denominator factor R.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, the leading P-by-NQ part of this array contains
the leading P-by-NQ part of the matrix (C+D*F)*Z,
the state/output matrix of the numerator factor Q.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading P-by-M part of this array must contain the
input/output matrix. D represents also the input/output
matrix of the numerator factor Q.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
NQ (output) INTEGER
The order of the resulting factors Q and R.
Generally, NQ = N - NS, where NS is the number of
uncontrollable eigenvalues outside the stability region.
NR (output) INTEGER
The order of the minimal realization of the factor R.
Generally, NR is the number of controllable eigenvalues
of A outside the stability region (the number of modified
eigenvalues).
CR (output) DOUBLE PRECISION array, dimension (LDCR,N)
The leading M-by-NQ part of this array contains the
leading M-by-NQ part of the feedback matrix F*Z, which
moves the eigenvalues of A lying outside the ALPHA-stable
region to values which are on the ALPHA-stability
boundary. The last NR columns of this matrix form the
state/output matrix of a minimal realization of the
denominator factor R.
LDCR INTEGER
The leading dimension of array CR. LDCR >= MAX(1,M).
DR (output) DOUBLE PRECISION array, dimension (LDDR,M)
The leading M-by-M part of this array contains an
identity matrix representing the input/output matrix
of the denominator factor R.
LDDR INTEGER
The leading dimension of array DR. LDDR >= MAX(1,M).
Tolerances
TOL DOUBLE PRECISION
The absolute tolerance level below which the elements of
B are considered zero (used for controllability tests).
If the user sets TOL <= 0, then an implicitly computed,
default tolerance, defined by TOLDEF = N*EPS*NORM(B),
is used instead, where EPS is the machine precision
(see LAPACK Library routine DLAMCH) and NORM(B) denotes
the 1-norm of B.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The dimension of working array DWORK.
LWORK >= MAX( 1, N*(N+5), 5*M, 4*P ).
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= K: K violations of the numerical stability condition
NORM(F) <= 10*NORM(A)/NORM(B) occured during the
assignment of eigenvalues.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction of A to a real Schur form failed;
= 2: a failure was detected during the ordering of the
real Schur form of A, or in the iterative process
for reordering the eigenvalues of Z'*(A + B*F)*Z
along the diagonal.
Method
The subroutine is based on the factorization algorithm of [1].References
[1] Varga A.
Coprime factors model reduction method based on
square-root balancing-free techniques.
System Analysis, Modelling and Simulation,
vol. 11, pp. 303-311, 1993.
Numerical Aspects
3 The algorithm requires no more than 14N floating point operations.Further Comments
NoneExample
Program Text
* SB08FD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDCR, LDD, LDDR
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDCR = MMAX, LDD = PMAX, LDDR = MMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( NMAX*( NMAX + 5 ), 5*MMAX,
$ 4*PMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, IWARN, J, M, N, NQ, NR, P
CHARACTER*1 DICO
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), ALPHA(2), B(LDB,MMAX), C(LDC,NMAX),
$ CR(LDCR,NMAX), D(LDD,MMAX), DR(LDDR,MMAX),
$ DWORK(LDWORK)
* .. External Subroutines ..
EXTERNAL SB08FD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, ALPHA(1), TOL, DICO
ALPHA(2) = ALPHA(1)
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1, N ), I = 1, N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1, M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1, N ), I = 1, P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1, M ), I = 1, P )
* Find a RCF for (A,B,C,D).
CALL SB08FD( DICO, N, M, P, ALPHA, A, LDA, B, LDB, C,
$ LDC, D, LDD, NQ, NR, CR, LDCR, DR, LDDR,
$ TOL, DWORK, LDWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NQ
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1, NQ )
20 CONTINUE
IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NQ
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1, M )
40 CONTINUE
IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1, NQ )
60 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 70 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1, M )
70 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99986 )
DO 80 I = NQ-NR+1, NQ
WRITE ( NOUT, FMT = 99995 )
$ ( A(I,J), J = NQ-NR+1, NQ )
80 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99985 )
DO 90 I = NQ-NR+1, NQ
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1, M )
90 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99984 )
DO 100 I = 1, M
WRITE ( NOUT, FMT = 99995 )
$ ( CR(I,J), J = NQ-NR+1, NQ )
100 CONTINUE
WRITE ( NOUT, FMT = 99983 )
DO 110 I = 1, M
WRITE ( NOUT, FMT = 99995 ) ( DR(I,J), J = 1, M )
110 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB08FD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB08FD = ',I2)
99996 FORMAT (/' The numerator state dynamics matrix AQ is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The numerator input/state matrix BQ is ')
99992 FORMAT (/' The numerator state/output matrix CQ is ')
99991 FORMAT (/' The numerator input/output matrix DQ is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99986 FORMAT (/' The denominator state dynamics matrix AR is ')
99985 FORMAT (/' The denominator input/state matrix BR is ')
99984 FORMAT (/' The denominator state/output matrix CR is ')
99983 FORMAT (/' The denominator input/output matrix DR is ')
END
Program Data
SB08FD EXAMPLE PROGRAM DATA (Continuous system) 7 2 3 -1.0 1.E-10 C -0.04165 0.0000 4.9200 0.4920 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 0.0000 0.0000 0.5450 0.0000 0.0000 0.0000 0.0545 0.0000 0.0000 0.0000 0.0000 0.0000 -0.49200 0.004165 0.0000 4.9200 0.0000 0.0000 0.0000 0.0000 0.5210 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 12.500 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000Program Results
SB08FD EXAMPLE PROGRAM RESULTS The numerator state dynamics matrix AQ is -1.4178 -5.1682 3.2450 -0.2173 0.0564 -3.2129 -3.6183 0.9109 -1.4178 -2.1262 0.1231 0.0805 -0.4392 -0.2528 0.0000 0.0000 -13.1627 0.0608 -0.0218 2.3461 5.8272 0.0000 0.0000 0.0000 -3.5957 -3.3373 1.3622 -3.6083 0.0000 0.0000 0.0000 0.0000 -12.4245 -9.8634 8.1191 0.0000 0.0000 0.0000 0.0000 0.0000 -1.0000 -0.0135 0.0000 0.0000 0.0000 0.0000 0.0000 1.7393 -1.0000 The numerator input/state matrix BQ is 5.0302 -0.0063 0.7078 -0.0409 -11.3663 0.0051 0.1760 0.5879 -0.0265 12.2119 1.0104 1.3262 0.4474 -2.2388 The numerator state/output matrix CQ is -0.8659 0.2787 -0.3432 0.0020 0.0000 0.2026 0.1172 0.0797 -0.3951 0.0976 -0.0292 0.0062 0.7676 0.4879 -0.0165 -0.0645 0.0097 0.8032 -0.1602 0.3050 -0.4812 The numerator input/output matrix DQ is 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 The denominator state dynamics matrix AR is -1.0000 -0.0135 1.7393 -1.0000 The denominator input/state matrix BR is 1.0104 1.3262 0.4474 -2.2388 The denominator state/output matrix CR is -0.1091 -0.4653 -0.7055 0.4766 The denominator input/output matrix DR is 1.0000 0.0000 0.0000 1.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB08GD.html 0000664 0000000 0000000 00000013377 14560147231 0020354 0 ustar 00root root 0000000 0000000
Purpose
To construct the state-space representation for the system
G = (A,B,C,D) from the factors Q = (AQR,BQ,CQR,DQ) and
R = (AQR,BR,CQR,DR) of its left coprime factorization
-1
G = R * Q,
where G, Q and R are the corresponding transfer-function matrices.
Specification
SUBROUTINE SB08GD( N, M, P, A, LDA, B, LDB, C, LDC, D, LDD, BR,
$ LDBR, DR, LDDR, IWORK, DWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDBR, LDC, LDD, LDDR, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), BR(LDBR,*), C(LDC,*),
$ D(LDD,*), DR(LDDR,*), DWORK(*)
INTEGER IWORK(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. Also the number of rows of the
matrices B and BR and the number of columns of the matrix
C. N represents the order of the systems Q and R. N >= 0.
M (input) INTEGER
The dimension of input vector, i.e. the number of columns
of the matrices B and D. M >= 0.
P (input) INTEGER
The dimension of output vector, i.e. the number of rows of
the matrices C, D and DR and the number of columns of the
matrices BR and DR. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix AQR of the systems
Q and R.
On exit, the leading N-by-N part of this array contains
the state dynamics matrix of the system G.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input/state matrix BQ of the system Q.
On exit, the leading N-by-M part of this array contains
the input/state matrix of the system G.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix CQR of the systems
Q and R.
On exit, the leading P-by-N part of this array contains
the state/output matrix of the system G.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the input/output matrix DQ of the system Q.
On exit, the leading P-by-M part of this array contains
the input/output matrix of the system G.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
BR (input) DOUBLE PRECISION array, dimension (LDBR,P)
The leading N-by-P part of this array must contain the
input/state matrix BR of the system R.
LDBR INTEGER
The leading dimension of array BR. LDBR >= MAX(1,N).
DR (input/output) DOUBLE PRECISION array, dimension (LDDR,P)
On entry, the leading P-by-P part of this array must
contain the input/output matrix DR of the system R.
On exit, the leading P-by-P part of this array contains
the LU factorization of the matrix DR, as computed by
LAPACK Library routine DGETRF.
LDDR INTEGER
The leading dimension of array DR. LDDR >= MAX(1,P).
Workspace
IWORK INTEGER array, dimension (P)
DWORK DOUBLE PRECISION array, dimension (MAX(1,4*P))
On exit, DWORK(1) contains an estimate of the reciprocal
condition number of the matrix DR.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the matrix DR is singular;
= 2: the matrix DR is numerically singular (warning);
the calculations continued.
Method
The subroutine computes the matrices of the state-space
representation G = (A,B,C,D) by using the formulas:
-1 -1
A = AQR - BR * DR * CQR, C = DR * CQR,
-1 -1
B = BQ - BR * DR * DQ, D = DR * DQ.
References
[1] Varga A.
Coprime factors model reduction method based on
square-root balancing-free techniques.
System Analysis, Modelling and Simulation,
vol. 11, pp. 303-311, 1993.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB08HD.html 0000664 0000000 0000000 00000013420 14560147231 0020342 0 ustar 00root root 0000000 0000000
Purpose
To construct the state-space representation for the system
G = (A,B,C,D) from the factors Q = (AQR,BQR,CQ,DQ) and
R = (AQR,BQR,CR,DR) of its right coprime factorization
-1
G = Q * R ,
where G, Q and R are the corresponding transfer-function matrices.
Specification
SUBROUTINE SB08HD( N, M, P, A, LDA, B, LDB, C, LDC, D, LDD, CR,
$ LDCR, DR, LDDR, IWORK, DWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDCR, LDD, LDDR, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), CR(LDCR,*),
$ D(LDD,*), DR(LDDR,*), DWORK(*)
INTEGER IWORK(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. Also the number of rows of the
matrix B and the number of columns of the matrices C and
CR. N represents the order of the systems Q and R.
N >= 0.
M (input) INTEGER
The dimension of input vector. Also the number of columns
of the matrices B, D and DR and the number of rows of the
matrices CR and DR. M >= 0.
P (input) INTEGER
The dimension of output vector. Also the number of rows
of the matrices C and D. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix AQR of the systems
Q and R.
On exit, the leading N-by-N part of this array contains
the state dynamics matrix of the system G.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input/state matrix BQR of the systems Q and R.
On exit, the leading N-by-M part of this array contains
the input/state matrix of the system G.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix CQ of the system Q.
On exit, the leading P-by-N part of this array contains
the state/output matrix of the system G.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the input/output matrix DQ of the system Q.
On exit, the leading P-by-M part of this array contains
the input/output matrix of the system G.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
CR (input) DOUBLE PRECISION array, dimension (LDCR,N)
The leading M-by-N part of this array must contain the
state/output matrix CR of the system R.
LDCR INTEGER
The leading dimension of array CR. LDCR >= MAX(1,M).
DR (input/output) DOUBLE PRECISION array, dimension (LDDR,M)
On entry, the leading M-by-M part of this array must
contain the input/output matrix DR of the system R.
On exit, the leading M-by-M part of this array contains
the LU factorization of the matrix DR, as computed by
LAPACK Library routine DGETRF.
LDDR INTEGER
The leading dimension of array DR. LDDR >= MAX(1,M).
Workspace
IWORK INTEGER array, dimension (M)
DWORK DOUBLE PRECISION array, dimension (MAX(1,4*M))
On exit, DWORK(1) contains an estimate of the reciprocal
condition number of the matrix DR.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the matrix DR is singular;
= 2: the matrix DR is numerically singular (warning);
the calculations continued.
Method
The subroutine computes the matrices of the state-space
representation G = (A,B,C,D) by using the formulas:
-1 -1
A = AQR - BQR * DR * CR, B = BQR * DR ,
-1 -1
C = CQ - DQ * DR * CR, D = DQ * DR .
References
[1] Varga A.
Coprime factors model reduction method based on
square-root balancing-free techniques.
System Analysis, Modelling and Simulation,
vol. 11, pp. 303-311, 1993.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB08MD.html 0000664 0000000 0000000 00000023014 14560147231 0020347 0 ustar 00root root 0000000 0000000
Purpose
To compute a real polynomial E(s) such that
(a) E(-s) * E(s) = A(-s) * A(s) and
(b) E(s) is stable - that is, all the zeros of E(s) have
non-positive real parts,
which corresponds to computing the spectral factorization of the
real polynomial A(s) arising from continuous optimality problems.
The input polynomial may be supplied either in the form
A(s) = a(0) + a(1) * s + ... + a(DA) * s**DA
or as
B(s) = A(-s) * A(s)
= b(0) + b(1) * s**2 + ... + b(DA) * s**(2*DA) (1)
Specification
SUBROUTINE SB08MD( ACONA, DA, A, RES, E, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER ACONA
INTEGER DA, INFO, LDWORK
DOUBLE PRECISION RES
C .. Array Arguments ..
DOUBLE PRECISION A(*), DWORK(*), E(*)
Arguments
Mode Parameters
ACONA CHARACTER*1
Indicates whether the coefficients of A(s) or B(s) =
A(-s) * A(s) are to be supplied as follows:
= 'A': The coefficients of A(s) are to be supplied;
= 'B': The coefficients of B(s) are to be supplied.
Input/Output Parameters
DA (input) INTEGER
The degree of the polynomials A(s) and E(s). DA >= 0.
A (input/output) DOUBLE PRECISION array, dimension (DA+1)
On entry, this array must contain either the coefficients
of the polynomial A(s) in increasing powers of s if
ACONA = 'A', or the coefficients of the polynomial B(s) in
increasing powers of s**2 (see equation (1)) if ACONA =
'B'.
On exit, this array contains the coefficients of the
polynomial B(s) in increasing powers of s**2.
RES (output) DOUBLE PRECISION
An estimate of the accuracy with which the coefficients of
the polynomial E(s) have been computed (see also METHOD
and NUMERICAL ASPECTS).
E (output) DOUBLE PRECISION array, dimension (DA+1)
The coefficients of the spectral factor E(s) in increasing
powers of s.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK. LDWORK >= 5*DA+5.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if on entry, A(I) = 0.0, for I = 1,2,...,DA+1.
= 2: if on entry, ACONA = 'B' but the supplied
coefficients of the polynomial B(s) are not the
coefficients of A(-s) * A(s) for some real A(s);
in this case, RES and E are unassigned;
= 3: if the iterative process (see METHOD) has failed to
converge in 30 iterations;
= 4: if the last computed iterate (see METHOD) is
unstable. If ACONA = 'B', then the supplied
coefficients of the polynomial B(s) may not be the
coefficients of A(-s) * A(s) for some real A(s).
Method
_ _
Let A(s) be the conjugate polynomial of A(s), i.e., A(s) = A(-s).
The method used by the routine is based on applying the
Newton-Raphson iteration to the function
_ _
F(e) = A * A - e * e,
which leads to the iteration formulae (see [1]):
_(i) (i) _(i) (i) _ )
q * x + x * q = 2 A * A )
) for i = 0, 1, 2,...
(i+1) (i) (i) )
q = (q + x )/2 )
(0) DA
Starting from q = (1 + s) (which has no zeros in the closed
(1) (2) (3)
right half-plane), the sequence of iterates q , q , q ,...
converges to a solution of F(e) = 0 which has no zeros in the
open right half-plane.
The iterates satisfy the following conditions:
(i)
(a) q is a stable polynomial (no zeros in the closed right
half-plane) and
(i) (i-1)
(b) q (1) <= q (1).
(i-1) (i)
The iterative process stops with q , (where i <= 30) if q
violates either (a) or (b), or if the condition
_(i) (i) _
(c) RES = ||(q q - A A)|| < tol,
is satisfied, where || . || denotes the largest coefficient of
_(i) (i) _
the polynomial (q q - A A) and tol is an estimate of the
_(i) (i)
rounding error in the computed coefficients of q q . If there
is no convergence after 30 iterations then the routine returns
with the Error Indicator (INFO) set to 3, and the value of RES may
indicate whether or not the last computed iterate is close to the
solution.
If ACONA = 'B', then it is possible that the equation e(-s) *
e(s) = B(s) has no real solution, which will be the case if A(1)
< 0 or if ( -1)**DA * A(DA+1) < 0.
References
[1] Vostry, Z.
New Algorithm for Polynomial Spectral Factorization with
Quadratic Convergence II.
Kybernetika, 12, pp. 248-259, 1976.
Numerical Aspects
The conditioning of the problem depends upon the distance of the zeros of A(s) from the imaginary axis and on their multiplicity. For a well-conditioned problem the accuracy of the computed coefficients of E(s) is of the order of RES. However, for problems with zeros near the imaginary axis or with multiple zeros, the value of RES may be an overestimate of the true accuracy.Further Comments
In order for the problem e(-s) * e(s) = B(s) to have a real solution e(s), it is necessary and sufficient that B(j*omega) >= 0 for any purely imaginary argument j*omega (see [1]).Example
Program Text
* SB08MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER DAMAX
PARAMETER ( DAMAX = 10 )
INTEGER LDWORK
PARAMETER ( LDWORK = 5*DAMAX+5 )
* .. Local Scalars ..
DOUBLE PRECISION RES
INTEGER DA, I, INFO
CHARACTER*1 ACONA
* .. Local Arrays ..
DOUBLE PRECISION A(DAMAX+1), DWORK(LDWORK), E(DAMAX+1)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL SB08MD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
READ ( NIN, FMT = '()' )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = * ) DA, ACONA
IF ( DA.LE.-1 .OR. DA.GT.DAMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) DA
ELSE
READ ( NIN, FMT = * ) ( A(I), I = 1,DA+1 )
* Compute the spectral factorization of the given polynomial.
CALL SB08MD( ACONA, DA, A, RES, E, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( LSAME( ACONA, 'A' ) ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 0, DA
WRITE ( NOUT, FMT = 99995 ) 2*I, A(I+1)
20 CONTINUE
WRITE ( NOUT, FMT = * )
END IF
WRITE ( NOUT, FMT = 99996 )
DO 40 I = 0, DA
WRITE ( NOUT, FMT = 99995 ) I, E(I+1)
40 CONTINUE
WRITE ( NOUT, FMT = 99994 ) RES
END IF
END IF
*
STOP
*
99999 FORMAT (' SB08MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB08MD = ',I2)
99997 FORMAT (' The coefficients of the polynomial B(s) are ',//' powe',
$ 'r of s coefficient ')
99996 FORMAT (' The coefficients of the spectral factor E(s) are ',
$ //' power of s coefficient ')
99995 FORMAT (2X,I5,9X,F9.4)
99994 FORMAT (/' RES = ',1P,E8.1)
99993 FORMAT (/' DA is out of range.',/' DA = ',I5)
END
Program Data
SB08MD EXAMPLE PROGRAM DATA 3 A 8.0 -6.0 -3.0 1.0Program Results
SB08MD EXAMPLE PROGRAM RESULTS
The coefficients of the polynomial B(s) are
power of s coefficient
0 64.0000
2 -84.0000
4 21.0000
6 -1.0000
The coefficients of the spectral factor E(s) are
power of s coefficient
0 8.0000
1 14.0000
2 7.0000
3 1.0000
RES = 2.7E-15
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB08ND.html 0000664 0000000 0000000 00000023051 14560147231 0020351 0 ustar 00root root 0000000 0000000
Purpose
To compute a real polynomial E(z) such that
(a) E(1/z) * E(z) = A(1/z) * A(z) and
(b) E(z) is stable - that is, E(z) has no zeros with modulus
greater than 1,
which corresponds to computing the spectral factorization of the
real polynomial A(z) arising from discrete optimality problems.
The input polynomial may be supplied either in the form
A(z) = a(0) + a(1) * z + ... + a(DA) * z**DA
or as
B(z) = A(1/z) * A(z)
= b(0) + b(1) * (z + 1/z) + ... + b(DA) * (z**DA + 1/z**DA)
(1)
Specification
SUBROUTINE SB08ND( ACONA, DA, A, RES, E, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER ACONA
INTEGER DA, INFO, LDWORK
DOUBLE PRECISION RES
C .. Array Arguments ..
DOUBLE PRECISION A(*), DWORK(*), E(*)
Arguments
Mode Parameters
ACONA CHARACTER*1
Indicates whether the coefficients of A(z) or B(z) =
A(1/z) * A(z) are to be supplied as follows:
= 'A': The coefficients of A(z) are to be supplied;
= 'B': The coefficients of B(z) are to be supplied.
Input/Output Parameters
DA (input) INTEGER
The degree of the polynomials A(z) and E(z). DA >= 0.
A (input/output) DOUBLE PRECISION array, dimension (DA+1)
On entry, if ACONA = 'A', this array must contain the
coefficients of the polynomial A(z) in increasing powers
of z, and if ACONA = 'B', this array must contain the
coefficients b ,b ,...,b of the polynomial B(z) in
0 1 DA
equation (1). That is, A(i) = b for i = 1,2,...,DA+1.
i-1
On exit, this array contains the coefficients of the
polynomial B(z) in eqation (1). Specifically, A(i)
contains b , for i = 1,2,...DA+1.
i-1
RES (output) DOUBLE PRECISION
An estimate of the accuracy with which the coefficients of
the polynomial E(z) have been computed (see also METHOD
and NUMERICAL ASPECTS).
E (output) DOUBLE PRECISION array, dimension (DA+1)
The coefficients of the spectral factor E(z) in increasing
powers of z.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK. LDWORK >= 5*DA+5.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 2: if on entry, ACONA = 'B' but the supplied
coefficients of the polynomial B(z) are not the
coefficients of A(1/z) * A(z) for some real A(z);
in this case, RES and E are unassigned;
= 3: if the iterative process (see METHOD) has failed to
converge in 30 iterations;
= 4: if the last computed iterate (see METHOD) is
unstable. If ACONA = 'B', then the supplied
coefficients of the polynomial B(z) may not be the
coefficients of A(1/z) * A(z) for some real A(z).
Method
_ _
Let A(z) be the conjugate polynomial of A(z), i.e., A(z) = A(1/z).
The method used by the routine is based on applying the
Newton-Raphson iteration to the function
_ _
F(e) = A * A - e * e,
which leads to the iteration formulae (see [1] and [2])
_(i) (i) _(i) (i) _ )
q * x + x * q = 2 A * A )
) for i = 0, 1, 2,...
(i+1) (i) (i) )
q = (q + x )/2 )
The iteration starts from
(0) DA
q (z) = (b(0) + b(1) * z + ... + b(DA) * z ) / SQRT( b(0))
which is a Hurwitz polynomial that has no zeros in the closed unit
(i)
circle (see [2], Theorem 3). Then lim q = e, the convergence is
uniform and e is a Hurwitz polynomial.
The iterates satisfy the following conditions:
(i)
(a) q has no zeros in the closed unit circle,
(i) (i-1)
(b) q <= q and
0 0
DA (i) 2 DA 2
(c) SUM (q ) - SUM (A ) >= 0.
k=0 k k=0 k
(i)
The iterative process stops if q violates (a), (b) or (c),
or if the condition
_(i) (i) _
(d) RES = ||(q q - A A)|| < tol,
is satisfied, where || . || denotes the largest coefficient of
_(i) (i) _
the polynomial (q q - A A) and tol is an estimate of the
_(i) (i)
rounding error in the computed coefficients of q q . If
(i-1)
condition (a) or (b) is violated then q is taken otherwise
(i)
q is used. Thus the computed reciprocal polynomial E(z) = z**DA
* q(1/z) is stable. If there is no convergence after 30 iterations
then the routine returns with the Error Indicator (INFO) set to 3,
and the value of RES may indicate whether or not the last computed
iterate is close to the solution.
(0)
If ACONA = 'B', then it is possible that q is not a Hurwitz
polynomial, in which case the equation e(1/z) * e(z) = B(z) has no
real solution (see [2], Theorem 3).
References
[1] Kucera, V.
Discrete Linear Control, The polynomial Approach.
John Wiley & Sons, Chichester, 1979.
[2] Vostry, Z.
New Algorithm for Polynomial Spectral Factorization with
Quadratic Convergence I.
Kybernetika, 11, pp. 415-422, 1975.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
* SB08ND EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER DAMAX
PARAMETER ( DAMAX = 10 )
INTEGER LDWORK
PARAMETER ( LDWORK = 5*DAMAX+5 )
* .. Local Scalars ..
DOUBLE PRECISION RES
INTEGER DA, I, INFO
CHARACTER*1 ACONA
* .. Local Arrays ..
DOUBLE PRECISION A(DAMAX+1), DWORK(LDWORK), E(DAMAX+1)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL SB08ND
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
READ ( NIN, FMT = '()' )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = * ) DA, ACONA
IF ( DA.LE.-1 .OR. DA.GT.DAMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) DA
ELSE
READ ( NIN, FMT = * ) ( A(I), I = 1,DA+1 )
* Compute the spectral factorization of the given polynomial.
CALL SB08ND( ACONA, DA, A, RES, E, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( LSAME( ACONA, 'A' ) ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 0, DA
WRITE ( NOUT, FMT = 99995 ) I, A(I+1)
20 CONTINUE
WRITE ( NOUT, FMT = * )
END IF
WRITE ( NOUT, FMT = 99996 )
DO 40 I = 0, DA
WRITE ( NOUT, FMT = 99995 ) I, E(I+1)
40 CONTINUE
WRITE ( NOUT, FMT = 99994 ) RES
END IF
END IF
*
STOP
*
99999 FORMAT (' SB08ND EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB08ND = ',I2)
99997 FORMAT (' The coefficients of the polynomial B(z) are ',//' powe',
$ 'r of z coefficient ')
99996 FORMAT (' The coefficients of the spectral factor E(z) are ',
$ //' power of z coefficient ')
99995 FORMAT (2X,I5,9X,F9.4)
99994 FORMAT (/' RES = ',1P,E8.1)
99993 FORMAT (/' DA is out of range.',/' DA = ',I5)
END
Program Data
SB08ND EXAMPLE PROGRAM DATA 2 A 2.0 4.5 1.0Program Results
SB08ND EXAMPLE PROGRAM RESULTS
The coefficients of the polynomial B(z) are
power of z coefficient
0 25.2500
1 13.5000
2 2.0000
The coefficients of the spectral factor E(z) are
power of z coefficient
0 0.5000
1 3.0000
2 4.0000
RES = 4.4E-16
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB09MD.html 0000664 0000000 0000000 00000020422 14560147231 0020350 0 ustar 00root root 0000000 0000000
Purpose
To compare two multivariable sequences M1(k) and M2(k) for k = 1,2,...,N, and evaluate their closeness. Each of the parameters M1(k) and M2(k) is an NC by NB matrix.Specification
SUBROUTINE SB09MD( N, NC, NB, H1, LDH1, H2, LDH2, SS, LDSS, SE,
$ LDSE, PRE, LDPRE, TOL, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDH1, LDH2, LDPRE, LDSE, LDSS, N, NB, NC
DOUBLE PRECISION TOL
C .. Array Arguments ..
DOUBLE PRECISION H1(LDH1,*), H2(LDH2,*), PRE(LDPRE,*),
$ SE(LDSE,*), SS(LDSS,*)
Arguments
Input/Output Parameters
N (input) INTEGER
The number of parameters. N >= 0.
NC (input) INTEGER
The number of rows in M1(k) and M2(k). NC >= 0.
NB (input) INTEGER
The number of columns in M1(k) and M2(k). NB >= 0.
H1 (input) DOUBLE PRECISION array, dimension (LDH1,N*NB)
The leading NC-by-N*NB part of this array must contain
the multivariable sequence M1(k), where k = 1,2,...,N.
Each parameter M1(k) is an NC-by-NB matrix, whose
(i,j)-th element must be stored in H1(i,(k-1)*NB+j) for
i = 1,2,...,NC and j = 1,2,...,NB.
LDH1 INTEGER
The leading dimension of array H1. LDH1 >= MAX(1,NC).
H2 (input) DOUBLE PRECISION array, dimension (LDH2,N*NB)
The leading NC-by-N*NB part of this array must contain
the multivariable sequence M2(k), where k = 1,2,...,N.
Each parameter M2(k) is an NC-by-NB matrix, whose
(i,j)-th element must be stored in H2(i,(k-1)*NB+j) for
i = 1,2,...,NC and j = 1,2,...,NB.
LDH2 INTEGER
The leading dimension of array H2. LDH2 >= MAX(1,NC).
SS (output) DOUBLE PRECISION array, dimension (LDSS,NB)
The leading NC-by-NB part of this array contains the
matrix SS.
LDSS INTEGER
The leading dimension of array SS. LDSS >= MAX(1,NC).
SE (output) DOUBLE PRECISION array, dimension (LDSE,NB)
The leading NC-by-NB part of this array contains the
quadratic error matrix SE.
LDSE INTEGER
The leading dimension of array SE. LDSE >= MAX(1,NC).
PRE (output) DOUBLE PRECISION array, dimension (LDPRE,NB)
The leading NC-by-NB part of this array contains the
percentage relative error matrix PRE.
LDPRE INTEGER
The leading dimension of array PRE. LDPRE >= MAX(1,NC).
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in the computation of the error
matrices SE and PRE. If the user sets TOL to be less than
EPS then the tolerance is taken as EPS, where EPS is the
machine precision (see LAPACK Library routine DLAMCH).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The (i,j)-th element of the matrix SS is defined by:
N 2
SS = SUM M1 (k) . (1)
ij k=1 ij
The (i,j)-th element of the quadratic error matrix SE is defined
by:
N 2
SE = SUM (M1 (k) - M2 (k)) . (2)
ij k=1 ij ij
The (i,j)-th element of the percentage relative error matrix PRE
is defined by:
PRE = 100 x SQRT( SE / SS ). (3)
ij ij ij
The following precautions are taken by the routine to guard
against underflow and overflow:
(i) if ABS( M1 (k) ) > 1/TOL or ABS( M1 (k) - M2 (k) ) > 1/TOL,
ij ij ij
then SE and SS are set to 1/TOL and PRE is set to 1; and
ij ij ij
(ii) if ABS( SS ) <= TOL, then PRE is set to 100.
ij ij
Numerical Aspects
The algorithm requires approximately
2xNBxNCx(N+1) multiplications/divisions,
4xNBxNCxN additions/subtractions and
NBxNC square roots.
Further Comments
NoneExample
Program Text
* SB09MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, NCMAX, NBMAX
PARAMETER ( NMAX = 20, NCMAX = 20, NBMAX = 20 )
INTEGER LDH1, LDH2, LDSS, LDSE, LDPRE
PARAMETER ( LDH1 = NCMAX, LDH2 = NCMAX, LDSS = NCMAX,
$ LDSE = NCMAX, LDPRE = NCMAX )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, J, N, NB, NC
* .. Local Arrays ..
DOUBLE PRECISION H1(LDH1,NMAX*NBMAX), H2(LDH2,NMAX*NBMAX),
$ PRE(LDPRE,NBMAX), SE(LDSE,NBMAX), SS(LDSS,NBMAX)
* .. External Subroutines ..
EXTERNAL SB09MD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, NC, NB, TOL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE IF ( NB.LT.0 .OR. NB.GT.NBMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) NB
ELSE IF ( NC.LT.0 .OR. NC.GT.NCMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) NC
ELSE
READ ( NIN, FMT = * ) ( ( H1(I,J), I = 1,NC ), J = 1,N*NB )
READ ( NIN, FMT = * ) ( ( H2(I,J), I = 1,NC ), J = 1,N*NB )
* Compare the given sequences and evaluate their closeness.
CALL SB09MD( N, NC, NB, H1, LDH1, H2, LDH2, SS, LDSS, SE, LDSE,
$ PRE, LDPRE, TOL, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, NC
WRITE ( NOUT, FMT = 99996 ) ( SS(I,J), J = 1,NB )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 40 I = 1, NC
WRITE ( NOUT, FMT = 99996 ) ( SE(I,J), J = 1,NB )
40 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 60 I = 1, NC
WRITE ( NOUT, FMT = 99996 ) ( PRE(I,J), J = 1,NB )
60 CONTINUE
END IF
END IF
STOP
*
99999 FORMAT (' SB09MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB09MD = ',I2)
99997 FORMAT (' The sum-of-squares matrix SS is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The quadratic error matrix SE is ')
99994 FORMAT (/' The percentage relative error matrix PRE is ')
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' NB is out of range.',/' NB = ',I5)
99991 FORMAT (/' NC is out of range.',/' NC = ',I5)
END
Program Data
SB09MD EXAMPLE PROGRAM DATA 2 2 2 0.0 1.3373 0.1205 0.6618 -0.3372 -0.4062 1.6120 0.9299 0.7429 1.1480 -0.1837 0.8843 -0.4947 -0.4616 1.4674 0.6028 0.9524Program Results
SB09MD EXAMPLE PROGRAM RESULTS The sum-of-squares matrix SS is 1.9534 1.3027 2.6131 0.6656 The quadratic error matrix SE is 0.0389 0.1565 0.1134 0.0687 The percentage relative error matrix PRE is 14.1125 34.6607 20.8363 32.1262
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB10AD.html 0000664 0000000 0000000 00000035373 14560147231 0020337 0 ustar 00root root 0000000 0000000
Purpose
To compute the matrices of an H-infinity optimal n-state
controller
| AK | BK |
K = |----|----|,
| CK | DK |
using modified Glover's and Doyle's 1988 formulas, for the system
| A | B1 B2 | | A | B |
P = |----|---------| = |---|---|
| C1 | D11 D12 | | C | D |
| C2 | D21 D22 |
and for the estimated minimal possible value of gamma with respect
to GTOL, where B2 has as column size the number of control inputs
(NCON) and C2 has as row size the number of measurements (NMEAS)
being provided to the controller, and then to compute the matrices
of the closed-loop system
| AC | BC |
G = |----|----|,
| CC | DC |
if the stabilizing controller exists.
It is assumed that
(A1) (A,B2) is stabilizable and (C2,A) is detectable,
(A2) D12 is full column rank and D21 is full row rank,
(A3) | A-j*omega*I B2 | has full column rank for all omega,
| C1 D12 |
(A4) | A-j*omega*I B1 | has full row rank for all omega.
| C2 D21 |
Specification
SUBROUTINE SB10AD( JOB, N, M, NP, NCON, NMEAS, GAMMA, A, LDA,
$ B, LDB, C, LDC, D, LDD, AK, LDAK, BK, LDBK, CK,
$ LDCK, DK, LDDK, AC, LDAC, BC, LDBC, CC, LDCC,
$ DC, LDDC, RCOND, GTOL, ACTOL, IWORK, LIWORK,
$ DWORK, LDWORK, BWORK, LBWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, JOB, LBWORK, LDA, LDAC, LDAK, LDB, LDBC,
$ LDBK, LDC, LDCC, LDCK, LDD, LDDC, LDDK, LDWORK,
$ LIWORK, M, N, NCON, NMEAS, NP
DOUBLE PRECISION ACTOL, GAMMA, GTOL
C .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), AC( LDAC, * ), AK( LDAK, * ),
$ B( LDB, * ), BC( LDBC, * ), BK( LDBK, * ),
$ C( LDC, * ), CC( LDCC, * ), CK( LDCK, * ),
$ D( LDD, * ), DC( LDDC, * ), DK( LDDK, * ),
$ DWORK( * ), RCOND( 4 )
Arguments
Input/Output Parameters
JOB (input) INTEGER
Indicates the strategy for reducing the GAMMA value, as
follows:
= 1: Use bisection method for decreasing GAMMA from GAMMA
to GAMMAMIN until the closed-loop system leaves
stability.
= 2: Scan from GAMMA to 0 trying to find the minimal GAMMA
for which the closed-loop system retains stability.
= 3: First bisection, then scanning.
= 4: Find suboptimal controller only.
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
NP (input) INTEGER
The row size of the matrix C. NP >= 0.
NCON (input) INTEGER
The number of control inputs (M2). M >= NCON >= 0,
NP-NMEAS >= NCON.
NMEAS (input) INTEGER
The number of measurements (NP2). NP >= NMEAS >= 0,
M-NCON >= NMEAS.
GAMMA (input/output) DOUBLE PRECISION
The initial value of gamma on input. It is assumed that
gamma is sufficiently large so that the controller is
admissible. GAMMA >= 0.
On output it contains the minimal estimated gamma.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
system input matrix B.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading NP-by-N part of this array must contain the
system output matrix C.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,NP).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading NP-by-M part of this array must contain the
system input/output matrix D.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1,NP).
AK (output) DOUBLE PRECISION array, dimension (LDAK,N)
The leading N-by-N part of this array contains the
controller state matrix AK.
LDAK INTEGER
The leading dimension of the array AK. LDAK >= max(1,N).
BK (output) DOUBLE PRECISION array, dimension (LDBK,NMEAS)
The leading N-by-NMEAS part of this array contains the
controller input matrix BK.
LDBK INTEGER
The leading dimension of the array BK. LDBK >= max(1,N).
CK (output) DOUBLE PRECISION array, dimension (LDCK,N)
The leading NCON-by-N part of this array contains the
controller output matrix CK.
LDCK INTEGER
The leading dimension of the array CK.
LDCK >= max(1,NCON).
DK (output) DOUBLE PRECISION array, dimension (LDDK,NMEAS)
The leading NCON-by-NMEAS part of this array contains the
controller input/output matrix DK.
LDDK INTEGER
The leading dimension of the array DK.
LDDK >= max(1,NCON).
AC (output) DOUBLE PRECISION array, dimension (LDAC,2*N)
The leading 2*N-by-2*N part of this array contains the
closed-loop system state matrix AC.
LDAC INTEGER
The leading dimension of the array AC.
LDAC >= max(1,2*N).
BC (output) DOUBLE PRECISION array, dimension (LDBC,M-NCON)
The leading 2*N-by-(M-NCON) part of this array contains
the closed-loop system input matrix BC.
LDBC INTEGER
The leading dimension of the array BC.
LDBC >= max(1,2*N).
CC (output) DOUBLE PRECISION array, dimension (LDCC,2*N)
The leading (NP-NMEAS)-by-2*N part of this array contains
the closed-loop system output matrix CC.
LDCC INTEGER
The leading dimension of the array CC.
LDCC >= max(1,NP-NMEAS).
DC (output) DOUBLE PRECISION array, dimension (LDDC,M-NCON)
The leading (NP-NMEAS)-by-(M-NCON) part of this array
contains the closed-loop system input/output matrix DC.
LDDC INTEGER
The leading dimension of the array DC.
LDDC >= max(1,NP-NMEAS).
RCOND (output) DOUBLE PRECISION array, dimension (4)
For the last successful step:
RCOND(1) contains the reciprocal condition number of the
control transformation matrix;
RCOND(2) contains the reciprocal condition number of the
measurement transformation matrix;
RCOND(3) contains an estimate of the reciprocal condition
number of the X-Riccati equation;
RCOND(4) contains an estimate of the reciprocal condition
number of the Y-Riccati equation.
Tolerances
GTOL DOUBLE PRECISION
Tolerance used for controlling the accuracy of GAMMA
and its distance to the estimated minimal possible
value of GAMMA.
If GTOL <= 0, then a default value equal to sqrt(EPS)
is used, where EPS is the relative machine precision.
ACTOL DOUBLE PRECISION
Upper bound for the poles of the closed-loop system
used for determining if it is stable.
ACTOL <= 0 for stable systems.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK INTEGER
The dimension of the array IWORK.
LIWORK >= max(2*max(N,M-NCON,NP-NMEAS,NCON,NMEAS),N*N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal
value of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= LW1 + max(1,LW2,LW3,LW4,LW5 + MAX(LW6,LW7)),
where
LW1 = N*M + NP*N + NP*M + M2*M2 + NP2*NP2;
LW2 = max( ( N + NP1 + 1 )*( N + M2 ) +
max( 3*( N + M2 ) + N + NP1, 5*( N + M2 ) ),
( N + NP2 )*( N + M1 + 1 ) +
max( 3*( N + NP2 ) + N + M1, 5*( N + NP2 ) ),
M2 + NP1*NP1 + max( NP1*max( N, M1 ),
3*M2 + NP1, 5*M2 ),
NP2 + M1*M1 + max( max( N, NP1 )*M1,
3*NP2 + M1, 5*NP2 ) );
LW3 = max( ND1*M1 + max( 4*min( ND1, M1 ) + max( ND1,M1 ),
6*min( ND1, M1 ) ),
NP1*ND2 + max( 4*min( NP1, ND2 ) +
max( NP1,ND2 ),
6*min( NP1, ND2 ) ) );
LW4 = 2*M*M + NP*NP + 2*M*N + M*NP + 2*N*NP;
LW5 = 2*N*N + M*N + N*NP;
LW6 = max( M*M + max( 2*M1, 3*N*N +
max( N*M, 10*N*N + 12*N + 5 ) ),
NP*NP + max( 2*NP1, 3*N*N +
max( N*NP, 10*N*N + 12*N + 5 ) ));
LW7 = M2*NP2 + NP2*NP2 + M2*M2 +
max( ND1*ND1 + max( 2*ND1, ( ND1 + ND2 )*NP2 ),
ND2*ND2 + max( 2*ND2, ND2*M2 ), 3*N,
N*( 2*NP2 + M2 ) +
max( 2*N*M2, M2*NP2 +
max( M2*M2 + 3*M2, NP2*( 2*NP2 +
M2 + max( NP2, N ) ) ) ) );
M1 = M - M2, NP1 = NP - NP2,
ND1 = NP1 - M2, ND2 = M1 - NP2.
For good performance, LDWORK must generally be larger.
BWORK LOGICAL array, dimension (LBWORK)
LBWORK INTEGER
The dimension of the array BWORK. LBWORK >= 2*N.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the matrix | A-j*omega*I B2 | had not full
| C1 D12 |
column rank in respect to the tolerance EPS;
= 2: if the matrix | A-j*omega*I B1 | had not full row
| C2 D21 |
rank in respect to the tolerance EPS;
= 3: if the matrix D12 had not full column rank in
respect to the tolerance SQRT(EPS);
= 4: if the matrix D21 had not full row rank in respect
to the tolerance SQRT(EPS);
= 5: if the singular value decomposition (SVD) algorithm
did not converge (when computing the SVD of one of
the matrices |A B2 |, |A B1 |, D12 or D21);
|C1 D12| |C2 D21|
= 6: if the controller is not admissible (too small value
of gamma);
= 7: if the X-Riccati equation was not solved
successfully (the controller is not admissible or
there are numerical difficulties);
= 8: if the Y-Riccati equation was not solved
successfully (the controller is not admissible or
there are numerical difficulties);
= 9: if the determinant of Im2 + Tu*D11HAT*Ty*D22 is
zero [3];
= 10: if there are numerical problems when estimating
singular values of D1111, D1112, D1111', D1121';
= 11: if the matrices Inp2 - D22*DK or Im2 - DK*D22
are singular to working precision;
= 12: if a stabilizing controller cannot be found.
Method
The routine implements the Glover's and Doyle's 1988 formulas [1], [2], modified to improve the efficiency as described in [3]. JOB = 1: It tries with a decreasing value of GAMMA, starting with the given, and with the newly obtained controller estimates of the closed-loop system. If it is stable, (i.e., max(eig(AC)) < ACTOL) the iterations can be continued until the given tolerance between GAMMA and the estimated GAMMAMIN is reached. Otherwise, in the next step GAMMA is increased. The step in the all next iterations is step = step/2. The closed-loop system is obtained by the formulas given in [2]. JOB = 2: The same as for JOB = 1, but with non-varying step till GAMMA = 0, step = max(0.1, GTOL). JOB = 3: Combines the JOB = 1 and JOB = 2 cases for a quicker procedure. JOB = 4: Suboptimal controller for current GAMMA only.References
[1] Glover, K. and Doyle, J.C.
State-space formulae for all stabilizing controllers that
satisfy an Hinf norm bound and relations to risk sensitivity.
Systems and Control Letters, vol. 11, pp. 167-172, 1988.
[2] Balas, G.J., Doyle, J.C., Glover, K., Packard, A., and
Smith, R.
mu-Analysis and Synthesis Toolbox.
The MathWorks Inc., Natick, MA, 1995.
[3] Petkov, P.Hr., Gu, D.W., and Konstantinov, M.M.
Fortran 77 routines for Hinf and H2 design of continuous-time
linear control systems.
Rep. 98-14, Department of Engineering, Leicester University,
Leicester, U.K., 1998.
Numerical Aspects
The accuracy of the result depends on the condition numbers of the input and output transformations and on the condition numbers of the two Riccati equations, as given by the values of RCOND(1), RCOND(2), RCOND(3) and RCOND(4), respectively. This approach by estimating the closed-loop system and checking its poles seems to be reliable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB10DD.html 0000664 0000000 0000000 00000042240 14560147231 0020331 0 ustar 00root root 0000000 0000000
Purpose
To compute the matrices of an H-infinity (sub)optimal n-state
controller
| AK | BK |
K = |----|----|,
| CK | DK |
for the discrete-time system
| A | B1 B2 | | A | B |
P = |----|---------| = |---|---|
| C1 | D11 D12 | | C | D |
| C2 | D21 D22 |
and for a given value of gamma, where B2 has as column size the
number of control inputs (NCON) and C2 has as row size the number
of measurements (NMEAS) being provided to the controller.
It is assumed that
(A1) (A,B2) is stabilizable and (C2,A) is detectable,
(A2) D12 is full column rank and D21 is full row rank,
j*Theta
(A3) | A-e *I B2 | has full column rank for all
| C1 D12 |
0 <= Theta < 2*Pi ,
j*Theta
(A4) | A-e *I B1 | has full row rank for all
| C2 D21 |
0 <= Theta < 2*Pi .
Specification
SUBROUTINE SB10DD( N, M, NP, NCON, NMEAS, GAMMA, A, LDA, B, LDB,
$ C, LDC, D, LDD, AK, LDAK, BK, LDBK, CK, LDCK,
$ DK, LDDK, X, LDX, Z, LDZ, RCOND, TOL, IWORK,
$ DWORK, LDWORK, BWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD,
$ LDDK, LDWORK, LDX, LDZ, M, N, NCON, NMEAS, NP
DOUBLE PRECISION GAMMA, TOL
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
$ BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
$ D( LDD, * ), DK( LDDK, * ), DWORK( * ),
$ RCOND( * ), X( LDX, * ), Z( LDZ, * )
LOGICAL BWORK( * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
NP (input) INTEGER
The row size of the matrix C. NP >= 0.
NCON (input) INTEGER
The number of control inputs (M2). M >= NCON >= 0,
NP-NMEAS >= NCON.
NMEAS (input) INTEGER
The number of measurements (NP2). NP >= NMEAS >= 0,
M-NCON >= NMEAS.
GAMMA (input) DOUBLE PRECISION
The value of gamma. It is assumed that gamma is
sufficiently large so that the controller is admissible.
GAMMA > 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
system input matrix B.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading NP-by-N part of this array must contain the
system output matrix C.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,NP).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading NP-by-M part of this array must contain the
system input/output matrix D.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1,NP).
AK (output) DOUBLE PRECISION array, dimension (LDAK,N)
The leading N-by-N part of this array contains the
controller state matrix AK.
LDAK INTEGER
The leading dimension of the array AK. LDAK >= max(1,N).
BK (output) DOUBLE PRECISION array, dimension (LDBK,NMEAS)
The leading N-by-NMEAS part of this array contains the
controller input matrix BK.
LDBK INTEGER
The leading dimension of the array BK. LDBK >= max(1,N).
CK (output) DOUBLE PRECISION array, dimension (LDCK,N)
The leading NCON-by-N part of this array contains the
controller output matrix CK.
LDCK INTEGER
The leading dimension of the array CK.
LDCK >= max(1,NCON).
DK (output) DOUBLE PRECISION array, dimension (LDDK,NMEAS)
The leading NCON-by-NMEAS part of this array contains the
controller input/output matrix DK.
LDDK INTEGER
The leading dimension of the array DK.
LDDK >= max(1,NCON).
X (output) DOUBLE PRECISION array, dimension (LDX,N)
The leading N-by-N part of this array contains the matrix
X, solution of the X-Riccati equation.
LDX INTEGER
The leading dimension of the array X. LDX >= max(1,N).
Z (output) DOUBLE PRECISION array, dimension (LDZ,N)
The leading N-by-N part of this array contains the matrix
Z, solution of the Z-Riccati equation.
LDZ INTEGER
The leading dimension of the array Z. LDZ >= max(1,N).
RCOND (output) DOUBLE PRECISION array, dimension (8)
RCOND contains estimates of the reciprocal condition
numbers of the matrices which are to be inverted and
estimates of the reciprocal condition numbers of the
Riccati equations which have to be solved during the
computation of the controller. (See the description of
the algorithm in [2].)
RCOND(1) contains the reciprocal condition number of the
matrix R3;
RCOND(2) contains the reciprocal condition number of the
matrix R1 - R2'*inv(R3)*R2;
RCOND(3) contains the reciprocal condition number of the
matrix V21;
RCOND(4) contains the reciprocal condition number of the
matrix St3;
RCOND(5) contains the reciprocal condition number of the
matrix V12;
RCOND(6) contains the reciprocal condition number of the
matrix Im2 + DKHAT*D22
RCOND(7) contains the reciprocal condition number of the
X-Riccati equation;
RCOND(8) contains the reciprocal condition number of the
Z-Riccati equation.
Tolerances
TOL DOUBLE PRECISION
Tolerance used in neglecting the small singular values
in rank determination. If TOL <= 0, then a default value
equal to 1000*EPS is used, where EPS is the relative
machine precision.
Workspace
IWORK INTEGER array, dimension (max(2*max(M2,N),M,M2+NP2,N*N))
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal
LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= max(LW1,LW2,LW3,LW4), where
LW1 = (N+NP1+1)*(N+M2) + max(3*(N+M2)+N+NP1,5*(N+M2));
LW2 = (N+NP2)*(N+M1+1) + max(3*(N+NP2)+N+M1,5*(N+NP2));
LW3 = 13*N*N + 2*M*M + N*(8*M+NP2) + M1*(M2+NP2) + 6*N +
max(14*N+23,16*N,2*N+M,3*M);
LW4 = 13*N*N + M*M + (8*N+M+M2+2*NP2)*(M2+NP2) + 6*N +
N*(M+NP2) + max(14*N+23,16*N,2*N+M2+NP2,3*(M2+NP2));
For good performance, LDWORK must generally be larger.
Denoting Q = max(M1,M2,NP1,NP2), an upper bound is
max((N+Q)*(N+Q+6),13*N*N + M*M + 2*Q*Q + N*(M+Q) +
max(M*(M+7*N),2*Q*(8*N+M+2*Q)) + 6*N +
max(14*N+23,16*N,2*N+max(M,2*Q),3*max(M,2*Q)).
BWORK LOGICAL array, dimension (2*N)
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
j*Theta
= 1: if the matrix | A-e *I B2 | had not full
| C1 D12 |
column rank;
j*Theta
= 2: if the matrix | A-e *I B1 | had not full
| C2 D21 |
row rank;
= 3: if the matrix D12 had not full column rank;
= 4: if the matrix D21 had not full row rank;
= 5: if the controller is not admissible (too small value
of gamma);
= 6: if the X-Riccati equation was not solved
successfully (the controller is not admissible or
there are numerical difficulties);
= 7: if the Z-Riccati equation was not solved
successfully (the controller is not admissible or
there are numerical difficulties);
= 8: if the matrix Im2 + DKHAT*D22 is singular.
= 9: if the singular value decomposition (SVD) algorithm
did not converge (when computing the SVD of one of
the matrices |A B2 |, |A B1 |, D12 or D21).
|C1 D12| |C2 D21|
Method
The routine implements the method presented in [1].References
[1] Green, M. and Limebeer, D.J.N.
Linear Robust Control.
Prentice-Hall, Englewood Cliffs, NJ, 1995.
[2] Petkov, P.Hr., Gu, D.W., and Konstantinov, M.M.
Fortran 77 routines for Hinf and H2 design of linear
discrete-time control systems.
Report 99-8, Department of Engineering, Leicester University,
April 1999.
Numerical Aspects
With approaching the minimum value of gamma some of the matrices which are to be inverted tend to become ill-conditioned and the X- or Z-Riccati equation may also become ill-conditioned which may deteriorate the accuracy of the result. (The corresponding reciprocal condition numbers are given in the output array RCOND.)Further Comments
NoneExample
Program Text
* SB10DD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 10, MMAX = 10, PMAX = 10 )
INTEGER LDA, LDB, LDC, LDD, LDAK, LDBK, LDCK, LDDK, LDX,
$ LDZ
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX, LDD = PMAX,
$ LDAK = NMAX, LDBK = NMAX, LDCK = PMAX,
$ LDDK = PMAX, LDX = NMAX, LDZ = NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MAX( 2*MAX( MMAX, NMAX ),
$ MMAX + PMAX, NMAX*NMAX ) )
INTEGER MPMX
PARAMETER ( MPMX = MAX( MMAX, PMAX ) )
INTEGER LDWORK
PARAMETER ( LDWORK =
$ MAX( ( NMAX + MPMX )*( NMAX + MPMX + 6 ),
$ 13*NMAX*NMAX + MMAX*MMAX + 2*MPMX*MPMX +
$ NMAX*( MMAX + MPMX ) +
$ MAX( MMAX*( MMAX + 7*NMAX ),
$ 2*MPMX*( 8*NMAX + MMAX + 2*MPMX ) )
$ + 6*NMAX +
$ MAX( 14*NMAX + 23, 16*NMAX,
$ 2*NMAX + MAX( MMAX, 2*MPMX ),
$ 3*MAX( MMAX, 2*MPMX ) ) ) )
* .. Local Scalars ..
DOUBLE PRECISION GAMMA, TOL
INTEGER I, INFO, J, M, N, NCON, NMEAS, NP
* .. Local Arrays ..
LOGICAL BWORK(2*NMAX)
INTEGER IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), AK(LDA,NMAX), B(LDB,MMAX),
$ BK(LDBK,PMAX), C(LDC,NMAX), CK(LDCK,NMAX),
$ D(LDD,MMAX), DK(LDDK,PMAX), X(LDX,NMAX),
$ Z(LDZ,NMAX), DWORK(LDWORK), RCOND( 8 )
* .. External Subroutines ..
EXTERNAL SB10DD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, NP, NCON, NMEAS
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE IF ( NP.LT.0 .OR. NP.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) NP
ELSE IF ( NCON.LT.0 .OR. NCON.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) NCON
ELSE IF ( NMEAS.LT.0 .OR. NMEAS.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) NMEAS
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,NP )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,NP )
READ ( NIN, FMT = * ) GAMMA, TOL
CALL SB10DD( N, M, NP, NCON, NMEAS, GAMMA, A, LDA, B, LDB,
$ C, LDC, D, LDD, AK, LDAK, BK, LDBK, CK, LDCK,
$ DK, LDDK, X, LDX, Z, LDZ, RCOND, TOL, IWORK,
$ DWORK, LDWORK, BWORK, INFO )
IF ( INFO.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99992 ) ( AK(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99992 ) ( BK(I,J), J = 1,NMEAS )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 30 I = 1, NCON
WRITE ( NOUT, FMT = 99992 ) ( CK(I,J), J = 1,N )
30 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 40 I = 1, NCON
WRITE ( NOUT, FMT = 99992 ) ( DK(I,J), J = 1,NMEAS )
40 CONTINUE
WRITE( NOUT, FMT = 99993 )
WRITE( NOUT, FMT = 99991 ) ( RCOND(I), I = 1, 8 )
ELSE
WRITE( NOUT, FMT = 99998 ) INFO
END IF
END IF
STOP
*
99999 FORMAT (' SB10DD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (/' INFO on exit from SB10DD =',I2)
99997 FORMAT (/' The controller state matrix AK is'/)
99996 FORMAT (/' The controller input matrix BK is'/)
99995 FORMAT (/' The controller output matrix CK is'/)
99994 FORMAT (/' The controller matrix DK is'/)
99993 FORMAT (/' The estimated condition numbers are'/)
99992 FORMAT (10(1X,F8.4))
99991 FORMAT ( 5(1X,D12.5))
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' NP is out of range.',/' NP = ',I5)
99987 FORMAT (/' NCON is out of range.',/' NCON = ',I5)
99986 FORMAT (/' NMEAS is out of range.',/' NMEAS = ',I5)
END
Program Data
SB10DD EXAMPLE PROGRAM DATA 6 5 5 2 2 -0.7 0.0 0.3 0.0 -0.5 -0.1 -0.6 0.2 -0.4 -0.3 0.0 0.0 -0.5 0.7 -0.1 0.0 0.0 -0.8 -0.7 0.0 0.0 -0.5 -1.0 0.0 0.0 0.3 0.6 -0.9 0.1 -0.4 0.5 -0.8 0.0 0.0 0.2 -0.9 -1.0 -2.0 -2.0 1.0 0.0 1.0 0.0 1.0 -2.0 1.0 -3.0 -4.0 0.0 2.0 -2.0 1.0 -2.0 1.0 0.0 -1.0 0.0 1.0 -2.0 0.0 3.0 1.0 0.0 3.0 -1.0 -2.0 1.0 -1.0 2.0 -2.0 0.0 -3.0 -3.0 0.0 1.0 -1.0 1.0 0.0 0.0 2.0 0.0 -4.0 0.0 -2.0 1.0 -3.0 0.0 0.0 3.0 1.0 0.0 1.0 -2.0 1.0 0.0 -2.0 1.0 -1.0 -2.0 0.0 0.0 0.0 1.0 0.0 1.0 0.0 2.0 -1.0 -3.0 0.0 1.0 0.0 1.0 0.0 1.0 -1.0 0.0 0.0 1.0 2.0 1.0 111.294 0.00000001Program Results
SB10DD EXAMPLE PROGRAM RESULTS The controller state matrix AK is -18.0030 52.0376 26.0831 -0.4271 -40.9022 18.0857 18.8203 -57.6244 -29.0938 0.5870 45.3309 -19.8644 -26.5994 77.9693 39.0368 -1.4020 -60.1129 26.6910 -21.4163 62.1719 30.7507 -0.9201 -48.6221 21.8351 -0.8911 4.2787 2.3286 -0.2424 -3.0376 1.2169 -5.3286 16.1955 8.4824 -0.2489 -12.2348 5.1590 The controller input matrix BK is 16.9788 14.1648 -18.9215 -15.6726 25.2046 21.2848 20.1122 16.8322 1.4104 1.2040 5.3181 4.5149 The controller output matrix CK is -9.1941 27.5165 13.7364 -0.3639 -21.5983 9.6025 3.6490 -10.6194 -5.2772 0.2432 8.1108 -3.6293 The controller matrix DK is 9.0317 7.5348 -3.4006 -2.8219 The estimated condition numbers are 0.24960D+00 0.98548D+00 0.99186D+00 0.63733D-05 0.48625D+00 0.29430D-01 0.56942D-02 0.12470D-01
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB10ED.html 0000664 0000000 0000000 00000041111 14560147231 0020326 0 ustar 00root root 0000000 0000000
Purpose
To compute the matrices of the H2 optimal n-state controller
| AK | BK |
K = |----|----|
| CK | DK |
for the discrete-time system
| A | B1 B2 | | A | B |
P = |----|---------| = |---|---| ,
| C1 | 0 D12 | | C | D |
| C2 | D21 D22 |
where B2 has as column size the number of control inputs (NCON)
and C2 has as row size the number of measurements (NMEAS) being
provided to the controller.
It is assumed that
(A1) (A,B2) is stabilizable and (C2,A) is detectable,
(A2) D12 is full column rank and D21 is full row rank,
j*Theta
(A3) | A-e *I B2 | has full column rank for all
| C1 D12 |
0 <= Theta < 2*Pi ,
j*Theta
(A4) | A-e *I B1 | has full row rank for all
| C2 D21 |
0 <= Theta < 2*Pi .
Specification
SUBROUTINE SB10ED( N, M, NP, NCON, NMEAS, A, LDA, B, LDB, C, LDC,
$ D, LDD, AK, LDAK, BK, LDBK, CK, LDCK, DK, LDDK,
$ RCOND, TOL, IWORK, DWORK, LDWORK, BWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD,
$ LDDK, LDWORK, M, N, NCON, NMEAS, NP
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
$ BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
$ D( LDD, * ), DK( LDDK, * ), DWORK( * ),
$ RCOND( * )
LOGICAL BWORK( * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
NP (input) INTEGER
The row size of the matrix C. NP >= 0.
NCON (input) INTEGER
The number of control inputs (M2). M >= NCON >= 0,
NP-NMEAS >= NCON.
NMEAS (input) INTEGER
The number of measurements (NP2). NP >= NMEAS >= 0,
M-NCON >= NMEAS.
A (input/worksp.) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A.
This array is modified internally, but it is restored on
exit.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
system input matrix B.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading NP-by-N part of this array must contain the
system output matrix C.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,NP).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading NP-by-M part of this array must contain the
system input/output matrix D.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1,NP).
AK (output) DOUBLE PRECISION array, dimension (LDAK,N)
The leading N-by-N part of this array contains the
controller state matrix AK.
LDAK INTEGER
The leading dimension of the array AK. LDAK >= max(1,N).
BK (output) DOUBLE PRECISION array, dimension (LDBK,NMEAS)
The leading N-by-NMEAS part of this array contains the
controller input matrix BK.
LDBK INTEGER
The leading dimension of the array BK. LDBK >= max(1,N).
CK (output) DOUBLE PRECISION array, dimension (LDCK,N)
The leading NCON-by-N part of this array contains the
controller output matrix CK.
LDCK INTEGER
The leading dimension of the array CK.
LDCK >= max(1,NCON).
DK (output) DOUBLE PRECISION array, dimension (LDDK,NMEAS)
The leading NCON-by-NMEAS part of this array contains the
controller input/output matrix DK.
LDDK INTEGER
The leading dimension of the array DK.
LDDK >= max(1,NCON).
RCOND (output) DOUBLE PRECISION array, dimension (7)
RCOND contains estimates the reciprocal condition
numbers of the matrices which are to be inverted and the
reciprocal condition numbers of the Riccati equations
which have to be solved during the computation of the
controller. (See the description of the algorithm in [2].)
RCOND(1) contains the reciprocal condition number of the
control transformation matrix TU;
RCOND(2) contains the reciprocal condition number of the
measurement transformation matrix TY;
RCOND(3) contains the reciprocal condition number of the
matrix Im2 + B2'*X2*B2;
RCOND(4) contains the reciprocal condition number of the
matrix Ip2 + C2*Y2*C2';
RCOND(5) contains the reciprocal condition number of the
X-Riccati equation;
RCOND(6) contains the reciprocal condition number of the
Y-Riccati equation;
RCOND(7) contains the reciprocal condition number of the
matrix Im2 + DKHAT*D22 .
Tolerances
TOL DOUBLE PRECISION
Tolerance used for controlling the accuracy of the
transformations applied for diagonalizing D12 and D21,
and for checking the nonsingularity of the matrices to be
inverted. If TOL <= 0, then a default value equal to
sqrt(EPS) is used, where EPS is the relative machine
precision.
Workspace
IWORK INTEGER array, dimension (max(2*M2,2*N,N*N,NP2))
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal
LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= N*M + NP*(N+M) + M2*M2 + NP2*NP2 +
max(1,LW1,LW2,LW3,LW4,LW5,LW6), where
LW1 = (N+NP1+1)*(N+M2) + max(3*(N+M2)+N+NP1,5*(N+M2)),
LW2 = (N+NP2)*(N+M1+1) + max(3*(N+NP2)+N+M1,5*(N+NP2)),
LW3 = M2 + NP1*NP1 + max(NP1*max(N,M1),3*M2+NP1,5*M2),
LW4 = NP2 + M1*M1 + max(max(N,NP1)*M1,3*NP2+M1,5*NP2),
LW5 = 2*N*N+max(1,14*N*N+6*N+max(14*N+23,16*N),M2*(N+M2+
max(3,M1)),NP2*(N+NP2+3)),
LW6 = max(N*M2,N*NP2,M2*NP2,M2*M2+4*M2),
with M1 = M - M2 and NP1 = NP - NP2.
For good performance, LDWORK must generally be larger.
Denoting Q = max(M1,M2,NP1,NP2), an upper bound is
2*Q*(3*Q+2*N)+max(1,(N+Q)*(N+Q+6),Q*(Q+max(N,Q,5)+1),
2*N*N+max(1,14*N*N+6*N+max(14*N+23,16*N),
Q*(N+Q+max(Q,3)))).
BWORK LOGICAL array, dimension (2*N)
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
j*Theta
= 1: if the matrix | A-e *I B2 | had not full
| C1 D12 |
column rank in respect to the tolerance EPS;
j*Theta
= 2: if the matrix | A-e *I B1 | had not full
| C2 D21 |
row rank in respect to the tolerance EPS;
= 3: if the matrix D12 had not full column rank in
respect to the tolerance TOL;
= 4: if the matrix D21 had not full row rank in respect
to the tolerance TOL;
= 5: if the singular value decomposition (SVD) algorithm
did not converge (when computing the SVD of one of
the matrices |A-I B2 |, |A-I B1 |, D12 or D21).
|C1 D12| |C2 D21|
= 6: if the X-Riccati equation was not solved
successfully;
= 7: if the matrix Im2 + B2'*X2*B2 is not positive
definite, or it is numerically singular (with
respect to the tolerance TOL);
= 8: if the Y-Riccati equation was not solved
successfully;
= 9: if the matrix Ip2 + C2*Y2*C2' is not positive
definite, or it is numerically singular (with
respect to the tolerance TOL);
=10: if the matrix Im2 + DKHAT*D22 is singular, or its
estimated condition number is larger than or equal
to 1/TOL.
Method
The routine implements the formulas given in [1].References
[1] Zhou, K., Doyle, J.C., and Glover, K.
Robust and Optimal Control.
Prentice-Hall, Upper Saddle River, NJ, 1996.
[2] Petkov, P.Hr., Gu, D.W., and Konstantinov, M.M.
Fortran 77 routines for Hinf and H2 design of linear
discrete-time control systems.
Report 99-8, Department of Engineering, Leicester University,
April 1999.
Numerical Aspects
The accuracy of the result depends on the condition numbers of the matrices which are to be inverted and on the condition numbers of the matrix Riccati equations which are to be solved in the computation of the controller. (The corresponding reciprocal condition numbers are given in the output array RCOND.)Further Comments
NoneExample
Program Text
* SB10ED EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 10, MMAX = 10, PMAX = 10 )
INTEGER LDA, LDB, LDC, LDD, LDAK, LDBK, LDCK, LDDK
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX, LDD = PMAX,
$ LDAK = NMAX, LDBK = NMAX, LDCK = PMAX,
$ LDDK = PMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MAX( 2*MMAX, PMAX, 2*NMAX,
$ NMAX*NMAX ) )
INTEGER MPMX
PARAMETER ( MPMX = MAX( MMAX, PMAX ) )
INTEGER LDWORK
PARAMETER ( LDWORK = 2*MPMX*( 3*MPMX + 2*NMAX ) +
$ MAX( ( NMAX + MPMX )*( NMAX + MPMX + 6 ),
$ MPMX*( MPMX + MAX( NMAX, MPMX, 5 ) + 1 ),
$ 2*NMAX*NMAX + MAX( 14*NMAX*NMAX + 6*NMAX +
$ MAX( 14*NMAX + 23, 16*NMAX ),
$ MPMX*( NMAX + MPMX + MAX( MPMX, 3 ) ) ) ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, J, M, N, NCON, NMEAS, NP
* .. Local Arrays ..
LOGICAL BWORK(2*NMAX)
INTEGER IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), AK(LDA,NMAX), B(LDB,MMAX),
$ BK(LDBK,MMAX), C(LDC,NMAX), CK(LDCK,NMAX),
$ D(LDD,MMAX), DK(LDDK,MMAX), DWORK(LDWORK),
$ RCOND( 8 )
* .. External Subroutines ..
EXTERNAL SB10ED
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, NP, NCON, NMEAS
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE IF ( NP.LT.0 .OR. NP.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) NP
ELSE IF ( NCON.LT.0 .OR. NCON.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) NCON
ELSE IF ( NMEAS.LT.0 .OR. NMEAS.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) NMEAS
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,NP )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,NP )
READ ( NIN, FMT = * ) TOL
CALL SB10ED( N, M, NP, NCON, NMEAS, A, LDA, B, LDB,
$ C, LDC, D, LDD, AK, LDAK, BK, LDBK, CK, LDCK,
$ DK, LDDK, RCOND, TOL, IWORK, DWORK, LDWORK,
$ BWORK, INFO )
IF ( INFO.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99992 ) ( AK(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99992 ) ( BK(I,J), J = 1,NMEAS )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 30 I = 1, NCON
WRITE ( NOUT, FMT = 99992 ) ( CK(I,J), J = 1,N )
30 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 40 I = 1, NCON
WRITE ( NOUT, FMT = 99992 ) ( DK(I,J), J = 1,NMEAS )
40 CONTINUE
WRITE( NOUT, FMT = 99993 )
WRITE( NOUT, FMT = 99991 ) ( RCOND(I), I = 1, 7 )
ELSE
WRITE( NOUT, FMT = 99998 ) INFO
END IF
END IF
STOP
*
99999 FORMAT (' SB10ED EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (/' INFO on exit from SB10ED =',I2)
99997 FORMAT (' The controller state matrix AK is'/)
99996 FORMAT (/' The controller input matrix BK is'/)
99995 FORMAT (/' The controller output matrix CK is'/)
99994 FORMAT (/' The controller matrix DK is'/)
99993 FORMAT (/' The estimated condition numbers are'/)
99992 FORMAT (10(1X,F8.4))
99991 FORMAT ( 5(1X,D12.5))
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' NCON is out of range.',/' NCON = ',I5)
99986 FORMAT (/' NMEAS is out of range.',/' NMEAS = ',I5)
END
Program Data
SB10ED EXAMPLE PROGRAM DATA 6 5 5 2 2 -0.7 0.0 0.3 0.0 -0.5 -0.1 -0.6 0.2 -0.4 -0.3 0.0 0.0 -0.5 0.7 -0.1 0.0 0.0 -0.8 -0.7 0.0 0.0 -0.5 -1.0 0.0 0.0 0.3 0.6 -0.9 0.1 -0.4 0.5 -0.8 0.0 0.0 0.2 -0.9 -1.0 -2.0 -2.0 1.0 0.0 1.0 0.0 1.0 -2.0 1.0 -3.0 -4.0 0.0 2.0 -2.0 1.0 -2.0 1.0 0.0 -1.0 0.0 1.0 -2.0 0.0 3.0 1.0 0.0 3.0 -1.0 -2.0 1.0 -1.0 2.0 -2.0 0.0 -3.0 -3.0 0.0 1.0 -1.0 1.0 0.0 0.0 2.0 0.0 -4.0 0.0 -2.0 1.0 -3.0 0.0 0.0 3.0 1.0 0.0 1.0 -2.0 1.0 0.0 -2.0 1.0 -1.0 -2.0 0.0 0.0 0.0 1.0 0.0 1.0 0.0 2.0 -1.0 -3.0 0.0 1.0 0.0 1.0 0.0 1.0 -1.0 0.0 0.0 1.0 2.0 1.0 0.00000001Program Results
SB10ED EXAMPLE PROGRAM RESULTS The controller state matrix AK is -0.0551 -2.1891 -0.6607 -0.2532 0.6674 -1.0044 -1.0379 2.3804 0.5031 0.3960 -0.6605 1.2673 -0.0876 -2.1320 -0.4701 -1.1461 1.2927 -1.5116 -0.1358 -2.1237 -0.9560 -0.7144 0.6673 -0.7957 0.4900 0.0895 0.2634 -0.2354 0.1623 -0.2663 0.1672 -0.4163 0.2871 -0.1983 0.4944 -0.6967 The controller input matrix BK is -0.5985 -0.5464 0.5285 0.6087 -0.7600 -0.4472 -0.7288 -0.6090 0.0532 0.0658 -0.0663 0.0059 The controller output matrix CK is 0.2500 -1.0200 -0.3371 -0.2733 0.2747 -0.4444 0.0654 0.2095 0.0632 0.2089 -0.1895 0.1834 The controller matrix DK is -0.2181 -0.2070 0.1094 0.1159 The estimated condition numbers are 0.10000D+01 0.10000D+01 0.25207D+00 0.83985D-01 0.48628D-02 0.55015D-03 0.49886D+00
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB10FD.html 0000664 0000000 0000000 00000045606 14560147231 0020344 0 ustar 00root root 0000000 0000000
Purpose
To compute the matrices of an H-infinity (sub)optimal n-state
controller
| AK | BK |
K = |----|----|,
| CK | DK |
using modified Glover's and Doyle's 1988 formulas, for the system
| A | B1 B2 | | A | B |
P = |----|---------| = |---|---|
| C1 | D11 D12 | | C | D |
| C2 | D21 D22 |
and for a given value of gamma, where B2 has as column size the
number of control inputs (NCON) and C2 has as row size the number
of measurements (NMEAS) being provided to the controller.
It is assumed that
(A1) (A,B2) is stabilizable and (C2,A) is detectable,
(A2) D12 is full column rank and D21 is full row rank,
(A3) | A-j*omega*I B2 | has full column rank for all omega,
| C1 D12 |
(A4) | A-j*omega*I B1 | has full row rank for all omega.
| C2 D21 |
Specification
SUBROUTINE SB10FD( N, M, NP, NCON, NMEAS, GAMMA, A, LDA, B, LDB,
$ C, LDC, D, LDD, AK, LDAK, BK, LDBK, CK, LDCK,
$ DK, LDDK, RCOND, TOL, IWORK, DWORK, LDWORK,
$ BWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD,
$ LDDK, LDWORK, M, N, NCON, NMEAS, NP
DOUBLE PRECISION GAMMA, TOL
C .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
$ BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
$ D( LDD, * ), DK( LDDK, * ), DWORK( * ),
$ RCOND( 4 )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
NP (input) INTEGER
The row size of the matrix C. NP >= 0.
NCON (input) INTEGER
The number of control inputs (M2). M >= NCON >= 0,
NP-NMEAS >= NCON.
NMEAS (input) INTEGER
The number of measurements (NP2). NP >= NMEAS >= 0,
M-NCON >= NMEAS.
GAMMA (input) DOUBLE PRECISION
The value of gamma. It is assumed that gamma is
sufficiently large so that the controller is admissible.
GAMMA >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
system input matrix B.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading NP-by-N part of this array must contain the
system output matrix C.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,NP).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading NP-by-M part of this array must contain the
system input/output matrix D.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1,NP).
AK (output) DOUBLE PRECISION array, dimension (LDAK,N)
The leading N-by-N part of this array contains the
controller state matrix AK.
LDAK INTEGER
The leading dimension of the array AK. LDAK >= max(1,N).
BK (output) DOUBLE PRECISION array, dimension (LDBK,NMEAS)
The leading N-by-NMEAS part of this array contains the
controller input matrix BK.
LDBK INTEGER
The leading dimension of the array BK. LDBK >= max(1,N).
CK (output) DOUBLE PRECISION array, dimension (LDCK,N)
The leading NCON-by-N part of this array contains the
controller output matrix CK.
LDCK INTEGER
The leading dimension of the array CK.
LDCK >= max(1,NCON).
DK (output) DOUBLE PRECISION array, dimension (LDDK,NMEAS)
The leading NCON-by-NMEAS part of this array contains the
controller input/output matrix DK.
LDDK INTEGER
The leading dimension of the array DK.
LDDK >= max(1,NCON).
RCOND (output) DOUBLE PRECISION array, dimension (4)
RCOND(1) contains the reciprocal condition number of the
control transformation matrix;
RCOND(2) contains the reciprocal condition number of the
measurement transformation matrix;
RCOND(3) contains an estimate of the reciprocal condition
number of the X-Riccati equation;
RCOND(4) contains an estimate of the reciprocal condition
number of the Y-Riccati equation.
Tolerances
TOL DOUBLE PRECISION
Tolerance used for controlling the accuracy of the applied
transformations for computing the normalized form in
SLICOT Library routine SB10PD. Transformation matrices
whose reciprocal condition numbers are less than TOL are
not allowed. If TOL <= 0, then a default value equal to
sqrt(EPS) is used, where EPS is the relative machine
precision.
Workspace
IWORK INTEGER array, dimension (LIWORK), where
LIWORK = max(2*max(N,M-NCON,NP-NMEAS,NCON),N*N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal
LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= N*M + NP*(N+M) + M2*M2 + NP2*NP2 +
max(1,LW1,LW2,LW3,LW4,LW5,LW6), where
LW1 = (N+NP1+1)*(N+M2) + max(3*(N+M2)+N+NP1,5*(N+M2)),
LW2 = (N+NP2)*(N+M1+1) + max(3*(N+NP2)+N+M1,5*(N+NP2)),
LW3 = M2 + NP1*NP1 + max(NP1*max(N,M1),3*M2+NP1,5*M2),
LW4 = NP2 + M1*M1 + max(max(N,NP1)*M1,3*NP2+M1,5*NP2),
LW5 = 2*N*N + N*(M+NP) +
max(1,M*M + max(2*M1,3*N*N+max(N*M,10*N*N+12*N+5)),
NP*NP + max(2*NP1,3*N*N +
max(N*NP,10*N*N+12*N+5))),
LW6 = 2*N*N + N*(M+NP) +
max(1, M2*NP2 + NP2*NP2 + M2*M2 +
max(D1*D1 + max(2*D1, (D1+D2)*NP2),
D2*D2 + max(2*D2, D2*M2), 3*N,
N*(2*NP2 + M2) +
max(2*N*M2, M2*NP2 +
max(M2*M2+3*M2, NP2*(2*NP2+
M2+max(NP2,N)))))),
with D1 = NP1 - M2, D2 = M1 - NP2,
NP1 = NP - NP2, M1 = M - M2.
For good performance, LDWORK must generally be larger.
Denoting Q = max(M1,M2,NP1,NP2), an upper bound is
2*Q*(3*Q+2*N)+max(1,(N+Q)*(N+Q+6),Q*(Q+max(N,Q,5)+1),
2*N*(N+2*Q)+max(1,4*Q*Q+
max(2*Q,3*N*N+max(2*N*Q,10*N*N+12*N+5)),
Q*(3*N+3*Q+max(2*N,4*Q+max(N,Q))))).
BWORK LOGICAL array, dimension (2*N)
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the matrix | A-j*omega*I B2 | had not full
| C1 D12 |
column rank in respect to the tolerance EPS;
= 2: if the matrix | A-j*omega*I B1 | had not full row
| C2 D21 |
rank in respect to the tolerance EPS;
= 3: if the matrix D12 had not full column rank in
respect to the tolerance TOL;
= 4: if the matrix D21 had not full row rank in respect
to the tolerance TOL;
= 5: if the singular value decomposition (SVD) algorithm
did not converge (when computing the SVD of one of
the matrices |A B2 |, |A B1 |, D12 or D21).
|C1 D12| |C2 D21|
= 6: if the controller is not admissible (too small value
of gamma);
= 7: if the X-Riccati equation was not solved
successfully (the controller is not admissible or
there are numerical difficulties);
= 8: if the Y-Riccati equation was not solved
successfully (the controller is not admissible or
there are numerical difficulties);
= 9: if the determinant of Im2 + Tu*D11HAT*Ty*D22 is
zero [3].
Method
The routine implements the Glover's and Doyle's 1988 formulas [1], [2] modified to improve the efficiency as described in [3].References
[1] Glover, K. and Doyle, J.C.
State-space formulae for all stabilizing controllers that
satisfy an Hinf norm bound and relations to risk sensitivity.
Systems and Control Letters, vol. 11, pp. 167-172, 1988.
[2] Balas, G.J., Doyle, J.C., Glover, K., Packard, A., and
Smith, R.
mu-Analysis and Synthesis Toolbox.
The MathWorks Inc., Natick, Mass., 1995.
[3] Petkov, P.Hr., Gu, D.W., and Konstantinov, M.M.
Fortran 77 routines for Hinf and H2 design of continuous-time
linear control systems.
Rep. 98-14, Department of Engineering, Leicester University,
Leicester, U.K., 1998.
Numerical Aspects
The accuracy of the result depends on the condition numbers of the input and output transformations and on the condition numbers of the two Riccati equations, as given by the values of RCOND(1), RCOND(2), RCOND(3) and RCOND(4), respectively.Further Comments
NoneExample
Program Text
* SB10FD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX, N2MAX
PARAMETER ( NMAX = 10, MMAX = 10, PMAX = 10, N2MAX = 20 )
INTEGER LDA, LDB, LDC, LDD, LDAK, LDBK, LDCK, LDDK,
$ LDAC, LDBC, LDCC, LDDC
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX, LDD = PMAX,
$ LDAK = NMAX, LDBK = NMAX, LDCK = MMAX,
$ LDDK = MMAX, LDAC = 2*NMAX, LDBC = 2*NMAX,
$ LDCC = PMAX, LDDC = PMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MAX( 2*MAX( NMAX, MMAX, PMAX ),
$ NMAX*NMAX ) )
INTEGER MPMX
PARAMETER ( MPMX = MAX( MMAX, PMAX ) )
INTEGER LDWORK
PARAMETER ( LDWORK = 2*MPMX*( 3*MPMX + 2*NMAX ) +
$ MAX( ( NMAX + MPMX )*( NMAX + MPMX + 6 ),
$ MPMX*( MPMX + MAX( NMAX, MPMX, 5 ) + 1 ),
$ 2*NMAX*( NMAX + 2*MPMX ) +
$ MAX( 4*MPMX*MPMX + MAX( 2*MPMX, 3*NMAX*NMAX +
$ MAX( 2*NMAX*MPMX, 10*NMAX*NMAX+12*NMAX+5 ) ),
$ MPMX*( 3*NMAX + 3*MPMX +
$ MAX( 2*NMAX, 4*MPMX +
$ MAX( NMAX, MPMX ) ) ) ) ) )
* .. Local Scalars ..
INTEGER SDIM
LOGICAL SELECT
DOUBLE PRECISION GAMMA, TOL
INTEGER I, INFO1, INFO2, INFO3, J, M, N, NCON, NMEAS, NP
* .. Local Arrays ..
LOGICAL BWORK(N2MAX)
INTEGER IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), AK(LDAK,NMAX), AC(LDAC,N2MAX),
$ B(LDB,MMAX), BK(LDBK,PMAX), BC(LDBC,MMAX),
$ C(LDC,NMAX), CK(LDCK,NMAX), CC(LDCC,N2MAX),
$ D(LDD,MMAX), DK(LDDK,PMAX), DC(LDDC,MMAX),
$ DWORK(LDWORK), RCOND( 4 )
* .. External Subroutines ..
EXTERNAL SB10FD, SB10LD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, NP, NCON, NMEAS
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) N
ELSE IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) M
ELSE IF ( NP.LT.0 .OR. NP.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99985 ) NP
ELSE IF ( NCON.LT.0 .OR. NCON.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99984 ) NCON
ELSE IF ( NMEAS.LT.0 .OR. NMEAS.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99983 ) NMEAS
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,NP )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,NP )
READ ( NIN, FMT = * ) GAMMA, TOL
* Compute the suboptimal controller
CALL SB10FD( N, M, NP, NCON, NMEAS, GAMMA, A, LDA, B, LDB,
$ C, LDC, D, LDD, AK, LDAK, BK, LDBK, CK, LDCK,
$ DK, LDDK, RCOND, TOL, IWORK, DWORK, LDWORK,
$ BWORK, INFO1 )
*
IF ( INFO1.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99989 ) ( AK(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99989 ) ( BK(I,J), J = 1,NMEAS )
20 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 30 I = 1, NCON
WRITE ( NOUT, FMT = 99989 ) ( CK(I,J), J = 1,N )
30 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NCON
WRITE ( NOUT, FMT = 99989 ) ( DK(I,J), J = 1,NMEAS )
40 CONTINUE
WRITE( NOUT, FMT = 99992 )
WRITE( NOUT, FMT = 99988 ) ( RCOND(I), I = 1, 4 )
* Compute the closed-loop matrices
CALL SB10LD(N, M, NP, NCON, NMEAS, A, LDA, B, LDB, C, LDC,
$ D, LDD, AK, LDAK, BK, LDBK, CK, LDCK, DK, LDDK,
$ AC, LDAC, BC, LDBC, CC, LDCC, DC, LDDC, IWORK,
$ DWORK, LDWORK, INFO2 )
*
IF ( INFO2.EQ.0 ) THEN
* Compute the closed-loop poles
CALL DGEES( 'N','N', SELECT, 2*N, AC, LDAC, SDIM,
$ DWORK(1), DWORK(2*N+1), DWORK, 2*N,
$ DWORK(4*N+1), LDWORK-4*N, BWORK, INFO3)
*
IF( INFO3.EQ.0 ) THEN
WRITE( NOUT, FMT = 99991 )
WRITE( NOUT, FMT = 99988 ) (DWORK(I), I =1, 2*N)
WRITE( NOUT, FMT = 99990 )
WRITE( NOUT, FMT = 99988 ) (DWORK(2*N+I), I =1, 2*N)
ELSE
WRITE( NOUT, FMT = 99996 ) INFO3
END IF
ELSE
WRITE( NOUT, FMT = 99997 ) INFO2
END IF
ELSE
WRITE( NOUT, FMT = 99998 ) INFO1
END IF
END IF
STOP
*
99999 FORMAT (' SB10FD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (/' INFO on exit from SB10FD =',I2)
99997 FORMAT (/' INFO on exit from SB10LD =',I2)
99996 FORMAT (' The controller state matrix AK is'/)
99995 FORMAT (/' The controller input matrix BK is'/)
99994 FORMAT (/' The controller output matrix CK is'/)
99993 FORMAT (/' The controller matrix DK is'/)
99992 FORMAT (/' The estimated condition numbers are'/)
99991 FORMAT (/' The real parts of the closed-loop system poles are'/)
99990 FORMAT (/' The imaginary parts of the closed-loop system',
$ ' poles are'/)
99989 FORMAT (10(1X,F8.4))
99988 FORMAT ( 5(1X,D12.5))
99987 FORMAT (/' N is out of range.',/' N = ',I5)
99986 FORMAT (/' M is out of range.',/' M = ',I5)
99985 FORMAT (/' N is out of range.',/' N = ',I5)
99984 FORMAT (/' NCON is out of range.',/' NCON = ',I5)
99983 FORMAT (/' NMEAS is out of range.',/' NMEAS = ',I5)
END
Program Data
SB10FD EXAMPLE PROGRAM DATA 6 5 5 2 2 -1.0 0.0 4.0 5.0 -3.0 -2.0 -2.0 4.0 -7.0 -2.0 0.0 3.0 -6.0 9.0 -5.0 0.0 2.0 -1.0 -8.0 4.0 7.0 -1.0 -3.0 0.0 2.0 5.0 8.0 -9.0 1.0 -4.0 3.0 -5.0 8.0 0.0 2.0 -6.0 -3.0 -4.0 -2.0 1.0 0.0 2.0 0.0 1.0 -5.0 2.0 -5.0 -7.0 0.0 7.0 -2.0 4.0 -6.0 1.0 1.0 -2.0 -3.0 9.0 -8.0 0.0 5.0 1.0 -2.0 3.0 -6.0 -2.0 1.0 -1.0 2.0 -4.0 0.0 -3.0 -3.0 0.0 5.0 -1.0 1.0 1.0 -7.0 5.0 0.0 -8.0 2.0 -2.0 9.0 -3.0 4.0 0.0 3.0 7.0 0.0 1.0 -2.0 1.0 -6.0 -2.0 1.0 -2.0 -3.0 0.0 0.0 0.0 4.0 0.0 1.0 0.0 5.0 -3.0 -4.0 0.0 1.0 0.0 1.0 0.0 1.0 -3.0 0.0 0.0 1.0 7.0 1.0 15.0 0.00000001Program Results
SB10FD EXAMPLE PROGRAM RESULTS The controller state matrix AK is -2.8043 14.7367 4.6658 8.1596 0.0848 2.5290 4.6609 3.2756 -3.5754 -2.8941 0.2393 8.2920 -15.3127 23.5592 -7.1229 2.7599 5.9775 -2.0285 -22.0691 16.4758 12.5523 -16.3602 4.4300 -3.3168 30.6789 -3.9026 -1.3868 26.2357 -8.8267 10.4860 -5.7429 0.0577 10.8216 -11.2275 1.5074 -10.7244 The controller input matrix BK is -0.1581 -0.0793 -0.9237 -0.5718 0.7984 0.6627 0.1145 0.1496 -0.6743 -0.2376 0.0196 -0.7598 The controller output matrix CK is -0.2480 -0.1713 -0.0880 0.1534 0.5016 -0.0730 2.8810 -0.3658 1.3007 0.3945 1.2244 2.5690 The controller matrix DK is 0.0554 0.1334 -0.3195 0.0333 The estimated condition numbers are 0.10000D+01 0.10000D+01 0.11241D-01 0.80492D-03 The real parts of the closed-loop system poles are -0.10731D+03 -0.66556D+02 -0.38269D+02 -0.38269D+02 -0.20089D+02 -0.62557D+01 -0.62557D+01 -0.32405D+01 -0.32405D+01 -0.17178D+01 -0.41466D+01 -0.76437D+01 The imaginary parts of the closed-loop system poles are 0.00000D+00 0.00000D+00 0.13114D+02 -0.13114D+02 0.00000D+00 0.12961D+02 -0.12961D+02 0.67998D+01 -0.67998D+01 0.00000D+00 0.00000D+00 0.00000D+00
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB10HD.html 0000664 0000000 0000000 00000034155 14560147231 0020343 0 ustar 00root root 0000000 0000000
Purpose
To compute the matrices of the H2 optimal n-state controller
| AK | BK |
K = |----|----|
| CK | DK |
for the system
| A | B1 B2 | | A | B |
P = |----|---------| = |---|---| ,
| C1 | 0 D12 | | C | D |
| C2 | D21 D22 |
where B2 has as column size the number of control inputs (NCON)
and C2 has as row size the number of measurements (NMEAS) being
provided to the controller.
It is assumed that
(A1) (A,B2) is stabilizable and (C2,A) is detectable,
(A2) The block D11 of D is zero,
(A3) D12 is full column rank and D21 is full row rank.
Specification
SUBROUTINE SB10HD( N, M, NP, NCON, NMEAS, A, LDA, B, LDB, C, LDC,
$ D, LDD, AK, LDAK, BK, LDBK, CK, LDCK, DK, LDDK,
$ RCOND, TOL, IWORK, DWORK, LDWORK, BWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD,
$ LDDK, LDWORK, M, N, NCON, NMEAS, NP
DOUBLE PRECISION TOL
C .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
$ BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
$ D( LDD, * ), DK( LDDK, * ), DWORK( * ),
$ RCOND( 4 )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
NP (input) INTEGER
The row size of the matrix C. NP >= 0.
NCON (input) INTEGER
The number of control inputs (M2). M >= NCON >= 0,
NP-NMEAS >= NCON.
NMEAS (input) INTEGER
The number of measurements (NP2). NP >= NMEAS >= 0,
M-NCON >= NMEAS.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
system input matrix B.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading NP-by-N part of this array must contain the
system output matrix C.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,NP).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading NP-by-M part of this array must contain the
system input/output matrix D.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1,NP).
AK (output) DOUBLE PRECISION array, dimension (LDAK,N)
The leading N-by-N part of this array contains the
controller state matrix AK.
LDAK INTEGER
The leading dimension of the array AK. LDAK >= max(1,N).
BK (output) DOUBLE PRECISION array, dimension (LDBK,NMEAS)
The leading N-by-NMEAS part of this array contains the
controller input matrix BK.
LDBK INTEGER
The leading dimension of the array BK. LDBK >= max(1,N).
CK (output) DOUBLE PRECISION array, dimension (LDCK,N)
The leading NCON-by-N part of this array contains the
controller output matrix CK.
LDCK INTEGER
The leading dimension of the array CK.
LDCK >= max(1,NCON).
DK (output) DOUBLE PRECISION array, dimension (LDDK,NMEAS)
The leading NCON-by-NMEAS part of this array contains the
controller input/output matrix DK.
LDDK INTEGER
The leading dimension of the array DK.
LDDK >= max(1,NCON).
RCOND (output) DOUBLE PRECISION array, dimension (4)
RCOND(1) contains the reciprocal condition number of the
control transformation matrix;
RCOND(2) contains the reciprocal condition number of the
measurement transformation matrix;
RCOND(3) contains an estimate of the reciprocal condition
number of the X-Riccati equation;
RCOND(4) contains an estimate of the reciprocal condition
number of the Y-Riccati equation.
Tolerances
TOL DOUBLE PRECISION
Tolerance used for controlling the accuracy of the applied
transformations for computing the normalized form in
SLICOT Library routine SB10UD. Transformation matrices
whose reciprocal condition numbers are less than TOL are
not allowed. If TOL <= 0, then a default value equal to
sqrt(EPS) is used, where EPS is the relative machine
precision.
Workspace
IWORK INTEGER array, dimension (max(2*N,N*N))
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal
LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= N*M + NP*(N+M) + M2*M2 + NP2*NP2 +
max(max(M2 + NP1*NP1 +
max(NP1*N,3*M2+NP1,5*M2),
NP2 + M1*M1 +
max(M1*N,3*NP2+M1,5*NP2),
N*M2,NP2*N,NP2*M2,1),
N*(14*N+12+M2+NP2)+5),
where M1 = M - M2 and NP1 = NP - NP2.
For good performance, LDWORK must generally be larger.
Denoting Q = max(M1,M2,NP1,NP2), an upper bound is
2*Q*(3*Q+2*N)+max(1,Q*(Q+max(N,5)+1),N*(14*N+12+2*Q)+5).
BWORK LOGICAL array, dimension (2*N)
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the matrix D12 had not full column rank in
respect to the tolerance TOL;
= 2: if the matrix D21 had not full row rank in respect
to the tolerance TOL;
= 3: if the singular value decomposition (SVD) algorithm
did not converge (when computing the SVD of one of
the matrices D12 or D21).
= 4: if the X-Riccati equation was not solved
successfully;
= 5: if the Y-Riccati equation was not solved
successfully.
Method
The routine implements the formulas given in [1], [2].References
[1] Zhou, K., Doyle, J.C., and Glover, K.
Robust and Optimal Control.
Prentice-Hall, Upper Saddle River, NJ, 1996.
[2] Balas, G.J., Doyle, J.C., Glover, K., Packard, A., and
Smith, R.
mu-Analysis and Synthesis Toolbox.
The MathWorks Inc., Natick, Mass., 1995.
Numerical Aspects
The accuracy of the result depends on the condition numbers of the input and output transformations and on the condition numbers of the two Riccati equations, as given by the values of RCOND(1), RCOND(2), RCOND(3) and RCOND(4), respectively.Further Comments
NoneExample
Program Text
* SB10HD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 10, MMAX = 10, PMAX = 10 )
INTEGER LDA, LDB, LDC, LDD, LDAK, LDBK, LDCK, LDDK
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX, LDD = PMAX,
$ LDAK = NMAX, LDBK = NMAX, LDCK = PMAX,
$ LDDK = PMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MAX( 2*NMAX, NMAX*NMAX ) )
INTEGER MPMX
PARAMETER ( MPMX = MAX( MMAX, PMAX ) )
INTEGER LDWORK
PARAMETER ( LDWORK = 2*MPMX*( 2*NMAX + 3*MPMX ) +
$ MAX( MPMX*( MPMX + MAX( NMAX, 5 ) + 1 ),
$ NMAX*( 14*NMAX + 12 + 2*MPMX ) + 5 ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, J, M, N, NCON, NMEAS, NP
* .. Local Arrays ..
LOGICAL BWORK(2*NMAX)
INTEGER IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), AK(LDA,NMAX), B(LDB,MMAX),
$ BK(LDBK,MMAX), C(LDC,NMAX), CK(LDCK,NMAX),
$ D(LDD,MMAX), DK(LDDK,MMAX), DWORK(LDWORK),
$ RCOND( 4 )
* .. External Subroutines ..
EXTERNAL SB10HD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, NP, NCON, NMEAS
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE IF ( NP.LT.0 .OR. NP.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) NP
ELSE IF ( NCON.LT.0 .OR. NCON.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) NCON
ELSE IF ( NMEAS.LT.0 .OR. NMEAS.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) NMEAS
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,NP )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,NP )
READ ( NIN, FMT = * ) TOL
* Compute the optimal H2 controller
CALL SB10HD( N, M, NP, NCON, NMEAS, A, LDA, B, LDB,
$ C, LDC, D, LDD, AK, LDAK, BK, LDBK, CK, LDCK,
$ DK, LDDK, RCOND, TOL, IWORK, DWORK, LDWORK,
$ BWORK, INFO )
*
IF ( INFO.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99992 ) ( AK(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99992 ) ( BK(I,J), J = 1,NMEAS )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 30 I = 1, NCON
WRITE ( NOUT, FMT = 99992 ) ( CK(I,J), J = 1,N )
30 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 40 I = 1, NCON
WRITE ( NOUT, FMT = 99992 ) ( DK(I,J), J = 1,NMEAS )
40 CONTINUE
WRITE( NOUT, FMT = 99993 )
WRITE( NOUT, FMT = 99991 ) ( RCOND(I), I = 1, 4 )
ELSE
WRITE( NOUT, FMT = 99998 ) INFO
END IF
END IF
STOP
*
99999 FORMAT (' SB10HD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (/' INFO on exit from SB10HD =',I2)
99997 FORMAT (' The controller state matrix AK is'/)
99996 FORMAT (/' The controller input matrix BK is'/)
99995 FORMAT (/' The controller output matrix CK is'/)
99994 FORMAT (/' The controller matrix DK is'/)
99993 FORMAT (/' The estimated condition numbers are'/)
99992 FORMAT (6(1X,F10.4))
99991 FORMAT (5(1X,D12.5))
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' NCON is out of range.',/' NCON = ',I5)
99986 FORMAT (/' NMEAS is out of range.',/' NMEAS = ',I5)
END
Program Data
SB10HD EXAMPLE PROGRAM DATA 6 5 5 2 2 -1.0 0.0 4.0 5.0 -3.0 -2.0 -2.0 4.0 -7.0 -2.0 0.0 3.0 -6.0 9.0 -5.0 0.0 2.0 -1.0 -8.0 4.0 7.0 -1.0 -3.0 0.0 2.0 5.0 8.0 -9.0 1.0 -4.0 3.0 -5.0 8.0 0.0 2.0 -6.0 -3.0 -4.0 -2.0 1.0 0.0 2.0 0.0 1.0 -5.0 2.0 -5.0 -7.0 0.0 7.0 -2.0 4.0 -6.0 1.0 1.0 -2.0 -3.0 9.0 -8.0 0.0 5.0 1.0 -2.0 3.0 -6.0 -2.0 1.0 -1.0 2.0 -4.0 0.0 -3.0 -3.0 0.0 5.0 -1.0 1.0 1.0 -7.0 5.0 0.0 -8.0 2.0 -2.0 9.0 -3.0 4.0 0.0 3.0 7.0 0.0 1.0 -2.0 1.0 -6.0 -2.0 0.0 0.0 0.0 -4.0 -1.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 1.0 3.0 1.0 0.0 1.0 -3.0 -2.0 0.0 1.0 7.0 1.0 0.00000001Program Results
SB10HD EXAMPLE PROGRAM RESULTS
The controller state matrix AK is
88.0015 -145.7298 -46.2424 82.2168 -45.2996 -31.1407
25.7489 -31.4642 -12.4198 9.4625 -3.5182 2.7056
54.3008 -102.4013 -41.4968 50.8412 -20.1286 -26.7191
108.1006 -198.0785 -45.4333 70.3962 -25.8591 -37.2741
-115.8900 226.1843 47.2549 -47.8435 -12.5004 34.7474
59.0362 -101.8471 -20.1052 36.7834 -16.1063 -26.4309
The controller input matrix BK is
3.7345 3.4758
-0.3020 0.6530
3.4735 4.0499
4.3198 7.2755
-3.9424 -10.5942
2.1784 2.5048
The controller output matrix CK is
-2.3346 3.2556 0.7150 -0.9724 0.6962 0.4074
7.6899 -8.4558 -2.9642 7.0365 -4.2844 0.1390
The controller matrix DK is
0.0000 0.0000
0.0000 0.0000
The estimated condition numbers are
0.23570D+00 0.26726D+00 0.22747D-01 0.21130D-02
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB10ID.html 0000664 0000000 0000000 00000027473 14560147231 0020351 0 ustar 00root root 0000000 0000000
Purpose
To compute the matrices of the positive feedback controller
| Ak | Bk |
K = |----|----|
| Ck | Dk |
for the shaped plant
| A | B |
G = |---|---|
| C | D |
in the McFarlane/Glover Loop Shaping Design Procedure.
Specification
SUBROUTINE SB10ID( N, M, NP, A, LDA, B, LDB, C, LDC, D, LDD,
$ FACTOR, NK, AK, LDAK, BK, LDBK, CK, LDCK,
$ DK, LDDK, RCOND, IWORK, DWORK, LDWORK, BWORK,
$ INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD,
$ LDDK, LDWORK, M, N, NK, NP
DOUBLE PRECISION FACTOR
C .. Array Arguments ..
INTEGER IWORK( * )
LOGICAL BWORK( * )
DOUBLE PRECISION A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
$ BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
$ D( LDD, * ), DK( LDDK, * ), DWORK( * ),
$ RCOND( 2 )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the plant. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
NP (input) INTEGER
The row size of the matrix C. NP >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A of the shaped plant.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
system input matrix B of the shaped plant.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading NP-by-N part of this array must contain the
system output matrix C of the shaped plant.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,NP).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading NP-by-M part of this array must contain the
system matrix D of the shaped plant.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1,NP).
FACTOR (input) DOUBLE PRECISION
= 1 implies that an optimal controller is required;
> 1 implies that a suboptimal controller is required,
achieving a performance FACTOR less than optimal.
FACTOR >= 1.
NK (output) INTEGER
The order of the positive feedback controller. NK <= N.
AK (output) DOUBLE PRECISION array, dimension (LDAK,N)
The leading NK-by-NK part of this array contains the
controller state matrix Ak.
LDAK INTEGER
The leading dimension of the array AK. LDAK >= max(1,N).
BK (output) DOUBLE PRECISION array, dimension (LDBK,NP)
The leading NK-by-NP part of this array contains the
controller input matrix Bk.
LDBK INTEGER
The leading dimension of the array BK. LDBK >= max(1,N).
CK (output) DOUBLE PRECISION array, dimension (LDCK,N)
The leading M-by-NK part of this array contains the
controller output matrix Ck.
LDCK INTEGER
The leading dimension of the array CK. LDCK >= max(1,M).
DK (output) DOUBLE PRECISION array, dimension (LDDK,NP)
The leading M-by-NP part of this array contains the
controller matrix Dk.
LDDK INTEGER
The leading dimension of the array DK. LDDK >= max(1,M).
RCOND (output) DOUBLE PRECISION array, dimension (2)
RCOND(1) contains an estimate of the reciprocal condition
number of the X-Riccati equation;
RCOND(2) contains an estimate of the reciprocal condition
number of the Z-Riccati equation.
Workspace
IWORK INTEGER array, dimension (max(2*N,N*N,M,NP))
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= 4*N*N + M*M + NP*NP + 2*M*N + N*NP + 4*N +
max( 6*N*N + 5 + max(1,4*N*N+8*N), N*NP + 2*N ).
For good performance, LDWORK must generally be larger.
An upper bound of LDWORK in the above formula is
LDWORK >= 10*N*N + M*M + NP*NP + 2*M*N + 2*N*NP + 4*N +
5 + max(1,4*N*N+8*N).
BWORK LOGICAL array, dimension (2*N)
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the X-Riccati equation is not solved successfully;
= 2: the Z-Riccati equation is not solved successfully;
= 3: the iteration to compute eigenvalues or singular
values failed to converge;
= 4: the matrix Ip - D*Dk is singular;
= 5: the matrix Im - Dk*D is singular;
= 6: the closed-loop system is unstable.
Method
The routine implements the formulas given in [1].References
[1] McFarlane, D. and Glover, K.
A loop shaping design procedure using H_infinity synthesis.
IEEE Trans. Automat. Control, vol. AC-37, no. 6, pp. 759-769,
1992.
Numerical Aspects
The accuracy of the results depends on the conditioning of the two Riccati equations solved in the controller design (see the output parameter RCOND).Further Comments
NoneExample
Program Text
* SB10ID EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 10, MMAX = 10, PMAX = 10 )
INTEGER LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD, LDDK
PARAMETER ( LDA = NMAX, LDAK = NMAX, LDB = NMAX,
$ LDBK = NMAX, LDC = PMAX, LDCK = MMAX,
$ LDD = PMAX, LDDK = MMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MAX( 2*NMAX, NMAX*NMAX, MMAX, PMAX ) )
INTEGER LDWORK
PARAMETER ( LDWORK = 4*NMAX*NMAX + MMAX*MMAX + PMAX*PMAX +
$ 2*MMAX*NMAX + NMAX*PMAX + 4*NMAX +
$ MAX( 10*NMAX*NMAX + 8*NMAX + 5,
$ NMAX*PMAX + 2*NMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION FACTOR
INTEGER I, INFO, J, M, N, NK, NP
* .. Local Arrays ..
LOGICAL BWORK(2*NMAX)
INTEGER IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), AK(LDA,NMAX), B(LDB,MMAX),
$ BK(LDBK,PMAX), C(LDC,NMAX), CK(LDCK,NMAX),
$ D(LDD,MMAX), DK(LDDK,PMAX), DWORK(LDWORK),
$ RCOND( 2 )
* .. External Subroutines ..
EXTERNAL SB10ID
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, NP
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE IF ( NP.LT.0 .OR. NP.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) NP
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,NP )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,NP )
READ ( NIN, FMT = * ) FACTOR
CALL SB10ID( N, M, NP, A, LDA, B, LDB, C, LDC, D, LDD,
$ FACTOR, NK, AK, LDAK, BK, LDBK, CK, LDCK,
$ DK, LDDK, RCOND, IWORK, DWORK, LDWORK,
$ BWORK, INFO )
IF ( INFO.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, NK
WRITE ( NOUT, FMT = 99992 ) ( AK(I,J), J = 1,NK )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NK
WRITE ( NOUT, FMT = 99992 ) ( BK(I,J), J = 1,NP )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 30 I = 1, M
WRITE ( NOUT, FMT = 99992 ) ( CK(I,J), J = 1,NK )
30 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 40 I = 1, M
WRITE ( NOUT, FMT = 99992 ) ( DK(I,J), J = 1,NP )
40 CONTINUE
WRITE( NOUT, FMT = 99993 )
WRITE( NOUT, FMT = 99991 ) ( RCOND(I), I = 1, 2 )
ELSE
WRITE( NOUT, FMT = 99998 ) INFO
END IF
END IF
STOP
*
99999 FORMAT (' SB10ID EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (/' INFO on exit from SB10ID =',I2)
99997 FORMAT (/' The controller state matrix AK is'/)
99996 FORMAT (/' The controller input matrix BK is'/)
99995 FORMAT (/' The controller output matrix CK is'/)
99994 FORMAT (/' The controller matrix DK is'/)
99993 FORMAT (/' The estimated condition numbers are'/)
99992 FORMAT (10(1X,F9.4))
99991 FORMAT ( 2(1X,D12.5))
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' NP is out of range.',/' NP = ',I5)
END
Program Data
SB10ID EXAMPLE PROGRAM DATA 6 2 3 -1.0 0.0 4.0 5.0 -3.0 -2.0 -2.0 4.0 -7.0 -2.0 0.0 3.0 -6.0 9.0 -5.0 0.0 2.0 -1.0 -8.0 4.0 7.0 -1.0 -3.0 0.0 2.0 5.0 8.0 -9.0 1.0 -4.0 3.0 -5.0 8.0 0.0 2.0 -6.0 -3.0 -4.0 2.0 0.0 -5.0 -7.0 4.0 -6.0 -3.0 9.0 1.0 -2.0 1.0 -1.0 2.0 -4.0 0.0 -3.0 -3.0 0.0 5.0 -1.0 1.0 1.0 -7.0 5.0 0.0 -8.0 2.0 -2.0 1.0 -2.0 0.0 4.0 5.0 -3.0 1.0Program Results
SB10ID EXAMPLE PROGRAM RESULTS
The controller state matrix AK is
-39.0671 9.9293 22.2322 -27.4113 43.8655
-6.6117 3.0006 11.0878 -11.4130 15.4269
33.6805 -6.6934 -23.9953 14.1438 -33.4358
-32.3191 9.7316 25.4033 -24.0473 42.0517
-44.1655 18.7767 34.8873 -42.4369 50.8437
The controller input matrix BK is
-10.2905 -16.5382 -10.9782
-4.3598 -8.7525 -5.1447
6.5962 1.8975 6.2316
-9.8770 -14.7041 -11.8778
-9.6726 -22.7309 -18.2692
The controller output matrix CK is
-0.6647 -0.0599 -1.0376 0.5619 1.7297
-8.4202 3.9573 7.3094 -7.6283 10.6768
The controller matrix DK is
0.8466 0.4979 -0.6993
-1.2226 -4.8689 -4.5056
The estimated condition numbers are
0.13861D-01 0.90541D-02
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB10JD.html 0000664 0000000 0000000 00000011572 14560147231 0020343 0 ustar 00root root 0000000 0000000
Purpose
To convert the descriptor state-space system
E*dx/dt = A*x + B*u
y = C*x + D*u
into regular state-space form
dx/dt = Ad*x + Bd*u
y = Cd*x + Dd*u .
Specification
SUBROUTINE SB10JD( N, M, NP, A, LDA, B, LDB, C, LDC, D, LDD, E,
$ LDE, NSYS, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDD, LDE, LDWORK, M, N,
$ NP, NSYS
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), DWORK( * ), E( LDE, * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the descriptor system. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
NP (input) INTEGER
The row size of the matrix C. NP >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state matrix A of the descriptor system.
On exit, the leading NSYS-by-NSYS part of this array
contains the state matrix Ad of the converted system.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input matrix B of the descriptor system.
On exit, the leading NSYS-by-M part of this array
contains the input matrix Bd of the converted system.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading NP-by-N part of this array must
contain the output matrix C of the descriptor system.
On exit, the leading NP-by-NSYS part of this array
contains the output matrix Cd of the converted system.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,NP).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading NP-by-M part of this array must
contain the matrix D of the descriptor system.
On exit, the leading NP-by-M part of this array contains
the matrix Dd of the converted system.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1,NP).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain the matrix E of the descriptor system.
On exit, this array contains no useful information.
LDE INTEGER
The leading dimension of the array E. LDE >= max(1,N).
NSYS (output) INTEGER
The order of the converted state-space system.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= max( 1, 2*N*N + 2*N + N*MAX( 5, N + M + NP ) ).
For good performance, LDWORK must generally be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the iteration for computing singular value
decomposition did not converge.
Method
The routine performs the transformations described in [1].References
[1] Chiang, R.Y. and Safonov, M.G.
Robust Control Toolbox User's Guide.
The MathWorks Inc., Natick, Mass., 1992.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the matrices of the positive feedback controller
| Ak | Bk |
K = |----|----|
| Ck | Dk |
for the shaped plant
| A | B |
G = |---|---|
| C | 0 |
in the Discrete-Time Loop Shaping Design Procedure.
Specification
SUBROUTINE SB10KD( N, M, NP, A, LDA, B, LDB, C, LDC, FACTOR,
$ AK, LDAK, BK, LDBK, CK, LDCK, DK, LDDK, RCOND,
$ IWORK, DWORK, LDWORK, BWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDDK,
$ LDWORK, M, N, NP
DOUBLE PRECISION FACTOR
C .. Array Arguments ..
INTEGER IWORK( * )
LOGICAL BWORK( * )
DOUBLE PRECISION A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
$ BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
$ DK( LDDK, * ), DWORK( * ), RCOND( 4 )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the plant. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
NP (input) INTEGER
The row size of the matrix C. NP >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A of the shaped plant.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
system input matrix B of the shaped plant.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading NP-by-N part of this array must contain the
system output matrix C of the shaped plant.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,NP).
FACTOR (input) DOUBLE PRECISION
= 1 implies that an optimal controller is required;
> 1 implies that a suboptimal controller is required
achieving a performance FACTOR less than optimal.
FACTOR >= 1.
AK (output) DOUBLE PRECISION array, dimension (LDAK,N)
The leading N-by-N part of this array contains the
controller state matrix Ak.
LDAK INTEGER
The leading dimension of the array AK. LDAK >= max(1,N).
BK (output) DOUBLE PRECISION array, dimension (LDBK,NP)
The leading N-by-NP part of this array contains the
controller input matrix Bk.
LDBK INTEGER
The leading dimension of the array BK. LDBK >= max(1,N).
CK (output) DOUBLE PRECISION array, dimension (LDCK,N)
The leading M-by-N part of this array contains the
controller output matrix Ck.
LDCK INTEGER
The leading dimension of the array CK. LDCK >= max(1,M).
DK (output) DOUBLE PRECISION array, dimension (LDDK,NP)
The leading M-by-NP part of this array contains the
controller matrix Dk.
LDDK INTEGER
The leading dimension of the array DK. LDDK >= max(1,M).
RCOND (output) DOUBLE PRECISION array, dimension (4)
RCOND(1) contains an estimate of the reciprocal condition
number of the linear system of equations from
which the solution of the P-Riccati equation is
obtained;
RCOND(2) contains an estimate of the reciprocal condition
number of the linear system of equations from
which the solution of the Q-Riccati equation is
obtained;
RCOND(3) contains an estimate of the reciprocal condition
number of the linear system of equations from
which the solution of the X-Riccati equation is
obtained;
RCOND(4) contains an estimate of the reciprocal condition
number of the matrix Rx + Bx'*X*Bx (see the
comments in the code).
Workspace
IWORK INTEGER array, dimension (2*max(N,NP+M))
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= 15*N*N + 6*N +
max( 14*N+23, 16*N, 2*N+NP+M, 3*(NP+M) ) +
max( N*N, 11*N*NP + 2*M*M + 8*NP*NP + 8*M*N +
4*M*NP + NP ).
For good performance, LDWORK must generally be larger.
BWORK LOGICAL array, dimension (2*N)
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the P-Riccati equation is not solved successfully;
= 2: the Q-Riccati equation is not solved successfully;
= 3: the X-Riccati equation is not solved successfully;
= 4: the iteration to compute eigenvalues failed to
converge;
= 5: the matrix Rx + Bx'*X*Bx is singular;
= 6: the closed-loop system is unstable.
Method
The routine implements the method presented in [1].References
[1] McFarlane, D. and Glover, K.
A loop shaping design procedure using H_infinity synthesis.
IEEE Trans. Automat. Control, vol. AC-37, no. 6, pp. 759-769,
1992.
Numerical Aspects
The accuracy of the results depends on the conditioning of the two Riccati equations solved in the controller design. For better conditioning it is advised to take FACTOR > 1.Further Comments
NoneExample
Program Text
* SB10KD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 10, MMAX = 10, PMAX = 10 )
INTEGER LDA, LDAK, LDB, LDBK, LDC, LDCK, LDDK
PARAMETER ( LDA = NMAX, LDAK = NMAX, LDB = NMAX,
$ LDBK = NMAX, LDC = PMAX, LDCK = MMAX,
$ LDDK = MMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = 2*MAX( NMAX, MMAX + PMAX ) )
INTEGER LDWORK
PARAMETER ( LDWORK = 15*NMAX*NMAX + 6*NMAX +
$ MAX( 14*NMAX + 23, 16*NMAX,
$ 2*NMAX+PMAX+MMAX,
$ 3*(PMAX+MMAX) ) +
$ MAX( NMAX*NMAX,
$ 11*NMAX*PMAX + 2*MMAX*MMAX +
$ 8*PMAX*PMAX + 8*MMAX*NMAX +
$ 4*MMAX*PMAX + PMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION FACTOR
INTEGER I, INFO, J, M, N, NP
* .. Local Arrays ..
LOGICAL BWORK(2*NMAX)
INTEGER IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), AK(LDA,NMAX), B(LDB,MMAX),
$ BK(LDBK,PMAX), C(LDC,NMAX), CK(LDCK,NMAX),
$ DK(LDDK,PMAX), DWORK(LDWORK), RCOND(4)
* .. External Subroutines ..
EXTERNAL SB10KD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, NP
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE IF ( NP.LT.0 .OR. NP.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) NP
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,NP )
READ ( NIN, FMT = * ) FACTOR
CALL SB10KD( N, M, NP, A, LDA, B, LDB, C, LDC, FACTOR, AK,
$ LDAK, BK, LDBK, CK, LDCK, DK, LDDK, RCOND,
$ IWORK, DWORK, LDWORK, BWORK, INFO )
IF ( INFO.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99992 ) ( AK(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99992 ) ( BK(I,J), J = 1,NP )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 30 I = 1, M
WRITE ( NOUT, FMT = 99992 ) ( CK(I,J), J = 1,N )
30 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 40 I = 1, M
WRITE ( NOUT, FMT = 99992 ) ( DK(I,J), J = 1,NP )
40 CONTINUE
WRITE( NOUT, FMT = 99993 )
WRITE( NOUT, FMT = 99991 ) ( RCOND(I), I = 1, 4 )
ELSE
WRITE( NOUT, FMT = 99998 ) INFO
END IF
END IF
STOP
*
99999 FORMAT (' SB10KD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (/' INFO on exit from SB10KD =',I2)
99997 FORMAT (/' The controller state matrix AK is'/)
99996 FORMAT (/' The controller input matrix BK is'/)
99995 FORMAT (/' The controller output matrix CK is'/)
99994 FORMAT (/' The controller matrix DK is'/)
99993 FORMAT (/' The estimated condition numbers are'/)
99992 FORMAT (10(1X,F8.4))
99991 FORMAT ( 5(1X,D12.5))
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' NP is out of range.',/' NP = ',I5)
END
Program Data
SB10KD EXAMPLE PROGRAM DATA 6 2 2 0.2 0.0 0.3 0.0 -0.3 -0.1 -0.3 0.2 -0.4 -0.3 0.0 0.0 -0.1 0.1 -0.1 0.0 0.0 -0.3 0.1 0.0 0.0 -0.1 -0.1 0.0 0.0 0.3 0.6 0.2 0.1 -0.4 0.2 -0.4 0.0 0.0 0.2 -0.2 -1.0 -2.0 1.0 3.0 -3.0 -4.0 1.0 -2.0 0.0 1.0 1.0 5.0 1.0 -1.0 2.0 -2.0 0.0 -3.0 -3.0 0.0 1.0 -1.0 1.0 -1.0 1.1Program Results
SB10KD EXAMPLE PROGRAM RESULTS The controller state matrix AK is 0.0337 0.0222 0.0858 0.1264 -0.1872 0.1547 0.4457 0.0668 -0.2255 -0.3204 -0.4548 -0.0691 -0.2419 -0.2506 -0.0982 -0.1321 -0.0130 -0.0838 -0.4402 0.3654 -0.0335 -0.2444 0.6366 -0.6469 -0.3623 0.3854 0.4162 0.4502 0.0065 0.1261 -0.0121 -0.4377 0.0604 0.2265 -0.3389 0.4542 The controller input matrix BK is 0.0931 -0.0269 -0.0872 0.1599 0.0956 -0.1469 -0.1728 0.0129 0.2022 -0.1154 0.2419 -0.1737 The controller output matrix CK is -0.3677 0.2188 0.0403 -0.0854 0.3564 -0.3535 0.1624 -0.0708 0.0058 0.0606 -0.2163 0.1802 The controller matrix DK is -0.0857 -0.0246 0.0460 0.0074 The estimated condition numbers are 0.11269D-01 0.17596D-01 0.18225D+00 0.75968D-03
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB10LD.html 0000664 0000000 0000000 00000016247 14560147231 0020351 0 ustar 00root root 0000000 0000000
Purpose
To compute the matrices of the closed-loop system
| AC | BC |
G = |----|----|,
| CC | DC |
from the matrices of the open-loop system
| A | B |
P = |---|---|
| C | D |
and the matrices of the controller
| AK | BK |
K = |----|----|.
| CK | DK |
Specification
SUBROUTINE SB10LD( N, M, NP, NCON, NMEAS, A, LDA, B, LDB, C, LDC,
$ D, LDD, AK, LDAK, BK, LDBK, CK, LDCK, DK, LDDK,
$ AC, LDAC, BC, LDBC, CC, LDCC, DC, LDDC, IWORK,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDAC, LDAK, LDB, LDBC, LDBK, LDC,
$ LDCC, LDCK, LDD, LDDC, LDDK, LDWORK, M, N,
$ NCON, NMEAS, NP
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), AC( LDAC, * ), AK( LDAK, * ),
$ B( LDB, * ), BC( LDBC, * ), BK( LDBK, * ),
$ C( LDC, * ), CC( LDCC, * ), CK( LDCK, * ),
$ D( LDD, * ), DC( LDDC, * ), DK( LDDK, * ),
$ DWORK( * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
NP (input) INTEGER
The row size of the matrix C. NP >= 0.
NCON (input) INTEGER
The number of control inputs (M2). M >= NCON >= 0.
NP-NMEAS >= NCON.
NMEAS (input) INTEGER
The number of measurements (NP2). NP >= NMEAS >= 0.
M-NCON >= NMEAS.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
system input matrix B.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading NP-by-N part of this array must contain the
system output matrix C.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,NP).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading NP-by-M part of this array must contain the
system input/output matrix D.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1,NP).
AK (input) DOUBLE PRECISION array, dimension (LDAK,N)
The leading N-by-N part of this array must contain the
controller state matrix AK.
LDAK INTEGER
The leading dimension of the array AK. LDAK >= max(1,N).
BK (input) DOUBLE PRECISION array, dimension (LDBK,NMEAS)
The leading N-by-NMEAS part of this array must contain the
controller input matrix BK.
LDBK INTEGER
The leading dimension of the array BK. LDBK >= max(1,N).
CK (input) DOUBLE PRECISION array, dimension (LDCK,N)
The leading NCON-by-N part of this array must contain the
controller output matrix CK.
LDCK INTEGER
The leading dimension of the array CK.
LDCK >= max(1,NCON).
DK (input) DOUBLE PRECISION array, dimension (LDDK,NMEAS)
The leading NCON-by-NMEAS part of this array must contain
the controller input/output matrix DK.
LDDK INTEGER
The leading dimension of the array DK.
LDDK >= max(1,NCON).
AC (output) DOUBLE PRECISION array, dimension (LDAC,2*N)
The leading 2*N-by-2*N part of this array contains the
closed-loop system state matrix AC.
LDAC INTEGER
The leading dimension of the array AC.
LDAC >= max(1,2*N).
BC (output) DOUBLE PRECISION array, dimension (LDBC,M-NCON)
The leading 2*N-by-(M-NCON) part of this array contains
the closed-loop system input matrix BC.
LDBC INTEGER
The leading dimension of the array BC.
LDBC >= max(1,2*N).
CC (output) DOUBLE PRECISION array, dimension (LDCC,2*N)
The leading (NP-NMEAS)-by-2*N part of this array contains
the closed-loop system output matrix CC.
LDCC INTEGER
The leading dimension of the array CC.
LDCC >= max(1,NP-NMEAS).
DC (output) DOUBLE PRECISION array, dimension (LDDC,M-NCON)
The leading (NP-NMEAS)-by-(M-NCON) part of this array
contains the closed-loop system input/output matrix DC.
LDDC INTEGER
The leading dimension of the array DC.
LDDC >= max(1,NP-NMEAS).
Workspace
IWORK INTEGER array, dimension (2*max(NCON,NMEAS))
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal
LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= 2*M*M+NP*NP+2*M*N+M*NP+2*N*NP.
For good performance, LDWORK must generally be larger.
Error Indicactor
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the matrix Inp2 - D22*DK is singular to working
precision;
= 2: if the matrix Im2 - DK*D22 is singular to working
precision.
Method
The routine implements the formulas given in [1].References
[1] Balas, G.J., Doyle, J.C., Glover, K., Packard, A., and
Smith, R.
mu-Analysis and Synthesis Toolbox.
The MathWorks Inc., Natick, Mass., 1995.
Numerical Aspects
The accuracy of the result depends on the condition numbers of the matrices Inp2 - D22*DK and Im2 - DK*D22.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To perform the D-step in the D-K iteration. It handles continuous-time case.Specification
SUBROUTINE SB10MD( NC, MP, LENDAT, F, ORD, MNB, NBLOCK, ITYPE,
$ QUTOL, A, LDA, B, LDB, C, LDC, D, LDD, OMEGA,
$ TOTORD, AD, LDAD, BD, LDBD, CD, LDCD, DD, LDDD,
$ MJU, IWORK, LIWORK, DWORK, LDWORK, ZWORK,
$ LZWORK, INFO )
C .. Scalar Arguments ..
INTEGER F, INFO, LDA, LDAD, LDB, LDBD, LDC, LDCD, LDD,
$ LDDD, LDWORK, LENDAT, LIWORK, LZWORK, MNB, MP,
$ NC, ORD, TOTORD
DOUBLE PRECISION QUTOL
C .. Array Arguments ..
INTEGER ITYPE(*), IWORK(*), NBLOCK(*)
DOUBLE PRECISION A(LDA, *), AD(LDAD, *), B(LDB, *), BD(LDBD, *),
$ C(LDC, *), CD(LDCD, *), D(LDD, *), DD(LDDD, *),
$ DWORK(*), MJU(*), OMEGA(*)
COMPLEX*16 ZWORK(*)
Arguments
Input/Output Parameters
NC (input) INTEGER
The order of the matrix A. NC >= 0.
MP (input) INTEGER
The order of the matrix D. MP >= 0.
LENDAT (input) INTEGER
The length of the vector OMEGA. LENDAT >= 2.
F (input) INTEGER
The number of the measurements and controls, i.e.,
the size of the block I_f in the D-scaling system.
F >= 0.
ORD (input/output) INTEGER
The MAX order of EACH block in the fitting procedure.
ORD <= LENDAT-1.
On exit, if ORD < 1 then ORD = 1.
MNB (input) INTEGER
The number of diagonal blocks in the block structure of
the uncertainty, and the length of the vectors NBLOCK
and ITYPE. 1 <= MNB <= MP.
NBLOCK (input) INTEGER array, dimension (MNB)
The vector of length MNB containing the block structure
of the uncertainty. NBLOCK(I), I = 1:MNB, is the size of
each block.
ITYPE (input) INTEGER array, dimension (MNB)
The vector of length MNB indicating the type of each
block.
For I = 1 : MNB,
ITYPE(I) = 1 indicates that the corresponding block is a
real block. IN THIS CASE ONLY MJU(JW) WILL BE ESTIMATED
CORRECTLY, BUT NOT D(S)!
ITYPE(I) = 2 indicates that the corresponding block is a
complex block. THIS IS THE ONLY ALLOWED VALUE NOW!
NBLOCK(I) must be equal to 1 if ITYPE(I) is equal to 1.
QUTOL (input) DOUBLE PRECISION
The acceptable mean relative error between the D(jw) and
the frequency responce of the estimated block
[ADi,BDi;CDi,DDi]. When it is reached, the result is
taken as good enough.
A good value is QUTOL = 2.0.
If QUTOL < 0 then only mju(jw) is being estimated,
not D(s).
A (input/output) DOUBLE PRECISION array, dimension (LDA,NC)
On entry, the leading NC-by-NC part of this array must
contain the A matrix of the closed-loop system.
On exit, if MP > 0, the leading NC-by-NC part of this
array contains an upper Hessenberg matrix similar to A.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,NC).
B (input/output) DOUBLE PRECISION array, dimension (LDB,MP)
On entry, the leading NC-by-MP part of this array must
contain the B matrix of the closed-loop system.
On exit, the leading NC-by-MP part of this array contains
the transformed B matrix corresponding to the Hessenberg
form of A.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,NC).
C (input/output) DOUBLE PRECISION array, dimension (LDC,NC)
On entry, the leading MP-by-NC part of this array must
contain the C matrix of the closed-loop system.
On exit, the leading MP-by-NC part of this array contains
the transformed C matrix corresponding to the Hessenberg
form of A.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,MP).
D (input) DOUBLE PRECISION array, dimension (LDD,MP)
The leading MP-by-MP part of this array must contain the
D matrix of the closed-loop system.
LDD INTEGER
The leading dimension of the array D. LDD >= MAX(1,MP).
OMEGA (input) DOUBLE PRECISION array, dimension (LENDAT)
The vector with the frequencies.
TOTORD (output) INTEGER
The TOTAL order of the D-scaling system.
TOTORD is set to zero, if QUTOL < 0.
AD (output) DOUBLE PRECISION array, dimension (LDAD,MP*ORD)
The leading TOTORD-by-TOTORD part of this array contains
the A matrix of the D-scaling system.
Not referenced if QUTOL < 0.
LDAD INTEGER
The leading dimension of the array AD.
LDAD >= MAX(1,MP*ORD), if QUTOL >= 0;
LDAD >= 1, if QUTOL < 0.
BD (output) DOUBLE PRECISION array, dimension (LDBD,MP+F)
The leading TOTORD-by-(MP+F) part of this array contains
the B matrix of the D-scaling system.
Not referenced if QUTOL < 0.
LDBD INTEGER
The leading dimension of the array BD.
LDBD >= MAX(1,MP*ORD), if QUTOL >= 0;
LDBD >= 1, if QUTOL < 0.
CD (output) DOUBLE PRECISION array, dimension (LDCD,MP*ORD)
The leading (MP+F)-by-TOTORD part of this array contains
the C matrix of the D-scaling system.
Not referenced if QUTOL < 0.
LDCD INTEGER
The leading dimension of the array CD.
LDCD >= MAX(1,MP+F), if QUTOL >= 0;
LDCD >= 1, if QUTOL < 0.
DD (output) DOUBLE PRECISION array, dimension (LDDD,MP+F)
The leading (MP+F)-by-(MP+F) part of this array contains
the D matrix of the D-scaling system.
Not referenced if QUTOL < 0.
LDDD INTEGER
The leading dimension of the array DD.
LDDD >= MAX(1,MP+F), if QUTOL >= 0;
LDDD >= 1, if QUTOL < 0.
MJU (output) DOUBLE PRECISION array, dimension (LENDAT)
The vector with the upper bound of the structured
singular value (mju) for each frequency in OMEGA.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK INTEGER
The length of the array IWORK.
LIWORK >= MAX( NC, 4*MNB-2, MP, 2*ORD+1 ), if QUTOL >= 0;
LIWORK >= MAX( NC, 4*MNB-2, MP ), if QUTOL < 0.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK, DWORK(2) returns the optimal value of LZWORK,
and DWORK(3) returns an estimate of the minimum reciprocal
of the condition numbers (with respect to inversion) of
the generated Hessenberg matrices.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( 3, LWM, LWD ), where
LWM = LWA + MAX( NC + MAX( NC, MP-1 ),
2*MP*MP*MNB - MP*MP + 9*MNB*MNB +
MP*MNB + 11*MP + 33*MNB - 11 );
LWD = LWB + MAX( 2, LW1, LW2, LW3, LW4, 2*ORD ),
if QUTOL >= 0;
LWD = 0, if QUTOL < 0;
LWA = MP*LENDAT + 2*MNB + MP - 1;
LWB = LENDAT*(MP + 2) + ORD*(ORD + 2) + 1;
LW1 = 2*LENDAT + 4*HNPTS; HNPTS = 2048;
LW2 = LENDAT + 6*HNPTS; MN = MIN( 2*LENDAT, 2*ORD+1 );
LW3 = 2*LENDAT*(2*ORD + 1) + MAX( 2*LENDAT, 2*ORD + 1 ) +
MAX( MN + 6*ORD + 4, 2*MN + 1 );
LW4 = MAX( ORD*ORD + 5*ORD, 6*ORD + 1 + MIN( 1, ORD ) ).
ZWORK COMPLEX*16 array, dimension (LZWORK)
LZWORK INTEGER
The length of the array ZWORK.
LZWORK >= MAX( LZM, LZD ), where
LZM = MAX( MP*MP + NC*MP + NC*NC + 2*NC,
6*MP*MP*MNB + 13*MP*MP + 6*MNB + 6*MP - 3 );
LZD = MAX( LENDAT*(2*ORD + 3), ORD*ORD + 3*ORD + 1 ),
if QUTOL >= 0;
LZD = 0, if QUTOL < 0.
Error Indicator
INFO (output) INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if one or more values w in OMEGA are (close to
some) poles of the closed-loop system, i.e., the
matrix jw*I - A is (numerically) singular;
= 2: the block sizes must be positive integers;
= 3: the sum of block sizes must be equal to MP;
= 4: the size of a real block must be equal to 1;
= 5: the block type must be either 1 or 2;
= 6: errors in solving linear equations or in matrix
inversion;
= 7: errors in computing eigenvalues or singular values.
= 1i: INFO on exit from SB10YD is i. (1i means 10 + i.)
Method
I. First, W(jw) for the given closed-loop system is being
estimated.
II. Now, AB13MD SLICOT subroutine can obtain the D(jw) scaling
system with respect to NBLOCK and ITYPE, and colaterally,
mju(jw).
If QUTOL < 0 then the estimations stop and the routine exits.
III. Now that we have D(jw), SB10YD subroutine can do block-by-
block fit. For each block it tries with an increasing order
of the fit, starting with 1 until the
(mean quadratic error + max quadratic error)/2
between the Dii(jw) and the estimated frequency responce
of the block becomes less than or equal to the routine
argument QUTOL, or the order becomes equal to ORD.
IV. Arrange the obtained blocks in the AD, BD, CD and DD
matrices and estimate the total order of D(s), TOTORD.
V. Add the system I_f to the system obtained in IV.
References
[1] Balas, G., Doyle, J., Glover, K., Packard, A. and Smith, R.
Mu-analysis and Synthesis toolbox - User's Guide,
The Mathworks Inc., Natick, MA, USA, 1998.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB10PD.html 0000664 0000000 0000000 00000020245 14560147231 0020346 0 ustar 00root root 0000000 0000000
Purpose
To reduce the matrices D12 and D21 of the linear time-invariant
system
| A | B1 B2 | | A | B |
P = |----|---------| = |---|---|
| C1 | D11 D12 | | C | D |
| C2 | D21 D22 |
to unit diagonal form, to transform the matrices B, C, and D11 to
satisfy the formulas in the computation of an H2 and H-infinity
(sub)optimal controllers and to check the rank conditions.
Specification
SUBROUTINE SB10PD( N, M, NP, NCON, NMEAS, A, LDA, B, LDB, C, LDC,
$ D, LDD, TU, LDTU, TY, LDTY, RCOND, TOL, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDD, LDTU, LDTY, LDWORK,
$ M, N, NCON, NMEAS, NP
DOUBLE PRECISION TOL
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), DWORK( * ), RCOND( 2 ),
$ TU( LDTU, * ), TY( LDTY, * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
NP (input) INTEGER
The row size of the matrix C. NP >= 0.
NCON (input) INTEGER
The number of control inputs (M2). M >= NCON >= 0,
NP-NMEAS >= NCON.
NMEAS (input) INTEGER
The number of measurements (NP2). NP >= NMEAS >= 0,
M-NCON >= NMEAS.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the system input matrix B.
On exit, the leading N-by-M part of this array contains
the transformed system input matrix B.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading NP-by-N part of this array must
contain the system output matrix C.
On exit, the leading NP-by-N part of this array contains
the transformed system output matrix C.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,NP).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading NP-by-M part of this array must
contain the system input/output matrix D. The
NMEAS-by-NCON trailing submatrix D22 is not referenced.
On exit, the leading (NP-NMEAS)-by-(M-NCON) part of this
array contains the transformed submatrix D11.
The transformed submatrices D12 = [ 0 Im2 ]' and
D21 = [ 0 Inp2 ] are not stored. The corresponding part
of this array contains no useful information.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1,NP).
TU (output) DOUBLE PRECISION array, dimension (LDTU,M2)
The leading M2-by-M2 part of this array contains the
control transformation matrix TU.
LDTU INTEGER
The leading dimension of the array TU. LDTU >= max(1,M2).
TY (output) DOUBLE PRECISION array, dimension (LDTY,NP2)
The leading NP2-by-NP2 part of this array contains the
measurement transformation matrix TY.
LDTY INTEGER
The leading dimension of the array TY.
LDTY >= max(1,NP2).
RCOND (output) DOUBLE PRECISION array, dimension (2)
RCOND(1) contains the reciprocal condition number of the
control transformation matrix TU;
RCOND(2) contains the reciprocal condition number of the
measurement transformation matrix TY.
RCOND is set even if INFO = 3 or INFO = 4; if INFO = 3,
then RCOND(2) was not computed, but it is set to 0.
Tolerances
TOL DOUBLE PRECISION
Tolerance used for controlling the accuracy of the applied
transformations. Transformation matrices TU and TY whose
reciprocal condition numbers are less than TOL are not
allowed. If TOL <= 0, then a default value equal to
sqrt(EPS) is used, where EPS is the relative machine
precision.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal
LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= MAX(1,LW1,LW2,LW3,LW4), where
LW1 = (N+NP1+1)*(N+M2) + MAX(3*(N+M2)+N+NP1,5*(N+M2)),
LW2 = (N+NP2)*(N+M1+1) + MAX(3*(N+NP2)+N+M1,5*(N+NP2)),
LW3 = M2 + NP1*NP1 + MAX(NP1*MAX(N,M1),3*M2+NP1,5*M2),
LW4 = NP2 + M1*M1 + MAX(MAX(N,NP1)*M1,3*NP2+M1,5*NP2),
with M1 = M - M2 and NP1 = NP - NP2.
For good performance, LDWORK must generally be larger.
Denoting Q = MAX(M1,M2,NP1,NP2), an upper bound is
MAX(1,(N+Q)*(N+Q+6),Q*(Q+MAX(N,Q,5)+1).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the matrix | A B2 | had not full column rank
| C1 D12 |
in respect to the tolerance EPS;
= 2: if the matrix | A B1 | had not full row rank in
| C2 D21 |
respect to the tolerance EPS;
= 3: if the matrix D12 had not full column rank in
respect to the tolerance TOL;
= 4: if the matrix D21 had not full row rank in respect
to the tolerance TOL;
= 5: if the singular value decomposition (SVD) algorithm
did not converge (when computing the SVD of one of
the matrices |A B2 |, |A B1 |, D12 or D21).
|C1 D12| |C2 D21|
Method
The routine performs the transformations described in [2].References
[1] Glover, K. and Doyle, J.C.
State-space formulae for all stabilizing controllers that
satisfy an Hinf norm bound and relations to risk sensitivity.
Systems and Control Letters, vol. 11, pp. 167-172, 1988.
[2] Balas, G.J., Doyle, J.C., Glover, K., Packard, A., and
Smith, R.
mu-Analysis and Synthesis Toolbox.
The MathWorks Inc., Natick, Mass., 1995.
Numerical Aspects
The precision of the transformations can be controlled by the condition numbers of the matrices TU and TY as given by the values of RCOND(1) and RCOND(2), respectively. An error return with INFO = 3 or INFO = 4 will be obtained if the condition number of TU or TY, respectively, would exceed 1/TOL.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the state feedback and the output injection
matrices for an H-infinity (sub)optimal n-state controller,
using Glover's and Doyle's 1988 formulas, for the system
| A | B1 B2 | | A | B |
P = |----|---------| = |---|---|
| C1 | D11 D12 | | C | D |
| C2 | D21 D22 |
and for a given value of gamma, where B2 has as column size the
number of control inputs (NCON) and C2 has as row size the number
of measurements (NMEAS) being provided to the controller.
It is assumed that
(A1) (A,B2) is stabilizable and (C2,A) is detectable,
(A2) D12 is full column rank with D12 = | 0 | and D21 is
| I |
full row rank with D21 = | 0 I | as obtained by the
subroutine SB10PD,
(A3) | A-j*omega*I B2 | has full column rank for all omega,
| C1 D12 |
(A4) | A-j*omega*I B1 | has full row rank for all omega.
| C2 D21 |
Specification
SUBROUTINE SB10QD( N, M, NP, NCON, NMEAS, GAMMA, A, LDA, B, LDB,
$ C, LDC, D, LDD, F, LDF, H, LDH, X, LDX, Y, LDY,
$ XYCOND, IWORK, DWORK, LDWORK, BWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDD, LDF, LDH, LDWORK,
$ LDX, LDY, M, N, NCON, NMEAS, NP
DOUBLE PRECISION GAMMA
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), DWORK( * ), F( LDF, * ),
$ H( LDH, * ), X( LDX, * ), XYCOND( 2 ),
$ Y( LDY, * )
LOGICAL BWORK( * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
NP (input) INTEGER
The row size of the matrix C. NP >= 0.
NCON (input) INTEGER
The number of control inputs (M2). M >= NCON >= 0,
NP-NMEAS >= NCON.
NMEAS (input) INTEGER
The number of measurements (NP2). NP >= NMEAS >= 0,
M-NCON >= NMEAS.
GAMMA (input) DOUBLE PRECISION
The value of gamma. It is assumed that gamma is
sufficiently large so that the controller is admissible.
GAMMA >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
system input matrix B.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading NP-by-N part of this array must contain the
system output matrix C.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,NP).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading NP-by-M part of this array must contain the
system input/output matrix D.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1,NP).
F (output) DOUBLE PRECISION array, dimension (LDF,N)
The leading M-by-N part of this array contains the state
feedback matrix F.
LDF INTEGER
The leading dimension of the array F. LDF >= max(1,M).
H (output) DOUBLE PRECISION array, dimension (LDH,NP)
The leading N-by-NP part of this array contains the output
injection matrix H.
LDH INTEGER
The leading dimension of the array H. LDH >= max(1,N).
X (output) DOUBLE PRECISION array, dimension (LDX,N)
The leading N-by-N part of this array contains the matrix
X, solution of the X-Riccati equation.
LDX INTEGER
The leading dimension of the array X. LDX >= max(1,N).
Y (output) DOUBLE PRECISION array, dimension (LDY,N)
The leading N-by-N part of this array contains the matrix
Y, solution of the Y-Riccati equation.
LDY INTEGER
The leading dimension of the array Y. LDY >= max(1,N).
XYCOND (output) DOUBLE PRECISION array, dimension (2)
XYCOND(1) contains an estimate of the reciprocal condition
number of the X-Riccati equation;
XYCOND(2) contains an estimate of the reciprocal condition
number of the Y-Riccati equation.
Workspace
IWORK INTEGER array, dimension
(max(2*max(N,M-NCON,NP-NMEAS),N*N))
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal
LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= max(1,M*M + max(2*M1,3*N*N +
max(N*M,10*N*N+12*N+5)),
NP*NP + max(2*NP1,3*N*N +
max(N*NP,10*N*N+12*N+5))),
where M1 = M - M2 and NP1 = NP - NP2.
For good performance, LDWORK must generally be larger.
Denoting Q = MAX(M1,M2,NP1,NP2), an upper bound is
max(1,4*Q*Q+max(2*Q,3*N*N + max(2*N*Q,10*N*N+12*N+5))).
BWORK LOGICAL array, dimension (2*N)
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the controller is not admissible (too small value
of gamma);
= 2: if the X-Riccati equation was not solved
successfully (the controller is not admissible or
there are numerical difficulties);
= 3: if the Y-Riccati equation was not solved
successfully (the controller is not admissible or
there are numerical difficulties).
Method
The routine implements the Glover's and Doyle's formulas [1],[2] modified as described in [3]. The X- and Y-Riccati equations are solved with condition and accuracy estimates [4].References
[1] Glover, K. and Doyle, J.C.
State-space formulae for all stabilizing controllers that
satisfy an Hinf norm bound and relations to risk sensitivity.
Systems and Control Letters, vol. 11, pp. 167-172, 1988.
[2] Balas, G.J., Doyle, J.C., Glover, K., Packard, A., and
Smith, R.
mu-Analysis and Synthesis Toolbox.
The MathWorks Inc., Natick, Mass., 1995.
[3] Petkov, P.Hr., Gu, D.W., and Konstantinov, M.M.
Fortran 77 routines for Hinf and H2 design of continuous-time
linear control systems.
Rep. 98-14, Department of Engineering, Leicester University,
Leicester, U.K., 1998.
[4] Petkov, P.Hr., Konstantinov, M.M., and Mehrmann, V.
DGRSVX and DMSRIC: Fortan 77 subroutines for solving
continuous-time matrix algebraic Riccati equations with
condition and accuracy estimates.
Preprint SFB393/98-16, Fak. f. Mathematik, Tech. Univ.
Chemnitz, May 1998.
Numerical Aspects
The precision of the solution of the matrix Riccati equations can be controlled by the values of the condition numbers XYCOND(1) and XYCOND(2) of these equations.Further Comments
The Riccati equations are solved by the Schur approach implementing condition and accuracy estimates.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the matrices of an H-infinity (sub)optimal controller
| AK | BK |
K = |----|----|,
| CK | DK |
from the state feedback matrix F and output injection matrix H as
determined by the SLICOT Library routine SB10QD.
Specification
SUBROUTINE SB10RD( N, M, NP, NCON, NMEAS, GAMMA, A, LDA, B, LDB,
$ C, LDC, D, LDD, F, LDF, H, LDH, TU, LDTU, TY,
$ LDTY, X, LDX, Y, LDY, AK, LDAK, BK, LDBK, CK,
$ LDCK, DK, LDDK, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD,
$ LDDK, LDF, LDH, LDTU, LDTY, LDWORK, LDX, LDY,
$ M, N, NCON, NMEAS, NP
DOUBLE PRECISION GAMMA
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
$ BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
$ D( LDD, * ), DK( LDDK, * ), DWORK( * ),
$ F( LDF, * ), H( LDH, * ), TU( LDTU, * ),
$ TY( LDTY, * ), X( LDX, * ), Y( LDY, * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
NP (input) INTEGER
The row size of the matrix C. NP >= 0.
NCON (input) INTEGER
The number of control inputs (M2). M >= NCON >= 0.
NP-NMEAS >= NCON.
NMEAS (input) INTEGER
The number of measurements (NP2). NP >= NMEAS >= 0.
M-NCON >= NMEAS.
GAMMA (input) DOUBLE PRECISION
The value of gamma. It is assumed that gamma is
sufficiently large so that the controller is admissible.
GAMMA >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
system input matrix B.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading NP-by-N part of this array must contain the
system output matrix C.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,NP).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading NP-by-M part of this array must contain the
system input/output matrix D.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1,NP).
F (input) DOUBLE PRECISION array, dimension (LDF,N)
The leading M-by-N part of this array must contain the
state feedback matrix F.
LDF INTEGER
The leading dimension of the array F. LDF >= max(1,M).
H (input) DOUBLE PRECISION array, dimension (LDH,NP)
The leading N-by-NP part of this array must contain the
output injection matrix H.
LDH INTEGER
The leading dimension of the array H. LDH >= max(1,N).
TU (input) DOUBLE PRECISION array, dimension (LDTU,M2)
The leading M2-by-M2 part of this array must contain the
control transformation matrix TU, as obtained by the
SLICOT Library routine SB10PD.
LDTU INTEGER
The leading dimension of the array TU. LDTU >= max(1,M2).
TY (input) DOUBLE PRECISION array, dimension (LDTY,NP2)
The leading NP2-by-NP2 part of this array must contain the
measurement transformation matrix TY, as obtained by the
SLICOT Library routine SB10PD.
LDTY INTEGER
The leading dimension of the array TY.
LDTY >= max(1,NP2).
X (input) DOUBLE PRECISION array, dimension (LDX,N)
The leading N-by-N part of this array must contain the
matrix X, solution of the X-Riccati equation, as obtained
by the SLICOT Library routine SB10QD.
LDX INTEGER
The leading dimension of the array X. LDX >= max(1,N).
Y (input) DOUBLE PRECISION array, dimension (LDY,N)
The leading N-by-N part of this array must contain the
matrix Y, solution of the Y-Riccati equation, as obtained
by the SLICOT Library routine SB10QD.
LDY INTEGER
The leading dimension of the array Y. LDY >= max(1,N).
AK (output) DOUBLE PRECISION array, dimension (LDAK,N)
The leading N-by-N part of this array contains the
controller state matrix AK.
LDAK INTEGER
The leading dimension of the array AK. LDAK >= max(1,N).
BK (output) DOUBLE PRECISION array, dimension (LDBK,NMEAS)
The leading N-by-NMEAS part of this array contains the
controller input matrix BK.
LDBK INTEGER
The leading dimension of the array BK. LDBK >= max(1,N).
CK (output) DOUBLE PRECISION array, dimension (LDCK,N)
The leading NCON-by-N part of this array contains the
controller output matrix CK.
LDCK INTEGER
The leading dimension of the array CK.
LDCK >= max(1,NCON).
DK (output) DOUBLE PRECISION array, dimension (LDDK,NMEAS)
The leading NCON-by-NMEAS part of this array contains the
controller input/output matrix DK.
LDDK INTEGER
The leading dimension of the array DK.
LDDK >= max(1,NCON).
Workspace
IWORK INTEGER array, dimension (LIWORK), where
LIWORK = max(2*(max(NP,M)-M2-NP2,M2,N),NP2)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal
LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= max(1, M2*NP2 + NP2*NP2 + M2*M2 +
max(D1*D1 + max(2*D1, (D1+D2)*NP2),
D2*D2 + max(2*D2, D2*M2), 3*N,
N*(2*NP2 + M2) +
max(2*N*M2, M2*NP2 +
max(M2*M2+3*M2, NP2*(2*NP2+
M2+max(NP2,N))))))
where D1 = NP1 - M2, D2 = M1 - NP2,
NP1 = NP - NP2, M1 = M - M2.
For good performance, LDWORK must generally be larger.
Denoting Q = max(M1,M2,NP1,NP2), an upper bound is
max( 1, Q*(3*Q + 3*N + max(2*N, 4*Q + max(Q, N)))).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the controller is not admissible (too small value
of gamma);
= 2: if the determinant of Im2 + Tu*D11HAT*Ty*D22 is zero.
Method
The routine implements the Glover's and Doyle's formulas [1],[2].References
[1] Glover, K. and Doyle, J.C.
State-space formulae for all stabilizing controllers that
satisfy an Hinf norm bound and relations to risk sensitivity.
Systems and Control Letters, vol. 11, pp. 167-172, 1988.
[2] Balas, G.J., Doyle, J.C., Glover, K., Packard, A., and
Smith, R.
mu-Analysis and Synthesis Toolbox.
The MathWorks Inc., Natick, Mass., 1995.
Numerical Aspects
The accuracy of the result depends on the condition numbers of the input and output transformations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the matrices of the H2 optimal controller
| AK | BK |
K = |----|----|,
| CK | DK |
for the normalized discrete-time system
| A | B1 B2 | | A | B |
P = |----|---------| = |---|---|
| C1 | D11 D12 | | C | D |
| C2 | D21 0 |
where B2 has as column size the number of control inputs (NCON)
and C2 has as row size the number of measurements (NMEAS) being
provided to the controller.
It is assumed that
(A1) (A,B2) is stabilizable and (C2,A) is detectable,
(A2) D12 is full column rank with D12 = | 0 | and D21 is
| I |
full row rank with D21 = | 0 I | as obtained by the
SLICOT Library routine SB10PD,
j*Theta
(A3) | A-e *I B2 | has full column rank for all
| C1 D12 |
0 <= Theta < 2*Pi ,
j*Theta
(A4) | A-e *I B1 | has full row rank for all
| C2 D21 |
0 <= Theta < 2*Pi .
Specification
SUBROUTINE SB10SD( N, M, NP, NCON, NMEAS, A, LDA, B, LDB, C, LDC,
$ D, LDD, AK, LDAK, BK, LDBK, CK, LDCK, DK, LDDK,
$ X, LDX, Y, LDY, RCOND, TOL, IWORK, DWORK,
$ LDWORK, BWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD,
$ LDDK, LDWORK, LDX, LDY, M, N, NCON, NMEAS, NP
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
$ BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
$ D( LDD, * ), DK( LDDK, * ), DWORK( * ),
$ RCOND( * ), X( LDX, * ), Y( LDY, * )
LOGICAL BWORK( * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
NP (input) INTEGER
The row size of the matrix C. NP >= 0.
NCON (input) INTEGER
The number of control inputs (M2). M >= NCON >= 0,
NP-NMEAS >= NCON.
NMEAS (input) INTEGER
The number of measurements (NP2). NP >= NMEAS >= 0,
M-NCON >= NMEAS.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
system input matrix B.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading NP-by-N part of this array must contain the
system output matrix C.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,NP).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading NP-by-M part of this array must contain the
system input/output matrix D. Only the leading
(NP-NP2)-by-(M-M2) submatrix D11 is used.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1,NP).
AK (output) DOUBLE PRECISION array, dimension (LDAK,N)
The leading N-by-N part of this array contains the
controller state matrix AK.
LDAK INTEGER
The leading dimension of the array AK. LDAK >= max(1,N).
BK (output) DOUBLE PRECISION array, dimension (LDBK,NMEAS)
The leading N-by-NMEAS part of this array contains the
controller input matrix BK.
LDBK INTEGER
The leading dimension of the array BK. LDBK >= max(1,N).
CK (output) DOUBLE PRECISION array, dimension (LDCK,N)
The leading NCON-by-N part of this array contains the
controller output matrix CK.
LDCK INTEGER
The leading dimension of the array CK.
LDCK >= max(1,NCON).
DK (output) DOUBLE PRECISION array, dimension (LDDK,NMEAS)
The leading NCON-by-NMEAS part of this array contains the
controller input/output matrix DK.
LDDK INTEGER
The leading dimension of the array DK.
LDDK >= max(1,NCON).
X (output) DOUBLE PRECISION array, dimension (LDX,N)
The leading N-by-N part of this array contains the matrix
X, solution of the X-Riccati equation.
LDX INTEGER
The leading dimension of the array X. LDX >= max(1,N).
Y (output) DOUBLE PRECISION array, dimension (LDY,N)
The leading N-by-N part of this array contains the matrix
Y, solution of the Y-Riccati equation.
LDY INTEGER
The leading dimension of the array Y. LDY >= max(1,N).
RCOND (output) DOUBLE PRECISION array, dimension (4)
RCOND contains estimates of the reciprocal condition
numbers of the matrices which are to be inverted and the
reciprocal condition numbers of the Riccati equations
which have to be solved during the computation of the
controller. (See the description of the algorithm in [2].)
RCOND(1) contains the reciprocal condition number of the
matrix Im2 + B2'*X2*B2;
RCOND(2) contains the reciprocal condition number of the
matrix Ip2 + C2*Y2*C2';
RCOND(3) contains the reciprocal condition number of the
X-Riccati equation;
RCOND(4) contains the reciprocal condition number of the
Y-Riccati equation.
Tolerances
TOL DOUBLE PRECISION
Tolerance used in determining the nonsingularity of the
matrices which must be inverted. If TOL <= 0, then a
default value equal to sqrt(EPS) is used, where EPS is the
relative machine precision.
Workspace
IWORK INTEGER array, dimension (max(M2,2*N,N*N,NP2))
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal
LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= max(1, 14*N*N+6*N+max(14*N+23,16*N),
M2*(N+M2+max(3,M1)), NP2*(N+NP2+3)),
where M1 = M - M2.
For good performance, LDWORK must generally be larger.
BWORK LOGICAL array, dimension (2*N)
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the X-Riccati equation was not solved
successfully;
= 2: if the matrix Im2 + B2'*X2*B2 is not positive
definite, or it is numerically singular (with
respect to the tolerance TOL);
= 3: if the Y-Riccati equation was not solved
successfully;
= 4: if the matrix Ip2 + C2*Y2*C2' is not positive
definite, or it is numerically singular (with
respect to the tolerance TOL).
Method
The routine implements the formulas given in [1]. The X- and Y-Riccati equations are solved with condition estimates.References
[1] Zhou, K., Doyle, J.C., and Glover, K.
Robust and Optimal Control.
Prentice-Hall, Upper Saddle River, NJ, 1996.
[2] Petkov, P.Hr., Gu, D.W., and Konstantinov, M.M.
Fortran 77 routines for Hinf and H2 design of linear
discrete-time control systems.
Report 99-8, Department of Engineering, Leicester University,
April 1999.
Numerical Aspects
The accuracy of the result depends on the condition numbers of the matrices which are to be inverted and on the condition numbers of the matrix Riccati equations which are to be solved in the computation of the controller. (The corresponding reciprocal condition numbers are given in the output array RCOND.)Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the matrices of the H2 optimal discrete-time controller
| AK | BK |
K = |----|----|,
| CK | DK |
from the matrices of the controller for the normalized system,
as determined by the SLICOT Library routine SB10SD.
Specification
SUBROUTINE SB10TD( N, M, NP, NCON, NMEAS, D, LDD, TU, LDTU, TY,
$ LDTY, AK, LDAK, BK, LDBK, CK, LDCK, DK, LDDK,
$ RCOND, TOL, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDAK, LDBK, LDCK, LDD, LDDK, LDTU, LDTY,
$ LDWORK, M, N, NCON, NMEAS, NP
DOUBLE PRECISION RCOND, TOL
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION AK( LDAK, * ), BK( LDBK, * ), CK( LDCK, * ),
$ D( LDD, * ), DK( LDDK, * ), DWORK( * ),
$ TU( LDTU, * ), TY( LDTY, * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
NP (input) INTEGER
The row size of the matrix C. NP >= 0.
NCON (input) INTEGER
The number of control inputs (M2). M >= NCON >= 0.
NP-NMEAS >= NCON.
NMEAS (input) INTEGER
The number of measurements (NP2). NP >= NMEAS >= 0.
M-NCON >= NMEAS.
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading NP-by-M part of this array must contain the
system input/output matrix D. Only the trailing
NMEAS-by-NCON submatrix D22 is used.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1,NP).
TU (input) DOUBLE PRECISION array, dimension (LDTU,M2)
The leading M2-by-M2 part of this array must contain the
control transformation matrix TU, as obtained by the
SLICOT Library routine SB10PD.
LDTU INTEGER
The leading dimension of the array TU. LDTU >= max(1,M2).
TY (input) DOUBLE PRECISION array, dimension (LDTY,NP2)
The leading NP2-by-NP2 part of this array must contain the
measurement transformation matrix TY, as obtained by the
SLICOT Library routine SB10PD.
LDTY INTEGER
The leading dimension of the array TY.
LDTY >= max(1,NP2).
AK (input/output) DOUBLE PRECISION array, dimension (LDAK,N)
On entry, the leading N-by-N part of this array must
contain controller state matrix for the normalized system
as obtained by the SLICOT Library routine SB10SD.
On exit, the leading N-by-N part of this array contains
controller state matrix AK.
LDAK INTEGER
The leading dimension of the array AK. LDAK >= max(1,N).
BK (input/output) DOUBLE PRECISION array, dimension
(LDBK,NMEAS)
On entry, the leading N-by-NMEAS part of this array must
contain controller input matrix for the normalized system
as obtained by the SLICOT Library routine SB10SD.
On exit, the leading N-by-NMEAS part of this array
contains controller input matrix BK.
LDBK INTEGER
The leading dimension of the array BK. LDBK >= max(1,N).
CK (input/output) DOUBLE PRECISION array, dimension (LDCK,N)
On entry, the leading NCON-by-N part of this array must
contain controller output matrix for the normalized
system as obtained by the SLICOT Library routine SB10SD.
On exit, the leading NCON-by-N part of this array contains
controller output matrix CK.
LDCK INTEGER
The leading dimension of the array CK.
LDCK >= max(1,NCON).
DK (input/output) DOUBLE PRECISION array, dimension
(LDDK,NMEAS)
On entry, the leading NCON-by-NMEAS part of this array
must contain controller matrix DK for the normalized
system as obtained by the SLICOT Library routine SB10SD.
On exit, the leading NCON-by-NMEAS part of this array
contains controller input/output matrix DK.
LDDK INTEGER
The leading dimension of the array DK.
LDDK >= max(1,NCON).
RCOND (output) DOUBLE PRECISION
RCOND contains an estimate of the reciprocal condition
number of the matrix Im2 + DKHAT*D22 which must be
inverted in the computation of the controller.
Tolerances
TOL DOUBLE PRECISION
Tolerance used in determining the nonsingularity of the
matrix which must be inverted. If TOL <= 0, then a default
value equal to sqrt(EPS) is used, where EPS is the
relative machine precision.
Workspace
IWORK INTEGER array, dimension (2*M2)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= max(N*M2,N*NP2,M2*NP2,M2*M2+4*M2).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the matrix Im2 + DKHAT*D22 is singular, or the
estimated condition number is larger than or equal
to 1/TOL.
Method
The routine implements the formulas given in [1].References
[1] Zhou, K., Doyle, J.C., and Glover, K.
Robust and Optimal Control.
Prentice-Hall, Upper Saddle River, NJ, 1996.
[2] Petkov, P.Hr., Gu, D.W., and Konstantinov, M.M.
Fortran 77 routines for Hinf and H2 design of linear
discrete-time control systems.
Report 99-8, Department of Engineering, Leicester University,
April 1999.
Numerical Aspects
The accuracy of the result depends on the condition numbers of the input and output transformations and of the matrix Im2 + DKHAT*D22.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To reduce the matrices D12 and D21 of the linear time-invariant
system
| A | B1 B2 | | A | B |
P = |----|---------| = |---|---|
| C1 | 0 D12 | | C | D |
| C2 | D21 D22 |
to unit diagonal form, and to transform the matrices B and C to
satisfy the formulas in the computation of the H2 optimal
controller.
Specification
SUBROUTINE SB10UD( N, M, NP, NCON, NMEAS, B, LDB, C, LDC, D, LDD,
$ TU, LDTU, TY, LDTY, RCOND, TOL, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDB, LDC, LDD, LDTU, LDTY, LDWORK, M, N,
$ NCON, NMEAS, NP
DOUBLE PRECISION TOL
C .. Array Arguments ..
DOUBLE PRECISION B( LDB, * ), C( LDC, * ), D( LDD, * ),
$ DWORK( * ), RCOND( 2 ), TU( LDTU, * ),
$ TY( LDTY, * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
NP (input) INTEGER
The row size of the matrix C. NP >= 0.
NCON (input) INTEGER
The number of control inputs (M2). M >= NCON >= 0,
NP-NMEAS >= NCON.
NMEAS (input) INTEGER
The number of measurements (NP2). NP >= NMEAS >= 0,
M-NCON >= NMEAS.
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the system input matrix B.
On exit, the leading N-by-M part of this array contains
the transformed system input matrix B.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading NP-by-N part of this array must
contain the system output matrix C.
On exit, the leading NP-by-N part of this array contains
the transformed system output matrix C.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,NP).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading NP-by-M part of this array must
contain the system input/output matrix D.
The (NP-NMEAS)-by-(M-NCON) leading submatrix D11 is not
referenced.
On exit, the trailing NMEAS-by-NCON part (in the leading
NP-by-M part) of this array contains the transformed
submatrix D22.
The transformed submatrices D12 = [ 0 Im2 ]' and
D21 = [ 0 Inp2 ] are not stored. The corresponding part
of this array contains no useful information.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1,NP).
TU (output) DOUBLE PRECISION array, dimension (LDTU,M2)
The leading M2-by-M2 part of this array contains the
control transformation matrix TU.
LDTU INTEGER
The leading dimension of the array TU. LDTU >= max(1,M2).
TY (output) DOUBLE PRECISION array, dimension (LDTY,NP2)
The leading NP2-by-NP2 part of this array contains the
measurement transformation matrix TY.
LDTY INTEGER
The leading dimension of the array TY.
LDTY >= max(1,NP2).
RCOND (output) DOUBLE PRECISION array, dimension (2)
RCOND(1) contains the reciprocal condition number of the
control transformation matrix TU;
RCOND(2) contains the reciprocal condition number of the
measurement transformation matrix TY.
RCOND is set even if INFO = 1 or INFO = 2; if INFO = 1,
then RCOND(2) was not computed, but it is set to 0.
Tolerances
TOL DOUBLE PRECISION
Tolerance used for controlling the accuracy of the applied
transformations. Transformation matrices TU and TY whose
reciprocal condition numbers are less than TOL are not
allowed. If TOL <= 0, then a default value equal to
sqrt(EPS) is used, where EPS is the relative machine
precision.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal
LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= MAX( M2 + NP1*NP1 + MAX(NP1*N,3*M2+NP1,5*M2),
NP2 + M1*M1 + MAX(M1*N,3*NP2+M1,5*NP2),
N*M2, NP2*N, NP2*M2, 1 )
where M1 = M - M2 and NP1 = NP - NP2.
For good performance, LDWORK must generally be larger.
Denoting Q = MAX(M1,M2,NP1,NP2), an upper bound is
MAX(1,Q*(Q+MAX(N,5)+1)).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the matrix D12 had not full column rank in
respect to the tolerance TOL;
= 2: if the matrix D21 had not full row rank in respect
to the tolerance TOL;
= 3: if the singular value decomposition (SVD) algorithm
did not converge (when computing the SVD of D12 or
D21).
Method
The routine performs the transformations described in [1], [2].References
[1] Zhou, K., Doyle, J.C., and Glover, K.
Robust and Optimal Control.
Prentice-Hall, Upper Saddle River, NJ, 1996.
[2] Balas, G.J., Doyle, J.C., Glover, K., Packard, A., and
Smith, R.
mu-Analysis and Synthesis Toolbox.
The MathWorks Inc., Natick, Mass., 1995.
Numerical Aspects
The precision of the transformations can be controlled by the condition numbers of the matrices TU and TY as given by the values of RCOND(1) and RCOND(2), respectively. An error return with INFO = 1 or INFO = 2 will be obtained if the condition number of TU or TY, respectively, would exceed 1/TOL.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the state feedback and the output injection
matrices for an H2 optimal n-state controller for the system
| A | B1 B2 | | A | B |
P = |----|---------| = |---|---|
| C1 | 0 D12 | | C | D |
| C2 | D21 D22 |
where B2 has as column size the number of control inputs (NCON)
and C2 has as row size the number of measurements (NMEAS) being
provided to the controller.
It is assumed that
(A1) (A,B2) is stabilizable and (C2,A) is detectable,
(A2) D12 is full column rank with D12 = | 0 | and D21 is
| I |
full row rank with D21 = | 0 I | as obtained by the
SLICOT Library routine SB10UD. Matrix D is not used
explicitly.
Specification
SUBROUTINE SB10VD( N, M, NP, NCON, NMEAS, A, LDA, B, LDB, C, LDC,
$ F, LDF, H, LDH, X, LDX, Y, LDY, XYCOND, IWORK,
$ DWORK, LDWORK, BWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDF, LDH, LDWORK, LDX,
$ LDY, M, N, NCON, NMEAS, NP
C .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ DWORK( * ), F( LDF, * ), H( LDH, * ),
$ X( LDX, * ), XYCOND( 2 ), Y( LDY, * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
NP (input) INTEGER
The row size of the matrix C. NP >= 0.
NCON (input) INTEGER
The number of control inputs (M2). M >= NCON >= 0,
NP-NMEAS >= NCON.
NMEAS (input) INTEGER
The number of measurements (NP2). NP >= NMEAS >= 0,
M-NCON >= NMEAS.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
system input matrix B.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading NP-by-N part of this array must contain the
system output matrix C.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,NP).
F (output) DOUBLE PRECISION array, dimension (LDF,N)
The leading NCON-by-N part of this array contains the
state feedback matrix F.
LDF INTEGER
The leading dimension of the array F. LDF >= max(1,NCON).
H (output) DOUBLE PRECISION array, dimension (LDH,NMEAS)
The leading N-by-NMEAS part of this array contains the
output injection matrix H.
LDH INTEGER
The leading dimension of the array H. LDH >= max(1,N).
X (output) DOUBLE PRECISION array, dimension (LDX,N)
The leading N-by-N part of this array contains the matrix
X, solution of the X-Riccati equation.
LDX INTEGER
The leading dimension of the array X. LDX >= max(1,N).
Y (output) DOUBLE PRECISION array, dimension (LDY,N)
The leading N-by-N part of this array contains the matrix
Y, solution of the Y-Riccati equation.
LDY INTEGER
The leading dimension of the array Y. LDY >= max(1,N).
XYCOND (output) DOUBLE PRECISION array, dimension (2)
XYCOND(1) contains an estimate of the reciprocal condition
number of the X-Riccati equation;
XYCOND(2) contains an estimate of the reciprocal condition
number of the Y-Riccati equation.
Workspace
IWORK INTEGER array, dimension (max(2*N,N*N))
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal
LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= 13*N*N + 12*N + 5.
For good performance, LDWORK must generally be larger.
BWORK LOGICAL array, dimension (2*N)
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the X-Riccati equation was not solved
successfully;
= 2: if the Y-Riccati equation was not solved
successfully.
Method
The routine implements the formulas given in [1], [2]. The X- and Y-Riccati equations are solved with condition and accuracy estimates [3].References
[1] Zhou, K., Doyle, J.C., and Glover, K.
Robust and Optimal Control.
Prentice-Hall, Upper Saddle River, NJ, 1996.
[2] Balas, G.J., Doyle, J.C., Glover, K., Packard, A., and
Smith, R.
mu-Analysis and Synthesis Toolbox.
The MathWorks Inc., Natick, Mass., 1995.
[3] Petkov, P.Hr., Konstantinov, M.M., and Mehrmann, V.
DGRSVX and DMSRIC: Fortan 77 subroutines for solving
continuous-time matrix algebraic Riccati equations with
condition and accuracy estimates.
Preprint SFB393/98-16, Fak. f. Mathematik, Tech. Univ.
Chemnitz, May 1998.
Numerical Aspects
The precision of the solution of the matrix Riccati equations can be controlled by the values of the condition numbers XYCOND(1) and XYCOND(2) of these equations.Further Comments
The Riccati equations are solved by the Schur approach implementing condition and accuracy estimates.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the matrices of the H2 optimal controller
| AK | BK |
K = |----|----|,
| CK | DK |
from the state feedback matrix F and output injection matrix H as
determined by the SLICOT Library routine SB10VD.
Specification
SUBROUTINE SB10WD( N, M, NP, NCON, NMEAS, A, LDA, B, LDB, C, LDC,
$ D, LDD, F, LDF, H, LDH, TU, LDTU, TY, LDTY,
$ AK, LDAK, BK, LDBK, CK, LDCK, DK, LDDK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD,
$ LDDK, LDF, LDH, LDTU, LDTY, M, N, NCON, NMEAS,
$ NP
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
$ BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
$ D( LDD, * ), DK( LDDK, * ), F( LDF, * ),
$ H( LDH, * ), TU( LDTU, * ), TY( LDTY, * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
NP (input) INTEGER
The row size of the matrix C. NP >= 0.
NCON (input) INTEGER
The number of control inputs (M2). M >= NCON >= 0.
NP-NMEAS >= NCON.
NMEAS (input) INTEGER
The number of measurements (NP2). NP >= NMEAS >= 0.
M-NCON >= NMEAS.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
system input matrix B. Only the submatrix
B2 = B(:,M-M2+1:M) is used.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading NP-by-N part of this array must contain the
system output matrix C. Only the submatrix
C2 = C(NP-NP2+1:NP,:) is used.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,NP).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading NP-by-M part of this array must contain the
system input/output matrix D. Only the submatrix
D22 = D(NP-NP2+1:NP,M-M2+1:M) is used.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1,NP).
F (input) DOUBLE PRECISION array, dimension (LDF,N)
The leading NCON-by-N part of this array must contain the
state feedback matrix F.
LDF INTEGER
The leading dimension of the array F. LDF >= max(1,NCON).
H (input) DOUBLE PRECISION array, dimension (LDH,NMEAS)
The leading N-by-NMEAS part of this array must contain the
output injection matrix H.
LDH INTEGER
The leading dimension of the array H. LDH >= max(1,N).
TU (input) DOUBLE PRECISION array, dimension (LDTU,M2)
The leading M2-by-M2 part of this array must contain the
control transformation matrix TU, as obtained by the
SLICOT Library routine SB10UD.
LDTU INTEGER
The leading dimension of the array TU. LDTU >= max(1,M2).
TY (input) DOUBLE PRECISION array, dimension (LDTY,NP2)
The leading NP2-by-NP2 part of this array must contain the
measurement transformation matrix TY, as obtained by the
SLICOT Library routine SB10UD.
LDTY INTEGER
The leading dimension of the array TY.
LDTY >= max(1,NP2).
AK (output) DOUBLE PRECISION array, dimension (LDAK,N)
The leading N-by-N part of this array contains the
controller state matrix AK.
LDAK INTEGER
The leading dimension of the array AK. LDAK >= max(1,N).
BK (output) DOUBLE PRECISION array, dimension (LDBK,NMEAS)
The leading N-by-NMEAS part of this array contains the
controller input matrix BK.
LDBK INTEGER
The leading dimension of the array BK. LDBK >= max(1,N).
CK (output) DOUBLE PRECISION array, dimension (LDCK,N)
The leading NCON-by-N part of this array contains the
controller output matrix CK.
LDCK INTEGER
The leading dimension of the array CK.
LDCK >= max(1,NCON).
DK (output) DOUBLE PRECISION array, dimension (LDDK,NMEAS)
The leading NCON-by-NMEAS part of this array contains the
controller input/output matrix DK.
LDDK INTEGER
The leading dimension of the array DK.
LDDK >= max(1,NCON).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine implements the formulas given in [1], [2].References
[1] Zhou, K., Doyle, J.C., and Glover, K.
Robust and Optimal Control.
Prentice-Hall, Upper Saddle River, NJ, 1996.
[2] Balas, G.J., Doyle, J.C., Glover, K., Packard, A., and
Smith, R.
mu-Analysis and Synthesis Toolbox.
The MathWorks Inc., Natick, Mass., 1995.
Numerical Aspects
The accuracy of the result depends on the condition numbers of the input and output transformations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To fit a supplied frequency response data with a stable, minimum phase SISO (single-input single-output) system represented by its matrices A, B, C, D. It handles both discrete- and continuous-time cases.Specification
SUBROUTINE SB10YD( DISCFL, FLAG, LENDAT, RFRDAT, IFRDAT, OMEGA, N,
$ A, LDA, B, C, D, TOL, IWORK, DWORK, LDWORK,
$ ZWORK, LZWORK, INFO )
C .. Scalar Arguments ..
INTEGER DISCFL, FLAG, INFO, LDA, LDWORK, LENDAT,
$ LZWORK, N
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA, *), B(*), C(*), D(*), DWORK(*),
$ IFRDAT(*), OMEGA(*), RFRDAT(*)
COMPLEX*16 ZWORK(*)
Arguments
Input/Output Parameters
DISCFL (input) INTEGER
Indicates the type of the system, as follows:
= 0: continuous-time system;
= 1: discrete-time system.
FLAG (input) INTEGER
If FLAG = 0, then the system zeros and poles are not
constrained.
If FLAG = 1, then the system zeros and poles will have
negative real parts in the continuous-time case, or moduli
less than 1 in the discrete-time case. Consequently, FLAG
must be equal to 1 in mu-synthesis routines.
LENDAT (input) INTEGER
The length of the vectors RFRDAT, IFRDAT and OMEGA.
LENDAT >= 2.
RFRDAT (input) DOUBLE PRECISION array, dimension (LENDAT)
The real part of the frequency data to be fitted.
IFRDAT (input) DOUBLE PRECISION array, dimension (LENDAT)
The imaginary part of the frequency data to be fitted.
OMEGA (input) DOUBLE PRECISION array, dimension (LENDAT)
The frequencies corresponding to RFRDAT and IFRDAT.
These values must be nonnegative and monotonically
increasing. Additionally, for discrete-time systems
they must be between 0 and PI.
N (input/output) INTEGER
On entry, the desired order of the system to be fitted.
N <= LENDAT-1.
On exit, the order of the obtained system. The value of N
could only be modified if N > 0 and FLAG = 1.
A (output) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array contains the
matrix A. If FLAG = 1, then A is in an upper Hessenberg
form, and corresponds to a minimal realization.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (output) DOUBLE PRECISION array, dimension (N)
The computed vector B.
C (output) DOUBLE PRECISION array, dimension (N)
The computed vector C. If FLAG = 1, the first N-1 elements
are zero (for the exit value of N).
D (output) DOUBLE PRECISION array, dimension (1)
The computed scalar D.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used for determining the effective
rank of matrices. If the user sets TOL > 0, then the given
value of TOL is used as a lower bound for the reciprocal
condition number; a (sub)matrix whose estimated condition
number is less than 1/TOL is considered to be of full
rank. If the user sets TOL <= 0, then an implicitly
computed, default tolerance, defined by TOLDEF = SIZE*EPS,
is used instead, where SIZE is the product of the matrix
dimensions, and EPS is the machine precision (see LAPACK
Library routine DLAMCH).
Workspace
IWORK INTEGER array, dimension (max(2,2*N+1))
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK and DWORK(2) contains the optimal value of
LZWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK = max( 2, LW1, LW2, LW3, LW4 ), where
LW1 = 2*LENDAT + 4*HNPTS; HNPTS = 2048;
LW2 = LENDAT + 6*HNPTS;
MN = min( 2*LENDAT, 2*N+1 )
LW3 = 2*LENDAT*(2*N+1) + max( 2*LENDAT, 2*N+1 ) +
max( MN + 6*N + 4, 2*MN + 1 ), if N > 0;
LW3 = 4*LENDAT + 5 , if N = 0;
LW4 = max( N*N + 5*N, 6*N + 1 + min( 1,N ) ), if FLAG = 1;
LW4 = 0, if FLAG = 0.
For optimum performance LDWORK should be larger.
ZWORK COMPLEX*16 array, dimension (LZWORK)
LZWORK INTEGER
The length of the array ZWORK.
LZWORK = LENDAT*(2*N+3), if N > 0;
LZWORK = LENDAT, if N = 0.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the discrete --> continuous transformation cannot
be made;
= 2: if the system poles cannot be found;
= 3: if the inverse system cannot be found, i.e., D is
(close to) zero;
= 4: if the system zeros cannot be found;
= 5: if the state-space representation of the new
transfer function T(s) cannot be found;
= 6: if the continuous --> discrete transformation cannot
be made.
Method
First, if the given frequency data are corresponding to a continuous-time system, they are changed to a discrete-time system using a bilinear transformation with a scaled alpha. Then, the magnitude is obtained from the supplied data. Then, the frequency data are linearly interpolated around the unit-disc. Then, Oppenheim and Schafer complex cepstrum method is applied to get frequency data corresponding to a stable, minimum- phase system. This is done in the following steps: - Obtain LOG (magnitude) - Obtain IFFT of the result (DG01MD SLICOT subroutine); - halve the data at 0; - Obtain FFT of the halved data (DG01MD SLICOT subroutine); - Obtain EXP of the result. Then, the new frequency data are interpolated back to the original frequency. Then, based on these newly obtained data, the system matrices A, B, C, D are constructed; the very identification is performed by Least Squares Method using DGELSY LAPACK subroutine. If needed, a discrete-to-continuous time transformation is applied on the system matrices by AB04MD SLICOT subroutine. Finally, if requested, the poles and zeros of the system are checked. If some of them have positive real parts in the continuous-time case (or are not inside the unit disk in the complex plane in the discrete-time case), they are exchanged with their negatives (or reciprocals, respectively), to preserve the frequency response, while getting a minimum phase and stable system. This is done by SB10ZP SLICOT subroutine.References
[1] Oppenheim, A.V. and Schafer, R.W.
Discrete-Time Signal Processing.
Prentice-Hall Signal Processing Series, 1989.
[2] Balas, G., Doyle, J., Glover, K., Packard, A., and Smith, R.
Mu-analysis and Synthesis toolbox - User's Guide,
The Mathworks Inc., Natick, MA, USA, 1998.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the matrices of the positive feedback controller
| Ak | Bk |
K = |----|----|
| Ck | Dk |
for the shaped plant
| A | B |
G = |---|---|
| C | D |
in the Discrete-Time Loop Shaping Design Procedure.
Specification
SUBROUTINE SB10ZD( N, M, NP, A, LDA, B, LDB, C, LDC, D, LDD,
$ FACTOR, AK, LDAK, BK, LDBK, CK, LDCK, DK,
$ LDDK, RCOND, TOL, IWORK, DWORK, LDWORK, BWORK,
$ INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD,
$ LDDK, LDWORK, M, N, NP
DOUBLE PRECISION FACTOR, TOL
C .. Array Arguments ..
INTEGER IWORK( * )
LOGICAL BWORK( * )
DOUBLE PRECISION A ( LDA, * ), AK( LDAK, * ), B ( LDB, * ),
$ BK( LDBK, * ), C ( LDC, * ), CK( LDCK, * ),
$ D ( LDD, * ), DK( LDDK, * ), DWORK( * ),
$ RCOND( 6 )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the plant. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
NP (input) INTEGER
The row size of the matrix C. NP >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A of the shaped plant.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
system input matrix B of the shaped plant.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading NP-by-N part of this array must contain the
system output matrix C of the shaped plant.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,NP).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading NP-by-M part of this array must contain the
system input/output matrix D of the shaped plant.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1,NP).
FACTOR (input) DOUBLE PRECISION
= 1 implies that an optimal controller is required
(not recommended);
> 1 implies that a suboptimal controller is required
achieving a performance FACTOR less than optimal.
FACTOR >= 1.
AK (output) DOUBLE PRECISION array, dimension (LDAK,N)
The leading N-by-N part of this array contains the
controller state matrix Ak.
LDAK INTEGER
The leading dimension of the array AK. LDAK >= max(1,N).
BK (output) DOUBLE PRECISION array, dimension (LDBK,NP)
The leading N-by-NP part of this array contains the
controller input matrix Bk.
LDBK INTEGER
The leading dimension of the array BK. LDBK >= max(1,N).
CK (output) DOUBLE PRECISION array, dimension (LDCK,N)
The leading M-by-N part of this array contains the
controller output matrix Ck.
LDCK INTEGER
The leading dimension of the array CK. LDCK >= max(1,M).
DK (output) DOUBLE PRECISION array, dimension (LDDK,NP)
The leading M-by-NP part of this array contains the
controller matrix Dk.
LDDK INTEGER
The leading dimension of the array DK. LDDK >= max(1,M).
RCOND (output) DOUBLE PRECISION array, dimension (6)
RCOND(1) contains an estimate of the reciprocal condition
number of the linear system of equations from
which the solution of the P-Riccati equation is
obtained;
RCOND(2) contains an estimate of the reciprocal condition
number of the linear system of equations from
which the solution of the Q-Riccati equation is
obtained;
RCOND(3) contains an estimate of the reciprocal condition
number of the matrix (gamma^2-1)*In - P*Q;
RCOND(4) contains an estimate of the reciprocal condition
number of the matrix Rx + Bx'*X*Bx;
RCOND(5) contains an estimate of the reciprocal condition
^
number of the matrix Ip + D*Dk;
RCOND(6) contains an estimate of the reciprocal condition
^
number of the matrix Im + Dk*D.
Tolerances
TOL DOUBLE PRECISION
Tolerance used for checking the nonsingularity of the
matrices to be inverted. If TOL <= 0, then a default value
equal to sqrt(EPS) is used, where EPS is the relative
machine precision. TOL < 1.
Workspace
IWORK INTEGER array, dimension (2*max(N,M+NP))
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= 16*N*N + 5*M*M + 7*NP*NP + 6*M*N + 7*M*NP +
7*N*NP + 6*N + 2*(M + NP) +
max(14*N+23,16*N,2*M-1,2*NP-1).
For good performance, LDWORK must generally be larger.
BWORK LOGICAL array, dimension (2*N)
Error Indicator
INFO (output) INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the P-Riccati equation is not solved successfully;
= 2: the Q-Riccati equation is not solved successfully;
= 3: the iteration to compute eigenvalues or singular
values failed to converge;
= 4: the matrix (gamma^2-1)*In - P*Q is singular;
= 5: the matrix Rx + Bx'*X*Bx is singular;
^
= 6: the matrix Ip + D*Dk is singular;
^
= 7: the matrix Im + Dk*D is singular;
= 8: the matrix Ip - D*Dk is singular;
= 9: the matrix Im - Dk*D is singular;
= 10: the closed-loop system is unstable.
Method
The routine implements the formulas given in [1].References
[1] Gu, D.-W., Petkov, P.H., and Konstantinov, M.M.
On discrete H-infinity loop shaping design procedure routines.
Technical Report 00-6, Dept. of Engineering, Univ. of
Leicester, UK, 2000.
Numerical Aspects
The accuracy of the results depends on the conditioning of the two Riccati equations solved in the controller design. For better conditioning it is advised to take FACTOR > 1.Further Comments
NoneExample
Program Text
* SB10ZD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, NMAX, PMAX
PARAMETER ( MMAX = 10, NMAX = 10, PMAX = 10 )
INTEGER LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD, LDDK
PARAMETER ( LDA = NMAX, LDAK = NMAX, LDB = NMAX,
$ LDBK = NMAX, LDC = PMAX, LDCK = MMAX,
$ LDD = PMAX, LDDK = MMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = 2*MAX( NMAX, MMAX + PMAX ) )
INTEGER LDWORK
PARAMETER ( LDWORK = 16*NMAX*NMAX + 5*MMAX*MMAX +
$ 7*PMAX*PMAX + 6*MMAX*NMAX +
$ 7*MMAX*PMAX + 7*NMAX*PMAX + 6*NMAX +
$ 2*( MMAX + PMAX ) +
$ MAX( 14*NMAX + 23, 16*NMAX,
$ 2*MMAX - 1, 2*PMAX - 1 ) )
* .. Local Scalars ..
DOUBLE PRECISION FACTOR, TOL
INTEGER I, INFO, J, M, N, NP
* .. Local Arrays ..
LOGICAL BWORK(2*NMAX)
INTEGER IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), AK(LDAK,NMAX), B(LDB,MMAX),
$ BK(LDBK,PMAX), C(LDC,NMAX), CK(LDCK,NMAX),
$ D(LDD,MMAX), DK(LDDK,PMAX), DWORK(LDWORK),
$ RCOND( 6 )
* .. External Subroutines ..
EXTERNAL SB10ZD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, NP
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE IF ( NP.LT.0 .OR. NP.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) NP
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,NP )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,NP )
READ ( NIN, FMT = * ) FACTOR, TOL
CALL SB10ZD( N, M, NP, A, LDA, B, LDB, C, LDC, D, LDD, FACTOR,
$ AK, LDAK, BK, LDBK, CK, LDCK, DK, LDDK, RCOND,
$ TOL, IWORK, DWORK, LDWORK, BWORK, INFO )
IF ( INFO.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99992 ) ( AK(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99992 ) ( BK(I,J), J = 1,NP )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 30 I = 1, M
WRITE ( NOUT, FMT = 99992 ) ( CK(I,J), J = 1,N )
30 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 40 I = 1, M
WRITE ( NOUT, FMT = 99992 ) ( DK(I,J), J = 1,NP )
40 CONTINUE
WRITE( NOUT, FMT = 99993 )
WRITE( NOUT, FMT = 99991 ) ( RCOND(I), I = 1,6 )
ELSE
WRITE( NOUT, FMT = 99998 ) INFO
END IF
END IF
STOP
*
99999 FORMAT (' SB10ZD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (/' INFO on exit from SB10ZD =',I2)
99997 FORMAT (/' The controller state matrix AK is'/)
99996 FORMAT (/' The controller input matrix BK is'/)
99995 FORMAT (/' The controller output matrix CK is'/)
99994 FORMAT (/' The controller matrix DK is'/)
99993 FORMAT (/' The estimated condition numbers are'/)
99992 FORMAT (10(1X,F8.4))
99991 FORMAT ( 5(1X,D12.5))
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' NP is out of range.',/' NP = ',I5)
END
Program Data
SB10LD EXAMPLE PROGRAM DATA 6 2 3 0.2 0.0 3.0 0.0 -0.3 -0.1 -3.0 0.2 -0.4 -0.3 0.0 0.0 -0.1 0.1 -1.0 0.0 0.0 -3.0 1.0 0.0 0.0 -1.0 -1.0 0.0 0.0 0.3 0.6 2.0 0.1 -0.4 0.2 -4.0 0.0 0.0 0.2 -2.0 -1.0 -2.0 1.0 3.0 -3.0 -4.0 1.0 -2.0 0.0 1.0 1.0 5.0 1.0 -1.0 2.0 -2.0 0.0 -3.0 -3.0 0.0 1.0 -1.0 1.0 -1.0 2.0 4.0 -3.0 0.0 5.0 1.0 10.0 -6.0 -7.0 8.0 2.0 -4.0 1.1 0.0Program Results
SB10ZD EXAMPLE PROGRAM RESULTS The controller state matrix AK is 1.0128 0.5101 -0.1546 1.1300 3.3759 0.4911 -2.1257 -1.4517 -0.4486 0.3493 -1.5506 -1.4296 -1.0930 -0.6026 -0.1344 0.2253 -1.5625 -0.6762 0.3207 0.1698 0.2376 -1.1781 -0.8705 0.2896 0.5017 0.9006 0.0668 2.3613 0.2049 0.3703 1.0787 0.6703 0.2783 -0.7213 0.4918 0.7435 The controller input matrix BK is 0.4132 0.3112 -0.8077 0.2140 0.4253 0.1811 -0.0710 0.0807 0.3558 -0.0121 -0.2019 0.0249 0.1047 0.1399 -0.0457 -0.2542 -0.3472 0.0523 The controller output matrix CK is -0.0372 -0.0456 -0.0040 0.0962 -0.2059 -0.0571 0.1999 0.2994 0.1335 -0.0251 -0.3108 0.2048 The controller matrix DK is 0.0629 -0.0022 0.0363 -0.0228 0.0195 0.0600 The estimated condition numbers are 0.27949D-03 0.66679D-03 0.45677D-01 0.23433D-07 0.68495D-01 0.76854D-01
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB10ZP.html 0000664 0000000 0000000 00000012445 14560147231 0020377 0 ustar 00root root 0000000 0000000
Purpose
To transform a SISO (single-input single-output) system [A,B;C,D] by mirroring its unstable poles and zeros in the boundary of the stability domain, thus preserving the frequency response of the system, but making it stable and minimum phase. Specifically, for a continuous-time system, the positive real parts of its poles and zeros are exchanged with their negatives. Discrete-time systems are first converted to continuous-time systems using a bilinear transformation, and finally converted back.Specification
SUBROUTINE SB10ZP( DISCFL, N, A, LDA, B, C, D, IWORK, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER DISCFL, INFO, LDA, LDWORK, N
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( * ), C( * ), D( * ), DWORK( * )
Arguments
Input/Output Parameters
DISCFL (input) INTEGER
Indicates the type of the system, as follows:
= 0: continuous-time system;
= 1: discrete-time system.
N (input/output) INTEGER
On entry, the order of the original system. N >= 0.
On exit, the order of the transformed, minimal system.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original system matrix A.
On exit, the leading N-by-N part of this array contains
the transformed matrix A, in an upper Hessenberg form.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (N)
On entry, this array must contain the original system
vector B.
On exit, this array contains the transformed vector B.
C (input/output) DOUBLE PRECISION array, dimension (N)
On entry, this array must contain the original system
vector C.
On exit, this array contains the transformed vector C.
The first N-1 elements are zero (for the exit value of N).
D (input/output) DOUBLE PRECISION array, dimension (1)
On entry, this array must contain the original system
scalar D.
On exit, this array contains the transformed scalar D.
Workspace
IWORK INTEGER array, dimension (max(2,N+1))
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= max(N*N + 5*N, 6*N + 1 + min(1,N)).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the discrete --> continuous transformation cannot
be made;
= 2: if the system poles cannot be found;
= 3: if the inverse system cannot be found, i.e., D is
(close to) zero;
= 4: if the system zeros cannot be found;
= 5: if the state-space representation of the new
transfer function T(s) cannot be found;
= 6: if the continuous --> discrete transformation cannot
be made.
Method
First, if the system is discrete-time, it is transformed to continuous-time using alpha = beta = 1 in the bilinear transformation implemented in the SLICOT routine AB04MD. Then the eigenvalues of A, i.e., the system poles, are found. Then, the inverse of the original system is found and its poles, i.e., the system zeros, are evaluated. The obtained system poles Pi and zeros Zi are checked and if a positive real part is detected, it is exchanged by -Pi or -Zi. Then the polynomial coefficients of the transfer function T(s) = Q(s)/P(s) are found. The state-space representation of T(s) is then obtained. The system matrices B, C, D are scaled so that the transformed system has the same system gain as the original system. If the original system is discrete-time, then the result (which is continuous-time) is converted back to discrete-time.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute a reduced order controller (Acr,Bcr,Ccr,Dcr) for an
original state-space controller representation (Ac,Bc,Cc,Dc) by
using the frequency-weighted square-root or balancing-free
square-root Balance & Truncate (B&T) or Singular Perturbation
Approximation (SPA) model reduction methods. The algorithm tries
to minimize the norm of the frequency-weighted error
||V*(K-Kr)*W||
where K and Kr are the transfer-function matrices of the original
and reduced order controllers, respectively. V and W are special
frequency-weighting transfer-function matrices constructed
to enforce closed-loop stability and/or closed-loop performance.
If G is the transfer-function matrix of the open-loop system, then
the following weightings V and W can be used:
-1
(a) V = (I-G*K) *G, W = I - to enforce closed-loop stability;
-1
(b) V = I, W = (I-G*K) *G - to enforce closed-loop stability;
-1 -1
(c) V = (I-G*K) *G, W = (I-G*K) - to enforce closed-loop
stability and performance.
G has the state space representation (A,B,C,D).
If K is unstable, only the ALPHA-stable part of K is reduced.
Specification
SUBROUTINE SB16AD( DICO, JOBC, JOBO, JOBMR, WEIGHT, EQUIL, ORDSEL,
$ N, M, P, NC, NCR, ALPHA, A, LDA, B, LDB,
$ C, LDC, D, LDD, AC, LDAC, BC, LDBC, CC, LDCC,
$ DC, LDDC, NCS, HSVC, TOL1, TOL2, IWORK, DWORK,
$ LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, EQUIL, JOBC, JOBO, JOBMR, ORDSEL, WEIGHT
INTEGER INFO, IWARN, LDA, LDAC, LDB, LDBC, LDC, LDCC,
$ LDD, LDDC, LDWORK, M, N, NC, NCR, NCS, P
DOUBLE PRECISION ALPHA, TOL1, TOL2
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), AC(LDAC,*), B(LDB,*), BC(LDBC,*),
$ C(LDC,*), CC(LDCC,*), D(LDD,*), DC(LDDC,*),
$ DWORK(*), HSVC(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the original controller as follows:
= 'C': continuous-time controller;
= 'D': discrete-time controller.
JOBC CHARACTER*1
Specifies the choice of frequency-weighted controllability
Grammian as follows:
= 'S': choice corresponding to standard Enns' method [1];
= 'E': choice corresponding to the stability enhanced
modified Enns' method of [2].
JOBO CHARACTER*1
Specifies the choice of frequency-weighted observability
Grammian as follows:
= 'S': choice corresponding to standard Enns' method [1];
= 'E': choice corresponding to the stability enhanced
modified combination method of [2].
JOBMR CHARACTER*1
Specifies the model reduction approach to be used
as follows:
= 'B': use the square-root B&T method;
= 'F': use the balancing-free square-root B&T method;
= 'S': use the square-root SPA method;
= 'P': use the balancing-free square-root SPA method.
WEIGHT CHARACTER*1
Specifies the type of frequency-weighting, as follows:
= 'N': no weightings are used (V = I, W = I);
= 'O': stability enforcing left (output) weighting
-1
V = (I-G*K) *G is used (W = I);
= 'I': stability enforcing right (input) weighting
-1
W = (I-G*K) *G is used (V = I);
= 'P': stability and performance enforcing weightings
-1 -1
V = (I-G*K) *G , W = (I-G*K) are used.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily
equilibrate the triplets (A,B,C) and (Ac,Bc,Cc) as
follows:
= 'S': perform equilibration (scaling);
= 'N': do not perform equilibration.
ORDSEL CHARACTER*1
Specifies the order selection method as follows:
= 'F': the resulting order NCR is fixed;
= 'A': the resulting order NCR is automatically
determined on basis of the given tolerance TOL1.
Input/Output Parameters
N (input) INTEGER
The order of the open-loop system state-space
representation, i.e., the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NC (input) INTEGER
The order of the controller state-space representation,
i.e., the order of the matrix AC. NC >= 0.
NCR (input/output) INTEGER
On entry with ORDSEL = 'F', NCR is the desired order of
the resulting reduced order controller. 0 <= NCR <= NC.
On exit, if INFO = 0, NCR is the order of the resulting
reduced order controller. For a controller with NCU
ALPHA-unstable eigenvalues and NCS ALPHA-stable
eigenvalues (NCU+NCS = NC), NCR is set as follows:
if ORDSEL = 'F', NCR is equal to
NCU+MIN(MAX(0,NCR-NCU),NCMIN), where NCR is the desired
order on entry, NCMIN is the number of frequency-weighted
Hankel singular values greater than NCS*EPS*S1, EPS is the
machine precision (see LAPACK Library Routine DLAMCH) and
S1 is the largest Hankel singular value (computed in
HSVC(1)); NCR can be further reduced to ensure
HSVC(NCR-NCU) > HSVC(NCR+1-NCU);
if ORDSEL = 'A', NCR is the sum of NCU and the number of
Hankel singular values greater than MAX(TOL1,NCS*EPS*S1).
ALPHA (input) DOUBLE PRECISION
Specifies the ALPHA-stability boundary for the eigenvalues
of the state dynamics matrix AC. For a continuous-time
controller (DICO = 'C'), ALPHA <= 0 is the boundary value
for the real parts of eigenvalues; for a discrete-time
controller (DICO = 'D'), 0 <= ALPHA <= 1 represents the
boundary value for the moduli of eigenvalues.
The ALPHA-stability domain does not include the boundary.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A of the open-loop
system.
On exit, if INFO = 0 and EQUIL = 'S', the leading N-by-N
part of this array contains the scaled state dynamics
matrix of the open-loop system.
If EQUIL = 'N', this array is unchanged on exit.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input/state matrix B of the open-loop system.
On exit, if INFO = 0 and EQUIL = 'S', the leading N-by-M
part of this array contains the scaled input/state matrix
of the open-loop system.
If EQUIL = 'N', this array is unchanged on exit.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C of the open-loop system.
On exit, if INFO = 0 and EQUIL = 'S', the leading P-by-N
part of this array contains the scaled state/output matrix
of the open-loop system.
If EQUIL = 'N', this array is unchanged on exit.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading P-by-M part of this array must contain the
input/output matrix D of the open-loop system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
AC (input/output) DOUBLE PRECISION array, dimension (LDAC,NC)
On entry, the leading NC-by-NC part of this array must
contain the state dynamics matrix Ac of the original
controller.
On exit, if INFO = 0, the leading NCR-by-NCR part of this
array contains the state dynamics matrix Acr of the
reduced controller. The resulting Ac has a
block-diagonal form with two blocks.
For a system with NCU ALPHA-unstable eigenvalues and
NCS ALPHA-stable eigenvalues (NCU+NCS = NC), the leading
NCU-by-NCU block contains the unreduced part of Ac
corresponding to the ALPHA-unstable eigenvalues.
The trailing (NCR+NCS-NC)-by-(NCR+NCS-NC) block contains
the reduced part of Ac corresponding to ALPHA-stable
eigenvalues.
LDAC INTEGER
The leading dimension of array AC. LDAC >= MAX(1,NC).
BC (input/output) DOUBLE PRECISION array, dimension (LDBC,P)
On entry, the leading NC-by-P part of this array must
contain the input/state matrix Bc of the original
controller.
On exit, if INFO = 0, the leading NCR-by-P part of this
array contains the input/state matrix Bcr of the reduced
controller.
LDBC INTEGER
The leading dimension of array BC. LDBC >= MAX(1,NC).
CC (input/output) DOUBLE PRECISION array, dimension (LDCC,NC)
On entry, the leading M-by-NC part of this array must
contain the state/output matrix Cc of the original
controller.
On exit, if INFO = 0, the leading M-by-NCR part of this
array contains the state/output matrix Ccr of the reduced
controller.
LDCC INTEGER
The leading dimension of array CC. LDCC >= MAX(1,M).
DC (input/output) DOUBLE PRECISION array, dimension (LDDC,P)
On entry, the leading M-by-P part of this array must
contain the input/output matrix Dc of the original
controller.
On exit, if INFO = 0, the leading M-by-P part of this
array contains the input/output matrix Dcr of the reduced
controller.
LDDC INTEGER
The leading dimension of array DC. LDDC >= MAX(1,M).
NCS (output) INTEGER
The dimension of the ALPHA-stable part of the controller.
HSVC (output) DOUBLE PRECISION array, dimension (NC)
If INFO = 0, the leading NCS elements of this array
contain the frequency-weighted Hankel singular values,
ordered decreasingly, of the ALPHA-stable part of the
controller.
Tolerances
TOL1 DOUBLE PRECISION
If ORDSEL = 'A', TOL1 contains the tolerance for
determining the order of the reduced controller.
For model reduction, the recommended value is
TOL1 = c*S1, where c is a constant in the
interval [0.00001,0.001], and S1 is the largest
frequency-weighted Hankel singular value of the
ALPHA-stable part of the original controller
(computed in HSVC(1)).
If TOL1 <= 0 on entry, the used default value is
TOL1 = NCS*EPS*S1, where NCS is the number of
ALPHA-stable eigenvalues of Ac and EPS is the machine
precision (see LAPACK Library Routine DLAMCH).
If ORDSEL = 'F', the value of TOL1 is ignored.
TOL2 DOUBLE PRECISION
The tolerance for determining the order of a minimal
realization of the ALPHA-stable part of the given
controller. The recommended value is TOL2 = NCS*EPS*S1.
This value is used by default if TOL2 <= 0 on entry.
If TOL2 > 0 and ORDSEL = 'A', then TOL2 <= TOL1.
Workspace
IWORK INTEGER array, dimension (MAX(1,LIWRK1,LIWRK2))
LIWRK1 = 0, if JOBMR = 'B';
LIWRK1 = NC, if JOBMR = 'F';
LIWRK1 = 2*NC, if JOBMR = 'S' or 'P';
LIWRK2 = 0, if WEIGHT = 'N';
LIWRK2 = 2*(M+P), if WEIGHT = 'O', 'I', or 'P'.
On exit, if INFO = 0, IWORK(1) contains NCMIN, the order
of the computed minimal realization of the stable part of
the controller.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 2*NC*NC + MAX( 1, LFREQ, LSQRED ),
where
LFREQ = (N+NC)*(N+NC+2*M+2*P)+
MAX((N+NC)*(N+NC+MAX(N+NC,M,P)+7), (M+P)*(M+P+4))
if WEIGHT = 'I' or 'O' or 'P';
LFREQ = NC*(MAX(M,P)+5) if WEIGHT = 'N' and EQUIL = 'N';
LFREQ = MAX(N,NC*(MAX(M,P)+5)) if WEIGHT = 'N' and
EQUIL = 'S';
LSQRED = MAX( 1, 2*NC*NC+5*NC );
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: with ORDSEL = 'F', the selected order NCR is greater
than NSMIN, the sum of the order of the
ALPHA-unstable part and the order of a minimal
realization of the ALPHA-stable part of the given
controller; in this case, the resulting NCR is set
equal to NSMIN;
= 2: with ORDSEL = 'F', the selected order NCR
corresponds to repeated singular values for the
ALPHA-stable part of the controller, which are
neither all included nor all excluded from the
reduced model; in this case, the resulting NCR is
automatically decreased to exclude all repeated
singular values;
= 3: with ORDSEL = 'F', the selected order NCR is less
than the order of the ALPHA-unstable part of the
given controller. In this case NCR is set equal to
the order of the ALPHA-unstable part.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the closed-loop system is not well-posed;
its feedthrough matrix is (numerically) singular;
= 2: the computation of the real Schur form of the
closed-loop state matrix failed;
= 3: the closed-loop state matrix is not stable;
= 4: the solution of a symmetric eigenproblem failed;
= 5: the computation of the ordered real Schur form of Ac
failed;
= 6: the separation of the ALPHA-stable/unstable
diagonal blocks failed because of very close
eigenvalues;
= 7: the computation of Hankel singular values failed.
Method
Let K be the transfer-function matrix of the original linear
controller
d[xc(t)] = Ac*xc(t) + Bc*y(t)
u(t) = Cc*xc(t) + Dc*y(t), (1)
where d[xc(t)] is dxc(t)/dt for a continuous-time system and
xc(t+1) for a discrete-time system. The subroutine SB16AD
determines the matrices of a reduced order controller
d[z(t)] = Acr*z(t) + Bcr*y(t)
u(t) = Ccr*z(t) + Dcr*y(t), (2)
such that the corresponding transfer-function matrix Kr minimizes
the norm of the frequency-weighted error
V*(K-Kr)*W, (3)
where V and W are special stable transfer-function matrices
chosen to enforce stability and/or performance of the closed-loop
system [3] (see description of the parameter WEIGHT).
The following procedure is used to reduce K in conjunction
with the frequency-weighted balancing approach of [2]
(see also [3]):
1) Decompose additively K, of order NC, as
K = K1 + K2,
such that K1 has only ALPHA-stable poles and K2, of order NCU,
has only ALPHA-unstable poles.
2) Compute for K1 a B&T or SPA frequency-weighted approximation
K1r of order NCR-NCU using the frequency-weighted balancing
approach of [1] in conjunction with accuracy enhancing
techniques specified by the parameter JOBMR.
3) Assemble the reduced model Kr as
Kr = K1r + K2.
For the reduction of the ALPHA-stable part, several accuracy
enhancing techniques can be employed (see [2] for details).
If JOBMR = 'B', the square-root B&T method of [1] is used.
If JOBMR = 'F', the balancing-free square-root version of the
B&T method [1] is used.
If JOBMR = 'S', the square-root version of the SPA method [2,3]
is used.
If JOBMR = 'P', the balancing-free square-root version of the
SPA method [2,3] is used.
For each of these methods, two left and right truncation matrices
are determined using the Cholesky factors of an input
frequency-weighted controllability Grammian P and an output
frequency-weighted observability Grammian Q.
P and Q are determined as the leading NC-by-NC diagonal blocks
of the controllability Grammian of K*W and of the
observability Grammian of V*K. Special techniques developed in [2]
are used to compute the Cholesky factors of P and Q directly
(see also SLICOT Library routine SB16AY).
The frequency-weighted Hankel singular values HSVC(1), ....,
HSVC(NC) are computed as the square roots of the eigenvalues
of the product P*Q.
References
[1] Enns, D.
Model reduction with balanced realizations: An error bound
and a frequency weighted generalization.
Proc. 23-th CDC, Las Vegas, pp. 127-132, 1984.
[2] Varga, A. and Anderson, B.D.O.
Square-root balancing-free methods for frequency-weighted
balancing related model reduction.
(report in preparation)
[3] Anderson, B.D.O and Liu, Y.
Controller reduction: concepts and approaches.
IEEE Trans. Autom. Control, Vol. 34, pp. 802-812, 1989.
Numerical Aspects
The implemented methods rely on accuracy enhancing square-root techniques.Further Comments
NoneExample
Program Text
* SB16AD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX, NCMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20,
$ NCMAX = 20 )
INTEGER MPMAX, NNCMAX
PARAMETER ( MPMAX = MMAX + PMAX, NNCMAX = NMAX + NCMAX )
INTEGER LDA, LDB, LDC, LDD, LDAC, LDBC, LDCC, LDDC
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX, LDAC = NCMAX, LDBC = NCMAX,
$ LDCC = PMAX, LDDC = PMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = 2*MAX( NCMAX, MPMAX ) )
INTEGER LDWORK
PARAMETER ( LDWORK = 2*NCMAX*NCMAX +
$ NNCMAX*( NNCMAX + 2*MPMAX ) +
$ MAX( NNCMAX*( NNCMAX +
$ MAX( NNCMAX, MMAX, PMAX ) + 7 ),
$ MPMAX*( MPMAX + 4 ) ) )
* .. Local Scalars ..
DOUBLE PRECISION ALPHA, TOL1, TOL2
INTEGER I, INFO, IWARN, J, M, N, NCR, NCS, NC, P
CHARACTER*1 DICO, EQUIL, JOBC, JOBO, JOBMR, ORDSEL, WEIGHT
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK), HSVC(NMAX),
$ AC(LDAC,NCMAX), BC(LDBC,PMAX), CC(LDCC,NMAX),
$ DC(LDDC,PMAX)
INTEGER IWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL SB16AD
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, NC, NCR, ALPHA, TOL1, TOL2, DICO,
$ JOBC, JOBO, JOBMR, WEIGHT, EQUIL, ORDSEL
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1, N )
IF( P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
IF( NC.LT.0 .OR. NC.GT.NCMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) NC
ELSE
IF( NC.GT.0 ) THEN
READ ( NIN, FMT = * )
$ ( ( AC(I,J), J = 1,NC ), I = 1,NC )
READ ( NIN, FMT = * )
$ ( ( BC(I,J), J = 1,P ), I = 1, NC )
READ ( NIN, FMT = * )
$ ( ( CC(I,J), J = 1,NC ), I = 1,M )
END IF
READ ( NIN, FMT = * )
$ ( ( DC(I,J), J = 1,P ), I = 1,M )
END IF
* Find a reduced ssr for (AC,BC,CC,DC).
CALL SB16AD( DICO, JOBC, JOBO, JOBMR, WEIGHT, EQUIL,
$ ORDSEL, N, M, P, NC, NCR, ALPHA, A, LDA,
$ B, LDB, C, LDC, D, LDD, AC, LDAC, BC, LDBC,
$ CC, LDCC, DC, LDDC, NCS, HSVC, TOL1, TOL2,
$ IWORK, DWORK, LDWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF( IWARN.NE.0) WRITE ( NOUT, FMT = 99984 ) IWARN
WRITE ( NOUT, FMT = 99997 ) NCR
WRITE ( NOUT, FMT = 99987 )
WRITE ( NOUT, FMT = 99995 ) ( HSVC(J), J = 1, NCS )
IF( NCR.GT.0 ) WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NCR
WRITE ( NOUT, FMT = 99995 ) ( AC(I,J), J = 1,NCR )
20 CONTINUE
IF( NCR.GT.0 ) WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NCR
WRITE ( NOUT, FMT = 99995 ) ( BC(I,J), J = 1,P )
40 CONTINUE
IF( NCR.GT.0 ) WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, M
WRITE ( NOUT, FMT = 99995 ) ( CC(I,J), J = 1,NCR )
60 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 70 I = 1, M
WRITE ( NOUT, FMT = 99995 ) ( DC(I,J), J = 1,P )
70 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB16AD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB16AD = ',I2)
99997 FORMAT (/' The order of reduced controller = ',I2)
99996 FORMAT (/' The reduced controller state dynamics matrix Ac is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The reduced controller input/state matrix Bc is ')
99992 FORMAT (/' The reduced controller state/output matrix Cc is ')
99991 FORMAT (/' The reduced controller input/output matrix Dc is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99987 FORMAT (/' The Hankel singular values of weighted ALPHA-stable',
$ ' part are')
99986 FORMAT (/' NC is out of range.',/' NC = ',I5)
99984 FORMAT (' IWARN on exit from SB16AD = ',I2)
END
Program Data
SB16AD EXAMPLE PROGRAM DATA (Continuous system)
3 1 1 3 2 0.0 0.1E0 0.0 C S S F I N F
-1. 0. 4.
0. 2. 0.
0. 0. -3.
1.
1.
1.
1. 1. 1.
0.
-26.4000 6.4023 4.3868
32.0000 0 0
0 8.0000 0
-16
0
0
9.2994 1.1624 0.1090
0
Program Results
SB16AD EXAMPLE PROGRAM RESULTS The order of reduced controller = 2 The Hankel singular values of weighted ALPHA-stable part are 3.8253 0.2005 The reduced controller state dynamics matrix Ac is 9.1900 0.0000 0.0000 -34.5297 The reduced controller input/state matrix Bc is -11.9593 86.3137 The reduced controller state/output matrix Cc is 2.8955 -1.3566 The reduced controller input/output matrix Dc is 0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB16AY.html 0000664 0000000 0000000 00000027261 14560147231 0020367 0 ustar 00root root 0000000 0000000
Purpose
To compute for given state-space representations (A,B,C,D) and (Ac,Bc,Cc,Dc) of the transfer-function matrices of the open-loop system G and feedback controller K, respectively, the Cholesky factors of the frequency-weighted controllability and observability Grammians corresponding to a frequency-weighted model reduction problem. The controller must stabilize the closed-loop system. The state matrix Ac must be in a block-diagonal real Schur form Ac = diag(Ac1,Ac2), where Ac1 contains the unstable eigenvalues of Ac and Ac2 contains the stable eigenvalues of Ac.Specification
SUBROUTINE SB16AY( DICO, JOBC, JOBO, WEIGHT, N, M, P, NC, NCS,
$ A, LDA, B, LDB, C, LDC, D, LDD,
$ AC, LDAC, BC, LDBC, CC, LDCC, DC, LDDC,
$ SCALEC, SCALEO, S, LDS, R, LDR,
$ IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, JOBC, JOBO, WEIGHT
INTEGER INFO, LDA, LDAC, LDB, LDBC, LDC, LDCC, LDD, LDDC,
$ LDR, LDS, LDWORK, M, N, NC, NCS, P
DOUBLE PRECISION SCALEC, SCALEO
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), AC(LDAC,*), B(LDB,*), BC(LDBC,*),
$ C(LDC,*), CC(LDCC,*), D(LDD,*), DC(LDDC,*),
$ DWORK(*), R(LDR,*), S(LDS,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the systems as follows:
= 'C': G and K are continuous-time systems;
= 'D': G and K are discrete-time systems.
JOBC CHARACTER*1
Specifies the choice of frequency-weighted controllability
Grammian as follows:
= 'S': choice corresponding to standard Enns' method [1];
= 'E': choice corresponding to the stability enhanced
modified Enns' method of [2].
JOBO CHARACTER*1
Specifies the choice of frequency-weighted observability
Grammian as follows:
= 'S': choice corresponding to standard Enns' method [1];
= 'E': choice corresponding to the stability enhanced
modified combination method of [2].
WEIGHT CHARACTER*1
Specifies the type of frequency-weighting, as follows:
= 'N': no weightings are used (V = I, W = I);
= 'O': stability enforcing left (output) weighting
-1
V = (I-G*K) *G is used (W = I);
= 'I': stability enforcing right (input) weighting
-1
W = (I-G*K) *G is used (V = I);
= 'P': stability and performance enforcing weightings
-1 -1
V = (I-G*K) *G , W = (I-G*K) are used.
Input/Output Parameters
N (input) INTEGER
The order of the open-loop system state-space
representation, i.e., the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NC (input) INTEGER
The order of the controller state-space representation,
i.e., the order of the matrix AC. NC >= 0.
NCS (input) INTEGER
The dimension of the stable part of the controller, i.e.,
the order of matrix Ac2. NC >= NCS >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
state matrix A of the system with the transfer-function
matrix G.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
input/state matrix B.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array must contain the
state/output matrix C.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading P-by-M part of this array must contain the
input/output matrix D of the open-loop system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
AC (input) DOUBLE PRECISION array, dimension (LDAC,NC)
The leading NC-by-NC part of this array must contain
the state dynamics matrix Ac of the controller in a
block diagonal real Schur form Ac = diag(Ac1,Ac2), where
Ac1 is (NC-NCS)-by-(NC-NCS) and contains the unstable
eigenvalues of Ac, and Ac2 is NCS-by-NCS and contains
the stable eigenvalues of Ac.
LDAC INTEGER
The leading dimension of array AC. LDAC >= MAX(1,NC).
BC (input) DOUBLE PRECISION array, dimension (LDBC,P)
The leading NC-by-P part of this array must contain
the input/state matrix Bc of the controller.
LDBC INTEGER
The leading dimension of array BC. LDBC >= MAX(1,NC).
CC (input) DOUBLE PRECISION array, dimension (LDCC,NC)
The leading M-by-NC part of this array must contain
the state/output matrix Cc of the controller.
LDCC INTEGER
The leading dimension of array CC. LDCC >= MAX(1,M).
DC (input) DOUBLE PRECISION array, dimension (LDDC,P)
The leading M-by-P part of this array must contain
the input/output matrix Dc of the controller.
LDDC INTEGER
The leading dimension of array DC. LDDC >= MAX(1,M).
SCALEC (output) DOUBLE PRECISION
Scaling factor for the controllability Grammian.
See METHOD.
SCALEO (output) DOUBLE PRECISION
Scaling factor for the observability Grammian. See METHOD.
S (output) DOUBLE PRECISION array, dimension (LDS,NCS)
The leading NCS-by-NCS upper triangular part of this array
contains the Cholesky factor S of the frequency-weighted
controllability Grammian P = S*S'. See METHOD.
LDS INTEGER
The leading dimension of array S. LDS >= MAX(1,NCS).
R (output) DOUBLE PRECISION array, dimension (LDR,NCS)
The leading NCS-by-NCS upper triangular part of this array
contains the Cholesky factor R of the frequency-weighted
observability Grammian Q = R'*R. See METHOD.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,NCS).
Workspace
IWORK INTEGER array, dimension (LIWRK)
LIWRK = 0, if WEIGHT = 'N';
LIWRK = 2(M+P), if WEIGHT = 'O', 'I', or 'P'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( 1, LFREQ ),
where
LFREQ = (N+NC)*(N+NC+2*M+2*P)+
MAX((N+NC)*(N+NC+MAX(N+NC,M,P)+7), (M+P)*(M+P+4))
if WEIGHT = 'I' or 'O' or 'P';
LFREQ = NCS*(MAX(M,P)+5) if WEIGHT = 'N'.
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the closed-loop system is not well-posed;
its feedthrough matrix is (numerically) singular;
= 2: the computation of the real Schur form of the
closed-loop state matrix failed;
= 3: the closed-loop state matrix is not stable;
= 4: the solution of a symmetric eigenproblem failed;
= 5: the NCS-by-NCS trailing part Ac2 of the state
matrix Ac is not stable or not in a real Schur form.
Method
If JOBC = 'S', the controllability Grammian P is determined as
follows:
- if WEIGHT = 'O' or 'N', P satisfies for a continuous-time
controller the Lyapunov equation
Ac2*P + P*Ac2' + scalec^2*Bc*Bc' = 0
and for a discrete-time controller
Ac2*P*Ac2' - P + scalec^2*Bc*Bc' = 0;
- if WEIGHT = 'I' or 'P', let Pi be the solution of the
continuous-time Lyapunov equation
Ai*Pi + Pi*Ai' + scalec^2*Bi*Bi' = 0
or of the discrete-time Lyapunov equation
Ai*Pi*Ai' - Pi + scalec^2*Bi*Bi' = 0,
where Ai and Bi are the state and input matrices of a special
state-space realization of the input frequency weight (see [2]);
P results as the trailing NCS-by-NCS part of Pi partitioned as
Pi = ( * * ).
( * P )
If JOBC = 'E', a modified controllability Grammian P1 >= P is
determined to guarantee stability for a modified Enns' method [2].
If JOBO = 'S', the observability Grammian Q is determined as
follows:
- if WEIGHT = 'I' or 'N', Q satisfies for a continuous-time
controller the Lyapunov equation
Ac2'*Q + Q*Ac2 + scaleo^2*Cc'*Cc = 0
and for a discrete-time controller
Ac2'*Q*Ac2 - Q + scaleo^2*Cc'*Cc = 0;
- if WEIGHT = 'O' or 'P', let Qo be the solution of the
continuous-time Lyapunov equation
Ao'*Qo + Qo*Ao + scaleo^2*Co'*Co = 0
or of the discrete-time Lyapunov equation
Ao'*Qo*Ao - Qo + scaleo^2*Co'*Co = 0,
where Ao and Co are the state and output matrices of a
special state-space realization of the output frequency weight
(see [2]); if WEIGHT = 'O', Q results as the leading NCS-by-NCS
part of Qo partitioned as
Qo = ( Q * )
( * * )
while if WEIGHT = 'P', Q results as the trailing NCS-by-NCS
part of Qo partitioned as
Qo = ( * * ).
( * Q )
If JOBO = 'E', a modified observability Grammian Q1 >= Q is
determined to guarantee stability for a modified Enns' method [2].
The routine computes directly the Cholesky factors S and R
such that P = S*S' and Q = R'*R according to formulas
developed in [2].
References
[1] Enns, D.
Model reduction with balanced realizations: An error bound
and a frequency weighted generalization.
Proc. CDC, Las Vegas, pp. 127-132, 1984.
[2] Varga, A. and Anderson, B.D.O.
Frequency-weighted balancing related controller reduction.
Proceedings of the 15th IFAC World Congress, July 21-26, 2002,
Barcelona, Spain, Vol.15, Part 1, 2002-07-21.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute, for a given open-loop model (A,B,C,D), and for given state feedback gain F and full observer gain G, such that A+B*F and A+G*C are stable, a reduced order controller model (Ac,Bc,Cc,Dc) using a coprime factorization based controller reduction approach. For reduction, either the square-root or the balancing-free square-root versions of the Balance & Truncate (B&T) or Singular Perturbation Approximation (SPA) model reduction methods are used in conjunction with stable coprime factorization techniques.Specification
SUBROUTINE SB16BD( DICO, JOBD, JOBMR, JOBCF, EQUIL, ORDSEL,
$ N, M, P, NCR, A, LDA, B, LDB, C, LDC, D, LDD,
$ F, LDF, G, LDG, DC, LDDC, HSV, TOL1, TOL2,
$ IWORK, DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, EQUIL, JOBCF, JOBD, JOBMR, ORDSEL
INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, LDDC,
$ LDF, LDG, LDWORK, M, N, NCR, P
DOUBLE PRECISION TOL1, TOL2
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DC(LDDC,*), DWORK(*), F(LDF,*), G(LDG,*), HSV(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the open-loop system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOBD CHARACTER*1
Specifies whether or not a non-zero matrix D appears
in the given state space model:
= 'D': D is present;
= 'Z': D is assumed a zero matrix.
JOBMR CHARACTER*1
Specifies the model reduction approach to be used
as follows:
= 'B': use the square-root B&T method;
= 'F': use the balancing-free square-root B&T method;
= 'S': use the square-root SPA method;
= 'P': use the balancing-free square-root SPA method.
JOBCF CHARACTER*1
Specifies whether left or right coprime factorization is
to be used as follows:
= 'L': use left coprime factorization;
= 'R': use right coprime factorization.
EQUIL CHARACTER*1
Specifies whether the user wishes to perform a
preliminary equilibration before performing
order reduction as follows:
= 'S': perform equilibration (scaling);
= 'N': do not perform equilibration.
ORDSEL CHARACTER*1
Specifies the order selection method as follows:
= 'F': the resulting controller order NCR is fixed;
= 'A': the resulting controller order NCR is
automatically determined on basis of the given
tolerance TOL1.
Input/Output Parameters
N (input) INTEGER
The order of the open-loop state-space representation,
i.e., the order of the matrix A. N >= 0.
N also represents the order of the original state-feedback
controller.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NCR (input/output) INTEGER
On entry with ORDSEL = 'F', NCR is the desired order of
the resulting reduced order controller. 0 <= NCR <= N.
On exit, if INFO = 0, NCR is the order of the resulting
reduced order controller. NCR is set as follows:
if ORDSEL = 'F', NCR is equal to MIN(NCR,NMIN), where NCR
is the desired order on entry, and NMIN is the order of a
minimal realization of an extended system Ge (see METHOD);
NMIN is determined as the number of
Hankel singular values greater than N*EPS*HNORM(Ge),
where EPS is the machine precision (see LAPACK Library
Routine DLAMCH) and HNORM(Ge) is the Hankel norm of the
extended system (computed in HSV(1));
if ORDSEL = 'A', NCR is equal to the number of Hankel
singular values greater than MAX(TOL1,N*EPS*HNORM(Ge)).
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state dynamics matrix A.
On exit, if INFO = 0, the leading NCR-by-NCR part of this
array contains the state dynamics matrix Ac of the reduced
controller.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must
contain the original input/state matrix B.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array must
contain the original state/output matrix C.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
If JOBD = 'D', the leading P-by-M part of this
array must contain the system direct input/output
transmission matrix D.
The array D is not referenced if JOBD = 'Z'.
LDD INTEGER
The leading dimension of array D.
LDD >= MAX(1,P), if JOBD = 'D';
LDD >= 1, if JOBD = 'Z'.
F (input/output) DOUBLE PRECISION array, dimension (LDF,N)
On entry, the leading M-by-N part of this array must
contain a stabilizing state feedback matrix.
On exit, if INFO = 0, the leading M-by-NCR part of this
array contains the state/output matrix Cc of the reduced
controller.
LDF INTEGER
The leading dimension of array F. LDF >= MAX(1,M).
G (input/output) DOUBLE PRECISION array, dimension (LDG,P)
On entry, the leading N-by-P part of this array must
contain a stabilizing observer gain matrix.
On exit, if INFO = 0, the leading NCR-by-P part of this
array contains the input/state matrix Bc of the reduced
controller.
LDG INTEGER
The leading dimension of array G. LDG >= MAX(1,N).
DC (output) DOUBLE PRECISION array, dimension (LDDC,P)
If INFO = 0, the leading M-by-P part of this array
contains the input/output matrix Dc of the reduced
controller.
LDDC INTEGER
The leading dimension of array DC. LDDC >= MAX(1,M).
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, it contains the N Hankel singular values
of the extended system ordered decreasingly (see METHOD).
Tolerances
TOL1 DOUBLE PRECISION
If ORDSEL = 'A', TOL1 contains the tolerance for
determining the order of the reduced extended system.
For model reduction, the recommended value is
TOL1 = c*HNORM(Ge), where c is a constant in the
interval [0.00001,0.001], and HNORM(Ge) is the
Hankel norm of the extended system (computed in HSV(1)).
The value TOL1 = N*EPS*HNORM(Ge) is used by default if
TOL1 <= 0 on entry, where EPS is the machine precision
(see LAPACK Library Routine DLAMCH).
If ORDSEL = 'F', the value of TOL1 is ignored.
TOL2 DOUBLE PRECISION
The tolerance for determining the order of a minimal
realization of the coprime factorization controller
(see METHOD). The recommended value is
TOL2 = N*EPS*HNORM(Ge) (see METHOD).
This value is used by default if TOL2 <= 0 on entry.
If TOL2 > 0 and ORDSEL = 'A', then TOL2 <= TOL1.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK = 0, if ORDSEL = 'F' and NCR = N.
Otherwise,
LIWORK = MAX(PM,M), if JOBCF = 'L',
LIWORK = MAX(PM,P), if JOBCF = 'R', where
PM = 0, if JOBMR = 'B',
PM = N, if JOBMR = 'F',
PM = MAX(1,2*N), if JOBMR = 'S' or 'P'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= P*N, if ORDSEL = 'F' and NCR = N. Otherwise,
LDWORK >= (N+M)*(M+P) + MAX(LWR,4*M), if JOBCF = 'L',
LDWORK >= (N+P)*(M+P) + MAX(LWR,4*P), if JOBCF = 'R',
where LWR = MAX(1,N*(2*N+MAX(N,M+P)+5)+N*(N+1)/2).
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: with ORDSEL = 'F', the selected order NCR is
greater than the order of a minimal
realization of the controller.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction of A+G*C to a real Schur form
failed;
= 2: the matrix A+G*C is not stable (if DICO = 'C'),
or not convergent (if DICO = 'D');
= 3: the computation of Hankel singular values failed;
= 4: the reduction of A+B*F to a real Schur form
failed;
= 5: the matrix A+B*F is not stable (if DICO = 'C'),
or not convergent (if DICO = 'D').
Method
Let be the linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t), (1)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system, and let Go(d) be the open-loop
transfer-function matrix
-1
Go(d) = C*(d*I-A) *B + D .
Let F and G be the state feedback and observer gain matrices,
respectively, chosen so that A+B*F and A+G*C are stable matrices.
The controller has a transfer-function matrix K(d) given by
-1
K(d) = F*(d*I-A-B*F-G*C-G*D*F) *G .
The closed-loop transfer-function matrix is given by
-1
Gcl(d) = Go(d)(I+K(d)Go(d)) .
K(d) can be expressed as a left coprime factorization (LCF),
-1
K(d) = M_left(d) *N_left(d) ,
or as a right coprime factorization (RCF),
-1
K(d) = N_right(d)*M_right(d) ,
where M_left(d), N_left(d), N_right(d), and M_right(d) are
stable transfer-function matrices.
The subroutine SB16BD determines the matrices of a reduced
controller
d[z(t)] = Ac*z(t) + Bc*y(t)
u(t) = Cc*z(t) + Dc*y(t), (2)
with the transfer-function matrix Kr as follows:
(1) If JOBCF = 'L', the extended system
Ge(d) = [ N_left(d) M_left(d) ] is reduced to
Ger(d) = [ N_leftr(d) M_leftr(d) ] by using either the
B&T or SPA methods. The reduced order controller Kr(d)
is computed as
-1
Kr(d) = M_leftr(d) *N_leftr(d) ;
(2) If JOBCF = 'R', the extended system
Ge(d) = [ N_right(d) ] is reduced to
[ M_right(d) ]
Ger(d) = [ N_rightr(d) ] by using either the
[ M_rightr(d) ]
B&T or SPA methods. The reduced order controller Kr(d)
is computed as
-1
Kr(d) = N_rightr(d)* M_rightr(d) .
If ORDSEL = 'A', the order of the controller is determined by
computing the number of Hankel singular values greater than
the given tolerance TOL1. The Hankel singular values are
the square roots of the eigenvalues of the product of
the controllability and observability Grammians of the
extended system Ge.
If JOBMR = 'B', the square-root B&T method of [1] is used.
If JOBMR = 'F', the balancing-free square-root version of the
B&T method [1] is used.
If JOBMR = 'S', the square-root version of the SPA method [2,3]
is used.
If JOBMR = 'P', the balancing-free square-root version of the
SPA method [2,3] is used.
References
[1] Tombs, M.S. and Postlethwaite, I.
Truncated balanced realization of stable, non-minimal
state-space systems.
Int. J. Control, Vol. 46, pp. 1319-1330, 1987.
[2] Varga, A.
Efficient minimal realization procedure based on balancing.
Proc. of IMACS/IFAC Symp. MCTS, Lille, France, May 1991,
A. El Moudui, P. Borne, S. G. Tzafestas (Eds.), Vol. 2,
pp. 42-46, 1991.
[3] Varga, A.
Coprime factors model reduction method based on square-root
balancing-free techniques.
System Analysis, Modelling and Simulation, Vol. 11,
pp. 303-311, 1993.
[4] Liu, Y., Anderson, B.D.O. and Ly, O.L.
Coprime factorization controller reduction with Bezout
identity induced frequency weighting.
Automatica, vol. 26, pp. 233-249, 1990.
Numerical Aspects
The implemented methods rely on accuracy enhancing square-root or
balancing-free square-root techniques.
3
The algorithms require less than 30N floating point operations.
Further Comments
NoneExample
Program Text
* SB16BD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDD, LDDC, LDF, LDG
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX, LDDC = MMAX, LDF = MMAX, LDG = NMAX
$ )
INTEGER LDWORK, LIWORK, MAXMP, MPMAX
PARAMETER ( LIWORK = 2*NMAX, MAXMP = MAX( MMAX, PMAX ),
$ MPMAX = MMAX + PMAX )
PARAMETER ( LDWORK = ( NMAX + MAXMP )*MPMAX +
$ MAX ( NMAX*( 2*NMAX +
$ MAX( NMAX, MPMAX ) + 5 )
$ + ( NMAX*( NMAX + 1 ) )/2,
$ 4*MAXMP ) )
CHARACTER DICO, EQUIL, JOBCF, JOBD, JOBMR, ORDSEL
INTEGER I, INFO, IWARN, J, M, N, NCR, P
DOUBLE PRECISION TOL1, TOL2
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DC(LDDC,PMAX), DWORK(LDWORK),
$ F(LDF,NMAX), G(LDG,PMAX), HSV(NMAX)
INTEGER IWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL SB16BD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, NCR, TOL1, TOL2,
$ DICO, JOBD, JOBMR, JOBCF, EQUIL, ORDSEL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
READ ( NIN, FMT = * ) ( ( F(I,J), J = 1,N ), I = 1,M )
READ ( NIN, FMT = * ) ( ( G(I,J), J = 1,P ), I = 1,N )
* Find a reduced ssr for (A,B,C,D).
CALL SB16BD( DICO, JOBD, JOBMR, JOBCF, EQUIL, ORDSEL, N,
$ M, P, NCR, A, LDA, B, LDB, C, LDC, D, LDD,
$ F, LDF, G, LDG, DC, LDDC, HSV, TOL1, TOL2,
$ IWORK, DWORK, LDWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NCR
WRITE ( NOUT, FMT = 99987 )
WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1,N )
IF( NCR.GT.0 ) WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NCR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NCR )
20 CONTINUE
IF( NCR.GT.0 ) WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NCR
WRITE ( NOUT, FMT = 99995 ) ( G(I,J), J = 1,P )
40 CONTINUE
IF( NCR.GT.0 ) WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, M
WRITE ( NOUT, FMT = 99995 ) ( F(I,J), J = 1,NCR )
60 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 80 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( DC(I,J), J = 1,M )
80 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB16BD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB16BD = ',I2)
99997 FORMAT (' The order of reduced controller = ',I2)
99996 FORMAT (/' The reduced controller state dynamics matrix Ac is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The reduced controller input/state matrix Bc is ')
99992 FORMAT (/' The reduced controller state/output matrix Cc is ')
99991 FORMAT (/' The reduced controller input/output matrix Dc is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99987 FORMAT (/' The Hankel singular values of extended system are:')
END
Program Data
SB16BD EXAMPLE PROGRAM DATA (Continuous system)
8 1 1 4 0.1E0 0.0 C D F L S F
0 1.0000 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 -0.0150 0.7650 0 0 0 0
0 0 -0.7650 -0.0150 0 0 0 0
0 0 0 0 -0.0280 1.4100 0 0
0 0 0 0 -1.4100 -0.0280 0 0
0 0 0 0 0 0 -0.0400 1.850
0 0 0 0 0 0 -1.8500 -0.040
0.0260
-0.2510
0.0330
-0.8860
-4.0170
0.1450
3.6040
0.2800
-.996 -.105 0.261 .009 -.001 -.043 0.002 -0.026
0.0
4.4721e-002 6.6105e-001 4.6986e-003 3.6014e-001 1.0325e-001 -3.7541e-002 -4.2685e-002 3.2873e-002
4.1089e-001
8.6846e-002
3.8523e-004
-3.6194e-003
-8.8037e-003
8.4205e-003
1.2349e-003
4.2632e-003
Program Results
SB16BD EXAMPLE PROGRAM RESULTS The order of reduced controller = 4 The Hankel singular values of extended system are: 4.9078 4.8745 3.8455 3.7811 1.2289 1.1785 0.5176 0.1148 The reduced controller state dynamics matrix Ac is 0.5946 -0.7336 0.1914 -0.3368 0.5960 -0.0184 -0.1088 0.0207 1.2253 0.2043 0.1009 -1.4948 -0.0330 -0.0243 1.3440 0.0035 The reduced controller input/state matrix Bc is 0.0015 -0.0202 0.0159 -0.0544 The reduced controller state/output matrix Cc is 0.3534 0.0274 0.0337 -0.0320 The reduced controller input/output matrix Dc is 0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB16CD.html 0000664 0000000 0000000 00000047540 14560147231 0020346 0 ustar 00root root 0000000 0000000
Purpose
To compute, for a given open-loop model (A,B,C,D), and for given state feedback gain F and full observer gain G, such that A+B*F and A+G*C are stable, a reduced order controller model (Ac,Bc,Cc) using a coprime factorization based controller reduction approach. For reduction of coprime factors, a stability enforcing frequency-weighted model reduction is performed using either the square-root or the balancing-free square-root versions of the Balance & Truncate (B&T) model reduction method.Specification
SUBROUTINE SB16CD( DICO, JOBD, JOBMR, JOBCF, ORDSEL, N, M, P, NCR,
$ A, LDA, B, LDB, C, LDC, D, LDD, F, LDF, G, LDG,
$ HSV, TOL, IWORK, DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, JOBCF, JOBD, JOBMR, ORDSEL
INTEGER INFO, IWARN, LDA, LDB, LDC, LDD,
$ LDF, LDG, LDWORK, M, N, NCR, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), F(LDF,*), G(LDG,*), HSV(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the open-loop system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOBD CHARACTER*1
Specifies whether or not a non-zero matrix D appears
in the given state space model, as follows:
= 'D': D is present;
= 'Z': D is assumed a zero matrix.
JOBMR CHARACTER*1
Specifies the model reduction approach to be used
as follows:
= 'B': use the square-root B&T method;
= 'F': use the balancing-free square-root B&T method.
JOBCF CHARACTER*1
Specifies whether left or right coprime factorization
of the controller is to be used as follows:
= 'L': use left coprime factorization;
= 'R': use right coprime factorization.
ORDSEL CHARACTER*1
Specifies the order selection method as follows:
= 'F': the resulting controller order NCR is fixed;
= 'A': the resulting controller order NCR is
automatically determined on basis of the given
tolerance TOL.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation, i.e.
the order of the matrix A. N >= 0.
N also represents the order of the original state-feedback
controller.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NCR (input/output) INTEGER
On entry with ORDSEL = 'F', NCR is the desired order of
the resulting reduced order controller. 0 <= NCR <= N.
On exit, if INFO = 0, NCR is the order of the resulting
reduced order controller. NCR is set as follows:
if ORDSEL = 'F', NCR is equal to MIN(NCR,NCRMIN), where
NCR is the desired order on entry, and NCRMIN is the
number of Hankel-singular values greater than N*EPS*S1,
where EPS is the machine precision (see LAPACK Library
Routine DLAMCH) and S1 is the largest Hankel singular
value (computed in HSV(1)); NCR can be further reduced
to ensure HSV(NCR) > HSV(NCR+1);
if ORDSEL = 'A', NCR is equal to the number of Hankel
singular values greater than MAX(TOL,N*EPS*S1).
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state dynamics matrix A.
On exit, if INFO = 0, the leading NCR-by-NCR part of this
array contains the state dynamics matrix Ac of the reduced
controller.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the open-loop system input/state matrix B.
On exit, this array is overwritten with a NCR-by-M
B&T approximation of the matrix B.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the open-loop system state/output matrix C.
On exit, this array is overwritten with a P-by-NCR
B&T approximation of the matrix C.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
On entry, if JOBD = 'D', the leading P-by-M part of this
array must contain the system direct input/output
transmission matrix D.
The array D is not referenced if JOBD = 'Z'.
LDD INTEGER
The leading dimension of array D.
LDD >= MAX(1,P), if JOBD = 'D';
LDD >= 1, if JOBD = 'Z'.
F (input/output) DOUBLE PRECISION array, dimension (LDF,N)
On entry, the leading M-by-N part of this array must
contain a stabilizing state feedback matrix.
On exit, if INFO = 0, the leading M-by-NCR part of this
array contains the output/state matrix Cc of the reduced
controller.
LDF INTEGER
The leading dimension of array F. LDF >= MAX(1,M).
G (input/output) DOUBLE PRECISION array, dimension (LDG,P)
On entry, the leading N-by-P part of this array must
contain a stabilizing observer gain matrix.
On exit, if INFO = 0, the leading NCR-by-P part of this
array contains the input/state matrix Bc of the reduced
controller.
LDG INTEGER
The leading dimension of array G. LDG >= MAX(1,N).
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, HSV contains the N frequency-weighted
Hankel singular values ordered decreasingly (see METHOD).
Tolerances
TOL DOUBLE PRECISION
If ORDSEL = 'A', TOL contains the tolerance for
determining the order of reduced controller.
The recommended value is TOL = c*S1, where c is a constant
in the interval [0.00001,0.001], and S1 is the largest
Hankel singular value (computed in HSV(1)).
The value TOL = N*EPS*S1 is used by default if
TOL <= 0 on entry, where EPS is the machine precision
(see LAPACK Library Routine DLAMCH).
If ORDSEL = 'F', the value of TOL is ignored.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK = 0, if JOBMR = 'B';
LIWORK = N, if JOBMR = 'F'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 2*N*N + MAX( 1, 2*N*N + 5*N, N*MAX(M,P),
N*(N + MAX(N,MP) + MIN(N,MP) + 6)),
where MP = M, if JOBCF = 'L';
MP = P, if JOBCF = 'R'.
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: with ORDSEL = 'F', the selected order NCR is
greater than the order of a minimal realization
of the controller;
= 2: with ORDSEL = 'F', the selected order NCR
corresponds to repeated singular values, which are
neither all included nor all excluded from the
reduced controller. In this case, the resulting NCR
is set automatically to the largest value such that
HSV(NCR) > HSV(NCR+1).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: eigenvalue computation failure;
= 2: the matrix A+G*C is not stable;
= 3: the matrix A+B*F is not stable;
= 4: the Lyapunov equation for computing the
observability Grammian is (nearly) singular;
= 5: the Lyapunov equation for computing the
controllability Grammian is (nearly) singular;
= 6: the computation of Hankel singular values failed.
Method
Let be the linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t), (1)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system, and let Go(d) be the open-loop
transfer-function matrix
-1
Go(d) = C*(d*I-A) *B + D .
Let F and G be the state feedback and observer gain matrices,
respectively, chosen such that A+BF and A+GC are stable matrices.
The controller has a transfer-function matrix K(d) given by
-1
K(d) = F*(d*I-A-B*F-G*C-G*D*F) *G .
The closed-loop transfer function matrix is given by
-1
Gcl(d) = Go(d)(I+K(d)Go(d)) .
K(d) can be expressed as a left coprime factorization (LCF)
-1
K(d) = M_left(d) *N_left(d),
or as a right coprime factorization (RCF)
-1
K(d) = N_right(d)*M_right(d) ,
where M_left(d), N_left(d), N_right(d), and M_right(d) are
stable transfer-function matrices.
The subroutine SB16CD determines the matrices of a reduced
controller
d[z(t)] = Ac*z(t) + Bc*y(t)
u(t) = Cc*z(t), (2)
with the transfer-function matrix Kr, using the following
stability enforcing approach proposed in [1]:
(1) If JOBCF = 'L', the frequency-weighted approximation problem
is solved
min||[M_left(d)-M_leftr(d) N_left(d)-N_leftr(d)][-Y(d)]|| ,
[ X(d)]
where
-1
G(d) = Y(d)*X(d)
is a RCF of the open-loop system transfer-function matrix.
The B&T model reduction technique is used in conjunction
with the method proposed in [1].
(2) If JOBCF = 'R', the frequency-weighted approximation problem
is solved
min || [ -U(d) V(d) ] [ N_right(d)-N_rightr(d) ] || ,
[ M_right(d)-M_rightr(d) ]
where
-1
G(d) = V(d) *U(d)
is a LCF of the open-loop system transfer-function matrix.
The B&T model reduction technique is used in conjunction
with the method proposed in [1].
If ORDSEL = 'A', the order of the controller is determined by
computing the number of Hankel singular values greater than
the given tolerance TOL. The Hankel singular values are
the square roots of the eigenvalues of the product of
two frequency-weighted Grammians P and Q, defined as follows.
If JOBCF = 'L', then P is the controllability Grammian of a system
of the form (A+BF,B,*,*), and Q is the observability Grammian of a
system of the form (A+GC,*,F,*). This choice corresponds to an
input frequency-weighted order reduction of left coprime
factors [1].
If JOBCF = 'R', then P is the controllability Grammian of a system
of the form (A+BF,G,*,*), and Q is the observability Grammian of a
system of the form (A+GC,*,C,*). This choice corresponds to an
output frequency-weighted order reduction of right coprime
factors [1].
For the computation of truncation matrices, the B&T approach
is used in conjunction with accuracy enhancing techniques.
If JOBMR = 'B', the square-root B&T method of [2,4] is used.
If JOBMR = 'F', the balancing-free square-root version of the
B&T method [3,4] is used.
References
[1] Liu, Y., Anderson, B.D.O. and Ly, O.L.
Coprime factorization controller reduction with Bezout
identity induced frequency weighting.
Automatica, vol. 26, pp. 233-249, 1990.
[2] Tombs, M.S. and Postlethwaite I.
Truncated balanced realization of stable, non-minimal
state-space systems.
Int. J. Control, Vol. 46, pp. 1319-1330, 1987.
[3] Varga, A.
Efficient minimal realization procedure based on balancing.
Proc. of IMACS/IFAC Symp. MCTS, Lille, France, May 1991,
A. El Moudui, P. Borne, S. G. Tzafestas (Eds.), Vol. 2,
pp. 42-46, 1991.
[4] Varga, A.
Coprime factors model reduction method based on square-root
balancing-free techniques.
System Analysis, Modelling and Simulation, Vol. 11,
pp. 303-311, 1993.
Numerical Aspects
The implemented methods rely on accuracy enhancing square-root or
balancing-free square-root techniques.
3
The algorithms require less than 30N floating point operations.
Further Comments
NoneExample
Program Text
* SB16CD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDD, LDDC, LDF, LDG
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX, LDDC = MMAX, LDF = MMAX, LDG = NMAX )
INTEGER LDWORK, LIWORK, MPMAX
PARAMETER ( LIWORK = 2*NMAX, MPMAX = MAX( MMAX, PMAX ) )
PARAMETER ( LDWORK = 2*NMAX*NMAX +
$ MAX( 2*NMAX*NMAX + 5*NMAX,
$ NMAX*( NMAX + MAX( NMAX, MPMAX )
$ + MIN( NMAX, MPMAX ) + 6 ) )
$ )
CHARACTER DICO, JOBCF, JOBD, JOBMR, ORDSEL
INTEGER I, INFO, IWARN, J, M, N, NCR, P
DOUBLE PRECISION TOL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK),
$ F(LDF,NMAX), G(LDG,PMAX), HSV(NMAX)
INTEGER IWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL SB16CD
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, NCR, TOL,
$ DICO, JOBD, JOBMR, JOBCF, ORDSEL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
READ ( NIN, FMT = * ) ( ( F(I,J), J = 1,N ), I = 1,M )
READ ( NIN, FMT = * ) ( ( G(I,J), J = 1,P ), I = 1,N )
* Find a reduced ssr for (A,B,C,D).
CALL SB16CD( DICO, JOBD, JOBMR, JOBCF, ORDSEL, N, M, P,
$ NCR, A, LDA, B, LDB, C, LDC, D, LDD, F, LDF,
$ G, LDG, HSV, TOL, IWORK, DWORK, LDWORK,
$ IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NCR
WRITE ( NOUT, FMT = 99987 )
WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1,N )
IF( NCR.GT.0 ) WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NCR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NCR )
20 CONTINUE
IF( NCR.GT.0 ) WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NCR
WRITE ( NOUT, FMT = 99995 ) ( G(I,J), J = 1,P )
40 CONTINUE
IF( NCR.GT.0 ) WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, M
WRITE ( NOUT, FMT = 99995 ) ( F(I,J), J = 1,NCR )
60 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB16CD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB16CD = ',I2)
99997 FORMAT (' The order of reduced controller = ',I2)
99996 FORMAT (/' The reduced controller state dynamics matrix Ac is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The reduced controller input/state matrix Bc is ')
99992 FORMAT (/' The reduced controller state/output matrix Cc is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99987 FORMAT (/' The frequency-weighted Hankel singular values are:')
END
Program Data
SB16CD EXAMPLE PROGRAM DATA (Continuous system)
8 1 1 2 0.1E0 C D F R F
0 1.0000 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 -0.0150 0.7650 0 0 0 0
0 0 -0.7650 -0.0150 0 0 0 0
0 0 0 0 -0.0280 1.4100 0 0
0 0 0 0 -1.4100 -0.0280 0 0
0 0 0 0 0 0 -0.0400 1.850
0 0 0 0 0 0 -1.8500 -0.040
0.0260
-0.2510
0.0330
-0.8860
-4.0170
0.1450
3.6040
0.2800
-.996 -.105 0.261 .009 -.001 -.043 0.002 -0.026
0.0
4.472135954999638e-002 6.610515358414598e-001 4.698598960657579e-003 3.601363251422058e-001 1.032530880771415e-001 -3.754055214487997e-002 -4.268536964759344e-002 3.287284547842979e-002
4.108939884667451e-001
8.684600000000012e-002
3.852317308197148e-004
-3.619366874815911e-003
-8.803722876359955e-003
8.420521094001852e-003
1.234944428038507e-003
4.263205617645322e-003
Program Results
SB16CD EXAMPLE PROGRAM RESULTS The order of reduced controller = 2 The frequency-weighted Hankel singular values are: 3.3073 0.7274 0.1124 0.0784 0.0242 0.0182 0.0101 0.0094 The reduced controller state dynamics matrix Ac is -0.4334 0.4884 -0.1950 -0.1093 The reduced controller input/state matrix Bc is -0.4231 -0.1785 The reduced controller state/output matrix Cc is -0.0326 -0.2307
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SB16CY.html 0000664 0000000 0000000 00000017315 14560147231 0020370 0 ustar 00root root 0000000 0000000
Purpose
To compute, for a given open-loop model (A,B,C,0), and for given state feedback gain F and full observer gain G, such that A+B*F and A+G*C are stable, the Cholesky factors Su and Ru of a controllability Grammian P = Su*Su' and of an observability Grammian Q = Ru'*Ru corresponding to a frequency-weighted model reduction of the left or right coprime factors of the state-feedback controller.Specification
SUBROUTINE SB16CY( DICO, JOBCF, N, M, P, A, LDA, B, LDB, C, LDC,
$ F, LDF, G, LDG, SCALEC, SCALEO, S, LDS, R, LDR,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, JOBCF
INTEGER INFO, LDA, LDB, LDC, LDF, LDG, LDR, LDS, LDWORK,
$ M, N, P
DOUBLE PRECISION SCALEC, SCALEO
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
$ F(LDF,*), G(LDG,*), R(LDR,*), S(LDS,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the open-loop system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOBCF CHARACTER*1
Specifies whether a left or right coprime factorization
of the state-feedback controller is to be used as follows:
= 'L': use a left coprime factorization;
= 'R': use a right coprime factorization.
Input/Output Parameters
N (input) INTEGER
The order of the open-loop state-space representation,
i.e., the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
state matrix A of the open-loop system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
input/state matrix B of the open-loop system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array must contain the
state/output matrix C of the open-loop system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
F (input) DOUBLE PRECISION array, dimension (LDF,N)
The leading M-by-N part of this array must contain a
stabilizing state feedback matrix.
LDF INTEGER
The leading dimension of array F. LDF >= MAX(1,M).
G (input) DOUBLE PRECISION array, dimension (LDG,P)
The leading N-by-P part of this array must contain a
stabilizing observer gain matrix.
LDG INTEGER
The leading dimension of array G. LDG >= MAX(1,N).
SCALEC (output) DOUBLE PRECISION
Scaling factor for the controllability Grammian.
See METHOD.
SCALEO (output) DOUBLE PRECISION
Scaling factor for the observability Grammian.
See METHOD.
S (output) DOUBLE PRECISION array, dimension (LDS,N)
The leading N-by-N upper triangular part of this array
contains the Cholesky factor Su of frequency-weighted
cotrollability Grammian P = Su*Su'. See METHOD.
LDS INTEGER
The leading dimension of the array S. LDS >= MAX(1,N).
R (output) DOUBLE PRECISION array, dimension (LDR,N)
The leading N-by-N upper triangular part of this array
contains the Cholesky factor Ru of the frequency-weighted
observability Grammian Q = Ru'*Ru. See METHOD.
LDR INTEGER
The leading dimension of the array R. LDR >= MAX(1,N).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1, N*(N + MAX(N,M) + MIN(N,M) + 6)),
if JOBCF = 'L';
LDWORK >= MAX(1, N*(N + MAX(N,P) + MIN(N,P) + 6)),
if JOBCF = 'R'.
For optimum performance LDWORK should be larger.
An upper bound for both cases is
LDWORK >= MAX(1, N*(N + MAX(N,M,P) + 7)).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: eigenvalue computation failure;
= 2: the matrix A+G*C is not stable;
= 3: the matrix A+B*F is not stable;
= 4: the Lyapunov equation for computing the
observability Grammian is (nearly) singular;
= 5: the Lyapunov equation for computing the
controllability Grammian is (nearly) singular.
Method
In accordance with the type of the coprime factorization
of the controller (left or right), the Cholesky factors Su and Ru
of the frequency-weighted controllability Grammian P = Su*Su' and
of the frequency-weighted observability Grammian Q = Ru'*Ru are
computed by solving appropriate Lyapunov or Stein equations [1].
If JOBCF = 'L' and DICO = 'C', P and Q are computed as the
solutions of the following Lyapunov equations:
(A+B*F)*P + P*(A+B*F)' + scalec^2*B*B' = 0, (1)
(A+G*C)'*Q + Q*(A+G*C) + scaleo^2*F'*F = 0. (2)
If JOBCF = 'L' and DICO = 'D', P and Q are computed as the
solutions of the following Stein equations:
(A+B*F)*P*(A+B*F)' - P + scalec^2*B*B' = 0, (3)
(A+G*C)'*Q*(A+G*C) - Q + scaleo^2*F'*F = 0. (4)
If JOBCF = 'R' and DICO = 'C', P and Q are computed as the
solutions of the following Lyapunov equations:
(A+B*F)*P + P*(A+B*F)' + scalec^2*G*G' = 0, (5)
(A+G*C)'*Q + Q*(A+G*C) + scaleo^2*C'*C = 0. (6)
If JOBCF = 'R' and DICO = 'D', P and Q are computed as the
solutions of the following Stein equations:
(A+B*F)*P*(A+B*F)' - P + scalec^2*G*G' = 0, (7)
(A+G*C)'*Q*(A+G*C) - Q + scaleo^2*C'*C = 0. (8)
References
[1] Liu, Y., Anderson, B.D.O. and Ly, O.L.
Coprime factorization controller reduction with Bezout
identity induced frequency weighting.
Automatica, vol. 26, pp. 233-249, 1990.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve for X either the continuous-time algebraic Riccati
equation
-1
Q + A'XE + E'XA - (L+E'XB)R (L+E'XB)' = 0 , (1)
or the discrete-time algebraic Riccati equation
-1
E'XE = A'XA - (L+A'XB)(R + B'XB) (L+A'XB)' + Q , (2)
where A, E, B, Q, R, and L are N-by-N, N-by-N, N-by-M, N-by-N,
M-by-M and N-by-M matrices, respectively, such that Q = C'C,
R = D'D and L = C'D; X is an N-by-N symmetric matrix.
The routine also returns the computed values of the closed-loop
spectrum of the system, i.e., the stable eigenvalues
lambda(1),...,lambda(N) of the pencil (A - BF,E), where F is
the optimal gain matrix,
-1
F = R (L+E'XB)' , for (1),
and
-1
F = (R+B'XB) (L+A'XB)' , for (2).
-1
Optionally, matrix G = BR B' may be given instead of B and R.
Other options include the case with Q and/or R given in a
factored form, Q = C'C, R = D'D, and with L a zero matrix.
The routine uses the method of deflating subspaces, based on
reordering the eigenvalues in a generalized Schur matrix pair.
It is assumed that E is nonsingular, but this condition is not
checked. Note that the definition (1) of the continuous-time
algebraic Riccati equation, and the formula for the corresponding
optimal gain matrix, require R to be nonsingular, but the
associated linear quadratic optimal problem could have a unique
solution even when matrix R is singular, under mild assumptions
(see METHOD). The routine SG02AD works accordingly in this case.
Specification
SUBROUTINE SG02AD( DICO, JOBB, FACT, UPLO, JOBL, SCAL, SORT, ACC,
$ N, M, P, A, LDA, E, LDE, B, LDB, Q, LDQ, R,
$ LDR, L, LDL, RCONDU, X, LDX, ALFAR, ALFAI,
$ BETA, S, LDS, T, LDT, U, LDU, TOL, IWORK,
$ DWORK, LDWORK, BWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER ACC, DICO, FACT, JOBB, JOBL, SCAL, SORT, UPLO
INTEGER INFO, IWARN, LDA, LDB, LDE, LDL, LDQ, LDR, LDS,
$ LDT, LDU, LDWORK, LDX, M, N, P
DOUBLE PRECISION RCONDU, TOL
C .. Array Arguments ..
LOGICAL BWORK(*)
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), ALFAI(*), ALFAR(*), B(LDB,*), BETA(*),
$ DWORK(*), E(LDE,*), L(LDL,*), Q(LDQ,*),
$ R(LDR,*), S(LDS,*), T(LDT,*), U(LDU,*), X(LDX,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of Riccati equation to be solved as
follows:
= 'C': Equation (1), continuous-time case;
= 'D': Equation (2), discrete-time case.
JOBB CHARACTER*1
Specifies whether or not the matrix G is given, instead
of the matrices B and R, as follows:
= 'B': B and R are given;
= 'G': G is given.
FACT CHARACTER*1
Specifies whether or not the matrices Q and/or R (if
JOBB = 'B') are factored, as follows:
= 'N': Not factored, Q and R are given;
= 'C': C is given, and Q = C'C;
= 'D': D is given, and R = D'D;
= 'B': Both factors C and D are given, Q = C'C, R = D'D.
UPLO CHARACTER*1
If JOBB = 'G', or FACT = 'N', specifies which triangle of
the matrices G, or Q and R, is stored, as follows:
= 'U': Upper triangle is stored;
= 'L': Lower triangle is stored.
JOBL CHARACTER*1
Specifies whether or not the matrix L is zero, as follows:
= 'Z': L is zero;
= 'N': L is nonzero.
JOBL is not used if JOBB = 'G' and JOBL = 'Z' is assumed.
SLICOT Library routine SB02MT should be called just before
SG02AD, for obtaining the results when JOBB = 'G' and
JOBL = 'N'.
SCAL CHARACTER*1
If JOBB = 'B', specifies whether or not a scaling strategy
should be used to scale Q, R, and L, as follows:
= 'G': General scaling should be used;
= 'N': No scaling should be used.
SCAL is not used if JOBB = 'G'.
SORT CHARACTER*1
Specifies which eigenvalues should be obtained in the top
of the generalized Schur form, as follows:
= 'S': Stable eigenvalues come first;
= 'U': Unstable eigenvalues come first.
ACC CHARACTER*1
Specifies whether or not iterative refinement should be
used to solve the system of algebraic equations giving
the solution matrix X, as follows:
= 'R': Use iterative refinement;
= 'N': Do not use iterative refinement.
Input/Output Parameters
N (input) INTEGER
The actual state dimension, i.e., the order of the
matrices A, E, Q, and X, and the number of rows of the
matrices B and L. N >= 0.
M (input) INTEGER
The number of system inputs. If JOBB = 'B', M is the
order of the matrix R, and the number of columns of the
matrix B. M >= 0.
M is not used if JOBB = 'G'.
P (input) INTEGER
The number of system outputs. If FACT = 'C' or 'D' or 'B',
P is the number of rows of the matrices C and/or D.
P >= 0.
Otherwise, P is not used.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
state matrix A of the descriptor system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
E (input) DOUBLE PRECISION array, dimension (LDE,N)
The leading N-by-N part of this array must contain the
matrix E of the descriptor system.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,*)
If JOBB = 'B', the leading N-by-M part of this array must
contain the input matrix B of the system.
If JOBB = 'G', the leading N-by-N upper triangular part
(if UPLO = 'U') or lower triangular part (if UPLO = 'L')
of this array must contain the upper triangular part or
lower triangular part, respectively, of the matrix
-1
G = BR B'. The strictly lower triangular part (if
UPLO = 'U') or strictly upper triangular part (if
UPLO = 'L') is not referenced.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
Q (input) DOUBLE PRECISION array, dimension (LDQ,N)
If FACT = 'N' or 'D', the leading N-by-N upper triangular
part (if UPLO = 'U') or lower triangular part (if UPLO =
'L') of this array must contain the upper triangular part
or lower triangular part, respectively, of the symmetric
state weighting matrix Q. The strictly lower triangular
part (if UPLO = 'U') or strictly upper triangular part (if
UPLO = 'L') is not referenced.
If FACT = 'C' or 'B', the leading P-by-N part of this
array must contain the output matrix C of the system.
If JOBB = 'B' and SCAL = 'G', then Q is modified
internally, but is restored on exit.
LDQ INTEGER
The leading dimension of array Q.
LDQ >= MAX(1,N) if FACT = 'N' or 'D';
LDQ >= MAX(1,P) if FACT = 'C' or 'B'.
R (input) DOUBLE PRECISION array, dimension (LDR,*)
If FACT = 'N' or 'C', the leading M-by-M upper triangular
part (if UPLO = 'U') or lower triangular part (if UPLO =
'L') of this array must contain the upper triangular part
or lower triangular part, respectively, of the symmetric
input weighting matrix R. The strictly lower triangular
part (if UPLO = 'U') or strictly upper triangular part (if
UPLO = 'L') is not referenced.
If FACT = 'D' or 'B', the leading P-by-M part of this
array must contain the direct transmission matrix D of the
system.
If JOBB = 'B' and SCAL = 'G', then R is modified
internally, but is restored on exit.
If JOBB = 'G', this array is not referenced.
LDR INTEGER
The leading dimension of array R.
LDR >= MAX(1,M) if JOBB = 'B' and FACT = 'N' or 'C';
LDR >= MAX(1,P) if JOBB = 'B' and FACT = 'D' or 'B';
LDR >= 1 if JOBB = 'G'.
L (input) DOUBLE PRECISION array, dimension (LDL,*)
If JOBL = 'N' and JOBB = 'B', the leading N-by-M part of
this array must contain the cross weighting matrix L.
If JOBB = 'B' and SCAL = 'G', then L is modified
internally, but is restored on exit.
If JOBL = 'Z' or JOBB = 'G', this array is not referenced.
LDL INTEGER
The leading dimension of array L.
LDL >= MAX(1,N) if JOBL = 'N' and JOBB = 'B';
LDL >= 1 if JOBL = 'Z' or JOBB = 'G'.
RCONDU (output) DOUBLE PRECISION
If N > 0 and INFO = 0 or INFO = 7, an estimate of the
reciprocal of the condition number (in the 1-norm) of
the N-th order system of algebraic equations from which
the solution matrix X is obtained.
X (output) DOUBLE PRECISION array, dimension (LDX,N)
If INFO = 0, the leading N-by-N part of this array
contains the solution matrix X of the problem.
LDX INTEGER
The leading dimension of array X. LDX >= MAX(1,N).
ALFAR (output) DOUBLE PRECISION array, dimension (2*N)
ALFAI (output) DOUBLE PRECISION array, dimension (2*N)
BETA (output) DOUBLE PRECISION array, dimension (2*N)
The generalized eigenvalues of the 2N-by-2N matrix pair,
ordered as specified by SORT (if INFO = 0, or INFO >= 5).
For instance, if SORT = 'S', the leading N elements of
these arrays contain the closed-loop spectrum of the
system. Specifically,
lambda(k) = [ALFAR(k)+j*ALFAI(k)]/BETA(k) for
k = 1,2,...,N.
S (output) DOUBLE PRECISION array, dimension (LDS,*)
The leading 2N-by-2N part of this array contains the
ordered real Schur form S of the first matrix in the
reduced matrix pencil associated to the optimal problem,
corresponding to the scaled Q, R, and L, if JOBB = 'B'
and SCAL = 'G'. That is,
(S S )
( 11 12)
S = ( ),
(0 S )
( 22)
where S , S and S are N-by-N matrices.
11 12 22
Array S must have 2*N+M columns if JOBB = 'B', and 2*N
columns, otherwise.
LDS INTEGER
The leading dimension of array S.
LDS >= MAX(1,2*N+M) if JOBB = 'B';
LDS >= MAX(1,2*N) if JOBB = 'G'.
T (output) DOUBLE PRECISION array, dimension (LDT,2*N)
The leading 2N-by-2N part of this array contains the
ordered upper triangular form T of the second matrix in
the reduced matrix pencil associated to the optimal
problem, corresponding to the scaled Q, R, and L, if
JOBB = 'B' and SCAL = 'G'. That is,
(T T )
( 11 12)
T = ( ),
(0 T )
( 22)
where T , T and T are N-by-N matrices.
11 12 22
LDT INTEGER
The leading dimension of array T.
LDT >= MAX(1,2*N+M) if JOBB = 'B';
LDT >= MAX(1,2*N) if JOBB = 'G'.
U (output) DOUBLE PRECISION array, dimension (LDU,2*N)
The leading 2N-by-2N part of this array contains the right
transformation matrix U which reduces the 2N-by-2N matrix
pencil to the ordered generalized real Schur form (S,T).
That is,
(U U )
( 11 12)
U = ( ),
(U U )
( 21 22)
where U , U , U and U are N-by-N matrices.
11 12 21 22
If JOBB = 'B' and SCAL = 'G', then U corresponds to the
scaled pencil. If a basis for the stable deflating
subspace of the original problem is needed, then the
submatrix U must be multiplied by the scaling factor
21
contained in DWORK(4).
LDU INTEGER
The leading dimension of array U. LDU >= MAX(1,2*N).
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used to test for near singularity of
the original matrix pencil, specifically of the triangular
M-by-M factor obtained during the reduction process. If
the user sets TOL > 0, then the given value of TOL is used
as a lower bound for the reciprocal condition number of
that matrix; a matrix whose estimated condition number is
less than 1/TOL is considered to be nonsingular. If the
user sets TOL <= 0, then a default tolerance, defined by
TOLDEF = EPS, is used instead, where EPS is the machine
precision (see LAPACK Library routine DLAMCH).
This parameter is not referenced if JOBB = 'G'.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK >= MAX(1,M,2*N) if JOBB = 'B';
LIWORK >= MAX(1,2*N) if JOBB = 'G'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK. If JOBB = 'B' and N > 0, DWORK(2) returns the
reciprocal of the condition number of the M-by-M bottom
right lower triangular matrix obtained while compressing
the matrix pencil of order 2N+M to obtain a pencil of
order 2N. If ACC = 'R', and INFO = 0 or INFO = 7, DWORK(3)
returns the reciprocal pivot growth factor (see SLICOT
Library routine MB02PD) for the LU factorization of the
coefficient matrix of the system of algebraic equations
giving the solution matrix X; if DWORK(3) is much
less than 1, then the computed X and RCONDU could be
unreliable. If INFO = 0 or INFO = 7, DWORK(4) returns the
scaling factor used to scale Q, R, and L. DWORK(4) is set
to 1 if JOBB = 'G' or SCAL = 'N'.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(7*(2*N+1)+16,16*N), if JOBB = 'G';
LDWORK >= MAX(7*(2*N+1)+16,16*N,2*N+M,3*M), if JOBB = 'B'.
For optimum performance LDWORK should be larger.
BWORK LOGICAL array, dimension (2*N)
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: the computed solution may be inaccurate due to poor
scaling or eigenvalues too close to the boundary of
the stability domain (the imaginary axis, if
DICO = 'C', or the unit circle, if DICO = 'D').
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the computed extended matrix pencil is singular,
possibly due to rounding errors;
= 2: if the QZ algorithm failed;
= 3: if reordering of the generalized eigenvalues failed;
= 4: if after reordering, roundoff changed values of
some complex eigenvalues so that leading eigenvalues
in the generalized Schur form no longer satisfy the
stability condition; this could also be caused due
to scaling;
= 5: if the computed dimension of the solution does not
equal N;
= 6: if the spectrum is too close to the boundary of
the stability domain;
= 7: if a singular matrix was encountered during the
computation of the solution matrix X.
Method
The routine uses a variant of the method of deflating subspaces
proposed by van Dooren [1]. See also [2], [3], [4].
It is assumed that E is nonsingular, the triple (E,A,B) is
strongly stabilizable and detectable (see [3]); if, in addition,
- [ Q L ]
R := [ ] >= 0 ,
[ L' R ]
then the pencils
discrete-time continuous-time
|A 0 B| |E 0 0| |A 0 B| |E 0 0|
|Q -E' L| - z |0 -A' 0| , |Q A' L| - s |0 -E' 0| , (3)
|L' 0 R| |0 -B' 0| |L' B' R| |0 0 0|
are dichotomic, i.e., they have no eigenvalues on the boundary of
the stability domain. The above conditions are sufficient for
regularity of these pencils. A necessary condition is that
rank([ B' L' R']') = m.
Under these assumptions the algebraic Riccati equation is known to
have a unique non-negative definite solution.
The first step in the method of deflating subspaces is to form the
extended matrices in (3), of order 2N + M. Next, these pencils are
compressed to a form of order 2N (see [1])
lambda x A - B .
f f
This generalized eigenvalue problem is then solved using the QZ
algorithm and the stable deflating subspace Ys is determined.
If [Y1'|Y2']' is a basis for Ys, then the required solution is
-1
X = Y2 x Y1 .
References
[1] Van Dooren, P.
A Generalized Eigenvalue Approach for Solving Riccati
Equations.
SIAM J. Sci. Stat. Comp., 2, pp. 121-135, 1981.
[2] Arnold, III, W.F. and Laub, A.J.
Generalized Eigenproblem Algorithms and Software for
Algebraic Riccati Equations.
Proc. IEEE, 72, 1746-1754, 1984.
[3] Mehrmann, V.
The Autonomous Linear Quadratic Control Problem. Theory and
Numerical Solution.
Lect. Notes in Control and Information Sciences, vol. 163,
Springer-Verlag, Berlin, 1991.
[4] Sima, V.
Algorithms for Linear-Quadratic Optimization.
Pure and Applied Mathematics: A Series of Monographs and
Textbooks, vol. 200, Marcel Dekker, Inc., New York, 1996.
Numerical Aspects
This routine is particularly suited for systems where the matrix R is ill-conditioned, or even singular.Further Comments
To obtain a stabilizing solution of the algebraic Riccati equations set SORT = 'S'. The routine can also compute the anti-stabilizing solutions of the algebraic Riccati equations, by specifying SORT = 'U'.Example
Program Text
* SG02AD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER NMAX2M, NMAX2, NMMAX
PARAMETER ( NMAX2M = 2*NMAX+MMAX, NMAX2 = 2*NMAX,
$ NMMAX = MAX(NMAX,MMAX) )
INTEGER LDA, LDB, LDE, LDL, LDQ, LDR, LDS, LDT, LDU, LDX
PARAMETER ( LDA = NMAX, LDB = NMAX, LDE = NMAX, LDL = NMAX,
$ LDQ = MAX(NMAX,PMAX), LDR = MAX(MMAX,PMAX),
$ LDS = NMAX2M, LDT = NMAX2M, LDU = NMAX2,
$ LDX = NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MAX(MMAX,NMAX2) )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX(14*NMAX+23,16*NMAX,2*NMAX+MMAX,
$ 3*MMAX) )
INTEGER LBWORK
PARAMETER ( LBWORK = NMAX2 )
* .. Local Scalars ..
DOUBLE PRECISION RCONDU, TOL
INTEGER I, INFO, IWARN, J, M, N, P
CHARACTER*1 ACC, DICO, FACT, JOBB, JOBL, SCAL, SORT, UPLO
LOGICAL LJOBB
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), ALFAI(NMAX2), ALFAR(NMAX2),
$ B(LDB,NMMAX), BETA(NMAX2), DWORK(LDWORK),
$ E(LDE,NMAX), L(LDL,MMAX), Q(LDQ,NMAX),
$ R(LDR,MMAX), S(LDS,NMAX2M), T(LDT,NMAX2),
$ U(LDU,NMAX2), X(LDX,NMAX)
INTEGER IWORK(LIWORK)
LOGICAL BWORK(LBWORK)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL SG02AD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL, DICO, JOBB, FACT, UPLO, JOBL,
$ SCAL, SORT, ACC
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) M
ELSE
LJOBB = LSAME( JOBB, 'B' )
IF ( LJOBB ) THEN
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,N ), I = 1,N )
END IF
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) P
ELSE
IF ( LSAME( FACT, 'N' ) .OR. LSAME( FACT, 'D' ) ) THEN
READ ( NIN, FMT = * )
$ ( ( Q(I,J), J = 1,N ), I = 1,N )
ELSE
READ ( NIN, FMT = * )
$ ( ( Q(I,J), J = 1,N ), I = 1,P )
END IF
IF ( LJOBB ) THEN
IF ( LSAME( FACT, 'N' ) .OR. LSAME( FACT, 'C' ) ) THEN
READ ( NIN, FMT = * )
$ ( ( R(I,J), J = 1,M ), I = 1,M )
ELSE
READ ( NIN, FMT = * )
$ ( ( R(I,J), J = 1,M ), I = 1,P )
END IF
IF ( LSAME( JOBL, 'N' ) )
$ READ ( NIN, FMT = * )
$ ( ( L(I,J), J = 1,M ), I = 1,N )
END IF
* Find the solution matrix X.
CALL SG02AD( DICO, JOBB, FACT, UPLO, JOBL, SCAL, SORT,
$ ACC, N, M, P, A, LDA, E, LDE, B, LDB, Q,
$ LDQ, R, LDR, L, LDL, RCONDU, X, LDX, ALFAR,
$ ALFAI, BETA, S, LDS, T, LDT, U, LDU, TOL,
$ IWORK, DWORK, LDWORK, BWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( X(I,J), J = 1,N )
20 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SG02AD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SG02AD = ',I2)
99997 FORMAT (' The solution matrix X is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' N is out of range.',/' N = ',I5)
99994 FORMAT (/' M is out of range.',/' M = ',I5)
99993 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
SG02AD EXAMPLE PROGRAM DATA 2 1 3 0.0 C B B U Z N S N 0.0 1.0 0.0 0.0 1.0 0.0 0.0 1.0 0.0 1.0 1.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 1.0Program Results
SG02AD EXAMPLE PROGRAM RESULTS The solution matrix X is 1.7321 1.0000 1.0000 1.7321
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SG02CV.html 0000664 0000000 0000000 00000021730 14560147231 0020361 0 ustar 00root root 0000000 0000000
Purpose
To compute the residual matrix R for a continuous-time or
discrete-time "reduced" Lyapunov equation, using the formulas
R = op(A)'*X + X*op(A) + Q,
or
R = op(A)'*X*op(E) + op(E)'*X*op(A) + Q,
in the continuous-time case, or the formulas
R = op(A)'*X*op(A) - X + Q,
or
R = op(A)'*X*op(A) - op(E)'*X*op(E) + Q,
in the discrete-time case, where X and Q are symmetric matrices,
A is in upper real Schur form, E is upper triangular, and op(W) is
op(W) = W or op(W) = W'.
Optionally, the Frobenius norms of the product terms defining the
denominator of the relative residual are also computed. The norms
of Q and X are not computed.
Specification
SUBROUTINE SG02CV( DICO, JOB, JOBE, UPLO, TRANS, N, A, LDA, E,
$ LDE, X, LDX, R, LDR, NORMS, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER DICO, JOB, JOBE, TRANS, UPLO
INTEGER INFO, LDA, LDE, LDR, LDWORK, LDX, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), E(LDE,*), NORMS(*),
$ R(LDR,*), X(LDX,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the Lyapunov equation, as follows:
= 'C': continuous-time Lyapunov equation;
= 'D': discrete-time Lyapunov equation.
JOB CHARACTER*1
Specifies which results must be computed, as follows:
= 'R': The matrix R only must be computed;
= 'N': The matrix R and the norms must be computed;
= 'B': The matrix R and the norms must be computed.
JOBE CHARACTER*1
Specifies whether E is a general or an identity matrix,
as follows:
= 'G': The matrix E is general and is given;
= 'I': The matrix E is assumed identity and is not given.
UPLO CHARACTER*1
Specifies which triangles of the symmetric matrices X and
Q are given, as follows:
= 'U': The upper triangular part is given;
= 'L': The lower triangular part is given.
TRANS CHARACTER*1
Specifies the form of op(W) to be used in the formulas
above, as follows:
= 'N': op(W) = W;
= 'T': op(W) = W';
= 'C': op(W) = W'.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, E, Q, X, and R. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N upper part of this array must contain
the upper real Schur matrix A.
If TRANS = 'N' and (DICO = 'D' or (JOB = 'R' and
JOBE = 'G')), the entries 3, 4,..., N of the first column
are modified internally, but are restored on exit.
Otherwise, the part of this array below the first
subdiagonal is not referenced.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
E (input) DOUBLE PRECISION array, dimension (LDE,*)
If JOBE = 'G', the leading N-by-N upper triangular part of
this array must contain the upper triangular matrix E.
The strictly lower triangular part of this array is not
referenced.
If JOBE = 'I', this array is not referenced.
LDE INTEGER
The leading dimension of array E.
LDE >= MAX(1,N), if JOBE = 'G';
LDE >= 1, if JOBE = 'I'.
X (input/works.) DOUBLE PRECISION array, dimension (LDX,N)
On entry, if UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix X and the strictly
lower triangular part of the array is not referenced.
On entry, if UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix X and the strictly
upper triangular part of the array is not referenced.
If DICO = 'D' or (JOB = 'R' and JOBE = 'G'), the diagonal
elements of this array are modified internally, but they
are restored on exit.
LDX INTEGER
The leading dimension of array X. LDX >= MAX(1,N).
R (input/output) DOUBLE PRECISION array, dimension (LDR,*)
On entry, the leading N-by-N upper or lower triangular
part (depending on UPLO) of this array must contain the
upper or lower triangular part, respectively, of the
matrix Q. The other strictly triangular part is not
referenced.
On exit, the leading N-by-N upper or lower triangular
part (depending on UPLO) of this array contains the upper
or lower triangular part, respectively, of the matrix R.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
NORMS (output) DOUBLE PRECISION array, dimension (LN)
If JOB = 'N' or JOB = 'B', LN = 1 or 2, if (DICO = 'C' or
JOBE = 'I'), or (DICO = 'D' and JOBE = 'G'), respectively.
If DICO = 'C',
NORMS(1) contains the Frobenius norm of the matrix
op(A)'*X (or of X*op(A)), if JOBE = 'I', or of the matrix
op(A)'*X*op(E) (or of op(E)'*X*op(A)), if JOBE = 'G'.
If DICO = 'D',
NORMS(1) contains the Frobenius norm of the matrix
op(A)'*X*op(A);
if JOBE = 'G', NORMS(2) contains the Frobenius norm of the
matrix op(E)'*X*op(E).
If JOB <> 'N', this array is not referenced.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = -17 or if LDWORK = -2 on input, then
DWORK(1) returns the minimum value of LDWORK.
On exit, if INFO = 0, or if LDWORK = -1 on input, then
DWORK(1) returns the optimal value of LDWORK.
LDWORK The length of the array DWORK. LDWORK >= MAX(v,1), with v
specified in the following table, where
a = 1, if JOBE = 'G';
a = 0, if JOBE = 'I'.
DICO JOB v
----------------------------
'C' 'R' a*N*N
'C' 'N','B' N*N
----------------------------
'D' 'R' N*N
'D' 'N','B' 2*N*N
----------------------------
If LDWORK = -1, an optimal workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
If LDWORK = -2, a minimal workspace query is assumed; the
routine only calculates the minimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The matrix expressions are efficiently evaluated, using symmetry. If JOB = 'N' or JOB = 'B', then: If DICO = 'C', the matrices op(op(A)'*X*op(E)) or op(X*op(A)), are efficiently computed. If DICO = 'D', the matrices op(A)'*X*op(A) and op(E)'*X*op(E), if JOBE = 'G', are efficiently computed. The results are used to evaluate R and the norms. If JOB = 'R', then the needed parts of the intermediate results are obtained and used to evaluate R.Numerical Aspects
The calculations are backward stable.
The algorithm requires approximately a*N^3 operations, where ^
denotes the power operator, and
a = 1, if DICO = 'C' and JOB <> 'R' and JOBE = 'G';
a = 1/2, otherwise.
An "operation" includes a multiplication, an addition, and some
address calculations.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the residual matrix R for a continuous-time or
discrete-time Riccati equation and/or the "closed-loop system"
matrix op(C), using the formulas
R = op(A)'*X + X*op(A) +/- X*G*X + Q,
C = op(A) +/- G*X,
or
R = op(A)'*X*op(E) + op(E)'*X*op(A) +/- op(E)'*X*G*X*op(E) + Q,
C = op(A) +/- G*X*op(E),
or
R = op(A)'*X*op(E) + op(E)'*X*op(A) +/- H*K + Q,
C = op(A) +/- B*K,
in the continuous-time case, or the formulas
R = op(A)'*X*op(A) - X +/- op(A)'*X*G*X*op(A) + Q,
C = op(A) +/- G*X*op(A),
or
R = op(A)'*X*op(A) - op(E)'*X*op(E) +/- op(A)'*X*G*X*op(A) + Q,
C = op(A) +/- G*X*op(A),
or
R = op(A)'*X*op(A) - op(E)'*X*op(E) +/- H*K + Q,
C = op(A) +/- B*K,
in the discrete-time case, where X, G, and Q are symmetric
matrices, A, E, H, K, B are general matrices, and op(W) is one of
op(W) = W or op(W) = W'.
_-1
Instead of the symmetric N-by-N matrix G, G = B*R *B', the N-by-M
_-1
matrix D, D = B*L , such that G = D*D', may be given on entry.
_ _ _ _
The matrix R, with R = L'*L, is a weighting matrix of the optimal
_ _
problem, if DICO = 'C', or it is R = B'*X*B + Rd, if DICO = 'D',
_ _ _
with Rd a similar weighting matrix; L is a Cholesky factor of R,
_ _
if R is positive definite. If R is not positive definite, which
may happen in the discrete-time case, a UdU' or LdL' factorization
is used to compute the matrices H and K. If M = 0, the residual
matrix of a (generalized) Lyapunov or Stein equation is computed.
To this end, set JOBG = 'D' and JOB = 'R' (since op(C) = A in this
case).
Optionally, the quadratic term in the formulas for R is specified
as H*K, where
H = L + op(E)'*X*B, if DICO = 'C', or
H = L + op(A)'*X*B, if DICO = 'D', and
_-1
K = R *H',
with L an N-by-M matrix. This is useful, e.g., for DICO = 'D',
_ _
when L <> 0 and/or Rd is singular, hence R might be numerically
indefinite; it might be indefinite in the first iterations of
Newton's algorithm. Depending on JOB, part or all of the matrices
H, K, and B should be given in such a case.
_
If R is positive definite, the quadratic term can be specified
as F*F', and the second term in the formulas for C is D*F', where
_-1
F = H*L .
The matrices F and/or D should be given. This option is not useful
when L = 0, unless F and D are available. If DICO = 'C', the
computational problem with L <> 0 is equivalent with one with
L = 0 after replacing
_-1 _-1
A := A - B*R* L', Q := Q - L*R* L'.
_ _
These formulas, with R replaced by Rd, can also be used in the
_
discrete-time case, if Rd is nonsingular and well-conditioned with
respect to inversion.
Optionally, the Frobenius norms of the product terms defining the
denominator of the relative residual are also computed. The norms
of Q and X are not computed.
Specification
SUBROUTINE SG02CW( DICO, JOB, JOBE, FLAG, JOBG, UPLO, TRANS, N, M,
$ A, LDA, E, LDE, G, LDG, X, LDX, F, LDF, K, LDK,
$ XE, LDXE, R, LDR, C, LDC, NORMS, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER DICO, FLAG, JOB, JOBE, JOBG, TRANS, UPLO
INTEGER INFO, LDA, LDC, LDE, LDF, LDG, LDK, LDR, LDWORK,
$ LDX, LDXE, M, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), C(LDC,*), DWORK(*), E(LDE,*),
$ F(LDF,*), G(LDG,*), K(LDK,*), NORMS(*),
$ R(LDR,*), X(LDX,*), XE(LDXE,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the Riccati equation, as follows:
= 'C': continuous-time algebraic Riccati equation;
= 'D': discrete-time algebraic Riccati equation.
JOB CHARACTER*1
Specifies which results must be computed, as follows:
= 'A': Both (all) matrices R and C must be computed;
= 'R': The matrix R only must be computed;
= 'C': The matrix C only must be computed;
= 'N': The matrices R and C and the norms must be
computed;
= 'B': The matrix R and the norms must be computed.
JOBE CHARACTER*1
Specifies whether E is a general or an identity matrix,
as follows:
= 'G': The matrix E is general and is given;
= 'I': The matrix E is assumed identity and is not given.
FLAG CHARACTER*1
Specifies which sign is used, as follows:
= 'P': The plus sign is used;
= 'M': The minus sign is used.
JOBG CHARACTER*1
Specifies how the quadratic term in the formulas for R is
defined, as follows:
= 'G': The matrix G is given;
= 'D': The matrix D is given;
= 'F': The matrix F is given;
= 'H': The matrices H and K are given.
UPLO CHARACTER*1
Specifies which triangles of the symmetric matrices X, G
(if JOBG = 'G'), and Q (if JOB <> 'C') are given, as
follows:
= 'U': The upper triangular part is given;
= 'L': The lower triangular part is given.
TRANS CHARACTER*1
Specifies the form of op(W) to be used in the formulas
above, as follows:
= 'N': op(W) = W;
= 'T': op(W) = W';
= 'C': op(W) = W'.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, E, Q, X, C and R. N >= 0.
M (input) INTEGER
If JOBG <> 'G', the number of columns of the matrices D,
F, and/or B, H, and K'. M >= 0.
If JOBG = 'G', the value of M is meaningless.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
matrix A.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
E (input) DOUBLE PRECISION array, dimension (LDE,*)
If JOBE = 'G' and (JOB <> 'C' or (DICO = 'C' and
(JOBG = 'G' or JOBG = 'D'))), the leading N-by-N part of
this array must contain the matrix E.
If JOBE = 'I' or (JOB = 'C' and (DICO = 'D' or
JOBG = 'F' or JOBG = 'H')), this array is not referenced.
LDE INTEGER
The leading dimension of array E.
LDE >= MAX(1,N), if JOBE = 'G' and (JOB <> 'C' or
(DICO = 'C' and (JOBG = 'G' or
JOBG = 'D')));
LDE >= 1, if JOBE = 'I' or (JOB = 'C' and
(DICO = 'D' or JOBG = 'F' or
JOBG = 'H')).
G (input/works.) DOUBLE PRECISION array, dimension (LDG,*)
If JOBG = 'G', the leading N-by-N upper or lower
triangular part (depending on UPLO) of this array must
contain the upper or lower triangular part, respectively,
of the matrix G. The other strictly triangular part is not
referenced. If DICO = 'D', (JOB = 'R' or JOB = 'B'), and
JOBG = 'G', the diagonal elements of this array are
modified internally, but are restored on exit.
If JOBG = 'D' or (JOBG = 'F' and JOB <> 'R' and
JOB <> 'B'), the leading N-by-M part of this array must
contain the matrix D, so that G = D*D'.
If JOBG = 'H' and JOB <> 'R' and JOB <> 'B', the leading
N-by-M part of this array must contain the matrix B.
If (JOBG = 'F' or JOBG = 'H') and JOB = 'R' or JOB = 'B',
this array is not referenced.
LDG INTEGER
The leading dimension of array G.
LDG >= MAX(1,N), if JOBG = 'G' or JOBG = 'D' or
(JOB <> 'R' and JOB <> 'B');
LDG >= 1, if (JOBG = 'F' or JOBG = 'H') and
(JOB = 'R' or JOB = 'B').
X (input/works.) DOUBLE PRECISION array, dimension (LDX,N)
The leading N-by-N part of this array must contain the
symmetric matrix X, and it is unchanged on exit.
If DICO = 'D', JOBE = 'G' and JOB <> 'C', the diagonal
elements of this array are modified internally, but they
are restored on exit.
The full matrix X should be input if DICO = 'C',
JOBE = 'I', and the conditions in the lines of the table
below are satisfied
JOBG JOB LDWORK
----------------------------------------------
'F','H' 'A','R' LDWORK < N*N
'G' 'A','R','N' LDWORK < 2*N*N
'G' 'C' LDWORK < N*N
'G' 'B' LDWORK < 3*N*N
'D' 'R' (M<=N, LDWORK < N*N) or
(M> N, LDWORK < 3*N*N)
'D' 'A' (M<=N, LDWORK < N*N) or
(LDWORK >= N*N and
LDWORK < 2*N*N)
----------------------------------------------
For all the other cases, including when the optimal length
of the workspace array DWORK is used, only the relevant
upper or lower triangular part (depending on UPLO) of this
array must be input, and the other strictly triangular
part is not referenced.
LDX INTEGER
The leading dimension of array X. LDX >= MAX(1,N).
F (input) DOUBLE PRECISION array, dimension (LDF,*)
If JOBG = 'F', the leading N-by-M part of this array must
contain the matrix F.
If JOBG = 'H', the leading N-by-M part of this array must
contain the matrix H.
If JOBG = 'G' or JOBG = 'D', this array is not referenced.
LDF INTEGER
The leading dimension of array F.
LDF >= MAX(1,N), if JOBG = 'F' or JOBG = 'H';
LDF >= 1, if JOBG = 'G' or JOBG = 'D'.
K (input) DOUBLE PRECISION array, dimension (LDK,*)
If JOBG = 'H', the leading M-by-N part of this array must
contain the matrix K.
If JOBG <> 'H', this array is not referenced.
LDK INTEGER
The leading dimension of array K.
LDK >= MAX(1,M), if JOBG = 'H';
LDK >= 1, if JOBG <> 'H'.
XE (input) DOUBLE PRECISION array, dimension (LDXE,*)
If (JOBG = 'F' or JOBG = 'H'), JOB <> 'C', DICO = 'C', and
JOBE = 'G', the leading N-by-N part of this array must
contain the matrix product X*E, if TRANS = 'N', or E*X, if
TRANS = 'T' or 'C'.
If (JOBG = 'F' or JOBG = 'H'), JOB <> 'C', and DICO = 'D',
the leading N-by-N part of this array must contain the
matrix product X*A, if TRANS = 'N', or A*X, if TRANS = 'T'
or 'C'.
These matrix products are needed for computing F or H.
If JOBG = 'G' or JOBG = 'D' or JOB = 'C' or (DICO = 'C'
and JOBE = 'I') this array is not referenced.
LDXE INTEGER
The leading dimension of array XE.
LDXE >= MAX(1,N), if (JOBG = 'F' or JOBG = 'H'),
JOB <> 'C', and either DICO = 'C' and
JOBE = 'G', or DICO = 'D';
LDXE >= 1, if JOBG = 'G' or JOBG = 'D' or JOB = 'C'
or (DICO = 'C' and JOBE = 'I').
R (input/output) DOUBLE PRECISION array, dimension (LDR,*)
On entry, if JOB <> 'C', the leading N-by-N upper or lower
triangular part (depending on UPLO) of this array must
contain the upper or lower triangular part, respectively,
of the matrix Q. The other strictly triangular part is not
referenced.
On exit, if JOB <> 'C' and INFO = 0, the leading N-by-N
upper or lower triangular part (depending on UPLO) of this
array contains the upper or lower triangular part,
respectively, of the matrix R.
If JOB = 'C', this array is not referenced.
LDR INTEGER
The leading dimension of array R.
LDR >= MAX(1,N), if JOB <> 'C';
LDR >= 1, if JOB = 'C'.
C (output) DOUBLE PRECISION array, dimension (LDC,*)
If JOB <> 'R' and JOB <> 'B' and INFO = 0, the leading
N-by-N part of this array contains the matrix op(C).
If JOB = 'R' or JOB = 'B', this array is not referenced.
LDC INTEGER
The leading dimension of array C.
LDC >= MAX(1,N), if JOB <> 'R' and JOB <> 'B';
LDC >= 1, if JOB = 'R' or JOB = 'B'.
NORMS (output) DOUBLE PRECISION array, dimension (LN)
If JOB = 'N' or JOB = 'B', LN = 2 or 3, if (DICO = 'C' or
JOBE = 'I'), or (DICO = 'D' and JOBE = 'G'), respectively.
If DICO = 'C',
NORMS(1) contains the Frobenius norm of the matrix
op(A)'*X (or of X*op(A)), if JOBE = 'I', or of the matrix
op(A)'*X*op(E) (or of op(E)'*X*op(A)), if JOBE = 'G';
NORMS(2) contains the Frobenius norm of the matrix
product X*G*X, if JOBE = 'I', or of the matrix product
V = op(E)'*X*G*X*op(E), if JOBE = 'G' (for JOBG = 'G' or
JOBG = 'D'), or of V = F*F', if JOBG = 'F', or of V = H*K,
if JOBG = 'H'.
If DICO = 'D',
NORMS(1) contains the Frobenius norm of the matrix
op(A)'*X*op(A);
NORMS(2) contains the Frobenius norm of the matrix product
V = op(A)'*X*G*X*op(A), if JOBG = 'G' or JOBG = 'D', or of
V = F*F', if JOBG = 'F', or of V = H*K, if JOBG = 'H';
if JOBE = 'G', NORMS(3) contains the Frobenius norm of the
matrix product op(E)'*X*op(E).
If JOB <> 'N' and JOB <> 'B', this array is not
referenced.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = -30, or if LDWORK = -2 on input, then
DWORK(1) returns the minimum value of LDWORK.
On exit, if INFO = 0, or if LDWORK = -1 on input, then
DWORK(1) returns the optimal value of LDWORK.
LDWORK The length of the array DWORK. LDWORK >= MAX(v,1), with v
specified in the following table, where
a = 1, if JOBE = 'G';
a = 0, if JOBE = 'I'.
DICO JOBG JOB v
-----------------------------------------------
'C' 'F','H' 'A','C','R' 0
'C' 'F','H' 'N' a*N*N
'C' 'F','H' 'B' N*N
'C' 'G' 'A','C' a*N*N
'C' 'G' 'N','R' (a+1)*N*N
'C' 'G' 'B' (a+2)*N*N
'C' 'D' 'A' N*MIN(M,(a+1)*N)
'C' 'D' 'C' N*MIN(N,M)
'C' 'D' 'N' N*(N+MIN(a*N,M))
'C' 'D' 'B' N*(N+MIN(N+a*N,M))
'C' 'D' 'R' N*MIN(a*N+M,(a+2)*N)
-----------------------------------------------
'D' 'F','H' 'A','C' 0
'D' 'F','H' 'N','R' a*N*N
'D' 'F','H' 'B' (a+1)*N*N
'D' 'G' 'A','C' N*N
'D' 'G' 'N','R' 2*N*N
'D' 'G' 'B' 3*N*N
'D' 'D' 'A','N' N*MIN(MAX(N,M),2*N)
'D' 'D' 'B' N*(N+MAX(N,M))
'D' 'D' 'C' N*MIN(N,M)
'D' 'D' 'R' N*MIN(3*N,N+M)
-----------------------------------------------
If LDWORK = -1, an optimal workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
This evaluation assumes that only the specified triangle
of the array X is always used, and the other strict
triangle is not referenced.
If LDWORK = -2, a minimal workspace query is assumed; the
routine only calculates the minimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
This evaluation assumes that full matrix is given in the
array X, when needed (see the table at the description of
the array X).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The matrix expressions are efficiently evaluated, using symmetry, common matrix subexpressions, and proper order of matrix multiplications. If JOB = 'N' or JOB = 'B', then: If DICO = 'C', the matrices op(op(A)'*X*op(E)) or op(X*op(A)), and V = op(E)'*X*G*X*op(E) or V = F*F' or V = H*K, are efficiently computed. If DICO = 'D', the matrices op(A)'*X*op(A), V = op(A)'*X*G*X*op(A) or V = F*F' or V = H*K, and op(E)'*X*op(E), if JOBE = 'G', are efficiently computed. The results are used to evaluate R, op(C) (if JOB = 'N'), and the norms. If JOB <> 'N', then the needed parts of the intermediate results are obtained and used to evaluate R and/or op(C).Numerical Aspects
The calculations are backward stable.
3 2
The algorithm requires approximately (a+b)N + cN M operations,
where,
a = 0, if JOBE = 'I',
a = 1, if JOBE = 'G' and (DICO = 'C' or
(DICO = 'D' and JOB = 'C')),
a = 1.5, if JOBE = 'G' and DICO = 'D',
and b and c are implicitly defined below. Specifically, the effort
is approximately as follows (using ^ to denote the power operator)
For DICO = 'C':
JOBG JOB
C R A, N, B
'G' (a+1)*N^3 (a+2)*N^3 (a+2)*N^3,(a+2.5)*N^3
'D' (a+2)*N^2*M (a+1)*N^3+1.5*N^2*M (a+1)*N^3+2.5*N^2*M
'F','H' N^2*M N^3+0.5*N^2*M N^3+1.5*N^2*M
For DICO = 'D':
JOBG JOB
C R A, N, B
'G' 2*N^3 (a+3 )*N^3 (a+3 )*N^3
'D' 3*N^2*M (a+1.5)*N^3+1.5*N^2*M (a+1.5)*N^3+2.5*N^2*M
'F','H' N^2*M (a+0.5)*N^3+0.5*N^2*M (a+0.5)*N^3+1.5*N^2*M
For JOBG <> 'G' and JOB = 'B', the effort reduces by N^2*M in
both tables.
An "operation" includes a multiplication, an addition, and some
address calculations.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SG02CX.html 0000664 0000000 0000000 00000026275 14560147231 0020374 0 ustar 00root root 0000000 0000000
Purpose
To find the line search parameter alpha minimizing the Frobenius
norm (in a MATLAB-style notation)
P(alpha) := norm(R(X+alpha*S), 'fro')
= norm((1-alpha)*R(X) +/- alpha^2*V, 'fro'),
where R(X) is the residual of a (generalized) continuous-time
algebraic Riccati equation
0 = op(A)'*X + X*op(A) +/- X*G*X + Q =: R(X),
or
0 = op(A)'*X*op(E) + op(E)'*X*op(A) +/- op(E)'*X*G*X*op(E) + Q
=: R(X),
V = op(E)'*S*G*S*op(E), and op(W) is either W or W'. The matrix S
is the Newton step.
_-1
Instead of the symmetric N-by-N matrix G, G = B*R *B', the N-by-M
-1
matrix D, D = B*L , such that G = D*D', may be given on entry.
_ _
The matrix R, R = L'*L, is a weighting matrix of the optimal
problem, and L is its (Cholesky) factor.
Optionally, V is specified as V = H*K, or V = F*F', but F or H and
K must be evaluated in S. See the SLICOT Library routine SG02CW
description for more details.
Specification
SUBROUTINE SG02CX( JOBE, FLAG, JOBG, UPLO, TRANS, N, M, E, LDE, R,
$ LDR, S, LDS, G, LDG, ALPHA, RNORM, DWORK,
$ LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER FLAG, JOBE, JOBG, TRANS, UPLO
INTEGER INFO, IWARN, LDE, LDG, LDR, LDS, LDWORK, M, N
DOUBLE PRECISION ALPHA, RNORM
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), E(LDE,*), G(LDG,*), R(LDR,*), S(LDS,*)
Arguments
Mode Parameters
JOBE CHARACTER*1
Specifies whether E is a general or an identity matrix,
as follows:
= 'G': The matrix E is general and is given;
= 'I': The matrix E is assumed identity and is not given.
FLAG CHARACTER*1
Specifies which sign is used, as follows:
= 'P': The plus sign is used;
= 'M': The minus sign is used.
JOBG CHARACTER*1
Specifies how the matrix product V is defined, as follows:
= 'G': The matrix G is given: V = op(E)'*S*G*S*op(E);
= 'D': The matrix D is given: V = op(E)'*S*D*D'*S*op(E);
= 'F': The matrix F is given: V = F*F';
= 'H': The matrices H and K are given: V = H*K.
UPLO CHARACTER*1
Specifies which triangles of the symmetric matrices R, G,
if JOBG = 'G', and S, if JOBG = 'G' or JOBG = 'D', are
given, as follows:
= 'U': The upper triangular part is given;
= 'L': The lower triangular part is given.
TRANS CHARACTER*1
Specifies the form of op(W) to be used in the matrix
multiplication, as follows:
= 'N': op(W) = W;
= 'T': op(W) = W';
= 'C': op(W) = W'.
Input/Output Parameters
N (input) INTEGER
The order of the matrices E, R, and S. N >= 0.
M (input) INTEGER
If JOBG <> 'G', the number of columns of the matrices D,
F, or K'. M >= 0.
If JOBG = 'G', the value of M is meaningless.
E (input) DOUBLE PRECISION array, dimension (LDE,*)
If JOBE = 'G' and (JOBG = 'G' or JOBG = 'D'), the leading
N-by-N part of this array must contain the matrix E.
If JOBE = 'I' or JOBG = 'F' or JOBG = 'H', this array is
not referenced.
LDE INTEGER
The leading dimension of array E.
LDE >= MAX(1,N), if JOBE = 'G' and (JOBG = 'G' or
JOBG = 'D');
LDE >= 1, if JOBE = 'I' or JOBG = 'F' or
JOBG = 'H'.
R (input) DOUBLE PRECISION array, dimension (LDR,N)
The leading N-by-N upper or lower triangular part
(depending on UPLO) of this array must contain the upper
or lower triangular part, respectively, of the matrix
R(X), the residual of the algebraic Riccati equation.
The other strictly triangular part is not referenced.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
S (input) DOUBLE PRECISION array, dimension (LDS,*)
If JOBG = 'G' or JOBG = 'D', the leading N-by-N part of
this array must contain the symmetric Newton step
matrix S. If JOBE = 'I', the full matrix must be given.
Otherwise, it is sufficient to input only the triangular
part specified by UPLO, and the remaining strictly
triangular part is not referenced.
If JOBG = 'F', this array is not referenced.
If JOBG = 'H', the leading M-by-N part of this array must
contain the matrix K.
LDS INTEGER
The leading dimension of array S.
LDS >= MAX(1,N), if JOBG = 'G' or JOBG = 'D';
LDS >= 1, if JOBG = 'F';
LDS >= MAX(1,M), if JOBG = 'H'.
G (input/works.) DOUBLE PRECISION array, dimension (LDG,*)
If JOBG = 'G', the leading N-by-N upper or lower
triangular part (depending on UPLO) of this array must
contain the upper or lower triangular part, respectively,
of the matrix G. The other strictly triangular part is not
referenced. The diagonal elements of this array are
modified internally, but are restored on exit.
If JOBG = 'D', the leading N-by-M part of this array must
contain the matrix D, so that G = D*D'.
If JOBG = 'F', the leading N-by-M part of this array must
contain the matrix F.
If JOBG = 'H', leading N-by-M part of this array must
contain the matrix H.
LDG INTEGER
The leading dimension of array G. LDG >= MAX(1,N).
ALPHA (output) DOUBLE PRECISION
If INFO = 0, ALPHA contains the real number alpha which
minimizes P(alpha) = norm(R(X+alpha*S), 'fro') in the
interval [0,2].
If INFO = 1 or IWARN = 2, ALPHA is set equal to 1.
RNORM (output) DOUBLE PRECISION
On exit, if INFO >= 0, RNORM contains the Frobenius norm
of the residual R(X+alpha*S).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if LDWORK = -1 on input, then DWORK(1) returns
the optimal value of LDWORK.
On exit, if LDWORK = -2 on input or INFO = -19, then
DWORK(1) returns the minimal value of LDWORK.
On exit, if INFO = 0, the leading N-by-N upper or lower
triangular part (depending on UPLO) of this array contains
the corresponding triangular part of the matrix V.
LDWORK The length of the array DWORK.
LDWORK >= N*N + MAX( 2*N*N, 51 ),
if JOBG = 'G' and JOBE = 'G';
LDWORK >= N*N + MAX( N*N, 51 ),
if JOBG = 'G' and JOBE = 'I',
or JOBG = 'F', or JOBG = 'H';
LDWORK >= N*N + MAX( MAX( N*N, 51 ), MIN( 2*N*N, N*M ) ),
if JOBG = 'D' and JOBE = 'G';
LDWORK >= N*N + MAX( N*N, N*M, 51 ),
if JOBG = 'D' and JOBE = 'I'.
For M <= N, the last two formulas simplify to
LDWORK >= N*N + MAX( N*N, 51).
If LDWORK = -1, an optimal workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
If LDWORK = -2, a minimal workspace query is assumed; the
routine only calculates the minimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
Warning Indicator
IWARN INTEGER
= 0: no warnings;
= 2: no optimal line search parameter t := alpha in [0,2]
was found; t = 1 was set.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -k, the k-th argument had an illegal
value;
= 1: an error occurred during the call of the SLICOT
Library routine MC01XD: the eigenvalues computation
for the 3-by-3 (generalized) eigenproblem failed.
ALPHA and RNORM are set as specified above.
Method
The matrix V is computed with the suitable formula, and used to set up a third order polynomial whose roots in [0,2], if any, are candidates for the solution of the minimum residual problem. The roots of the polynomial are computed by solving an equivalent 3-by-3 (generalized) eigenproblem.References
[1] Benner, P.
Contributions to the Numerical Solution of Algebraic
Riccati Equations and Related Eigenvalue Problems.
Fakultat fur Mathematik, Technische Universitat Chemnitz-
Zwickau, D-09107 Chemnitz, Germany, Feb. 1997.
Numerical Aspects
The calculations are backward stable. The computational effort is of the order of c*N**2*M operations. Here, M = N, if JOBG = 'G', and the coefficient c varies between 0.5 and 2.5, depending on JOBE and JOBG. (An "operation" includes a multiplication, an addition, and some address calculations.) The computed value of norm(R(X+alpha*S),'fro'), returned in RNORM, could be inaccurate if R(X) is small, since then subtraction cancellation could appear in the updating formula which is used. (This can happen, e.g., when solving a Riccati equation by Newton's method with line search, since then the sequence of R(.) tends to zero.) In such a case, it is better to recompute the residual from the data.Further Comments
The routine does not ckeck if the matrix S is zero, when a quick return with ALPHA = 1 is possible. With suitable input arguments E and G, this routine may also be used for discrete-time algebraic Riccati equations. (Matrix E must be the current closed-loop matrix.)Example
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SG02ND.html 0000664 0000000 0000000 00000064105 14560147231 0020355 0 ustar 00root root 0000000 0000000
Purpose
To compute the optimal gain matrix K for the problem of optimal
control given by
-1
K = (R + B'XB) (B'Xop(A) + L') (1)
in the discrete-time case and
-1
K = R (B'Xop(E) + L') (2)
in the continuous-time case, where A, E, B and L are N-by-N,
N-by-N, N-by-M, and N-by-M matrices, respectively; R and X are
M-by-M and N-by-N symmetric matrices, respectively, and op(W) is
either W or W'. Matrix op(K) defines the feedback gain matrix, if
op(W) = W, and the estimator matrix, if op(W) = W'. The formulas
above are also useful in Newton's algorithms for solving algebraic
Riccati equations, when X is the current iterate.
Optionally, matrix R may be specified in a factored form, and L
may be zero. If R or R + B'XB (for DICO = 'C', or DICO = 'D',
respectively), is positive definite, let C be its Cholesky factor
(denoted, e.g., C = chol(R), for DICO = 'C'). Optionally, the
matrix H, defined by
H = op(E)'XB + L, if DICO = 'C', or
H = op(A)'XB + L, if DICO = 'D', (3)
is returned on exit, besides K; if C exists, the matrix F, defined
by FC = H may be optionally returned, instead of K and H. The
matrix F or the pair of matrices H and K may be used for computing
the residual matrix for an (approximate) solution of an algebraic
Riccati equation (see SLICOT Library routine SG02CW).
Specification
SUBROUTINE SG02ND( DICO, JOBE, JOB, JOBX, FACT, UPLO, JOBL, TRANS,
$ N, M, P, A, LDA, E, LDE, B, LDB, R, LDR, IPIV,
$ L, LDL, X, LDX, RNORM, K, LDK, H, LDH, XE,
$ LDXE, OUFACT, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, FACT, JOB, JOBE, JOBL, JOBX, TRANS, UPLO
INTEGER INFO, LDA, LDB, LDE, LDH, LDK, LDL, LDR, LDWORK,
$ LDX, LDXE, M, N, P
DOUBLE PRECISION RNORM
C .. Array Arguments ..
INTEGER IPIV(*), IWORK(*), OUFACT(2)
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), E(LDE,*),
$ H(LDH,*), K(LDK,*), L(LDL,*), R(LDR,*),
$ X(LDX,*), XE(LDXE,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the equation from which K is to be determined,
as follows:
= 'D': Equation (1), discrete-time case;
= 'C': Equation (2), continuous-time case.
JOBE CHARACTER*1
Specifies whether E is a general or an identity matrix,
as follows:
= 'G': The matrix E is general and is given;
= 'I': The matrix E is assumed identity and is not given.
This parameter is not relevant for DICO = 'D'.
JOB CHARACTER*1
Specifies what should be computed, as follows:
= 'K': Compute and return the matrix K only;
= 'H': Compute and return both matrices H and K;
= 'F': Compute the matrix F, if possible; otherwise,
compute and return H and K;
= 'D': Compute and return both matrices H and K, when
B and L have previously been transformed using
SLICOT Library routines SB02MT or SB02MX, which
returned OUFACT = 1. This is useful for computing
K in (2), since then K is the solution of CK = H'.
In this case, FACT should be set to 'C', and the
array R must contain the Cholesky factor of
R + B'XB, if DICO = 'D';
= 'C': Compute and return the matrix F, when B and L have
previously been transformed using SB02MT or
SB02MX, which returned OUFACT = 1. In this case,
FACT should be set to 'C', and the array R must
contain the Cholesky factor of R + B'XB, if
DICO = 'D'.
JOB should not be set to 'F' if FACT = 'U'.
JOBX CHARACTER*1
Specifies whether the matrix op(Xop(E)), if DICO = 'C', or
op(Xop(A)), if DICO = 'D', must be computed, as follows:
= 'C': Compute and return the coresponding matrix;
= 'N': Do not compute that matrix.
This parameter is not relevant for DICO = 'C' and
JOBE = 'I'.
FACT CHARACTER*1
Specifies how the matrix R is given (factored or not), as
follows:
= 'N': Array R contains the matrix R;
= 'D': Array R contains a P-by-M matrix D, where R = D'D;
= 'C': Array R contains the Cholesky factor of R;
= 'U': Array R contains the symmetric indefinite UdU' or
LdL' factorization of R. This option is not
available for DICO = 'D'.
UPLO CHARACTER*1
Specifies which triangle of the possibly factored matrix R
(or R + B'XB, on exit) is or should be stored, as follows:
= 'U': Upper triangle is stored;
= 'L': Lower triangle is stored.
JOBL CHARACTER*1
Specifies whether or not the matrix L is zero, as follows:
= 'Z': L is zero;
= 'N': L is nonzero.
TRANS CHARACTER*1
Specifies the form of op(W) to be used in the matrix
multiplication, as follows:
= 'N': op(W) = W;
= 'T': op(W) = W';
= 'C': op(W) = W'.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A and X. N >= 0.
No computations are performed if MIN(N,M) = 0.
M (input) INTEGER
The order of the matrix R and the number of columns of the
matrices B and L. M >= 0.
P (input) INTEGER
The number of rows of the matrix D.
P >= M for DICO = 'C';
P >= 0 for DICO = 'D'.
This parameter is relevant only for FACT = 'D'.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
If DICO = 'D', the leading N-by-N part of this array must
contain the state matrix A of the system.
If DICO = 'C', this array is not referenced.
LDA INTEGER
The leading dimension of array A.
LDA >= MAX(1,N) if DICO = 'D';
LDA >= 1 if DICO = 'C'.
E (input) DOUBLE PRECISION array, dimension (LDE,*)
If JOBE = 'G' and DICO = 'C', the leading N-by-N part of
this array must contain the matrix E.
If JOBE = 'I' or DICO = 'D', this array is not referenced.
LDE INTEGER
The leading dimension of array E.
LDE >= MAX(1,N), if JOBE = 'G' and DICO = 'C';
LDE >= 1, if JOBE = 'I' or DICO = 'D'.
B (input/worksp.) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
input matrix B of the system, transformed by SB02MT or
SB02MX, if JOB = 'D' or JOB = 'C'.
If DICO = 'D' and FACT = 'D' or 'C', the contents of this
array is destroyed. Specifically, if, on exit,
OUFACT(2) = 1, this array contains chol(X)*B, and if
OUFACT(2) = 2 and INFO < M+2, but INFO >= 0, its trailing
part (in the first N rows) contains the submatrix of
sqrt(V)*U'B corresponding to the non-negligible, positive
eigenvalues of X, where V and U are the matrices with the
eigenvalues and eigenvectors of X.
Otherwise, B is unchanged on exit.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
R (input/output) DOUBLE PRECISION array, dimension (LDR,M)
On entry, if FACT = 'N', the leading M-by-M upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array must contain the upper
triangular part or lower triangular part, respectively,
of the symmetric input weighting matrix R.
On entry, if FACT = 'D', the leading P-by-M part of this
array must contain the direct transmission matrix D of the
system.
On entry, if FACT = 'C', the leading M-by-M upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array must contain the Cholesky
factor of the positive definite input weighting matrix R
(as produced by LAPACK routine DPOTRF).
On entry, if DICO = 'C' and FACT = 'U', the leading M-by-M
upper triangular part (if UPLO = 'U') or lower triangular
part (if UPLO = 'L') of this array must contain the
factors of the UdU' or LdL' factorization, respectively,
of the symmetric indefinite input weighting matrix R (as
produced by LAPACK routine DSYTRF).
The strictly lower triangular part (if UPLO = 'U') or
strictly upper triangular part (if UPLO = 'L') of this
array is used as workspace (filled in by symmetry with the
other strictly triangular part of R, of R+B'XB, or of the
result, if DICO = 'C', DICO = 'D' (if FACT = 'N', in both
cases), or (DICO = 'D' and (FACT = 'D' or FACT = 'C') and
UPLO = 'L'), respectively.
On exit, if OUFACT(1) = 1, and INFO = 0 (or INFO = M+1),
the leading M-by-M upper triangular part (if UPLO = 'U')
or lower triangular part (if UPLO = 'L') of this array
contains the Cholesky factor of the given input weighting
matrix R (for DICO = 'C'), or that of the matrix R + B'XB
(for DICO = 'D').
On exit, if OUFACT(1) = 2, and INFO = 0 (or INFO = M+1),
the leading M-by-M upper triangular part (if UPLO = 'U')
or lower triangular part (if UPLO = 'L') of this array
contains the factors of the UdU' or LdL' factorization,
respectively, of the given input weighting matrix
(for DICO = 'C'), or that of the matrix R + B'XB
(for DICO = 'D' and FACT = 'N').
On exit R is unchanged if FACT = 'U' or N = 0.
LDR INTEGER.
The leading dimension of the array R.
LDR >= MAX(1,M) if FACT <> 'D';
LDR >= MAX(1,M,P) if FACT = 'D'.
IPIV (input/output) INTEGER array, dimension (M)
On entry, if FACT = 'U', this array must contain details
of the interchanges performed and the block structure of
the d factor in the UdU' or LdL' factorization of matrix R
(as produced by LAPACK routine DSYTRF).
On exit, if OUFACT(1) = 2, this array contains details of
the interchanges performed and the block structure of the
d factor in the UdU' or LdL' factorization of matrix R or
R + B'XB, as produced by LAPACK routine DSYTRF.
This array is not referenced if FACT = 'D', or FACT = 'C',
or N = 0.
L (input) DOUBLE PRECISION array, dimension (LDL,M)
If JOBL = 'N', the leading N-by-M part of this array must
contain the cross weighting matrix L, transformed by
SB02MT or SB02MX, if JOB = 'D' or JOB = 'C'.
If JOBL = 'Z', this array is not referenced.
LDL INTEGER
The leading dimension of array L.
LDL >= MAX(1,N) if JOBL = 'N';
LDL >= 1 if JOBL = 'Z'.
X (input/output) DOUBLE PRECISION array, dimension (LDX,N)
On entry, the leading N-by-N part of this array must
contain the (approximate) solution matrix X of the
algebraic Riccati equation as produced by SLICOT Library
routines SB02MD or SB02OD (or SG02CD). Matrix X is assumed
non-negative definite if DICO = 'D', FACT <> 'N',
JOB <> 'D' and JOB <> 'C'. The full matrix X must be given
on input in this case.
For minimal workspace, full matrix X must also be given if
((JOBX = 'C', DICO = 'D', FACT = 'N', and M > N), or
(JOBX = 'N', ((DICO = 'C' or FACT = 'N'), (DICO = 'D' or
JOBE = 'I') or N >= M, or LDWORK < N*N) and (DICO = 'D'
or JOBE = 'G' or JOB = 'K'))) and LDWORK < N*M.
(Simpler, but more demanding conditions are the following:
((JOBX = 'C', DICO = 'D', FACT = 'N', and M > N), or
(JOBX = 'N', (DICO = 'D' or ((DICO = 'C', JOBE = 'G') or
JOB = 'K'))), LDWORK < N*N.)
For optimal workspace, full matrix X is not needed in any
of the cases described above for minimal workspace.
On exit, if DICO = 'D', FACT = 'D' or FACT = 'C', and
OUFACT(2) = 1, the N-by-N upper triangular part
(if UPLO = 'U') or lower triangular part (if UPLO = 'L')
of this array contains the Cholesky factor of the given
matrix X, which is found to be positive definite.
On exit, if DICO = 'D', FACT = 'D' or 'C', OUFACT(2) = 2,
and INFO <> M+2 (but INFO >= 0), the leading N-by-N part
of this array contains the matrix of orthonormal
eigenvectors of X.
On exit X is unchanged if DICO = 'C' or FACT = 'N'.
LDX INTEGER
The leading dimension of array X. LDX >= MAX(1,N).
RNORM (input) DOUBLE PRECISION
If FACT = 'U', this parameter must contain the 1-norm of
the original matrix R (before factoring it).
Otherwise, this parameter is not used.
K (output) DOUBLE PRECISION array, dimension (LDK,N)
If JOB = 'K' or JOB = 'H' or JOB = 'D' or OUFACT(1) = 2,
the leading M-by-N part of this array contains the gain
matrix K.
LDK INTEGER
The leading dimension of array K. LDK >= MAX(1,M).
H (output) DOUBLE PRECISION array, dimension (LDH,*)
If JOB = 'H' or JOB = 'D' or (JOB = 'F' and
OUFACT(1) = 2), the leading N-by-M part of this array
contains the matrix H.
If JOB = 'C' or (JOB = 'F' and OUFACT(1) = 1), the leading
N-by-M part of this array contains the matrix F.
If JOB = 'K', this array is not referenced.
LDH INTEGER
The leading dimension of array H.
LDH >= MAX(1,N), if JOB <> 'K';
LDH >= 1, if JOB = 'K'.
XE (output) DOUBLE PRECISION array, dimension (LDXE,*)
If JOBX = 'C', DICO = 'C', and JOBE = 'G', the leading
N-by-N part of this array contains the matrix product X*E,
if TRANS = 'N', or E*X, if TRANS = 'T' or TRANS = 'C'.
If JOBX = 'C' and DICO = 'D', the leading N-by-N part of
this array contains the matrix product X*A, if
TRANS = 'N', or A*X, if TRANS = 'T' or TRANS = 'C'.
These matrix products may be needed for computing the
residual matrix for an (approximate) solution of a Riccati
equation (see SLICOT Library routine SG02CW).
If JOBX = 'N' or (DICO = 'C' and JOBE = 'I'), this array
is not referenced.
LDXE INTEGER
The leading dimension of array XE.
LDXE >= MAX(1,N), if JOBX = 'C', and either DICO = 'C' and
JOBE = 'G', or DICO = 'D';
LDXE >= 1, if JOBX = 'N' or (DICO = 'C' and
JOBE = 'I').
OUFACT (output) INTEGER array, dimension (2)
Information about the factorization finally used.
OUFACT(1) = 1: Cholesky factorization of R (or R + B'XB)
has been used;
OUFACT(1) = 2: UdU' (if UPLO = 'U') or LdL' (if UPLO =
'L') factorization of R (or R + B'XB)
has been used;
OUFACT(2) = 1: Cholesky factorization of X has been used;
OUFACT(2) = 2: Spectral factorization of X has been used.
The value of OUFACT(2) is not set for DICO = 'C' or for
DICO = 'D' and FACT = 'N'.
This array is not set if N = 0 or M = 0.
Workspace
IWORK INTEGER array, dimension (M)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or LDWORK = -1, DWORK(1) returns the
optimal value of LDWORK, and for LDWORK set as specified
below, DWORK(2) contains the reciprocal condition number
of the matrix R (for DICO = 'C') or of R + B'XB (for
DICO = 'D'), if FACT = 'N' or FACT = 'U' or OUFACT(1) = 2,
or of its Cholesky factor, if FACT = 'C' or FACT = 'D' and
OUFACT(1) = 1; DWORK(2) is set to 1 if N = 0.
On exit, if LDWORK = -2 on input or INFO = -35, then
DWORK(1) returns the minimal value of LDWORK.
If on exit INFO = 0, and OUFACT(2) = 2, then DWORK(3),...,
DWORK(N+2) contain the eigenvalues of X, in ascending
order.
LDWORK INTEGER
Dimension of working array DWORK.
Let a = N, if JOBX = 'N' and (DICO = 'D' or JOBE = 'G');
a = 0, otherwise. Then
LDWORK >= max(2,2*M,a) if FACT = 'U';
LDWORK >= max(2,3*M,4*N+1) if FACT = 'D' or
(FACT = 'C' and JOB <> 'C'
and JOB <> 'D'), DICO = 'D';
LDWORK >= max(2,3*M,a) otherwise.
For optimum performance LDWORK should be larger.
If LDWORK = -1, an optimal workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
If LDWORK = -2, a minimal workspace query is assumed; the
routine only calculates the minimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= i: if the i-th element of the d factor is exactly zero;
the UdU' (or LdL') factorization has been completed,
but the block diagonal matrix d is exactly singular;
= M+1: if the matrix R (if DICO = 'C'), or R + B'XB
(if DICO = 'D') is numerically singular (to working
precision);
= M+2: if one or more of the eigenvalues of X has not
converged;
= M+3: if the matrix X is indefinite and updating the
triangular factorization failed.
If INFO > M+1, call the routine again with an appropriate,
unfactored matrix R.
Method
The (optimal) gain matrix K is obtained as the solution to the
system of linear equations
(R + B'XB) * K = B'Xop(A) + L'
in the discrete-time case and
R * K = B'Xop(E) + L'
in the continuous-time case, with R replaced by D'D if FACT = 'D'.
If FACT = 'N', Cholesky factorization is tried first, but
if the coefficient matrix is not positive definite, then UdU' (or
LdL') factorization is used. If FACT <> 'N', the factored form
of R is taken into account. The discrete-time case then involves
updating of a triangular factorization of R (or D'D); Cholesky or
symmetric spectral factorization of X is employed to avoid
squaring of the condition number of the matrix. When D is given,
its QR factorization is determined, and the triangular factor is
used as described above.
Numerical Aspects
The algorithm consists of numerically stable steps.
3 2
For DICO = 'C' and JOBE = 'I', it requires O(m + mn ) floating
2
point operations if FACT = 'N' and O(mn ) floating point
operations, otherwise.
For DICO = 'D' or JOBE = 'G', the operation counts are similar,
3
but additional O(n ) floating point operations may be needed in
the worst case.
These estimates assume that M <= N.
Further Comments
NoneExample
Program Text
* SG02ND EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER NMAX2
PARAMETER ( NMAX2 = 2*NMAX )
INTEGER LDA, LDB, LDC, LDF, LDH, LDL, LDR, LDS, LDT, LDU,
$ LDX, LDXE
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX, LDL = NMAX,
$ LDR = MAX(MMAX,PMAX), LDS = NMAX2+MMAX,
$ LDT = NMAX2+MMAX, LDU = NMAX2, LDX = NMAX,
$ LDF = MMAX, LDH = NMAX, LDXE = NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( NMAX+3*MMAX+2, 14*NMAX+23,
$ 16*NMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL, RCOND, RNORM
INTEGER I, INFO1, INFO2, J, M, N, P
CHARACTER*1 DICO, FACT, JOB, JOBB, JOBE, JOBL, JOBX, SORT,
$ TRANS, UPLO
* .. Local Arrays ..
DOUBLE PRECISION AE(LDA,NMAX), ALFAI(2*NMAX), ALFAR(2*NMAX),
$ B(LDB,MMAX), BETA(2*NMAX), C(LDC,NMAX),
$ DWORK(LDWORK), F(LDF,NMAX), H(LDH,MMAX),
$ L(LDL,MMAX), R(LDR,MMAX), S(LDS,NMAX2+MMAX),
$ T(LDT,NMAX2), U(LDU,NMAX2), X(LDX,NMAX),
$ XE(LDXE,NMAX)
INTEGER IPIV(LIWORK), IWORK(LIWORK), OUFACT(2)
LOGICAL BWORK(NMAX2)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL SG02ND, SB02OD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL, DICO, JOBE, JOB, JOBX, FACT,
$ JOBL, UPLO, TRANS
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( AE(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
IF ( LSAME( FACT, 'D' ) ) THEN
READ ( NIN, FMT = * ) ( ( R(I,J), J = 1,M ), I = 1,P )
ELSE
READ ( NIN, FMT = * ) ( ( R(I,J), J = 1,M ), I = 1,M )
END IF
IF ( LSAME( JOBL, 'N' ) )
$ READ ( NIN, FMT = * ) ( ( L(I,J), J = 1,M ), I = 1,N )
* Find the solution matrix X.
JOBB = 'B'
SORT = 'S'
CALL SB02OD( DICO, JOBB, 'Both', UPLO, JOBL, SORT, N, M,
$ P, AE, LDA, B, LDB, C, LDC, R, LDR, L, LDL,
$ RCOND, X, LDX, ALFAR, ALFAI, BETA, S, LDS,
$ T, LDT, U, LDU, TOL, IWORK, DWORK, LDWORK,
$ BWORK, INFO1 )
*
IF ( INFO1.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO1
ELSE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99994 ) ( X(I,J), J = 1,N )
20 CONTINUE
* Compute the optimal feedback matrix F.
CALL SG02ND( DICO, JOBE, JOB, JOBX, FACT, UPLO, JOBL,
$ TRANS, N, M, P, AE, LDA, AE, LDA, B, LDB,
$ R, LDR, IPIV, L, LDL, X, LDX, RNORM, F,
$ LDF, H, LDH, XE, LDXE, OUFACT, IWORK,
$ DWORK, LDWORK, INFO2 )
*
IF ( INFO2.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO2
ELSE
WRITE ( NOUT, FMT = 99995 )
DO 40 I = 1, M
WRITE ( NOUT, FMT = 99994 ) ( F(I,J), J = 1,N )
40 CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SG02ND EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB02OD = ',I2)
99997 FORMAT (' INFO on exit from SG02ND = ',I2)
99996 FORMAT (' The solution matrix X is ')
99995 FORMAT (/' The optimal feedback matrix F is ')
99994 FORMAT (20(1X,F8.4))
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' M is out of range.',/' M = ',I5)
99991 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
SG02ND EXAMPLE PROGRAM DATA 2 1 3 0.0 D I K N N Z U N 2.0 -1.0 1.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0Program Results
SG02ND EXAMPLE PROGRAM RESULTS The solution matrix X is 1.0000 0.0000 0.0000 1.0000 The optimal feedback matrix F is 2.0000 -1.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SG03AD.html 0000664 0000000 0000000 00000045073 14560147231 0020344 0 ustar 00root root 0000000 0000000
Purpose
To solve for X either the generalized continuous-time Lyapunov
equation
T T
op(A) X op(E) + op(E) X op(A) = SCALE * Y, (1)
or the generalized discrete-time Lyapunov equation
T T
op(A) X op(A) - op(E) X op(E) = SCALE * Y, (2)
where op(M) is either M or M**T for M = A, E and the right hand
side Y is symmetric. A, E, Y, and the solution X are N-by-N
matrices. SCALE is an output scale factor, set to avoid overflow
in X.
Estimates of the separation and the relative forward error norm
are provided.
Specification
SUBROUTINE SG03AD( DICO, JOB, FACT, TRANS, UPLO, N, A, LDA, E,
$ LDE, Q, LDQ, Z, LDZ, X, LDX, SCALE, SEP, FERR,
$ ALPHAR, ALPHAI, BETA, IWORK, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER DICO, FACT, JOB, TRANS, UPLO
DOUBLE PRECISION FERR, SCALE, SEP
INTEGER INFO, LDA, LDE, LDQ, LDWORK, LDX, LDZ, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), ALPHAI(*), ALPHAR(*), BETA(*),
$ DWORK(*), E(LDE,*), Q(LDQ,*), X(LDX,*),
$ Z(LDZ,*)
INTEGER IWORK(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies which type of the equation is considered:
= 'C': Continuous-time equation (1);
= 'D': Discrete-time equation (2).
JOB CHARACTER*1
Specifies if the solution is to be computed and if the
separation is to be estimated:
= 'X': Compute the solution only;
= 'S': Estimate the separation only;
= 'B': Compute the solution and estimate the separation.
FACT CHARACTER*1
Specifies whether the generalized real Schur
factorization of the pencil A - lambda * E is supplied
on entry or not:
= 'N': Factorization is not supplied;
= 'F': Factorization is supplied.
TRANS CHARACTER*1
Specifies whether the transposed equation is to be solved
or not:
= 'N': op(A) = A, op(E) = E;
= 'T': op(A) = A**T, op(E) = E**T.
UPLO CHARACTER*1
Specifies whether the lower or the upper triangle of the
array X is needed on input:
= 'L': Only the lower triangle is needed on input;
= 'U': Only the upper triangle is needed on input.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, if FACT = 'F', then the leading N-by-N upper
Hessenberg part of this array must contain the
generalized Schur factor A_s of the matrix A (see
definition (3) in section METHOD). A_s must be an upper
quasitriangular matrix. The elements below the upper
Hessenberg part of the array A are not referenced.
If FACT = 'N', then the leading N-by-N part of this
array must contain the matrix A.
On exit, the leading N-by-N part of this array contains
the generalized Schur factor A_s of the matrix A. (A_s is
an upper quasitriangular matrix.)
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, if FACT = 'F', then the leading N-by-N upper
triangular part of this array must contain the
generalized Schur factor E_s of the matrix E (see
definition (4) in section METHOD). The elements below the
upper triangular part of the array E are not referenced.
If FACT = 'N', then the leading N-by-N part of this
array must contain the coefficient matrix E of the
equation.
On exit, the leading N-by-N part of this array contains
the generalized Schur factor E_s of the matrix E. (E_s is
an upper triangular matrix.)
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,N).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if FACT = 'F', then the leading N-by-N part of
this array must contain the orthogonal matrix Q from
the generalized Schur factorization (see definitions (3)
and (4) in section METHOD).
If FACT = 'N', Q need not be set on entry.
On exit, the leading N-by-N part of this array contains
the orthogonal matrix Q from the generalized Schur
factorization.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= MAX(1,N).
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
On entry, if FACT = 'F', then the leading N-by-N part of
this array must contain the orthogonal matrix Z from
the generalized Schur factorization (see definitions (3)
and (4) in section METHOD).
If FACT = 'N', Z need not be set on entry.
On exit, the leading N-by-N part of this array contains
the orthogonal matrix Z from the generalized Schur
factorization.
LDZ INTEGER
The leading dimension of the array Z. LDZ >= MAX(1,N).
X (input/output) DOUBLE PRECISION array, dimension (LDX,N)
On entry, if JOB = 'B' or 'X', then the leading N-by-N
part of this array must contain the right hand side matrix
Y of the equation. Either the lower or the upper
triangular part of this array is needed (see mode
parameter UPLO).
If JOB = 'S', X is not referenced.
On exit, if JOB = 'B' or 'X', and INFO = 0, 3, or 4, then
the leading N-by-N part of this array contains the
solution matrix X of the equation.
If JOB = 'S', X is not referenced.
LDX INTEGER
The leading dimension of the array X. LDX >= MAX(1,N).
SCALE (output) DOUBLE PRECISION
The scale factor set to avoid overflow in X.
(0 < SCALE <= 1)
SEP (output) DOUBLE PRECISION
If JOB = 'S' or JOB = 'B', and INFO = 0, 3, or 4, then
SEP contains an estimate of the separation of the
Lyapunov operator.
FERR (output) DOUBLE PRECISION
If JOB = 'B', and INFO = 0, 3, or 4, then FERR contains an
estimated forward error bound for the solution X. If XTRUE
is the true solution, FERR estimates the relative error
in the computed solution, measured in the Frobenius norm:
norm(X - XTRUE) / norm(XTRUE)
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
BETA (output) DOUBLE PRECISION array, dimension (N)
If FACT = 'N' and INFO = 0, 3, or 4, then
(ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, are the
eigenvalues of the matrix pencil A - lambda * E.
If FACT = 'F', ALPHAR, ALPHAI, and BETA are not
referenced.
Workspace
IWORK INTEGER array, dimension (N**2)
IWORK is not referenced if JOB = 'X'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK. The following table
contains the minimal work space requirements depending
on the choice of JOB and FACT.
JOB FACT | LDWORK
-------------------+-------------------
'X' 'F' | MAX(1,N)
'X' 'N' | MAX(1,4*N)
'B', 'S' 'F' | MAX(1,2*N**2)
'B', 'S' 'N' | MAX(1,2*N**2,4*N)
For optimum performance, LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message related to LDWORK is issued by
XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: FACT = 'F' and the matrix contained in the upper
Hessenberg part of the array A is not in upper
quasitriangular form;
= 2: FACT = 'N' and the pencil A - lambda * E cannot be
reduced to generalized Schur form: LAPACK routine
DGEGS (or DGGES) has failed to converge;
= 3: DICO = 'D' and the pencil A - lambda * E has a
pair of reciprocal eigenvalues. That is, lambda_i =
1/lambda_j for some i and j, where lambda_i and
lambda_j are eigenvalues of A - lambda * E. Hence,
equation (2) is singular; perturbed values were
used to solve the equation (but the matrices A and
E are unchanged);
= 4: DICO = 'C' and the pencil A - lambda * E has a
degenerate pair of eigenvalues. That is, lambda_i =
-lambda_j for some i and j, where lambda_i and
lambda_j are eigenvalues of A - lambda * E. Hence,
equation (1) is singular; perturbed values were
used to solve the equation (but the matrices A and
E are unchanged).
Method
A straightforward generalization [3] of the method proposed by
Bartels and Stewart [1] is utilized to solve (1) or (2).
First the pencil A - lambda * E is reduced to real generalized
Schur form A_s - lambda * E_s by means of orthogonal
transformations (QZ-algorithm):
A_s = Q**T * A * Z (upper quasitriangular) (3)
E_s = Q**T * E * Z (upper triangular). (4)
If FACT = 'F', this step is omitted. Assuming SCALE = 1 and
defining
( Z**T * Y * Z : TRANS = 'N'
Y_s = <
( Q**T * Y * Q : TRANS = 'T'
( Q**T * X * Q if TRANS = 'N'
X_s = < (5)
( Z**T * X * Z if TRANS = 'T'
leads to the reduced Lyapunov equation
T T
op(A_s) X_s op(E_s) + op(E_s) X_s op(A_s) = Y_s, (6)
or
T T
op(A_s) X_s op(A_s) - op(E_s) X_s op(E_s) = Y_s, (7)
which are equivalent to (1) or (2), respectively. The solution X_s
of (6) or (7) is computed via block back substitution (if TRANS =
'N') or block forward substitution (if TRANS = 'T'), where the
block order is at most 2. (See [1] and [3] for details.)
Equation (5) yields the solution matrix X.
For fast computation the estimates of the separation and the
forward error are gained from (6) or (7) rather than (1) or
(2), respectively. We consider (6) and (7) as special cases of the
generalized Sylvester equation
R * X * S + U * X * V = Y, (8)
whose separation is defined as follows
sep = sep(R,S,U,V) = min || R * X * S + U * X * V || .
||X|| = 1 F
F
Equation (8) is equivalent to the system of linear equations
K * vec(X) = (kron(S**T,R) + kron(V**T,U)) * vec(X) = vec(Y),
where kron is the Kronecker product of two matrices and vec
is the mapping that stacks the columns of a matrix. If K is
nonsingular then
sep = 1 / ||K**(-1)|| .
2
We estimate ||K**(-1)|| by a method devised by Higham [2]. Note
that this method yields an estimation for the 1-norm but we use it
as an approximation for the 2-norm. Estimates for the forward
error norm are provided by
FERR = 2 * EPS * ||A_s|| * ||E_s|| / sep
F F
in the continuous-time case (1) and
FERR = EPS * ( ||A_s|| **2 + ||E_s|| **2 ) / sep
F F
in the discrete-time case (2).
The reciprocal condition number, RCOND, of the Lyapunov equation
can be estimated by FERR/EPS.
References
[1] Bartels, R.H., Stewart, G.W.
Solution of the equation A X + X B = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
[2] Higham, N.J.
FORTRAN codes for estimating the one-norm of a real or complex
matrix, with applications to condition estimation.
A.C.M. Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, 1988.
[3] Penzl, T.
Numerical solution of generalized Lyapunov equations.
Advances in Comp. Math., vol. 8, pp. 33-48, 1998.
Numerical Aspects
The number of flops required by the routine is given by the
following table. Note that we count a single floating point
arithmetic operation as one flop. c is an integer number of modest
size (say 4 or 5).
| FACT = 'F' FACT = 'N'
-----------+------------------------------------------
JOB = 'B' | (26+8*c)/3 * N**3 (224+8*c)/3 * N**3
JOB = 'S' | 8*c/3 * N**3 (198+8*c)/3 * N**3
JOB = 'X' | 26/3 * N**3 224/3 * N**3
The algorithm is backward stable if the eigenvalues of the pencil
A - lambda * E are real. Otherwise, linear systems of order at
most 4 are involved into the computation. These systems are solved
by Gauss elimination with complete pivoting. The loss of stability
of the Gauss elimination with complete pivoting is rarely
encountered in practice.
The Lyapunov equation may be very ill-conditioned. In particular,
if DICO = 'D' and the pencil A - lambda * E has a pair of almost
reciprocal eigenvalues, or DICO = 'C' and the pencil has an almost
degenerate pair of eigenvalues, then the Lyapunov equation will be
ill-conditioned. Perturbed values were used to solve the equation.
Ill-conditioning can be detected by a very small value of the
reciprocal condition number RCOND.
Further Comments
NoneExample
Program Text
* SG03AD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA, LDE, LDQ, LDX, LDZ
PARAMETER ( LDA = NMAX, LDE = NMAX, LDQ = NMAX,
$ LDX = NMAX, LDZ = NMAX )
INTEGER LIWORK, LDWORK
PARAMETER ( LIWORK = NMAX**2,
$ LDWORK = MAX( 2*NMAX**2, 4*NMAX ) )
* .. Local Scalars ..
CHARACTER*1 DICO, FACT, JOB, TRANS, UPLO
DOUBLE PRECISION FERR, SCALE, SEP
INTEGER I, INFO, J, N
* .. Local Arrays ..
INTEGER IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), ALPHAI(NMAX), ALPHAR(NMAX),
$ BETA(NMAX), DWORK(LDWORK), E(LDE,NMAX),
$ Q(LDQ,NMAX), X(LDX,NMAX), Z(LDZ,NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL SG03AD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, JOB, DICO, FACT, TRANS, UPLO
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
IF ( LSAME ( FACT, 'F' ) ) THEN
READ ( NIN, FMT = * ) ( ( Q(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( Z(I,J), J = 1,N ), I = 1,N )
END IF
IF ( .NOT.LSAME ( JOB, 'S' ) )
$ READ ( NIN, FMT = * ) ( ( X(I,J), J = 1,N ), I = 1,N )
* Find the solution matrix X and the scalar SEP.
CALL SG03AD( DICO, JOB, FACT, TRANS, UPLO, N, A, LDA, E, LDE,
$ Q, LDQ, Z, LDZ, X, LDX, SCALE, SEP, FERR, ALPHAR,
$ ALPHAI, BETA, IWORK, DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( LSAME ( JOB, 'B' ) .OR. LSAME ( JOB, 'S' ) ) THEN
WRITE ( NOUT, FMT = 99997 ) SEP
WRITE ( NOUT, FMT = 99996 ) FERR
END IF
IF ( LSAME ( JOB, 'B' ) .OR. LSAME ( JOB, 'X' ) ) THEN
WRITE ( NOUT, FMT = 99995 ) SCALE
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99994 ) ( X(I,J), J = 1,N )
20 CONTINUE
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SG03AD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SG03AD = ',I2)
99997 FORMAT (' SEP = ',D8.2)
99996 FORMAT (' FERR = ',D8.2)
99995 FORMAT (' SCALE = ',D8.2,//' The solution matrix X is ')
99994 FORMAT (20(1X,F8.4))
99993 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
SG03AD EXAMPLE PROGRAM DATA 3 B C N N U 3.0 1.0 1.0 1.0 3.0 0.0 1.0 0.0 2.0 1.0 3.0 0.0 3.0 2.0 1.0 1.0 0.0 1.0 -64.0 -73.0 -28.0 0.0 -70.0 -25.0 0.0 0.0 -18.0Program Results
SG03AD EXAMPLE PROGRAM RESULTS SEP = 0.29D+00 FERR = 0.40D-13 SCALE = 0.10D+01 The solution matrix X is -2.0000 -1.0000 0.0000 -1.0000 -3.0000 -1.0000 0.0000 -1.0000 -3.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SG03AX.html 0000664 0000000 0000000 00000011444 14560147231 0020363 0 ustar 00root root 0000000 0000000
Purpose
To solve for X either the reduced generalized discrete-time
Lyapunov equation
T T
A * X * A - E * X * E = SCALE * Y (1)
or
T T
A * X * A - E * X * E = SCALE * Y (2)
where the right hand side Y is symmetric. A, E, Y, and the
solution X are N-by-N matrices. The pencil A - lambda * E must be
in generalized Schur form (A upper quasitriangular, E upper
triangular). SCALE is an output scale factor, set to avoid
overflow in X.
Specification
SUBROUTINE SG03AX( TRANS, N, A, LDA, E, LDE, X, LDX, SCALE, INFO )
C .. Scalar Arguments ..
CHARACTER TRANS
DOUBLE PRECISION SCALE
INTEGER INFO, LDA, LDE, LDX, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), E(LDE,*), X(LDX,*)
Arguments
Mode Parameters
TRANS CHARACTER*1
Specifies whether the transposed equation is to be solved
or not:
= 'N': Solve equation (1);
= 'T': Solve equation (2).
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N upper Hessenberg part of this array
must contain the quasitriangular matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
E (input) DOUBLE PRECISION array, dimension (LDE,N)
The leading N-by-N upper triangular part of this array
must contain the matrix E.
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,N).
X (input/output) DOUBLE PRECISION array, dimension (LDX,N)
On entry, the leading N-by-N part of this array must
contain the right hand side matrix Y of the equation. Only
the upper triangular part of this matrix need be given.
On exit, the leading N-by-N part of this array contains
the solution matrix X of the equation.
LDX INTEGER
The leading dimension of the array X. LDX >= MAX(1,N).
SCALE (output) DOUBLE PRECISION
The scale factor set to avoid overflow in X.
(0 < SCALE <= 1)
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: equation is (almost) singular to working precision;
perturbed values were used to solve the equation
(but the matrices A and E are unchanged).
Method
The solution X of (1) or (2) is computed via block back substitution or block forward substitution, respectively. (See [1] and [2] for details.)References
[1] Bartels, R.H., Stewart, G.W.
Solution of the equation A X + X B = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
[2] Penzl, T.
Numerical solution of generalized Lyapunov equations.
Advances in Comp. Math., vol. 8, pp. 33-48, 1998.
Numerical Aspects
8/3 * N**3 flops are required by the routine. Note that we count a single floating point arithmetic operation as one flop. The algorithm is backward stable if the eigenvalues of the pencil A - lambda * E are real. Otherwise, linear systems of order at most 4 are involved into the computation. These systems are solved by Gauss elimination with complete pivoting. The loss of stability of the Gauss elimination with complete pivoting is rarely encountered in practice.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve for X either the reduced generalized continuous-time
Lyapunov equation
T T
A * X * E + E * X * A = SCALE * Y (1)
or
T T
A * X * E + E * X * A = SCALE * Y (2)
where the right hand side Y is symmetric. A, E, Y, and the
solution X are N-by-N matrices. The pencil A - lambda * E must be
in generalized Schur form (A upper quasitriangular, E upper
triangular). SCALE is an output scale factor, set to avoid
overflow in X.
Specification
SUBROUTINE SG03AY( TRANS, N, A, LDA, E, LDE, X, LDX, SCALE, INFO )
C .. Scalar Arguments ..
CHARACTER TRANS
DOUBLE PRECISION SCALE
INTEGER INFO, LDA, LDE, LDX, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), E(LDE,*), X(LDX,*)
Arguments
Mode Parameters
TRANS CHARACTER*1
Specifies whether the transposed equation is to be solved
or not:
= 'N': Solve equation (1);
= 'T': Solve equation (2).
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N upper Hessenberg part of this array
must contain the quasitriangular matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
E (input) DOUBLE PRECISION array, dimension (LDE,N)
The leading N-by-N upper triangular part of this array
must contain the matrix E.
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,N).
X (input/output) DOUBLE PRECISION array, dimension (LDX,N)
On entry, the leading N-by-N part of this array must
contain the right hand side matrix Y of the equation. Only
the upper triangular part of this matrix need be given.
On exit, the leading N-by-N part of this array contains
the solution matrix X of the equation.
LDX INTEGER
The leading dimension of the array X. LDX >= MAX(1,N).
SCALE (output) DOUBLE PRECISION
The scale factor set to avoid overflow in X.
(0 < SCALE <= 1)
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: equation is (almost) singular to working precision;
perturbed values were used to solve the equation
(but the matrices A and E are unchanged).
Method
The solution X of (1) or (2) is computed via block back substitution or block forward substitution, respectively. (See [1] and [2] for details.)References
[1] Bartels, R.H., Stewart, G.W.
Solution of the equation A X + X B = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
[2] Penzl, T.
Numerical solution of generalized Lyapunov equations.
Advances in Comp. Math., vol. 8, pp. 33-48, 1998.
Numerical Aspects
8/3 * N**3 flops are required by the routine. Note that we count a single floating point arithmetic operation as one flop. The algorithm is backward stable if the eigenvalues of the pencil A - lambda * E are real. Otherwise, linear systems of order at most 4 are involved into the computation. These systems are solved by Gauss elimination with complete pivoting. The loss of stability of the Gauss elimination with complete pivoting is rarely encountered in practice.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the Cholesky factor U of the matrix X,
T
X = op(U) * op(U),
which is the solution of either the generalized c-stable
continuous-time Lyapunov equation
T T
op(A) * X * op(E) + op(E) * X * op(A)
2 T
= - SCALE * op(B) * op(B), (1)
or the generalized d-stable discrete-time Lyapunov equation
T T
op(A) * X * op(A) - op(E) * X * op(E)
2 T
= - SCALE * op(B) * op(B), (2)
without first finding X and without the need to form the matrix
op(B)**T * op(B).
op(K) is either K or K**T for K = A, B, E, U. A and E are N-by-N
matrices, op(B) is an M-by-N matrix. The resulting matrix U is an
N-by-N upper triangular matrix with non-negative entries on its
main diagonal. SCALE is an output scale factor set to avoid
overflow in U.
In the continuous-time case (1) the pencil A - lambda * E must be
c-stable (that is, all eigenvalues must have negative real parts).
In the discrete-time case (2) the pencil A - lambda * E must be
d-stable (that is, the moduli of all eigenvalues must be smaller
than one).
Specification
SUBROUTINE SG03BD( DICO, FACT, TRANS, N, M, A, LDA, E, LDE, Q,
$ LDQ, Z, LDZ, B, LDB, SCALE, ALPHAR, ALPHAI,
$ BETA, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
DOUBLE PRECISION SCALE
INTEGER INFO, LDA, LDB, LDE, LDQ, LDWORK, LDZ, M, N
CHARACTER DICO, FACT, TRANS
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), ALPHAI(*), ALPHAR(*), B(LDB,*),
$ BETA(*), DWORK(*), E(LDE,*), Q(LDQ,*), Z(LDZ,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies which type of the equation is considered:
= 'C': Continuous-time equation (1);
= 'D': Discrete-time equation (2).
FACT CHARACTER*1
Specifies whether the generalized real Schur
factorization of the pencil A - lambda * E is supplied on
entry or not:
= 'N': Factorization is not supplied;
= 'F': Factorization is supplied.
TRANS CHARACTER*1
Specifies whether the transposed equation is to be solved
or not:
= 'N': op(A) = A, op(E) = E;
= 'T': op(A) = A**T, op(E) = E**T.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
M (input) INTEGER
The number of rows in the matrix op(B). M >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, if FACT = 'F', then the leading N-by-N upper
Hessenberg part of this array must contain the generalized
Schur factor A_s of the matrix A (see definition (3) in
section METHOD). A_s must be an upper quasitriangular
matrix. The elements below the upper Hessenberg part of
the array A are not referenced.
If FACT = 'N', then the leading N-by-N part of this array
must contain the matrix A.
On exit, if FACT = 'N', the leading N-by-N upper
Hessenberg part of this array contains the generalized
Schur factor A_s of the matrix A. (A_s is an upper
quasitriangular matrix.) If FACT = 'F', the leading N-by-N
upper triangular part of this array is unchanged.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, if FACT = 'F', then the leading N-by-N upper
triangular part of this array must contain the generalized
Schur factor E_s of the matrix E (see definition (4) in
section METHOD). E_s must be an upper triangular matrix.
The elements below the upper triangular part of the array
E are not referenced.
If FACT = 'N', then the leading N-by-N part of this array
must contain the coefficient matrix E of the equation.
On exit, if FACT = 'N', the leading N-by-N upper
triangular part of this array contains the generalized
Schur factor E_s of the matrix E. (E_s is an upper
triangular matrix.) If FACT = 'F', the leading N-by-N
upper triangular part of this array is unchanged.
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,N).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if FACT = 'F', then the leading N-by-N part of
this array must contain the orthogonal matrix Q from the
generalized Schur factorization (see definitions (3) and
(4) in section METHOD), or an identity matrix (if the
original equation has upper triangular matrices A and E).
If FACT = 'N', Q need not be set on entry.
On exit, if FACT = 'N', the leading N-by-N part of this
array contains the orthogonal matrix Q from the
generalized Schur factorization. If FACT = 'F', this array
is unchanged.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= MAX(1,N).
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
On entry, if FACT = 'F', then the leading N-by-N part of
this array must contain the orthogonal matrix Z from the
generalized Schur factorization (see definitions (3) and
(4) in section METHOD), or an identity matrix (if the
original equation has upper triangular matrices A and E).
If FACT = 'N', Z need not be set on entry.
On exit, if FACT = 'N', the leading N-by-N part of this
array contains the orthogonal matrix Z from the
generalized Schur factorization. If FACT = 'F', this array
is unchanged.
LDZ INTEGER
The leading dimension of the array Z. LDZ >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,N1)
On entry, if TRANS = 'T', the leading N-by-M part of this
array must contain the matrix B and N1 >= MAX(M,N).
If TRANS = 'N', the leading M-by-N part of this array
must contain the matrix B and N1 >= N.
On exit, if INFO = 0, the leading N-by-N part of this
array contains the Cholesky factor U of the solution
matrix X of the problem, X = op(U)**T * op(U).
If M = 0 and N > 0, then U is set to zero.
LDB INTEGER
The leading dimension of the array B.
If TRANS = 'T', LDB >= MAX(1,N).
If TRANS = 'N', LDB >= MAX(1,M,N).
SCALE (output) DOUBLE PRECISION
The scale factor set to avoid overflow in U.
0 < SCALE <= 1.
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
BETA (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, 3, 5, 6, or 7, then
(ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j = 1, ... ,N, are the
eigenvalues of the matrix pencil A - lambda * E.
All BETA(j) are non-negative numbers.
ALPHAR and ALPHAI will be always less than and usually
comparable with norm(A) in magnitude, and BETA always less
than and usually comparable with norm(B).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
On exit, if INFO = -21, DWORK(1) returns the minimum value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= MAX(1,4*N,6*N-6), if FACT = 'N';
LDWORK >= MAX(1,2*N,6*N-6), if FACT = 'F'.
For good performance, LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message related to LDWORK is issued by
XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the pencil A - lambda * E is (nearly) singular;
perturbed values were used to solve the equation
(but the reduced (quasi)triangular matrices A and E
are unchanged);
= 2: FACT = 'F' and the matrix contained in the upper
Hessenberg part of the array A is not in upper
quasitriangular form;
= 3: FACT = 'F' and there is a 2-by-2 block on the main
diagonal of the pencil A_s - lambda * E_s whose
eigenvalues are not conjugate complex;
= 4: FACT = 'N' and the pencil A - lambda * E cannot be
reduced to generalized Schur form: LAPACK routine
DGEGS (or DGGES) has failed to converge;
= 5: DICO = 'C' and the pencil A - lambda * E is not
c-stable;
= 6: DICO = 'D' and the pencil A - lambda * E is not
d-stable;
= 7: the LAPACK routine DSYEVX utilized to factorize M3
failed to converge in the discrete-time case (see
section METHOD for SLICOT Library routine SG03BU).
This error is unlikely to occur.
Method
An extension [2] of Hammarling's method [1] to generalized
Lyapunov equations is utilized to solve (1) or (2).
First the pencil A - lambda * E is reduced to real generalized
Schur form A_s - lambda * E_s by means of orthogonal
transformations (QZ-algorithm):
A_s = Q**T * A * Z (upper quasitriangular), (3)
E_s = Q**T * E * Z (upper triangular). (4)
If the pencil A - lambda * E has already been factorized prior to
calling the routine, however, then the factors A_s, E_s, Q and Z
may be supplied and the initial factorization omitted.
Depending on the parameters TRANS and M, the N-by-N upper
triangular matrix B_s is defined as follows. In any case Q_B is
an M-by-M orthogonal matrix, which need not be accumulated.
1. If TRANS = 'N' and M < N, B_s is the upper triangular matrix
from the QR-factorization
( Q_B O ) ( B * Z )
( ) * B_s = ( ),
( O I ) ( O )
where the O's are zero matrices of proper size and I is the
identity matrix of order N-M.
2. If TRANS = 'N' and M >= N, B_s is the upper triangular matrix
from the (rectangular) QR-factorization
( B_s )
Q_B * ( ) = B * Z,
( O )
where O is the (M-N)-by-N zero matrix.
3. If TRANS = 'T' and M < N, B_s is the upper triangular matrix
from the RQ-factorization
( Q_B O )
(B_s O ) * ( ) = ( Q**T * B O ).
( O I )
4. If TRANS = 'T' and M >= N, B_s is the upper triangular matrix
from the (rectangular) RQ-factorization
( B_s O ) * Q_B = Q**T * B,
where O is the N-by-(M-N) zero matrix.
Assuming SCALE = 1, the transformation of A, E and B described
above leads to the reduced continuous-time equation
T T
op(A_s) op(U_s) op(U_s) op(E_s)
T T
+ op(E_s) op(U_s) op(U_s) op(A_s)
T
= - op(B_s) op(B_s) (5)
or to the reduced discrete-time equation
T T
op(A_s) op(U_s) op(U_s) op(A_s)
T T
- op(E_s) op(U_s) op(U_s) op(E_s)
T
= - op(B_s) op(B_s). (6)
For brevity we restrict ourself to equation (5) and the case
TRANS = 'N'. The other three cases can be treated in a similar
fashion.
We use the following partitioning for the matrices A_s, E_s, B_s,
and U_s
( A11 A12 ) ( E11 E12 )
A_s = ( ), E_s = ( ),
( 0 A22 ) ( 0 E22 )
( B11 B12 ) ( U11 U12 )
B_s = ( ), U_s = ( ). (7)
( 0 B22 ) ( 0 U22 )
The size of the (1,1)-blocks is 1-by-1 (iff A_s(2,1) = 0.0) or
2-by-2.
We compute U11, U12**T, and U22 in three steps.
Step I:
From (5) and (7) we get the 1-by-1 or 2-by-2 equation
T T T T
A11 * U11 * U11 * E11 + E11 * U11 * U11 * A11
T
= - B11 * B11.
For brevity, details are omitted here. See [2]. The technique
for computing U11 is similar to those applied to standard
Lyapunov equations in Hammarling's algorithm ([1], section 6).
Furthermore, the auxiliary matrices M1 and M2 defined as
follows
-1 -1
M1 = U11 * A11 * E11 * U11
-1 -1
M2 = B11 * E11 * U11
are computed in a numerically reliable way.
Step II:
The generalized Sylvester equation
T T T T
A22 * U12 + E22 * U12 * M1 =
T T T T T
- B12 * M2 - A12 * U11 - E12 * U11 * M1
is solved for U12**T.
Step III:
It can be shown that
T T T T
A22 * U22 * U22 * E22 + E22 * U22 * U22 * A22 =
T T
- B22 * B22 - y * y (8)
holds, where y is defined as
T T T T T T
y = B12 - ( E12 * U11 + E22 * U12 ) * M2 .
If B22_tilde is the square triangular matrix arising from the
(rectangular) QR-factorization
( B22_tilde ) ( B22 )
Q_B_tilde * ( ) = ( ),
( O ) ( y**T )
where Q_B_tilde is an orthogonal matrix of order N, then
T T T
- B22 * B22 - y * y = - B22_tilde * B22_tilde.
Replacing the right hand side in (8) by the term
- B22_tilde**T * B22_tilde leads to a reduced generalized
Lyapunov equation of lower dimension compared to (5).
The recursive application of the steps I to III yields the
solution U_s of the equation (5).
It remains to compute the solution matrix U of the original
problem (1) or (2) from the matrix U_s. To this end we transform
the solution back (with respect to the transformation that led
from (1) to (5) (from (2) to (6)) and apply the QR-factorization
(RQ-factorization). The upper triangular solution matrix U is
obtained by
Q_U * U = U_s * Q**T (if TRANS = 'N'),
or
U * Q_U = Z * U_s (if TRANS = 'T'),
where Q_U is an N-by-N orthogonal matrix. Again, the orthogonal
matrix Q_U need not be accumulated.
References
[1] Hammarling, S.J.
Numerical solution of the stable, non-negative definite
Lyapunov equation.
IMA J. Num. Anal., 2, pp. 303-323, 1982.
[2] Penzl, T.
Numerical solution of generalized Lyapunov equations.
Advances in Comp. Math., vol. 8, pp. 33-48, 1998.
Numerical Aspects
The number of flops required by the routine is given by the
following table. Note that we count a single floating point
arithmetic operation as one flop.
| FACT = 'F' FACT = 'N'
---------+--------------------------------------------------
M <= N | (13*N**3+6*M*N**2 (211*N**3+6*M*N**2
| +6*M**2*N-2*M**3)/3 +6*M**2*N-2*M**3)/3
|
M > N | (11*N**3+12*M*N**2)/3 (209*N**3+12*M*N**2)/3
Further Comments
The Lyapunov equation may be very ill-conditioned. In particular, if DICO = 'D' and the pencil A - lambda * E has a pair of almost reciprocal eigenvalues, or DICO = 'C' and the pencil has an almost degenerate pair of eigenvalues, then the Lyapunov equation will be ill-conditioned. Perturbed values were used to solve the equation. A condition estimate can be obtained from the routine SG03AD. When setting the error indicator INFO, the routine does not test for near instability in the equation but only for exact instability.Example
Program Text
* SG03BD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA, LDB, LDE, LDQ, LDZ
PARAMETER ( LDA = NMAX, LDB = NMAX, LDE = NMAX,
$ LDQ = NMAX, LDZ = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( 1, 4*NMAX, 6*NMAX-6 ) )
* .. Local Scalars ..
CHARACTER*1 DICO, FACT, TRANS
DOUBLE PRECISION SCALE
INTEGER I, INFO, J, N, M
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), ALPHAI(NMAX), ALPHAR(NMAX),
$ B(LDB,NMAX), BETA(NMAX), DWORK(LDWORK),
$ E(LDE,NMAX), Q(LDQ,NMAX), Z(LDZ,NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL SG03BD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, DICO, FACT, TRANS
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) N
ELSE IF ( M.LT.0 .OR. M.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) M
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( FACT, 'F' ) ) THEN
READ ( NIN, FMT = * ) ( ( Q(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( Z(I,J), J = 1,N ), I = 1,N )
END IF
IF ( LSAME( FACT, 'T' ) ) THEN
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,N ), I = 1,M )
END IF
* Find the Cholesky factor U of the solution matrix.
CALL SG03BD( DICO, FACT, TRANS, N, M, A, LDA, E, LDE, Q, LDQ,
$ Z, LDZ, B, LDB, SCALE, ALPHAR, ALPHAI, BETA,
$ DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) SCALE
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( B(I,J), J = 1,N )
20 CONTINUE
END IF
END IF
STOP
*
99999 FORMAT (' SG03BD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SG03BD = ',I2)
99997 FORMAT (' SCALE = ',F8.4,//' The Cholesky factor U of the solution
$ matrix is')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' N is out of range.',/' N = ',I5)
99994 FORMAT (/' M is out of range.',/' M = ',I5)
END
Program Data
SG03BD EXAMPLE PROGRAM DATA 3 1 C N N -1.0 3.0 -4.0 0.0 5.0 -2.0 -4.0 4.0 1.0 2.0 1.0 3.0 2.0 0.0 1.0 4.0 5.0 1.0 2.0 -1.0 7.0Program Results
SG03BD EXAMPLE PROGRAM RESULTS SCALE = 1.0000 The Cholesky factor U of the solution matrix is 1.6003 -0.4418 -0.1523 0.0000 0.6795 -0.2499 0.0000 0.0000 0.2041
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/SG03BR.html 0000664 0000000 0000000 00000004232 14560147231 0020353 0 ustar 00root root 0000000 0000000
Purpose
To compute the parameters for the complex Givens rotation
( C SR+SI*I ) ( XR+XI*I ) ( ZR+ZI*I )
( ) * ( ) = ( )
( -SR+SI*I C ) ( YR+YI*I ) ( 0 )
where C, SR, SI, XR, XI, YR, YI, ZR, ZI are real numbers, I is the
imaginary unit, I = SQRT(-1), and C**2 + |SR+SI*I|**2 = 1.
Specification
SUBROUTINE SG03BR( XR, XI, YR, YI, C, SR, SI, ZR, ZI )
C .. Scalar Arguments ..
DOUBLE PRECISION C, SI, SR, XI, XR, YI, YR, ZI, ZR
Arguments
Input/Output Parameters
XR, XI, (input) DOUBLE PRECISION
YR, YI (input) DOUBLE PRECISION
The given real scalars XR, XI, YR, YI.
C, (output) DOUBLE PRECISION
SR, SI, (output) DOUBLE PRECISION
ZR, ZI (output) DOUBLE PRECISION
The computed real scalars C, SR, SI, ZR, ZI defining the
complex Givens rotation and Z = ZR+ZI*I.
Numerical Aspects
The subroutine avoids unnecessary overflow.Further Comments
In the interest of speed, this routine does not check the input for errors.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the Cholesky factor U of the matrix X, X = U**H * U or
X = U * U**H, which is the solution of the generalized d-stable
discrete-time Lyapunov equation
H H 2 H
A * X * A - E * X * E = - SCALE * B * B, (1)
or the conjugate transposed equation
H H 2 H
A * X * A - E * X * E = - SCALE * B * B , (2)
respectively, where A, E, B, and U are complex N-by-N matrices.
The Cholesky factor U of the solution is computed without first
finding X. The pencil A - lambda * E must be in complex
generalized Schur form (A and E are upper triangular and the
diagonal elements of E are non-negative real numbers). Moreover,
it must be d-stable, i.e., the moduli of its eigenvalues must be
less than one. B must be an upper triangular matrix with real
non-negative entries on its main diagonal.
The resulting matrix U is upper triangular. The entries on its
main diagonal are non-negative. SCALE is an output scale factor
set to avoid overflow in U.
Specification
SUBROUTINE SG03BS( TRANS, N, A, LDA, E, LDE, B, LDB, SCALE, DWORK,
$ ZWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TRANS
DOUBLE PRECISION SCALE
INTEGER INFO, LDA, LDB, LDE, N
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*)
COMPLEX*16 A(LDA,*), B(LDB,*), E(LDE,*), ZWORK(*)
Arguments
Mode Parameters
TRANS CHARACTER*1
Specifies whether equation (1) or equation (2) is to be
solved:
= 'N': Solve equation (1);
= 'C': Solve equation (2).
Input/Output Parameters
N (input) INTEGER
The order of the matrices. N >= 0.
A (input/workspace) COMPLEX*16 array, dimension (LDA,N)
The leading N-by-N upper triangular part of this array
must contain the triangular matrix A. The lower triangular
part is used as workspace, but the diagonal is restored.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
E (input/workspace) COMPLEX*16 array, dimension (LDE,N)
The leading N-by-N upper triangular part of this array
must contain the triangular matrix E. If TRANS = 'N', the
strictly lower triangular part is used as workspace.
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,N).
B (input/output) COMPLEX*16 array, dimension (LDB,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the matrix B.
On exit, the leading N-by-N upper triangular part of this
array contains the solution matrix U.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
SCALE (output) DOUBLE PRECISION
The scale factor set to avoid overflow in U.
0 < SCALE <= 1.
Workspace
DWORK DOUBLE PRECISION array, dimension LDWORK, where
LDWORK = 0, if N <= 1;
LDWORK = MAX(N-1,10), if N > 1.
ZWORK COMPLEX*16, dimension MAX(3*N-3,0)
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 3: the pencil A - lambda * E is not stable, i.e., there
there are eigenvalues outside the open unit circle;
= 4: the LAPACK routine ZSTEIN utilized to factorize M3
failed to converge. This error is unlikely to occur.
Method
The method used by the routine is an extension of Hammarling's
algorithm [1] to generalized Lyapunov equations. The real case is
described in [2].
We present the method for solving equation (1). Equation (2) can
be treated in a similar fashion. For simplicity, assume SCALE = 1.
Since all matrices A, E, B, and U are upper triangular, we use the
following partitioning
( A11 A12 ) ( E11 E12 )
A = ( ), E = ( ),
( 0 A22 ) ( 0 E22 )
( B11 B12 ) ( U11 U12 )
B = ( ), U = ( ), (3)
( 0 B22 ) ( 0 U22 )
where the size of the (1,1)-blocks is 1-by-1.
We compute U11, U12**H and U22 in three steps.
Step I:
From (1) and (3) we get the 1-by-1 equation
H H H H H
A11 * U11 * U11 * A11 - E11 * U11 * U11 * E11 = -B11 * B11.
For brevity, details are omitted here. The technique for
computing U11 is similar to those applied to standard Lyapunov
equations in Hammarling's algorithm ([1], section 5).
Furthermore, the auxiliary scalars M1 and M2 defined as follows
M1 = A11 / E11 , M2 = B11 / E11 / U11 ,
are computed in a numerically reliable way.
Step II:
We solve for U12**H the linear system of equations, with
scaling to prevent overflow,
H H H
( M1 * A22 - E22 ) U12 =
H H H
= - M2 * B12 + U11 * ( E12 - M1 * A12 ) .
Step III:
One can show that
H H H H
A22 * U22 * U22 * A22 - E22 * U22 * U22 * E22 =
H H
- B22 * B22 - y * y (4)
holds, where y is defined as follows
H H H H
y = ( B12 U11 * A12 + A22 * U12 ) * M3EV,
where M3EV is a matrix which fulfils
( I - M2*M2 -M2*M1**H ) H
M3 = ( ) = M3EV * M3EV .
( -M1*M2 I - M1*M1**H )
M3 is positive semidefinite and its rank is equal to 1.
Therefore, a matrix M3EV can be found by solving the Hermitian
eigenvalue problem for M3 such that y consists of one column.
If B22_tilde is the square triangular matrix arising from the
QR-factorization
( B22_tilde ) ( B22 )
Q * ( ) = ( ),
( 0 ) ( y**H )
then
H H H
- B22 * B22 - y * y = - B22_tilde * B22_tilde.
Replacing the right hand side in (4) by the term
- B22_tilde**H * B22_tilde leads to a generalized Lyapunov
equation like (1), but of dimension N-1.
The solution U of the equation (1) can be obtained by recursive
application of the steps I to III.
References
[1] Hammarling, S.J.
Numerical solution of the stable, non-negative definite
Lyapunov equation.
IMA J. Num. Anal., 2, pp. 303-323, 1982.
[2] Penzl, T.
Numerical solution of generalized Lyapunov equations.
Advances in Comp. Math., vol. 8, pp. 33-48, 1998.
Numerical Aspects
The routine requires 2*N**3 flops. Note that we count a single floating point arithmetic operation as one flop.Further Comments
The Lyapunov equation may be very ill-conditioned. In particular, if the pencil A - lambda * E has a pair of almost reciprocal eigenvalues, then the Lyapunov equation will be ill-conditioned. Perturbed values were used to solve the equation. A condition estimate can be obtained from the routine SG03AD.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the Cholesky factor U of the matrix X, X = U**H * U or
X = U * U**H, which is the solution of the generalized c-stable
continuous-time Lyapunov equation
H H 2 H
A * X * E + E * X * A = - SCALE * B * B, (1)
or the conjugate transposed equation
H H 2 H
A * X * E + E * X * A = - SCALE * B * B , (2)
respectively, where A, E, B, and U are complex N-by-N matrices.
The Cholesky factor U of the solution is computed without first
finding X. The pencil A - lambda * E must be in complex
generalized Schur form (A and E are upper triangular and the
diagonal elements of E are non-negative real numbers). Moreover,
it must be c-stable, i.e., its eigenvalues must have negative real
parts. B must be an upper triangular matrix with real non-negative
entries on its main diagonal.
The resulting matrix U is upper triangular. The entries on its
main diagonal are non-negative. SCALE is an output scale factor
set to avoid overflow in U.
Specification
SUBROUTINE SG03BT( TRANS, N, A, LDA, E, LDE, B, LDB, SCALE, DWORK,
$ ZWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TRANS
DOUBLE PRECISION SCALE
INTEGER INFO, LDA, LDB, LDE, N
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*)
COMPLEX*16 A(LDA,*), B(LDB,*), E(LDE,*), ZWORK(*)
Arguments
Mode Parameters
TRANS CHARACTER*1
Specifies whether equation (1) or equation (2) is to be
solved:
= 'N': Solve equation (1);
= 'C': Solve equation (2).
Input/Output Parameters
N (input) INTEGER
The order of the matrices. N >= 0.
A (input/workspace) COMPLEX*16 array, dimension (LDA,N)
The leading N-by-N upper triangular part of this array
must contain the triangular matrix A. The lower triangular
part is used as workspace, but the diagonal is restored.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
E (input/workspace) COMPLEX*16 array, dimension (LDE,N)
The leading N-by-N upper triangular part of this array
must contain the triangular matrix E. If TRANS = 'N', the
strictly lower triangular part is used as workspace.
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,N).
B (input/output) COMPLEX*16 array, dimension (LDB,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the matrix B.
On exit, the leading N-by-N upper triangular part of this
array contains the solution matrix U.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
SCALE (output) DOUBLE PRECISION
The scale factor set to avoid overflow in U.
0 < SCALE <= 1.
Workspace
DWORK DOUBLE PRECISION array, dimension MAX(N-1,0) ZWORK COMPLEX*16, dimension MAX(3*N-3,0)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 3: the pencil A - lambda * E is not stable, i.e., there
is an eigenvalue without a negative real part.
Method
The method used by the routine is an extension of Hammarling's
algorithm [1] to generalized Lyapunov equations. The real case is
described in [2].
We present the method for solving equation (1). Equation (2) can
be treated in a similar fashion. For simplicity, assume SCALE = 1.
Since all matrices A, E, B, and U are upper triangular, we use the
following partitioning
( A11 A12 ) ( E11 E12 )
A = ( ), E = ( ),
( 0 A22 ) ( 0 E22 )
( B11 B12 ) ( U11 U12 )
B = ( ), U = ( ), (3)
( 0 B22 ) ( 0 U22 )
where the size of the (1,1)-blocks is 1-by-1.
We compute U11, U12**H and U22 in three steps.
Step I:
From (1) and (3) we get the 1-by-1 equation
H H H H H
A11 * U11 * U11 * E11 + E11 * U11 * U11 * A11 = -B11 * B11.
For brevity, details are omitted here. The technique for
computing U11 is similar to those applied to standard Lyapunov
equations in Hammarling's algorithm ([1], section 5).
Furthermore, the auxiliary scalars M1 and M2 defined as follows
M1 = A11 / E11 , M2 = B11 / E11 / U11 ,
are computed in a numerically reliable way.
Step II:
We solve for U12**H the linear system of equations, with
scaling to prevent overflow,
H H H
( A22 + M1 * E22 ) U12 =
H H H
= - M2 * B12 - U11 * ( A12 + M1 * E12 ) .
Step III:
One can show that
H H H H
A22 * U22 * U22 * E22 + E22 * U22 * U22 * A22 =
H H
- B22 * B22 - y * y (4)
holds, where y is defined as follows
H H H H
y = B12 - M2 * ( U11 * E12 + E22 * U12 ).
If B22_tilde is the square triangular matrix arising from the
QR-factorization
( B22_tilde ) ( B22 )
Q * ( ) = ( ),
( 0 ) ( y**H )
then
H H H
- B22 * B22 - y * y = - B22_tilde * B22_tilde.
Replacing the right hand side in (4) by the term
- B22_tilde**H * B22_tilde leads to a generalized Lyapunov
equation like (1), but of dimension N-1.
The solution U of the equation (1) can be obtained by recursive
application of the steps I to III.
References
[1] Hammarling, S.J.
Numerical solution of the stable, non-negative definite
Lyapunov equation.
IMA J. Num. Anal., 2, pp. 303-323, 1982.
[2] Penzl, T.
Numerical solution of generalized Lyapunov equations.
Advances in Comp. Math., vol. 8, pp. 33-48, 1998.
Numerical Aspects
The routine requires 2*N**3 flops. Note that we count a single floating point arithmetic operation as one flop.Further Comments
The Lyapunov equation may be very ill-conditioned. In particular, if the pencil A - lambda * E has a pair of almost degenerate eigenvalues, then the Lyapunov equation will be ill-conditioned. Perturbed values were used to solve the equation. A condition estimate can be obtained from the routine SG03AD.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the Cholesky factor U of the matrix X, X = U**T * U or
X = U * U**T, which is the solution of the generalized d-stable
discrete-time Lyapunov equation
T T 2 T
A * X * A - E * X * E = - SCALE * B * B, (1)
or the transposed equation
T T 2 T
A * X * A - E * X * E = - SCALE * B * B , (2)
respectively, where A, E, B, and U are real N-by-N matrices. The
Cholesky factor U of the solution is computed without first
finding X. The pencil A - lambda * E must be in generalized Schur
form ( A upper quasitriangular, E upper triangular ). Moreover, it
must be d-stable, i.e., the moduli of its eigenvalues must be less
than one. B must be an upper triangular matrix with non-negative
entries on its main diagonal.
The resulting matrix U is upper triangular. The entries on its
main diagonal are non-negative. SCALE is an output scale factor
set to avoid overflow in U.
Specification
SUBROUTINE SG03BU( TRANS, N, A, LDA, E, LDE, B, LDB, SCALE, DWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER TRANS
DOUBLE PRECISION SCALE
INTEGER INFO, LDA, LDB, LDE, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), E(LDE,*)
Arguments
Mode Parameters
TRANS CHARACTER*1
Specifies whether equation (1) or equation (2) is to be
solved:
= 'N': Solve equation (1);
= 'T': Solve equation (2).
Input/Output Parameters
N (input) INTEGER
The order of the matrices. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N upper Hessenberg part of this array
must contain the quasitriangular matrix A. The elements
below the upper Hessenberg part are not referenced.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
E (input) DOUBLE PRECISION array, dimension (LDE,N)
The leading N-by-N upper triangular part of this array
must contain the triangular matrix E. The elements below
the main diagonal are not referenced.
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the matrix B.
On exit, the leading N-by-N upper triangular part of this
array contains the solution matrix U. The elements below
the main diagonal are not referenced.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
SCALE (output) DOUBLE PRECISION
The scale factor set to avoid overflow in U.
0 < SCALE <= 1.
Workspace
DWORK DOUBLE PRECISION array, dimension (6*N-6)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the generalized Sylvester equation to be solved in
step II (see METHOD) is (nearly) singular to working
precision; perturbed values were used to solve the
equation (but the matrices A and E are unchanged);
= 2: the generalized Schur form of the pencil
A - lambda * E contains a 2-by-2 main diagonal block
whose eigenvalues are not a pair of complex
conjugate numbers;
= 3: the pencil A - lambda * E is not d-stable, i.e.,
there are eigenvalues outside the open unit circle;
= 4: the LAPACK routine DSYEVX utilized to factorize M3
failed to converge. This error is unlikely to occur.
Method
The method [2] used by the routine is an extension of Hammarling's
algorithm [1] to generalized Lyapunov equations.
We present the method for solving equation (1). Equation (2) can
be treated in a similar fashion. For simplicity, assume SCALE = 1.
The matrix A is an upper quasitriangular matrix, i.e., it is a
block triangular matrix with square blocks on the main diagonal
and the block order at most 2. We use the following partitioning
for the matrices A, E, B and the solution matrix U
( A11 A12 ) ( E11 E12 )
A = ( ), E = ( ),
( 0 A22 ) ( 0 E22 )
( B11 B12 ) ( U11 U12 )
B = ( ), U = ( ). (3)
( 0 B22 ) ( 0 U22 )
The size of the (1,1)-blocks is 1-by-1 (iff A(2,1) = 0.0) or
2-by-2.
We compute U11, U12**T, and U22 in three steps.
Step I:
From (1) and (3) we get the 1-by-1 or 2-by-2 equation
T T T T
A11 * U11 * U11 * A11 - E11 * U11 * U11 * E11
T
= - B11 * B11.
For brevity, details are omitted here. The technique for
computing U11 is similar to those applied to standard Lyapunov
equations in Hammarling's algorithm ([1], section 6).
Furthermore, the auxiliary matrices M1 and M2 defined as
follows
-1 -1
M1 = U11 * A11 * E11 * U11 ,
-1 -1
M2 = B11 * E11 * U11 ,
are computed in a numerically reliable way.
Step II:
We solve for U12**T the generalized Sylvester equation
T T T T
A22 * U12 * M1 - E22 * U12
T T T T T
= - B12 * M2 + E12 * U11 - A12 * U11 * M1.
Step III:
One can show that
T T T T
A22 * U22 * U22 * A22 - E22 * U22 * U22 * E22 =
T T
- B22 * B22 - y * y (4)
holds, where y is defined as follows
T T T T
w = A12 * U11 + A22 * U12 ,
T
y = ( B12 w ) * M3EV,
where M3EV is a matrix which fulfils
( I-M2*M2**T -M2*M1**T ) T
M3 = ( ) = M3EV * M3EV .
( -M1*M2**T I-M1*M1**T )
M3 is positive semidefinite and its rank is equal to the size
of U11. Therefore, a matrix M3EV can be found by solving the
symmetric eigenvalue problem for M3 such that y consists of
either 1 or 2 rows.
If B22_tilde is the square triangular matrix arising from the
QR-factorization
( B22_tilde ) ( B22 )
Q * ( ) = ( ),
( 0 ) ( y**T )
then
T T T
- B22 * B22 - y * y = - B22_tilde * B22_tilde.
Replacing the right hand side in (4) by the term
- B22_tilde**T * B22_tilde leads to a generalized Lyapunov
equation of lower dimension compared to (1).
The solution U of the equation (1) can be obtained by recursive
application of the steps I to III.
References
[1] Hammarling, S.J.
Numerical solution of the stable, non-negative definite
Lyapunov equation.
IMA J. Num. Anal., 2, pp. 303-323, 1982.
[2] Penzl, T.
Numerical solution of generalized Lyapunov equations.
Advances in Comp. Math., vol. 8, pp. 33-48, 1998.
Numerical Aspects
The routine requires 2*N**3 flops. Note that we count a single floating point arithmetic operation as one flop.Further Comments
The Lyapunov equation may be very ill-conditioned. In particular, if the pencil A - lambda * E has a pair of almost reciprocal eigenvalues, then the Lyapunov equation will be ill-conditioned. Perturbed values were used to solve the equation. A condition estimate can be obtained from the routine SG03AD. When setting the error indicator INFO, the routine does not test for near instability in the equation but only for exact instability.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the Cholesky factor U of the matrix X, X = U**T * U or
X = U * U**T, which is the solution of the generalized c-stable
continuous-time Lyapunov equation
T T 2 T
A * X * E + E * X * A = - SCALE * B * B, (1)
or the transposed equation
T T 2 T
A * X * E + E * X * A = - SCALE * B * B , (2)
respectively, where A, E, B, and U are real N-by-N matrices. The
Cholesky factor U of the solution is computed without first
finding X. The pencil A - lambda * E must be in generalized Schur
form ( A upper quasitriangular, E upper triangular ). Moreover, it
must be c-stable, i.e., its eigenvalues must have negative real
parts. B must be an upper triangular matrix with non-negative
entries on its main diagonal.
The resulting matrix U is upper triangular. The entries on its
main diagonal are non-negative. SCALE is an output scale factor
set to avoid overflow in U.
Specification
SUBROUTINE SG03BV( TRANS, N, A, LDA, E, LDE, B, LDB, SCALE, DWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER TRANS
DOUBLE PRECISION SCALE
INTEGER INFO, LDA, LDB, LDE, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), E(LDE,*)
Arguments
Mode Parameters
TRANS CHARACTER*1
Specifies whether equation (1) or equation (2) is to be
solved:
= 'N': Solve equation (1);
= 'T': Solve equation (2).
Input/Output Parameters
N (input) INTEGER
The order of the matrices. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N upper Hessenberg part of this array
must contain the quasitriangular matrix A. The elements
below the upper Hessenberg part are not referenced.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
E (input) DOUBLE PRECISION array, dimension (LDE,N)
The leading N-by-N upper triangular part of this array
must contain the triangular matrix E. The elements below
the main diagonal are not referenced.
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the matrix B.
On exit, the leading N-by-N upper triangular part of this
array contains the solution matrix U. The elements below
the main diagonal are not referenced.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
SCALE (output) DOUBLE PRECISION
The scale factor set to avoid overflow in U.
0 < SCALE <= 1.
Workspace
DWORK DOUBLE PRECISION array, dimension (6*N-6)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the generalized Sylvester equation to be solved in
step II (see METHOD) is (nearly) singular to working
precision; perturbed values were used to solve the
equation (but the matrices A and E are unchanged);
= 2: the generalized Schur form of the pencil
A - lambda * E contains a 2-by-2 main diagonal block
whose eigenvalues are not a pair of complex
conjugate numbers;
= 3: the pencil A - lambda * E is not stable, i.e., there
is an eigenvalue with zero or positive real part.
Method
The method [2] used by the routine is an extension of Hammarling's
algorithm [1] to generalized Lyapunov equations.
We present the method for solving equation (1). Equation (2) can
be treated in a similar fashion. For simplicity, assume SCALE = 1.
The matrix A is an upper quasitriangular matrix, i.e., it is a
block triangular matrix with square blocks on the main diagonal
and the block order at most 2. We use the following partitioning
for the matrices A, E, B and the solution matrix U
( A11 A12 ) ( E11 E12 )
A = ( ), E = ( ),
( 0 A22 ) ( 0 E22 )
( B11 B12 ) ( U11 U12 )
B = ( ), U = ( ). (3)
( 0 B22 ) ( 0 U22 )
The size of the (1,1)-blocks is 1-by-1 (iff A(2,1) = 0.0) or
2-by-2.
We compute U11, U12**T, and U22 in three steps.
Step I:
From (1) and (3) we get the 1-by-1 or 2-by-2 equation
T T T T
A11 * U11 * U11 * E11 + E11 * U11 * U11 * A11
T
= - B11 * B11.
For brevity, details are omitted here. The technique for
computing U11 is similar to those applied to standard Lyapunov
equations in Hammarling's algorithm ([1], section 6).
Furthermore, the auxiliary matrices M1 and M2 defined as
follows
-1 -1
M1 = U11 * A11 * E11 * U11 ,
-1 -1
M2 = B11 * E11 * U11 ,
are computed in a numerically reliable way.
Step II:
We solve for U12**T the generalized Sylvester equation
T T T T
A22 * U12 + E22 * U12 * M1
T T T T T
= - B12 * M2 - A12 * U11 - E12 * U11 * M1.
Step III:
One can show that
T T T T
A22 * U22 * U22 * E22 + E22 * U22 * U22 * A22 =
T T
- B22 * B22 - y * y (4)
holds, where y is defined as follows
T T T T
w = E12 * U11 + E22 * U12 ,
T T
y = B12 - w * M2 .
If B22_tilde is the square triangular matrix arising from the
QR-factorization
( B22_tilde ) ( B22 )
Q * ( ) = ( ),
( 0 ) ( y**T )
then
T T T
- B22 * B22 - y * y = - B22_tilde * B22_tilde.
Replacing the right hand side in (4) by the term
- B22_tilde**T * B22_tilde leads to a generalized Lyapunov
equation of lower dimension compared to (1).
The solution U of the equation (1) can be obtained by recursive
application of the steps I to III.
References
[1] Hammarling, S.J.
Numerical solution of the stable, non-negative definite
Lyapunov equation.
IMA J. Num. Anal., 2, pp. 303-323, 1982.
[2] Penzl, T.
Numerical solution of generalized Lyapunov equations.
Advances in Comp. Math., vol. 8, pp. 33-48, 1998.
Numerical Aspects
The routine requires 2*N**3 flops. Note that we count a single floating point arithmetic operation as one flop.Further Comments
The Lyapunov equation may be very ill-conditioned. In particular, if the pencil A - lambda * E has a pair of almost degenerate eigenvalues, then the Lyapunov equation will be ill-conditioned. Perturbed values were used to solve the equation. A condition estimate can be obtained from the routine SG03AD. When setting the error indicator INFO, the routine does not test for near instability in the equation but only for exact instability.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve for X the generalized Sylvester equation
T T
A * X * C + E * X * D = SCALE * Y, (1)
or the transposed equation
T T
A * X * C + E * X * D = SCALE * Y, (2)
where A and E are real M-by-M matrices, C and D are real N-by-N
matrices, X and Y are real M-by-N matrices. N is either 1 or 2.
The pencil A - lambda * E must be in generalized real Schur form
(A upper quasitriangular, E upper triangular). SCALE is an output
scale factor, set to avoid overflow in X.
Specification
SUBROUTINE SG03BW( TRANS, M, N, A, LDA, C, LDC, E, LDE, D, LDD, X,
$ LDX, SCALE, INFO )
C .. Scalar Arguments ..
CHARACTER TRANS
DOUBLE PRECISION SCALE
INTEGER INFO, LDA, LDC, LDD, LDE, LDX, M, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), C(LDC,*), D(LDD,*), E(LDE,*), X(LDX,*)
Arguments
Mode Parameters
TRANS CHARACTER*1
Specifies whether the transposed equation is to be solved
or not:
= 'N': Solve equation (1);
= 'T': Solve equation (2).
Input/Output Parameters
M (input) INTEGER
The order of the matrices A and E. M >= 0.
N (input) INTEGER
The order of the matrices C and D. N = 1 or N = 2.
A (input) DOUBLE PRECISION array, dimension (LDA,M)
The leading M-by-M part of this array must contain the
upper quasitriangular matrix A. The elements below the
upper Hessenberg part are not referenced.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,M).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading N-by-N part of this array must contain the
matrix C.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,N).
E (input) DOUBLE PRECISION array, dimension (LDE,M)
The leading M-by-M part of this array must contain the
upper triangular matrix E. The elements below the main
diagonal are not referenced.
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,M).
D (input) DOUBLE PRECISION array, dimension (LDD,N)
The leading N-by-N part of this array must contain the
matrix D.
LDD INTEGER
The leading dimension of the array D. LDD >= MAX(1,N).
X (input/output) DOUBLE PRECISION array, dimension (LDX,N)
On entry, the leading M-by-N part of this array must
contain the right hand side matrix Y.
On exit, the leading M-by-N part of this array contains
the solution matrix X.
LDX INTEGER
The leading dimension of the array X. LDX >= MAX(1,M).
SCALE (output) DOUBLE PRECISION
The scale factor set to avoid overflow in X.
0 < SCALE <= 1.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the generalized Sylvester equation is (nearly)
singular to working precision; perturbed values
were used to solve the equation (but the matrices
A, C, D, and E are unchanged).
Method
The method used by the routine is based on a generalization of the algorithm due to Bartels and Stewart [1]. See also [2] and [3] for details.References
[1] Bartels, R.H., Stewart, G.W.
Solution of the equation A X + X B = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
[2] Gardiner, J.D., Laub, A.J., Amato, J.J., Moler, C.B.
Solution of the Sylvester Matrix Equation
A X B**T + C X D**T = E.
A.C.M. Trans. Math. Soft., vol. 18, no. 2, pp. 223-231, 1992.
[3] Penzl, T.
Numerical solution of generalized Lyapunov equations.
Advances in Comp. Math., vol. 8, pp. 33-48, 1998.
Numerical Aspects
The routine requires about 2 * N * M**2 flops. Note that we count a single floating point arithmetic operation as one flop. The algorithm is backward stable if the eigenvalues of the pencil A - lambda * E are real. Otherwise, linear systems of order at most 4 are involved into the computation. These systems are solved by Gauss elimination with complete pivoting. The loss of stability of the Gauss elimination with complete pivoting is rarely encountered in practice.Further Comments
When near singularity is detected, perturbed values are used to solve the equation (but the given matrices are unchanged).Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To solve for X = op(U)**T * op(U) either the generalized c-stable
continuous-time Lyapunov equation
T T
op(A) * X * op(E) + op(E) * X * op(A)
2 T
= - SCALE * op(B) * op(B), (1)
or the generalized d-stable discrete-time Lyapunov equation
T T
op(A) * X * op(A) - op(E) * X * op(E)
2 T
= - SCALE * op(B) * op(B), (2)
where op(K) is either K or K**T for K = A, B, E, U. The Cholesky
factor U of the solution is computed without first finding X.
Furthermore, the auxiliary matrices
-1 -1
M1 := op(U) * op(A) * op(E) * op(U)
-1 -1
M2 := op(B) * op(E) * op(U)
are computed in a numerically reliable way.
The matrices A, B, E, M1, M2, and U are real 2-by-2 matrices. The
pencil A - lambda * E must have a pair of complex conjugate
eigenvalues. The eigenvalues must be in the open right half plane
(in the continuous-time case) or inside the unit circle (in the
discrete-time case). The matrices E and B are upper triangular.
The resulting matrix U is upper triangular. The entries on its
main diagonal are non-negative. SCALE is an output scale factor
set to avoid overflow in U.
Specification
SUBROUTINE SG03BX( DICO, TRANS, A, LDA, E, LDE, B, LDB, U, LDU,
$ SCALE, M1, LDM1, M2, LDM2, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, TRANS
DOUBLE PRECISION SCALE
INTEGER INFO, LDA, LDB, LDE, LDM1, LDM2, LDU
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), E(LDE,*), M1(LDM1,*),
$ M2(LDM2,*), U(LDU,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies whether the continuous-time or the discrete-time
equation is to be solved:
= 'C': Solve continuous-time equation (1);
= 'D': Solve discrete-time equation (2).
TRANS CHARACTER*1
Specifies whether the transposed equation is to be solved
or not:
= 'N': op(K) = K, K = A, B, E, U;
= 'T': op(K) = K**T, K = A, B, E, U.
Input/Output Parameters
A (input) DOUBLE PRECISION array, dimension (LDA,2)
The leading 2-by-2 part of this array must contain the
matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= 2.
E (input) DOUBLE PRECISION array, dimension (LDE,2)
The leading 2-by-2 upper triangular part of this array
must contain the matrix E.
LDE INTEGER
The leading dimension of the array E. LDE >= 2.
B (input) DOUBLE PRECISION array, dimension (LDB,2)
The leading 2-by-2 upper triangular part of this array
must contain the matrix B.
LDB INTEGER
The leading dimension of the array B. LDB >= 2.
U (output) DOUBLE PRECISION array, dimension (LDU,2)
The leading 2-by-2 part of this array contains the upper
triangular matrix U.
LDU INTEGER
The leading dimension of the array U. LDU >= 2.
SCALE (output) DOUBLE PRECISION
The scale factor set to avoid overflow in U.
0 < SCALE <= 1.
M1 (output) DOUBLE PRECISION array, dimension (LDM1,2)
The leading 2-by-2 part of this array contains the
matrix M1.
LDM1 INTEGER
The leading dimension of the array M1. LDM1 >= 2.
M2 (output) DOUBLE PRECISION array, dimension (LDM2,2)
The leading 2-by-2 part of this array contains the
matrix M2.
LDM2 INTEGER
The leading dimension of the array M2. LDM2 >= 2.
Error Indicator
INFO INTEGER
= 0: successful exit;
= 2: the eigenvalues of the pencil A - lambda * E are not
a pair of complex conjugate numbers;
= 3: the eigenvalues of the pencil A - lambda * E are
not in the open right half plane (in the continuous-
time case) or inside the unit circle (in the
discrete-time case);
= 4: the LAPACK routine ZSTEIN utilized to factorize M3
(see SLICOT Library routine SG03BS) failed to
converge. This error is unlikely to occur.
Method
The method used by the routine is based on a generalization of the method due to Hammarling ([1], section 6) for Lyapunov equations of order 2. A more detailed description is given in [2].References
[1] Hammarling, S.J.
Numerical solution of the stable, non-negative definite
Lyapunov equation.
IMA J. Num. Anal., 2, pp. 303-323, 1982.
[2] Penzl, T.
Numerical solution of generalized Lyapunov equations.
Advances in Comp. Math., vol. 8, pp. 33-48, 1998.
Further Comments
If the solution matrix U is singular, the matrices M1 and M2 are properly set (see [1], equation (6.21)).Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the parameters for the complex Givens rotation
( CR-CI*I SR-SI*I ) ( XR+XI*I ) ( Z )
( ) * ( ) = ( ),
( -SR-SI*I CR+CI*I ) ( YR+YI*I ) ( 0 )
where CR, CI, SR, SI, XR, XI, YR, YI are real numbers and I is the
imaginary unit, I = SQRT(-1). Z is a non-negative real number.
Specification
SUBROUTINE SG03BY( XR, XI, YR, YI, CR, CI, SR, SI, Z )
C .. Scalar Arguments ..
DOUBLE PRECISION CI, CR, SI, SR, XI, XR, YI, YR, Z
Arguments
Input/Output Parameters
XR, XI, (input) DOUBLE PRECISION
YR, YI (input) DOUBLE PRECISION
The given real scalars XR, XI, YR, YI.
CR, CI, (output) DOUBLE PRECISION
SR, SI, (output) DOUBLE PRECISION
Z (output) DOUBLE PRECISION
The computed real scalars CR, CI, SR, SI, Z, defining the
complex Givens rotation and Z.
Numerical Aspects
The subroutine avoids unnecessary overflow.Further Comments
In the interest of speed, this routine does not check the input for errors.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the Cholesky factor U of the matrix X,
H
X = op(U) * op(U),
which is the solution of either the generalized c-stable
continuous-time Lyapunov equation
H H
op(A) * X * op(E) + op(E) * X * op(A)
2 H
= - SCALE * op(B) * op(B), (1)
or the generalized d-stable discrete-time Lyapunov equation
H H
op(A) * X * op(A) - op(E) * X * op(E)
2 H
= - SCALE * op(B) * op(B), (2)
without first finding X and without the need to form the matrix
op(B)**H * op(B).
op(K) is either K or K**H for K = A, B, E, U. A and E are N-by-N
matrices, op(B) is an M-by-N matrix. The resulting matrix U is an
N-by-N upper triangular matrix with non-negative entries on its
main diagonal. SCALE is an output scale factor set to avoid
overflow in U.
In the continuous-time case (1) the pencil A - lambda * E must be
c-stable (that is, all eigenvalues must have negative real parts).
In the discrete-time case (2) the pencil A - lambda * E must be
d-stable (that is, the moduli of all eigenvalues must be smaller
than one).
Specification
SUBROUTINE SG03BZ( DICO, FACT, TRANS, N, M, A, LDA, E, LDE, Q,
$ LDQ, Z, LDZ, B, LDB, SCALE, ALPHA, BETA, DWORK,
$ ZWORK, LZWORK, INFO )
C .. Scalar Arguments ..
DOUBLE PRECISION SCALE
INTEGER INFO, LDA, LDB, LDE, LDQ, LDZ, LZWORK, M, N
CHARACTER DICO, FACT, TRANS
C .. Array Arguments ..
COMPLEX*16 A(LDA,*), ALPHA(*), B(LDB,*), BETA(*), E(LDE,*),
$ Q(LDQ,*), Z(LDZ,*), ZWORK(*)
DOUBLE PRECISION DWORK(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies which type of the equation is considered:
= 'C': Continuous-time equation (1);
= 'D': Discrete-time equation (2).
FACT CHARACTER*1
Specifies whether the generalized (complex) Schur
factorization of the pencil A - lambda * E is supplied on
entry or not:
= 'N': Factorization is not supplied;
= 'F': Factorization is supplied.
TRANS CHARACTER*1
Specifies whether the conjugate transposed equation is to
be solved or not:
= 'N': op(A) = A, op(E) = E;
= 'C': op(A) = A**H, op(E) = E**H.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
M (input) INTEGER
The number of rows in the matrix op(B). M >= 0.
If M = 0, A and E are unchanged on exit, and Q, Z, ALPHA
and BETA are not set.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, if FACT = 'F', then the leading N-by-N upper
triangular part of this array must contain the generalized
Schur factor A_s of the matrix A (see definition (3) in
section METHOD). A_s must be an upper triangular matrix.
The elements below the upper triangular part of the array
A are used as workspace.
If FACT = 'N', then the leading N-by-N part of this array
must contain the matrix A.
On exit, if FACT = 'N', the leading N-by-N upper
triangular part of this array contains the generalized
Schur factor A_s of the matrix A. (A_s is an upper
triangular matrix.) If FACT = 'F', the leading N-by-N
upper triangular part of this array is unchanged.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
E (input/output) COMPLEX*16 array, dimension (LDE,N)
On entry, if FACT = 'F', then the leading N-by-N upper
triangular part of this array must contain the generalized
Schur factor E_s of the matrix E (see definition (4) in
section METHOD). E_s must be an upper triangular matrix.
The elements below the upper triangular part of the array
E are used as workspace.
If FACT = 'N', then the leading N-by-N part of this array
must contain the coefficient matrix E of the equation.
On exit, if FACT = 'N', the leading N-by-N upper
triangular part of this array contains the generalized
Schur factor E_s of the matrix E. (E_s is an upper
triangular matrix.) If FACT = 'F', the leading N-by-N
upper triangular part of this array is unchanged.
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,N).
Q (input/output) COMPLEX*16 array, dimension (LDQ,N)
On entry, if FACT = 'F', then the leading N-by-N part of
this array must contain the unitary matrix Q from the
generalized Schur factorization (see definitions (3) and
(4) in section METHOD), or an identity matrix (if the
original equation has upper triangular matrices A and E).
If FACT = 'N', Q need not be set on entry.
On exit, if FACT = 'N', the leading N-by-N part of this
array contains the unitary matrix Q from the generalized
Schur factorization. If FACT = 'F', this array is
unchanged.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= MAX(1,N).
Z (input/output) COMPLEX*16 array, dimension (LDZ,N)
On entry, if FACT = 'F', then the leading N-by-N part of
this array must contain the unitary matrix Z from the
generalized Schur factorization (see definitions (3) and
(4) in section METHOD), or an identity matrix (if the
original equation has upper triangular matrices A and E).
If FACT = 'N', Z need not be set on entry.
On exit, if FACT = 'N', the leading N-by-N part of this
array contains the unitary matrix Z from the generalized
Schur factorization. If FACT = 'F', this array is
unchanged.
LDZ INTEGER
The leading dimension of the array Z. LDZ >= MAX(1,N).
B (input/output) COMPLEX*16 array, dimension (LDB,N1)
On entry, if TRANS = 'C', the leading N-by-M part of this
array must contain the matrix B and N1 >= MAX(M,N).
If TRANS = 'N', the leading M-by-N part of this array
must contain the matrix B and N1 >= N.
On exit, if INFO = 0, the leading N-by-N part of this
array contains the Cholesky factor U of the solution
matrix X of the problem, X = op(U)**H * op(U).
If M = 0 and N > 0, then U is set to zero.
LDB INTEGER
The leading dimension of the array B.
If TRANS = 'C', LDB >= MAX(1,N).
If TRANS = 'N', LDB >= MAX(1,M,N).
SCALE (output) DOUBLE PRECISION
The scale factor set to avoid overflow in U.
0 < SCALE <= 1.
ALPHA (output) COMPLEX*16 arrays, dimension (N)
BETA If INFO = 0, 5, 6, or 7, then ALPHA(j)/BETA(j),
j = 1, ... , N, are the eigenvalues of the matrix pencil
A - lambda * E (the diagonals of the complex Schur form).
All BETA(j) are non-negative real numbers.
ALPHA will be always less than and usually comparable with
norm(A) in magnitude, and BETA always less than and
usually comparable with norm(B).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK), where
LDWORK = 0, if MIN(M,N) = 0 or
FACT = 'F' and N <= 1; else,
LDWORK = N-1, if FACT = 'F' and DICO = 'C';
LDWORK = MAX(N-1,10), if FACT = 'F' and DICO = 'D';
LDWORK = 8*N, if FACT = 'N'.
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) returns the optimal value
of LZWORK.
On exit, if INFO = -21, ZWORK(1) returns the minimum value
of LZWORK.
LZWORK INTEGER
The dimension of the array ZWORK.
LZWORK >= MAX(1,3*N-3,2*N).
For good performance, LZWORK should be larger.
If LZWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the ZWORK
array, returns this value as the first entry of the ZWORK
array, and no error message related to LZWORK is issued by
XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 4: FACT = 'N' and the pencil A - lambda * E cannot be
reduced to generalized Schur form: LAPACK routine
ZGGES has failed to converge;
= 5: DICO = 'C' and the pencil A - lambda * E is not
c-stable;
= 6: DICO = 'D' and the pencil A - lambda * E is not
d-stable;
= 7: the LAPACK routine ZSTEIN utilized to factorize M3
failed to converge in the discrete-time case (see
section METHOD for SLICOT Library routine SG03BS).
This error is unlikely to occur.
Method
An extension [2] of Hammarling's method [1] to generalized
Lyapunov equations is utilized to solve (1) or (2).
First the pencil A - lambda * E is reduced to complex generalized
Schur form A_s - lambda * E_s by means of unitary transformations
(QZ-algorithm):
A_s = Q**H * A * Z (upper triangular), (3)
E_s = Q**H * E * Z (upper triangular). (4)
If the pencil A - lambda * E has already been factorized prior to
calling the routine, however, then the factors A_s, E_s, Q and Z
may be supplied and the initial factorization omitted.
Depending on the parameters TRANS and M, the N-by-N upper
triangular matrix B_s is defined as follows. In any case Q_B is
an M-by-M unitary matrix, which need not be accumulated.
1. If TRANS = 'N' and M < N, B_s is the upper triangular matrix
from the QR-factorization
( Q_B O ) ( B * Z )
( ) * B_s = ( ),
( O I ) ( O )
where the O's are zero matrices of proper size and I is the
identity matrix of order N-M.
2. If TRANS = 'N' and M >= N, B_s is the upper triangular matrix
from the (rectangular) QR-factorization
( B_s )
Q_B * ( ) = B * Z,
( O )
where O is the (M-N)-by-N zero matrix.
3. If TRANS = 'C' and M < N, B_s is the upper triangular matrix
from the RQ-factorization
( Q_B O )
(B_s O ) * ( ) = ( Q**H * B O ).
( O I )
4. If TRANS = 'C' and M >= N, B_s is the upper triangular matrix
from the (rectangular) RQ-factorization
( B_s O ) * Q_B = Q**H * B,
where O is the N-by-(M-N) zero matrix.
Assuming SCALE = 1, the transformation of A, E and B described
above leads to the reduced continuous-time equation
H H
op(A_s) op(U_s) op(U_s) op(E_s)
H H
+ op(E_s) op(U_s) op(U_s) op(A_s)
H
= - op(B_s) op(B_s) (5)
or to the reduced discrete-time equation
H H
op(A_s) op(U_s) op(U_s) op(A_s)
H H
- op(E_s) op(U_s) op(U_s) op(E_s)
H
= - op(B_s) op(B_s). (6)
For brevity we restrict ourself to equation (5) and the case
TRANS = 'N'. The other three cases can be treated in a similar
fashion.
We use the following partitioning for the matrices A_s, E_s, B_s,
and U_s
( A11 A12 ) ( E11 E12 )
A_s = ( ), E_s = ( ),
( 0 A22 ) ( 0 E22 )
( B11 B12 ) ( U11 U12 )
B_s = ( ), U_s = ( ). (7)
( 0 B22 ) ( 0 U22 )
The size of the (1,1)-blocks is 1-by-1.
We compute U11, U12**H, and U22 in three steps.
Step I:
From (5) and (7) we get the 1-by-1 equation
H H H H
A11 * U11 * U11 * E11 + E11 * U11 * U11 * A11
H
= - B11 * B11.
For brevity, details are omitted here. See [2]. The technique
for computing U11 is similar to those applied to standard
Lyapunov equations in Hammarling's algorithm ([1], section 5).
Furthermore, the auxiliary scalars M1 and M2 defined as follows
M1 = A11 / E11 ,
M2 = B11 / E11 / U11 ,
are computed in a numerically reliable way.
Step II:
The generalized Sylvester equation
H H H H
A22 * U12 + E22 * U12 * M1 =
H H H H H
- B12 * M2 - A12 * U11 - E12 * U11 * M1
is solved for U12**H, as a linear system of order N-1.
Step III:
It can be shown that
H H H H
A22 * U22 * U22 * E22 + E22 * U22 * U22 * A22 =
H H
- B22 * B22 - y * y (8)
holds, where y is defined as
H H H H H
y = B12 - ( E12 * U11 + E22 * U12 ) * M2 .
If B22_tilde is the square triangular matrix arising from the
(rectangular) QR-factorization
( B22_tilde ) ( B22 )
Q_B_tilde * ( ) = ( ),
( O ) ( y**H )
where Q_B_tilde is a unitary matrix of order N, then
H H H
- B22 * B22 - y * y = - B22_tilde * B22_tilde.
Replacing the right hand side in (8) by the term
- B22_tilde**H * B22_tilde leads to a reduced generalized
Lyapunov equation like (5), but of dimension N-1.
The recursive application of the steps I to III yields the
solution U_s of the equation (5).
It remains to compute the solution matrix U of the original
problem (1) or (2) from the matrix U_s. To this end we transform
the solution back (with respect to the transformation that led
from (1) to (5) (from (2) to (6)) and apply the QR-factorization
(RQ-factorization). The upper triangular solution matrix U is
obtained by
Q_U * U = U_s * Q**H (if TRANS = 'N'),
or
U * Q_U = Z * U_s (if TRANS = 'C'),
where Q_U is an N-by-N unitary matrix. Again, the unitary matrix
Q_U need not be accumulated.
References
[1] Hammarling, S.J.
Numerical solution of the stable, non-negative definite
Lyapunov equation.
IMA J. Num. Anal., 2, pp. 303-323, 1982.
[2] Penzl, T.
Numerical solution of generalized Lyapunov equations.
Advances in Comp. Math., vol. 8, pp. 33-48, 1998.
Numerical Aspects
The number of flops required by the routine is given by the
following table. Note that we count a single floating point
arithmetic operation as one flop.
| FACT = 'F' FACT = 'N'
---------+--------------------------------------------------
M <= N | (13*N**3+6*M*N**2 (211*N**3+6*M*N**2
| +6*M**2*N-2*M**3)/3 +6*M**2*N-2*M**3)/3
|
M > N | (11*N**3+12*M*N**2)/3 (209*N**3+12*M*N**2)/3
Further Comments
The Lyapunov equation may be very ill-conditioned. In particular, if DICO = 'D' and the pencil A - lambda * E has a pair of almost reciprocal eigenvalues, or DICO = 'C' and the pencil has an almost degenerate pair of eigenvalues, then the Lyapunov equation will be ill-conditioned. Perturbed values were used to solve the equation. A condition estimate can be obtained from the routine SG03AD.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TB01ID.html 0000664 0000000 0000000 00000025327 14560147231 0020346 0 ustar 00root root 0000000 0000000
Purpose
To reduce the 1-norm of a system matrix
S = ( A B )
( C 0 )
corresponding to the triple (A,B,C), by balancing. This involves
a diagonal similarity transformation inv(D)*A*D applied
iteratively to A to make the rows and columns of
-1
diag(D,I) * S * diag(D,I)
as close in norm as possible.
The balancing can be performed optionally on the following
particular system matrices
S = A, S = ( A B ) or S = ( A )
( C )
Specification
SUBROUTINE TB01ID( JOB, N, M, P, MAXRED, A, LDA, B, LDB, C, LDC,
$ SCALE, INFO )
C .. Scalar Arguments ..
CHARACTER JOB
INTEGER INFO, LDA, LDB, LDC, M, N, P
DOUBLE PRECISION MAXRED
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ SCALE( * )
Arguments
Mode Parameters
JOB CHARACTER*1
Indicates which matrices are involved in balancing, as
follows:
= 'A': All matrices are involved in balancing;
= 'B': B and A matrices are involved in balancing;
= 'C': C and A matrices are involved in balancing;
= 'N': B and C matrices are not involved in balancing.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A, the number of rows of matrix B
and the number of columns of matrix C.
N represents the dimension of the state vector. N >= 0.
M (input) INTEGER.
The number of columns of matrix B.
M represents the dimension of input vector. M >= 0.
P (input) INTEGER.
The number of rows of matrix C.
P represents the dimension of output vector. P >= 0.
MAXRED (input/output) DOUBLE PRECISION
On entry, the maximum allowed reduction in the 1-norm of
S (in an iteration) if zero rows or columns are
encountered.
If MAXRED > 0.0, MAXRED must be larger than one (to enable
the norm reduction).
If MAXRED <= 0.0, then the value 10.0 for MAXRED is
used.
On exit, if the 1-norm of the given matrix S is non-zero,
the ratio between the 1-norm of the given matrix and the
1-norm of the balanced matrix.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the system state matrix A.
On exit, the leading N-by-N part of this array contains
the balanced matrix inv(D)*A*D.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, if M > 0, the leading N-by-M part of this array
must contain the system input matrix B.
On exit, if M > 0, the leading N-by-M part of this array
contains the balanced matrix inv(D)*B.
The array B is not referenced if M = 0.
LDB INTEGER
The leading dimension of the array B.
LDB >= MAX(1,N) if M > 0.
LDB >= 1 if M = 0.
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, if P > 0, the leading P-by-N part of this array
must contain the system output matrix C.
On exit, if P > 0, the leading P-by-N part of this array
contains the balanced matrix C*D.
The array C is not referenced if P = 0.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,P).
SCALE (output) DOUBLE PRECISION array, dimension (N)
The scaling factors applied to S. If D(j) is the scaling
factor applied to row and column j, then SCALE(j) = D(j),
for j = 1,...,N.
Error Indicator
INFO INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Balancing consists of applying a diagonal similarity
transformation
-1
diag(D,I) * S * diag(D,I)
to make the 1-norms of each row of the first N rows of S and its
corresponding column nearly equal.
Information about the diagonal matrix D is returned in the vector
SCALE.
References
[1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
* TB01ID EXAMPLE PROGRAM TEXT.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX )
* .. Local Scalars ..
CHARACTER*1 JOB
INTEGER I, INFO, J, M, N, P
DOUBLE PRECISION MAXRED
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ SCALE(NMAX)
* .. External Subroutines ..
EXTERNAL TB01ID, UD01MD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, JOB, MAXRED
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LT.0 .OR. P.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Balance system matrix S.
CALL TB01ID( JOB, N, M, P, MAXRED, A, LDA, B, LDB, C,
$ LDC, SCALE, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
CALL UD01MD( N, N, 5, NOUT, A, LDA,
$ 'The balanced matrix A', INFO )
IF ( M.GT.0 )
$ CALL UD01MD( N, M, 5, NOUT, B, LDB,
$ 'The balanced matrix B', INFO )
IF ( P.GT.0 )
$ CALL UD01MD( P, N, 5, NOUT, C, LDC,
$ 'The balanced matrix C', INFO )
CALL UD01MD( 1, N, 5, NOUT, SCALE, 1,
$ 'The scaling vector SCALE', INFO )
WRITE ( NOUT, FMT = 99994 ) MAXRED
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TB01ID EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TB01ID = ',I2)
99994 FORMAT (/' MAXRED is ',E13.4)
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' M is out of range.',/' M = ',I5)
99991 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
TB01ID EXAMPLE PROGRAM DATA
5 2 5 A 0.0
0.0 1.0000e+000 0.0 0.0 0.0
-1.5800e+006 -1.2570e+003 0.0 0.0 0.0
3.5410e+014 0.0 -1.4340e+003 0.0 -5.3300e+011
0.0 0.0 0.0 0.0 1.0000e+000
0.0 0.0 0.0 -1.8630e+004 -1.4820e+000
0.0 0.0
1.1030e+002 0.0
0.0 0.0
0.0 0.0
0.0 8.3330e-003
1.0000e+000 0.0 0.0 0.0 0.0
0.0 0.0 1.0000e+000 0.0 0.0
0.0 0.0 0.0 1.0000e+000 0.0
6.6640e-001 0.0 -6.2000e-013 0.0 0.0
0.0 0.0 -1.0000e-003 1.8960e+006 1.5080e+002
Program Results
TB01ID EXAMPLE PROGRAM RESULTS
The balanced matrix A ( 5X 5)
1 2 3 4 5
1 0.0000000D+00 0.1000000D+05 0.0000000D+00 0.0000000D+00 0.0000000D+00
2 -0.1580000D+03 -0.1257000D+04 0.0000000D+00 0.0000000D+00 0.0000000D+00
3 0.3541000D+05 0.0000000D+00 -0.1434000D+04 0.0000000D+00 -0.5330000D+03
4 0.0000000D+00 0.0000000D+00 0.0000000D+00 0.0000000D+00 0.1000000D+03
5 0.0000000D+00 0.0000000D+00 0.0000000D+00 -0.1863000D+03 -0.1482000D+01
The balanced matrix B ( 5X 2)
1 2
1 0.0000000D+00 0.0000000D+00
2 0.1103000D+04 0.0000000D+00
3 0.0000000D+00 0.0000000D+00
4 0.0000000D+00 0.0000000D+00
5 0.0000000D+00 0.8333000D+02
The balanced matrix C ( 5X 5)
1 2 3 4 5
1 0.1000000D-04 0.0000000D+00 0.0000000D+00 0.0000000D+00 0.0000000D+00
2 0.0000000D+00 0.0000000D+00 0.1000000D+06 0.0000000D+00 0.0000000D+00
3 0.0000000D+00 0.0000000D+00 0.0000000D+00 0.1000000D-05 0.0000000D+00
4 0.6664000D-05 0.0000000D+00 -0.6200000D-07 0.0000000D+00 0.0000000D+00
5 0.0000000D+00 0.0000000D+00 -0.1000000D+03 0.1896000D+01 0.1508000D-01
The scaling vector SCALE ( 1X 5)
1 2 3 4 5
1 0.1000000D-04 0.1000000D+00 0.1000000D+06 0.1000000D-05 0.1000000D-03
MAXRED is 0.3488E+10
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TB01IZ.html 0000664 0000000 0000000 00000030576 14560147231 0020376 0 ustar 00root root 0000000 0000000
Purpose
To reduce the 1-norm of a system matrix
S = ( A B )
( C 0 )
corresponding to the triple (A,B,C), by balancing. This involves
a diagonal similarity transformation inv(D)*A*D applied
iteratively to A to make the rows and columns of
-1
diag(D,I) * S * diag(D,I)
as close in norm as possible.
The balancing can be performed optionally on the following
particular system matrices
S = A, S = ( A B ) or S = ( A )
( C )
Specification
SUBROUTINE TB01IZ( JOB, N, M, P, MAXRED, A, LDA, B, LDB, C, LDC,
$ SCALE, INFO )
C .. Scalar Arguments ..
CHARACTER JOB
INTEGER INFO, LDA, LDB, LDC, M, N, P
DOUBLE PRECISION MAXRED
C .. Array Arguments ..
COMPLEX*16 A( LDA, * ), B( LDB, * ), C( LDC, * )
DOUBLE PRECISION SCALE( * )
Arguments
Mode Parameters
JOB CHARACTER*1
Indicates which matrices are involved in balancing, as
follows:
= 'A': All matrices are involved in balancing;
= 'B': B and A matrices are involved in balancing;
= 'C': C and A matrices are involved in balancing;
= 'N': B and C matrices are not involved in balancing.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A, the number of rows of matrix B
and the number of columns of matrix C.
N represents the dimension of the state vector. N >= 0.
M (input) INTEGER.
The number of columns of matrix B.
M represents the dimension of input vector. M >= 0.
P (input) INTEGER.
The number of rows of matrix C.
P represents the dimension of output vector. P >= 0.
MAXRED (input/output) DOUBLE PRECISION
On entry, the maximum allowed reduction in the 1-norm of
S (in an iteration) if zero rows or columns are
encountered.
If MAXRED > 0.0, MAXRED must be larger than one (to enable
the norm reduction).
If MAXRED <= 0.0, then the value 10.0 for MAXRED is
used.
On exit, if the 1-norm of the given matrix S is non-zero,
the ratio between the 1-norm of the given matrix and the
1-norm of the balanced matrix.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the system state matrix A.
On exit, the leading N-by-N part of this array contains
the balanced matrix inv(D)*A*D.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input/output) COMPLEX*16 array, dimension (LDB,M)
On entry, if M > 0, the leading N-by-M part of this array
must contain the system input matrix B.
On exit, if M > 0, the leading N-by-M part of this array
contains the balanced matrix inv(D)*B.
The array B is not referenced if M = 0.
LDB INTEGER
The leading dimension of the array B.
LDB >= MAX(1,N) if M > 0.
LDB >= 1 if M = 0.
C (input/output) COMPLEX*16 array, dimension (LDC,N)
On entry, if P > 0, the leading P-by-N part of this array
must contain the system output matrix C.
On exit, if P > 0, the leading P-by-N part of this array
contains the balanced matrix C*D.
The array C is not referenced if P = 0.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,P).
SCALE (output) DOUBLE PRECISION array, dimension (N)
The scaling factors applied to S. If D(j) is the scaling
factor applied to row and column j, then SCALE(j) = D(j),
for j = 1,...,N.
Error Indicator
INFO INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Balancing consists of applying a diagonal similarity
transformation
-1
diag(D,I) * S * diag(D,I)
to make the 1-norms of each row of the first N rows of S and its
corresponding column nearly equal.
Information about the diagonal matrix D is returned in the vector
SCALE.
References
[1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
* TB01IZ EXAMPLE PROGRAM TEXT.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX )
* .. Local Scalars ..
CHARACTER*1 JOB
INTEGER I, INFO, J, M, N, P
DOUBLE PRECISION MAXRED
* .. Local Arrays ..
COMPLEX*16 A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX)
DOUBLE PRECISION SCALE(NMAX)
* .. External Subroutines ..
EXTERNAL TB01IZ, UD01MD, UD01MZ
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, JOB, MAXRED
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LT.0 .OR. P.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Balance system matrix S.
CALL TB01IZ( JOB, N, M, P, MAXRED, A, LDA, B, LDB, C,
$ LDC, SCALE, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
CALL UD01MZ( N, N, 3, NOUT, A, LDA,
$ 'The balanced matrix A', INFO )
IF ( M.GT.0 )
$ CALL UD01MZ( N, M, 3, NOUT, B, LDB,
$ 'The balanced matrix B', INFO )
IF ( P.GT.0 )
$ CALL UD01MZ( P, N, 3, NOUT, C, LDC,
$ 'The balanced matrix C', INFO )
CALL UD01MD( 1, N, 5, NOUT, SCALE, 1,
$ 'The scaling vector SCALE', INFO )
WRITE ( NOUT, FMT = 99994 ) MAXRED
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TB01IZ EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TB01IZ = ',I2)
99994 FORMAT (/' MAXRED is ',E13.4)
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' M is out of range.',/' M = ',I5)
99991 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
TB01IZ EXAMPLE PROGRAM DATA
5 2 5 A 0.0
(0.0,0.0) (1.0000e+000,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0)
(-1.5800e+006,0.0) (-1.2570e+003,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0)
(3.5410e+014,0.0) (0.0,0.0) (-1.4340e+003,0.0) (0.0,0.0) (-5.3300e+011,0.0)
(0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (1.0000e+000,0.0)
(0.0,0.0) (0.0,0.0) (0.0,0.0) (-1.8630e+004,0.0) (-1.4820e+000,0.0)
(0.0,0.0) (0.0,0.0)
(1.1030e+002,0.0) (0.0,0.0)
(0.0,0.0) (0.0,0.0)
(0.0,0.0) (0.0,0.0)
(0.0,0.0) (8.3330e-003,0.0)
(1.0000e+000,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0)
(0.0,0.0) (0.0,0.0) (1.0000e+000,0.0) (0.0,0.0) (0.0,0.0)
(0.0,0.0) (0.0,0.0) (0.0,0.0) (1.0000e+000,0.0) (0.0,0.0)
(6.6640e-001,0.0) (0.0,0.0) (-6.2000e-013,0.0) (0.0,0.0) (0.0,0.0)
(0.0,0.0) (0.0,0.0) (-1.0000e-003,0.0) (1.8960e+006,0.0) (1.5080e+002,0.0)
Program Results
TB01IZ EXAMPLE PROGRAM RESULTS
The balanced matrix A ( 5X 5)
1 2 3
1 0.0000000D+00 +0.0000000D+00i 0.1000000D+05 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i
2 -0.1580000D+03 +0.0000000D+00i -0.1257000D+04 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i
3 0.3541000D+05 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i -0.1434000D+04 +0.0000000D+00i
4 0.0000000D+00 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i
5 0.0000000D+00 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i
4 5
1 0.0000000D+00 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i
2 0.0000000D+00 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i
3 0.0000000D+00 +0.0000000D+00i -0.5330000D+03 +0.0000000D+00i
4 0.0000000D+00 +0.0000000D+00i 0.1000000D+03 +0.0000000D+00i
5 -0.1863000D+03 +0.0000000D+00i -0.1482000D+01 +0.0000000D+00i
The balanced matrix B ( 5X 2)
1 2
1 0.0000000D+00 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i
2 0.1103000D+04 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i
3 0.0000000D+00 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i
4 0.0000000D+00 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i
5 0.0000000D+00 +0.0000000D+00i 0.8333000D+02 +0.0000000D+00i
The balanced matrix C ( 5X 5)
1 2 3
1 0.1000000D-04 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i
2 0.0000000D+00 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i 0.1000000D+06 +0.0000000D+00i
3 0.0000000D+00 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i
4 0.6664000D-05 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i -0.6200000D-07 +0.0000000D+00i
5 0.0000000D+00 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i -0.1000000D+03 +0.0000000D+00i
4 5
1 0.0000000D+00 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i
2 0.0000000D+00 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i
3 0.1000000D-05 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i
4 0.0000000D+00 +0.0000000D+00i 0.0000000D+00 +0.0000000D+00i
5 0.1896000D+01 +0.0000000D+00i 0.1508000D-01 +0.0000000D+00i
The scaling vector SCALE ( 1X 5)
1 2 3 4 5
1 0.1000000D-04 0.1000000D+00 0.1000000D+06 0.1000000D-05 0.1000000D-03
MAXRED is 0.3488E+10
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TB01KD.html 0000664 0000000 0000000 00000032731 14560147231 0020345 0 ustar 00root root 0000000 0000000
Purpose
To compute an additive spectral decomposition of the transfer- function matrix of the system (A,B,C) by reducing the system state-matrix A to a block-diagonal form. The system matrices are transformed as A <-- inv(U)*A*U, B <--inv(U)*B and C <-- C*U. The leading diagonal block of the resulting A has eigenvalues in a suitably defined domain of interest.Specification
SUBROUTINE TB01KD( DICO, STDOM, JOBA, N, M, P, ALPHA, A, LDA, B,
$ LDB, C, LDC, NDIM, U, LDU, WR, WI, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, JOBA, STDOM
INTEGER INFO, LDA, LDB, LDC, LDU, LDWORK, M, N, NDIM, P
DOUBLE PRECISION ALPHA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), U(LDU,*),
$ WI(*), WR(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
STDOM CHARACTER*1
Specifies whether the domain of interest is of stability
type (left part of complex plane or inside of a circle)
or of instability type (right part of complex plane or
outside of a circle) as follows:
= 'S': stability type domain;
= 'U': instability type domain.
JOBA CHARACTER*1
Specifies the shape of the state dynamics matrix on entry
as follows:
= 'S': A is in an upper real Schur form;
= 'G': A is a general square dense matrix.
Input/Output Parameters
N (input) INTEGER
The order of the state-space representation,
i.e. the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs, or of columns of B. M >= 0.
P (input) INTEGER
The number of system outputs, or of rows of C. P >= 0.
ALPHA (input) DOUBLE PRECISION.
Specifies the boundary of the domain of interest for the
eigenvalues of A. For a continuous-time system
(DICO = 'C'), ALPHA is the boundary value for the real
parts of eigenvalues, while for a discrete-time system
(DICO = 'D'), ALPHA >= 0 represents the boundary value for
the moduli of eigenvalues.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the unreduced state dynamics matrix A.
If JOBA = 'S' then A must be a matrix in real Schur form.
On exit, the leading N-by-N part of this array contains a
block diagonal matrix inv(U) * A * U with two diagonal
blocks in real Schur form with the elements below the
first subdiagonal set to zero.
The leading NDIM-by-NDIM block of A has eigenvalues in the
domain of interest and the trailing (N-NDIM)-by-(N-NDIM)
block has eigenvalues outside the domain of interest.
The domain of interest for lambda(A), the eigenvalues
of A, is defined by the parameters ALPHA, DICO and STDOM
as follows:
For a continuous-time system (DICO = 'C'):
Real(lambda(A)) < ALPHA if STDOM = 'S';
Real(lambda(A)) > ALPHA if STDOM = 'U';
For a discrete-time system (DICO = 'D'):
Abs(lambda(A)) < ALPHA if STDOM = 'S';
Abs(lambda(A)) > ALPHA if STDOM = 'U'.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input matrix B.
On exit, the leading N-by-M part of this array contains
the transformed input matrix inv(U) * B.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed output matrix C * U.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
NDIM (output) INTEGER
The number of eigenvalues of A lying inside the domain of
interest for eigenvalues.
U (output) DOUBLE PRECISION array, dimension (LDU,N)
The leading N-by-N part of this array contains the
transformation matrix used to reduce A to the block-
diagonal form. The first NDIM columns of U span the
invariant subspace of A corresponding to the eigenvalues
of its leading diagonal block. The last N-NDIM columns
of U span the reducing subspace of A corresponding to
the eigenvalues of the trailing diagonal block of A.
LDU INTEGER
The leading dimension of array U. LDU >= max(1,N).
WR, WI (output) DOUBLE PRECISION arrays, dimension (N)
WR and WI contain the real and imaginary parts,
respectively, of the computed eigenvalues of A. The
eigenvalues will be in the same order that they appear on
the diagonal of the output real Schur form of A. Complex
conjugate pairs of eigenvalues will appear consecutively
with the eigenvalue having the positive imaginary part
first.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The dimension of working array DWORK.
LDWORK >= MAX(1,N) if JOBA = 'S';
LDWORK >= MAX(1,3*N) if JOBA = 'G'.
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the QR algorithm failed to compute all the
eigenvalues of A;
= 2: a failure occured during the ordering of the real
Schur form of A;
= 3: the separation of the two diagonal blocks failed
because of very close eigenvalues.
Method
A similarity transformation U is determined that reduces the
system state-matrix A to a block-diagonal form (with two diagonal
blocks), so that the leading diagonal block of the resulting A has
eigenvalues in a specified domain of the complex plane. The
determined transformation is applied to the system (A,B,C) as
A <-- inv(U)*A*U, B <-- inv(U)*B and C <-- C*U.
References
[1] Safonov, M.G., Jonckheere, E.A., Verma, M., Limebeer, D.J.N.
Synthesis of positive real multivariable feedback systems.
Int. J. Control, pp. 817-842, 1987.
Numerical Aspects
3 The algorithm requires about 14N floating point operations.Further Comments
NoneExample
Program Text
* TB01KD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDU
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDU = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 3*NMAX )
* .. Local Scalars ..
CHARACTER*1 DICO, JOBA, STDOM
INTEGER I, INFO, J, M, N, NDIM, P
DOUBLE PRECISION ALPHA
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ DWORK(LDWORK), U(LDU,NMAX), WI(NMAX), WR(NMAX)
* .. External Subroutines ..
EXTERNAL TB01KD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, ALPHA, DICO, STDOM, JOBA
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find the transformed ssr for (A,B,C).
CALL TB01KD( DICO, STDOM, JOBA, N, M, P, ALPHA, A, LDA,
$ B, LDB, C, LDC, NDIM, U, LDU, WR, WI, DWORK,
$ LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99987 ) NDIM
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99994 ) WR(I), WI(I)
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
60 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 70 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( U(I,J), J = 1,N )
70 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TB01KD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TB01KD = ',I2)
99997 FORMAT (/' The eigenvalues of state dynamics matrix A are ')
99996 FORMAT (/' The transformed state dynamics matrix inv(U)*A*U is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT ( ' (',F8.4,', ',F8.4,' )')
99993 FORMAT (/' The transformed input/state matrix inv(U)*B is ')
99992 FORMAT (/' The transformed state/output matrix C*U is ')
99991 FORMAT (/' The similarity transformation matrix U is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99987 FORMAT (' The number of eigenvalues in the domain of interest =',
$ I5 )
END
Program Data
TB01KD EXAMPLE PROGRAM DATA (Continuous system)
5 2 3 -1.0 C U G
-0.04165 4.9200 -4.9200 0 0
-1.387944 -3.3300 0 0 0
0.5450 0 0 -0.5450 0
0 0 4.9200 -0.04165 4.9200
0 0 0 -1.387944 -3.3300
0 0
3.3300 0
0 0
0 0
0 3.3300
1 0 0 0 0
0 0 1 0 0
0 0 0 1 0
Program Results
TB01KD EXAMPLE PROGRAM RESULTS The number of eigenvalues in the domain of interest = 2 The eigenvalues of state dynamics matrix A are ( -0.7483, 2.9940 ) ( -0.7483, -2.9940 ) ( -1.6858, 2.0311 ) ( -1.6858, -2.0311 ) ( -1.8751, 0.0000 ) The transformed state dynamics matrix inv(U)*A*U is -0.7483 -8.6406 0.0000 0.0000 0.0000 1.0374 -0.7483 0.0000 0.0000 0.0000 0.0000 0.0000 -1.6858 5.5669 0.0000 0.0000 0.0000 -0.7411 -1.6858 0.0000 0.0000 0.0000 0.0000 0.0000 -1.8751 The transformed input/state matrix inv(U)*B is 2.0240 -2.0240 -1.1309 1.1309 -0.8621 -0.8621 2.1912 2.1912 -1.5555 1.5555 The transformed state/output matrix C*U is 0.6864 -0.0987 0.6580 0.2589 0.9650 -0.0471 0.6873 0.0000 0.0000 -0.5609 -0.6864 0.0987 0.6580 0.2589 -0.9650 The similarity transformation matrix U is 0.6864 -0.0987 0.6580 0.2589 0.9650 -0.1665 -0.5041 -0.2589 0.6580 -0.9205 -0.0471 0.6873 0.0000 0.0000 -0.5609 -0.6864 0.0987 0.6580 0.2589 -0.9650 0.1665 0.5041 -0.2589 0.6580 0.9205
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TB01KX.html 0000664 0000000 0000000 00000013574 14560147231 0020375 0 ustar 00root root 0000000 0000000
Purpose
To compute an additive spectral decomposition of the transfer-
function matrix of the system (A,B,C) by reducing the system
state-matrix A to a block-diagonal form. It is assumed that A is
in a real Schur form, and the leading diagonal block of order NDIM
has eigenvalues distinct from those of the trailing diagonal
block. The system matrices are transformed as
A <-- V*A*U, B <--V*B and C <-- C*U, where V = inv(U),
preserving the spectra of the two diagonal blocks.
Specification
SUBROUTINE TB01KX( N, M, P, NDIM, A, LDA, B, LDB, C, LDC, U, LDU,
$ V, LDV, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDU, LDV, M, N, NDIM, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), U(LDU,*), V(LDV,*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the state-space representation, i.e., the
order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs, or of columns of B. M >= 0.
P (input) INTEGER
The number of system outputs, or of rows of C. P >= 0.
NDIM (input) INTEGER
The dimension of the leading diagonal block of A having
eigenvalues distinct from those of the trailing diagonal
block. 0 <= NDIM <= N.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A in real Schur form.
On exit, the leading N-by-N part of this array contains a
block diagonal matrix inv(U) * A * U with two diagonal
blocks in real Schur form, with the elements below the
first subdiagonal set to zero. The leading block has
dimension NDIM-by-NDIM.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input matrix B.
On exit, the leading N-by-M part of this array contains
the transformed input matrix inv(U) * B.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed output matrix C * U.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,P).
U (input/output) DOUBLE PRECISION array, dimension (LDU,N)
On entry, the leading N-by-N part of this array must
contain an initial transformation matrix U.
On exit, the leading N-by-N part of this array contains
the transformation matrix used to reduce A to the block-
diagonal form. The first NDIM columns of U span the
invariant subspace of A corresponding to the eigenvalues
of its leading diagonal block. The last N-NDIM columns of
U span the reducing subspace of A corresponding to the
eigenvalues of the trailing diagonal block of A.
LDU INTEGER
The leading dimension of the array U. LDU >= max(1,N).
V (output) DOUBLE PRECISION array, dimension (LDV,N)
The leading N-by-N part of this array contains the
inverse of the transformation matrix U used to reduce A
to the block-diagonal form.
LDV INTEGER
The leading dimension of the array V. LDV >= max(1,N).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: the separation of the two diagonal blocks failed
because of very close eigenvalues.
Method
A similarity transformation U is determined that reduces the given
system state-matrix A to a block-diagonal form (with two diagonal
blocks), so that the eigenvalues of the leading diagonal block of
the resulting A are preserved. The determined transformation is
applied to the system (A,B,C) as
A <-- inv(U)*A*U, B <-- inv(U)*B and C <-- C*U.
References
[1] Safonov, M.G., Jonckheere, E.A., Verma, M., Limebeer, D.J.N.
Synthesis of positive real multivariable feedback systems.
Int. J. Control, pp. 817-842, 1987.
Numerical Aspects
3 The algorithm requires about N /2 + NDIM*(N-NDIM)*(2*N+M+P) floating point operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To reduce the system state matrix A to an ordered upper real Schur form by using an orthogonal similarity transformation A <-- U'*A*U and to apply the transformation to the matrices B and C: B <-- U'*B and C <-- C*U. The leading block of the resulting A has eigenvalues in a suitably defined domain of interest.Specification
SUBROUTINE TB01LD( DICO, STDOM, JOBA, N, M, P, ALPHA, A, LDA, B,
$ LDB, C, LDC, NDIM, U, LDU, WR, WI, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, JOBA, STDOM
INTEGER INFO, LDA, LDB, LDC, LDU, LDWORK, M, N, NDIM, P
DOUBLE PRECISION ALPHA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), U(LDU,*),
$ WI(*), WR(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
STDOM CHARACTER*1
Specifies whether the domain of interest is of stability
type (left part of complex plane or inside of a circle)
or of instability type (right part of complex plane or
outside of a circle) as follows:
= 'S': stability type domain;
= 'U': instability type domain.
JOBA CHARACTER*1
Specifies the shape of the state dynamics matrix on entry
as follows:
= 'S': A is in an upper real Schur form;
= 'G': A is a general square dense matrix.
Input/Output Parameters
N (input) INTEGER
The order of the state-space representation,
i.e. the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs, or of columns of B. M >= 0.
P (input) INTEGER
The number of system outputs, or of rows of C. P >= 0.
ALPHA (input) DOUBLE PRECISION.
Specifies the boundary of the domain of interest for the
eigenvalues of A. For a continuous-time system
(DICO = 'C'), ALPHA is the boundary value for the real
parts of eigenvalues, while for a discrete-time system
(DICO = 'D'), ALPHA >= 0 represents the boundary value
for the moduli of eigenvalues.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the unreduced state dynamics matrix A.
If JOBA = 'S' then A must be a matrix in real Schur form.
On exit, the leading N-by-N part of this array contains
the ordered real Schur matrix U' * A * U with the elements
below the first subdiagonal set to zero.
The leading NDIM-by-NDIM part of A has eigenvalues in the
domain of interest and the trailing (N-NDIM)-by-(N-NDIM)
part has eigenvalues outside the domain of interest.
The domain of interest for lambda(A), the eigenvalues
of A, is defined by the parameters ALPHA, DICO and STDOM
as follows:
For a continuous-time system (DICO = 'C'):
Real(lambda(A)) < ALPHA if STDOM = 'S';
Real(lambda(A)) > ALPHA if STDOM = 'U';
For a discrete-time system (DICO = 'D'):
Abs(lambda(A)) < ALPHA if STDOM = 'S';
Abs(lambda(A)) > ALPHA if STDOM = 'U'.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input matrix B.
On exit, the leading N-by-M part of this array contains
the transformed input matrix U' * B.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed output matrix C * U.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
NDIM (output) INTEGER
The number of eigenvalues of A lying inside the domain of
interest for eigenvalues.
U (output) DOUBLE PRECISION array, dimension (LDU,N)
The leading N-by-N part of this array contains the
orthogonal transformation matrix used to reduce A to the
real Schur form and/or to reorder the diagonal blocks of
real Schur form of A. The first NDIM columns of U form
an orthogonal basis for the invariant subspace of A
corresponding to the first NDIM eigenvalues.
LDU INTEGER
The leading dimension of array U. LDU >= max(1,N).
WR, WI (output) DOUBLE PRECISION arrays, dimension (N)
WR and WI contain the real and imaginary parts,
respectively, of the computed eigenvalues of A. The
eigenvalues will be in the same order that they appear on
the diagonal of the output real Schur form of A. Complex
conjugate pairs of eigenvalues will appear consecutively
with the eigenvalue having the positive imaginary part
first.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The dimension of working array DWORK.
LDWORK >= MAX(1,N) if JOBA = 'S';
LDWORK >= MAX(1,3*N) if JOBA = 'G'.
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the QR algorithm failed to compute all the
eigenvalues of A;
= 2: a failure occured during the ordering of the real
Schur form of A.
Method
Matrix A is reduced to an ordered upper real Schur form using an orthogonal similarity transformation A <-- U'*A*U. This transformation is determined so that the leading block of the resulting A has eigenvalues in a suitably defined domain of interest. Then, the transformation is applied to the matrices B and C: B <-- U'*B and C <-- C*U.Numerical Aspects
3 The algorithm requires about 14N floating point operations.Further Comments
NoneExample
Program Text
* TB01LD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDU
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDU = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 3*NMAX )
* .. Local Scalars ..
CHARACTER*1 DICO, JOBA, STDOM
INTEGER I, INFO, J, M, N, NDIM, P
DOUBLE PRECISION ALPHA
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ DWORK(LDWORK), U(LDU,NMAX), WI(NMAX), WR(NMAX)
* .. External Subroutines ..
EXTERNAL TB01LD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, ALPHA, DICO, STDOM, JOBA
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find the transformed ssr for (A,B,C).
CALL TB01LD( DICO, STDOM, JOBA, N, M, P, ALPHA,
$ A, LDA, B, LDB, C, LDC, NDIM, U, LDU,
$ WR, WI, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99987 ) NDIM
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99994 ) WR(I), WI(I)
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
60 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 70 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( U(I,J), J = 1,N )
70 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TB01LD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TB01LD = ',I2)
99997 FORMAT (/' The eigenvalues of state dynamics matrix A are ')
99996 FORMAT (/' The transformed state dynamics matrix U''*A*U is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT ( ' (',F8.4,', ',F8.4,' )')
99993 FORMAT (/' The transformed input/state matrix U''*B is ')
99992 FORMAT (/' The transformed state/output matrix C*U is ')
99991 FORMAT (/' The similarity transformation matrix U is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99987 FORMAT (/' The number of eigenvalues in the domain of interest =',
$ I5 )
END
Program Data
TB01LD EXAMPLE PROGRAM DATA (Continuous system)
5 2 3 -1.0 C U G
-0.04165 4.9200 -4.9200 0 0
-1.387944 -3.3300 0 0 0
0.5450 0 0 -0.5450 0
0 0 4.9200 -0.04165 4.9200
0 0 0 -1.387944 -3.3300
0 0
3.3300 0
0 0
0 0
0 3.3300
1 0 0 0 0
0 0 1 0 0
0 0 0 1 0
Program Results
TB01LD EXAMPLE PROGRAM RESULTS The number of eigenvalues in the domain of interest = 2 The eigenvalues of state dynamics matrix A are ( -0.7483, 2.9940 ) ( -0.7483, -2.9940 ) ( -1.6858, 2.0311 ) ( -1.6858, -2.0311 ) ( -1.8751, 0.0000 ) The transformed state dynamics matrix U'*A*U is -0.7483 -8.6406 0.0000 0.0000 1.1745 1.0374 -0.7483 0.0000 0.0000 -2.1164 0.0000 0.0000 -1.6858 5.5669 0.0000 0.0000 0.0000 -0.7411 -1.6858 0.0000 0.0000 0.0000 0.0000 0.0000 -1.8751 The transformed input/state matrix U'*B is -0.5543 0.5543 -1.6786 1.6786 -0.8621 -0.8621 2.1912 2.1912 -1.5555 1.5555 The transformed state/output matrix C*U is 0.6864 -0.0987 0.6580 0.2589 -0.1381 -0.0471 0.6873 0.0000 0.0000 -0.7249 -0.6864 0.0987 0.6580 0.2589 0.1381 The similarity transformation matrix U is 0.6864 -0.0987 0.6580 0.2589 -0.1381 -0.1665 -0.5041 -0.2589 0.6580 -0.4671 -0.0471 0.6873 0.0000 0.0000 -0.7249 -0.6864 0.0987 0.6580 0.2589 0.1381 0.1665 0.5041 -0.2589 0.6580 0.4671
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TB01MD.html 0000664 0000000 0000000 00000024426 14560147231 0020351 0 ustar 00root root 0000000 0000000
Purpose
To reduce the pair (B,A) to upper or lower controller Hessenberg form using (and optionally accumulating) unitary state-space transformations.Specification
SUBROUTINE TB01MD( JOBU, UPLO, N, M, A, LDA, B, LDB, U, LDU,
$ DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBU, UPLO
INTEGER INFO, LDA, LDB, LDU, M, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), U(LDU,*)
Arguments
Mode Parameters
JOBU CHARACTER*1
Indicates whether the user wishes to accumulate in a
matrix U the unitary state-space transformations for
reducing the system, as follows:
= 'N': Do not form U;
= 'I': U is initialized to the unit matrix and the
unitary transformation matrix U is returned;
= 'U': The given matrix U is updated by the unitary
transformations used in the reduction.
UPLO CHARACTER*1
Indicates whether the user wishes the pair (B,A) to be
reduced to upper or lower controller Hessenberg form as
follows:
= 'U': Upper controller Hessenberg form;
= 'L': Lower controller Hessenberg form.
Input/Output Parameters
N (input) INTEGER
The actual state dimension, i.e. the order of the
matrix A. N >= 0.
M (input) INTEGER
The actual input dimension, i.e. the number of columns of
the matrix B. M >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state transition matrix A to be transformed.
On exit, the leading N-by-N part of this array contains
the transformed state transition matrix U' * A * U.
The annihilated elements are set to zero.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input matrix B to be transformed.
On exit, the leading N-by-M part of this array contains
the transformed input matrix U' * B.
The annihilated elements are set to zero.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
U (input/output) DOUBLE PRECISION array, dimension (LDU,*)
On entry, if JOBU = 'U', then the leading N-by-N part of
this array must contain a given matrix U (e.g. from a
previous call to another SLICOT routine), and on exit, the
leading N-by-N part of this array contains the product of
the input matrix U and the state-space transformation
matrix which reduces the given pair to controller
Hessenberg form.
On exit, if JOBU = 'I', then the leading N-by-N part of
this array contains the matrix of accumulated unitary
similarity transformations which reduces the given pair
to controller Hessenberg form.
If JOBU = 'N', the array U is not referenced and can be
supplied as a dummy array (i.e. set parameter LDU = 1 and
declare this array to be U(1,1) in the calling program).
LDU INTEGER
The leading dimension of array U. If JOBU = 'U' or
JOBU = 'I', LDU >= MAX(1,N); if JOBU = 'N', LDU >= 1.
Workspace
DWORK DOUBLE PRECISION array, dimension (MAX(N,M-1))Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine computes a unitary state-space transformation U, which
reduces the pair (B,A) to one of the following controller
Hessenberg forms:
|* . . . *|* . . . . . . *|
| . .|. .|
| . .|. .|
| . .|. .|
[U'B|U'AU] = | *|. .| N
| |* .|
| | . .|
| | . .|
| | . .|
| | * . . *|
M N
if UPLO = 'U', or
|* . . * | |
|. . | |
|. . | |
|. . | |
[U'AU|U'B] = |. *| | N
|. .|* |
|. .|. . |
|. .|. . |
|. .|. . |
|* . . . . . . *|* . . . *|
N M
if UPLO = 'L'.
If M >= N, then the matrix U'B is trapezoidal and U'AU is full.
If M = 0, but N > 0, the array A is unchanged on exit.
References
[1] Van Dooren, P. and Verhaegen, M.H.G.
On the use of unitary state-space transformations.
In : Contemporary Mathematics on Linear Algebra and its Role
in Systems Theory, 47, AMS, Providence, 1985.
Numerical Aspects
The algorithm requires O((N + M) x N**2) operations and is backward stable (see [1]).Further Comments
NoneExample
Program Text
* TB01MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX
PARAMETER ( NMAX = 20, MMAX = 20 )
INTEGER LDA, LDB, LDU, LDWORK
PARAMETER ( LDA = NMAX, LDB = NMAX, LDU = NMAX,
$ LDWORK = NMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, M, N
CHARACTER*1 JOBU, UPLO
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), U(LDU,NMAX),
$ DWORK(LDWORK)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL TB01MD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, JOBU, UPLO
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), I = 1,N ), J = 1,N )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,N ), J = 1,M )
IF ( LSAME( JOBU, 'U' ) )
$ READ ( NIN, FMT = * ) ( ( U(I,J), J = 1,N ), I = 1,N )
* Reduce the pair (B,A) to controller Hessenberg form.
CALL TB01MD( JOBU, UPLO, N, M, A, LDA, B, LDB, U, LDU,
$ DWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( A(I,J), J = 1,N )
60 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 80 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( B(I,J), J = 1,M )
80 CONTINUE
IF ( LSAME( JOBU, 'I' ).OR.LSAME( JOBU, 'U' ) ) THEN
WRITE ( NOUT, FMT = 99994 )
DO 100 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( U(I,J), J = 1,N )
100 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TB01MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TB01MD = ',I2)
99997 FORMAT (' The transformed state transition matrix is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The transformed input matrix is ')
99994 FORMAT (/' The transformation matrix that reduces (B,A) to contr',
$ 'oller Hessenberg form is ')
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' M is out of range.',/' M = ',I5)
END
Program Data
TB01MD EXAMPLE PROGRAM DATA 6 3 N U 35.0 1.0 6.0 26.0 19.0 24.0 3.0 32.0 7.0 21.0 23.0 25.0 31.0 9.0 2.0 22.0 27.0 20.0 8.0 28.0 33.0 17.0 10.0 15.0 30.0 5.0 34.0 12.0 14.0 16.0 4.0 36.0 29.0 13.0 18.0 11.0 1.0 5.0 11.0 -1.0 4.0 11.0 -5.0 1.0 9.0 -11.0 -4.0 5.0 -19.0 -11.0 -1.0 -29.0 -20.0 -9.0Program Results
TB01MD EXAMPLE PROGRAM RESULTS The transformed state transition matrix is 60.3649 58.8853 5.0480 -5.4406 2.1382 -7.3870 54.5832 33.1865 36.5234 6.3272 -3.1377 8.8154 17.6406 21.4501 -13.5942 0.5417 1.6926 0.0786 -9.0567 10.7202 0.3531 1.5444 -1.2846 24.6407 0.0000 6.8796 -20.1372 -2.6440 2.4983 -21.8071 0.0000 0.0000 0.0000 0.0000 0.0000 27.0000 The transformed input matrix is -16.8819 -8.8260 13.9202 0.0000 13.8240 39.9205 0.0000 0.0000 4.1928 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TB01ND.html 0000664 0000000 0000000 00000024122 14560147231 0020343 0 ustar 00root root 0000000 0000000
Purpose
To reduce the pair (A,C) to lower or upper observer Hessenberg form using (and optionally accumulating) unitary state-space transformations.Specification
SUBROUTINE TB01ND( JOBU, UPLO, N, P, A, LDA, C, LDC, U, LDU,
$ DWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDC, LDU, N, P
CHARACTER JOBU, UPLO
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), C(LDC,*), DWORK(*), U(LDU,*)
Arguments
Mode Parameters
JOBU CHARACTER*1
Indicates whether the user wishes to accumulate in a
matrix U the unitary state-space transformations for
reducing the system, as follows:
= 'N': Do not form U;
= 'I': U is initialized to the unit matrix and the
unitary transformation matrix U is returned;
= 'U': The given matrix U is updated by the unitary
transformations used in the reduction.
UPLO CHARACTER*1
Indicates whether the user wishes the pair (A,C) to be
reduced to upper or lower observer Hessenberg form as
follows:
= 'U': Upper observer Hessenberg form;
= 'L': Lower observer Hessenberg form.
Input/Output Parameters
N (input) INTEGER
The actual state dimension, i.e. the order of the
matrix A. N >= 0.
P (input) INTEGER
The actual output dimension, i.e. the number of rows of
the matrix C. 0 <= P <= N.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state transition matrix A to be transformed.
On exit, the leading N-by-N part of this array contains
the transformed state transition matrix U' * A * U.
The annihilated elements are set to zero.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the output matrix C to be transformed.
On exit, the leading P-by-N part of this array contains
the transformed output matrix C * U.
The annihilated elements are set to zero.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
U (input/output) DOUBLE PRECISION array, dimension (LDU,*)
On entry, if JOBU = 'U', then the leading N-by-N part of
this array must contain a given matrix U (e.g. from a
previous call to another SLICOT routine), and on exit, the
leading N-by-N part of this array contains the product of
the input matrix U and the state-space transformation
matrix which reduces the given pair to observer Hessenberg
form.
On exit, if JOBU = 'I', then the leading N-by-N part of
this array contains the matrix of accumulated unitary
similarity transformations which reduces the given pair
to observer Hessenberg form.
If JOBU = 'N', the array U is not referenced and can be
supplied as a dummy array (i.e. set parameter LDU = 1 and
declare this array to be U(1,1) in the calling program).
LDU INTEGER
The leading dimension of array U. If JOBU = 'U' or
JOBU = 'I', LDU >= MAX(1,N); if JOBU = 'N', LDU >= 1.
Workspace
DWORK DOUBLE PRECISION array, dimension (MAX(N,P-1))Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine computes a unitary state-space transformation U, which
reduces the pair (A,C) to one of the following observer Hessenberg
forms:
N
|* . . . . . . *|
|. .|
|. .|
|. .| N
|* .|
|U'AU| | . .|
|----| = | . .|
|CU | | * . . . *|
-------------------
| * . . *|
| . .| P
| . .|
| *|
if UPLO = 'U', or
N
|* |
|. . |
|. . | P
|* . . * |
|CU | -------------------
|----| = |* . . . * |
|U'AU| |. . |
|. . |
|. *|
|. .| N
|. .|
|. .|
|* . . . . . . *|
if UPLO = 'L'.
If P >= N, then the matrix CU is trapezoidal and U'AU is full.
If P = 0, but N > 0, the array A is unchanged on exit.
References
[1] Van Dooren, P. and Verhaegen, M.H.G.
On the use of unitary state-space transformations.
In : Contemporary Mathematics on Linear Algebra and its Role
in Systems Theory, 47, AMS, Providence, 1985.
Numerical Aspects
The algorithm requires O((N + P) x N**2) operations and is backward stable (see [1]).Further Comments
NoneExample
Program Text
* TB01ND EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, PMAX
PARAMETER ( NMAX = 20, PMAX = 20 )
INTEGER LDA, LDC, LDU, LDWORK
PARAMETER ( LDA = NMAX, LDC = PMAX, LDU = NMAX,
$ LDWORK = NMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, N, P
CHARACTER*1 JOBU, UPLO
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), C(LDC,NMAX), U(LDU,NMAX),
$ DWORK(LDWORK)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL TB01ND
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, P, JOBU, UPLO
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), I = 1,N ), J = 1,N )
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
IF ( LSAME( JOBU, 'U' ) )
$ READ ( NIN, FMT = * ) ( ( U(I,J), J = 1,N ), I = 1,N )
* Reduce the pair (A,C) to observer Hessenberg form.
CALL TB01ND( JOBU, UPLO, N, P, A, LDA, C, LDC, U, LDU,
$ DWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( A(I,J), J = 1,N )
60 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 80 I = 1, P
WRITE ( NOUT, FMT = 99996 ) ( C(I,J), J = 1,N )
80 CONTINUE
IF ( LSAME( JOBU, 'I' ).OR.LSAME( JOBU, 'U' ) ) THEN
WRITE ( NOUT, FMT = 99994 )
DO 100 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( U(I,J), J = 1,N )
100 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TB01ND EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TB01ND = ',I2)
99997 FORMAT (' The transformed state transition matrix is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The transformed output matrix is ')
99994 FORMAT (/' The transformation matrix that reduces (A,C) to obser',
$ 'ver Hessenberg form is ')
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
TB01ND EXAMPLE PROGRAM DATA 5 3 N U 15.0 21.0 -3.0 3.0 9.0 20.0 1.0 2.0 8.0 9.0 4.0 1.0 7.0 13.0 14.0 5.0 6.0 12.0 13.0 -6.0 5.0 11.0 17.0 -7.0 -1.0 7.0 -1.0 3.0 -6.0 -3.0 4.0 5.0 6.0 -2.0 -3.0 9.0 8.0 5.0 2.0 1.0Program Results
TB01ND EXAMPLE PROGRAM RESULTS The transformed state transition matrix is 7.1637 -0.9691 -16.5046 0.2869 0.9205 -2.3285 11.5431 -8.7471 3.4122 -3.7118 -10.5440 -7.6032 -0.3215 3.6571 -0.4335 -3.6845 5.6449 0.5906 -15.6996 17.4267 0.0000 -6.4260 1.5591 14.4317 32.3143 The transformed output matrix is 0.0000 0.0000 7.6585 5.2973 -4.1576 0.0000 0.0000 0.0000 5.8305 -7.4837 0.0000 0.0000 0.0000 0.0000 -13.2288
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TB01PD.html 0000664 0000000 0000000 00000027151 14560147231 0020352 0 ustar 00root root 0000000 0000000
Purpose
To find a reduced (controllable, observable, or minimal) state- space representation (Ar,Br,Cr) for any original state-space representation (A,B,C). The matrix Ar is in upper block Hessenberg form.Specification
SUBROUTINE TB01PD( JOB, EQUIL, N, M, P, A, LDA, B, LDB, C, LDC,
$ NR, TOL, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER EQUIL, JOB
INTEGER INFO, LDA, LDB, LDC, LDWORK, M, N, NR, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*)
Arguments
Mode Parameters
JOB CHARACTER*1
Indicates whether the user wishes to remove the
uncontrollable and/or unobservable parts as follows:
= 'M': Remove both the uncontrollable and unobservable
parts to get a minimal state-space representation;
= 'C': Remove the uncontrollable part only to get a
controllable state-space representation;
= 'O': Remove the unobservable part only to get an
observable state-space representation.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily balance
the triplet (A,B,C) as follows:
= 'S': Perform balancing (scaling);
= 'N': Do not perform balancing.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation, i.e.
the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state dynamics matrix A.
On exit, the leading NR-by-NR part of this array contains
the upper block Hessenberg state dynamics matrix Ar of a
minimal, controllable, or observable realization for the
original system, depending on the value of JOB, JOB = 'M',
JOB = 'C', or JOB = 'O', respectively.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M),
if JOB = 'C', or (LDB,MAX(M,P)), otherwise.
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B; if JOB = 'M',
or JOB = 'O', the remainder of the leading N-by-MAX(M,P)
part is used as internal workspace.
On exit, the leading NR-by-M part of this array contains
the transformed input/state matrix Br of a minimal,
controllable, or observable realization for the original
system, depending on the value of JOB, JOB = 'M',
JOB = 'C', or JOB = 'O', respectively.
If JOB = 'C', only the first IWORK(1) rows of B are
nonzero.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C; if JOB = 'M',
or JOB = 'O', the remainder of the leading MAX(M,P)-by-N
part is used as internal workspace.
On exit, the leading P-by-NR part of this array contains
the transformed state/output matrix Cr of a minimal,
controllable, or observable realization for the original
system, depending on the value of JOB, JOB = 'M',
JOB = 'C', or JOB = 'O', respectively.
If JOB = 'M', or JOB = 'O', only the last IWORK(1) columns
(in the first NR columns) of C are nonzero.
LDC INTEGER
The leading dimension of array C.
LDC >= MAX(1,M,P) if N > 0.
LDC >= 1 if N = 0.
NR (output) INTEGER
The order of the reduced state-space representation
(Ar,Br,Cr) of a minimal, controllable, or observable
realization for the original system, depending on
JOB = 'M', JOB = 'C', or JOB = 'O'.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in rank determination when
transforming (A, B, C). If the user sets TOL > 0, then
the given value of TOL is used as a lower bound for the
reciprocal condition number (see the description of the
argument RCOND in the SLICOT routine MB03OD); a
(sub)matrix whose estimated condition number is less than
1/TOL is considered to be of full rank. If the user sets
TOL <= 0, then an implicitly computed, default tolerance
(determined by the SLICOT routine TB01UD) is used instead.
Workspace
IWORK INTEGER array, dimension (N+MAX(M,P))
On exit, if INFO = 0, the first nonzero elements of
IWORK(1:N) return the orders of the diagonal blocks of A.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1, N + MAX(N, 3*M, 3*P)).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
If JOB = 'M', the matrices A and B are operated on by orthogonal similarity transformations (made up of products of Householder transformations) so as to produce an upper block Hessenberg matrix A1 and a matrix B1 with all but its first rank(B) rows zero; this separates out the controllable part of the original system. Applying the same algorithm to the dual of this subsystem, therefore separates out the controllable and observable (i.e. minimal) part of the original system representation, with the final Ar upper block Hessenberg (after using pertransposition). If JOB = 'C', or JOB = 'O', only the corresponding part of the above procedure is applied.References
[1] Van Dooren, P.
The Generalized Eigenstructure Problem in Linear System
Theory. (Algorithm 1)
IEEE Trans. Auto. Contr., AC-26, pp. 111-129, 1981.
Numerical Aspects
3 The algorithm requires 0(N ) operations and is backward stable.Further Comments
NoneExample
Program Text
* TB01PD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER MAXMP
PARAMETER ( MAXMP = MAX( MMAX, PMAX ) )
INTEGER LDA, LDB, LDC
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = MAXMP )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX+MAXMP )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX+MAX( NMAX, 3*MAXMP ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, J, M, N, NR, P
CHARACTER JOB, EQUIL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MAXMP), C(LDC,NMAX),
$ DWORK(LDWORK)
INTEGER IWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL TB01PD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL, JOB, EQUIL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,N ), J = 1,M )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find a minimal ssr for (A,B,C).
CALL TB01PD( JOB, EQUIL, N, M, P, A, LDA, B, LDB, C, LDC,
$ NR, TOL, IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NR
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
60 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TB01PD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TB01PD = ',I2)
99997 FORMAT (' The order of the minimal realization = ',I2)
99996 FORMAT (/' The transformed state dynamics matrix of a minimal re',
$ 'alization is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The transformed input/state matrix of a minimal reali',
$ 'zation is ')
99992 FORMAT (/' The transformed state/output matrix of a minimal real',
$ 'ization is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
TB01PD EXAMPLE PROGRAM DATA 3 1 2 0.0 M N 1.0 2.0 0.0 4.0 -1.0 0.0 0.0 0.0 1.0 1.0 0.0 1.0 0.0 1.0 -1.0 0.0 0.0 1.0Program Results
TB01PD EXAMPLE PROGRAM RESULTS The order of the minimal realization = 3 The transformed state dynamics matrix of a minimal realization is 1.0000 -1.4142 1.4142 -2.8284 0.0000 1.0000 2.8284 1.0000 0.0000 The transformed input/state matrix of a minimal realization is -1.0000 0.7071 0.7071 The transformed state/output matrix of a minimal realization is 0.0000 0.0000 -1.4142 0.0000 0.7071 0.7071
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TB01PX.html 0000664 0000000 0000000 00000034230 14560147231 0020372 0 ustar 00root root 0000000 0000000
Purpose
To find a reduced (controllable, observable, or minimal) state- space representation (Ar,Br,Cr) for any original state-space representation (A,B,C). The matrix Ar is in an upper block Hessenberg staircase form.Specification
SUBROUTINE TB01PX( JOB, EQUIL, N, M, P, A, LDA, B, LDB, C, LDC,
$ NR, INFRED, TOL, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER EQUIL, JOB
INTEGER INFO, LDA, LDB, LDC, LDWORK, M, N, NR, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER INFRED(*), IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*)
Arguments
Mode Parameters
JOB CHARACTER*1
Indicates whether the user wishes to remove the
uncontrollable and/or unobservable parts as follows:
= 'M': Remove both the uncontrollable and unobservable
parts to get a minimal state-space representation;
= 'C': Remove the uncontrollable part only to get a
controllable state-space representation;
= 'O': Remove the unobservable part only to get an
observable state-space representation.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily balance
the triplet (A,B,C) as follows:
= 'S': Perform balancing (scaling);
= 'N': Do not perform balancing.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation, i.e.
the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state dynamics matrix A.
On exit, if INFRED(1) >= 0 and/or INFRED(2) >= 0, the
leading NR-by-NR part of this array contains the upper
block Hessenberg state dynamics matrix Ar of a minimal,
controllable, or observable realization for the original
system, depending on the value of JOB, JOB = 'M',
JOB = 'C', or JOB = 'O', respectively.
The block structure of the resulting staircase form is
contained in the leading INFRED(4) elements of IWORK.
If INFRED(1:2) < 0, then A contains the original matrix.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M),
if JOB = 'C', or (LDB,MAX(M,P)), otherwise.
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B; if JOB = 'M',
or JOB = 'O', the remainder of the leading N-by-MAX(M,P)
part is used as internal workspace.
On exit, if INFRED(1) >= 0 and/or INFRED(2) >= 0, the
leading NR-by-M part of this array contains the
input/state matrix Br of a minimal, controllable, or
observable realization for the original system, depending
on the value of JOB, JOB = 'M', JOB = 'C', or JOB = 'O',
respectively. If JOB = 'C', only the first IWORK(1) rows
of B are nonzero.
If INFRED(1:2) < 0, then B contains the original matrix.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C; if JOB = 'M',
or JOB = 'O', the remainder of the leading MAX(M,P)-by-N
part is used as internal workspace.
On exit, if INFRED(1) >= 0 and/or INFRED(2) >= 0, the
leading P-by-NR part of this array contains the
state/output matrix Cr of a minimal, controllable, or
observable realization for the original system, depending
on the value of JOB, JOB = 'M', JOB = 'C', or JOB = 'O',
respectively. If JOB = 'M', or JOB = 'O', only the last
IWORK(1) columns (in the first NR columns) of C are
nonzero.
If INFRED(1:2) < 0, then C contains the original matrix.
LDC INTEGER
The leading dimension of the array C.
LDC >= MAX(1,M,P), if N > 0.
LDC >= 1, if N = 0.
NR (output) INTEGER
The order of the reduced state-space representation
(Ar,Br,Cr) of a minimal, controllable, or observable
realization for the original system, depending on
JOB = 'M', JOB = 'C', or JOB = 'O'.
INFRED (output) INTEGER array, dimension 4
This array contains information on the performed reduction
and on structure of resulting system matrices, as follows:
INFRED(k) >= 0 (k = 1 or 2) if Phase k of the reduction
(see METHOD) has been performed. In this
case, INFRED(k) is the achieved order
reduction in Phase k.
INFRED(k) < 0 (k = 1 or 2) if Phase k was not performed.
This can also appear when Phase k was
tried, but did not reduce the order, if
enough workspace is provided for saving the
system matrices (see LDWORK description).
INFRED(3) - the number of nonzero subdiagonals of A.
INFRED(4) - the number of blocks in the resulting
staircase form at the last performed
reduction phase. The block dimensions are
contained in the first INFRED(4) elements
of IWORK.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in rank determinations when
transforming (A, B, C). If the user sets TOL > 0, then
the given value of TOL is used as a lower bound for the
reciprocal condition number (see the description of the
argument RCOND in the SLICOT routine MB03OD); a
(sub)matrix whose estimated condition number is less than
1/TOL is considered to be of full rank. If the user sets
TOL <= 0, then an implicitly computed, default tolerance
(determined by the SLICOT routine TB01UD) is used instead.
Workspace
IWORK INTEGER array, dimension (c*N+MAX(M,P)), where
c = 2, if JOB = 'M', and c = 1, otherwise.
On exit, if INFO = 0, the first INFRED(4) elements of
IWORK return the orders of the diagonal blocks of A.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= 1, and if N > 0,
LDWORK >= N + MAX(N, 3*M, 3*P).
For optimum performance LDWORK should be larger.
If LDWORK >= MAX(1, N + MAX(N, 3*M, 3*P) + N*(N+M+P) ),
then more accurate results are to be expected by accepting
only those reductions phases (see METHOD), where effective
order reduction occurs. This is achieved by saving the
system matrices before each phase and restoring them if
no order reduction took place in that phase.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The order reduction is performed in two phases: Phase 1: If JOB = 'M' or 'C', the pair (A,B) is reduced by orthogonal similarity transformations to the controllability staircase form (see [1]) and a controllable realization (Ac,Bc,Cc) is extracted. Ac results in an upper block Hessenberg form. Phase 2: If JOB = 'M' or 'O', the same algorithm is applied to the dual of the controllable realization (Ac,Bc,Cc), or to the dual of the original system, respectively, to extract an observable realization (Ar,Br,Cr). If JOB = 'M', the resulting realization is also controllable, and thus minimal. Ar results in an upper block Hessenberg form.References
[1] Van Dooren, P.
The Generalized Eigenstructure Problem in Linear System
Theory. (Algorithm 1)
IEEE Trans. Auto. Contr., AC-26, pp. 111-129, 1981.
Numerical Aspects
3 The algorithm requires 0(N ) operations and is backward stable.Further Comments
NoneExample
Program Text
* TB01PX EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER MAXMP
PARAMETER ( MAXMP = MAX( MMAX, PMAX ) )
INTEGER LDA, LDB, LDC
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = MAXMP )
INTEGER LIWORK
PARAMETER ( LIWORK = 2*NMAX+MAXMP )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX+MAX( NMAX, 3*MAXMP +
$ NMAX*( NMAX+MMAX+PMAX ) ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, J, M, N, NR, P
CHARACTER JOB, EQUIL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MAXMP), C(LDC,NMAX),
$ DWORK(LDWORK)
INTEGER INFRED(4), IWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL TB01PX
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL, JOB, EQUIL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,N ), J = 1,M )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find a minimal ssr for (A,B,C).
CALL TB01PX( JOB, EQUIL, N, M, P, A, LDA, B, LDB, C, LDC,
$ NR, INFRED, TOL, IWORK, DWORK, LDWORK, INFO)
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NR
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
60 CONTINUE
WRITE ( NOUT, FMT = 99994 ) ( INFRED(I), I = 1,4 )
IF ( INFRED(4).GT.0 )
$ WRITE ( NOUT, FMT = 99991 ) ( IWORK(I),
$ I = 1,INFRED(4) )
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TB01PX EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TB01PX = ',I2)
99997 FORMAT (' The order of the minimal realization = ',I2)
99996 FORMAT (/' The transformed state dynamics matrix of a minimal re',
$ 'alization is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' Information on the performed reduction is ', 4I5)
99993 FORMAT (/' The transformed input/state matrix of a minimal reali',
$ 'zation is ')
99992 FORMAT (/' The transformed state/output matrix of a minimal real',
$ 'ization is ')
99991 FORMAT (/' The block dimensions are ',/ 20I5)
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
TB01PX EXAMPLE PROGRAM DATA 3 1 2 0.0 M N 1.0 2.0 0.0 4.0 -1.0 0.0 0.0 0.0 1.0 1.0 0.0 1.0 0.0 1.0 -1.0 0.0 0.0 1.0Program Results
TB01PX EXAMPLE PROGRAM RESULTS The order of the minimal realization = 3 The transformed state dynamics matrix of a minimal realization is 1.0000 2.0000 0.0000 4.0000 -1.0000 0.0000 0.0000 0.0000 1.0000 The transformed input/state matrix of a minimal realization is 1.0000 0.0000 1.0000 The transformed state/output matrix of a minimal realization is 0.0000 1.0000 -1.0000 0.0000 0.0000 1.0000 Information on the performed reduction is -1 -1 2 0
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TB01TD.html 0000664 0000000 0000000 00000026205 14560147231 0020355 0 ustar 00root root 0000000 0000000
Purpose
To reduce a given state-space representation (A,B,C,D) to balanced form by means of state permutations and state, input and output scalings.Specification
SUBROUTINE TB01TD( N, M, P, A, LDA, B, LDB, C, LDC, D, LDD, LOW,
$ IGH, SCSTAT, SCIN, SCOUT, DWORK, INFO )
C .. Scalar Arguments ..
INTEGER IGH, INFO, LDA, LDB, LDC, LDD, LOW, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), SCIN(*), SCOUT(*), SCSTAT(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the state-space representation, i.e. the
order of the original state dynamics matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state dynamics matrix A.
On exit, the leading N-by-N part of this array contains
the balanced state dynamics matrix A.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, the leading N-by-M part of this array contains
the balanced input/state matrix B.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, the leading P-by-N part of this array contains
the balanced state/output matrix C.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the original direct transmission matrix D.
On exit, the leading P-by-M part of this array contains
the scaled direct transmission matrix D.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
LOW (output) INTEGER
The index of the lower end of the balanced submatrix of A.
IGH (output) INTEGER
The index of the upper end of the balanced submatrix of A.
SCSTAT (output) DOUBLE PRECISION array, dimension (N)
This array contains the information defining the
similarity transformations used to permute and balance
the state dynamics matrix A, as returned from the LAPACK
library routine DGEBAL.
SCIN (output) DOUBLE PRECISION array, dimension (M)
Contains the scalars used to scale the system inputs so
that the columns of the final matrix B have norms roughly
equal to the column sums of the balanced matrix A
(see FURTHER COMMENTS).
The j-th input of the balanced state-space representation
is SCIN(j)*(j-th column of the permuted and balanced
input/state matrix B).
SCOUT (output) DOUBLE PRECISION array, dimension (P)
Contains the scalars used to scale the system outputs so
that the rows of the final matrix C have norms roughly
equal to the row sum of the balanced matrix A.
The i-th output of the balanced state-space representation
is SCOUT(i)*(i-th row of the permuted and balanced
state/ouput matrix C).
Workspace
DWORK DOUBLE PRECISION array, dimension (N)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Similarity transformations are used to permute the system states
and balance the corresponding row and column sum norms of a
submatrix of the state dynamics matrix A. These operations are
also applied to the input/state matrix B and the system inputs
are then scaled (see parameter SCIN) so that the columns of the
final matrix B have norms roughly equal to the column sum norm of
the balanced matrix A (see FURTHER COMMENTS).
The above operations are also applied to the matrix C, and the
system outputs are then scaled (see parameter SCOUT) so that the
rows of the final matrix C have norms roughly equal to the row sum
norm of the balanced matrix A (see FURTHER COMMENTS).
Finally, the (I,J)-th element of the direct transmission matrix D
is scaled as
D(I,J) = D(I,J)*(1.0/SCIN(J))*SCOUT(I), where I = 1,2,...,P
and J = 1,2,...,M.
Scaling performed to balance the row/column sum norms is by
integer powers of the machine base so as to avoid introducing
rounding errors.
References
[1] Wilkinson, J.H. and Reinsch, C.
Handbook for Automatic Computation, (Vol II, Linear Algebra).
Springer-Verlag, 1971, (contribution II/11).
Numerical Aspects
3 The algorithm requires 0(N ) operations and is backward stable.Further Comments
The columns (rows) of the final matrix B (matrix C) have norms
'roughly' equal to the column (row) sum norm of the balanced
matrix A, i.e.
size/BASE < abssum <= size
where
BASE = the base of the arithmetic used on the computer, which
can be obtained from the LAPACK Library routine
DLAMCH;
size = column or row sum norm of the balanced matrix A;
abssum = column sum norm of the balanced matrix B or row sum
norm of the balanced matrix C.
The routine is BASE dependent.
Example
Program Text
* TB01TD EXAMPLE PROGRAM TEXT.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDD
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX )
* .. Local Scalars ..
INTEGER I, INFO, IGH, J, LOW, M, N, P
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(NMAX), SCIN(MMAX),
$ SCOUT(PMAX), SCSTAT(NMAX)
* .. External Subroutines ..
EXTERNAL TB01TD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,N ), J = 1,M )
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
* Balance the state-space representation (A,B,C,D).
CALL TB01TD( N, M, P, A, LDA, B, LDB, C, LDC, D, LDD,
$ LOW, IGH, SCSTAT, SCIN, SCOUT, DWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) LOW, IGH
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
60 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 80 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1,M )
80 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TB01TD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TB01TD = ',I2)
99997 FORMAT (' LOW = ',I2,' IGH = ',I2,/)
99996 FORMAT (' The balanced state dynamics matrix A is ')
99995 FORMAT (20(1X,F9.4))
99994 FORMAT (/' The balanced input/state matrix B is ')
99993 FORMAT (/' The balanced state/output matrix C is ')
99992 FORMAT (/' The scaled direct transmission matrix D is ')
99991 FORMAT (/' N is out of range.',/' N = ',I5)
99990 FORMAT (/' M is out of range.',/' M = ',I5)
99989 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
TB01TD EXAMPLE PROGRAM DATA 5 2 2 0.0 0.0 1.0 4.0 5.0 50.0 10.0 1.0 0.0 0.0 0.0 0.0 90.0 10.0 0.0 0.0 1.0 1.0 1.0 1.0 100.0 0.0 0.0 0.0 70.0 0.0 2.0 0.0 1.0 2.0 0.0 20.0 100.0 1.0 0.0 1.0 0.0 0.0 1.0 0.0 1.0 1.0 0.0 2.0 1.0 1.0 1.0 1.0 1.0Program Results
TB01TD EXAMPLE PROGRAM RESULTS
LOW = 1 IGH = 5
The balanced state dynamics matrix A is
0.0000 0.0000 1.0000 4.0000 40.0000
6.2500 10.0000 0.1250 0.0000 0.0000
0.0000 0.0000 90.0000 10.0000 0.0000
0.0000 8.0000 1.0000 1.0000 8.0000
12.5000 0.0000 0.0000 0.0000 70.0000
The balanced input/state matrix B is
0.0000 0.0000
16.0000 2.5000
0.0000 100.0000
64.0000 1.0000
16.0000 0.0000
The balanced state/output matrix C is
32.0000 0.0000 0.0000 32.0000 0.0000
4.0000 32.0000 0.0000 8.0000 32.0000
The scaled direct transmission matrix D is
2048.0000 32.0000
256.0000 4.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TB01UD.html 0000664 0000000 0000000 00000035760 14560147231 0020364 0 ustar 00root root 0000000 0000000
Purpose
To find a controllable realization for the linear time-invariant
multi-input system
dX/dt = A * X + B * U,
Y = C * X,
where A, B, and C are N-by-N, N-by-M, and P-by-N matrices,
respectively, and A and B are reduced by this routine to
orthogonal canonical form using (and optionally accumulating)
orthogonal similarity transformations, which are also applied
to C. Specifically, the system (A, B, C) is reduced to the
triplet (Ac, Bc, Cc), where Ac = Z' * A * Z, Bc = Z' * B,
Cc = C * Z, with
[ Acont * ] [ Bcont ]
Ac = [ ], Bc = [ ],
[ 0 Auncont ] [ 0 ]
and
[ A11 A12 . . . A1,p-1 A1p ] [ B1 ]
[ A21 A22 . . . A2,p-1 A2p ] [ 0 ]
[ 0 A32 . . . A3,p-1 A3p ] [ 0 ]
Acont = [ . . . . . . . ], Bc = [ . ],
[ . . . . . . ] [ . ]
[ . . . . . ] [ . ]
[ 0 0 . . . Ap,p-1 App ] [ 0 ]
where the blocks B1, A21, ..., Ap,p-1 have full row ranks and
p is the controllability index of the pair. The size of the
block Auncont is equal to the dimension of the uncontrollable
subspace of the pair (A, B).
Specification
SUBROUTINE TB01UD( JOBZ, N, M, P, A, LDA, B, LDB, C, LDC, NCONT,
$ INDCON, NBLK, Z, LDZ, TAU, TOL, IWORK, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBZ
INTEGER INDCON, INFO, LDA, LDB, LDC, LDWORK, LDZ, M, N,
$ NCONT, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), TAU(*),
$ Z(LDZ,*)
INTEGER IWORK(*), NBLK(*)
Arguments
Mode Parameters
JOBZ CHARACTER*1
Indicates whether the user wishes to accumulate in a
matrix Z the orthogonal similarity transformations for
reducing the system, as follows:
= 'N': Do not form Z and do not store the orthogonal
transformations;
= 'F': Do not form Z, but store the orthogonal
transformations in the factored form;
= 'I': Z is initialized to the unit matrix and the
orthogonal transformation matrix Z is returned.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation,
i.e. the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs, or of columns of B. M >= 0.
P (input) INTEGER
The number of system outputs, or of rows of C. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state dynamics matrix A.
On exit, the leading NCONT-by-NCONT part contains the
upper block Hessenberg state dynamics matrix Acont in Ac,
given by Z' * A * Z, of a controllable realization for
the original system. The elements below the first block-
subdiagonal are set to zero. The leading N-by-N part
contains the matrix Ac.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input matrix B.
On exit, the leading NCONT-by-M part of this array
contains the transformed input matrix Bcont in Bc, given
by Z' * B, with all elements but the first block set to
zero. The leading N-by-M part contains the matrix Bc.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed output matrix Cc, given by C * Z.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
NCONT (output) INTEGER
The order of the controllable state-space representation.
INDCON (output) INTEGER
The controllability index of the controllable part of the
system representation.
NBLK (output) INTEGER array, dimension (N)
The leading INDCON elements of this array contain the
the orders of the diagonal blocks of Acont.
Z (output) DOUBLE PRECISION array, dimension (LDZ,N)
If JOBZ = 'I', then the leading N-by-N part of this
array contains the matrix of accumulated orthogonal
similarity transformations which reduces the given system
to orthogonal canonical form.
If JOBZ = 'F', the elements below the diagonal, with the
array TAU, represent the orthogonal transformation matrix
as a product of elementary reflectors. The transformation
matrix can then be obtained by calling the LAPACK Library
routine DORGQR.
If JOBZ = 'N', the array Z is not referenced and can be
supplied as a dummy array (i.e. set parameter LDZ = 1 and
declare this array to be Z(1,1) in the calling program).
LDZ INTEGER
The leading dimension of array Z. If JOBZ = 'I' or
JOBZ = 'F', LDZ >= MAX(1,N); if JOBZ = 'N', LDZ >= 1.
TAU (output) DOUBLE PRECISION array, dimension (N)
The elements of TAU contain the scalar factors of the
elementary reflectors used in the reduction of B and A.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in rank determination when
transforming (A, B). If the user sets TOL > 0, then
the given value of TOL is used as a lower bound for the
reciprocal condition number (see the description of the
argument RCOND in the SLICOT routine MB03OD); a
(sub)matrix whose estimated condition number is less than
1/TOL is considered to be of full rank. If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
defined by TOLDEF = N*N*EPS, is used instead, where EPS
is the machine precision (see LAPACK Library routine
DLAMCH).
Workspace
IWORK INTEGER array, dimension (M)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1, N, 3*M, P).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Matrix B is first QR-decomposed and the appropriate orthogonal similarity transformation applied to the matrix A. Leaving the first rank(B) states unchanged, the remaining lower left block of A is then QR-decomposed and the new orthogonal matrix, Q1, is also applied to the right of A to complete the similarity transformation. By continuing in this manner, a completely controllable state-space pair (Acont, Bcont) is found for the given (A, B), where Acont is upper block Hessenberg with each subdiagonal block of full row rank, and Bcont is zero apart from its (independent) first rank(B) rows. All orthogonal transformations determined in this process are also applied to the matrix C, from the right. NOTE that the system controllability indices are easily calculated from the dimensions of the blocks of Acont.References
[1] Konstantinov, M.M., Petkov, P.Hr. and Christov, N.D.
Orthogonal Invariants and Canonical Forms for Linear
Controllable Systems.
Proc. 8th IFAC World Congress, Kyoto, 1, pp. 49-54, 1981.
[2] Paige, C.C.
Properties of numerical algorithms related to computing
controllablity.
IEEE Trans. Auto. Contr., AC-26, pp. 130-138, 1981.
[3] Petkov, P.Hr., Konstantinov, M.M., Gu, D.W. and
Postlethwaite, I.
Optimal Pole Assignment Design of Linear Multi-Input Systems.
Leicester University, Report 99-11, May 1996.
Numerical Aspects
3 The algorithm requires 0(N ) operations and is backward stable.Further Comments
If the system matrices A and B are badly scaled, it would be useful to scale them with SLICOT routine TB01ID, before calling the routine.Example
Program Text
* TB01UD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDZ
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDZ = NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( NMAX, 3*MMAX, PMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, INDCON, J, M, N, NCONT, P
CHARACTER*1 JOBZ
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ DWORK(LDWORK), TAU(NMAX), Z(LDZ,NMAX)
INTEGER IWORK(LIWORK), NBLK(NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL TB01UD, DORGQR
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL, JOBZ
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,N ), J = 1,M )
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find a controllable ssr for the given system.
CALL TB01UD( JOBZ, N, M, P, A, LDA, B, LDB, C, LDC,
$ NCONT, INDCON, NBLK, Z, LDZ, TAU, TOL,
$ IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NCONT
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NCONT
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NCONT )
20 CONTINUE
WRITE ( NOUT, FMT = 99994 ) ( NBLK(I), I = 1,INDCON )
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NCONT
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
WRITE ( NOUT, FMT = 99987 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NCONT )
60 CONTINUE
WRITE ( NOUT, FMT = 99992 ) INDCON
IF ( LSAME( JOBZ, 'F' ) )
$ CALL DORGQR( N, N, N, Z, LDZ, TAU, DWORK, LDWORK,
$ INFO )
IF ( LSAME( JOBZ, 'F' ).OR.LSAME( JOBZ, 'I' ) ) THEN
WRITE ( NOUT, FMT = 99991 )
DO 80 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
80 CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TB01UD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TB01UD = ',I2)
99997 FORMAT (' The order of the controllable state-space representati',
$ 'on = ',I2)
99996 FORMAT (/' The transformed state dynamics matrix of a controllab',
$ 'le realization is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' and the dimensions of its diagonal blocks are ',
$ /20(1X,I2))
99993 FORMAT (/' The transformed input/state matrix B of a controllabl',
$ 'e realization is ')
99992 FORMAT (/' The controllability index of the transformed system r',
$ 'epresentation = ',I2)
99991 FORMAT (/' The similarity transformation matrix Z is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99987 FORMAT (/' The transformed output/state matrix C of a controlla',
$ 'ble realization is ')
END
Program Data
TB01UD EXAMPLE PROGRAM DATA 3 2 2 0.0 I -1.0 0.0 0.0 -2.0 -2.0 -2.0 -1.0 0.0 -3.0 1.0 0.0 0.0 0.0 2.0 1.0 0.0 2.0 1.0 1.0 0.0 0.0Program Results
TB01UD EXAMPLE PROGRAM RESULTS The order of the controllable state-space representation = 2 The transformed state dynamics matrix of a controllable realization is -3.0000 2.2361 0.0000 -1.0000 and the dimensions of its diagonal blocks are 2 The transformed input/state matrix B of a controllable realization is 0.0000 -2.2361 1.0000 0.0000 The transformed output/state matrix C of a controllable realization is -2.2361 0.0000 0.0000 1.0000 The controllability index of the transformed system representation = 1 The similarity transformation matrix Z is 0.0000 1.0000 0.0000 -0.8944 0.0000 -0.4472 -0.4472 0.0000 0.8944
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TB01UX.html 0000664 0000000 0000000 00000020504 14560147231 0020376 0 ustar 00root root 0000000 0000000
Purpose
To compute an orthogonal transformation matrix Z which reduces the
N-th order system (A,B,C) to the form
( Ano * ) ( Bno )
Z'*A*Z = ( ) , Z'*B = ( ) ,
( 0 Ao ) ( Bo )
C*Z = ( 0 Co ) ,
where the NOBSV-th order system (Ao,Bo,Co) is observable.
The matrix Ano of order N-NOBSV contains the unobservable
eigenvalues of A.
The pencil ( Ao-lambda*I ) has full column rank NOBSV for all
( Co )
lambda, and is in a staircase form, with
_ _ _ _
( Ak,k Ak,k-1 ... Ak,2 Ak,1 )
( _ _ _ _ )
( Ao ) = ( Ak-1,k Ak-1,k-1 ... Ak-1,2 Ak-1,1 ) , (1)
( Co ) ( : : ... _ : _ : )
( 0 0 ... A1,2 A1,1 )
( _ )
( 0 0 ... 0 A0,1 )
_
where Ai-1,i is a CTAU(i-1)-by-CTAU(i) full column rank matrix
(with CTAU(0) = P).
The orthogonal transformation Z, performed to reduce the system
matrices, can be optionally accumulated.
The reduced order system (Ao,Bo,Co) has the same transfer-function
matrix as the original system (A,B,C).
Specification
SUBROUTINE TB01UX( COMPZ, N, M, P, A, LDA, B, LDB, C, LDC, Z, LDZ,
$ NOBSV, NLBLCK, CTAU, TOL, IWORK, DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPZ
INTEGER INFO, LDA, LDB, LDC, LDZ, M, N, NLBLCK, NOBSV,
$ P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER CTAU( * ), IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ DWORK( * ), Z( LDZ, * )
Arguments
Mode Parameters
COMPZ CHARACTER*1
= 'N': do not compute Z;
= 'I': Z is initialized to the unit matrix, and the
orthogonal matrix Z is returned.
Input/Output Parameters
N (input) INTEGER
The dimension of the system state vector; also the order
of the square matrix A, the number of rows of the matrix B
and the number of columns of the matrix C. N >= 0.
M (input) INTEGER
The dimension of system input vector; also the number of
columns of the matrix B. M >= 0.
P (input) INTEGER
The dimension of system output vector; also the number of
rows of the matrix C. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the N-by-N state matrix A.
On exit, the leading N-by-N part of this array contains
the transformed state matrix Z'*A*Z,
( Ano * )
Z'*A*Z = ( ) ,
( 0 Ao )
where Ao is NOBSV-by-NOBSV and Ano is
(N-NOBSV)-by-(N-NOBSV).
The matrix ( Ao ) is in the observability staircase
( Co )
form (1).
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension
(LDB,MAX(M,P))
On entry, the leading N-by-M part of this array must
contain the N-by-M input matrix B.
On exit, the leading N-by-M part of this array contains
the transformed input matrix Z'*B.
LDB INTEGER
The leading dimension of the array B.
LDB >= MAX(1,N) if M > 0 or P > 0;
LDB >= 1 if M = 0 and P = 0.
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed matrix
C*Z = ( 0 Co ) ,
where Co is P-by-NOBSV.
The matrix ( Ao ) is in the observability staircase
( Co )
form (1).
LDC INTEGER
The leading dimension of the array C.
LDC >= MAX(1,M,P) if N > 0;
LDC >= 1 if N = 0.
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,*)
If COMPZ = 'N': Z is not referenced.
If COMPZ = 'I': on entry, Z need not be set;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix Z,
i.e., the product of the transformations
applied to A and C on the right.
LDZ INTEGER
The leading dimension of the array Z.
LDZ >= 1, if COMPZ = 'N';
LDZ >= MAX(1,N), if COMPZ = 'I'.
NOBSV (output) INTEGER
The order of the reduced matrix Ao, and the number of
columns of the reduced matrix Co; also, the order of the
observable part of the pair (C, A-lambda*I).
NLBLCK (output) INTEGER _
The number k, of full column rank blocks Ai-1,i in the
staircase form of the pencil (Ao-lambda*I) (see (1)).
( Co )
CTAU (output) INTEGER array, dimension (N)
CTAU(i), for i = 1, ..., NLBLCK, is the column dimension
_
of the full column rank block Ai-1,i in the staircase
form (1).
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in rank determinations when
transforming the pair (A,C). If the user sets TOL > 0,
then the given value of TOL is used as a lower bound for
reciprocal condition numbers in rank determinations; a
(sub)matrix whose estimated condition number is less than
1/TOL is considered to be of full rank. If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
defined by TOLDEF = N*N*EPS, is used instead, where EPS
is the machine precision (see LAPACK Library routine
DLAMCH). TOL < 1.
Workspace
IWORK INTEGER array, dimension (P)
DWORK DOUBLE PRECISION array, dimension (N+MAX(1, N, 3*P, M))
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The subroutine is based on the dual of the reduction algorithms of [1].References
[1] Varga, A.
Computation of Irreducible Generalized State-Space
Realizations.
Kybernetika, vol. 26, pp. 89-106, 1990.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( N**3 ) floating point operations.Further Comments
If the system matrices A and C are badly scaled, it is generally recommendable to scale them with the SLICOT routine TB01ID, before calling TG01UX.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To find a controllable realization for the linear time-invariant
multi-input system
dX/dt = A * X + B1 * U1 + B2 * U2,
Y = C * X,
where A, B1, B2 and C are N-by-N, N-by-M1, N-by-M2, and P-by-N
matrices, respectively, and A and [B1,B2] are reduced by this
routine to orthogonal canonical form using (and optionally
accumulating) orthogonal similarity transformations, which are
also applied to C. Specifically, the system (A, [B1,B2], C) is
reduced to the triplet (Ac, [Bc1,Bc2], Cc), where
Ac = Z' * A * Z, [Bc1,Bc2] = Z' * [B1,B2], Cc = C * Z, with
[ Acont * ] [ Bcont1, Bcont2 ]
Ac = [ ], [Bc1,Bc1] = [ ],
[ 0 Auncont ] [ 0 0 ]
and
[ A11 A12 . . . A1,p-2 A1,p-1 A1p ]
[ A21 A22 . . . A2,p-2 A2,p-1 A2p ]
[ A31 A32 . . . A3,p-2 A3,p-1 A3p ]
[ 0 A42 . . . A4,p-2 A4,p-1 A4p ]
Acont = [ . . . . . . . . ],
[ . . . . . . . ]
[ . . . . . . ]
[ 0 0 . . . Ap,p-2 Ap,p-1 App ]
[ B11 B12 ]
[ 0 B22 ]
[ 0 0 ]
[ 0 0 ]
[Bc1,Bc2] = [ . . ],
[ . . ]
[ . . ]
[ 0 0 ]
where the blocks B11, B22, A31, ..., Ap,p-2 have full row ranks and
p is the controllability index of the pair (A,[B1,B2]). The size of the
block Auncont is equal to the dimension of the uncontrollable
subspace of the pair (A,[B1,B2]).
Specification
SUBROUTINE TB01UY( JOBZ, N, M1, M2, P, A, LDA, B, LDB, C, LDC,
$ NCONT, INDCON, NBLK, Z, LDZ, TAU, TOL, IWORK,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBZ
INTEGER INDCON, INFO, LDA, LDB, LDC, LDWORK, LDZ, M1,
$ M2, N, NCONT, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), TAU(*),
$ Z(LDZ,*)
INTEGER IWORK(*), NBLK(*)
Arguments
Mode Parameters
JOBZ CHARACTER*1
Indicates whether the user wishes to accumulate in a
matrix Z the orthogonal similarity transformations for
reducing the system, as follows:
= 'N': Do not form Z and do not store the orthogonal
transformations;
= 'F': Do not form Z, but store the orthogonal
transformations in the factored form;
= 'I': Z is initialized to the unit matrix and the
orthogonal transformation matrix Z is returned.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation,
i.e., the order of the matrix A. N >= 0.
M1 (input) INTEGER
The number of system inputs in U1, or of columns of B1.
M1 >= 0.
M2 (input) INTEGER
The number of system inputs in U2, or of columns of B2.
M2 >= 0.
P (input) INTEGER
The number of system outputs, or of rows of C. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state dynamics matrix A.
On exit, the leading N-by-N part of this array contains
the transformed state dynamics matrix Ac = Z'*A*Z. The
leading NCONT-by-NCONT diagonal block of this matrix,
Acont, is the state dynamics matrix of a controllable
realization for the original system. The elements below
the second block-subdiagonal are set to zero.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension
(LDB,M1+M2)
On entry, the leading N-by-(M1+M2) part of this array must
contain the compound input matrix B = [B1,B2], where B1 is
N-by-M1 and B2 is N-by-M2.
On exit, the leading N-by-(M1+M2) part of this array
contains the transformed compound input matrix [Bc1,Bc2] =
Z'*[B1,B2]. The leading NCONT-by-(M1+M2) part of this
array, [Bcont1, Bcont2], is the compound input matrix of
a controllable realization for the original system.
All elements below the first block-diagonal are set to
zero.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed output matrix Cc, given by C * Z.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,P).
NCONT (output) INTEGER
The order of the controllable state-space representation.
INDCON (output) INTEGER
The controllability index of the controllable part of the
system representation.
NBLK (output) INTEGER array, dimension (2*N)
The leading INDCON elements of this array contain the
orders of the diagonal blocks of Acont. INDCON is always
an even number, and the INDCON/2 odd and even components
of NBLK have decreasing values, respectively.
Note that some elements of NBLK can be zero.
Z (output) DOUBLE PRECISION array, dimension (LDZ,N)
If JOBZ = 'I', then the leading N-by-N part of this
array contains the matrix of accumulated orthogonal
similarity transformations which reduces the given system
to orthogonal canonical form.
If JOBZ = 'F', the elements below the diagonal, with the
array TAU, represent the orthogonal transformation matrix
as a product of elementary reflectors. The transformation
matrix can then be obtained by calling the LAPACK Library
routine DORGQR.
If JOBZ = 'N', the array Z is not referenced and can be
supplied as a dummy array (i.e., set parameter LDZ = 1 and
declare this array to be Z(1,1) in the calling program).
LDZ INTEGER
The leading dimension of the array Z. If JOBZ = 'I' or
JOBZ = 'F', LDZ >= MAX(1,N); if JOBZ = 'N', LDZ >= 1.
TAU (output) DOUBLE PRECISION array, dimension (MIN(N,M1+M2))
The elements of TAU contain the scalar factors of the
elementary reflectors used in the reduction of [B1,B2]
and A.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in rank determinations when
transforming (A, [B1,B2]). If the user sets TOL > 0, then
the given value of TOL is used as a lower bound for the
reciprocal condition number (see the description of the
argument RCOND in the SLICOT routine MB03OD); a
(sub)matrix whose estimated condition number is less than
1/TOL is considered to be of full rank. If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
defined by TOLDEF = N*N*EPS, is used instead, where EPS
is the machine precision (see LAPACK Library routine
DLAMCH).
Workspace
IWORK INTEGER array, dimension (MAX(M1,M2))
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= 1, and
LDWORK >= MAX(N, 3*MAX(M1,M2), P), if MIN(N,M1+M2) > 0.
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message related to LDWORK is issued by
XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The implemented algorithm [1] represents a specialization of the controllability staircase algorithm of [2] to the special structure of the input matrix B = [B1,B2].References
[1] Varga, A.
Reliable algorithms for computing minimal dynamic covers.
Proc. CDC'2003, Hawaii, 2003.
[2] Varga, A.
Numerically stable algorithm for standard controllability
form determination.
Electronics Letters, vol. 17, pp. 74-75, 1981.
Numerical Aspects
3 The algorithm requires 0(N ) operations and is backward stable.Further Comments
If the system matrices A and B are badly scaled, it would be useful to scale them with SLICOT routine TB01ID, before calling the routine.Example
Program Text
* TB01UY EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDZ
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDZ = NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( NMAX, 3*MMAX, PMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, INDCON, J, M, M1, M2, N, NCONT, P
CHARACTER*1 JOBZ
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ DWORK(LDWORK), TAU(MIN(NMAX,MMAX)), Z(LDZ,NMAX)
INTEGER IWORK(LIWORK), NBLK(2*NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL DORGQR, TB01UY
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M1, M2, P, TOL, JOBZ
M = M1 + M2
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,N ), J = 1,M )
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find a controllable ssr for the given system.
CALL TB01UY( JOBZ, N, M1, M2, P, A, LDA, B, LDB, C, LDC,
$ NCONT, INDCON, NBLK, Z, LDZ, TAU, TOL,
$ IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NCONT
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NCONT
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NCONT )
20 CONTINUE
WRITE ( NOUT, FMT = 99994 ) ( NBLK(I), I = 1,INDCON )
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NCONT
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
WRITE ( NOUT, FMT = 99987 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NCONT )
60 CONTINUE
WRITE ( NOUT, FMT = 99992 ) INDCON
IF ( LSAME( JOBZ, 'F' ) )
$ CALL DORGQR( N, N, N, Z, LDZ, TAU, DWORK, LDWORK,
$ INFO )
IF ( LSAME( JOBZ, 'F' ).OR.LSAME( JOBZ, 'I' ) ) THEN
WRITE ( NOUT, FMT = 99991 )
DO 80 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
80 CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TB01UY EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TB01UY = ',I2)
99997 FORMAT (' The order of the controllable state-space representati',
$ 'on = ',I2)
99996 FORMAT (/' The transformed state dynamics matrix of a controllab',
$ 'le realization is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' and the dimensions of its diagonal blocks are ',
$ /20(1X,I2))
99993 FORMAT (/' The transformed input/state matrix B of a controllabl',
$ 'e realization is ')
99992 FORMAT (/' The controllability index of the transformed system r',
$ 'epresentation = ',I2)
99991 FORMAT (/' The similarity transformation matrix Z is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99987 FORMAT (/' The transformed output/state matrix C of a controlla',
$ 'ble realization is ')
END
Program Data
TB01UY EXAMPLE PROGRAM DATA 3 1 1 2 0.0 I -1.0 0.0 0.0 -2.0 -2.0 -2.0 -1.0 0.0 -3.0 1.0 0.0 0.0 0.0 2.0 1.0 0.0 2.0 1.0 1.0 0.0 0.0Program Results
TB01UY EXAMPLE PROGRAM RESULTS The order of the controllable state-space representation = 2 The transformed state dynamics matrix of a controllable realization is -1.0000 0.0000 2.2361 -3.0000 and the dimensions of its diagonal blocks are 1 1 The transformed input/state matrix B of a controllable realization is 1.0000 0.0000 0.0000 -2.2361 The transformed output/state matrix C of a controllable realization is 0.0000 -2.2361 1.0000 0.0000 The controllability index of the transformed system representation = 2 The similarity transformation matrix Z is 1.0000 0.0000 0.0000 0.0000 -0.8944 -0.4472 0.0000 -0.4472 0.8944
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TB01VD.html 0000664 0000000 0000000 00000016143 14560147231 0020357 0 ustar 00root root 0000000 0000000
Purpose
To convert the linear discrete-time system given as (A, B, C, D), with initial state x0, into the output normal form [1], with parameter vector THETA. The matrix A is assumed to be stable. The matrices A, B, C, D and the vector x0 are converted, so that on exit they correspond to the system defined by THETA.Specification
SUBROUTINE TB01VD( APPLY, N, M, L, A, LDA, B, LDB, C, LDC, D, LDD,
$ X0, THETA, LTHETA, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER APPLY
INTEGER INFO, L, LDA, LDB, LDC, LDD, LDWORK, LTHETA, M,
$ N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), THETA(*), X0(*)
Arguments
Mode Parameters
APPLY CHARACTER*1
Specifies whether or not the parameter vector should be
transformed using a bijective mapping, as follows:
= 'A' : apply the bijective mapping to the N vectors in
THETA corresponding to the matrices A and C;
= 'N' : do not apply the bijective mapping.
The transformation performed when APPLY = 'A' allows
to get rid of the constraints norm(THETAi) < 1, i = 1:N.
A call of the SLICOT Library routine TB01VY associated to
a call of TB01VD must use the same value of APPLY.
Input/Output Parameters
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
L (input) INTEGER
The number of system outputs. L >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the system state matrix A, assumed to be stable.
On exit, the leading N-by-N part of this array contains
the transformed system state matrix corresponding to the
output normal form with parameter vector THETA.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the system input matrix B.
On exit, the leading N-by-M part of this array contains
the transformed system input matrix corresponding to the
output normal form with parameter vector THETA.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading L-by-N part of this array must
contain the system output matrix C.
On exit, the leading L-by-N part of this array contains
the transformed system output matrix corresponding to the
output normal form with parameter vector THETA.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,L).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading L-by-M part of this array must contain the
system input/output matrix D.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,L).
X0 (input/output) DOUBLE PRECISION array, dimension (N)
On entry, this array must contain the initial state of the
system, x0.
On exit, this array contains the transformed initial state
of the system, corresponding to the output normal form
with parameter vector THETA.
THETA (output) DOUBLE PRECISION array, dimension (LTHETA)
The leading N*(L+M+1)+L*M part of this array contains the
parameter vector that defines a system (A, B, C, D, x0)
which is equivalent up to a similarity transformation to
the system given on entry. The parameters are:
THETA(1:N*L) : parameters for A, C;
THETA(N*L+1:N*(L+M)) : parameters for B;
THETA(N*(L+M)+1:N*(L+M)+L*M) : parameters for D;
THETA(N*(L+M)+L*M+1:N*(L+M+1)+L*M): parameters for x0.
LTHETA INTEGER
The length of array THETA. LTHETA >= N*(L+M+1)+L*M.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1, N*N*L + N*L + N,
N*N + MAX(N*N + N*MAX(N,L) + 6*N + MIN(N,L),
N*M)).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the Lyapunov equation A'*Q*A - Q = -scale^2*C'*C
could only be solved with scale = 0;
= 2: if matrix A is not discrete-time stable;
= 3: if the QR algorithm failed to converge for
matrix A.
Method
The matrices A and C are converted to output normal form.
First, the Lyapunov equation
A'*Q*A - Q = -scale^2*C'*C,
is solved in the Cholesky factor T, T'*T = Q, and then T is used
to get the transformation matrix.
The matrix B and the initial state x0 are transformed accordingly.
Then, the QR factorization of the transposed observability matrix
is computed, and the matrix Q is used to further transform the
system matrices. The parameters characterizing A and C are finally
obtained by applying a set of N orthogonal transformations.
References
[1] Peeters, R.L.M., Hanzon, B., and Olivi, M.
Balanced realizations of discrete-time stable all-pass
systems and the tangential Schur algorithm.
Proceedings of the European Control Conference,
31 August - 3 September 1999, Karlsruhe, Germany.
Session CP-6, Discrete-time Systems, 1999.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To convert the linear discrete-time system given as its output normal form [1], with parameter vector THETA, into the state-space representation (A, B, C, D), with the initial state x0.Specification
SUBROUTINE TB01VY( APPLY, N, M, L, THETA, LTHETA, A, LDA, B, LDB,
$ C, LDC, D, LDD, X0, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER APPLY
INTEGER INFO, L, LDA, LDB, LDC, LDD, LDWORK, LTHETA, M,
$ N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), THETA(*), X0(*)
Arguments
Mode Parameters
APPLY CHARACTER*1
Specifies whether or not the parameter vector should be
transformed using a bijective mapping, as follows:
= 'A' : apply the bijective mapping to the N vectors in
THETA corresponding to the matrices A and C;
= 'N' : do not apply the bijective mapping.
The transformation performed when APPLY = 'A' allows
to get rid of the constraints norm(THETAi) < 1, i = 1:N.
A call of the SLICOT Library routine TB01VD associated to
a call of TB01VY must use the same value of APPLY.
Input/Output Parameters
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
L (input) INTEGER
The number of system outputs. L >= 0.
THETA (input) DOUBLE PRECISION array, dimension (LTHETA)
The leading N*(L+M+1)+L*M part of this array must contain
the parameter vector that defines a system (A, B, C, D),
with the initial state x0. The parameters are:
THETA(1:N*L) : parameters for A, C;
THETA(N*L+1:N*(L+M)) : parameters for B;
THETA(N*(L+M)+1:N*(L+M)+L*M) : parameters for D;
THETA(N*(L+M)+L*M+1:N*(L+M+1)+L*M): parameters for x0.
LTHETA INTEGER
The length of array THETA. LTHETA >= N*(L+M+1)+L*M.
A (output) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array contains the system
state matrix corresponding to the output normal form with
parameter vector THETA.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (output) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array contains the system
input matrix corresponding to the output normal form with
parameter vector THETA.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (output) DOUBLE PRECISION array, dimension (LDC,N)
The leading L-by-N part of this array contains the system
output matrix corresponding to the output normal form with
parameter vector THETA.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,L).
D (output) DOUBLE PRECISION array, dimension (LDD,M)
The leading L-by-M part of this array contains the system
input/output matrix corresponding to the output normal
form with parameter vector THETA.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,L).
X0 (output) DOUBLE PRECISION array, dimension (N)
This array contains the initial state of the system, x0,
corresponding to the output normal form with parameter
vector THETA.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= N*(N+L+1).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The parameters characterizing A and C are used to build N orthogonal transformations, which are then applied to recover these matrices.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To reduce the system state matrix A to an upper real Schur form by using an orthogonal similarity transformation A <-- U'*A*U and to apply the transformation to the matrices B and C: B <-- U'*B and C <-- C*U.Specification
SUBROUTINE TB01WD( N, M, P, A, LDA, B, LDB, C, LDC, U, LDU,
$ WR, WI, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDU, LDWORK, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), U(LDU,*),
$ WI(*), WR(*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation,
i.e. the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs, or of columns of B. M >= 0.
P (input) INTEGER
The number of system outputs, or of rows of C. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state dynamics matrix A.
On exit, the leading N-by-N part of this array contains
the matrix U' * A * U in real Schur form. The elements
below the first subdiagonal are set to zero.
Note: A matrix is in real Schur form if it is upper
quasi-triangular with 1-by-1 and 2-by-2 blocks.
2-by-2 blocks are standardized in the form
[ a b ]
[ c a ]
where b*c < 0. The eigenvalues of such a block
are a +- sqrt(bc).
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input matrix B.
On exit, the leading N-by-M part of this array contains
the transformed input matrix U' * B.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed output matrix C * U.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
U (output) DOUBLE PRECISION array, dimension (LDU,N)
The leading N-by-N part of this array contains the
orthogonal transformation matrix used to reduce A to the
real Schur form. The columns of U are the Schur vectors of
matrix A.
LDU INTEGER
The leading dimension of array U. LDU >= max(1,N).
WR, WI (output) DOUBLE PRECISION arrays, dimension (N)
WR and WI contain the real and imaginary parts,
respectively, of the computed eigenvalues of A. The
eigenvalues will be in the same order that they appear on
the diagonal of the output real Schur form of A. Complex
conjugate pairs of eigenvalues will appear consecutively
with the eigenvalue having the positive imaginary part
first.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The dimension of working array DWORK. LWORK >= 3*N.
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, the QR algorithm failed to compute
all the eigenvalues; elements i+1:N of WR and WI
contain those eigenvalues which have converged;
U contains the matrix which reduces A to its
partially converged Schur form.
Method
Matrix A is reduced to a real Schur form using an orthogonal similarity transformation A <- U'*A*U. Then, the transformation is applied to the matrices B and C: B <-- U'*B and C <-- C*U.Numerical Aspects
3 The algorithm requires about 10N floating point operations.Further Comments
NoneExample
Program Text
* TB01WD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDU
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDU = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 3*NMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, M, N, P
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ DWORK(LDWORK), U(LDU,NMAX), WI(NMAX), WR(NMAX)
* .. External Subroutines ..
EXTERNAL TB01WD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find the transformed ssr for (A,B,C).
CALL TB01WD( N, M, P, A, LDA, B, LDB, C, LDC, U, LDU,
$ WR, WI, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99994 ) WR(I), WI(I)
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
60 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 70 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( U(I,J), J = 1,N )
70 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TB01WD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TB01WD = ',I2)
99997 FORMAT (' The eigenvalues of state dynamics matrix A are ')
99996 FORMAT (/' The transformed state dynamics matrix U''*A*U is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT ( ' (',F8.4,', ',F8.4,' )')
99993 FORMAT (/' The transformed input/state matrix U''*B is ')
99992 FORMAT (/' The transformed state/output matrix C*U is ')
99991 FORMAT (/' The similarity transformation matrix U is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
TB01WD EXAMPLE PROGRAM DATA (Continuous system)
5 2 3
-0.04165 4.9200 -4.9200 0 0
-1.387944 -3.3300 0 0 0
0.5450 0 0 -0.5450 0
0 0 4.9200 -0.04165 4.9200
0 0 0 -1.387944 -3.3300
0 0
3.3300 0
0 0
0 0
0 3.3300
1 0 0 0 0
0 0 1 0 0
0 0 0 1 0
Program Results
TB01WD EXAMPLE PROGRAM RESULTS The eigenvalues of state dynamics matrix A are ( -0.7483, 2.9940 ) ( -0.7483, -2.9940 ) ( -1.6858, 2.0311 ) ( -1.6858, -2.0311 ) ( -1.8751, 0.0000 ) The transformed state dynamics matrix U'*A*U is -0.7483 -8.6406 0.0000 0.0000 1.1745 1.0374 -0.7483 0.0000 0.0000 -2.1164 0.0000 0.0000 -1.6858 5.5669 0.0000 0.0000 0.0000 -0.7411 -1.6858 0.0000 0.0000 0.0000 0.0000 0.0000 -1.8751 The transformed input/state matrix U'*B is -0.5543 0.5543 -1.6786 1.6786 -0.8621 -0.8621 2.1912 2.1912 -1.5555 1.5555 The transformed state/output matrix C*U is 0.6864 -0.0987 0.6580 0.2589 -0.1381 -0.0471 0.6873 0.0000 0.0000 -0.7249 -0.6864 0.0987 0.6580 0.2589 0.1381 The similarity transformation matrix U is 0.6864 -0.0987 0.6580 0.2589 -0.1381 -0.1665 -0.5041 -0.2589 0.6580 -0.4671 -0.0471 0.6873 0.0000 0.0000 -0.7249 -0.6864 0.0987 0.6580 0.2589 0.1381 0.1665 0.5041 -0.2589 0.6580 0.4671
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TB01WX.html 0000664 0000000 0000000 00000023757 14560147231 0020415 0 ustar 00root root 0000000 0000000
Purpose
To reduce the system state matrix A to an upper Hessenberg form by using an orthogonal similarity transformation A <-- U'*A*U and to apply the transformation to the matrices B and C: B <-- U'*B and C <-- C*U.Specification
SUBROUTINE TB01WX( COMPU, N, M, P, A, LDA, B, LDB, C, LDC, U, LDU,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPU
INTEGER INFO, LDA, LDB, LDC, LDU, LDWORK, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), U(LDU,*)
Arguments
Mode Parameters
COMPU CHARACTER*1
= 'N': do not compute U;
= 'I': U is initialized to the unit matrix, and the
orthogonal matrix U is returned;
= 'U': U must contain an orthogonal matrix U1 on entry,
and the product U1*U is returned.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation,
i.e., the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs, or of columns of B. M >= 0.
P (input) INTEGER
The number of system outputs, or of rows of C. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state dynamics matrix A.
On exit, the leading N-by-N part of this array contains
the matrix U' * A * U in Hessenberg form. The elements
below the first subdiagonal are set to zero.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input matrix B.
On exit, the leading N-by-M part of this array contains
the transformed input matrix U' * B.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed output matrix C * U.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,P).
U (input/output) DOUBLE PRECISION array, dimension (LDU,*)
On entry, if COMPU = 'U', the leading N-by-N part of this
array must contain the given matrix U1. Otherwise, this
array need not be set on input.
On exit, if COMPU <> 'N', the leading N-by-N part of this
array contains the orthogonal transformation matrix used
to reduce A to the Hessenberg form (U1*U if COMPU = 'U').
If COMPU = 'N', this array is not referenced.
LDU INTEGER
The leading dimension of the array U.
LDU >= 1, if COMPU = 'N';
LDU >= max(1,N), if COMPU <> 'N'.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= 1, and if N > 0,
LDWORK >= N - 1 + MAX(N,M,P).
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message related to LDWORK is issued by
XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Matrix A is reduced to the Hessenberg form using an orthogonal similarity transformation A <- U'*A*U. Then, the transformation is applied to the matrices B and C: B <-- U'*B and C <-- C*U.Numerical Aspects
3 2
The algorithm requires about 5N /3 + N (M+P) floating point
3
operations, if COMPU = 'N'. Otherwise, 2N /3 additional operations
are needed.
Further Comments
NoneExample
Program Text
* TB01WX EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDU
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDU = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX - 1 + MAX( NMAX, MMAX, PMAX ) )
* .. Local Scalars ..
CHARACTER COMPU
INTEGER I, INFO, J, M, N, P
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ DWORK(LDWORK), U(LDU,NMAX)
* .. External Subroutines ..
EXTERNAL TB01WX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, COMPU
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find the transformed ssr for (A,B,C).
CALL TB01WX( COMPU, N, M, P, A, LDA, B, LDB, C, LDC, U,
$ LDU, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
60 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 70 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( U(I,J), J = 1,N )
70 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TB01WX EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TB01WX = ',I2)
99996 FORMAT (/' The transformed state dynamics matrix U''*A*U is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT ( ' (',F8.4,', ',F8.4,' )')
99993 FORMAT (/' The transformed input/state matrix U''*B is ')
99992 FORMAT (/' The transformed state/output matrix C*U is ')
99991 FORMAT (/' The similarity transformation matrix U is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
TB01WX EXAMPLE PROGRAM DATA (Continuous system)
5 2 3 I
-0.04165 4.9200 -4.9200 0 0
-1.387944 -3.3300 0 0 0
0.5450 0 0 -0.5450 0
0 0 4.9200 -0.04165 4.9200
0 0 0 -1.387944 -3.3300
0 0
3.3300 0
0 0
0 0
0 3.3300
1 0 0 0 0
0 0 1 0 0
0 0 0 1 0
Program Results
TB01WX EXAMPLE PROGRAM RESULTS The transformed state dynamics matrix U'*A*U is -0.0416 -6.3778 1.4826 -1.9856 1.2630 1.4911 -2.8851 -0.4353 0.8984 -0.5714 0.0000 -2.1254 1.6804 -4.9686 -1.7731 0.0000 0.0000 2.1880 -3.3545 -2.6069 0.0000 0.0000 0.0000 0.7554 -2.1424 The transformed input/state matrix U'*B is 0.0000 0.0000 -3.0996 0.0000 -0.6488 0.0000 0.8689 1.7872 -0.5527 2.8098 The transformed state/output matrix C*U is 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.3655 -0.4962 0.6645 -0.4227 0.0000 0.0000 -0.8461 -0.4498 0.2861 The similarity transformation matrix U is 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.9308 -0.1948 0.2609 -0.1660 0.0000 0.3655 -0.4962 0.6645 -0.4227 0.0000 0.0000 -0.8461 -0.4498 0.2861 0.0000 0.0000 0.0000 0.5367 0.8438
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TB01XD.html 0000664 0000000 0000000 00000012207 14560147231 0020356 0 ustar 00root root 0000000 0000000
Purpose
To apply a special transformation to a system given as a triple
(A,B,C),
A <-- P * A' * P, B <-- P * C', C <-- B' * P,
where P is a matrix with 1 on the secondary diagonal, and with 0
in the other entries. Matrix A can be specified as a band matrix.
Optionally, matrix D of the system can be transposed. This
transformation is actually a special similarity transformation of
the dual system.
Specification
SUBROUTINE TB01XD( JOBD, N, M, P, KL, KU, A, LDA, B, LDB, C, LDC,
$ D, LDD, INFO )
C .. Scalar Arguments ..
CHARACTER JOBD
INTEGER INFO, KL, KU, LDA, LDB, LDC, LDD, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * )
Arguments
Mode Parameters
JOBD CHARACTER*1
Specifies whether or not a non-zero matrix D appears in
the given state space model:
= 'D': D is present;
= 'Z': D is assumed a zero matrix.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A, the number of rows of matrix B
and the number of columns of matrix C.
N represents the dimension of the state vector. N >= 0.
M (input) INTEGER.
The number of columns of matrix B.
M represents the dimension of input vector. M >= 0.
P (input) INTEGER.
The number of rows of matrix C.
P represents the dimension of output vector. P >= 0.
KL (input) INTEGER
The number of subdiagonals of A to be transformed.
MAX( 0, N-1 ) >= KL >= 0.
KU (input) INTEGER
The number of superdiagonals of A to be transformed.
MAX( 0, N-1 ) >= KU >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the system state matrix A.
On exit, the leading N-by-N part of this array contains
the transformed (pertransposed) matrix P*A'*P.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension
(LDB,MAX(M,P))
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, the leading N-by-P part of this array contains
the dual input/state matrix P*C'.
LDB INTEGER
The leading dimension of the array B.
LDB >= MAX(1,N) if M > 0 or P > 0.
LDB >= 1 if M = 0 and P = 0.
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, the leading M-by-N part of this array contains
the dual state/output matrix B'*P.
LDC INTEGER
The leading dimension of array C.
LDC >= MAX(1,M,P) if N > 0.
LDC >= 1 if N = 0.
D (input/output) DOUBLE PRECISION array, dimension
(LDD,MAX(M,P))
On entry, if JOBD = 'D', the leading P-by-M part of this
array must contain the original direct transmission
matrix D.
On exit, if JOBD = 'D', the leading M-by-P part of this
array contains the transposed direct transmission matrix
D'. The array D is not referenced if JOBD = 'Z'.
LDD INTEGER
The leading dimension of array D.
LDD >= MAX(1,M,P) if JOBD = 'D'.
LDD >= 1 if JOBD = 'Z'.
Error Indicator
INFO INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The rows and/or columns of the matrices of the triplet (A,B,C) and, optionally, of the matrix D are swapped in a special way.Numerical Aspects
None.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To apply a special transformation to a system given as a triple
(A,B,C),
A <-- P * A' * P, B <-- P * C', C <-- B' * P,
where P is a matrix with 1 on the secondary diagonal, and with 0
in the other entries. Matrix A can be specified as a band matrix.
Optionally, matrix D of the system can be transposed. This
transformation is actually a special similarity transformation of
the dual system.
Specification
SUBROUTINE TB01XZ( JOBD, N, M, P, KL, KU, A, LDA, B, LDB, C, LDC,
$ D, LDD, INFO )
C .. Scalar Arguments ..
CHARACTER JOBD
INTEGER INFO, KL, KU, LDA, LDB, LDC, LDD, M, N, P
C .. Array Arguments ..
COMPLEX*16 A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * )
Arguments
Mode Parameters
JOBD CHARACTER*1
Specifies whether or not a non-zero matrix D appears in
the given state space model:
= 'D': D is present;
= 'Z': D is assumed a zero matrix.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A, the number of rows of matrix B
and the number of columns of matrix C.
N represents the dimension of the state vector. N >= 0.
M (input) INTEGER.
The number of columns of matrix B.
M represents the dimension of input vector. M >= 0.
P (input) INTEGER.
The number of rows of matrix C.
P represents the dimension of output vector. P >= 0.
KL (input) INTEGER
The number of subdiagonals of A to be transformed.
MAX( 0, N-1 ) >= KL >= 0.
KU (input) INTEGER
The number of superdiagonals of A to be transformed.
MAX( 0, N-1 ) >= KU >= 0.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the system state matrix A.
On exit, the leading N-by-N part of this array contains
the transformed (pertransposed) matrix P*A'*P.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) COMPLEX*16 array, dimension (LDB,MAX(M,P))
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, the leading N-by-P part of this array contains
the dual input/state matrix P*C'.
LDB INTEGER
The leading dimension of the array B.
LDB >= MAX(1,N) if M > 0 or P > 0.
LDB >= 1 if M = 0 and P = 0.
C (input/output) COMPLEX*16 array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, the leading M-by-N part of this array contains
the dual state/output matrix B'*P.
LDC INTEGER
The leading dimension of array C.
LDC >= MAX(1,M,P) if N > 0.
LDC >= 1 if N = 0.
D (input/output) COMPLEX*16 array, dimension (LDD,MAX(M,P))
On entry, if JOBD = 'D', the leading P-by-M part of this
array must contain the original direct transmission
matrix D.
On exit, if JOBD = 'D', the leading M-by-P part of this
array contains the transposed direct transmission matrix
D'. The array D is not referenced if JOBD = 'Z'.
LDD INTEGER
The leading dimension of array D.
LDD >= MAX(1,M,P) if JOBD = 'D'.
LDD >= 1 if JOBD = 'Z'.
Error Indicator
INFO INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The rows and/or columns of the matrices of the triplet (A,B,C) and, optionally, of the matrix D are swapped in a special way.Numerical Aspects
None.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To apply a special similarity transformation to a system given as
a triple (A,B,C),
A <-- P * A * P, B <-- P * B, C <-- C * P,
where P is a matrix with 1 on the secondary diagonal, and with 0
in the other entries.
Specification
SUBROUTINE TB01YD( N, M, P, A, LDA, B, LDB, C, LDC, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix A, the number of rows of matrix B
and the number of columns of matrix C.
N represents the dimension of the state vector. N >= 0.
M (input) INTEGER.
The number of columns of matrix B.
M represents the dimension of input vector. M >= 0.
P (input) INTEGER.
The number of rows of matrix C.
P represents the dimension of output vector. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the system state matrix A.
On exit, the leading N-by-N part of this array contains
the transformed matrix P*A*P.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the system input matrix B.
On exit, the leading N-by-M part of this array contains
the transformed matrix P*B.
LDB INTEGER
The leading dimension of the array B.
LDB >= MAX(1,N) if M > 0.
LDB >= 1 if M = 0.
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the system output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed matrix C*P.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,P).
Error Indicator
INFO INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The rows and/or columns of the matrices of the triplet (A,B,C) are swapped in a special way.Numerical Aspects
None.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To find a controllable realization for the linear time-invariant
single-input system
dX/dt = A * X + B * U,
Y = C * X,
where A is an N-by-N matrix, B is an N element vector, C is an
P-by-N matrix, and A and B are reduced by this routine to
orthogonal canonical form using (and optionally accumulating)
orthogonal similarity transformations, which are also applied
to C.
Specification
SUBROUTINE TB01ZD( JOBZ, N, P, A, LDA, B, C, LDC, NCONT, Z, LDZ,
$ TAU, TOL, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBZ
INTEGER INFO, LDA, LDC, LDWORK, LDZ, N, NCONT, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(*), C(LDC,*), DWORK(*), TAU(*),
$ Z(LDZ,*)
Arguments
Mode Parameters
JOBZ CHARACTER*1
Indicates whether the user wishes to accumulate in a
matrix Z the orthogonal similarity transformations for
reducing the system, as follows:
= 'N': Do not form Z and do not store the orthogonal
transformations;
= 'F': Do not form Z, but store the orthogonal
transformations in the factored form;
= 'I': Z is initialized to the unit matrix and the
orthogonal transformation matrix Z is returned.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation,
i.e. the order of the matrix A. N >= 0.
P (input) INTEGER
The number of system outputs, or of rows of C. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state dynamics matrix A.
On exit, the leading NCONT-by-NCONT upper Hessenberg
part of this array contains the canonical form of the
state dynamics matrix, given by Z' * A * Z, of a
controllable realization for the original system. The
elements below the first subdiagonal are set to zero.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (N)
On entry, the original input/state vector B.
On exit, the leading NCONT elements of this array contain
canonical form of the input/state vector, given by Z' * B,
with all elements but B(1) set to zero.
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the output/state matrix C.
On exit, the leading P-by-N part of this array contains
the transformed output/state matrix, given by C * Z, and
the leading P-by-NCONT part contains the output/state
matrix of the controllable realization.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
NCONT (output) INTEGER
The order of the controllable state-space representation.
Z (output) DOUBLE PRECISION array, dimension (LDZ,N)
If JOBZ = 'I', then the leading N-by-N part of this array
contains the matrix of accumulated orthogonal similarity
transformations which reduces the given system to
orthogonal canonical form.
If JOBZ = 'F', the elements below the diagonal, with the
array TAU, represent the orthogonal transformation matrix
as a product of elementary reflectors. The transformation
matrix can then be obtained by calling the LAPACK Library
routine DORGQR.
If JOBZ = 'N', the array Z is not referenced and can be
supplied as a dummy array (i.e. set parameter LDZ = 1 and
declare this array to be Z(1,1) in the calling program).
LDZ INTEGER
The leading dimension of array Z. If JOBZ = 'I' or
JOBZ = 'F', LDZ >= MAX(1,N); if JOBZ = 'N', LDZ >= 1.
TAU (output) DOUBLE PRECISION array, dimension (N)
The elements of TAU contain the scalar factors of the
elementary reflectors used in the reduction of B and A.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in determining the
controllability of (A,B). If the user sets TOL > 0, then
the given value of TOL is used as an absolute tolerance;
elements with absolute value less than TOL are considered
neglijible. If the user sets TOL <= 0, then an implicitly
computed, default tolerance, defined by
TOLDEF = N*EPS*MAX( NORM(A), NORM(B) ) is used instead,
where EPS is the machine precision (see LAPACK Library
routine DLAMCH).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,N,P).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The Householder matrix which reduces all but the first element of vector B to zero is found and this orthogonal similarity transformation is applied to the matrix A. The resulting A is then reduced to upper Hessenberg form by a sequence of Householder transformations. Finally, the order of the controllable state- space representation (NCONT) is determined by finding the position of the first sub-diagonal element of A which is below an appropriate zero threshold, either TOL or TOLDEF (see parameter TOL); if NORM(B) is smaller than this threshold, NCONT is set to zero, and no computations for reducing the system to orthogonal canonical form are performed. All orthogonal transformations determined in this process are also applied to the matrix C, from the right.References
[1] Konstantinov, M.M., Petkov, P.Hr. and Christov, N.D.
Orthogonal Invariants and Canonical Forms for Linear
Controllable Systems.
Proc. 8th IFAC World Congress, Kyoto, 1, pp. 49-54, 1981.
[2] Hammarling, S.J.
Notes on the use of orthogonal similarity transformations in
control.
NPL Report DITC 8/82, August 1982.
[3] Paige, C.C
Properties of numerical algorithms related to computing
controllability.
IEEE Trans. Auto. Contr., AC-26, pp. 130-138, 1981.
Numerical Aspects
3 The algorithm requires 0(N ) operations and is backward stable.Further Comments
NoneExample
Program Text
* TB01ZD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, PMAX
PARAMETER ( NMAX = 20, PMAX = 20 )
INTEGER LDA, LDC, LDZ
PARAMETER ( LDA = NMAX, LDC = PMAX, LDZ = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( NMAX, PMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, J, N, NCONT, P
CHARACTER*1 JOBZ
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(NMAX), C(LDC,NMAX), DWORK(LDWORK),
$ TAU(NMAX), Z(LDZ,NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL TB01ZD, DORGQR
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read in the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, P, TOL, JOBZ
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( B(I), I = 1,N )
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find a controllable realization for the given system.
CALL TB01ZD( JOBZ, N, P, A, LDA, B, C, LDC, NCONT, Z, LDZ,
$ TAU, TOL, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NCONT
DO 20 I = 1, NCONT
WRITE ( NOUT, FMT = 99994 ) ( A(I,J), J = 1,NCONT )
20 CONTINUE
WRITE ( NOUT, FMT = 99996 ) ( B(I), I = 1,NCONT )
WRITE ( NOUT, FMT = 99991 )
DO 30 I = 1, P
WRITE ( NOUT, FMT = 99994 ) ( C(I,J), J = 1,NCONT )
30 CONTINUE
IF ( LSAME( JOBZ, 'F' ) )
$ CALL DORGQR( N, N, N, Z, LDZ, TAU, DWORK, LDWORK,
$ INFO )
IF ( LSAME( JOBZ, 'F' ).OR.LSAME( JOBZ, 'I' ) ) THEN
WRITE ( NOUT, FMT = 99995 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99994 ) ( Z(I,J), J = 1,N )
40 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TB01ZD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TB01ZD = ',I2)
99997 FORMAT (' The order of the controllable state-space representati',
$ 'on = ',I2,//' The state dynamics matrix A of a controlla',
$ 'ble realization is ')
99996 FORMAT (/' The input/state vector B of a controllable realizatio',
$ 'n is ',/(1X,F8.4))
99995 FORMAT (/' The similarity transformation matrix Z is ')
99994 FORMAT (20(1X,F8.4))
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' P is out of range.',/' P = ',I5)
99991 FORMAT (/' The output/state matrix C of a controllable realizati',
$ 'on is ')
END
Program Data
TB01ZD EXAMPLE PROGRAM DATA 3 2 0.0 I 1.0 2.0 0.0 4.0 -1.0 0.0 0.0 0.0 1.0 1.0 0.0 1.0 0.0 2.0 1.0 1.0 0.0 0.0Program Results
TB01ZD EXAMPLE PROGRAM RESULTS The order of the controllable state-space representation = 3 The state dynamics matrix A of a controllable realization is 1.0000 1.4142 0.0000 2.8284 -1.0000 2.8284 0.0000 1.4142 1.0000 The input/state vector B of a controllable realization is -1.4142 0.0000 0.0000 The output/state matrix C of a controllable realization is -0.7071 -2.0000 0.7071 -0.7071 0.0000 -0.7071 The similarity transformation matrix Z is -0.7071 0.0000 -0.7071 0.0000 -1.0000 0.0000 -0.7071 0.0000 0.7071
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TB03AD.html 0000664 0000000 0000000 00000045251 14560147231 0020336 0 ustar 00root root 0000000 0000000
Purpose
To find a relatively prime left polynomial matrix representation
inv(P(s))*Q(s) or right polynomial matrix representation
Q(s)*inv(P(s)) with the same transfer matrix T(s) as that of a
given state-space representation, i.e.
inv(P(s))*Q(s) = Q(s)*inv(P(s)) = T(s) = C*inv(s*I-A)*B + D.
Specification
SUBROUTINE TB03AD( LERI, EQUIL, N, M, P, A, LDA, B, LDB, C, LDC,
$ D, LDD, NR, INDEX, PCOEFF, LDPCO1, LDPCO2,
$ QCOEFF, LDQCO1, LDQCO2, VCOEFF, LDVCO1, LDVCO2,
$ TOL, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER EQUIL, LERI
INTEGER INFO, LDA, LDB, LDC, LDD, LDPCO1, LDPCO2,
$ LDQCO1, LDQCO2, LDVCO1, LDVCO2, LDWORK, M, N,
$ NR, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER INDEX(*), IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), PCOEFF(LDPCO1,LDPCO2,*),
$ QCOEFF(LDQCO1,LDQCO2,*), VCOEFF(LDVCO1,LDVCO2,*)
Arguments
Mode Parameters
LERI CHARACTER*1
Indicates whether the left polynomial matrix
representation or the right polynomial matrix
representation is required as follows:
= 'L': A left matrix fraction is required;
= 'R': A right matrix fraction is required.
EQUIL CHARACTER*1
Specifies whether the user wishes to balance the triplet
(A,B,C), before computing a minimal state-space
representation, as follows:
= 'S': Perform balancing (scaling);
= 'N': Do not perform balancing.
Input/Output Parameters
N (input) INTEGER
The order of the state-space representation, i.e. the
order of the original state dynamics matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state dynamics matrix A.
On exit, the leading NR-by-NR part of this array contains
the upper block Hessenberg state dynamics matrix Amin of a
minimal realization for the original system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension
(LDB,MAX(M,P))
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B; the remainder
of the leading N-by-MAX(M,P) part is used as internal
workspace.
On exit, the leading NR-by-M part of this array contains
the transformed input/state matrix Bmin.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C; the remainder
of the leading MAX(M,P)-by-N part is used as internal
workspace.
On exit, the leading P-by-NR part of this array contains
the transformed state/output matrix Cmin.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,M,P).
D (input) DOUBLE PRECISION array, dimension (LDD,MAX(M,P))
The leading P-by-M part of this array must contain the
original direct transmission matrix D; the remainder of
the leading MAX(M,P)-by-MAX(M,P) part is used as internal
workspace.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,M,P).
NR (output) INTEGER
The order of the minimal state-space representation
(Amin,Bmin,Cmin).
INDEX (output) INTEGER array, dimension (P), if LERI = 'L', or
dimension (M), if LERI = 'R'.
If LERI = 'L', INDEX(I), I = 1,2,...,P, contains the
maximum degree of the polynomials in the I-th row of the
denominator matrix P(s) of the left polynomial matrix
representation.
These elements are ordered so that
INDEX(1) >= INDEX(2) >= ... >= INDEX(P).
If LERI = 'R', INDEX(I), I = 1,2,...,M, contains the
maximum degree of the polynomials in the I-th column of
the denominator matrix P(s) of the right polynomial
matrix representation.
These elements are ordered so that
INDEX(1) >= INDEX(2) >= ... >= INDEX(M).
PCOEFF (output) DOUBLE PRECISION array, dimension
(LDPCO1,LDPCO2,N+1)
If LERI = 'L' then porm = P, otherwise porm = M.
The leading porm-by-porm-by-kpcoef part of this array
contains the coefficients of the denominator matrix P(s),
where kpcoef = MAX(INDEX(I)) + 1.
PCOEFF(I,J,K) is the coefficient in s**(INDEX(iorj)-K+1)
of polynomial (I,J) of P(s), where K = 1,2,...,kpcoef; if
LERI = 'L' then iorj = I, otherwise iorj = J.
Thus for LERI = 'L', P(s) =
diag(s**INDEX(I))*(PCOEFF(.,.,1)+PCOEFF(.,.,2)/s+...).
LDPCO1 INTEGER
The leading dimension of array PCOEFF.
LDPCO1 >= MAX(1,P), if LERI = 'L';
LDPCO1 >= MAX(1,M), if LERI = 'R'.
LDPCO2 INTEGER
The second dimension of array PCOEFF.
LDPCO2 >= MAX(1,P), if LERI = 'L';
LDPCO2 >= MAX(1,M), if LERI = 'R'.
QCOEFF (output) DOUBLE PRECISION array, dimension
(LDQCO1,LDQCO2,N+1)
If LERI = 'L' then porp = M, otherwise porp = P.
If LERI = 'L', the leading porm-by-porp-by-kpcoef part
of this array contains the coefficients of the numerator
matrix Q(s).
If LERI = 'R', the leading porp-by-porm-by-kpcoef part
of this array contains the coefficients of the numerator
matrix Q(s).
QCOEFF(I,J,K) is defined as for PCOEFF(I,J,K).
LDQCO1 INTEGER
The leading dimension of array QCOEFF.
LDQCO1 >= MAX(1,P), if LERI = 'L';
LDQCO1 >= MAX(1,M,P), if LERI = 'R'.
LDQCO2 INTEGER
The second dimension of array QCOEFF.
LDQCO2 >= MAX(1,M), if LERI = 'L';
LDQCO2 >= MAX(1,M,P), if LERI = 'R'.
VCOEFF (output) DOUBLE PRECISION array, dimension
(LDVCO1,LDVCO2,N+1)
The leading porm-by-NR-by-kpcoef part of this array
contains the coefficients of the intermediate matrix V(s).
VCOEFF(I,J,K) is defined as for PCOEFF(I,J,K).
LDVCO1 INTEGER
The leading dimension of array VCOEFF.
LDVCO1 >= MAX(1,P), if LERI = 'L';
LDVCO1 >= MAX(1,M), if LERI = 'R'.
LDVCO2 INTEGER
The second dimension of array VCOEFF. LDVCO2 >= MAX(1,N).
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in rank determination when
transforming (A, B, C). If the user sets TOL > 0, then
the given value of TOL is used as a lower bound for the
reciprocal condition number (see the description of the
argument RCOND in the SLICOT routine MB03OD); a
(sub)matrix whose estimated condition number is less than
1/TOL is considered to be of full rank. If the user sets
TOL <= 0, then an implicitly computed, default tolerance
(determined by the SLICOT routine TB01UD) is used instead.
Workspace
IWORK INTEGER array, dimension (N+MAX(M,P))
On exit, if INFO = 0, the first nonzero elements of
IWORK(1:N) return the orders of the diagonal blocks of A.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1, N + MAX(N, 3*M, 3*P), PM*(PM + 2))
where PM = P, if LERI = 'L';
PM = M, if LERI = 'R'.
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if a singular matrix was encountered during the
computation of V(s);
= 2: if a singular matrix was encountered during the
computation of P(s).
Method
The method for a left matrix fraction will be described here: right matrix fractions are dealt with by constructing a left fraction for the dual of the original system. The first step is to obtain, by means of orthogonal similarity transformations, a minimal state-space representation (Amin,Bmin,Cmin,D) for the original system (A,B,C,D), where Amin is lower block Hessenberg with all its superdiagonal blocks upper triangular and Cmin has all but its first rank(C) columns zero. The number and dimensions of the blocks of Amin now immediately yield the row degrees of P(s) with P(s) row proper: furthermore, the P-by-NR polynomial matrix V(s) (playing a similar role to S(s) in Wolovich's Structure Theorem) can be calculated a column block at a time, in reverse order, from Amin. P(s) is then found as if it were the O-th column block of V(s) (using Cmin as well as Amin), while Q(s) = (V(s) * Bmin) + (P(s) * D). Finally, a special similarity transformation is used to put Amin in an upper block Hessenberg form.References
[1] Williams, T.W.C.
An Orthogonal Structure Theorem for Linear Systems.
Kingston Polytechnic Control Systems Research Group,
Internal Report 82/2, July 1982.
[2] Patel, R.V.
On Computing Matrix Fraction Descriptions and Canonical
Forms of Linear Time-Invariant Systems.
UMIST Control Systems Centre Report 489, 1980.
(Algorithms 1 and 2, extensively modified).
Numerical Aspects
3 The algorithm requires 0(N ) operations.Further Comments
NoneExample
Program Text
* TB03AD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER MAXMP
PARAMETER ( MAXMP = MAX( MMAX, PMAX ) )
INTEGER LDA, LDB, LDC, LDD, LDPCO1, LDPCO2, LDQCO1,
$ LDQCO2, LDVCO1, LDVCO2, NMAXP1
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = MAXMP,
$ LDD = MAXMP, LDPCO1 = MAXMP, LDPCO2 = MAXMP,
$ LDQCO1 = MAXMP, LDQCO2 = MAXMP, LDVCO1 = MAXMP,
$ LDVCO2 = NMAX, NMAXP1 = NMAX+1 )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX + MAXMP )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( NMAX + MAX( NMAX, 3*MAXMP ),
$ MAXMP*( MAXMP + 2 ) ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INDBLK, INFO, J, K, KPCOEF, M, N, NR, P, PORM,
$ PORP
CHARACTER*1 EQUIL, LERI
LOGICAL LLERI
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MAXMP), C(LDC,NMAX),
$ D(LDD,MAXMP), DWORK(LDWORK),
$ PCOEFF(LDPCO1,LDPCO2,NMAXP1),
$ QCOEFF(LDQCO1,LDQCO2,NMAXP1),
$ VCOEFF(LDVCO1,LDVCO2,NMAXP1)
INTEGER INDEX(MAXMP), IWORK(LIWORK)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL TB03AD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL, LERI, EQUIL
LLERI = LSAME( LERI, 'L' )
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99985 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
* Find the right pmr which is equivalent to the ssr
* C*inv(sI-A)*B+D.
CALL TB03AD( LERI, EQUIL, N, M, P, A, LDA, B, LDB, C,
$ LDC, D, LDD, NR, INDEX, PCOEFF, LDPCO1,
$ LDPCO2, QCOEFF, LDQCO1, LDQCO2, VCOEFF,
$ LDVCO1, LDVCO2, TOL, IWORK, DWORK, LDWORK,
$ INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NR
DO 20 I = 1, NR
WRITE ( NOUT, FMT = 99996 ) ( A(I,J), J = 1,NR )
20 CONTINUE
INDBLK = 0
DO 40 I = 1, N
IF ( IWORK(I).NE.0 ) INDBLK = INDBLK + 1
40 CONTINUE
WRITE ( NOUT, FMT = 99995 ) ( IWORK(I), I = 1,INDBLK )
WRITE ( NOUT, FMT = 99994 )
DO 60 I = 1, NR
WRITE ( NOUT, FMT = 99996 ) ( B(I,J), J = 1,M )
60 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 80 I = 1, P
WRITE ( NOUT, FMT = 99996 ) ( C(I,J), J = 1,NR )
80 CONTINUE
IF ( LLERI ) THEN
PORM = P
PORP = M
WRITE ( NOUT, FMT = 99992 ) INDBLK
ELSE
PORM = M
PORP = P
WRITE ( NOUT, FMT = 99991 ) INDBLK
END IF
WRITE ( NOUT, FMT = 99990 ) ( INDEX(I), I = 1,PORM )
KPCOEF = 0
DO 100 I = 1, PORM
KPCOEF = MAX( KPCOEF, INDEX(I) )
100 CONTINUE
KPCOEF = KPCOEF + 1
WRITE ( NOUT, FMT = 99989 )
DO 140 I = 1, PORM
DO 120 J = 1, PORM
WRITE ( NOUT, FMT = 99996 )
$ ( PCOEFF(I,J,K), K = 1,KPCOEF )
120 CONTINUE
140 CONTINUE
WRITE ( NOUT, FMT = 99988 )
IF ( LLERI ) THEN
DO 180 I = 1, PORM
DO 160 J = 1, PORP
WRITE ( NOUT, FMT = 99996 )
$ ( QCOEFF(I,J,K), K = 1,KPCOEF )
160 CONTINUE
180 CONTINUE
ELSE
DO 220 I = 1, PORP
DO 200 J = 1, PORM
WRITE ( NOUT, FMT = 99996 )
$ ( QCOEFF(I,J,K), K = 1,KPCOEF )
200 CONTINUE
220 CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TB03AD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TB03AD = ',I2)
99997 FORMAT (' The order of the minimal state-space representation = ',
$ I2,//' The transformed state dynamics matrix of a minimal',
$ ' realization is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' and the dimensions of its diagonal blocks are ',/20(I5)
$ )
99994 FORMAT (/' The transformed input/state matrix of a minimal reali',
$ 'zation is ')
99993 FORMAT (/' The transformed state/output matrix of a minimal real',
$ 'ization is ')
99992 FORMAT (/' The observability index of the transformed minimal sy',
$ 'stem representation = ',I2)
99991 FORMAT (/' The controllability index of the transformed minimal ',
$ 'system representation = ',I2)
99990 FORMAT (/' INDEX is ',/20(I5))
99989 FORMAT (/' The denominator matrix P(s) is ')
99988 FORMAT (/' The numerator matrix Q(s) is ')
99987 FORMAT (/' N is out of range.',/' N = ',I5)
99986 FORMAT (/' M is out of range.',/' M = ',I5)
99985 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
TB03AD EXAMPLE PROGRAM DATA 3 1 2 0.0 R N 1.0 2.0 0.0 4.0 -1.0 0.0 0.0 0.0 1.0 1.0 0.0 1.0 0.0 1.0 -1.0 0.0 0.0 1.0 0.0 1.0Program Results
TB03AD EXAMPLE PROGRAM RESULTS
The order of the minimal state-space representation = 3
The transformed state dynamics matrix of a minimal realization is
1.0000 -1.4142 0.0000
-2.8284 -1.0000 2.8284
0.0000 1.4142 1.0000
and the dimensions of its diagonal blocks are
1 1 1
The transformed input/state matrix of a minimal realization is
-1.4142
0.0000
0.0000
The transformed state/output matrix of a minimal realization is
0.7071 1.0000 0.7071
-0.7071 0.0000 -0.7071
The controllability index of the transformed minimal system representation = 3
INDEX is
3
The denominator matrix P(s) is
0.1768 -0.1768 -1.5910 1.5910
The numerator matrix Q(s) is
0.0000 -0.1768 0.7071 0.8839
0.1768 0.0000 -1.5910 0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TB04AD.html 0000664 0000000 0000000 00000042561 14560147231 0020340 0 ustar 00root root 0000000 0000000
Purpose
To find the transfer matrix T(s) of a given state-space representation (A,B,C,D). T(s) is expressed as either row or column polynomial vectors over monic least common denominator polynomials.Specification
SUBROUTINE TB04AD( ROWCOL, N, M, P, A, LDA, B, LDB, C, LDC, D,
$ LDD, NR, INDEX, DCOEFF, LDDCOE, UCOEFF, LDUCO1,
$ LDUCO2, TOL1, TOL2, IWORK, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER ROWCOL
INTEGER INFO, LDA, LDB, LDC, LDD, LDDCOE, LDUCO1,
$ LDUCO2, LDWORK, M, N, NR, P
DOUBLE PRECISION TOL1, TOL2
C .. Array Arguments ..
INTEGER INDEX(*), IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DCOEFF(LDDCOE,*), DWORK(*),
$ UCOEFF(LDUCO1,LDUCO2,*)
Arguments
Mode Parameters
ROWCOL CHARACTER*1
Indicates whether the transfer matrix T(s) is required
as rows or columns over common denominators as follows:
= 'R': T(s) is required as rows over common denominators;
= 'C': T(s) is required as columns over common
denominators.
Input/Output Parameters
N (input) INTEGER
The order of the state-space representation, i.e. the
order of the original state dynamics matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state dynamics matrix A.
On exit, the leading NR-by-NR part of this array contains
the upper block Hessenberg state dynamics matrix A of a
transformed representation for the original system: this
is completely controllable if ROWCOL = 'R', or completely
observable if ROWCOL = 'C'.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M),
if ROWCOL = 'R', and (LDB,MAX(M,P)) if ROWCOL = 'C'.
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B; if
ROWCOL = 'C', the remainder of the leading N-by-MAX(M,P)
part is used as internal workspace.
On exit, the leading NR-by-M part of this array contains
the transformed input/state matrix B.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C; if
ROWCOL = 'C', the remainder of the leading MAX(M,P)-by-N
part is used as internal workspace.
On exit, the leading P-by-NR part of this array contains
the transformed state/output matrix C.
LDC INTEGER
The leading dimension of array C.
LDC >= MAX(1,P) if ROWCOL = 'R';
LDC >= MAX(1,M,P) if ROWCOL = 'C'.
D (input) DOUBLE PRECISION array, dimension (LDD,M),
if ROWCOL = 'R', and (LDD,MAX(M,P)) if ROWCOL = 'C'.
The leading P-by-M part of this array must contain the
original direct transmission matrix D; if ROWCOL = 'C',
this array is modified internally, but restored on exit,
and the remainder of the leading MAX(M,P)-by-MAX(M,P)
part is used as internal workspace.
LDD INTEGER
The leading dimension of array D.
LDD >= MAX(1,P) if ROWCOL = 'R';
LDD >= MAX(1,M,P) if ROWCOL = 'C'.
NR (output) INTEGER
The order of the transformed state-space representation.
INDEX (output) INTEGER array, dimension (porm), where porm = P,
if ROWCOL = 'R', and porm = M, if ROWCOL = 'C'.
The degrees of the denominator polynomials.
DCOEFF (output) DOUBLE PRECISION array, dimension (LDDCOE,N+1)
The leading porm-by-kdcoef part of this array contains
the coefficients of each denominator polynomial, where
kdcoef = MAX(INDEX(I)) + 1.
DCOEFF(I,K) is the coefficient in s**(INDEX(I)-K+1) of
the I-th denominator polynomial, where K = 1,2,...,kdcoef.
LDDCOE INTEGER
The leading dimension of array DCOEFF.
LDDCOE >= MAX(1,P) if ROWCOL = 'R';
LDDCOE >= MAX(1,M) if ROWCOL = 'C'.
UCOEFF (output) DOUBLE PRECISION array, dimension
(LDUCO1,LDUCO2,N+1)
If ROWCOL = 'R' then porp = M, otherwise porp = P.
The leading porm-by-porp-by-kdcoef part of this array
contains the coefficients of the numerator matrix U(s).
UCOEFF(I,J,K) is the coefficient in s**(INDEX(iorj)-K+1)
of polynomial (I,J) of U(s), where K = 1,2,...,kdcoef;
if ROWCOL = 'R' then iorj = I, otherwise iorj = J.
Thus for ROWCOL = 'R', U(s) =
diag(s**INDEX(I))*(UCOEFF(.,.,1)+UCOEFF(.,.,2)/s+...).
LDUCO1 INTEGER
The leading dimension of array UCOEFF.
LDUCO1 >= MAX(1,P) if ROWCOL = 'R';
LDUCO1 >= MAX(1,M) if ROWCOL = 'C'.
LDUCO2 INTEGER
The second dimension of array UCOEFF.
LDUCO2 >= MAX(1,M) if ROWCOL = 'R';
LDUCO2 >= MAX(1,P) if ROWCOL = 'C'.
Tolerances
TOL1 DOUBLE PRECISION
The tolerance to be used in determining the i-th row of
T(s), where i = 1,2,...,porm. If the user sets TOL1 > 0,
then the given value of TOL1 is used as an absolute
tolerance; elements with absolute value less than TOL1 are
considered neglijible. If the user sets TOL1 <= 0, then
an implicitly computed, default tolerance, defined in
the SLICOT Library routine TB01ZD, is used instead.
TOL2 DOUBLE PRECISION
The tolerance to be used to separate out a controllable
subsystem of (A,B,C). If the user sets TOL2 > 0, then
the given value of TOL2 is used as a lower bound for the
reciprocal condition number (see the description of the
argument RCOND in the SLICOT routine MB03OD); a
(sub)matrix whose estimated condition number is less than
1/TOL2 is considered to be of full rank. If the user sets
TOL2 <= 0, then an implicitly computed, default tolerance,
defined in the SLICOT Library routine TB01UD, is used
instead.
Workspace
IWORK INTEGER array, dimension (N+MAX(M,P))
On exit, if INFO = 0, the first nonzero elements of
IWORK(1:N) return the orders of the diagonal blocks of A.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1, N*(N + 1) + MAX(N*MP + 2*N + MAX(N,MP),
3*MP, PM)),
where MP = M, PM = P, if ROWCOL = 'R';
MP = P, PM = M, if ROWCOL = 'C'.
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The method for transfer matrices factorized by rows will be described here: T(s) factorized by columns is dealt with by operating on the dual of the original system. Each row of T(s) is simply a single-output relatively left prime polynomial matrix representation, so can be calculated by applying a simplified version of the Orthogonal Structure Theorem to a minimal state-space representation for the corresponding row of the given system. A minimal state-space representation is obtained using the Orthogonal Canonical Form to first separate out a completely controllable one for the overall system and then, for each row in turn, applying it again to the resulting dual SIMO (single-input multi-output) system. Note that the elements of the transformed matrix A so calculated are individually scaled in a way which guarantees a monic denominator polynomial.References
[1] Williams, T.W.C.
An Orthogonal Structure Theorem for Linear Systems.
Control Systems Research Group, Kingston Polytechnic,
Internal Report 82/2, 1982.
Numerical Aspects
3 The algorithm requires 0(N ) operations.Further Comments
NoneExample
Program Text
* TB04AD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER MAXMP
PARAMETER ( MAXMP = MAX( MMAX, PMAX ) )
INTEGER LDA, LDB, LDC, LDD, LDDCOE, LDUCO1, LDUCO2,
$ NMAXP1
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = MAXMP,
$ LDD = MAXMP, LDDCOE = MAXMP, LDUCO1 = MAXMP,
$ LDUCO2 = MAXMP, NMAXP1 = NMAX+1 )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX + MAXMP )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX*( NMAX + 1 ) +
$ MAX( NMAX*MAXMP + 2*NMAX +
$ MAX( NMAX, MAXMP ), 3*MAXMP ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL1, TOL2
INTEGER I, II, IJ, INDBLK, INFO, J, JJ, KDCOEF, M, N,
$ NR, P, PORM, PORP
CHARACTER*1 ROWCOL
CHARACTER*132 ULINE
LOGICAL LROWCO
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MAXMP), C(LDC,NMAX),
$ D(LDD,MAXMP), DCOEFF(LDDCOE,NMAXP1),
$ DWORK(LDWORK), UCOEFF(LDUCO1,LDUCO2,NMAXP1)
INTEGER INDEX(MAXMP), IWORK(LIWORK)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL TB04AD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL1, TOL2, ROWCOL
LROWCO = LSAME( ROWCOL, 'R' )
ULINE(1:20) = ' '
DO 20 I = 21, 132
ULINE(I:I) = '-'
20 CONTINUE
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99985 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,N ), J = 1,M )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99984 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
* Find the transfer matrix T(s) of (A,B,C,D).
CALL TB04AD( ROWCOL, N, M, P, A, LDA, B, LDB, C, LDC, D,
$ LDD, NR, INDEX, DCOEFF, LDDCOE, UCOEFF,
$ LDUCO1, LDUCO2, TOL1, TOL2, IWORK, DWORK,
$ LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NR
DO 40 I = 1, NR
WRITE ( NOUT, FMT = 99996 ) ( A(I,J), J = 1,NR )
40 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 60 I = 1, NR
WRITE ( NOUT, FMT = 99996 ) ( B(I,J), J = 1,M )
60 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 80 I = 1, P
WRITE ( NOUT, FMT = 99996 ) ( C(I,J), J = 1,NR )
80 CONTINUE
INDBLK = 0
DO 100 I = 1, N
IF ( IWORK(I).NE.0 ) INDBLK = INDBLK + 1
100 CONTINUE
IF ( LROWCO ) THEN
PORM = P
PORP = M
WRITE ( NOUT, FMT = 99993 ) INDBLK,
$ ( IWORK(I), I = 1,INDBLK )
ELSE
PORM = M
PORP = P
WRITE ( NOUT, FMT = 99992 ) INDBLK,
$ ( IWORK(I), I = 1,INDBLK )
END IF
WRITE ( NOUT, FMT = 99991 ) ( INDEX(I), I = 1,PORM )
WRITE ( NOUT, FMT = 99990 )
KDCOEF = 0
DO 120 I = 1, PORM
KDCOEF = MAX( KDCOEF, INDEX(I) )
120 CONTINUE
KDCOEF = KDCOEF + 1
DO 160 II = 1, PORM
DO 140 JJ = 1, PORP
WRITE ( NOUT, FMT = 99989 ) II, JJ,
$ ( UCOEFF(II,JJ,IJ), IJ = 1,KDCOEF )
WRITE ( NOUT, FMT = 99988 ) ULINE(1:7*KDCOEF+21)
WRITE ( NOUT, FMT = 99987 )
$ ( DCOEFF(II,IJ), IJ = 1,KDCOEF )
140 CONTINUE
160 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TB04AD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TB04AD = ',I2)
99997 FORMAT (' The order of the transformed state-space representatio',
$ 'n = ',I2,//' The transformed state dynamics matrix A is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The transformed input/state matrix B is ')
99994 FORMAT (/' The transformed state/output matrix C is ')
99993 FORMAT (/' The controllability index of the transformed state-sp',
$ 'ace representation = ',I2,//' The dimensions of the diag',
$ 'onal blocks of the transformed A are ',/20(I5))
99992 FORMAT (/' The observability index of the transformed state-spac',
$ 'e representation = ',I2,//' The dimensions of the diagon',
$ 'al blocks of the transformed A are ',/20(I5))
99991 FORMAT (/' The degrees of the denominator polynomials are',/20(I5)
$ )
99990 FORMAT (/' The coefficients of polynomials in the transfer matri',
$ 'x T(s) are ')
99989 FORMAT (/' element (',I2,',',I2,') is ',20(1X,F6.2))
99988 FORMAT (1X,A)
99987 FORMAT (20X,20(1X,F6.2))
99986 FORMAT (/' N is out of range.',/' N = ',I5)
99985 FORMAT (/' M is out of range.',/' M = ',I5)
99984 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
TB04AD EXAMPLE PROGRAM DATA 3 2 2 0.0 0.0 R -1.0 0.0 0.0 0.0 -2.0 0.0 0.0 0.0 -3.0 0.0 1.0 -1.0 1.0 1.0 0.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 0.0 1.0Program Results
TB04AD EXAMPLE PROGRAM RESULTS
The order of the transformed state-space representation = 3
The transformed state dynamics matrix A is
-2.5000 -0.2887 -0.4082
-0.2887 -1.5000 -0.7071
-0.4082 -0.7071 -2.0000
The transformed input/state matrix B is
-1.4142 -0.7071
0.0000 1.2247
0.0000 0.0000
The transformed state/output matrix C is
0.0000 0.8165 1.1547
0.0000 1.6330 0.5774
The controllability index of the transformed state-space representation = 2
The dimensions of the diagonal blocks of the transformed A are
2 1
The degrees of the denominator polynomials are
2 3
The coefficients of polynomials in the transfer matrix T(s) are
element ( 1, 1) is 1.00 5.00 7.00 0.00
-----------------------------
1.00 5.00 6.00 0.00
element ( 1, 2) is 0.00 1.00 3.00 0.00
-----------------------------
1.00 5.00 6.00 0.00
element ( 2, 1) is 0.00 0.00 1.00 1.00
-----------------------------
1.00 6.00 11.00 6.00
element ( 2, 2) is 1.00 8.00 20.00 15.00
-----------------------------
1.00 6.00 11.00 6.00
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TB04BD.html 0000664 0000000 0000000 00000036653 14560147231 0020346 0 ustar 00root root 0000000 0000000
Purpose
To compute the transfer function matrix G of a state-space representation (A,B,C,D) of a linear time-invariant multivariable system, using the pole-zeros method. Each element of the transfer function matrix is returned in a cancelled, minimal form, with numerator and denominator polynomials stored either in increasing or decreasing order of the powers of the indeterminate.Specification
SUBROUTINE TB04BD( JOBD, ORDER, EQUIL, N, M, P, MD, A, LDA, B,
$ LDB, C, LDC, D, LDD, IGN, LDIGN, IGD, LDIGD,
$ GN, GD, TOL, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER EQUIL, JOBD, ORDER
DOUBLE PRECISION TOL
INTEGER INFO, LDA, LDB, LDC, LDD, LDIGD, LDIGN, LDWORK,
$ M, MD, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), GD(*), GN(*)
INTEGER IGD(LDIGD,*), IGN(LDIGN,*), IWORK(*)
Arguments
Mode Parameters
JOBD CHARACTER*1
Specifies whether or not a non-zero matrix D appears in
the given state-space model:
= 'D': D is present;
= 'Z': D is assumed to be a zero matrix.
ORDER CHARACTER*1
Specifies the order in which the polynomial coefficients
are stored, as follows:
= 'I': Increasing order of powers of the indeterminate;
= 'D': Decreasing order of powers of the indeterminate.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily
equilibrate the triplet (A,B,C) as follows:
= 'S': perform equilibration (scaling);
= 'N': do not perform equilibration.
Input/Output Parameters
N (input) INTEGER
The order of the system (A,B,C,D). N >= 0.
M (input) INTEGER
The number of the system inputs. M >= 0.
P (input) INTEGER
The number of the system outputs. P >= 0.
MD (input) INTEGER
The maximum degree of the polynomials in G, plus 1. An
upper bound for MD is N+1. MD >= 1.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state dynamics matrix A.
On exit, if EQUIL = 'S', the leading N-by-N part of this
array contains the balanced matrix inv(S)*A*S, as returned
by SLICOT Library routine TB01ID.
If EQUIL = 'N', this array is unchanged on exit.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input matrix B.
On exit, the contents of B are destroyed: all elements but
those in the first row are set to zero.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the output matrix C.
On exit, if EQUIL = 'S', the leading P-by-N part of this
array contains the balanced matrix C*S, as returned by
SLICOT Library routine TB01ID.
If EQUIL = 'N', this array is unchanged on exit.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
If JOBD = 'D', the leading P-by-M part of this array must
contain the matrix D.
If JOBD = 'Z', the array D is not referenced.
LDD INTEGER
The leading dimension of array D.
LDD >= MAX(1,P), if JOBD = 'D';
LDD >= 1, if JOBD = 'Z'.
IGN (output) INTEGER array, dimension (LDIGN,M)
The leading P-by-M part of this array contains the degrees
of the numerator polynomials in the transfer function
matrix G. Specifically, the (i,j) element of IGN contains
the degree of the numerator polynomial of the transfer
function G(i,j) from the j-th input to the i-th output.
LDIGN INTEGER
The leading dimension of array IGN. LDIGN >= max(1,P).
IGD (output) INTEGER array, dimension (LDIGD,M)
The leading P-by-M part of this array contains the degrees
of the denominator polynomials in the transfer function
matrix G. Specifically, the (i,j) element of IGD contains
the degree of the denominator polynomial of the transfer
function G(i,j).
LDIGD INTEGER
The leading dimension of array IGD. LDIGD >= max(1,P).
GN (output) DOUBLE PRECISION array, dimension (P*M*MD)
This array contains the coefficients of the numerator
polynomials, Num(i,j), of the transfer function matrix G.
The polynomials are stored in a column-wise order, i.e.,
Num(1,1), Num(2,1), ..., Num(P,1), Num(1,2), Num(2,2),
..., Num(P,2), ..., Num(1,M), Num(2,M), ..., Num(P,M);
MD memory locations are reserved for each polynomial,
hence, the (i,j) polynomial is stored starting from the
location ((j-1)*P+i-1)*MD+1. The coefficients appear in
increasing or decreasing order of the powers of the
indeterminate, according to ORDER.
GD (output) DOUBLE PRECISION array, dimension (P*M*MD)
This array contains the coefficients of the denominator
polynomials, Den(i,j), of the transfer function matrix G.
The polynomials are stored in the same way as the
numerator polynomials.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in determining the
controllability of a single-input system (A,b) or (A',c'),
where b and c' are columns in B and C' (C transposed). If
the user sets TOL > 0, then the given value of TOL is used
as an absolute tolerance; elements with absolute value
less than TOL are considered neglijible. If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
defined by TOLDEF = N*EPS*MAX( NORM(A), NORM(bc) ) is used
instead, where EPS is the machine precision (see LAPACK
Library routine DLAMCH), and bc denotes the currently used
column in B or C' (see METHOD).
Workspace
IWORK INTEGER array, dimension (N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1, N*(N+P) +
MAX( N + MAX( N,P ), N*(2*N+5)))
If N >= P, N >= 1, the formula above can be written as
LDWORK >= N*(3*N + P + 5).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the QR algorithm failed to converge when trying to
compute the zeros of a transfer function;
= 2: the QR algorithm failed to converge when trying to
compute the poles of a transfer function.
The errors INFO = 1 or 2 are unlikely to appear.
Method
The routine implements the pole-zero method proposed in [1]. This method is based on an algorithm for computing the transfer function of a single-input single-output (SISO) system. Let (A,b,c,d) be a SISO system. Its transfer function is computed as follows: 1) Find a controllable realization (Ac,bc,cc) of (A,b,c). 2) Find an observable realization (Ao,bo,co) of (Ac,bc,cc). 3) Compute the r eigenvalues of Ao (the poles of (Ao,bo,co)). 4) Compute the zeros of (Ao,bo,co,d). 5) Compute the gain of (Ao,bo,co,d). This algorithm can be implemented using only orthogonal transformations [1]. However, for better efficiency, the implementation in TB04BD uses one elementary transformation in Step 4 and r elementary transformations in Step 5 (to reduce an upper Hessenberg matrix to upper triangular form). These special elementary transformations are numerically stable in practice. In the multi-input multi-output (MIMO) case, the algorithm computes each element (i,j) of the transfer function matrix G, for i = 1 : P, and for j = 1 : M. For efficiency reasons, Step 1 is performed once for each value of j (each column of B). The matrices Ac and Ao result in Hessenberg form.References
[1] Varga, A. and Sima, V.
Numerically Stable Algorithm for Transfer Function Matrix
Evaluation.
Int. J. Control, vol. 33, nr. 6, pp. 1123-1133, 1981.
Numerical Aspects
The algorithm is numerically stable in practice and requires about 20*N**3 floating point operations at most, but usually much less.Further Comments
For maximum efficiency of index calculations, GN and GD are implemented as one-dimensional arrays.Example
Program Text
* TB04BD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX, MDMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20,
$ MDMAX = NMAX + 1 )
INTEGER PMNMAX
PARAMETER ( PMNMAX = PMAX*MMAX*MDMAX )
INTEGER LDA, LDB, LDC, LDD, LDIGD, LDIGN
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX, LDIGD = PMAX, LDIGN = PMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX*( NMAX + PMAX ) +
$ MAX( NMAX + MAX( NMAX, PMAX ),
$ NMAX*( 2*NMAX + 5 ) ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, IJ, INFO, J, K, M, MD, N, P
CHARACTER*1 JOBD, ORDER, EQUIL
CHARACTER*132 ULINE
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK), GD(PMNMAX),
$ GN(PMNMAX)
INTEGER IGD(LDIGD,MMAX), IGN(LDIGN,MMAX), IWORK(LIWORK)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL TB04BD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL, JOBD, ORDER, EQUIL
MD = N + 1
ULINE(1:20) = ' '
DO 20 I = 21, 132
ULINE(I:I) = '-'
20 CONTINUE
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,N ), J = 1,M )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
* Find the transfer matrix T(s) of (A,B,C,D).
CALL TB04BD( JOBD, ORDER, EQUIL, N, M, P, MD, A, LDA, B,
$ LDB, C, LDC, D, LDD, IGN, LDIGN, IGD, LDIGD,
$ GN, GD, TOL, IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( LSAME( ORDER, 'I' ) ) THEN
WRITE ( NOUT, FMT = 99997 )
ELSE
WRITE ( NOUT, FMT = 99996 )
END IF
WRITE ( NOUT, FMT = 99995 )
DO 60 J = 1, M
DO 40 I = 1, P
IJ = ( (J-1)*P + I-1 )*MD + 1
WRITE ( NOUT, FMT = 99994 ) I, J,
$ ( GN(K), K = IJ,IJ+IGN(I,J) )
WRITE ( NOUT, FMT = 99993 )
$ ULINE(1:7*(IGD(I,J)+1)+21)
WRITE ( NOUT, FMT = 99992 )
$ ( GD(K), K = IJ,IJ+IGD(I,J) )
40 CONTINUE
60 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TB04BD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TB04BD = ',I2)
99997 FORMAT (/' The polynomial coefficients appear in increasing',
$ ' order'/' of the powers of the indeterminate')
99996 FORMAT (/' The polynomial coefficients appear in decreasing',
$ ' order'/' of the powers of the indeterminate')
99995 FORMAT (/' The coefficients of polynomials in the transfer matri',
$ 'x T(s) are ')
99994 FORMAT (/' element (',I2,',',I2,') is ',20(1X,F6.2))
99993 FORMAT (1X,A)
99992 FORMAT (20X,20(1X,F6.2))
99991 FORMAT (/' N is out of range.',/' N = ',I5)
99990 FORMAT (/' M is out of range.',/' M = ',I5)
99989 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
TB04BD EXAMPLE PROGRAM DATA 3 2 2 0.0 D I N -1.0 0.0 0.0 0.0 -2.0 0.0 0.0 0.0 -3.0 0.0 1.0 -1.0 1.0 1.0 0.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 0.0 1.0Program Results
TB04BD EXAMPLE PROGRAM RESULTS
The polynomial coefficients appear in increasing order
of the powers of the indeterminate
The coefficients of polynomials in the transfer matrix T(s) are
element ( 1, 1) is 7.00 5.00 1.00
----------------------
6.00 5.00 1.00
element ( 2, 1) is 1.00
----------------------
6.00 5.00 1.00
element ( 1, 2) is 1.00
---------------
2.00 1.00
element ( 2, 2) is 5.00 5.00 1.00
----------------------
2.00 3.00 1.00
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TB04BV.html 0000664 0000000 0000000 00000013745 14560147231 0020365 0 ustar 00root root 0000000 0000000
Purpose
To separate the strictly proper part G0 from the constant part D of an P-by-M proper transfer function matrix G.Specification
SUBROUTINE TB04BV( ORDER, P, M, MD, IGN, LDIGN, IGD, LDIGD, GN,
$ GD, D, LDD, TOL, INFO )
C .. Scalar Arguments ..
CHARACTER ORDER
DOUBLE PRECISION TOL
INTEGER INFO, LDD, LDIGD, LDIGN, M, MD, P
C .. Array Arguments ..
DOUBLE PRECISION D(LDD,*), GD(*), GN(*)
INTEGER IGD(LDIGD,*), IGN(LDIGN,*)
Arguments
Mode Parameters
ORDER CHARACTER*1
Specifies the order in which the polynomial coefficients
of the transfer function matrix are stored, as follows:
= 'I': Increasing order of powers of the indeterminate;
= 'D': Decreasing order of powers of the indeterminate.
Input/Output Parameters
P (input) INTEGER
The number of the system outputs. P >= 0.
M (input) INTEGER
The number of the system inputs. M >= 0.
MD (input) INTEGER
The maximum degree of the polynomials in G, plus 1, i.e.,
MD = MAX(IGD(I,J)) + 1.
I,J
IGN (input/output) INTEGER array, dimension (LDIGN,M)
On entry, the leading P-by-M part of this array must
contain the degrees of the numerator polynomials in G:
the (i,j) element of IGN must contain the degree of the
numerator polynomial of the polynomial ratio G(i,j).
On exit, the leading P-by-M part of this array contains
the degrees of the numerator polynomials in G0.
LDIGN INTEGER
The leading dimension of array IGN. LDIGN >= max(1,P).
IGD (input) INTEGER array, dimension (LDIGD,M)
The leading P-by-M part of this array must contain the
degrees of the denominator polynomials in G (and G0):
the (i,j) element of IGD contains the degree of the
denominator polynomial of the polynomial ratio G(i,j).
LDIGD INTEGER
The leading dimension of array IGD. LDIGD >= max(1,P).
GN (input/output) DOUBLE PRECISION array, dimension (P*M*MD)
On entry, this array must contain the coefficients of the
numerator polynomials, Num(i,j), of the transfer function
matrix G. The polynomials are stored in a column-wise
order, i.e., Num(1,1), Num(2,1), ..., Num(P,1), Num(1,2),
Num(2,2), ..., Num(P,2), ..., Num(1,M), Num(2,M), ...,
Num(P,M); MD memory locations are reserved for each
polynomial, hence, the (i,j) polynomial is stored starting
from the location ((j-1)*P+i-1)*MD+1. The coefficients
appear in increasing or decreasing order of the powers
of the indeterminate, according to ORDER.
On exit, this array contains the coefficients of the
numerator polynomials of the strictly proper part G0 of
the transfer function matrix G, stored similarly.
GD (input) DOUBLE PRECISION array, dimension (P*M*MD)
This array must contain the coefficients of the
denominator polynomials, Den(i,j), of the transfer
function matrix G. The polynomials are stored as for the
numerator polynomials.
D (output) DOUBLE PRECISION array, dimension (LDD,M)
The leading P-by-M part of this array contains the
matrix D.
LDD INTEGER
The leading dimension of array D. LDD >= max(1,P).
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in determining the degrees of
the numerators Num0(i,j) of the strictly proper part of
the transfer function matrix G. If the user sets TOL > 0,
then the given value of TOL is used as an absolute
tolerance; the leading coefficients with absolute value
less than TOL are considered neglijible. If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
defined by TOLDEF = IGN(i,j)*EPS*NORM( Num(i,j) ) is used
instead, where EPS is the machine precision (see LAPACK
Library routine DLAMCH), and NORM denotes the infinity
norm (the maximum coefficient in absolute value).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the transfer function matrix is not proper;
= 2: if a denominator polynomial is null.
Method
The (i,j) entry of the real matrix D is zero, if the degree of Num(i,j), IGN(i,j), is less than the degree of Den(i,j), IGD(i,j), and it is given by the ratio of the leading coefficients of Num(i,j) and Den(i,j), if IGN(i,j) is equal to IGD(i,j), for i = 1 : P, and for j = 1 : M.Further Comments
For maximum efficiency of index calculations, GN and GD are implemented as one-dimensional arrays.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the sum of an P-by-M rational matrix G and a real P-by-M matrix D.Specification
SUBROUTINE TB04BW( ORDER, P, M, MD, IGN, LDIGN, IGD, LDIGD, GN,
$ GD, D, LDD, INFO )
C .. Scalar Arguments ..
CHARACTER ORDER
INTEGER INFO, LDD, LDIGD, LDIGN, M, MD, P
C .. Array Arguments ..
DOUBLE PRECISION D(LDD,*), GD(*), GN(*)
INTEGER IGD(LDIGD,*), IGN(LDIGN,*)
Arguments
Mode Parameters
ORDER CHARACTER*1
Specifies the order in which the polynomial coefficients
of the rational matrix are stored, as follows:
= 'I': Increasing order of powers of the indeterminate;
= 'D': Decreasing order of powers of the indeterminate.
Input/Output Parameters
P (input) INTEGER
The number of the system outputs. P >= 0.
M (input) INTEGER
The number of the system inputs. M >= 0.
MD (input) INTEGER
The maximum degree of the polynomials in G, plus 1, i.e.,
MD = MAX(IGN(I,J),IGD(I,J)) + 1.
I,J
IGN (input/output) INTEGER array, dimension (LDIGN,M)
On entry, the leading P-by-M part of this array must
contain the degrees of the numerator polynomials in G:
the (i,j) element of IGN must contain the degree of the
numerator polynomial of the polynomial ratio G(i,j).
On exit, the leading P-by-M part of this array contains
the degrees of the numerator polynomials in G + D.
LDIGN INTEGER
The leading dimension of array IGN. LDIGN >= max(1,P).
IGD (input) INTEGER array, dimension (LDIGD,M)
The leading P-by-M part of this array must contain the
degrees of the denominator polynomials in G (and G + D):
the (i,j) element of IGD contains the degree of the
denominator polynomial of the polynomial ratio G(i,j).
LDIGD INTEGER
The leading dimension of array IGD. LDIGD >= max(1,P).
GN (input/output) DOUBLE PRECISION array, dimension (P*M*MD)
On entry, this array must contain the coefficients of the
numerator polynomials, Num(i,j), of the rational matrix G.
The polynomials are stored in a column-wise order, i.e.,
Num(1,1), Num(2,1), ..., Num(P,1), Num(1,2), Num(2,2),
..., Num(P,2), ..., Num(1,M), Num(2,M), ..., Num(P,M);
MD memory locations are reserved for each polynomial,
hence, the (i,j) polynomial is stored starting from the
location ((j-1)*P+i-1)*MD+1. The coefficients appear in
increasing or decreasing order of the powers of the
indeterminate, according to ORDER.
On exit, this array contains the coefficients of the
numerator polynomials of the rational matrix G + D,
stored similarly.
GD (input) DOUBLE PRECISION array, dimension (P*M*MD)
This array must contain the coefficients of the
denominator polynomials, Den(i,j), of the rational
matrix G. The polynomials are stored as for the
numerator polynomials.
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading P-by-M part of this array must contain the
matrix D.
LDD INTEGER
The leading dimension of array D. LDD >= max(1,P).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The (i,j) entry of the real matrix D is added to the (i,j) entry of the matrix G, g(i,j), which is a ratio of two polynomials, for i = 1 : P, and for j = 1 : M. If g(i,j) = 0, it is assumed that its denominator is 1.Numerical Aspects
The algorithm is numerically stable.Further Comments
Often, the rational matrix G is found from a state-space representation (A,B,C), and D corresponds to the direct feedthrough matrix of the system. The sum G + D gives the transfer function matrix of the system (A,B,C,D). For maximum efficiency of index calculations, GN and GD are implemented as one-dimensional arrays.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the gain of a single-input single-output linear system,
given its state-space representation (A,b,c,d), and its poles and
zeros. The matrix A is assumed to be in an upper Hessenberg form.
The gain is computed using the formula
-1 IP IZ
g = (c*( S0*I - A ) *b + d)*Prod( S0 - Pi )/Prod( S0 - Zi ) ,
i=1 i=1 (1)
where Pi, i = 1 : IP, and Zj, j = 1 : IZ, are the poles and zeros,
respectively, and S0 is a real scalar different from all poles and
zeros.
Specification
SUBROUTINE TB04BX( IP, IZ, A, LDA, B, C, D, PR, PI, ZR, ZI, GAIN,
$ IWORK )
C .. Scalar Arguments ..
DOUBLE PRECISION D, GAIN
INTEGER IP, IZ, LDA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(*), C(*), PI(*), PR(*), ZI(*),
$ ZR(*)
INTEGER IWORK(*)
Arguments
Input/Output Parameters
IP (input) INTEGER
The number of the system poles. IP >= 0.
IZ (input) INTEGER
The number of the system zeros. IZ >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,IP)
On entry, the leading IP-by-IP part of this array must
contain the state dynamics matrix A in an upper Hessenberg
form. The elements below the second diagonal are not
referenced.
On exit, the leading IP-by-IP upper Hessenberg part of
this array contains the LU factorization of the matrix
A - S0*I, as computed by SLICOT Library routine MB02SD.
LDA INTEGER
The leading dimension of array A. LDA >= max(1,IP).
B (input/output) DOUBLE PRECISION array, dimension (IP)
On entry, this array must contain the system input
vector b.
On exit, this array contains the solution of the linear
system ( A - S0*I )x = b .
C (input) DOUBLE PRECISION array, dimension (IP)
This array must contain the system output vector c.
D (input) DOUBLE PRECISION
The variable must contain the system feedthrough scalar d.
PR (input) DOUBLE PRECISION array, dimension (IP)
This array must contain the real parts of the system
poles. Pairs of complex conjugate poles must be stored in
consecutive memory locations.
PI (input) DOUBLE PRECISION array, dimension (IP)
This array must contain the imaginary parts of the system
poles.
ZR (input) DOUBLE PRECISION array, dimension (IZ)
This array must contain the real parts of the system
zeros. Pairs of complex conjugate zeros must be stored in
consecutive memory locations.
ZI (input) DOUBLE PRECISION array, dimension (IZ)
This array must contain the imaginary parts of the system
zeros.
GAIN (output) DOUBLE PRECISION
The gain of the linear system (A,b,c,d), given by (1).
Workspace
IWORK INTEGER array, dimension (IP)
On exit, it contains the pivot indices; for 1 <= i <= IP,
row i of the matrix A - S0*I was interchanged with
row IWORK(i).
Method
The routine implements the method presented in [1]. A suitable value of S0 is chosen based on the system poles and zeros. Then, the LU factorization of the upper Hessenberg, nonsingular matrix A - S0*I is computed and used to solve the linear system in (1).References
[1] Varga, A. and Sima, V.
Numerically Stable Algorithm for Transfer Function Matrix
Evaluation.
Int. J. Control, vol. 33, nr. 6, pp. 1123-1133, 1981.
Numerical Aspects
The algorithm is numerically stable in practice and requires O(IP*IP) floating point operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the transfer function matrix G of a state-space representation (A,B,C,D) of a linear time-invariant multivariable system, using the pole-zeros method. The transfer function matrix is returned in a minimal pole-zero-gain form.Specification
SUBROUTINE TB04CD( JOBD, EQUIL, N, M, P, NPZ, A, LDA, B, LDB, C,
$ LDC, D, LDD, NZ, LDNZ, NP, LDNP, ZEROSR,
$ ZEROSI, POLESR, POLESI, GAINS, LDGAIN, TOL,
$ IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER EQUIL, JOBD
DOUBLE PRECISION TOL
INTEGER INFO, LDA, LDB, LDC, LDD, LDGAIN, LDNP, LDNZ,
$ LDWORK, M, N, NPZ, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), GAINS(LDGAIN,*), POLESI(*),
$ POLESR(*), ZEROSI(*), ZEROSR(*)
INTEGER IWORK(*), NP(LDNP,*), NZ(LDNZ,*)
Arguments
Mode Parameters
JOBD CHARACTER*1
Specifies whether or not a non-zero matrix D appears in
the given state-space model:
= 'D': D is present;
= 'Z': D is assumed to be a zero matrix.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily
equilibrate the triplet (A,B,C) as follows:
= 'S': perform equilibration (scaling);
= 'N': do not perform equilibration.
Input/Output Parameters
N (input) INTEGER
The order of the system (A,B,C,D). N >= 0.
M (input) INTEGER
The number of the system inputs. M >= 0.
P (input) INTEGER
The number of the system outputs. P >= 0.
NPZ (input) INTEGER
The maximum number of poles or zeros of the single-input
single-output channels in the system. An upper bound
for NPZ is N. NPZ >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state dynamics matrix A.
On exit, if EQUIL = 'S', the leading N-by-N part of this
array contains the balanced matrix inv(S)*A*S, as returned
by SLICOT Library routine TB01ID.
If EQUIL = 'N', this array is unchanged on exit.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input matrix B.
On exit, the contents of B are destroyed: all elements but
those in the first row are set to zero.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the output matrix C.
On exit, if EQUIL = 'S', the leading P-by-N part of this
array contains the balanced matrix C*S, as returned by
SLICOT Library routine TB01ID.
If EQUIL = 'N', this array is unchanged on exit.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
If JOBD = 'D', the leading P-by-M part of this array must
contain the matrix D.
If JOBD = 'Z', the array D is not referenced.
LDD INTEGER
The leading dimension of array D.
LDD >= MAX(1,P), if JOBD = 'D';
LDD >= 1, if JOBD = 'Z'.
NZ (output) INTEGER array, dimension (LDNZ,M)
The leading P-by-M part of this array contains the numbers
of zeros of the elements of the transfer function
matrix G. Specifically, the (i,j) element of NZ contains
the number of zeros of the transfer function G(i,j) from
the j-th input to the i-th output.
LDNZ INTEGER
The leading dimension of array NZ. LDNZ >= max(1,P).
NP (output) INTEGER array, dimension (LDNP,M)
The leading P-by-M part of this array contains the numbers
of poles of the elements of the transfer function
matrix G. Specifically, the (i,j) element of NP contains
the number of poles of the transfer function G(i,j).
LDNP INTEGER
The leading dimension of array NP. LDNP >= max(1,P).
ZEROSR (output) DOUBLE PRECISION array, dimension (P*M*NPZ)
This array contains the real parts of the zeros of the
transfer function matrix G. The real parts of the zeros
are stored in a column-wise order, i.e., for the transfer
functions (1,1), (2,1), ..., (P,1), (1,2), (2,2), ...,
(P,2), ..., (1,M), (2,M), ..., (P,M); NPZ memory locations
are reserved for each transfer function, hence, the real
parts of the zeros for the (i,j) transfer function
are stored starting from the location ((j-1)*P+i-1)*NPZ+1.
Pairs of complex conjugate zeros are stored in consecutive
memory locations. Note that only the first NZ(i,j) entries
are initialized for the (i,j) transfer function.
ZEROSI (output) DOUBLE PRECISION array, dimension (P*M*NPZ)
This array contains the imaginary parts of the zeros of
the transfer function matrix G, stored in a similar way
as the real parts of the zeros.
POLESR (output) DOUBLE PRECISION array, dimension (P*M*NPZ)
This array contains the real parts of the poles of the
transfer function matrix G, stored in the same way as
the zeros. Note that only the first NP(i,j) entries are
initialized for the (i,j) transfer function.
POLESI (output) DOUBLE PRECISION array, dimension (P*M*NPZ)
This array contains the imaginary parts of the poles of
the transfer function matrix G, stored in the same way as
the poles.
GAINS (output) DOUBLE PRECISION array, dimension (LDGAIN,M)
The leading P-by-M part of this array contains the gains
of the transfer function matrix G. Specifically,
GAINS(i,j) contains the gain of the transfer function
G(i,j).
LDGAIN INTEGER
The leading dimension of array GAINS. LDGAIN >= max(1,P).
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in determining the
controllability of a single-input system (A,b) or (A',c'),
where b and c' are columns in B and C' (C transposed). If
the user sets TOL > 0, then the given value of TOL is used
as an absolute tolerance; elements with absolute value
less than TOL are considered neglijible. If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
defined by TOLDEF = N*EPS*MAX( NORM(A), NORM(bc) ) is used
instead, where EPS is the machine precision (see LAPACK
Library routine DLAMCH), and bc denotes the currently used
column in B or C' (see METHOD).
Workspace
IWORK INTEGER array, dimension (N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1, N*(N+P) +
MAX( N + MAX( N,P ), N*(2*N+3)))
If N >= P, N >= 1, the formula above can be written as
LDWORK >= N*(3*N + P + 3).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the QR algorithm failed to converge when trying to
compute the zeros of a transfer function;
= 2: the QR algorithm failed to converge when trying to
compute the poles of a transfer function.
The errors INFO = 1 or 2 are unlikely to appear.
Method
The routine implements the pole-zero method proposed in [1]. This method is based on an algorithm for computing the transfer function of a single-input single-output (SISO) system. Let (A,b,c,d) be a SISO system. Its transfer function is computed as follows: 1) Find a controllable realization (Ac,bc,cc) of (A,b,c). 2) Find an observable realization (Ao,bo,co) of (Ac,bc,cc). 3) Compute the r eigenvalues of Ao (the poles of (Ao,bo,co)). 4) Compute the zeros of (Ao,bo,co,d). 5) Compute the gain of (Ao,bo,co,d). This algorithm can be implemented using only orthogonal transformations [1]. However, for better efficiency, the implementation in TB04CD uses one elementary transformation in Step 4 and r elementary transformations in Step 5 (to reduce an upper Hessenberg matrix to upper triangular form). These special elementary transformations are numerically stable in practice. In the multi-input multi-output (MIMO) case, the algorithm computes each element (i,j) of the transfer function matrix G, for i = 1 : P, and for j = 1 : M. For efficiency reasons, Step 1 is performed once for each value of j (each column of B). The matrices Ac and Ao result in Hessenberg form.References
[1] Varga, A. and Sima, V.
Numerically Stable Algorithm for Transfer Function Matrix
Evaluation.
Int. J. Control, vol. 33, nr. 6, pp. 1123-1133, 1981.
Numerical Aspects
The algorithm is numerically stable in practice and requires about 20*N**3 floating point operations at most, but usually much less.Further Comments
NoneExample
Program Text
* TB04CD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX, NPZMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20,
$ NPZMAX = NMAX )
INTEGER PMNMAX
PARAMETER ( PMNMAX = PMAX*MMAX*NPZMAX )
INTEGER LDA, LDB, LDC, LDD, LDGAIN, LDNP, LDNZ
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX, LDGAIN = PMAX, LDNP = PMAX,
$ LDNZ = PMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX*( NMAX + PMAX ) +
$ MAX( NMAX + MAX( NMAX, PMAX ),
$ NMAX*( 2*NMAX + 3 ) ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, IJ, INFO, J, K, M, N, NPZ, P
CHARACTER*1 JOBD, EQUIL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK), GAINS(LDGAIN,MMAX),
$ POLESI(PMNMAX), POLESR(PMNMAX), ZEROSI(PMNMAX),
$ ZEROSR(PMNMAX)
INTEGER IWORK(LIWORK), NP(LDNP,MMAX), NZ(LDNZ,MMAX)
* .. External Subroutines ..
EXTERNAL TB04CD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL, JOBD, EQUIL
NPZ = N
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,N ), J = 1,M )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
* Find the transfer matrix T(s) of (A,B,C,D) in the
* pole-zero-gain form.
CALL TB04CD( JOBD, EQUIL, N, M, P, NPZ, A, LDA, B, LDB,
$ C, LDC, D, LDD, NZ, LDNZ, NP, LDNP, ZEROSR,
$ ZEROSI, POLESR, POLESI, GAINS, LDGAIN, TOL,
$ IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 60 J = 1, M
DO 40 I = 1, P
IJ = ( (J-1)*P + I-1 )*NPZ + 1
IF ( NZ(I,J).EQ.0 ) THEN
WRITE ( NOUT, FMT = 99996 ) I, J
ELSE
WRITE ( NOUT, FMT = 99995 ) I, J,
$ ( ZEROSR(K), ZEROSI(K),
$ K = IJ,IJ+NZ(I,J)-1 )
END IF
WRITE ( NOUT, FMT = 99994 ) I, J,
$ ( POLESR(K), POLESI(K), K = IJ,IJ+NP(I,J)-1 )
WRITE ( NOUT, FMT = 99993 ) I, J, ( GAINS(I,J) )
40 CONTINUE
60 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TB04CD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TB04CD = ',I2)
99997 FORMAT (/' The poles, zeros and gains of the transfer matrix',
$ ' elements: ')
99996 FORMAT (/' no zeros for element (',I2,',',I2,')')
99995 FORMAT (/' zeros of element (',I2,',',I2,') are ',//
$ ' real part imag part '// (2X,F9.4,5X,F9.4))
99994 FORMAT (/' poles of element (',I2,',',I2,') are ',//
$ ' real part imag part '// (2X,F9.4,5X,F9.4))
99993 FORMAT (/' gain of element (',I2,',',I2,') is ', F9.4)
99992 FORMAT (/' N is out of range.',/' N = ',I5)
99991 FORMAT (/' M is out of range.',/' M = ',I5)
99990 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
TB04CD EXAMPLE PROGRAM DATA 3 2 2 0.0 D N -1.0 0.0 0.0 0.0 -2.0 0.0 0.0 0.0 -3.0 0.0 1.0 -1.0 1.0 1.0 0.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 0.0 1.0Program Results
TB04CD EXAMPLE PROGRAM RESULTS
The poles, zeros and gains of the transfer matrix elements:
zeros of element ( 1, 1) are
real part imag part
-2.5000 0.8660
-2.5000 -0.8660
poles of element ( 1, 1) are
real part imag part
-2.0000 0.0000
-3.0000 0.0000
gain of element ( 1, 1) is 1.0000
no zeros for element ( 2, 1)
poles of element ( 2, 1) are
real part imag part
-2.0000 0.0000
-3.0000 0.0000
gain of element ( 2, 1) is 1.0000
no zeros for element ( 1, 2)
poles of element ( 1, 2) are
real part imag part
-2.0000 0.0000
gain of element ( 1, 2) is 1.0000
zeros of element ( 2, 2) are
real part imag part
-3.6180 0.0000
-1.3820 0.0000
poles of element ( 2, 2) are
real part imag part
-1.0000 0.0000
-2.0000 0.0000
gain of element ( 2, 2) is 1.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TB05AD.html 0000664 0000000 0000000 00000032436 14560147231 0020341 0 ustar 00root root 0000000 0000000
Purpose
To find the complex frequency response matrix (transfer matrix)
G(freq) of the state-space representation (A,B,C) given by
-1
G(freq) = C * ((freq*I - A) ) * B
where A, B and C are real N-by-N, N-by-M and P-by-N matrices
respectively and freq is a complex scalar.
Specification
SUBROUTINE TB05AD( BALEIG, INITA, N, M, P, FREQ, A, LDA, B, LDB,
$ C, LDC, RCOND, G, LDG, EVRE, EVIM, HINVB,
$ LDHINV, IWORK, DWORK, LDWORK, ZWORK, LZWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER BALEIG, INITA
INTEGER INFO, LDA, LDB, LDC, LDG, LDHINV, LDWORK,
$ LZWORK, M, N, P
DOUBLE PRECISION RCOND
COMPLEX*16 FREQ
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), EVIM(*),
$ EVRE(*)
COMPLEX*16 ZWORK(*), G(LDG,*), HINVB(LDHINV,*)
Arguments
Mode Parameters
BALEIG CHARACTER*1
Determines whether the user wishes to balance matrix A
and/or compute its eigenvalues and/or estimate the
condition number of the problem as follows:
= 'N': The matrix A should not be balanced and neither
the eigenvalues of A nor the condition number
estimate of the problem are to be calculated;
= 'C': The matrix A should not be balanced and only an
estimate of the condition number of the problem
is to be calculated;
= 'B' or 'E' and INITA = 'G': The matrix A is to be
balanced and its eigenvalues calculated;
= 'A' and INITA = 'G': The matrix A is to be balanced,
and its eigenvalues and an estimate of the
condition number of the problem are to be
calculated.
INITA CHARACTER*1
Specifies whether or not the matrix A is already in upper
Hessenberg form as follows:
= 'G': The matrix A is a general matrix;
= 'H': The matrix A is in upper Hessenberg form and
neither balancing nor the eigenvalues of A are
required.
INITA must be set to 'G' for the first call to the
routine, unless the matrix A is already in upper
Hessenberg form and neither balancing nor the eigenvalues
of A are required. Thereafter, it must be set to 'H' for
all subsequent calls.
Input/Output Parameters
N (input) INTEGER
The number of states, i.e. the order of the state
transition matrix A. N >= 0.
M (input) INTEGER
The number of inputs, i.e. the number of columns in the
matrix B. M >= 0.
P (input) INTEGER
The number of outputs, i.e. the number of rows in the
matrix C. P >= 0.
FREQ (input) COMPLEX*16
The frequency freq at which the frequency response matrix
(transfer matrix) is to be evaluated.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state transition matrix A.
If INITA = 'G', then, on exit, the leading N-by-N part of
this array contains an upper Hessenberg matrix similar to
(via an orthogonal matrix consisting of a sequence of
Householder transformations) the original state transition
matrix A.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input/state matrix B.
If INITA = 'G', then, on exit, the leading N-by-M part of
this array contains the product of the transpose of the
orthogonal transformation matrix used to reduce A to upper
Hessenberg form and the original input/state matrix B.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
If INITA = 'G', then, on exit, the leading P-by-N part of
this array contains the product of the original output/
state matrix C and the orthogonal transformation matrix
used to reduce A to upper Hessenberg form.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
RCOND (output) DOUBLE PRECISION
If BALEIG = 'C' or BALEIG = 'A', then RCOND contains an
estimate of the reciprocal of the condition number of
matrix H with respect to inversion (see METHOD).
G (output) COMPLEX*16 array, dimension (LDG,M)
The leading P-by-M part of this array contains the
frequency response matrix G(freq).
LDG INTEGER
The leading dimension of array G. LDG >= MAX(1,P).
EVRE, (output) DOUBLE PRECISION arrays, dimension (N)
EVIM If INITA = 'G' and BALEIG = 'B' or 'E' or BALEIG = 'A',
then these arrays contain the real and imaginary parts,
respectively, of the eigenvalues of the matrix A.
Otherwise, these arrays are not referenced.
HINVB (output) COMPLEX*16 array, dimension (LDHINV,M)
The leading N-by-M part of this array contains the
-1
product H B.
LDHINV INTEGER
The leading dimension of array HINVB. LDHINV >= MAX(1,N).
Workspace
IWORK INTEGER array, dimension (N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1, N - 1 + MAX(N,M,P)),
if INITA = 'G' and BALEIG = 'N', or 'B', or 'E';
LDWORK >= MAX(1, N + MAX(N,M-1,P-1)),
if INITA = 'G' and BALEIG = 'C', or 'A';
LDWORK >= MAX(1, 2*N),
if INITA = 'H' and BALEIG = 'C', or 'A';
LDWORK >= 1, otherwise.
For optimum performance when INITA = 'G' LDWORK should be
larger.
ZWORK COMPLEX*16 array, dimension (LZWORK)
LZWORK INTEGER
The length of the array ZWORK.
LZWORK >= MAX(1,N*N+2*N), if BALEIG = 'C', or 'A';
LZWORK >= MAX(1,N*N), otherwise.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if more than 30*N iterations are required to
isolate all the eigenvalues of the matrix A; the
computations are continued;
= 2: if either FREQ is too near to an eigenvalue of the
matrix A, or RCOND is less than EPS, where EPS is
the machine precision (see LAPACK Library routine
DLAMCH).
Method
The matrix A is first balanced (if BALEIG = 'B' or 'E', or
BALEIG = 'A') and then reduced to upper Hessenberg form; the same
transformations are applied to the matrix B and the matrix C.
The complex Hessenberg matrix H = (freq*I - A) is then used
-1
to solve for C * H * B.
Depending on the input values of parameters BALEIG and INITA,
the eigenvalues of matrix A and the condition number of
matrix H with respect to inversion are also calculated.
References
[1] Laub, A.J.
Efficient Calculation of Frequency Response Matrices from
State-Space Models.
ACM TOMS, 12, pp. 26-33, 1986.
Numerical Aspects
3 The algorithm requires 0(N ) operations.Further Comments
NoneExample
Program Text
* TB05AD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDG, LDHINV
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX, LDG = PMAX,
$ LDHINV = NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 2*NMAX )
INTEGER LZWORK
PARAMETER ( LZWORK = NMAX*( NMAX+2 ) )
* .. Local Scalars ..
COMPLEX*16 FREQ
DOUBLE PRECISION RCOND
INTEGER I, INFO, J, M, N, P
CHARACTER*1 BALEIG, INITA
LOGICAL LBALBA, LBALEA, LBALEB, LBALEC, LINITA
* .. Local Arrays ..
COMPLEX*16 G(LDG,MMAX), HINVB(LDHINV,MMAX), ZWORK(LZWORK)
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ DWORK(LDWORK), EVIM(NMAX), EVRE(NMAX)
INTEGER IWORK(LIWORK)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL TB05AD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, FREQ, INITA, BALEIG
LBALEC = LSAME( BALEIG, 'C' )
LBALEB = LSAME( BALEIG, 'B' ) .OR. LSAME( BALEIG, 'E' )
LBALEA = LSAME( BALEIG, 'A' )
LBALBA = LBALEB.OR.LBALEA
LINITA = LSAME( INITA, 'G' )
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find the frequency response matrix of the ssr (A,B,C).
CALL TB05AD( BALEIG, INITA, N, M, P, FREQ, A, LDA, B,
$ LDB, C, LDC, RCOND, G, LDG, EVRE, EVIM,
$ HINVB, LDHINV, IWORK, DWORK, LDWORK, ZWORK,
$ LZWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( ( LBALEC ) .OR. ( LBALEA ) ) WRITE ( NOUT,
$ FMT = 99997 ) RCOND
IF ( ( LINITA ) .AND. ( LBALBA ) )
$ WRITE ( NOUT, FMT = 99996 )
$ ( EVRE(I), EVIM(I), I = 1,N )
WRITE ( NOUT, FMT = 99995 )
DO 20 I = 1, P
WRITE ( NOUT, FMT = 99994 ) ( G(I,J), J = 1,M )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99994 ) ( HINVB(I,J), J = 1,M )
40 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TB05AD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TB05AD = ',I2)
99997 FORMAT (' RCOND = ',F4.2)
99996 FORMAT (/' Eigenvalues of the state transmission matrix A are ',
$ /(1X,2F7.2,'*j'))
99995 FORMAT (/' The frequency response matrix G(freq) is ')
99994 FORMAT (20(' (',F5.2,',',F5.2,') ',:))
99993 FORMAT (/' H(inverse)*B is ')
99992 FORMAT (/' N is out of range.',/' N = ',I5)
99991 FORMAT (/' M is out of range.',/' M = ',I5)
99990 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
TB05AD EXAMPLE PROGRAM DATA 3 1 2 (0.0,0.5) G A 1.0 2.0 0.0 4.0 -1.0 0.0 0.0 0.0 1.0 1.0 0.0 1.0 1.0 0.0 -1.0 0.0 0.0 1.0Program Results
TB05AD EXAMPLE PROGRAM RESULTS
RCOND = 0.22
Eigenvalues of the state transmission matrix A are
3.00 0.00*j
-3.00 0.00*j
1.00 0.00*j
The frequency response matrix G(freq) is
( 0.69, 0.35)
(-0.80,-0.40)
H(inverse)*B is
(-0.11,-0.05)
(-0.43, 0.00)
(-0.80,-0.40)
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TC01OD.html 0000664 0000000 0000000 00000020563 14560147231 0020352 0 ustar 00root root 0000000 0000000
Purpose
To find the dual right (left) polynomial matrix representation of a given left (right) polynomial matrix representation, where the right and left polynomial matrix representations are of the form Q(s)*inv(P(s)) and inv(P(s))*Q(s) respectively.Specification
SUBROUTINE TC01OD( LERI, M, P, INDLIM, PCOEFF, LDPCO1, LDPCO2,
$ QCOEFF, LDQCO1, LDQCO2, INFO )
C .. Scalar Arguments ..
CHARACTER LERI
INTEGER INFO, INDLIM, LDPCO1, LDPCO2, LDQCO1, LDQCO2, M,
$ P
C .. Array Arguments ..
DOUBLE PRECISION PCOEFF(LDPCO1,LDPCO2,*), QCOEFF(LDQCO1,LDQCO2,*)
Arguments
Mode Parameters
LERI CHARACTER*1
Indicates whether a left or right matrix fraction is input
as follows:
= 'L': A left matrix fraction is input;
= 'R': A right matrix fraction is input.
Input/Output Parameters
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
INDLIM (input) INTEGER
The highest value of K for which PCOEFF(.,.,K) and
QCOEFF(.,.,K) are to be transposed.
K = kpcoef + 1, where kpcoef is the maximum degree of the
polynomials in P(s). INDLIM >= 1.
PCOEFF (input/output) DOUBLE PRECISION array, dimension
(LDPCO1,LDPCO2,INDLIM)
If LERI = 'L' then porm = P, otherwise porm = M.
On entry, the leading porm-by-porm-by-INDLIM part of this
array must contain the coefficients of the denominator
matrix P(s).
PCOEFF(I,J,K) is the coefficient in s**(INDLIM-K) of
polynomial (I,J) of P(s), where K = 1,2,...,INDLIM.
On exit, the leading porm-by-porm-by-INDLIM part of this
array contains the coefficients of the denominator matrix
P'(s) of the dual system.
LDPCO1 INTEGER
The leading dimension of array PCOEFF.
LDPCO1 >= MAX(1,P) if LERI = 'L',
LDPCO1 >= MAX(1,M) if LERI = 'R'.
LDPCO2 INTEGER
The second dimension of array PCOEFF.
LDPCO2 >= MAX(1,P) if LERI = 'L',
LDPCO2 >= MAX(1,M) if LERI = 'R'.
QCOEFF (input/output) DOUBLE PRECISION array, dimension
(LDQCO1,LDQCO2,INDLIM)
On entry, the leading P-by-M-by-INDLIM part of this array
must contain the coefficients of the numerator matrix
Q(s).
QCOEFF(I,J,K) is the coefficient in s**(INDLIM-K) of
polynomial (I,J) of Q(s), where K = 1,2,...,INDLIM.
On exit, the leading M-by-P-by-INDLIM part of the array
contains the coefficients of the numerator matrix Q'(s)
of the dual system.
LDQCO1 INTEGER
The leading dimension of array QCOEFF.
LDQCO1 >= MAX(1,M,P).
LDQCO2 INTEGER
The second dimension of array QCOEFF.
LDQCO2 >= MAX(1,M,P).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
If the given M-input/P-output left (right) polynomial matrix representation has numerator matrix Q(s) and denominator matrix P(s), its dual P-input/M-output right (left) polynomial matrix representation simply has numerator matrix Q'(s) and denominator matrix P'(s).References
None.Numerical Aspects
None.Further Comments
NoneExample
Program Text
* TC01OD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, PMAX, INDMAX
PARAMETER ( MMAX = 20, PMAX = 20, INDMAX = 20 )
INTEGER MAXMP
PARAMETER ( MAXMP = MAX( MMAX, PMAX ) )
INTEGER LDPCO1, LDPCO2, LDQCO1, LDQCO2
PARAMETER ( LDPCO1 = MAXMP, LDPCO2 = MAXMP,
$ LDQCO1 = MAXMP, LDQCO2 = MAXMP )
* .. Local Scalars ..
INTEGER I, INDLIM, INFO, J, K, M, P, PORM
CHARACTER*1 LERI
LOGICAL LLERI
* .. Local Arrays ..
DOUBLE PRECISION PCOEFF(LDPCO1,LDPCO2,INDMAX),
$ QCOEFF(LDQCO1,LDQCO2,INDMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL TC01OD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, P, INDLIM, LERI
LLERI = LSAME( LERI, 'L' )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) M
ELSE IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) P
ELSE IF ( INDLIM.LE.0 .OR. INDLIM.GT.INDMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) INDLIM
ELSE
PORM = P
IF ( .NOT.LLERI ) PORM = M
READ ( NIN, FMT = * )
$ ( ( ( PCOEFF(I,J,K), K = 1,INDLIM ), J = 1,PORM ),
$ I = 1,PORM )
READ ( NIN, FMT = * )
$ ( ( ( QCOEFF(I,J,K), K = 1,INDLIM ), J = 1,M ), I = 1,P )
* Find the dual right pmr of the given left pmr.
CALL TC01OD( LERI, M, P, INDLIM, PCOEFF, LDPCO1, LDPCO2,
$ QCOEFF, LDQCO1, LDQCO2, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 40 I = 1, PORM
DO 20 J = 1, PORM
WRITE ( NOUT, FMT = 99996 ) I, J,
$ ( PCOEFF(I,J,K), K = 1,INDLIM )
20 CONTINUE
40 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 80 I = 1, M
DO 60 J = 1, P
WRITE ( NOUT, FMT = 99996 ) I, J,
$ ( QCOEFF(I,J,K), K = 1,INDLIM )
60 CONTINUE
80 CONTINUE
END IF
END IF
STOP
*
99999 FORMAT (' TC01OD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TC01OD = ',I2)
99997 FORMAT (' The coefficients of the denominator matrix of the dual',
$ ' system are ')
99996 FORMAT (/' element (',I2,',',I2,') is ',20(1X,F6.2))
99995 FORMAT (//' The coefficients of the numerator matrix of the dual',
$ ' system are ')
99994 FORMAT (/' M is out of range.',/' M = ',I5)
99993 FORMAT (/' P is out of range.',/' P = ',I5)
99992 FORMAT (/' INDLIM is out of range.',/' INDLIM = ',I5)
END
Program Data
TC01OD EXAMPLE PROGRAM DATA 2 2 3 L 2.0 3.0 1.0 4.0 -1.0 -1.0 5.0 7.0 -6.0 3.0 2.0 2.0 6.0 -1.0 5.0 1.0 7.0 5.0 1.0 1.0 1.0 4.0 1.0 -1.0Program Results
TC01OD EXAMPLE PROGRAM RESULTS The coefficients of the denominator matrix of the dual system are element ( 1, 1) is 2.00 3.00 1.00 element ( 1, 2) is 5.00 7.00 -6.00 element ( 2, 1) is 4.00 -1.00 -1.00 element ( 2, 2) is 3.00 2.00 2.00 The coefficients of the numerator matrix of the dual system are element ( 1, 1) is 6.00 -1.00 5.00 element ( 1, 2) is 1.00 1.00 1.00 element ( 2, 1) is 1.00 7.00 5.00 element ( 2, 2) is 4.00 1.00 -1.00
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TC04AD.html 0000664 0000000 0000000 00000033626 14560147231 0020343 0 ustar 00root root 0000000 0000000
Purpose
To find a state-space representation (A,B,C,D) with the same
transfer matrix T(s) as that of a given left or right polynomial
matrix representation, i.e.
C*inv(sI-A)*B + D = T(s) = inv(P(s))*Q(s) = Q(s)*inv(P(s)).
Specification
SUBROUTINE TC04AD( LERI, M, P, INDEX, PCOEFF, LDPCO1, LDPCO2,
$ QCOEFF, LDQCO1, LDQCO2, N, RCOND, A, LDA, B,
$ LDB, C, LDC, D, LDD, IWORK, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER LERI
INTEGER INFO, LDA, LDB, LDC, LDD, LDPCO1, LDPCO2,
$ LDQCO1, LDQCO2, LDWORK, M, N, P
DOUBLE PRECISION RCOND
C .. Array Arguments ..
INTEGER INDEX(*), IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), PCOEFF(LDPCO1,LDPCO2,*),
$ QCOEFF(LDQCO1,LDQCO2,*)
Arguments
Mode Parameters
LERI CHARACTER*1
Indicates whether a left polynomial matrix representation
or a right polynomial matrix representation is input as
follows:
= 'L': A left matrix fraction is input;
= 'R': A right matrix fraction is input.
Input/Output Parameters
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
INDEX (input) INTEGER array, dimension (MAX(M,P))
If LERI = 'L', INDEX(I), I = 1,2,...,P, must contain the
maximum degree of the polynomials in the I-th row of the
denominator matrix P(s) of the given left polynomial
matrix representation.
If LERI = 'R', INDEX(I), I = 1,2,...,M, must contain the
maximum degree of the polynomials in the I-th column of
the denominator matrix P(s) of the given right polynomial
matrix representation.
PCOEFF (input) DOUBLE PRECISION array, dimension
(LDPCO1,LDPCO2,kpcoef), where kpcoef = MAX(INDEX(I)) + 1.
If LERI = 'L' then porm = P, otherwise porm = M.
The leading porm-by-porm-by-kpcoef part of this array must
contain the coefficients of the denominator matrix P(s).
PCOEFF(I,J,K) is the coefficient in s**(INDEX(iorj)-K+1)
of polynomial (I,J) of P(s), where K = 1,2,...,kpcoef; if
LERI = 'L' then iorj = I, otherwise iorj = J.
Thus for LERI = 'L', P(s) =
diag(s**INDEX(I))*(PCOEFF(.,.,1)+PCOEFF(.,.,2)/s+...).
If LERI = 'R', PCOEFF is modified by the routine but
restored on exit.
LDPCO1 INTEGER
The leading dimension of array PCOEFF.
LDPCO1 >= MAX(1,P) if LERI = 'L',
LDPCO1 >= MAX(1,M) if LERI = 'R'.
LDPCO2 INTEGER
The second dimension of array PCOEFF.
LDPCO2 >= MAX(1,P) if LERI = 'L',
LDPCO2 >= MAX(1,M) if LERI = 'R'.
QCOEFF (input) DOUBLE PRECISION array, dimension
(LDQCO1,LDQCO2,kpcoef)
If LERI = 'L' then porp = M, otherwise porp = P.
The leading porm-by-porp-by-kpcoef part of this array must
contain the coefficients of the numerator matrix Q(s).
QCOEFF(I,J,K) is defined as for PCOEFF(I,J,K).
If LERI = 'R', QCOEFF is modified by the routine but
restored on exit.
LDQCO1 INTEGER
The leading dimension of array QCOEFF.
LDQCO1 >= MAX(1,P) if LERI = 'L',
LDQCO1 >= MAX(1,M,P) if LERI = 'R'.
LDQCO2 INTEGER
The second dimension of array QCOEFF.
LDQCO2 >= MAX(1,M) if LERI = 'L',
LDQCO2 >= MAX(1,M,P) if LERI = 'R'.
N (output) INTEGER
The order of the resulting state-space representation.
porm
That is, N = SUM INDEX(I).
I=1
RCOND (output) DOUBLE PRECISION
The estimated reciprocal of the condition number of the
leading row (if LERI = 'L') or the leading column (if
LERI = 'R') coefficient matrix of P(s).
If RCOND is nearly zero, P(s) is nearly row or column
non-proper.
A (output) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array contains the state
dynamics matrix A.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (output) DOUBLE PRECISION array, dimension (LDB,MAX(M,P))
The leading N-by-M part of this array contains the
input/state matrix B; the remainder of the leading
N-by-MAX(M,P) part is used as internal workspace.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (output) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array contains the
state/output matrix C; the remainder of the leading
MAX(M,P)-by-N part is used as internal workspace.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,M,P).
D (output) DOUBLE PRECISION array, dimension (LDD,MAX(M,P))
The leading P-by-M part of this array contains the direct
transmission matrix D; the remainder of the leading
MAX(M,P)-by-MAX(M,P) part is used as internal workspace.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,M,P).
Workspace
IWORK INTEGER array, dimension (2*MAX(M,P))
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,MAX(M,P)*(MAX(M,P)+4)).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if P(s) is not row (if LERI = 'L') or column
(if LERI = 'R') proper. Consequently, no state-space
representation is calculated.
Method
The method for a left matrix fraction will be described here;
right matrix fractions are dealt with by obtaining the dual left
polynomial matrix representation and constructing an equivalent
state-space representation for this. The first step is to check
if the denominator matrix P(s) is row proper; if it is not then
the routine returns with the Error Indicator (INFO) set to 1.
Otherwise, Wolovich's Observable Structure Theorem is used to
construct a state-space representation (A,B,C,D) in observable
companion form. The sizes of the blocks of matrix A and matrix C
here are precisely the row degrees of P(s), while their
'non-trivial' columns are given easily from its coefficients.
Similarly, the matrix D is obtained from the leading coefficients
of P(s) and of the numerator matrix Q(s), while matrix B is given
by the relation Sbar(s)B = Q(s) - P(s)D, where Sbar(s) is a
polynomial matrix whose (j,k)(th) element is given by
j-u(k-1)-1
( s , j = u(k-1)+1,u(k-1)+2,....,u(k)
Sbar = (
j,k ( 0 , otherwise
k
u(k) = SUM d , k = 1,2,...,M and d ,d ,...,d are the
i=1 i 1 2 M
controllability indices. For convenience in solving this, C' and B
are initially set up to contain the coefficients of P(s) and Q(s),
respectively, stored by rows.
References
[1] Wolovich, W.A.
Linear Multivariate Systems, (Theorem 4.3.3).
Springer-Verlag, 1974.
Numerical Aspects
3 The algorithm requires 0(N ) operations.Further Comments
NoneExample
Program Text
* TC04AD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, PMAX, KPCMAX, NMAX
PARAMETER ( MMAX = 5, PMAX = 5, KPCMAX = 5, NMAX = 5 )
INTEGER MAXMP
PARAMETER ( MAXMP = MAX( MMAX, PMAX ) )
INTEGER LDPCO1, LDPCO2, LDQCO1, LDQCO2, LDA, LDB, LDC,
$ LDD
PARAMETER ( LDPCO1 = MAXMP, LDPCO2 = MAXMP,
$ LDQCO1 = MAXMP, LDQCO2 = MAXMP,
$ LDA = NMAX, LDB = NMAX, LDC = MAXMP,
$ LDD = MAXMP )
INTEGER LIWORK
PARAMETER ( LIWORK = 2*MAXMP )
INTEGER LDWORK
PARAMETER ( LDWORK = ( MAXMP )*( MAXMP+4 ) )
* .. Local Scalars ..
DOUBLE PRECISION RCOND
INTEGER I, INFO, J, K, KPCOEF, M, N, P, PORM, PORP
CHARACTER*1 LERI
LOGICAL LLERI
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MAXMP), C(LDC,NMAX),
$ D(LDD,MAXMP), PCOEFF(LDPCO1,LDPCO2,KPCMAX),
$ QCOEFF(LDQCO1,LDQCO2,KPCMAX), DWORK(LDWORK)
INTEGER INDEX(MAXMP), IWORK(LIWORK)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL TC04AD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, P, LERI
LLERI = LSAME( LERI, 'L' )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) M
ELSE IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) P
ELSE
PORM = P
IF ( .NOT.LLERI ) PORM = M
READ ( NIN, FMT = * ) ( INDEX(I), I = 1,PORM )
PORP = M
IF ( .NOT.LLERI ) PORP = P
KPCOEF = 0
DO 20 I = 1, PORM
KPCOEF = MAX( KPCOEF, INDEX(I) )
20 CONTINUE
KPCOEF = KPCOEF + 1
IF ( KPCOEF.LE.0 .OR. KPCOEF.GT.KPCMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) KPCOEF
ELSE
READ ( NIN, FMT = * )
$ ( ( ( PCOEFF(I,J,K), K = 1,KPCOEF ), J = 1,PORM ),
$ I = 1,PORM )
READ ( NIN, FMT = * )
$ ( ( ( QCOEFF(I,J,K), K = 1,KPCOEF ), J = 1,PORP ),
$ I = 1,PORM )
* Find a ssr of the given left pmr.
CALL TC04AD( LERI, M, P, INDEX, PCOEFF, LDPCO1, LDPCO2,
$ QCOEFF, LDQCO1, LDQCO2, N, RCOND, A, LDA, B,
$ LDB, C, LDC, D, LDD, IWORK, DWORK, LDWORK,
$ INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) N, RCOND
WRITE ( NOUT, FMT = 99996 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
40 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
60 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 80 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
80 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 100 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1,M )
100 CONTINUE
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TC04AD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TC04AD = ',I2)
99997 FORMAT (' The order of the resulting state-space representation ',
$ ' = ',I2,//' RCOND = ',F4.2)
99996 FORMAT (/' The state dynamics matrix A is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' The input/state matrix B is ')
99993 FORMAT (/' The state/output matrix C is ')
99992 FORMAT (/' The direct transmission matrix D is ')
99991 FORMAT (/' M is out of range.',/' M = ',I5)
99990 FORMAT (/' P is out of range.',/' P = ',I5)
99989 FORMAT (/' KPCOEF is out of range.',/' KPCOEF = ',I5)
END
Program Data
TC04AD EXAMPLE PROGRAM DATA 2 2 L 2 2 2.0 3.0 1.0 4.0 -1.0 -1.0 5.0 7.0 -6.0 3.0 2.0 2.0 6.0 -1.0 5.0 1.0 7.0 5.0 1.0 1.0 1.0 4.0 1.0 -1.0Program Results
TC04AD EXAMPLE PROGRAM RESULTS The order of the resulting state-space representation = 4 RCOND = 0.25 The state dynamics matrix A is 0.0000 0.5714 0.0000 -0.4286 1.0000 1.0000 0.0000 -1.0000 0.0000 -2.0000 0.0000 2.0000 0.0000 0.7857 1.0000 -1.7143 The input/state matrix B is 8.0000 3.8571 4.0000 4.0000 -9.0000 5.0000 4.0000 -5.0714 The state/output matrix C is 0.0000 -0.2143 0.0000 0.2857 0.0000 0.3571 0.0000 -0.1429 The direct transmission matrix D is -1.0000 0.9286 2.0000 -0.2143
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TC05AD.html 0000664 0000000 0000000 00000026406 14560147231 0020342 0 ustar 00root root 0000000 0000000
Purpose
To evaluate the transfer matrix T(s) of a left polynomial matrix representation [T(s) = inv(P(s))*Q(s)] or a right polynomial matrix representation [T(s) = Q(s)*inv(P(s))] at any specified complex frequency s = SVAL. This routine will calculate the standard frequency response matrix at frequency omega if SVAL is supplied as (0.0,omega).Specification
SUBROUTINE TC05AD( LERI, M, P, SVAL, INDEX, PCOEFF, LDPCO1,
$ LDPCO2, QCOEFF, LDQCO1, LDQCO2, RCOND, CFREQR,
$ LDCFRE, IWORK, DWORK, ZWORK, INFO )
C .. Scalar Arguments ..
CHARACTER LERI
INTEGER INFO, LDCFRE, LDPCO1, LDPCO2, LDQCO1, LDQCO2, M,
$ P
DOUBLE PRECISION RCOND
COMPLEX*16 SVAL
C .. Array Arguments ..
INTEGER INDEX(*), IWORK(*)
DOUBLE PRECISION DWORK(*), PCOEFF(LDPCO1,LDPCO2,*),
$ QCOEFF(LDQCO1,LDQCO2,*)
COMPLEX*16 CFREQR(LDCFRE,*), ZWORK(*)
Arguments
Mode Parameters
LERI CHARACTER*1
Indicates whether a left polynomial matrix representation
or a right polynomial matrix representation is to be used
to evaluate the transfer matrix as follows:
= 'L': A left matrix fraction is input;
= 'R': A right matrix fraction is input.
Input/Output Parameters
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
SVAL (input) COMPLEX*16
The frequency at which the transfer matrix or the
frequency respose matrix is to be evaluated.
For a standard frequency response set the real part
of SVAL to zero.
INDEX (input) INTEGER array, dimension (MAX(M,P))
If LERI = 'L', INDEX(I), I = 1,2,...,P, must contain the
maximum degree of the polynomials in the I-th row of the
denominator matrix P(s) of the given left polynomial
matrix representation.
If LERI = 'R', INDEX(I), I = 1,2,...,M, must contain the
maximum degree of the polynomials in the I-th column of
the denominator matrix P(s) of the given right polynomial
matrix representation.
PCOEFF (input) DOUBLE PRECISION array, dimension
(LDPCO1,LDPCO2,kpcoef), where kpcoef = MAX(INDEX(I)) + 1.
If LERI = 'L' then porm = P, otherwise porm = M.
The leading porm-by-porm-by-kpcoef part of this array must
contain the coefficients of the denominator matrix P(s).
PCOEFF(I,J,K) is the coefficient in s**(INDEX(iorj)-K+1)
of polynomial (I,J) of P(s), where K = 1,2,...,kpcoef; if
LERI = 'L' then iorj = I, otherwise iorj = J.
Thus for LERI = 'L', P(s) =
diag(s**INDEX(I))*(PCOEFF(.,.,1)+PCOEFF(.,.,2)/s+...).
If LERI = 'R', PCOEFF is modified by the routine but
restored on exit.
LDPCO1 INTEGER
The leading dimension of array PCOEFF.
LDPCO1 >= MAX(1,P) if LERI = 'L',
LDPCO1 >= MAX(1,M) if LERI = 'R'.
LDPCO2 INTEGER
The second dimension of array PCOEFF.
LDPCO2 >= MAX(1,P) if LERI = 'L',
LDPCO2 >= MAX(1,M) if LERI = 'R'.
QCOEFF (input) DOUBLE PRECISION array, dimension
(LDQCO1,LDQCO2,kpcoef)
If LERI = 'L' then porp = M, otherwise porp = P.
The leading porm-by-porp-by-kpcoef part of this array must
contain the coefficients of the numerator matrix Q(s).
QCOEFF(I,J,K) is defined as for PCOEFF(I,J,K).
If LERI = 'R', QCOEFF is modified by the routine but
restored on exit.
LDQCO1 INTEGER
The leading dimension of array QCOEFF.
LDQCO1 >= MAX(1,P) if LERI = 'L',
LDQCO1 >= MAX(1,M,P) if LERI = 'R'.
LDQCO2 INTEGER
The second dimension of array QCOEFF.
LDQCO2 >= MAX(1,M) if LERI = 'L',
LDQCO2 >= MAX(1,M,P) if LERI = 'R'.
RCOND (output) DOUBLE PRECISION
The estimated reciprocal of the condition number of the
denominator matrix P(SVAL).
If RCOND is nearly zero, SVAL is approximately a system
pole.
CFREQR (output) COMPLEX*16 array, dimension (LDCFRE,MAX(M,P))
The leading porm-by-porp part of this array contains the
frequency response matrix T(SVAL).
LDCFRE INTEGER
The leading dimension of array CFREQR.
LDCFRE >= MAX(1,P) if LERI = 'L',
LDCFRE >= MAX(1,M,P) if LERI = 'R'.
Workspace
IWORK INTEGER array, dimension (liwork)
where liwork = P, if LERI = 'L',
liwork = M, if LERI = 'R'.
DWORK DOUBLE PRECISION array, dimension (ldwork)
where ldwork = 2*P, if LERI = 'L',
ldwork = 2*M, if LERI = 'R'.
ZWORK COMPLEX*16 array, dimension (lzwork),
where lzwork = P*(P+2), if LERI = 'L',
lzwork = M*(M+2), if LERI = 'R'.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if P(SVAL) is exactly or nearly singular;
no frequency response is calculated.
Method
The method for a left matrix fraction will be described here; right matrix fractions are dealt with by obtaining the dual left fraction and calculating its frequency response (see SLICOT Library routine TC01OD). The first step is to calculate the complex value P(SVAL) of the denominator matrix P(s) at the desired frequency SVAL. If P(SVAL) is approximately singular, SVAL is approximately a pole of this system and so the frequency response matrix T(SVAL) is not calculated; in this case, the routine returns with the Error Indicator (INFO) set to 1. Otherwise, the complex value Q(SVAL) of the numerator matrix Q(s) at frequency SVAL is calculated in a similar way to P(SVAL), and the desired response matrix T(SVAL) = inv(P(SVAL))*Q(SVAL) is found by solving the corresponding system of complex linear equations.References
NoneNumerical Aspects
3 The algorithm requires 0(N ) operations.Further Comments
NoneExample
Program Text
* TC05AD EXAMPLE PROGRAM TEXT.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, PMAX, KPCMAX
PARAMETER ( MMAX = 20, PMAX = 20, KPCMAX = 20 )
INTEGER MAXMP
PARAMETER ( MAXMP = MAX( MMAX, PMAX ) )
INTEGER LDCFRE, LDPCO1, LDPCO2, LDQCO1, LDQCO2
PARAMETER ( LDCFRE = MAXMP, LDPCO1 = MAXMP,
$ LDPCO2 = MAXMP, LDQCO1 = MAXMP,
$ LDQCO2 = MAXMP )
INTEGER LDWORK
PARAMETER ( LDWORK = 2*MAXMP )
INTEGER LZWORK
PARAMETER ( LZWORK = ( MAXMP )*( MAXMP+2 ) )
* .. Local Scalars ..
COMPLEX*16 SVAL
DOUBLE PRECISION RCOND
INTEGER I, INFO, J, K, KPCOEF, M, P, PORM, PORP
CHARACTER*1 LERI
LOGICAL LLERI
* .. Local Arrays ..
COMPLEX*16 CFREQR(LDCFRE,MAXMP), ZWORK(LZWORK)
DOUBLE PRECISION DWORK(LDWORK), PCOEFF(LDPCO1,LDPCO2,KPCMAX),
$ QCOEFF(LDQCO1,LDQCO2,KPCMAX)
INTEGER INDEX(MAXMP), IWORK(MAXMP)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL TC05AD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, P, SVAL, LERI
LLERI = LSAME( LERI, 'L' )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) M
ELSE IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) P
ELSE
PORM = P
IF ( .NOT.LLERI ) PORM = M
READ ( NIN, FMT = * ) ( INDEX(I), I = 1,PORM )
PORP = M
IF ( .NOT.LLERI ) PORP = P
KPCOEF = 0
DO 20 I = 1, PORM
KPCOEF = MAX( KPCOEF, INDEX(I) )
20 CONTINUE
KPCOEF = KPCOEF + 1
IF ( KPCOEF.LE.0 .OR. KPCOEF.GT.KPCMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) KPCOEF
ELSE
READ ( NIN, FMT = * )
$ ( ( ( PCOEFF(I,J,K), K = 1,KPCOEF ), J = 1,PORM ),
$ I = 1,PORM )
READ ( NIN, FMT = * )
$ ( ( ( QCOEFF(I,J,K), K = 1,KPCOEF ), J = 1,PORP ),
$ I = 1,PORM )
* Find the standard frequency response matrix of left pmr
* at 0.5*j.
CALL TC05AD( LERI, M, P, SVAL, INDEX, PCOEFF, LDPCO1,
$ LDPCO2, QCOEFF, LDQCO1, LDQCO2, RCOND, CFREQR,
$ LDCFRE, IWORK, DWORK, ZWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) RCOND
DO 40 I = 1, PORM
WRITE ( NOUT, FMT = 99996 )
$ ( CFREQR(I,J), J = 1,PORP )
40 CONTINUE
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TC05AD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TC05AD = ',I2)
99997 FORMAT (' RCOND = ',F4.2,//' The frequency response matrix T(SVA',
$ 'L) is ')
99996 FORMAT (20(' (',F5.2,',',F5.2,') ',:))
99995 FORMAT (/' M is out of range.',/' M = ',I5)
99994 FORMAT (/' P is out of range.',/' P = ',I5)
99993 FORMAT (/' KPCOEF is out of range.',/' KPCOEF = ',I5)
END
Program Data
TC05AD EXAMPLE PROGRAM DATA 2 2 (0.0,0.5) L 2 2 2.0 3.0 1.0 4.0 -1.0 -1.0 5.0 7.0 -6.0 3.0 2.0 2.0 6.0 -1.0 5.0 1.0 7.0 5.0 1.0 1.0 1.0 4.0 1.0 -1.0Program Results
TC05AD EXAMPLE PROGRAM RESULTS RCOND = 0.19 The frequency response matrix T(SVAL) is (-0.25,-0.33) ( 0.26,-0.45) (-1.48, 0.35) (-2.25,-1.11)
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TD03AD.html 0000664 0000000 0000000 00000053266 14560147231 0020345 0 ustar 00root root 0000000 0000000
Purpose
To find a relatively prime left or right polynomial matrix representation for a proper transfer matrix T(s) given as either row or column polynomial vectors over common denominator polynomials, possibly with uncancelled common terms.Specification
SUBROUTINE TD03AD( ROWCOL, LERI, EQUIL, M, P, INDEXD, DCOEFF,
$ LDDCOE, UCOEFF, LDUCO1, LDUCO2, NR, A, LDA, B,
$ LDB, C, LDC, D, LDD, INDEXP, PCOEFF, LDPCO1,
$ LDPCO2, QCOEFF, LDQCO1, LDQCO2, VCOEFF, LDVCO1,
$ LDVCO2, TOL, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER EQUIL, LERI, ROWCOL
INTEGER INFO, LDA, LDB, LDC, LDD, LDDCOE, LDPCO1,
$ LDPCO2, LDQCO1, LDQCO2, LDUCO1, LDUCO2, LDVCO1,
$ LDVCO2, LDWORK, M, NR, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER INDEXD(*), INDEXP(*), IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DCOEFF(LDDCOE,*), DWORK(*),
$ PCOEFF(LDPCO1,LDPCO2,*),
$ QCOEFF(LDQCO1,LDQCO2,*),
$ UCOEFF(LDUCO1,LDUCO2,*), VCOEFF(LDVCO1,LDVCO2,*)
Arguments
Mode Parameters
ROWCOL CHARACTER*1
Indicates whether T(s) is to be factorized by rows or by
columns as follows:
= 'R': T(s) is factorized by rows;
= 'C': T(s) is factorized by columns.
LERI CHARACTER*1
Indicates whether a left or a right polynomial matrix
representation is required as follows:
= 'L': A left polynomial matrix representation
inv(P(s))*Q(s) is required;
= 'R': A right polynomial matrix representation
Q(s)*inv(P(s)) is required.
EQUIL CHARACTER*1
Specifies whether the user wishes to balance the triplet
(A,B,C), before computing a minimal state-space
representation, as follows:
= 'S': Perform balancing (scaling);
= 'N': Do not perform balancing.
Input/Output Parameters
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
INDEXD (input) INTEGER array, dimension (P), if ROWCOL = 'R', or
dimension (M), if ROWCOL = 'C'.
The leading pormd elements of this array must contain the
row degrees of the denominator polynomials in D(s).
pormd = P if the transfer matrix T(s) is given as row
polynomial vectors over denominator polynomials;
pormd = M if the transfer matrix T(s) is given as column
polynomial vectors over denominator polynomials.
DCOEFF (input) DOUBLE PRECISION array, dimension (LDDCOE,kdcoef),
where kdcoef = MAX(INDEXD(I)) + 1.
The leading pormd-by-kdcoef part of this array must
contain the coefficients of each denominator polynomial.
DCOEFF(I,K) is the coefficient in s**(INDEXD(I)-K+1) of
the I-th denominator polynomial in D(s), where K = 1,2,
...,kdcoef.
LDDCOE INTEGER
The leading dimension of array DCOEFF.
LDDCOE >= MAX(1,P), if ROWCOL = 'R';
LDDCOE >= MAX(1,M), if ROWCOL = 'C'.
UCOEFF (input) DOUBLE PRECISION array, dimension
(LDUCO1,LDUCO2,kdcoef)
The leading P-by-M-by-kdcoef part of this array must
contain the coefficients of the numerator matrix U(s);
if ROWCOL = 'C', this array is modified internally but
restored on exit, and the remainder of the leading
MAX(M,P)-by-MAX(M,P)-by-kdcoef part is used as internal
workspace.
UCOEFF(I,J,K) is the coefficient in s**(INDEXD(iorj)-K+1)
of polynomial (I,J) of U(s), where K = 1,2,...,kdcoef;
iorj = I if T(s) is given as row polynomial vectors over
denominator polynomials; iorj = J if T(s) is given as
column polynomial vectors over denominator polynomials.
Thus for ROWCOL = 'R', U(s) =
diag(s**INDEXD(I))*(UCOEFF(.,.,1)+UCOEFF(.,.,2)/s+...).
LDUCO1 INTEGER
The leading dimension of array UCOEFF.
LDUCO1 >= MAX(1,P), if ROWCOL = 'R';
LDUCO1 >= MAX(1,M,P), if ROWCOL = 'C'.
LDUCO2 INTEGER
The second dimension of array UCOEFF.
LDUCO2 >= MAX(1,M), if ROWCOL = 'R';
LDUCO2 >= MAX(1,M,P), if ROWCOL = 'C'.
NR (output) INTEGER
The order of the resulting minimal realization, i.e. the
order of the state dynamics matrix A.
A (output) DOUBLE PRECISION array, dimension (LDA,N),
pormd
where N = SUM INDEXD(I)
I=1
The leading NR-by-NR part of this array contains the upper
block Hessenberg state dynamics matrix A.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (output) DOUBLE PRECISION array, dimension (LDB,MAX(M,P))
The leading NR-by-M part of this array contains the
input/state matrix B; the remainder of the leading
N-by-MAX(M,P) part is used as internal workspace.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (output) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-NR part of this array contains the
state/output matrix C; the remainder of the leading
MAX(M,P)-by-N part is used as internal workspace.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,M,P).
D (output) DOUBLE PRECISION array, dimension (LDD,MAX(M,P))
The leading P-by-M part of this array contains the direct
transmission matrix D; the remainder of the leading
MAX(M,P)-by-MAX(M,P) part is used as internal workspace.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,M,P).
INDEXP (output) INTEGER array, dimension (P), if ROWCOL = 'R', or
dimension (M), if ROWCOL = 'C'.
The leading pormp elements of this array contain the
row (column if ROWCOL = 'C') degrees of the denominator
matrix P(s).
pormp = P if a left polynomial matrix representation
is requested; pormp = M if a right polynomial matrix
representation is requested.
These elements are ordered so that
INDEXP(1) >= INDEXP(2) >= ... >= INDEXP(pormp).
PCOEFF (output) DOUBLE PRECISION array, dimension
(LDPCO1,LDPCO2,N+1)
The leading pormp-by-pormp-by-kpcoef part of this array
contains the coefficients of the denominator matrix P(s),
where kpcoef = MAX(INDEXP(I)) + 1.
PCOEFF(I,J,K) is the coefficient in s**(INDEXP(iorj)-K+1)
of polynomial (I,J) of P(s), where K = 1,2,...,kpcoef;
iorj = I if a left polynomial matrix representation is
requested; iorj = J if a right polynomial matrix
representation is requested.
Thus for a left polynomial matrix representation, P(s) =
diag(s**INDEXP(I))*(PCOEFF(.,.,1)+PCOEFF(.,.,2)/s+...).
LDPCO1 INTEGER
The leading dimension of array PCOEFF.
LDPCO1 >= MAX(1,P), if ROWCOL = 'R';
LDPCO1 >= MAX(1,M), if ROWCOL = 'C'.
LDPCO2 INTEGER
The second dimension of array PCOEFF.
LDPCO2 >= MAX(1,P), if ROWCOL = 'R';
LDPCO2 >= MAX(1,M), if ROWCOL = 'C'.
QCOEFF (output) DOUBLE PRECISION array, dimension
(LDQCO1,LDQCO2,N+1)
The leading pormp-by-pormd-by-kpcoef part of this array
contains the coefficients of the numerator matrix Q(s).
QCOEFF(I,J,K) is defined as for PCOEFF(I,J,K).
LDQCO1 INTEGER
The leading dimension of array QCOEFF.
If LERI = 'L', LDQCO1 >= MAX(1,PM),
where PM = P, if ROWCOL = 'R';
PM = M, if ROWCOL = 'C'.
If LERI = 'R', LDQCO1 >= MAX(1,M,P).
LDQCO2 INTEGER
The second dimension of array QCOEFF.
If LERI = 'L', LDQCO2 >= MAX(1,MP),
where MP = M, if ROWCOL = 'R';
MP = P, if ROWCOL = 'C'.
If LERI = 'R', LDQCO2 >= MAX(1,M,P).
VCOEFF (output) DOUBLE PRECISION array, dimension
(LDVCO1,LDVCO2,N+1)
The leading pormp-by-NR-by-kpcoef part of this array
contains the coefficients of the intermediate matrix
V(s) as produced by SLICOT Library routine TB03AD.
LDVCO1 INTEGER
The leading dimension of array VCOEFF.
LDVCO1 >= MAX(1,P), if ROWCOL = 'R';
LDVCO1 >= MAX(1,M), if ROWCOL = 'C'.
LDVCO2 INTEGER
The second dimension of array VCOEFF. LDVCO2 >= MAX(1,N).
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in rank determination when
transforming (A, B, C). If the user sets TOL > 0, then
the given value of TOL is used as a lower bound for the
reciprocal condition number (see the description of the
argument RCOND in the SLICOT routine MB03OD); a
(sub)matrix whose estimated condition number is less than
1/TOL is considered to be of full rank. If the user sets
TOL <= 0, then an implicitly computed, default tolerance
(determined by the SLICOT routine TB01UD) is used instead.
Workspace
IWORK INTEGER array, dimension (N+MAX(M,P))
On exit, if INFO = 0, the first nonzero elements of
IWORK(1:N) return the orders of the diagonal blocks of A.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1, N + MAX(N, 3*M, 3*P), PM*(PM + 2))
where PM = P, if ROWCOL = 'R';
PM = M, if ROWCOL = 'C'.
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i (i <= k = pormd), then i is the first
integer I for which ABS( DCOEFF(I,1) ) is so small
that the calculations would overflow (see SLICOT
Library routine TD03AY); that is, the leading
coefficient of a polynomial is nearly zero; no
state-space representation or polynomial matrix
representation is calculated;
= k+1: if a singular matrix was encountered during the
computation of V(s);
= k+2: if a singular matrix was encountered during the
computation of P(s).
Method
The method for transfer matrices factorized by rows will be
described here; T(s) factorized by columns is dealt with by
operating on the dual T'(s). The description for T(s) is actually
the left polynomial matrix representation
T(s) = inv(D(s))*U(s),
where D(s) is diagonal with its (I,I)-th polynomial element of
degree INDEXD(I). The first step is to check whether the leading
coefficient of any polynomial element of D(s) is approximately
zero, if so the routine returns with INFO > 0. Otherwise,
Wolovich's Observable Structure Theorem is used to construct a
state-space representation in observable companion form which is
equivalent to the above polynomial matrix representation. The
method is particularly easy here due to the diagonal form of D(s).
This state-space representation is not necessarily controllable
(as D(s) and U(s) are not necessarily relatively left prime), but
it is in theory completely observable; however, its observability
matrix may be poorly conditioned, so it is treated as a general
state-space representation and SLICOT Library routine TB03AD is
used to separate out a minimal realization for T(s) from it by
means of orthogonal similarity transformations and then to
calculate a relatively prime (left or right) polynomial matrix
representation which is equivalent to this.
References
[1] Patel, R.V.
On Computing Matrix Fraction Descriptions and Canonical
Forms of Linear Time-Invariant Systems.
UMIST Control Systems Centre Report 489, 1980.
[2] Wolovich, W.A.
Linear Multivariable Systems, (Theorem 4.3.3).
Springer-Verlag, 1974.
Numerical Aspects
3 The algorithm requires 0(N ) operations.Further Comments
NoneExample
Program Text
* TD03AD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, PMAX, KDCMAX, NMAX
PARAMETER ( MMAX = 8, PMAX = 8, KDCMAX = 8, NMAX = 8 )
INTEGER MAXMP
PARAMETER ( MAXMP = MAX( MMAX, PMAX ) )
INTEGER LDA, LDB, LDC, LDD, LDDCOE, LDPCO1, LDPCO2,
$ LDQCO1, LDQCO2, LDUCO1, LDUCO2, LDVCO1, LDVCO2
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = MAXMP,
$ LDD = MAXMP, LDDCOE = MAXMP,
$ LDPCO1 = MAXMP, LDPCO2 = MAXMP,
$ LDQCO1 = MAXMP, LDQCO2 = MAXMP,
$ LDUCO1 = MAXMP, LDUCO2 = MAXMP,
$ LDVCO1 = MAXMP, LDVCO2 = NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX + MAXMP )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( NMAX + MAX( NMAX, 3*MAXMP ),
$ MAXMP*( MAXMP + 2 ) ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
CHARACTER*1 EQUIL, LERI, ROWCOL
INTEGER I, INDBLK, INFO, J, K, KDCOEF, M, MAXINP, N, NR,
$ P, PORMD, PORMP
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MAXMP), C(LDC,NMAX),
$ D(LDD,MAXMP), DCOEFF(LDDCOE,KDCMAX),
$ DWORK(LDWORK), PCOEFF(LDPCO1,LDPCO2,NMAX+1),
$ QCOEFF(LDQCO1,LDQCO2,NMAX+1),
$ UCOEFF(LDUCO1,LDUCO2,KDCMAX),
$ VCOEFF(LDVCO1,LDVCO2,NMAX+1)
INTEGER INDEXD(MAXMP), INDEXP(MAXMP), IWORK(LIWORK)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL TD03AD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, P, TOL, ROWCOL, LERI, EQUIL
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) M
ELSE IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99985 ) P
ELSE
PORMD = P
IF ( LSAME( ROWCOL, 'C' ) ) PORMD = M
PORMP = M
IF ( LSAME( LERI, 'R' ) ) PORMP = P
READ ( NIN, FMT = * ) ( INDEXD(I), I = 1,PORMD )
*
KDCOEF = 0
N = 0
DO 20 I = 1, PORMD
KDCOEF = MAX( KDCOEF, INDEXD(I) )
N = N + INDEXD(I)
20 CONTINUE
KDCOEF = KDCOEF + 1
*
IF ( KDCOEF.LE.0 .OR. KDCOEF.GT.KDCMAX ) THEN
WRITE ( NOUT, FMT = 99984 ) KDCOEF
ELSE
READ ( NIN, FMT = * )
$ ( ( DCOEFF(I,J), J = 1,KDCOEF ), I = 1,PORMD )
READ ( NIN, FMT = * )
$ ( ( ( UCOEFF(I,J,K), K = 1,KDCOEF ), J = 1,M ), I = 1,P )
* Find a relatively prime left pmr for the given transfer
* function.
CALL TD03AD( ROWCOL, LERI, EQUIL, M, P, INDEXD, DCOEFF,
$ LDDCOE, UCOEFF, LDUCO1, LDUCO2, NR, A, LDA, B,
$ LDB, C, LDC, D, LDD, INDEXP, PCOEFF, LDPCO1,
$ LDPCO2, QCOEFF, LDQCO1, LDQCO2, VCOEFF, LDVCO1,
$ LDVCO2, TOL, IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NR
DO 40 I = 1, NR
WRITE ( NOUT, FMT = 99996 ) ( A(I,J), J = 1,NR )
40 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 60 I = 1, NR
WRITE ( NOUT, FMT = 99996 ) ( B(I,J), J = 1,M )
60 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 80 I = 1, P
WRITE ( NOUT, FMT = 99996 ) ( C(I,J), J = 1,NR )
80 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 100 I = 1, P
WRITE ( NOUT, FMT = 99996 ) ( D(I,J), J = 1,M )
100 CONTINUE
INDBLK = 0
DO 120 I = 1, N
IF ( IWORK(I).NE.0 ) INDBLK = INDBLK + 1
120 CONTINUE
IF ( LSAME( LERI, 'L' ) ) THEN
WRITE ( NOUT, FMT = 99992 ) INDBLK,
$ ( IWORK(I), I = 1,INDBLK )
WRITE ( NOUT, FMT = 99990 ) ( INDEXP(I), I = 1,P )
ELSE
WRITE ( NOUT, FMT = 99991 ) INDBLK,
$ ( IWORK(I), I = 1,INDBLK )
WRITE ( NOUT, FMT = 99989 ) ( INDEXP(I), I = 1,M )
END IF
MAXINP = 0
DO 140 I = 1, PORMP
MAXINP = MAX( MAXINP, INDEXP(I) )
140 CONTINUE
MAXINP = MAXINP + 1
WRITE ( NOUT, FMT = 99988 )
DO 180 I = 1, PORMP
DO 160 J = 1, PORMP
WRITE ( NOUT, FMT = 99996 )
$ ( PCOEFF(I,J,K), K = 1,MAXINP )
160 CONTINUE
180 CONTINUE
WRITE ( NOUT, FMT = 99987 )
DO 220 I = 1, PORMP
DO 200 J = 1, PORMD
WRITE ( NOUT, FMT = 99996 )
$ ( QCOEFF(I,J,K), K = 1,MAXINP )
200 CONTINUE
220 CONTINUE
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TD03AD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TD03AD = ',I2)
99997 FORMAT (' The order of the resulting minimal realization = ',I2,
$ //' The state dynamics matrix A is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The input/state matrix B is ')
99994 FORMAT (/' The state/output matrix C is ')
99993 FORMAT (/' The direct transmission matrix D is ')
99992 FORMAT (/' The observability index of the minimal realization = ',
$ I2,//' The dimensions of the diagonal blocks of the state',
$ ' dynamics matrix are ',/20(I5))
99991 FORMAT (/' The controllability index of the minimal realization ',
$ '= ',I2,//' The dimensions of the diagonal blocks of the ',
$ 'state dynamics matrix are ',/20(I5))
99990 FORMAT (/' The row degrees of the denominator matrix P(s) are',
$ /20(I5))
99989 FORMAT (/' The column degrees of the denominator matrix P(s) are',
$ /20(I5))
99988 FORMAT (/' The denominator matrix P(s) is ')
99987 FORMAT (/' The numerator matrix Q(s) is ')
99986 FORMAT (/' M is out of range.',/' M = ',I5)
99985 FORMAT (/' P is out of range.',/' P = ',I5)
99984 FORMAT (/' KDCOEF is out of range.',/' KDCOEF = ',I5)
END
Program Data
TD01ND EXAMPLE PROGRAM DATA 2 2 0.0 R L N 3 3 1.0 6.0 11.0 6.0 1.0 6.0 11.0 6.0 1.0 6.0 12.0 7.0 0.0 1.0 4.0 3.0 0.0 0.0 1.0 1.0 1.0 8.0 20.0 15.0Program Results
TD03AD EXAMPLE PROGRAM RESULTS
The order of the resulting minimal realization = 3
The state dynamics matrix A is
0.5000 0.9478 10.1036
0.0000 -1.0000 0.0000
-0.8660 -0.6156 -5.5000
The input/state matrix B is
2.0000 12.5000
0.0000 -5.6273
0.0000 -2.0207
The state/output matrix C is
0.0000 0.0296 -0.5774
0.0000 -0.1481 -0.5774
The direct transmission matrix D is
1.0000 0.0000
0.0000 1.0000
The observability index of the minimal realization = 2
The dimensions of the diagonal blocks of the state dynamics matrix are
2 1
The row degrees of the denominator matrix P(s) are
2 1
The denominator matrix P(s) is
1.6667 4.3333 6.6667
0.3333 5.6667 5.3333
5.6273 5.6273 0.0000
-5.6273 -5.6273 0.0000
The numerator matrix Q(s) is
1.6667 4.3333 8.6667
0.3333 8.0000 16.6667
5.6273 5.6273 0.0000
-5.6273 -11.2546 0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TD04AD.html 0000664 0000000 0000000 00000036270 14560147231 0020342 0 ustar 00root root 0000000 0000000
Purpose
To find a minimal state-space representation (A,B,C,D) for a proper transfer matrix T(s) given as either row or column polynomial vectors over denominator polynomials, possibly with uncancelled common terms.Specification
SUBROUTINE TD04AD( ROWCOL, M, P, INDEX, DCOEFF, LDDCOE, UCOEFF,
$ LDUCO1, LDUCO2, NR, A, LDA, B, LDB, C, LDC, D,
$ LDD, TOL, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER ROWCOL
INTEGER INFO, LDA, LDB, LDC, LDD, LDDCOE, LDUCO1,
$ LDUCO2, LDWORK, M, NR, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER INDEX(*), IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DCOEFF(LDDCOE,*), DWORK(*),
$ UCOEFF(LDUCO1,LDUCO2,*)
Arguments
Mode Parameters
ROWCOL CHARACTER*1
Indicates whether the transfer matrix T(s) is given as
rows or columns over common denominators as follows:
= 'R': T(s) is given as rows over common denominators;
= 'C': T(s) is given as columns over common denominators.
Input/Output Parameters
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
INDEX (input) INTEGER array, dimension (porm), where porm = P,
if ROWCOL = 'R', and porm = M, if ROWCOL = 'C'.
This array must contain the degrees of the denominator
polynomials in D(s).
DCOEFF (input) DOUBLE PRECISION array, dimension (LDDCOE,kdcoef),
where kdcoef = MAX(INDEX(I)) + 1.
The leading porm-by-kdcoef part of this array must contain
the coefficients of each denominator polynomial.
DCOEFF(I,K) is the coefficient in s**(INDEX(I)-K+1) of the
I-th denominator polynomial in D(s), where
K = 1,2,...,kdcoef.
LDDCOE INTEGER
The leading dimension of array DCOEFF.
LDDCOE >= MAX(1,P) if ROWCOL = 'R';
LDDCOE >= MAX(1,M) if ROWCOL = 'C'.
UCOEFF (input) DOUBLE PRECISION array, dimension
(LDUCO1,LDUCO2,kdcoef)
The leading P-by-M-by-kdcoef part of this array must
contain the numerator matrix U(s); if ROWCOL = 'C', this
array is modified internally but restored on exit, and the
remainder of the leading MAX(M,P)-by-MAX(M,P)-by-kdcoef
part is used as internal workspace.
UCOEFF(I,J,K) is the coefficient in s**(INDEX(iorj)-K+1)
of polynomial (I,J) of U(s), where K = 1,2,...,kdcoef;
if ROWCOL = 'R' then iorj = I, otherwise iorj = J.
Thus for ROWCOL = 'R', U(s) =
diag(s**INDEX(I))*(UCOEFF(.,.,1)+UCOEFF(.,.,2)/s+...).
LDUCO1 INTEGER
The leading dimension of array UCOEFF.
LDUCO1 >= MAX(1,P) if ROWCOL = 'R';
LDUCO1 >= MAX(1,M,P) if ROWCOL = 'C'.
LDUCO2 INTEGER
The second dimension of array UCOEFF.
LDUCO2 >= MAX(1,M) if ROWCOL = 'R';
LDUCO2 >= MAX(1,M,P) if ROWCOL = 'C'.
NR (output) INTEGER
The order of the resulting minimal realization, i.e. the
order of the state dynamics matrix A.
A (output) DOUBLE PRECISION array, dimension (LDA,N),
porm
where N = SUM INDEX(I).
I=1
The leading NR-by-NR part of this array contains the upper
block Hessenberg state dynamics matrix A of a minimal
realization.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (output) DOUBLE PRECISION array, dimension (LDB,MAX(M,P))
The leading NR-by-M part of this array contains the
input/state matrix B of a minimal realization; the
remainder of the leading N-by-MAX(M,P) part is used as
internal workspace.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (output) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-NR part of this array contains the
state/output matrix C of a minimal realization; the
remainder of the leading MAX(M,P)-by-N part is used as
internal workspace.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,M,P).
D (output) DOUBLE PRECISION array, dimension (LDD,M),
if ROWCOL = 'R', and (LDD,MAX(M,P)) if ROWCOL = 'C'.
The leading P-by-M part of this array contains the direct
transmission matrix D; if ROWCOL = 'C', the remainder of
the leading MAX(M,P)-by-MAX(M,P) part is used as internal
workspace.
LDD INTEGER
The leading dimension of array D.
LDD >= MAX(1,P) if ROWCOL = 'R';
LDD >= MAX(1,M,P) if ROWCOL = 'C'.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in rank determination when
transforming (A, B, C). If the user sets TOL > 0, then
the given value of TOL is used as a lower bound for the
reciprocal condition number (see the description of the
argument RCOND in the SLICOT routine MB03OD); a
(sub)matrix whose estimated condition number is less than
1/TOL is considered to be of full rank. If the user sets
TOL <= 0, then an implicitly computed, default tolerance
(determined by the SLICOT routine TB01UD) is used instead.
Workspace
IWORK INTEGER array, dimension (N+MAX(M,P))
On exit, if INFO = 0, the first nonzero elements of
IWORK(1:N) return the orders of the diagonal blocks of A.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1, N + MAX(N, 3*M, 3*P)).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, then i is the first integer for which
ABS( DCOEFF(I,1) ) is so small that the calculations
would overflow (see SLICOT Library routine TD03AY);
that is, the leading coefficient of a polynomial is
nearly zero; no state-space representation is
calculated.
Method
The method for transfer matrices factorized by rows will be
described here: T(s) factorized by columns is dealt with by
operating on the dual T'(s). This description for T(s) is
actually the left polynomial matrix representation
T(s) = inv(D(s))*U(s),
where D(s) is diagonal with its (I,I)-th polynomial element of
degree INDEX(I). The first step is to check whether the leading
coefficient of any polynomial element of D(s) is approximately
zero; if so the routine returns with INFO > 0. Otherwise,
Wolovich's Observable Structure Theorem is used to construct a
state-space representation in observable companion form which
is equivalent to the above polynomial matrix representation.
The method is particularly easy here due to the diagonal form
of D(s). This state-space representation is not necessarily
controllable (as D(s) and U(s) are not necessarily relatively
left prime), but it is in theory completely observable; however,
its observability matrix may be poorly conditioned, so it is
treated as a general state-space representation and SLICOT
Library routine TB01PD is then called to separate out a minimal
realization from this general state-space representation by means
of orthogonal similarity transformations.
References
[1] Patel, R.V.
Computation of Minimal-Order State-Space Realizations and
Observability Indices using Orthogonal Transformations.
Int. J. Control, 33, pp. 227-246, 1981.
[2] Wolovich, W.A.
Linear Multivariable Systems, (Theorem 4.3.3).
Springer-Verlag, 1974.
Numerical Aspects
3 The algorithm requires 0(N ) operations.Further Comments
NoneExample
Program Text
* TD04AD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, PMAX, KDCMAX, NMAX
PARAMETER ( MMAX = 10, PMAX = 10, KDCMAX = 10, NMAX = 10 )
INTEGER MAXMP
PARAMETER ( MAXMP = MAX( MMAX, PMAX ) )
INTEGER LDDCOE, LDUCO1, LDUCO2, LDA, LDB, LDC, LDD
PARAMETER ( LDDCOE = MAXMP, LDUCO1 = MAXMP,
$ LDUCO2 = MAXMP, LDA = NMAX, LDB = NMAX,
$ LDC = MAXMP, LDD = MAXMP )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX + MAXMP )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX + MAX( NMAX, 3*MAXMP ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INDBLK, INFO, J, K, KDCOEF, M, N, NR, P, PORM
CHARACTER*1 ROWCOL
LOGICAL LROWCO
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MAXMP), C(LDC,NMAX),
$ D(LDD,MAXMP), DCOEFF(LDDCOE,KDCMAX),
$ DWORK(LDWORK), UCOEFF(LDUCO1,LDUCO2,KDCMAX)
INTEGER INDEX(MAXMP), IWORK(LIWORK)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL TD04AD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, P, TOL, ROWCOL
LROWCO = LSAME( ROWCOL, 'R' )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) M
ELSE IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) P
ELSE
PORM = P
IF ( .NOT.LROWCO ) PORM = M
READ ( NIN, FMT = * ) ( INDEX(I), I = 1,PORM )
*
N = 0
KDCOEF = 0
DO 20 I = 1, PORM
N = N + INDEX(I)
KDCOEF = MAX( KDCOEF, INDEX(I) )
20 CONTINUE
KDCOEF = KDCOEF + 1
*
IF ( KDCOEF.LE.0 .OR. KDCOEF.GT.KDCMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) KDCOEF
ELSE
READ ( NIN, FMT = * )
$ ( ( DCOEFF(I,J), J = 1,KDCOEF ), I = 1,PORM )
READ ( NIN, FMT = * )
$ ( ( ( UCOEFF(I,J,K), K = 1,KDCOEF ), J = 1,M ), I = 1,P )
* Find a minimal state-space representation (A,B,C,D).
CALL TD04AD( ROWCOL, M, P, INDEX, DCOEFF, LDDCOE, UCOEFF,
$ LDUCO1, LDUCO2, NR, A, LDA, B, LDB, C, LDC, D,
$ LDD, TOL, IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NR
DO 40 I = 1, NR
WRITE ( NOUT, FMT = 99996 ) ( A(I,J), J = 1,NR )
40 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 60 I = 1, NR
WRITE ( NOUT, FMT = 99996 ) ( B(I,J), J = 1,M )
60 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 80 I = 1, P
WRITE ( NOUT, FMT = 99996 ) ( C(I,J), J = 1,NR )
80 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 100 I = 1, P
WRITE ( NOUT, FMT = 99996 ) ( D(I,J), J = 1,M )
100 CONTINUE
INDBLK = 0
DO 120 I = 1, N
IF ( IWORK(I).NE.0 ) INDBLK = INDBLK + 1
120 CONTINUE
IF ( LROWCO ) THEN
WRITE ( NOUT, FMT = 99992 ) INDBLK,
$ ( IWORK(I), I = 1,INDBLK )
ELSE
WRITE ( NOUT, FMT = 99991 ) INDBLK,
$ ( IWORK(I), I = 1,INDBLK )
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TD04AD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TD04AD = ',I2)
99997 FORMAT (' The order of the minimal realization = ',I2,//' The st',
$ 'ate dynamics matrix A of a minimal realization is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The input/state matrix B of a minimal realization is ')
99994 FORMAT (/' The state/output matrix C of a minimal realization is '
$ )
99993 FORMAT (/' The direct transmission matrix D is ')
99992 FORMAT (/' The observability index of a minimal state-space repr',
$ 'esentation = ',I2,//' The dimensions of the diagonal blo',
$ 'cks of the state dynamics matrix are',/20(1X,I2))
99991 FORMAT (/' The controllability index of a minimal state-space re',
$ 'presentation = ',I2,//' The dimensions of the diagonal b',
$ 'locks of the state dynamics matrix are',/20(1X,I2))
99990 FORMAT (/' M is out of range.',/' M = ',I5)
99989 FORMAT (/' P is out of range.',/' P = ',I5)
99988 FORMAT (/' KDCOEF is out of range.',/' KDCOEF = ',I5)
END
Program Data
TD04AD EXAMPLE PROGRAM DATA 2 2 0.0 R 3 3 1.0 6.0 11.0 6.0 1.0 6.0 11.0 6.0 1.0 6.0 12.0 7.0 0.0 1.0 4.0 3.0 0.0 0.0 1.0 1.0 1.0 8.0 20.0 15.0Program Results
TD04AD EXAMPLE PROGRAM RESULTS The order of the minimal realization = 3 The state dynamics matrix A of a minimal realization is 0.5000 -0.8028 0.9387 4.4047 -2.3380 2.5076 -5.5541 1.6872 -4.1620 The input/state matrix B of a minimal realization is -0.2000 -1.2500 0.0000 -0.6097 0.0000 2.2217 The state/output matrix C of a minimal realization is 0.0000 -0.8679 0.2119 0.0000 0.0000 0.9002 The direct transmission matrix D is 1.0000 0.0000 0.0000 1.0000 The observability index of a minimal state-space representation = 2 The dimensions of the diagonal blocks of the state dynamics matrix are 2 1
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TD05AD.html 0000664 0000000 0000000 00000014541 14560147231 0020340 0 ustar 00root root 0000000 0000000
Purpose
Given a complex valued rational function of frequency (transfer function) G(jW) this routine will calculate its complex value or its magnitude and phase for a specified frequency value.Specification
SUBROUTINE TD05AD( UNITF, OUTPUT, NP1, MP1, W, A, B, VALR, VALI,
$ INFO )
C .. Scalar Arguments ..
CHARACTER OUTPUT, UNITF
INTEGER INFO, MP1, NP1
DOUBLE PRECISION VALI, VALR, W
C .. Array Arguments ..
DOUBLE PRECISION A(*), B(*)
Arguments
Mode Parameters
UNITF CHARACTER*1
Indicates the choice of frequency unit as follows:
= 'R': Input frequency W in radians/second;
= 'H': Input frequency W in hertz.
OUTPUT CHARACTER*1
Indicates the choice of co-ordinates for output as folows:
= 'C': Cartesian co-ordinates (output real and imaginary
parts of G(jW));
= 'P': Polar co-ordinates (output magnitude and phase
of G(jW)).
Input/Output Parameters
NP1 (input) INTEGER
The order of the denominator + 1, i.e. N + 1. NP1 >= 1.
MP1 (input) INTEGER
The order of the numerator + 1, i.e. M + 1. MP1 >= 1.
W (input) DOUBLE PRECISION
The frequency value W for which the transfer function is
to be evaluated.
A (input) DOUBLE PRECISION array, dimension (NP1)
This array must contain the vector of denominator
coefficients in ascending order of powers. That is, A(i)
must contain the coefficient of (jW)**(i-1) for i = 1,
2,...,NP1.
B (input) DOUBLE PRECISION array, dimension (MP1)
This array must contain the vector of numerator
coefficients in ascending order of powers. That is, B(i)
must contain the coefficient of (jW)**(i-1) for i = 1,
2,...,MP1.
VALR (output) DOUBLE PRECISION
If OUTPUT = 'C', VALR contains the real part of G(jW).
If OUTPUT = 'P', VALR contains the magnitude of G(jW)
in dBs.
VALI (output) DOUBLE PRECISION
If OUTPUT = 'C', VALI contains the imaginary part of
G(jW).
If OUTPUT = 'P', VALI contains the phase of G(jW) in
degrees.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the frequency value W is a pole of G(jW), or all
the coefficients of the A polynomial are zero.
Method
By substituting the values of A, B and W in the following
formula:
B(1)+B(2)*(jW)+B(3)*(jW)**2+...+B(MP1)*(jW)**(MP1-1)
G(jW) = ---------------------------------------------------.
A(1)+A(2)*(jW)+A(3)*(jW)**2+...+A(NP1)*(jW)**(NP1-1)
References
None.Numerical Aspects
The algorithm requires 0(N+M) operations.Further Comments
NoneExample
Program Text
* TD05AD EXAMPLE PROGRAM TEXT.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NP1MAX, MP1MAX
PARAMETER ( NP1MAX = 20, MP1MAX = 20 )
* .. Local Scalars ..
DOUBLE PRECISION VALI, VALR, W
INTEGER I, INFO, MP1, NP1
CHARACTER*1 UNITF, OUTPUT
* .. Local Arrays ..
DOUBLE PRECISION A(NP1MAX), B(MP1MAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL TD05AD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) NP1, MP1, W, UNITF, OUTPUT
IF ( NP1.LE.0 .OR. NP1.GT.NP1MAX ) THEN
WRITE ( NOUT, FMT = 99995 ) NP1
ELSE
READ ( NIN, FMT = * ) ( A(I), I = 1,NP1 )
IF ( MP1.LE.0 .OR. MP1.GT.MP1MAX ) THEN
WRITE ( NOUT, FMT = 99994 ) MP1
ELSE
READ ( NIN, FMT = * ) ( B(I), I = 1,MP1 )
* Find the real and imaginary parts of G(jW), where
* W = 1.0 radian.
CALL TD05AD( UNITF, OUTPUT, NP1, MP1, W, A, B, VALR, VALI,
$ INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( LSAME( OUTPUT, 'C' ) ) THEN
WRITE ( NOUT, FMT = 99997 ) VALR, VALI
ELSE
WRITE ( NOUT, FMT = 99996 ) VALR, VALI
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TD05AD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TD05AD = ',I2)
99997 FORMAT (' Complex value of G(jW) = ',F8.4,1X,F8.4,'*j')
99996 FORMAT (' Magnitude of G(jW) = ',F8.4,' dBs, Phase of G(jW) = ',
$ F8.4,' degrees ')
99995 FORMAT (/' NP1 is out of range.',/' NP1 = ',I5)
99994 FORMAT (/' MP1 is out of range.',/' MP1 = ',I5)
END
Program Data
TD05AD EXAMPLE PROGRAM DATA 6 4 1.0 R C 1.0 1.0 0.0 0.0 2.0 1.0 6.0 2.0 3.0 1.0Program Results
TD05AD EXAMPLE PROGRAM RESULTS Complex value of G(jW) = 0.8462 -0.2308*j
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TF01MD.html 0000664 0000000 0000000 00000021466 14560147231 0020356 0 ustar 00root root 0000000 0000000
Purpose
To compute the output sequence of a linear time-invariant open-loop system given by its discrete-time state-space model (A,B,C,D), where A is an N-by-N general matrix. The initial state vector x(1) must be supplied by the user.Specification
SUBROUTINE TF01MD( N, M, P, NY, A, LDA, B, LDB, C, LDC, D, LDD,
$ U, LDU, X, Y, LDY, DWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDD, LDU, LDY, M, N, NY, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), U(LDU,*), X(*), Y(LDY,*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NY (input) INTEGER
The number of output vectors y(k) to be computed.
NY >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
state matrix A of the system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
input matrix B of the system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array must contain the
output matrix C of the system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading P-by-M part of this array must contain the
direct link matrix D of the system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
U (input) DOUBLE PRECISION array, dimension (LDU,NY)
The leading M-by-NY part of this array must contain the
input vector sequence u(k), for k = 1,2,...,NY.
Specifically, the k-th column of U must contain u(k).
LDU INTEGER
The leading dimension of array U. LDU >= MAX(1,M).
X (input/output) DOUBLE PRECISION array, dimension (N)
On entry, this array must contain the initial state vector
x(1) which consists of the N initial states of the system.
On exit, this array contains the final state vector
x(NY+1) of the N states of the system at instant NY.
Y (output) DOUBLE PRECISION array, dimension (LDY,NY)
The leading P-by-NY part of this array contains the output
vector sequence y(1),y(2),...,y(NY) such that the k-th
column of Y contains y(k) (the outputs at instant k),
for k = 1,2,...,NY.
LDY INTEGER
The leading dimension of array Y. LDY >= MAX(1,P).
Workspace
DWORK DOUBLE PRECISION array, dimension (N)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Given an initial state vector x(1), the output vector sequence
y(1), y(2),..., y(NY) is obtained via the formulae
x(k+1) = A x(k) + B u(k)
y(k) = C x(k) + D u(k),
where each element y(k) is a vector of length P containing the
outputs at instant k and k = 1,2,...,NY.
References
[1] Luenberger, D.G.
Introduction to Dynamic Systems: Theory, Models and
Applications.
John Wiley & Sons, New York, 1979.
Numerical Aspects
The algorithm requires approximately (N + M) x (N + P) x NY multiplications and additions.Further Comments
NoneExample
Program Text
* TF01MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX, NYMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20, NYMAX = 20 )
INTEGER LDA, LDB, LDC, LDD, LDU, LDY
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX, LDD = PMAX,
$ LDU = MMAX, LDY = PMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, K, M, N, NY, P
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK), U(LDU,NYMAX),
$ X(NMAX), Y(LDY,NYMAX)
* .. External Subroutines ..
EXTERNAL TF01MD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, NY
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), I = 1,N ), J = 1,N )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,N ), J = 1,M )
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), I = 1,P ), J = 1,N )
READ ( NIN, FMT = * ) ( ( D(I,J), I = 1,P ), J = 1,M )
READ ( NIN, FMT = * ) ( X(I), I = 1,N )
IF ( NY.LE.0 .OR. NY.GT.NYMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) NY
ELSE
READ ( NIN, FMT = * )
$ ( ( U(I,J), I = 1,M ), J = 1,NY )
* Compute y(1),...,y(NY) of the given system.
CALL TF01MD( N, M, P, NY, A, LDA, B, LDB, C, LDC, D,
$ LDD, U, LDU, X, Y, LDY, DWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NY
DO 20 K = 1, NY
WRITE ( NOUT, FMT = 99996 ) K, Y(1,K)
WRITE ( NOUT, FMT = 99995 ) ( Y(J,K), J = 2,P )
20 CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TF01MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TF01MD = ',I2)
99997 FORMAT (' The output sequence Y(1),...,Y(',I2,') is',/)
99996 FORMAT (' Y(',I2,') : ',F8.4)
99995 FORMAT (9X,F8.4,/)
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' M is out of range.',/' M = ',I5)
99992 FORMAT (/' P is out of range.',/' P = ',I5)
99991 FORMAT (/' NY is out of range.',/' NY = ',I5)
END
Program Data
TF01MD EXAMPLE PROGRAM DATA 3 2 2 10 0.0000 -0.0700 0.0150 1.0000 0.8000 -0.1500 0.0000 0.0000 0.5000 0.0000 2.0000 1.0000 -1.0000 -0.1000 1.0000 0.0000 1.0000 0.0000 0.0000 1.0000 0.0000 1.0000 0.5000 0.0000 0.5000 1.0000 1.0000 1.0000 -0.6922 -1.4934 0.3081 -2.7726 2.0039 0.2614 -0.9160 -0.6030 1.2556 0.2951 -1.5734 1.5639 -0.9942 1.8957 0.8988 0.4118 -1.4893 -0.9344 1.2506 -0.0701Program Results
TF01MD EXAMPLE PROGRAM RESULTS
The output sequence Y(1),...,Y(10) is
Y( 1) : 0.3078
-0.0928
Y( 2) : -1.5125
1.2611
Y( 3) : -1.2577
3.4002
Y( 4) : -0.2947
-0.7060
Y( 5) : -0.5632
5.4532
Y( 6) : -1.0846
1.1846
Y( 7) : -1.2427
2.2286
Y( 8) : 1.8097
-1.9534
Y( 9) : 0.6685
-4.4965
Y(10) : -0.0896
1.1654
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TF01MX.html 0000664 0000000 0000000 00000011371 14560147231 0020374 0 ustar 00root root 0000000 0000000
Purpose
To compute the output sequence of a linear time-invariant
open-loop system given by its discrete-time state-space model
with an (N+P)-by-(N+M) general system matrix S,
( A B )
S = ( ) .
( C D )
The initial state vector x(1) must be supplied by the user.
The input and output trajectories are stored as in the SLICOT
Library routine TF01MY.
Specification
SUBROUTINE TF01MX( N, M, P, NY, S, LDS, U, LDU, X, Y, LDY,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDS, LDU, LDWORK, LDY, M, N, NY, P
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), S(LDS,*), U(LDU,*), X(*), Y(LDY,*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NY (input) INTEGER
The number of output vectors y(k) to be computed.
NY >= 0.
S (input) DOUBLE PRECISION array, dimension (LDS,N+M)
The leading (N+P)-by-(N+M) part of this array must contain
the system matrix S.
LDS INTEGER
The leading dimension of array S. LDS >= MAX(1,N+P).
U (input) DOUBLE PRECISION array, dimension (LDU,M)
The leading NY-by-M part of this array must contain the
input vector sequence u(k), for k = 1,2,...,NY.
Specifically, the k-th row of U must contain u(k)'.
LDU INTEGER
The leading dimension of array U. LDU >= MAX(1,NY).
X (input/output) DOUBLE PRECISION array, dimension (N)
On entry, this array must contain the initial state vector
x(1) which consists of the N initial states of the system.
On exit, this array contains the final state vector
x(NY+1) of the N states of the system at instant NY+1.
Y (output) DOUBLE PRECISION array, dimension (LDY,P)
The leading NY-by-P part of this array contains the output
vector sequence y(1),y(2),...,y(NY) such that the k-th
row of Y contains y(k)' (the outputs at instant k),
for k = 1,2,...,NY.
LDY INTEGER
The leading dimension of array Y. LDY >= MAX(1,NY).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 0, if MIN(N,P,NY) = 0; otherwise,
LDWORK >= N+P, if M = 0;
LDWORK >= 2*N+M+P, if M > 0.
For better performance, LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Given an initial state vector x(1), the output vector sequence
y(1), y(2),..., y(NY) is obtained via the formulae
( x(k+1) ) ( x(k) )
( ) = S ( ) ,
( y(k) ) ( u(k) )
where each element y(k) is a vector of length P containing the
outputs at instant k, and k = 1,2,...,NY.
References
[1] Luenberger, D.G.
Introduction to Dynamic Systems: Theory, Models and
Applications.
John Wiley & Sons, New York, 1979.
Numerical Aspects
The algorithm requires approximately (N + M) x (N + P) x NY multiplications and additions.Further Comments
The implementation exploits data locality as much as possible, given the workspace length.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the output sequence of a linear time-invariant open-loop system given by its discrete-time state-space model (A,B,C,D), where A is an N-by-N general matrix. The initial state vector x(1) must be supplied by the user. This routine differs from SLICOT Library routine TF01MD in the way the input and output trajectories are stored.Specification
SUBROUTINE TF01MY( N, M, P, NY, A, LDA, B, LDB, C, LDC, D, LDD,
$ U, LDU, X, Y, LDY, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDD, LDU, LDWORK, LDY, M,
$ N, NY, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), U(LDU,*), X(*), Y(LDY,*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NY (input) INTEGER
The number of output vectors y(k) to be computed.
NY >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
state matrix A of the system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
input matrix B of the system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array must contain the
output matrix C of the system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading P-by-M part of this array must contain the
direct link matrix D of the system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
U (input) DOUBLE PRECISION array, dimension (LDU,M)
The leading NY-by-M part of this array must contain the
input vector sequence u(k), for k = 1,2,...,NY.
Specifically, the k-th row of U must contain u(k)'.
LDU INTEGER
The leading dimension of array U. LDU >= MAX(1,NY).
X (input/output) DOUBLE PRECISION array, dimension (N)
On entry, this array must contain the initial state vector
x(1) which consists of the N initial states of the system.
On exit, this array contains the final state vector
x(NY+1) of the N states of the system at instant NY+1.
Y (output) DOUBLE PRECISION array, dimension (LDY,P)
The leading NY-by-P part of this array contains the output
vector sequence y(1),y(2),...,y(NY) such that the k-th
row of Y contains y(k)' (the outputs at instant k),
for k = 1,2,...,NY.
LDY INTEGER
The leading dimension of array Y. LDY >= MAX(1,NY).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,N).
For better performance, LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Given an initial state vector x(1), the output vector sequence
y(1), y(2),..., y(NY) is obtained via the formulae
x(k+1) = A x(k) + B u(k)
y(k) = C x(k) + D u(k),
where each element y(k) is a vector of length P containing the
outputs at instant k and k = 1,2,...,NY.
References
[1] Luenberger, D.G.
Introduction to Dynamic Systems: Theory, Models and
Applications.
John Wiley & Sons, New York, 1979.
Numerical Aspects
The algorithm requires approximately (N + M) x (N + P) x NY multiplications and additions.Further Comments
The implementation exploits data locality and uses BLAS 3 operations as much as possible, given the workspace length.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute the output sequence of a linear time-invariant open-loop system given by its discrete-time state-space model (A,B,C,D), where A is an N-by-N upper or lower Hessenberg matrix. The initial state vector x(1) must be supplied by the user.Specification
SUBROUTINE TF01ND( UPLO, N, M, P, NY, A, LDA, B, LDB, C, LDC, D,
$ LDD, U, LDU, X, Y, LDY, DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LDB, LDC, LDD, LDU, LDY, M, N, NY, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), U(LDU,*), X(*), Y(LDY,*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Indicates whether the user wishes to use an upper or lower
Hessenberg matrix as follows:
= 'U': Upper Hessenberg matrix;
= 'L': Lower Hessenberg matrix.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NY (input) INTEGER
The number of output vectors y(k) to be computed.
NY >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
If UPLO = 'U', the leading N-by-N upper Hessenberg part
of this array must contain the state matrix A of the
system.
If UPLO = 'L', the leading N-by-N lower Hessenberg part
of this array must contain the state matrix A of the
system.
The remainder of the leading N-by-N part is not
referenced.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
input matrix B of the system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array must contain the
output matrix C of the system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading P-by-M part of this array must contain the
direct link matrix D of the system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
U (input) DOUBLE PRECISION array, dimension (LDU,NY)
The leading M-by-NY part of this array must contain the
input vector sequence u(k), for k = 1,2,...,NY.
Specifically, the k-th column of U must contain u(k).
LDU INTEGER
The leading dimension of array U. LDU >= MAX(1,M).
X (input/output) DOUBLE PRECISION array, dimension (N)
On entry, this array must contain the initial state vector
x(1) which consists of the N initial states of the system.
On exit, this array contains the final state vector
x(NY+1) of the N states of the system at instant NY.
Y (output) DOUBLE PRECISION array, dimension (LDY,NY)
The leading P-by-NY part of this array contains the output
vector sequence y(1),y(2),...,y(NY) such that the k-th
column of Y contains y(k) (the outputs at instant k),
for k = 1,2,...,NY.
LDY INTEGER
The leading dimension of array Y. LDY >= MAX(1,P).
Workspace
DWORK DOUBLE PRECISION array, dimension (N)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Given an initial state vector x(1), the output vector sequence
y(1), y(2),..., y(NY) is obtained via the formulae
x(k+1) = A x(k) + B u(k)
y(k) = C x(k) + D u(k),
where each element y(k) is a vector of length P containing the
outputs at instant k and k = 1,2,...,NY.
References
[1] Luenberger, D.G.
Introduction to Dynamic Systems: Theory, Models and
Applications.
John Wiley & Sons, New York, 1979.
Numerical Aspects
The algorithm requires approximately ((N+M)xP + (N/2+M)xN) x NY multiplications and additions.Further Comments
The processing time required by this routine will be approximately half that required by the SLICOT Library routine TF01MD, which treats A as a general matrix.Example
Program Text
* TF01ND EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX, NYMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20, NYMAX = 20 )
INTEGER LDA, LDB, LDC, LDD, LDU, LDY
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX, LDU = MMAX, LDY = PMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX )
* .. Local Scalars ..
CHARACTER*1 UPLO
INTEGER I, INFO, J, K, M, N, NY, P
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK), U(LDU,NYMAX),
$ X(NMAX), Y(LDY,NYMAX)
* .. External Subroutines ..
EXTERNAL TF01ND
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, NY, UPLO
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), I = 1,N ), J = 1,N )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,N ), J = 1,M )
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), I = 1,P ), J = 1,N )
READ ( NIN, FMT = * ) ( ( D(I,J), I = 1,P ), J = 1,M )
READ ( NIN, FMT = * ) ( X(I), I = 1,N )
IF ( NY.LE.0 .OR. NY.GT.NYMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) NY
ELSE
READ ( NIN, FMT = * )
$ ( ( U(I,J), I = 1,M ), J = 1,NY )
* Compute y(1),...,y(NY) of the given system.
CALL TF01ND( UPLO, N, M, P, NY, A, LDA, B, LDB, C,
$ LDC, D, LDD, U, LDU, X, Y, LDY, DWORK,
$ INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NY
DO 20 K = 1, NY
WRITE ( NOUT, FMT = 99996 ) K, Y(1,K)
WRITE ( NOUT, FMT = 99995 ) ( Y(J,K), J = 2,P )
20 CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TF01ND EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TF01ND = ',I2)
99997 FORMAT (' The output sequence Y(1),...,Y(',I2,') is',/)
99996 FORMAT (' Y(',I2,') : ',F8.4)
99995 FORMAT (9X,F8.4,/)
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' M is out of range.',/' M = ',I5)
99992 FORMAT (/' P is out of range.',/' P = ',I5)
99991 FORMAT (/' NY is out of range.',/' NY = ',I5)
END
Program Data
TF01ND EXAMPLE PROGRAM DATA 3 2 2 10 U 0.0000 -0.0700 0.0000 1.0000 0.8000 -0.1500 0.0000 0.0000 0.5000 0.0000 2.0000 1.0000 -1.0000 -0.1000 1.0000 0.0000 1.0000 0.0000 0.0000 1.0000 0.0000 1.0000 0.5000 0.0000 0.5000 1.0000 1.0000 1.0000 -0.6922 -1.4934 0.3081 -2.7726 2.0039 0.2614 -0.9160 -0.6030 1.2556 0.2951 -1.5734 1.5639 -0.9942 1.8957 0.8988 0.4118 -1.4893 -0.9344 1.2506 -0.0701Program Results
TF01ND EXAMPLE PROGRAM RESULTS
The output sequence Y(1),...,Y(10) is
Y( 1) : 0.3078
-0.0928
Y( 2) : -1.5275
1.2611
Y( 3) : -1.3026
3.4002
Y( 4) : -0.3512
-0.7060
Y( 5) : -0.5922
5.4532
Y( 6) : -1.1693
1.1846
Y( 7) : -1.3029
2.2286
Y( 8) : 1.7529
-1.9534
Y( 9) : 0.6793
-4.4965
Y(10) : -0.0349
1.1654
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TF01OD.html 0000664 0000000 0000000 00000015250 14560147231 0020352 0 ustar 00root root 0000000 0000000
Purpose
To construct the block Hankel expansion T of a multivariable parameter sequence M(1),...,M(NR+NC-1), where each parameter M(k) is an NH1-by-NH2 block matrix and k = 1,2,...,(NR+NC-1).Specification
SUBROUTINE TF01OD( NH1, NH2, NR, NC, H, LDH, T, LDT, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDH, LDT, NC, NH1, NH2, NR
C .. Array Arguments ..
DOUBLE PRECISION H(LDH,*), T(LDT,*)
Arguments
Input/Output Parameters
NH1 (input) INTEGER
The number of rows in each parameter M(k). NH1 >= 0.
NH2 (input) INTEGER
The number of columns in each parameter M(k). NH2 >= 0.
NR (input) INTEGER
The number of parameters required in each column of the
block Hankel expansion matrix T. NR >= 0.
NC (input) INTEGER
The number of parameters required in each row of the
block Hankel expansion matrix T. NC >= 0.
H (input) DOUBLE PRECISION array, dimension
(LDH,(NR+NC-1)*NH2)
The leading NH1-by-(NR+NC-1)*NH2 part of this array must
contain the multivariable sequence M(k), where k = 1,2,
...,(NR+NC-1). Specifically, each parameter M(k) is an
NH1-by-NH2 matrix whose (i,j)-th element must be stored in
H(i,(k-1)*NH2+j) for i = 1,2,...,NH1 and j = 1,2,...,NH2.
LDH INTEGER
The leading dimension of array H. LDH >= MAX(1,NH1).
T (output) DOUBLE PRECISION array, dimension (LDT,NH2*NC)
The leading NH1*NR-by-NH2*NC part of this array contains
the block Hankel expansion of the multivariable sequence
M(k).
LDT INTEGER
The leading dimension of array T. LDT >= MAX(1,NH1*NR).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The NH1-by-NH2 dimensional parameters M(k) of a multivariable
sequence are arranged into a matrix T in Hankel form such that
| M(1) M(2) M(3) . . . M(NC) |
| |
| M(2) M(3) M(4) . . . M(NC+1) |
T = | . . . . |.
| . . . . |
| . . . . |
| |
| M(NR) M(NR+1) M(NR+2) . . . M(NR+NC-1)|
References
[1] Johvidov, J.S.
Hankel and Toeplitz Matrices and Forms: Algebraic Theory,
(translated by G.P.A. Thijsse, I. Gohberg, ed.).
Birkhaeuser, Boston, 1982.
Numerical Aspects
The time taken is approximately proportional to NH1 x NH2 x NR x NC.Further Comments
NoneExample
Program Text
* TF01OD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NH1MAX, NH2MAX, NRMAX, NCMAX
PARAMETER ( NH1MAX = 20, NH2MAX = 20, NRMAX = 20,
$ NCMAX = 20 )
INTEGER LDH, LDT
PARAMETER ( LDH = NH1MAX, LDT = NH1MAX*NRMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, NC, NCT, NH1, NH2, NR, NRT
* .. Local Arrays ..
DOUBLE PRECISION H(LDH,(NRMAX+NCMAX-1)*NH2MAX),
$ T(LDT,NH2MAX*NCMAX)
* .. External Subroutines ..
EXTERNAL TF01OD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) NH1, NH2, NR, NC
IF ( NH1.LE.0 .OR. NH1.GE.NH1MAX ) THEN
WRITE ( NOUT, FMT = 99995 ) NH1
ELSE IF ( NH2.LE.0 .OR. NH2.GT.NH2MAX ) THEN
WRITE ( NOUT, FMT = 99994 ) NH2
ELSE IF ( NR.LE.0 .OR. NR.GT.NRMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) NR
ELSE IF ( NC.LE.0 .OR. NC.GT.NCMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) NC
ELSE
READ ( NIN, FMT = * )
$ ( ( H(I,J), I = 1,NH1 ), J = 1,( NR+NC-1 )*NH2 )
* Construct the NRT by NCT block Hankel expansion of M(k).
CALL TF01OD( NH1, NH2, NR, NC, H, LDH, T, LDT, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
NRT = NH1*NR
NCT = NH2*NC
WRITE ( NOUT, FMT = 99997 ) NRT, NCT
DO 20 I = 1, NRT
WRITE ( NOUT, FMT = 99996 ) ( T(I,J), J = 1,NCT )
20 CONTINUE
END IF
END IF
STOP
*
99999 FORMAT (' TF01OD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TF01OD = ',I2)
99997 FORMAT (' The ',I2,' by ',I2,' matrix T is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' NH1 is out of range.',/' NH1 = ',I5)
99994 FORMAT (/' NH2 is out of range.',/' NH2 = ',I5)
99993 FORMAT (/' NR is out of range.',/' NR = ',I5)
99992 FORMAT (/' NC is out of range.',/' NC = ',I5)
END
Program Data
TF01OD EXAMPLE PROGRAM DATA 2 2 3 3 1.0647 -0.4282 -0.4922 -1.2072 -0.3043 0.6883 -0.0926 0.7167 -0.1844 -0.8507 0.4441 -0.0478 0.7195 0.0500 -0.3955 0.5674 1.3387 -0.2801 0.1073 -0.5315Program Results
TF01OD EXAMPLE PROGRAM RESULTS The 6 by 6 matrix T is 1.0647 -0.4922 -0.3043 -0.0926 -0.1844 0.4441 -0.4282 -1.2072 0.6883 0.7167 -0.8507 -0.0478 -0.3043 -0.0926 -0.1844 0.4441 0.7195 -0.3955 0.6883 0.7167 -0.8507 -0.0478 0.0500 0.5674 -0.1844 0.4441 0.7195 -0.3955 1.3387 0.1073 -0.8507 -0.0478 0.0500 0.5674 -0.2801 -0.5315
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TF01PD.html 0000664 0000000 0000000 00000015406 14560147231 0020356 0 ustar 00root root 0000000 0000000
Purpose
To construct the block Toeplitz expansion T of a multivariable parameter sequence M(1),...,M(NR+NC-1), where each parameter M(k) is an NH1-by-NH2 block matrix and k = 1,2,...,(NR+NC-1).Specification
SUBROUTINE TF01PD( NH1, NH2, NR, NC, H, LDH, T, LDT, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDH, LDT, NC, NH1, NH2, NR
C .. Array Arguments ..
DOUBLE PRECISION H(LDH,*), T(LDT,*)
Arguments
Input/Output Parameters
NH1 (input) INTEGER
The number of rows in each parameter M(k). NH1 >= 0.
NH2 (input) INTEGER
The number of columns in each parameter M(k). NH2 >= 0.
NR (input) INTEGER
The number of parameters required in each column of the
block Toeplitz expansion matrix T. NR >= 0.
NC (input) INTEGER
The number of parameters required in each row of the
block Toeplitz expansion matrix T. NC >= 0.
H (input) DOUBLE PRECISION array, dimension
(LDH,(NR+NC-1)*NH2)
The leading NH1-by-(NR+NC-1)*NH2 part of this array must
contain the multivariable sequence M(k), where k = 1,2,
...,(NR+NC-1). Specifically, each parameter M(k) is an
NH1-by-NH2 matrix whose (i,j)-th element must be stored in
H(i,(k-1)*NH2+j) for i = 1,2,...,NH1 and j = 1,2,...,NH2.
LDH INTEGER
The leading dimension of array H. LDH >= MAX(1,NH1).
T (output) DOUBLE PRECISION array, dimension (LDT,NH2*NC)
The leading NH1*NR-by-NH2*NC part of this array contains
the block Toeplitz expansion of the multivariable sequence
M(k).
LDT INTEGER
The leading dimension of array T. LDT >= MAX(1,NH1*NR).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The NH1-by-NH2 dimensional parameters M(k) of a multivariable
sequence are arranged into a matrix T in Toeplitz form such that
| M(NC) M(NC-1) M(NC-2) . . . M(1) |
| |
| M(NC+1) M(NC) M(NC-1) . . . M(2) |
T = | . . . . |.
| . . . . |
| . . . . |
| |
| M(NR+NC-1) M(NR+NC-2) M(NR+NC-3) . . . M(NR) |
References
[1] Johvidov, J.S.
Hankel and Toeplitz Matrices and Forms: Algebraic Theory,
(translated by G.P.A. Thijsse, I. Gohberg, ed.).
Birkhaeuser, Boston, 1982.
Numerical Aspects
The time taken is approximately proportional to NH1 x NH2 x NR x NC.Further Comments
NoneExample
Program Text
* TF01PD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NH1MAX, NH2MAX, NRMAX, NCMAX
PARAMETER ( NH1MAX = 20, NH2MAX = 20, NRMAX = 20,
$ NCMAX = 20 )
INTEGER LDH, LDT
PARAMETER ( LDH = NH1MAX, LDT = NH1MAX*NRMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, NC, NCT, NH1, NH2, NR, NRT
* .. Local Arrays ..
DOUBLE PRECISION H(LDH,(NRMAX+NCMAX-1)*NH2MAX),
$ T(LDT,NH2MAX*NCMAX)
* .. External Subroutines ..
EXTERNAL TF01PD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) NH1, NH2, NR, NC
IF ( NH1.LE.0 .OR. NH1.GE.NH1MAX ) THEN
WRITE ( NOUT, FMT = 99995 ) NH1
ELSE IF ( NH2.LE.0 .OR. NH2.GT.NH2MAX ) THEN
WRITE ( NOUT, FMT = 99994 ) NH2
ELSE IF ( NR.LE.0 .OR. NR.GT.NRMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) NR
ELSE IF ( NC.LE.0 .OR. NC.GT.NCMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) NC
ELSE
READ ( NIN, FMT = * )
$ ( ( H(I,J), I = 1,NH1 ), J = 1,( NR+NC-1 )*NH2 )
* Construct the NRT by NCT block Toeplitz expansion of M(k).
CALL TF01PD( NH1, NH2, NR, NC, H, LDH, T, LDT, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
NRT = NH1*NR
NCT = NH2*NC
WRITE ( NOUT, FMT = 99997 ) NRT, NCT
DO 20 I = 1, NRT
WRITE ( NOUT, FMT = 99996 ) ( T(I,J), J = 1,NCT )
20 CONTINUE
END IF
END IF
STOP
*
99999 FORMAT (' TF01PD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TF01PD = ',I2)
99997 FORMAT (' The ',I2,' by ',I2,' matrix T is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' NH1 is out of range.',/' NH1 = ',I5)
99994 FORMAT (/' NH2 is out of range.',/' NH2 = ',I5)
99993 FORMAT (/' NR is out of range.',/' NR = ',I5)
99992 FORMAT (/' NC is out of range.',/' NC = ',I5)
END
Program Data
TF01PD EXAMPLE PROGRAM DATA 2 2 3 3 1.0647 -0.4282 -0.4922 -1.2072 -0.3043 0.6883 -0.0926 0.7167 -0.1844 -0.8507 0.4441 -0.0478 0.7195 0.0500 -0.3955 0.5674 1.3387 -0.2801 0.1073 -0.5315Program Results
TF01PD EXAMPLE PROGRAM RESULTS The 6 by 6 matrix T is -0.1844 0.4441 -0.3043 -0.0926 1.0647 -0.4922 -0.8507 -0.0478 0.6883 0.7167 -0.4282 -1.2072 0.7195 -0.3955 -0.1844 0.4441 -0.3043 -0.0926 0.0500 0.5674 -0.8507 -0.0478 0.6883 0.7167 1.3387 0.1073 0.7195 -0.3955 -0.1844 0.4441 -0.2801 -0.5315 0.0500 0.5674 -0.8507 -0.0478
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TF01QD.html 0000664 0000000 0000000 00000023322 14560147231 0020353 0 ustar 00root root 0000000 0000000
Purpose
To compute N Markov parameters M(1), M(2),..., M(N) from a multivariable system whose transfer function matrix G(z) is given.Specification
SUBROUTINE TF01QD( NC, NB, N, IORD, AR, MA, H, LDH, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDH, N, NB, NC
C .. Array Arguments ..
INTEGER IORD(*)
DOUBLE PRECISION AR(*), H(LDH,*), MA(*)
Arguments
Input/Output Parameters
NC (input) INTEGER
The number of system outputs, i.e. the number of rows in
the transfer function matrix G(z). NC >= 0.
NB (input) INTEGER
The number of system inputs, i.e. the number of columns in
the transfer function matrix G(z). NB >= 0.
N (input) INTEGER
The number of Markov parameters M(k) to be computed.
N >= 0.
IORD (input) INTEGER array, dimension (NC*NB)
This array must contain the order r of the elements of the
transfer function matrix G(z), stored row by row.
For example, the order of the (i,j)-th element of G(z) is
given by IORD((i-1)xNB+j).
AR (input) DOUBLE PRECISION array, dimension (NA), where
NA = IORD(1) + IORD(2) + ... + IORD(NC*NB).
The leading NA elements of this array must contain the
denominator coefficients AR(1),...,AR(r) in equation (1)
of the (i,j)-th element of the transfer function matrix
G(z), stored row by row, i.e. in the order
(1,1),(1,2),...,(1,NB), (2,1),(2,2),...,(2,NB), ...,
(NC,1),(NC,2),...,(NC,NB). The coefficients must be given
in decreasing order of powers of z; the coefficient of the
highest order term is assumed to be equal to 1.
MA (input) DOUBLE PRECISION array, dimension (NA)
The leading NA elements of this array must contain the
numerator coefficients MA(1),...,MA(r) in equation (1)
of the (i,j)-th element of the transfer function matrix
G(z), stored row by row, i.e. in the order
(1,1),(1,2),...,(1,NB), (2,1),(2,2),...,(2,NB), ...,
(NC,1),(NC,2),...,(NC,NB). The coefficients must be given
in decreasing order of powers of z.
H (output) DOUBLE PRECISION array, dimension (LDH,N*NB)
The leading NC-by-N*NB part of this array contains the
multivariable Markov parameter sequence M(k), where each
parameter M(k) is an NC-by-NB matrix and k = 1,2,...,N.
The Markov parameters are stored such that H(i,(k-1)xNB+j)
contains the (i,j)-th element of M(k) for i = 1,2,...,NC
and j = 1,2,...,NB.
LDH INTEGER
The leading dimension of array H. LDH >= MAX(1,NC).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The (i,j)-th element of G(z), defining the particular I/O transfer
between output i and input j, has the following form:
-1 -2 -r
MA(1)z + MA(2)z + ... + MA(r)z
G (z) = ----------------------------------------. (1)
ij -1 -2 -r
1 + AR(1)z + AR(2)z + ... + AR(r)z
The (i,j)-th element of G(z) is defined by its order r, its r
moving average coefficients (= numerator) MA(1),...,MA(r) and its
r autoregressive coefficients (= denominator) AR(1),...,AR(r). The
coefficient of the constant term in the denominator is assumed to
be equal to 1.
The relationship between the (i,j)-th element of the Markov
parameters M(1),M(2),...,M(N) and the corresponding element of the
transfer function matrix G(z) is given by:
-1 -2 -k
G (z) = M (0) + M (1)z + M (2)z + ... + M (k)z + ...(2)
ij ij ij ij ij
Equating (1) and (2), we find that the relationship between the
(i,j)-th element of the Markov parameters M(k) and the ARMA
parameters AR(1),...,AR(r) and MA(1),...,MA(r) of the (i,j)-th
element of the transfer function matrix G(z) is as follows:
M (1) = MA(1),
ij
k-1
M (k) = MA(k) - SUM AR(p) x M (k-p) for 1 < k <= r and
ij p=1 ij
r
M (k+r) = - SUM AR(p) x M (k+r-p) for k > 0.
ij p=1 ij
From these expressions the Markov parameters M(k) are computed
element by element.
References
[1] Luenberger, D.G.
Introduction to Dynamic Systems: Theory, Models and
Applications.
John Wiley & Sons, New York, 1979.
Numerical Aspects
The computation of the (i,j)-th element of M(k) requires:
(k-1) multiplications and k additions if k <= r;
r multiplications and r additions if k > r.
Further Comments
NoneExample
Program Text
* TF01QD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, NAMAX, NBMAX, NCMAX
PARAMETER ( NMAX = 20, NAMAX = 20, NBMAX = 20, NCMAX = 20 )
INTEGER LDH
PARAMETER ( LDH = NCMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, K, L, N, NA, NASUM, NB, NC, NL, NORD
LOGICAL ERROR
* .. Local Arrays ..
DOUBLE PRECISION AR(NAMAX), H(LDH,NMAX*NBMAX), MA(NAMAX)
INTEGER IORD(NCMAX*NBMAX)
* .. External Subroutines ..
EXTERNAL TF01QD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, NA, NB, NC
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE IF ( NA.LE.0 .OR. NA.GT.NAMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) NA
ELSE IF ( NB.LE.0 .OR. NB.GT.NBMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) NB
ELSE IF ( NC.LE.0 .OR. NC.GT.NCMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) NC
ELSE
ERROR = .FALSE.
NL = 0
K = 1
NASUM = 0
DO 40 I = 1, NC
DO 20 J = 1, NB
READ ( NIN, FMT = * ) NORD
NASUM = NASUM + NORD
IF ( NA.GE.NASUM ) THEN
READ ( NIN, FMT = * ) ( MA(NL+L), L = 1,NORD )
READ ( NIN, FMT = * ) ( AR(NL+L), L = 1,NORD )
IORD(K) = NORD
K = K + 1
NL = NL + NORD
ELSE
WRITE ( NOUT, FMT = 99993 ) NA
ERROR = .TRUE.
END IF
20 CONTINUE
40 CONTINUE
IF ( .NOT. ERROR ) THEN
* Compute M(1),...,M(N) from the given transfer function
* matrix G(z).
CALL TF01QD( NC, NB, N, IORD, AR, MA, H, LDH, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) N
DO 80 K = 1, N
WRITE ( NOUT, FMT = 99996 ) K,
$ ( H(1,(K-1)*NB+J), J = 1,NB )
DO 60 I = 2, NC
WRITE ( NOUT, FMT = 99995 )
$ ( H(I,(K-1)*NB+J), J = 1,NB )
60 CONTINUE
80 CONTINUE
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TF01QD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TF01QD = ',I2)
99997 FORMAT (' The Markov Parameters M(1),...,M(',I1,') are ')
99996 FORMAT (/' M(',I1,') : ',20(1X,F8.4))
99995 FORMAT (8X,20(1X,F8.4))
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' NA is out of range.',/' NA = ',I5)
99992 FORMAT (/' NB is out of range.',/' NB = ',I5)
99991 FORMAT (/' NC is out of range.',/' NC = ',I5)
END
Program Data
TF01QD EXAMPLE PROGRAM DATA 8 10 2 2 2 1.0 -0.5 0.6 -0.2 1 1.0 -0.8 3 0.5 -0.4 0.3 0.8 0.4 0.1 4 1.0 0.5 -0.5 0.0 -0.8 0.6 0.0 -0.2Program Results
TF01QD EXAMPLE PROGRAM RESULTS
The Markov Parameters M(1),...,M(8) are
M(1) : 1.0000 1.0000
0.5000 1.0000
M(2) : -1.1000 0.8000
-0.8000 1.3000
M(3) : 0.8600 0.6400
0.7400 -0.0600
M(4) : -0.7360 0.5120
-0.3220 -0.8280
M(5) : 0.6136 0.4096
0.0416 -0.4264
M(6) : -0.5154 0.3277
0.0215 0.4157
M(7) : 0.4319 0.2621
-0.0017 0.5764
M(8) : -0.3622 0.2097
-0.0114 0.0461
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TF01RD.html 0000664 0000000 0000000 00000020145 14560147231 0020354 0 ustar 00root root 0000000 0000000
Purpose
To compute N Markov parameters M(1), M(2),..., M(N) from the parameters (A,B,C) of a linear time-invariant system, where each M(k) is an NC-by-NB matrix and k = 1,2,...,N. All matrices are treated as dense, and hence TF01RD is not intended for large sparse problems.Specification
SUBROUTINE TF01RD( NA, NB, NC, N, A, LDA, B, LDB, C, LDC, H, LDH,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDH, LDWORK, N, NA, NB, NC
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), H(LDH,*)
Arguments
Input/Output Parameters
NA (input) INTEGER
The order of the matrix A. NA >= 0.
NB (input) INTEGER
The number of system inputs. NB >= 0.
NC (input) INTEGER
The number of system outputs. NC >= 0.
N (input) INTEGER
The number of Markov parameters M(k) to be computed.
N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,NA)
The leading NA-by-NA part of this array must contain the
state matrix A of the system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,NA).
B (input) DOUBLE PRECISION array, dimension (LDB,NB)
The leading NA-by-NB part of this array must contain the
input matrix B of the system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,NA).
C (input) DOUBLE PRECISION array, dimension (LDC,NA)
The leading NC-by-NA part of this array must contain the
output matrix C of the system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,NC).
H (output) DOUBLE PRECISION array, dimension (LDH,N*NB)
The leading NC-by-N*NB part of this array contains the
multivariable parameters M(k), where each parameter M(k)
is an NC-by-NB matrix and k = 1,2,...,N. The Markov
parameters are stored such that H(i,(k-1)xNB+j) contains
the (i,j)-th element of M(k) for i = 1,2,...,NC and
j = 1,2,...,NB.
LDH INTEGER
The leading dimension of array H. LDH >= MAX(1,NC).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1, 2*NA*NC).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
For the linear time-invariant discrete-time system
x(k+1) = A x(k) + B u(k)
y(k) = C x(k) + D u(k),
the transfer function matrix G(z) is given by
-1
G(z) = C(zI-A) B + D
-1 -2 2 -3
= D + CB z + CAB z + CA B z + ... (1)
Using Markov parameters, G(z) can also be written as
-1 -2 -3
G(z) = M(0) + M(1)z + M(2)z + M(3)z + ... (2)
k-1
Equating (1) and (2), we find that M(0) = D and M(k) = C A B
for k > 0, from which the Markov parameters M(1),M(2)...,M(N) are
computed.
References
[1] Chen, C.T.
Introduction to Linear System Theory.
H.R.W. Series in Electrical Engineering, Electronics and
Systems, Holt, Rinehart and Winston Inc., London, 1970.
Numerical Aspects
The algorithm requires approximately (NA + NB) x NA x NC x N multiplications and additions.Further Comments
NoneExample
Program Text
* TF01RD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, NAMAX, NBMAX, NCMAX
PARAMETER ( NMAX = 20, NAMAX = 20, NBMAX = 20, NCMAX = 20 )
INTEGER LDA, LDB, LDC, LDH
PARAMETER ( LDA = NAMAX, LDB = NAMAX, LDC = NCMAX,
$ LDH = NCMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 2*NAMAX*NCMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, K, N, NA, NB, NC
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NAMAX), B(LDB,NBMAX), C(LDC,NAMAX),
$ H(LDH,NMAX*NBMAX), DWORK(LDWORK)
* .. External Subroutines ..
EXTERNAL TF01RD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, NA, NB, NC
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE IF ( NA.LE.0 .OR. NA.GT.NAMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) NA
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), I = 1,NA ), J = 1,NA )
IF ( NB.LE.0 .OR. NB.GT.NBMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) NB
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,NA ), J = 1,NB )
IF ( NC.LE.0 .OR. NC.GT.NCMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) NC
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), I = 1,NC ), J = 1,NA )
* Compute M(1),...,M(N) from the system (A,B,C).
CALL TF01RD( NA, NB, NC, N, A, LDA, B, LDB, C, LDC, H,
$ LDH, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) N
DO 40 K = 1, N
WRITE ( NOUT, FMT = 99996 ) K,
$ ( H(1,(K-1)*NB+J), J = 1,NB )
DO 20 I = 2, NC
WRITE ( NOUT, FMT = 99995 )
$ ( H(I,(K-1)*NB+J), J = 1,NB )
20 CONTINUE
40 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TF01RD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TF01RD = ',I2)
99997 FORMAT (' The Markov Parameters M(1),...,M(',I1,') are ')
99996 FORMAT (/' M(',I1,') : ',20(1X,F8.4))
99995 FORMAT (8X,20(1X,F8.4))
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' NA is out of range.',/' NA = ',I5)
99992 FORMAT (/' NB is out of range.',/' NB = ',I5)
99991 FORMAT (/' NC is out of range.',/' NC = ',I5)
END
Program Data
TF01RD EXAMPLE PROGRAM DATA 5 3 2 2 0.000 -0.070 0.015 1.000 0.800 -0.150 0.000 0.000 0.500 0.000 2.000 1.000 -1.000 -0.100 1.000 0.000 1.000 0.000 0.000 1.000 0.000Program Results
TF01RD EXAMPLE PROGRAM RESULTS
The Markov Parameters M(1),...,M(5) are
M(1) : 1.0000 1.0000
0.0000 -1.0000
M(2) : 0.2000 0.5000
2.0000 -0.1000
M(3) : -0.1100 0.2500
1.6000 -0.0100
M(4) : -0.2020 0.1250
1.1400 -0.0010
M(5) : -0.2039 0.0625
0.8000 -0.0001
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TG01AD.html 0000664 0000000 0000000 00000032026 14560147231 0020335 0 ustar 00root root 0000000 0000000
Purpose
To balance the matrices of the system pencil
S = ( A B ) - lambda ( E 0 ) := Q - lambda Z,
( C 0 ) ( 0 0 )
corresponding to the descriptor triple (A-lambda E,B,C),
by balancing. This involves diagonal similarity transformations
(Dl*A*Dr - lambda Dl*E*Dr, Dl*B, C*Dr) applied to the system
(A-lambda E,B,C) to make the rows and columns of system pencil
matrices
diag(Dl,I) * S * diag(Dr,I)
as close in norm as possible. Balancing may reduce the 1-norms
of the matrices of the system pencil S.
The balancing can be performed optionally on the following
particular system pencils
S = A-lambda E,
S = ( A-lambda E B ), or
S = ( A-lambda E ).
( C )
Specification
SUBROUTINE TG01AD( JOB, L, N, M, P, THRESH, A, LDA, E, LDE,
$ B, LDB, C, LDC, LSCALE, RSCALE, DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOB
INTEGER INFO, L, LDA, LDB, LDC, LDE, M, N, P
DOUBLE PRECISION THRESH
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ DWORK( * ), E( LDE, * ), LSCALE( * ),
$ RSCALE( * )
Arguments
Mode Parameters
JOB CHARACTER*1
Indicates which matrices are involved in balancing, as
follows:
= 'A': All matrices are involved in balancing;
= 'B': B, A and E matrices are involved in balancing;
= 'C': C, A and E matrices are involved in balancing;
= 'N': B and C matrices are not involved in balancing.
Input/Output Parameters
L (input) INTEGER
The number of rows of matrices A, B, and E. L >= 0.
N (input) INTEGER
The number of columns of matrices A, E, and C. N >= 0.
M (input) INTEGER
The number of columns of matrix B. M >= 0.
P (input) INTEGER
The number of rows of matrix C. P >= 0.
THRESH (input) DOUBLE PRECISION
Threshold value for magnitude of elements:
elements with magnitude less than or equal to
THRESH are ignored for balancing. THRESH >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading L-by-N part of this array must
contain the state dynamics matrix A.
On exit, the leading L-by-N part of this array contains
the balanced matrix Dl*A*Dr.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,L).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading L-by-N part of this array must
contain the descriptor matrix E.
On exit, the leading L-by-N part of this array contains
the balanced matrix Dl*E*Dr.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,L).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading L-by-M part of this array must
contain the input/state matrix B.
On exit, if M > 0, the leading L-by-M part of this array
contains the balanced matrix Dl*B.
The array B is not referenced if M = 0.
LDB INTEGER
The leading dimension of array B.
LDB >= MAX(1,L) if M > 0 or LDB >= 1 if M = 0.
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, if P > 0, the leading P-by-N part of this array
contains the balanced matrix C*Dr.
The array C is not referenced if P = 0.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
LSCALE (output) DOUBLE PRECISION array, dimension (L)
The scaling factors applied to S from left. If Dl(j) is
the scaling factor applied to row j, then
SCALE(j) = Dl(j), for j = 1,...,L.
RSCALE (output) DOUBLE PRECISION array, dimension (N)
The scaling factors applied to S from right. If Dr(j) is
the scaling factor applied to column j, then
SCALE(j) = Dr(j), for j = 1,...,N.
Workspace
DWORK DOUBLE PRECISION array, dimension (3*(L+N))Error Indicator
INFO INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Balancing consists of applying a diagonal similarity
transformation
-1
diag(Dl,I) * S * diag(Dr,I)
to make the 1-norms of each row of the first L rows of S and its
corresponding N columns nearly equal.
Information about the diagonal matrices Dl and Dr are returned in
the vectors LSCALE and RSCALE, respectively.
References
[1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
[2] R.C. Ward, R. C.
Balancing the generalized eigenvalue problem.
SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
* TG01AD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER LMAX, NMAX, MMAX, PMAX
PARAMETER ( LMAX = 20, NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDE
PARAMETER ( LDA = LMAX, LDB = LMAX, LDC = PMAX,
$ LDE = LMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( 1, 3*(LMAX+NMAX ) ) )
* .. Local Scalars ..
CHARACTER*1 JOBS
INTEGER I, INFO, J, L, M, N, P
DOUBLE PRECISION ABCNRM, ENORM, SABCNM, SENORM, THRESH
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ DWORK(LDWORK), E(LDE,NMAX), LSCALE(LMAX),
$ RSCALE(NMAX)
* .. External Functions ..
DOUBLE PRECISION DLANGE
EXTERNAL DLANGE
* .. External Subroutines ..
EXTERNAL TG01AD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) L, N, M, P, JOBS, THRESH
IF ( L.LT.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) L
ELSE
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,L )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,L )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,L )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Compute norms before scaling
ABCNRM = MAX( DLANGE( '1', L, N, A, LDA, DWORK ),
$ DLANGE( '1', L, M, B, LDB, DWORK ),
$ DLANGE( '1', P, N, C, LDC, DWORK ) )
ENORM = DLANGE( '1', L, N, E, LDE, DWORK )
* Find the transformed descriptor system
* (A-lambda E,B,C).
CALL TG01AD( JOBS, L, N, M, P, THRESH, A, LDA, E, LDE,
$ B, LDB, C, LDC, LSCALE, RSCALE, DWORK,
$ INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
SABCNM = MAX( DLANGE( '1', L, N, A, LDA, DWORK ),
$ DLANGE( '1', L, M, B, LDB, DWORK ),
$ DLANGE( '1', P, N, C, LDC, DWORK ) )
SENORM = DLANGE( '1', L, N, E, LDE, DWORK )
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 30 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
30 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 40 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
40 CONTINUE
WRITE ( NOUT, FMT = 99991 )
WRITE ( NOUT, FMT = 99995 ) ( LSCALE(I), I = 1,L )
WRITE ( NOUT, FMT = 99990 )
WRITE ( NOUT, FMT = 99995 ) ( RSCALE(J), J = 1,N )
WRITE ( NOUT, FMT = 99994 )
$ ABCNRM, SABCNM, ENORM, SENORM
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01AD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01AD = ',I2)
99997 FORMAT (/' The transformed state dynamics matrix Dl*A*Dr is ')
99996 FORMAT (/' The transformed descriptor matrix Dl*E*Dr is ')
99995 FORMAT (20(1X,F9.4))
99994 FORMAT (/' Norm of [ A B; C 0] =', 1PD10.3/
$ ' Norm of scaled [ A B; C 0] =', 1PD10.3/
$ ' Norm of E =', 1PD10.3/
$ ' Norm of scaled E =', 1PD10.3)
99993 FORMAT (/' The transformed input/state matrix Dl*B is ')
99992 FORMAT (/' The transformed state/output matrix C*Dr is ')
99991 FORMAT (/' The diagonal of left scaling matrix Dl is ')
99990 FORMAT (/' The diagonal of right scaling matrix Dr is ')
99989 FORMAT (/' L is out of range.',/' L = ',I5)
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
99986 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
TG01AD EXAMPLE PROGRAM DATA
4 4 2 2 A 0.0
-1 0 0 0.003
0 0 0.1000 0.02
100 10 0 0.4
0 0 0 0.0
1 0.2 0 0.0
0 1 0 0.01
300 90 6 0.3
0 0 20 0.0
10 0
0 0
0 1000
10000 10000
-0.1 0.0 0.001 0.0
0.0 0.01 -0.001 0.0001
Program Results
TG01AD EXAMPLE PROGRAM RESULTS
The transformed state dynamics matrix Dl*A*Dr is
-1.0000 0.0000 0.0000 0.3000
0.0000 0.0000 1.0000 2.0000
1.0000 0.1000 0.0000 0.4000
0.0000 0.0000 0.0000 0.0000
The transformed descriptor matrix Dl*E*Dr is
1.0000 0.2000 0.0000 0.0000
0.0000 1.0000 0.0000 1.0000
3.0000 0.9000 0.6000 0.3000
0.0000 0.0000 0.2000 0.0000
The transformed input/state matrix Dl*B is
100.0000 0.0000
0.0000 0.0000
0.0000 100.0000
100.0000 100.0000
The transformed state/output matrix C*Dr is
-0.0100 0.0000 0.0010 0.0000
0.0000 0.0010 -0.0010 0.0010
The diagonal of left scaling matrix Dl is
10.0000 10.0000 0.1000 0.0100
The diagonal of right scaling matrix Dr is
0.1000 0.1000 1.0000 10.0000
Norm of [ A B; C 0] = 1.100D+04
Norm of scaled [ A B; C 0] = 2.000D+02
Norm of E = 3.010D+02
Norm of scaled E = 4.000D+00
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TG01AZ.html 0000664 0000000 0000000 00000033465 14560147231 0020373 0 ustar 00root root 0000000 0000000
Purpose
To balance the matrices of the system pencil
S = ( A B ) - lambda ( E 0 ) := Q - lambda Z,
( C 0 ) ( 0 0 )
corresponding to the descriptor triple (A-lambda E,B,C),
by balancing. This involves diagonal similarity transformations
(Dl*A*Dr - lambda Dl*E*Dr, Dl*B, C*Dr) applied to the system
(A-lambda E,B,C) to make the rows and columns of system pencil
matrices
diag(Dl,I) * S * diag(Dr,I)
as close in norm as possible. Balancing may reduce the 1-norms
of the matrices of the system pencil S.
The balancing can be performed optionally on the following
particular system pencils
S = A-lambda E,
S = ( A-lambda E B ), or
S = ( A-lambda E ).
( C )
Specification
SUBROUTINE TG01AZ( JOB, L, N, M, P, THRESH, A, LDA, E, LDE,
$ B, LDB, C, LDC, LSCALE, RSCALE, DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOB
INTEGER INFO, L, LDA, LDB, LDC, LDE, M, N, P
DOUBLE PRECISION THRESH
C .. Array Arguments ..
COMPLEX*16 A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ E( LDE, * )
DOUBLE PRECISION DWORK( * ), LSCALE( * ), RSCALE( * )
Arguments
Mode Parameters
JOB CHARACTER*1
Indicates which matrices are involved in balancing, as
follows:
= 'A': All matrices are involved in balancing;
= 'B': B, A and E matrices are involved in balancing;
= 'C': C, A and E matrices are involved in balancing;
= 'N': B and C matrices are not involved in balancing.
Input/Output Parameters
L (input) INTEGER
The number of rows of matrices A, B, and E. L >= 0.
N (input) INTEGER
The number of columns of matrices A, E, and C. N >= 0.
M (input) INTEGER
The number of columns of matrix B. M >= 0.
P (input) INTEGER
The number of rows of matrix C. P >= 0.
THRESH (input) DOUBLE PRECISION
Threshold value for magnitude of elements:
elements with magnitude less than or equal to
THRESH are ignored for balancing. THRESH >= 0.
The magnitude is computed as the sum of the absolute
values of the real and imaginary parts.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the leading L-by-N part of this array must
contain the state dynamics matrix A.
On exit, the leading L-by-N part of this array contains
the balanced matrix Dl*A*Dr.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,L).
E (input/output) COMPLEX*16 array, dimension (LDE,N)
On entry, the leading L-by-N part of this array must
contain the descriptor matrix E.
On exit, the leading L-by-N part of this array contains
the balanced matrix Dl*E*Dr.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,L).
B (input/output) COMPLEX*16 array, dimension (LDB,M)
On entry, the leading L-by-M part of this array must
contain the input/state matrix B.
On exit, if M > 0, the leading L-by-M part of this array
contains the balanced matrix Dl*B.
The array B is not referenced if M = 0.
LDB INTEGER
The leading dimension of array B.
LDB >= MAX(1,L) if M > 0 or LDB >= 1 if M = 0.
C (input/output) COMPLEX*16 array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, if P > 0, the leading P-by-N part of this array
contains the balanced matrix C*Dr.
The array C is not referenced if P = 0.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
LSCALE (output) DOUBLE PRECISION array, dimension (L)
The scaling factors applied to S from left. If Dl(j) is
the scaling factor applied to row j, then
SCALE(j) = Dl(j), for j = 1,...,L.
RSCALE (output) DOUBLE PRECISION array, dimension (N)
The scaling factors applied to S from right. If Dr(j) is
the scaling factor applied to column j, then
SCALE(j) = Dr(j), for j = 1,...,N.
Workspace
DWORK DOUBLE PRECISION array, dimension (3*(L+N))Error Indicator
INFO INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Balancing consists of applying a diagonal similarity
transformation
-1
diag(Dl,I) * S * diag(Dr,I)
to make the 1-norms of each row of the first L rows of S and its
corresponding N columns nearly equal.
Information about the diagonal matrices Dl and Dr are returned in
the vectors LSCALE and RSCALE, respectively.
References
[1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
[2] R.C. Ward, R. C.
Balancing the generalized eigenvalue problem.
SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
* TG01AZ EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER LMAX, NMAX, MMAX, PMAX
PARAMETER ( LMAX = 20, NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDE
PARAMETER ( LDA = LMAX, LDB = LMAX, LDC = PMAX,
$ LDE = LMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( 1, 3*(LMAX+NMAX ) ) )
* .. Local Scalars ..
CHARACTER*1 JOBS
INTEGER I, INFO, J, L, M, N, P
DOUBLE PRECISION ABCNRM, ENORM, SABCNM, SENORM, THRESH
* .. Local Arrays ..
COMPLEX*16 A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ E(LDE,NMAX)
DOUBLE PRECISION DWORK(LDWORK), LSCALE(LMAX), RSCALE(NMAX)
* .. External Functions ..
DOUBLE PRECISION ZLANGE
EXTERNAL ZLANGE
* .. External Subroutines ..
EXTERNAL TG01AZ
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) L, N, M, P, JOBS, THRESH
IF ( L.LT.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) L
ELSE
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,L )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,L )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,L )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Compute norms before scaling
ABCNRM = MAX( ZLANGE( '1', L, N, A, LDA, DWORK ),
$ ZLANGE( '1', L, M, B, LDB, DWORK ),
$ ZLANGE( '1', P, N, C, LDC, DWORK ) )
ENORM = ZLANGE( '1', L, N, E, LDE, DWORK )
* Find the transformed descriptor system
* (A-lambda E,B,C).
CALL TG01AZ( JOBS, L, N, M, P, THRESH, A, LDA, E, LDE,
$ B, LDB, C, LDC, LSCALE, RSCALE, DWORK,
$ INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
SABCNM = MAX( ZLANGE( '1', L, N, A, LDA, DWORK ),
$ ZLANGE( '1', L, M, B, LDB, DWORK ),
$ ZLANGE( '1', P, N, C, LDC, DWORK ) )
SENORM = ZLANGE( '1', L, N, E, LDE, DWORK )
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 30 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
30 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 40 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
40 CONTINUE
WRITE ( NOUT, FMT = 99991 )
WRITE ( NOUT, FMT = 99985 ) ( LSCALE(I), I = 1,L )
WRITE ( NOUT, FMT = 99990 )
WRITE ( NOUT, FMT = 99985 ) ( RSCALE(J), J = 1,N )
WRITE ( NOUT, FMT = 99994 )
$ ABCNRM, SABCNM, ENORM, SENORM
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01AZ EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01AZ = ',I2)
99997 FORMAT (/' The transformed state dynamics matrix Dl*A*Dr is ')
99996 FORMAT (/' The transformed descriptor matrix Dl*E*Dr is ')
99995 FORMAT (20(1X,F9.4,SP,F9.4,S,'i '))
99994 FORMAT (/' Norm of [ A B; C 0] =', 1PD10.3/
$ ' Norm of scaled [ A B; C 0] =', 1PD10.3/
$ ' Norm of E =', 1PD10.3/
$ ' Norm of scaled E =', 1PD10.3)
99993 FORMAT (/' The transformed input/state matrix Dl*B is ')
99992 FORMAT (/' The transformed state/output matrix C*Dr is ')
99991 FORMAT (/' The diagonal of left scaling matrix Dl is ')
99990 FORMAT (/' The diagonal of right scaling matrix Dr is ')
99989 FORMAT (/' L is out of range.',/' L = ',I5)
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
99986 FORMAT (/' P is out of range.',/' P = ',I5)
99985 FORMAT (20(1X,F9.4))
END
Program Data
TG01AZ EXAMPLE PROGRAM DATA
4 4 2 2 A 0.0
(-1,0) (0,0) (0,0) (0.003,0)
(0,0) (0,0) (0.1000,0) (0.02,0)
(100,0) (10,0) (0,0) (0.4,0)
(0,0) (0,0) (0,0) (0.0,0)
(1,0) (0.2,0) (0,0) (0.0,0)
(0,0) (1,0) (0,0) ( 0.01,0)
(300,0) (90,0) (6,0) (0.3,0)
(0,0) (0,0) (20,0) (0.0,0)
(10,0) (0,0)
(0,0) (0,0)
(0,0) (1000,0)
(10000,0) (10000,0)
(-0.1,0) (0.0,0) (0.001,0) (0.0,0)
(0.0,0) (0.01,0) (-0.001,0) (0.0001,0)
Program Results
TG01AZ EXAMPLE PROGRAM RESULTS
The transformed state dynamics matrix Dl*A*Dr is
-1.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.3000 +0.0000i
0.0000 +0.0000i 0.0000 +0.0000i 1.0000 +0.0000i 2.0000 +0.0000i
1.0000 +0.0000i 0.1000 +0.0000i 0.0000 +0.0000i 0.4000 +0.0000i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i
The transformed descriptor matrix Dl*E*Dr is
1.0000 +0.0000i 0.2000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i
0.0000 +0.0000i 1.0000 +0.0000i 0.0000 +0.0000i 1.0000 +0.0000i
3.0000 +0.0000i 0.9000 +0.0000i 0.6000 +0.0000i 0.3000 +0.0000i
0.0000 +0.0000i 0.0000 +0.0000i 0.2000 +0.0000i 0.0000 +0.0000i
The transformed input/state matrix Dl*B is
100.0000 +0.0000i 0.0000 +0.0000i
0.0000 +0.0000i 0.0000 +0.0000i
0.0000 +0.0000i 100.0000 +0.0000i
100.0000 +0.0000i 100.0000 +0.0000i
The transformed state/output matrix C*Dr is
-0.0100 +0.0000i 0.0000 +0.0000i 0.0010 +0.0000i 0.0000 +0.0000i
0.0000 +0.0000i 0.0010 +0.0000i -0.0010 +0.0000i 0.0010 +0.0000i
The diagonal of left scaling matrix Dl is
10.0000 10.0000 0.1000 0.0100
The diagonal of right scaling matrix Dr is
0.1000 0.1000 1.0000 10.0000
Norm of [ A B; C 0] = 1.100D+04
Norm of scaled [ A B; C 0] = 2.000D+02
Norm of E = 3.010D+02
Norm of scaled E = 4.000D+00
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TG01BD.html 0000664 0000000 0000000 00000022547 14560147231 0020345 0 ustar 00root root 0000000 0000000
Purpose
To reduce the matrices A and E of the system pencil
S = ( A B ) - lambda ( E 0 ) ,
( C 0 ) ( 0 0 )
corresponding to the descriptor triple (A-lambda E,B,C),
to generalized upper Hessenberg form using orthogonal
transformations,
Q' * A * Z = H, Q' * E * Z = T,
where H is upper Hessenberg, T is upper triangular, Q and Z
are orthogonal, and ' means transpose. The corresponding
transformations, written compactly as diag(Q',I) * S * diag(Z,I),
are also applied to B and C, getting Q' * B and C * Z.
The orthogonal matrices Q and Z are determined as products of
Givens rotations. They may either be formed explicitly, or they
may be postmultiplied into input matrices Q1 and Z1, so that
Q1 * A * Z1' = (Q1*Q) * H * (Z1*Z)'
Q1 * E * Z1' = (Q1*Q) * T * (Z1*Z)'.
Specification
SUBROUTINE TG01BD( JOBE, COMPQ, COMPZ, N, M, P, ILO, IHI, A, LDA,
$ E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, COMPZ, JOBE
INTEGER IHI, ILO, INFO, LDA, LDB, LDC, LDE, LDQ,
$ LDWORK, LDZ, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ DWORK( * ), E( LDE, * ), Q( LDQ, * ),
$ Z( LDZ, * )
Arguments
Mode Parameters
JOBE CHARACTER*1
Specifies whether E is a general square or an upper
triangular matrix, as follows:
= 'G': E is a general square matrix;
= 'U': E is an upper triangular matrix.
COMPQ CHARACTER*1
Indicates what should be done with matrix Q, as follows:
= 'N': do not compute Q;
= 'I': Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= 'V': Q must contain an orthogonal matrix Q1 on entry,
and the product Q1*Q is returned.
COMPZ CHARACTER*1
Indicates what should be done with matrix Z, as follows:
= 'N': do not compute Z;
= 'I': Z is initialized to the unit matrix, and the
orthogonal matrix Z is returned;
= 'V': Z must contain an orthogonal matrix Z1 on entry,
and the product Z1*Z is returned.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, E, and the number of rows of
the matrix B. N >= 0.
M (input) INTEGER
The number of columns of the matrix B. M >= 0.
P (input) INTEGER
The number of rows of the matrix C. P >= 0.
ILO (input) INTEGER
IHI (input) INTEGER
It is assumed that A and E are already upper triangular in
rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI could
normally be set by a previous call to LAPACK Library
routine DGGBAL; otherwise they should be set to 1 and N,
respectively.
1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
If JOBE = 'U', the matrix E is assumed upper triangular.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A.
On exit, the leading N-by-N part of this array contains
the upper Hessenberg matrix H = Q' * A * Z. The elements
below the first subdiagonal are set to zero.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain the descriptor matrix E. If JOBE = 'U', this
matrix is assumed upper triangular.
On exit, the leading N-by-N part of this array contains
the upper triangular matrix T = Q' * E * Z. The elements
below the diagonal are set to zero.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input/state matrix B.
On exit, if M > 0, the leading N-by-M part of this array
contains the transformed matrix Q' * B.
The array B is not referenced if M = 0.
LDB INTEGER
The leading dimension of array B.
LDB >= MAX(1,N) if M > 0; LDB >= 1 if M = 0.
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, if P > 0, the leading P-by-N part of this array
contains the transformed matrix C * Z.
The array C is not referenced if P = 0.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
If COMPQ = 'N': Q is not referenced;
If COMPQ = 'I': on entry, Q need not be set, and on exit
it contains the orthogonal matrix Q,
where Q' is the product of the Givens
transformations which are applied to A,
E, and B on the left;
If COMPQ = 'V': on entry, Q must contain an orthogonal
matrix Q1, and on exit this is
overwritten by Q1*Q.
LDQ INTEGER
The leading dimension of array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1,N), if COMPQ = 'I' or 'V'.
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
If COMPZ = 'N': Z is not referenced;
If COMPZ = 'I': on entry, Z need not be set, and on exit
it contains the orthogonal matrix Z,
which is the product of the Givens
transformations applied to A, E, and C
on the right;
If COMPZ = 'V': on entry, Z must contain an orthogonal
matrix Z1, and on exit this is
overwritten by Z1*Z.
LDZ INTEGER
The leading dimension of array Z.
LDZ >= 1, if COMPZ = 'N';
LDZ >= MAX(1,N), if COMPZ = 'I' or 'V'.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= 1, if JOBE = 'U';
LDWORK >= MAX(1,IHI+1-ILO+MAX(NI,M)), if JOBE = 'G', where
NI = N+1-ILO, if COMPQ = 'N', and NI = N, otherwise.
For good performance, if JOBE = 'G', LDWORK must generally
be larger, LDWORK >= MAX(1,IHI+1-ILO+MAX(NI,M)*NB), where
NB is the optimal block size.
Error Indicator
INFO INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
First, this routine computes the QR factorization of E and applies the transformations to A, B, and possibly Q. Then, the routine reduces A to upper Hessenberg form, preserving E triangular, by an unblocked reduction [1], using two sequences of plane rotations applied alternately from the left and from the right. The corresponding transformations may be accumulated and/or applied to the matrices B and C. If JOBE = 'U', the initial reduction of E to upper triangular form is skipped. This routine is a modification and extension of the LAPACK Library routine DGGHRD [2].References
[1] Golub, G.H. and van Loan, C.F.
Matrix Computations. Third Edition.
M. D. Johns Hopkins University Press, Baltimore, 1996.
[2] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TG01CD.html 0000664 0000000 0000000 00000023663 14560147231 0020346 0 ustar 00root root 0000000 0000000
Purpose
To reduce the descriptor system pair (A-lambda E,B) to the QR-coordinate form by computing an orthogonal transformation matrix Q such that the transformed descriptor system pair (Q'*A-lambda Q'*E, Q'*B) has the descriptor matrix Q'*E in an upper trapezoidal form. The left orthogonal transformations performed to reduce E can be optionally accumulated.Specification
SUBROUTINE TG01CD( COMPQ, L, N, M, A, LDA, E, LDE, B, LDB, Q, LDQ,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ
INTEGER INFO, L, LDA, LDB, LDE, LDQ, LDWORK, M, N
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DWORK( * ),
$ E( LDE, * ), Q( LDQ, * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
= 'N': do not compute Q;
= 'I': Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= 'U': Q must contain an orthogonal matrix Q1 on entry,
and the product Q1*Q is returned.
Input/Output Parameters
L (input) INTEGER
The number of rows of matrices A, B, and E. L >= 0.
N (input) INTEGER
The number of columns of matrices A and E. N >= 0.
M (input) INTEGER
The number of columns of matrix B. M >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading L-by-N part of this array must
contain the state dynamics matrix A.
On exit, the leading L-by-N part of this array contains
the transformed matrix Q'*A.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,L).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading L-by-N part of this array must
contain the descriptor matrix E.
On exit, the leading L-by-N part of this array contains
the transformed matrix Q'*E in upper trapezoidal form,
i.e.
( E11 )
Q'*E = ( ) , if L >= N ,
( 0 )
or
Q'*E = ( E11 E12 ), if L < N ,
where E11 is an MIN(L,N)-by-MIN(L,N) upper triangular
matrix.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,L).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading L-by-M part of this array must
contain the input/state matrix B.
On exit, the leading L-by-M part of this array contains
the transformed matrix Q'*B.
LDB INTEGER
The leading dimension of array B.
LDB >= MAX(1,L) if M > 0 or LDB >= 1 if M = 0.
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,L)
If COMPQ = 'N': Q is not referenced.
If COMPQ = 'I': on entry, Q need not be set;
on exit, the leading L-by-L part of this
array contains the orthogonal matrix Q,
where Q' is the product of Householder
transformations which are applied to A,
E, and B on the left.
If COMPQ = 'U': on entry, the leading L-by-L part of this
array must contain an orthogonal matrix
Q1;
on exit, the leading L-by-L part of this
array contains the orthogonal matrix
Q1*Q.
LDQ INTEGER
The leading dimension of array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1,L), if COMPQ = 'U' or 'I'.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1, MIN(L,N) + MAX(L,N,M)).
For optimum performance
LWORK >= MAX(1, MIN(L,N) + MAX(L,N,M)*NB),
where NB is the optimal blocksize.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine computes the QR factorization of E to reduce it
to the upper trapezoidal form.
The transformations are also applied to the rest of system
matrices
A <- Q' * A , B <- Q' * B.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( L*L*N ) floating point operations.Further Comments
NoneExample
Program Text
* TG01CD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER LMAX, NMAX, MMAX
PARAMETER ( LMAX = 20, NMAX = 20, MMAX = 20)
INTEGER LDA, LDB, LDE, LDQ
PARAMETER ( LDA = LMAX, LDB = LMAX,
$ LDE = LMAX, LDQ = LMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MIN(LMAX,NMAX)+MAX(LMAX,NMAX,MMAX) )
* .. Local Scalars ..
CHARACTER*1 COMPQ
INTEGER I, INFO, J, L, M, N
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX),
$ DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,LMAX)
* .. External Subroutines ..
EXTERNAL TG01CD
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) L, N, M
COMPQ = 'I'
IF ( L.LT.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) L
ELSE
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,L )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,L )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,L )
* Find the transformed descriptor system pair
* (A-lambda E,B).
CALL TG01CD( COMPQ, L, N, M, A, LDA, E, LDE, B, LDB,
$ Q, LDQ, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 30 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
30 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,L )
40 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01CD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01CD = ',I2)
99997 FORMAT (/' The transformed state dynamics matrix Q''*A is ')
99996 FORMAT (/' The transformed descriptor matrix Q''*E is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' The transformed input/state matrix Q''*B is ')
99993 FORMAT (/' The left transformation matrix Q is ')
99992 FORMAT (/' L is out of range.',/' L = ',I5)
99991 FORMAT (/' N is out of range.',/' N = ',I5)
99990 FORMAT (/' M is out of range.',/' M = ',I5)
END
Program Data
TG01CD EXAMPLE PROGRAM DATA
4 4 2 0.0
-1 0 0 3
0 0 1 2
1 1 0 4
0 0 0 0
1 2 0 0
0 1 0 1
3 9 6 3
0 0 2 0
1 0
0 0
0 1
1 1
Program Results
TG01CD EXAMPLE PROGRAM RESULTS The transformed state dynamics matrix Q'*A is -0.6325 -0.9487 0.0000 -4.7434 -0.8706 -0.2176 -0.7255 -0.3627 -0.5203 -0.1301 0.3902 1.4307 -0.7559 -0.1890 0.5669 2.0788 The transformed descriptor matrix Q'*E is -3.1623 -9.1706 -5.6921 -2.8460 0.0000 -1.3784 -1.3059 -1.3784 0.0000 0.0000 -2.4279 0.0000 0.0000 0.0000 0.0000 0.0000 The transformed input/state matrix Q'*B is -0.3162 -0.9487 0.6529 -0.2176 -0.4336 -0.9538 1.1339 0.3780 The left transformation matrix Q is -0.3162 0.6529 0.3902 0.5669 0.0000 -0.7255 0.3902 0.5669 -0.9487 -0.2176 -0.1301 -0.1890 0.0000 0.0000 -0.8238 0.5669
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TG01DD.html 0000664 0000000 0000000 00000023526 14560147231 0020345 0 ustar 00root root 0000000 0000000
Purpose
To reduce the descriptor system pair (C,A-lambda E) to the RQ-coordinate form by computing an orthogonal transformation matrix Z such that the transformed descriptor system pair (C*Z,A*Z-lambda E*Z) has the descriptor matrix E*Z in an upper trapezoidal form. The right orthogonal transformations performed to reduce E can be optionally accumulated.Specification
SUBROUTINE TG01DD( COMPZ, L, N, P, A, LDA, E, LDE, C, LDC, Z, LDZ,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPZ
INTEGER INFO, L, LDA, LDC, LDE, LDWORK, LDZ, N, P
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), DWORK( * ),
$ E( LDE, * ), Z( LDZ, * )
Arguments
Mode Parameters
COMPZ CHARACTER*1
= 'N': do not compute Z;
= 'I': Z is initialized to the unit matrix, and the
orthogonal matrix Z is returned;
= 'U': Z must contain an orthogonal matrix Z1 on entry,
and the product Z1*Z is returned.
Input/Output Parameters
L (input) INTEGER
The number of rows of matrices A and E. L >= 0.
N (input) INTEGER
The number of columns of matrices A, E, and C. N >= 0.
P (input) INTEGER
The number of rows of matrix C. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading L-by-N part of this array must
contain the state dynamics matrix A.
On exit, the leading L-by-N part of this array contains
the transformed matrix A*Z.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,L).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading L-by-N part of this array must
contain the descriptor matrix E.
On exit, the leading L-by-N part of this array contains
the transformed matrix E*Z in upper trapezoidal form,
i.e.
( E11 )
E*Z = ( ) , if L >= N ,
( R )
or
E*Z = ( 0 R ), if L < N ,
where R is an MIN(L,N)-by-MIN(L,N) upper triangular
matrix.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,L).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed matrix C*Z.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
If COMPZ = 'N': Z is not referenced.
If COMPZ = 'I': on entry, Z need not be set;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix Z,
which is the product of Householder
transformations applied to A, E, and C
on the right.
If COMPZ = 'U': on entry, the leading N-by-N part of this
array must contain an orthogonal matrix
Z1;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix
Z1*Z.
LDZ INTEGER
The leading dimension of array Z.
LDZ >= 1, if COMPZ = 'N';
LDZ >= MAX(1,N), if COMPZ = 'U' or 'I'.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1, MIN(L,N) + MAX(L,N,P)).
For optimum performance
LWORK >= MAX(1, MIN(L,N) + MAX(L,N,P)*NB),
where NB is the optimal blocksize.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine computes the RQ factorization of E to reduce it
the upper trapezoidal form.
The transformations are also applied to the rest of system
matrices
A <- A * Z, C <- C * Z.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( L*N*N ) floating point operations.Further Comments
NoneExample
Program Text
* TG01DD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER LMAX, NMAX, PMAX
PARAMETER ( LMAX = 20, NMAX = 20, PMAX = 20)
INTEGER LDA, LDC, LDE, LDZ
PARAMETER ( LDA = LMAX, LDC = PMAX,
$ LDE = LMAX, LDZ = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MIN(LMAX,NMAX)+MAX(LMAX,NMAX,PMAX) )
* .. Local Scalars ..
CHARACTER*1 COMPZ
INTEGER I, INFO, J, L, N, P
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), C(LDC,NMAX),
$ DWORK(LDWORK), E(LDE,NMAX), Z(LDZ,NMAX)
* .. External Subroutines ..
EXTERNAL TG01DD
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) L, N, P
COMPZ = 'I'
IF ( L.LT.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) L
ELSE
IF( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,L )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,L )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find the transformed descriptor system pair
* (A-lambda E,B).
CALL TG01DD( COMPZ, L, N, P, A, LDA, E, LDE, C, LDC,
$ Z, LDZ, DWORK, LDWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 30 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
30 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
40 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01DD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01DD = ',I2)
99997 FORMAT (/' The transformed state dynamics matrix A*Z is ')
99996 FORMAT (/' The transformed descriptor matrix E*Z is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' The transformed input/state matrix C*Z is ')
99993 FORMAT (/' The right transformation matrix Z is ')
99992 FORMAT (/' L is out of range.',/' L = ',I5)
99991 FORMAT (/' N is out of range.',/' N = ',I5)
99990 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
TG01DD EXAMPLE PROGRAM DATA
4 4 2 0.0
-1 0 0 3
0 0 1 2
1 1 0 4
0 0 0 0
1 2 0 0
0 1 0 1
3 9 6 3
0 0 2 0
-1 0 1 0
0 1 -1 1
Program Results
TG01DD EXAMPLE PROGRAM RESULTS The transformed state dynamics matrix A*Z is 0.4082 3.0773 0.6030 0.0000 0.8165 1.7233 0.6030 -1.0000 2.0412 2.8311 2.4121 0.0000 0.0000 0.0000 0.0000 0.0000 The transformed descriptor matrix E*Z is 0.0000 -0.7385 2.1106 0.0000 0.0000 0.7385 1.2060 0.0000 0.0000 0.0000 9.9499 -6.0000 0.0000 0.0000 0.0000 -2.0000 The transformed input/state matrix C*Z is -0.8165 0.4924 -0.3015 -1.0000 0.0000 0.7385 1.2060 1.0000 The right transformation matrix Z is 0.8165 -0.4924 0.3015 0.0000 -0.4082 -0.1231 0.9045 0.0000 0.0000 0.0000 0.0000 -1.0000 0.4082 0.8616 0.3015 0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TG01ED.html 0000664 0000000 0000000 00000035174 14560147231 0020350 0 ustar 00root root 0000000 0000000
Purpose
To compute for the descriptor system (A-lambda E,B,C)
the orthogonal transformation matrices Q and Z such that the
transformed system (Q'*A*Z-lambda Q'*E*Z, Q'*B, C*Z) is in an
SVD (singular value decomposition) coordinate form with
the system matrices Q'*A*Z and Q'*E*Z in the form
( A11 A12 ) ( Er 0 )
Q'*A*Z = ( ) , Q'*E*Z = ( ) ,
( A21 A22 ) ( 0 0 )
where Er is an invertible diagonal matrix having on the diagonal
the decreasingly ordered nonzero singular values of E.
Optionally, the A22 matrix can be further reduced to the
SVD form
( Ar 0 )
A22 = ( ) ,
( 0 0 )
where Ar is an invertible diagonal matrix having on the diagonal
the decreasingly ordered nonzero singular values of A22.
The left and/or right orthogonal transformations performed
to reduce E and A22 are accumulated.
Specification
SUBROUTINE TG01ED( JOBA, L, N, M, P, A, LDA, E, LDE, B, LDB,
$ C, LDC, Q, LDQ, Z, LDZ, RANKE, RNKA22, TOL,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBA
INTEGER INFO, L, LDA, LDB, LDC, LDE, LDQ, LDWORK,
$ LDZ, M, N, P, RNKA22, RANKE
DOUBLE PRECISION TOL
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ DWORK( * ), E( LDE, * ), Q( LDQ, * ),
$ Z( LDZ, * )
Arguments
Mode Parameters
JOBA CHARACTER*1
= 'N': do not reduce A22;
= 'R': reduce A22 to an SVD form.
Input/Output Parameters
L (input) INTEGER
The number of rows of matrices A, B, and E. L >= 0.
N (input) INTEGER
The number of columns of matrices A, E, and C. N >= 0.
M (input) INTEGER
The number of columns of matrix B. M >= 0.
P (input) INTEGER
The number of rows of matrix C. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading L-by-N part of this array must
contain the state dynamics matrix A.
On exit, the leading L-by-N part of this array contains
the transformed matrix Q'*A*Z. If JOBA = 'R', this matrix
is in the form
( A11 * * )
Q'*A*Z = ( * Ar 0 ) ,
( * 0 0 )
where A11 is a RANKE-by-RANKE matrix and Ar is a
RNKA22-by-RNKA22 invertible diagonal matrix, with
decresingly ordered positive diagonal elements.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,L).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading L-by-N part of this array must
contain the descriptor matrix E.
On exit, the leading L-by-N part of this array contains
the transformed matrix Q'*E*Z.
( Er 0 )
Q'*E*Z = ( ) ,
( 0 0 )
where Er is a RANKE-by-RANKE invertible diagonal matrix
having on the diagonal the decreasingly ordered positive
singular values of E.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,L).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading L-by-M part of this array must
contain the input/state matrix B.
On exit, the leading L-by-M part of this array contains
the transformed matrix Q'*B.
LDB INTEGER
The leading dimension of array B.
LDB >= MAX(1,L) if M > 0 or LDB >= 1 if M = 0.
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed matrix C*Z.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
Q (output) DOUBLE PRECISION array, dimension (LDQ,L)
The leading L-by-L part of this array contains the
orthogonal matrix Q, which is the accumulated product of
transformations applied to A, E, and B on the left.
LDQ INTEGER
The leading dimension of array Q. LDQ >= MAX(1,L).
Z (output) DOUBLE PRECISION array, dimension (LDZ,N)
The leading N-by-N part of this array contains the
orthogonal matrix Z, which is the accumulated product of
transformations applied to A, E, and C on the right.
LDZ INTEGER
The leading dimension of array Z. LDZ >= MAX(1,N).
RANKE (output) INTEGER
The effective rank of matrix E, and thus also the order
of the invertible diagonal submatrix Er.
RANKE is computed as the number of singular values of E
greater than TOL*SVEMAX, where SVEMAX is the maximum
singular value of E.
RNKA22 (output) INTEGER
If JOBA = 'R', then RNKA22 is the effective rank of
matrix A22, and thus also the order of the invertible
diagonal submatrix Ar. RNKA22 is computed as the number
of singular values of A22 greater than TOL*SVAMAX,
where SVAMAX is an estimate of the maximum singular value
of A.
If JOBA = 'N', then RNKA22 is not referenced.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in determining the rank of E
and of A22. If TOL > 0, then singular values less than
TOL*SVMAX are treated as zero, where SVMAX is the maximum
singular value of E or an estimate of it for A and E.
If TOL <= 0, the default tolerance TOLDEF = EPS*L*N is
used instead, where EPS is the machine precision
(see LAPACK Library routine DLAMCH). TOL < 1.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,MIN(L,N) +
MAX(3*MIN(L,N)+MAX(L,N), 5*MIN(L,N), M, P)).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: the QR algorithm has failed to converge when computing
singular value decomposition. In this case INFO
specifies how many superdiagonals did not converge.
This failure is not likely to occur.
Method
The routine computes the singular value decomposition (SVD) of E,
in the form
( Er 0 )
E = Q * ( ) * Z'
( 0 0 )
and finds the largest RANKE-by-RANKE leading diagonal submatrix
Er whose condition number is less than 1/TOL. RANKE defines thus
the effective rank of matrix E.
If JOBA = 'R' the same reduction is performed on A22 in the
partitioned matrix
( A11 A12 )
Q'*A*Z = ( ) ,
( A21 A22 )
to obtain it in the form
( Ar 0 )
A22 = ( ) ,
( 0 0 )
with Ar an invertible diagonal matrix.
The accumulated transformations are also applied to the rest of
matrices
B <- Q' * B, C <- C * Z.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( L*L*N ) floating point operations.Further Comments
NoneExample
Program Text
* TG01ED EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER LMAX, NMAX, MMAX, PMAX
PARAMETER ( LMAX = 20, NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDE, LDQ, LDZ
PARAMETER ( LDA = LMAX, LDB = LMAX, LDC = PMAX,
$ LDE = LMAX, LDQ = LMAX, LDZ = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( 1, MIN( LMAX, NMAX ) +
$ MAX( MMAX, PMAX, 3*MIN( LMAX, NMAX ) +
$ MAX( LMAX, NMAX ),
$ 5*MIN( LMAX, NMAX ) ) ) )
* .. Local Scalars ..
CHARACTER*1 JOBA
INTEGER I, INFO, J, L, M, N, P, RANKE, RNKA22
DOUBLE PRECISION TOL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,LMAX),
$ Z(LDZ,NMAX)
* .. External Subroutines ..
EXTERNAL TG01ED
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) L, N, M, P, TOL
JOBA = 'R'
IF ( L.LT.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) L
ELSE
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,L )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,L )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,L )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find the transformed descriptor system
* (A-lambda E,B,C).
CALL TG01ED( JOBA, L, N, M, P, A, LDA, E, LDE, B, LDB,
$ C, LDC, Q, LDQ, Z, LDZ, RANKE, RNKA22,
$ TOL, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) RANKE, RNKA22
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 30 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
30 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 40 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
40 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 50 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,L )
50 CONTINUE
WRITE ( NOUT, FMT = 99990 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
60 CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01ED EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01ED = ',I2)
99997 FORMAT (/' The transformed state dynamics matrix Q''*A*Z is ')
99996 FORMAT (/' The transformed descriptor matrix Q''*E*Z is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (' Rank of matrix E =', I5/
$ ' Rank of matrix A22 =', I5)
99993 FORMAT (/' The transformed input/state matrix Q''*B is ')
99992 FORMAT (/' The transformed state/output matrix C*Z is ')
99991 FORMAT (/' The left transformation matrix Q is ')
99990 FORMAT (/' The right transformation matrix Z is ')
99989 FORMAT (/' L is out of range.',/' L = ',I5)
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
99986 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
TG01ED EXAMPLE PROGRAM DATA
4 4 2 2 0.0
-1 0 0 3
0 0 1 2
1 1 0 4
0 0 0 0
1 2 0 0
0 1 0 1
3 9 6 3
0 0 2 0
1 0
0 0
0 1
1 1
-1 0 1 0
0 1 -1 1
Program Results
TG01ED EXAMPLE PROGRAM RESULTS Rank of matrix E = 3 Rank of matrix A22 = 1 The transformed state dynamics matrix Q'*A*Z is 2.1882 -0.8664 -3.5097 -2.1353 -0.4569 -0.2146 1.9802 0.3531 -0.5717 -0.5245 -0.4591 0.4696 -0.4766 -0.5846 2.1414 0.3086 The transformed descriptor matrix Q'*E*Z is 11.8494 0.0000 0.0000 0.0000 0.0000 2.1302 0.0000 0.0000 0.0000 0.0000 1.0270 0.0000 0.0000 0.0000 0.0000 0.0000 The transformed input/state matrix Q'*B is -0.2396 -1.0668 -0.2656 -0.8393 -0.7657 -0.1213 1.1339 0.3780 The transformed state/output matrix C*Z is -0.2499 -1.0573 0.3912 -0.8165 -0.5225 1.3958 0.8825 0.0000 The left transformation matrix Q is -0.1534 0.5377 -0.6049 0.5669 -0.0872 0.2536 0.7789 0.5669 -0.9805 -0.0360 0.0395 -0.1890 -0.0863 -0.8033 -0.1608 0.5669 The right transformation matrix Z is -0.2612 0.2017 -0.4737 0.8165 -0.7780 0.4718 -0.0738 -0.4082 -0.5111 -0.8556 -0.0826 0.0000 -0.2556 0.0684 0.8737 0.4082
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TG01FD.html 0000664 0000000 0000000 00000042056 14560147231 0020346 0 ustar 00root root 0000000 0000000
Purpose
To compute for the descriptor system (A-lambda E,B,C)
the orthogonal transformation matrices Q and Z such that the
transformed system (Q'*A*Z-lambda Q'*E*Z, Q'*B, C*Z) is
in a SVD-like coordinate form with
( A11 A12 ) ( Er 0 )
Q'*A*Z = ( ) , Q'*E*Z = ( ) ,
( A21 A22 ) ( 0 0 )
where Er is an upper triangular invertible matrix.
Optionally, the A22 matrix can be further reduced to the form
( Ar X )
A22 = ( ) ,
( 0 0 )
with Ar an upper triangular invertible matrix, and X either a full
or a zero matrix.
The left and/or right orthogonal transformations performed
to reduce E and A22 can be optionally accumulated.
Specification
SUBROUTINE TG01FD( COMPQ, COMPZ, JOBA, L, N, M, P, A, LDA, E, LDE,
$ B, LDB, C, LDC, Q, LDQ, Z, LDZ, RANKE, RNKA22,
$ TOL, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, COMPZ, JOBA
INTEGER INFO, L, LDA, LDB, LDC, LDE, LDQ, LDWORK,
$ LDZ, M, N, P, RANKE, RNKA22
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ DWORK( * ), E( LDE, * ), Q( LDQ, * ),
$ Z( LDZ, * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
= 'N': do not compute Q;
= 'I': Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= 'U': Q must contain an orthogonal matrix Q1 on entry,
and the product Q1*Q is returned.
COMPZ CHARACTER*1
= 'N': do not compute Z;
= 'I': Z is initialized to the unit matrix, and the
orthogonal matrix Z is returned;
= 'U': Z must contain an orthogonal matrix Z1 on entry,
and the product Z1*Z is returned.
JOBA CHARACTER*1
= 'N': do not reduce A22;
= 'R': reduce A22 to a SVD-like upper triangular form.
= 'T': reduce A22 to an upper trapezoidal form.
Input/Output Parameters
L (input) INTEGER
The number of rows of matrices A, B, and E. L >= 0.
N (input) INTEGER
The number of columns of matrices A, E, and C. N >= 0.
M (input) INTEGER
The number of columns of matrix B. M >= 0.
P (input) INTEGER
The number of rows of matrix C. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading L-by-N part of this array must
contain the state dynamics matrix A.
On exit, the leading L-by-N part of this array contains
the transformed matrix Q'*A*Z. If JOBA = 'T', this matrix
is in the form
( A11 * * )
Q'*A*Z = ( * Ar X ) ,
( * 0 0 )
where A11 is a RANKE-by-RANKE matrix and Ar is a
RNKA22-by-RNKA22 invertible upper triangular matrix.
If JOBA = 'R' then A has the above form with X = 0.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,L).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading L-by-N part of this array must
contain the descriptor matrix E.
On exit, the leading L-by-N part of this array contains
the transformed matrix Q'*E*Z.
( Er 0 )
Q'*E*Z = ( ) ,
( 0 0 )
where Er is a RANKE-by-RANKE upper triangular invertible
matrix.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,L).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading L-by-M part of this array must
contain the input/state matrix B.
On exit, the leading L-by-M part of this array contains
the transformed matrix Q'*B.
LDB INTEGER
The leading dimension of array B.
LDB >= MAX(1,L) if M > 0 or LDB >= 1 if M = 0.
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed matrix C*Z.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,L)
If COMPQ = 'N': Q is not referenced.
If COMPQ = 'I': on entry, Q need not be set;
on exit, the leading L-by-L part of this
array contains the orthogonal matrix Q,
where Q' is the product of Householder
transformations which are applied to A,
E, and B on the left.
If COMPQ = 'U': on entry, the leading L-by-L part of this
array must contain an orthogonal matrix
Q1;
on exit, the leading L-by-L part of this
array contains the orthogonal matrix
Q1*Q.
LDQ INTEGER
The leading dimension of array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1,L), if COMPQ = 'U' or 'I'.
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
If COMPZ = 'N': Z is not referenced.
If COMPZ = 'I': on entry, Z need not be set;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix Z,
which is the product of Householder
transformations applied to A, E, and C
on the right.
If COMPZ = 'U': on entry, the leading N-by-N part of this
array must contain an orthogonal matrix
Z1;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix
Z1*Z.
LDZ INTEGER
The leading dimension of array Z.
LDZ >= 1, if COMPZ = 'N';
LDZ >= MAX(1,N), if COMPZ = 'U' or 'I'.
RANKE (output) INTEGER
The estimated rank of matrix E, and thus also the order
of the invertible upper triangular submatrix Er.
RNKA22 (output) INTEGER
If JOBA = 'R' or 'T', then RNKA22 is the estimated rank of
matrix A22, and thus also the order of the invertible
upper triangular submatrix Ar.
If JOBA = 'N', then RNKA22 is not referenced.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in determining the rank of E
and of A22. If the user sets TOL > 0, then the given
value of TOL is used as a lower bound for the
reciprocal condition numbers of leading submatrices
of R or R22 in the QR decompositions E * P = Q * R of E
or A22 * P22 = Q22 * R22 of A22.
A submatrix whose estimated condition number is less than
1/TOL is considered to be of full rank. If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
defined by TOLDEF = L*N*EPS, is used instead, where
EPS is the machine precision (see LAPACK Library routine
DLAMCH). TOL < 1.
Workspace
IWORK INTEGER array, dimension (N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( 1, N+P, MIN(L,N)+MAX(3*N-1,M,L) ).
For optimal performance, LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine computes a truncated QR factorization with column
pivoting of E, in the form
( E11 E12 )
E * P = Q * ( )
( 0 E22 )
and finds the largest RANKE-by-RANKE leading submatrix E11 whose
estimated condition number is less than 1/TOL. RANKE defines thus
the rank of matrix E. Further E22, being negligible, is set to
zero, and an orthogonal matrix Y is determined such that
( E11 E12 ) = ( Er 0 ) * Y .
The overal transformation matrix Z results as Z = P * Y' and the
resulting transformed matrices Q'*A*Z and Q'*E*Z have the form
( Er 0 ) ( A11 A12 )
E <- Q'* E * Z = ( ) , A <- Q' * A * Z = ( ) ,
( 0 0 ) ( A21 A22 )
where Er is an upper triangular invertible matrix.
If JOBA = 'R' the same reduction is performed on A22 to obtain it
in the form
( Ar 0 )
A22 = ( ) ,
( 0 0 )
with Ar an upper triangular invertible matrix.
If JOBA = 'T' then A22 is row compressed using the QR
factorization with column pivoting to the form
( Ar X )
A22 = ( )
( 0 0 )
with Ar an upper triangular invertible matrix.
The transformations are also applied to the rest of system
matrices
B <- Q' * B, C <- C * Z.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( L*L*N ) floating point operations.Further Comments
NoneExample
Program Text
* TG01FD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER LMAX, NMAX, MMAX, PMAX
PARAMETER ( LMAX = 20, NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDE, LDQ, LDZ
PARAMETER ( LDA = LMAX, LDB = LMAX, LDC = PMAX,
$ LDE = LMAX, LDQ = LMAX, LDZ = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( 1, PMAX,
$ MIN(LMAX,NMAX)+MAX( 3*NMAX, MMAX, LMAX ) ) )
* .. Local Scalars ..
CHARACTER*1 COMPQ, COMPZ, JOBA
INTEGER I, INFO, J, L, M, N, P, RANKE, RNKA22
DOUBLE PRECISION TOL
* .. Local Arrays ..
INTEGER IWORK(NMAX)
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,LMAX),
$ Z(LDZ,NMAX)
* .. External Subroutines ..
EXTERNAL TG01FD
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) L, N, M, P, TOL
COMPQ = 'I'
COMPZ = 'I'
JOBA = 'R'
IF ( L.LT.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) L
ELSE
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,L )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,L )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,L )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find the transformed descriptor system
* (A-lambda E,B,C).
CALL TG01FD( COMPQ, COMPZ, JOBA, L, N, M, P, A, LDA,
$ E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ,
$ RANKE, RNKA22, TOL, IWORK, DWORK, LDWORK,
$ INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) RANKE, RNKA22
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 30 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
30 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 40 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
40 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 50 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,L )
50 CONTINUE
WRITE ( NOUT, FMT = 99990 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
60 CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01FD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01FD = ',I2)
99997 FORMAT (/' The transformed state dynamics matrix Q''*A*Z is ')
99996 FORMAT (/' The transformed descriptor matrix Q''*E*Z is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (' Rank of matrix E =', I5/
$ ' Rank of matrix A22 =', I5)
99993 FORMAT (/' The transformed input/state matrix Q''*B is ')
99992 FORMAT (/' The transformed state/output matrix C*Z is ')
99991 FORMAT (/' The left transformation matrix Q is ')
99990 FORMAT (/' The right transformation matrix Z is ')
99989 FORMAT (/' L is out of range.',/' L = ',I5)
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
99986 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
TG01FD EXAMPLE PROGRAM DATA
4 4 2 2 0.0
-1 0 0 3
0 0 1 2
1 1 0 4
0 0 0 0
1 2 0 0
0 1 0 1
3 9 6 3
0 0 2 0
1 0
0 0
0 1
1 1
-1 0 1 0
0 1 -1 1
Program Results
TG01FD EXAMPLE PROGRAM RESULTS Rank of matrix E = 3 Rank of matrix A22 = 1 The transformed state dynamics matrix Q'*A*Z is 2.0278 0.1078 3.9062 -2.1571 -0.0980 0.2544 1.6053 -0.1269 0.2713 0.7760 -0.3692 -0.4853 0.0690 -0.5669 -2.1974 0.3086 The transformed descriptor matrix Q'*E*Z is 10.1587 5.8230 1.3021 0.0000 0.0000 -2.4684 -0.1896 0.0000 0.0000 0.0000 1.0338 0.0000 0.0000 0.0000 0.0000 0.0000 The transformed input/state matrix Q'*B is -0.2157 -0.9705 0.3015 0.9516 0.7595 0.0991 1.1339 0.3780 The transformed state/output matrix C*Z is 0.3651 -1.0000 -0.4472 -0.8165 -1.0954 1.0000 -0.8944 0.0000 The left transformation matrix Q is -0.2157 -0.5088 0.6109 0.5669 -0.1078 -0.2544 -0.7760 0.5669 -0.9705 0.1413 -0.0495 -0.1890 0.0000 0.8102 0.1486 0.5669 The right transformation matrix Z is -0.3651 0.0000 0.4472 0.8165 -0.9129 0.0000 0.0000 -0.4082 0.0000 -1.0000 0.0000 0.0000 -0.1826 0.0000 -0.8944 0.4082
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TG01FZ.html 0000664 0000000 0000000 00000044162 14560147231 0020374 0 ustar 00root root 0000000 0000000
Purpose
To compute for the descriptor system (A-lambda E,B,C)
the unitary transformation matrices Q and Z such that the
transformed system (Q'*A*Z-lambda Q'*E*Z, Q'*B, C*Z) is
in a SVD-like coordinate form with
( A11 A12 ) ( Er 0 )
Q'*A*Z = ( ) , Q'*E*Z = ( ) ,
( A21 A22 ) ( 0 0 )
where Er is an upper triangular invertible matrix, and ' denotes
the conjugate transpose. Optionally, the A22 matrix can be further
reduced to the form
( Ar X )
A22 = ( ) ,
( 0 0 )
with Ar an upper triangular invertible matrix, and X either a full
or a zero matrix.
The left and/or right unitary transformations performed
to reduce E and A22 can be optionally accumulated.
Specification
SUBROUTINE TG01FZ( COMPQ, COMPZ, JOBA, L, N, M, P, A, LDA, E, LDE,
$ B, LDB, C, LDC, Q, LDQ, Z, LDZ, RANKE, RNKA22,
$ TOL, IWORK, DWORK, ZWORK, LZWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, COMPZ, JOBA
INTEGER INFO, L, LDA, LDB, LDC, LDE, LDQ, LDZ, LZWORK,
$ M, N, P, RANKE, RNKA22
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK( * )
COMPLEX*16 A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ E( LDE, * ), Q( LDQ, * ), Z( LDZ, * ),
$ ZWORK( * )
DOUBLE PRECISION DWORK( * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
= 'N': do not compute Q;
= 'I': Q is initialized to the unit matrix, and the
unitary matrix Q is returned;
= 'U': Q must contain a unitary matrix Q1 on entry,
and the product Q1*Q is returned.
COMPZ CHARACTER*1
= 'N': do not compute Z;
= 'I': Z is initialized to the unit matrix, and the
unitary matrix Z is returned;
= 'U': Z must contain a unitary matrix Z1 on entry,
and the product Z1*Z is returned.
JOBA CHARACTER*1
= 'N': do not reduce A22;
= 'R': reduce A22 to a SVD-like upper triangular form.
= 'T': reduce A22 to an upper trapezoidal form.
Input/Output Parameters
L (input) INTEGER
The number of rows of matrices A, B, and E. L >= 0.
N (input) INTEGER
The number of columns of matrices A, E, and C. N >= 0.
M (input) INTEGER
The number of columns of matrix B. M >= 0.
P (input) INTEGER
The number of rows of matrix C. P >= 0.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the leading L-by-N part of this array must
contain the state dynamics matrix A.
On exit, the leading L-by-N part of this array contains
the transformed matrix Q'*A*Z. If JOBA = 'T', this matrix
is in the form
( A11 * * )
Q'*A*Z = ( * Ar X ) ,
( * 0 0 )
where A11 is a RANKE-by-RANKE matrix and Ar is a
RNKA22-by-RNKA22 invertible upper triangular matrix.
If JOBA = 'R' then A has the above form with X = 0.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,L).
E (input/output) COMPLEX*16 array, dimension (LDE,N)
On entry, the leading L-by-N part of this array must
contain the descriptor matrix E.
On exit, the leading L-by-N part of this array contains
the transformed matrix Q'*E*Z.
( Er 0 )
Q'*E*Z = ( ) ,
( 0 0 )
where Er is a RANKE-by-RANKE upper triangular invertible
matrix.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,L).
B (input/output) COMPLEX*16 array, dimension (LDB,M)
On entry, the leading L-by-M part of this array must
contain the input/state matrix B.
On exit, the leading L-by-M part of this array contains
the transformed matrix Q'*B.
LDB INTEGER
The leading dimension of array B.
LDB >= MAX(1,L) if M > 0 or LDB >= 1 if M = 0.
C (input/output) COMPLEX*16 array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed matrix C*Z.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
Q (input/output) COMPLEX*16 array, dimension (LDQ,L)
If COMPQ = 'N': Q is not referenced.
If COMPQ = 'I': on entry, Q need not be set;
on exit, the leading L-by-L part of this
array contains the unitary matrix Q,
where Q' is the product of Householder
transformations which are applied to A,
E, and B on the left.
If COMPQ = 'U': on entry, the leading L-by-L part of this
array must contain a unitary matrix Q1;
on exit, the leading L-by-L part of this
array contains the unitary matrix Q1*Q.
LDQ INTEGER
The leading dimension of array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1,L), if COMPQ = 'U' or 'I'.
Z (input/output) COMPLEX*16 array, dimension (LDZ,N)
If COMPZ = 'N': Z is not referenced.
If COMPZ = 'I': on entry, Z need not be set;
on exit, the leading N-by-N part of this
array contains the unitary matrix Z,
which is the product of Householder
transformations applied to A, E, and C
on the right.
If COMPZ = 'U': on entry, the leading N-by-N part of this
array must contain a unitary matrix Z1;
on exit, the leading N-by-N part of this
array contains the unitary matrix Z1*Z.
LDZ INTEGER
The leading dimension of array Z.
LDZ >= 1, if COMPZ = 'N';
LDZ >= MAX(1,N), if COMPZ = 'U' or 'I'.
RANKE (output) INTEGER
The estimated rank of matrix E, and thus also the order
of the invertible upper triangular submatrix Er.
RNKA22 (output) INTEGER
If JOBA = 'R' or 'T', then RNKA22 is the estimated rank of
matrix A22, and thus also the order of the invertible
upper triangular submatrix Ar.
If JOBA = 'N', then RNKA22 is not referenced.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in determining the rank of E
and of A22. If the user sets TOL > 0, then the given
value of TOL is used as a lower bound for the
reciprocal condition numbers of leading submatrices
of R or R22 in the QR decompositions E * P = Q * R of E
or A22 * P22 = Q22 * R22 of A22.
A submatrix whose estimated condition number is less than
1/TOL is considered to be of full rank. If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
defined by TOLDEF = L*N*EPS, is used instead, where
EPS is the machine precision (see LAPACK Library routine
DLAMCH). TOL < 1.
Workspace
IWORK INTEGER array, dimension (N)
DWORK DOUBLE PRECISION array, dimension (2*N)
ZWORK DOUBLE PRECISION array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) returns the optimal value
of LZWORK.
LZWORK INTEGER
The length of the array ZWORK.
LZWORK >= MAX( 1, N+P, MIN(L,N)+MAX(3*N-1,M,L) ).
For optimal performance, LZWORK should be larger.
If LZWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
ZWORK array, returns this value as the first entry of
the ZWORK array, and no error message related to LZWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine computes a truncated QR factorization with column
pivoting of E, in the form
( E11 E12 )
E * P = Q * ( )
( 0 E22 )
and finds the largest RANKE-by-RANKE leading submatrix E11 whose
estimated condition number is less than 1/TOL. RANKE defines thus
the rank of matrix E. Further E22, being negligible, is set to
zero, and a unitary matrix Y is determined such that
( E11 E12 ) = ( Er 0 ) * Y .
The overal transformation matrix Z results as Z = P * Y' and the
resulting transformed matrices Q'*A*Z and Q'*E*Z have the form
( Er 0 ) ( A11 A12 )
E <- Q'* E * Z = ( ) , A <- Q' * A * Z = ( ) ,
( 0 0 ) ( A21 A22 )
where Er is an upper triangular invertible matrix.
If JOBA = 'R' the same reduction is performed on A22 to obtain it
in the form
( Ar 0 )
A22 = ( ) ,
( 0 0 )
with Ar an upper triangular invertible matrix.
If JOBA = 'T' then A22 is row compressed using the QR
factorization with column pivoting to the form
( Ar X )
A22 = ( )
( 0 0 )
with Ar an upper triangular invertible matrix.
The transformations are also applied to the rest of system
matrices
B <- Q' * B, C <- C * Z.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( L*L*N ) floating point operations.Further Comments
NoneExample
Program Text
* TG01FZ EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER LMAX, NMAX, MMAX, PMAX
PARAMETER ( LMAX = 20, NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDE, LDQ, LDZ
PARAMETER ( LDA = LMAX, LDB = LMAX, LDC = PMAX,
$ LDE = LMAX, LDQ = LMAX, LDZ = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 2*NMAX )
INTEGER LZWORK
PARAMETER ( LZWORK = MAX( 1, NMAX+PMAX,
$ MIN(LMAX,NMAX)+MAX( 3*NMAX-1, MMAX, LMAX ) ) )
* .. Local Scalars ..
CHARACTER*1 COMPQ, COMPZ, JOBA
INTEGER I, INFO, J, L, M, N, P, RANKE, RNKA22
DOUBLE PRECISION TOL
* .. Local Arrays ..
INTEGER IWORK(NMAX)
COMPLEX*16 A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ E(LDE,NMAX), Q(LDQ,LMAX), Z(LDZ,NMAX),
$ ZWORK(LZWORK)
DOUBLE PRECISION DWORK(LDWORK)
* .. External Subroutines ..
EXTERNAL TG01FZ
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) L, N, M, P, TOL
COMPQ = 'I'
COMPZ = 'I'
JOBA = 'R'
IF ( L.LT.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) L
ELSE
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,L )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,L )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,L )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find the transformed descriptor system
* (A-lambda E,B,C).
CALL TG01FZ( COMPQ, COMPZ, JOBA, L, N, M, P, A, LDA,
$ E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ,
$ RANKE, RNKA22, TOL, IWORK, DWORK, ZWORK,
$ LZWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) RANKE, RNKA22
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 30 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
30 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 40 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
40 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 50 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,L )
50 CONTINUE
WRITE ( NOUT, FMT = 99990 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
60 CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01FZ EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01FZ = ',I2)
99997 FORMAT (/' The transformed state dynamics matrix Q''*A*Z is ')
99996 FORMAT (/' The transformed descriptor matrix Q''*E*Z is ')
99995 FORMAT (20(1X,F8.4,SP,F8.4,S,'i '))
99994 FORMAT (' Rank of matrix E =', I5/
$ ' Rank of matrix A22 =', I5)
99993 FORMAT (/' The transformed input/state matrix Q''*B is ')
99992 FORMAT (/' The transformed state/output matrix C*Z is ')
99991 FORMAT (/' The left transformation matrix Q is ')
99990 FORMAT (/' The right transformation matrix Z is ')
99989 FORMAT (/' L is out of range.',/' L = ',I5)
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
99986 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
TG01FZ EXAMPLE PROGRAM DATA
4 4 2 2 0.0
(-1,0) (0,0) (0,0) (3,0)
(0,0) (0,0) (1,0) (2,0)
(1,0) (1,0) (0,0) (4,0)
(0,0) (0,0) (0,0) (0,0)
(1,0) (2,0) (0,0) (0,0)
(0,0) (1,0) (0,0) (1,0)
(3,0) (9,0) (6,0) (3,0)
(0,0) (0,0) (2,0) (0,0)
(1,0) (0,0)
(0,0) (0,0)
(0,0) (1,0)
(1,0) (1,0)
(-1,0) (0,0) (1,0) (0,0)
(0,0) (1,0) (-1,0) (1,0)
Program Results
TG01FZ EXAMPLE PROGRAM RESULTS Rank of matrix E = 3 Rank of matrix A22 = 1 The transformed state dynamics matrix Q'*A*Z is 2.0278 +0.0000i 0.1078 +0.0000i 3.9062 +0.0000i -2.1571 +0.0000i -0.0980 +0.0000i 0.2544 +0.0000i 1.6053 +0.0000i -0.1269 +0.0000i 0.2713 +0.0000i 0.7760 +0.0000i -0.3692 +0.0000i -0.4853 +0.0000i 0.0690 +0.0000i -0.5669 +0.0000i -2.1974 +0.0000i 0.3086 +0.0000i The transformed descriptor matrix Q'*E*Z is 10.1587 +0.0000i 5.8230 +0.0000i 1.3021 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i -2.4684 +0.0000i -0.1896 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 1.0338 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i The transformed input/state matrix Q'*B is -0.2157 +0.0000i -0.9705 +0.0000i 0.3015 +0.0000i 0.9516 +0.0000i 0.7595 +0.0000i 0.0991 +0.0000i 1.1339 +0.0000i 0.3780 +0.0000i The transformed state/output matrix C*Z is 0.3651 +0.0000i -1.0000 +0.0000i -0.4472 +0.0000i -0.8165 +0.0000i -1.0954 +0.0000i 1.0000 +0.0000i -0.8944 +0.0000i 0.0000 +0.0000i The left transformation matrix Q is -0.2157 +0.0000i -0.5088 +0.0000i 0.6109 +0.0000i 0.5669 +0.0000i -0.1078 +0.0000i -0.2544 +0.0000i -0.7760 +0.0000i 0.5669 +0.0000i -0.9705 +0.0000i 0.1413 +0.0000i -0.0495 +0.0000i -0.1890 +0.0000i 0.0000 +0.0000i 0.8102 +0.0000i 0.1486 +0.0000i 0.5669 +0.0000i The right transformation matrix Z is -0.3651 +0.0000i 0.0000 +0.0000i 0.4472 +0.0000i 0.8165 +0.0000i -0.9129 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i -0.4082 +0.0000i 0.0000 +0.0000i -1.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i -0.1826 +0.0000i 0.0000 +0.0000i -0.8944 +0.0000i 0.4082 +0.0000i
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TG01GD.html 0000664 0000000 0000000 00000035060 14560147231 0020344 0 ustar 00root root 0000000 0000000
Purpose
To find a reduced descriptor representation (Ar-lambda*Er,Br,Cr) without non-dynamic modes for a descriptor representation (A-lambda*E,B,C). Optionally, the reduced descriptor system can be put into a standard form with the leading diagonal block of Er identity.Specification
SUBROUTINE TG01GD( JOBS, L, N, M, P, A, LDA, E, LDE, B, LDB,
$ C, LDC, D, LDD, LR, NR, RANKE, INFRED, TOL,
$ IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBS
INTEGER INFO, INFRED, L, LDA, LDB, LDC, LDD, LDE,
$ LDWORK, LR, M, N, NR, P, RANKE
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), E(LDE,*)
Arguments
Mode Parameters
JOBS CHARACTER*1
Indicates whether the user wishes to transform the leading
diagonal block of Er to an identity matrix, as follows:
= 'S': make Er with leading diagonal identity;
= 'D': keep Er unreduced or upper triangular.
Input/Output Parameters
L (input) INTEGER
The number of rows of the matrices A, E, and B;
also the number of differential equations. L >= 0.
N (input) INTEGER
The number of columns of the matrices A, E, and C;
also the dimension of descriptor state vector. N >= 0.
M (input) INTEGER
The number of columns of the matrix B;
also the dimension of the input vector. M >= 0.
P (input) INTEGER
The number of rows of the matrix C.
also the dimension of the output vector. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading L-by-N part of this array must
contain the state dynamics matrix A.
On exit, if NR < N, the leading LR-by-NR part of this
array contains the reduced order state matrix Ar of a
descriptor realization without non-dynamic modes.
Array A contains the original state dynamics matrix if
INFRED < 0.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,L).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading L-by-N part of this array must
contain the descriptor matrix E.
On exit, if INFRED >= 0, the leading LR-by-NR part of this
array contains the reduced order descriptor matrix Er of a
descriptor realization without non-dynamic modes.
In this case, only the leading RANKE-by-RANKE submatrix
of Er is nonzero and this submatrix is nonsingular and
upper triangular. Array E contains the original descriptor
matrix if INFRED < 0. If JOBS = 'S', then the leading
RANKE-by-RANKE submatrix results in an identity matrix.
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,L).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading L-by-M part of this array must
contain the input matrix B.
On exit, the leading LR-by-M part of this array contains
the reduced order input matrix Br of a descriptor
realization without non-dynamic modes. Array B contains
the original input matrix if INFRED < 0.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,L).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the output matrix C.
On exit, the leading P-by-NR part of this array contains
the reduced order output matrix Cr of a descriptor
realization without non-dynamic modes. Array C contains
the original output matrix if INFRED < 0.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the original feedthrough matrix D.
On exit, the leading P-by-M part of this array contains
the feedthrough matrix Dr of a reduced descriptor
realization without non-dynamic modes.
LDD INTEGER
The leading dimension of the array D. LDD >= MAX(1,P).
LR (output) INTEGER
The number of reduced differential equations.
NR (output) INTEGER
The dimension of the reduced descriptor state vector.
RANKE (output) INTEGER
The estimated rank of the matrix E.
INFRED (output) INTEGER
This parameter contains information on performed reduction
and on structure of resulting system matrices, as follows:
INFRED >= 0 the reduced system is in an SVD-like
coordinate form with Er upper triangular;
INFRED is the achieved order reduction.
INFRED < 0 no reduction achieved and the original
system has been restored.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in rank determinations when
transforming (A-lambda*E). If the user sets TOL > 0,
then the given value of TOL is used as a lower bound for
reciprocal condition numbers in rank determinations; a
(sub)matrix whose estimated condition number is less than
1/TOL is considered to be of full rank. If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
defined by TOLDEF = L*N*EPS, is used instead, where EPS
is the machine precision (see LAPACK Library routine
DLAMCH). TOL < 1.
Workspace
IWORK INTEGER array, dimension (N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 1, if MIN(L,N) = 0; otherwise,
LDWORK >= MAX( N+P, MIN(L,N)+MAX(3*N-1,M,L) ).
If LDWORK >= 2*L*N+L*M+N*P+
MAX( 1, N+P, MIN(L,N)+MAX(3*N-1,M,L) ) then
the original matrices are restored if no order reduction
is possible. This is achieved by saving system matrices
before reduction and restoring them if no order reduction
took place.
If LDWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message related to LDWORK is issued by
XERBLA. The optimal size does not necessarily include the
space needed for saving the original system matrices.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The subroutine elliminates the non-dynamics modes in two steps:
Step 1: Reduce the system to the SVD-like coordinate form
(Q'*A*Z-lambda*Q'*E*Z, Q'*B, C*Z) , where
( A11 A12 A13 ) ( E11 0 0 ) ( B1 )
Q'*A*Z = ( A21 A22 0 ), Q'*E*Z = ( 0 0 0 ), Q'*B = ( B2 ),
( A31 0 0 ) ( 0 0 0 ) ( B3 )
C*Z = ( C1 C2 C3 ),
where E11 and A22 are upper triangular invertible matrices.
Step 2: Compute the reduced system as (Ar-lambda*Er,Br,Cr,Dr),
where
( A11 - A12*inv(A22)*A21, A13 ) ( E11 0 )
Ar = ( ), Er = ( ),
( A31 0 ) ( 0 0 )
( B1 - A12*inv(A22)*B2 )
Br = ( ), Cr = ( C1 - C2*inv(A22)*A21, C3 ),
( B3 )
Dr = D - C2*inv(A22)*B2.
Step 3: If desired (JOBS = 'S'), reduce the descriptor system to
the standard form
Ar <- diag(inv(E11),I)*Ar; Br <- diag(inv(E11),I)*Br;
Er = diag(I,0).
If L = N and LR = NR = RANKE, then if Step 3 is performed,
the resulting system is a standard state space system.
Numerical Aspects
If L = N, the algorithm requires 0( N**3 ) floating point operations.Further Comments
NoneExample
Program Text
* TG01GD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER LMAX, NMAX, MMAX, PMAX
PARAMETER ( LMAX = 20, NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDD, LDE
PARAMETER ( LDA = LMAX, LDB = LMAX, LDC = PMAX,
$ LDD = PMAX, LDE = LMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MIN( LMAX, NMAX ) +
$ MAX( 3*NMAX - 1, MMAX, LMAX ) +
$ 2*LMAX*NMAX + LMAX*MMAX + PMAX*NMAX )
* .. Local Scalars ..
CHARACTER*1 JOBS
INTEGER I, INFO, INFRED, J, L, LR, M, N, NR, P, RANKE
DOUBLE PRECISION TOL
* .. Local Arrays ..
INTEGER IWORK(NMAX)
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK), E(LDE,NMAX)
* .. External Subroutines ..
EXTERNAL TG01GD
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) L, N, M, P, JOBS, TOL
IF ( L.LT.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) L
ELSE
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,L )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,L )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,L )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
* Find the reduced descriptor system
* (A-lambda E,B,C,D).
CALL TG01GD( JOBS, L, N, M, P, A, LDA, E, LDE, B, LDB,
$ C, LDC, D, LDD, LR, NR, RANKE, INFRED,
$ TOL, IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) RANKE
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, LR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, LR
WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,NR )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 30 I = 1, LR
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
30 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 40 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
40 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 50 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1,M )
50 CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01GD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01GD = ',I2)
99997 FORMAT (/' The reduced state dynamics matrix is ')
99996 FORMAT (/' The reduced descriptor matrix is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (' Rank of matrix E =', I5)
99993 FORMAT (/' The reduced input/state matrix is ')
99992 FORMAT (/' The reduced state/output matrix is ')
99991 FORMAT (/' The transformed feedthrough matrix is ')
99989 FORMAT (/' L is out of range.',/' L = ',I5)
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
99986 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
TG01GD EXAMPLE PROGRAM DATA
4 4 2 2 D 0.0
-1 0 0 3
0 0 1 2
1 1 0 4
0 0 0 0
1 2 0 0
0 1 0 1
3 9 6 3
0 0 2 0
1 0
0 0
0 1
1 1
-1 0 1 0
0 1 -1 1
1 0
1 1
Program Results
TG01GD EXAMPLE PROGRAM RESULTS Rank of matrix E = 3 The reduced state dynamics matrix is 2.5102 -3.8550 -11.4533 -0.0697 0.0212 0.7015 0.3798 -0.1156 -3.8250 The reduced descriptor matrix is 10.1587 5.8230 1.3021 0.0000 -2.4684 -0.1896 0.0000 0.0000 1.0338 The reduced input/state matrix is 7.7100 1.6714 0.7678 1.1070 2.5428 0.6935 The reduced state/output matrix is 0.5477 -2.5000 -6.2610 -1.0954 1.0000 -0.8944 The transformed feedthrough matrix is 4.0000 1.0000 1.0000 1.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TG01HD.html 0000664 0000000 0000000 00000056245 14560147231 0020355 0 ustar 00root root 0000000 0000000
Purpose
To compute orthogonal transformation matrices Q and Z which
reduce the N-th order descriptor system (A-lambda*E,B,C)
to the form
( Ac * ) ( Ec * ) ( Bc )
Q'*A*Z = ( ) , Q'*E*Z = ( ) , Q'*B = ( ) ,
( 0 Anc ) ( 0 Enc ) ( 0 )
C*Z = ( Cc Cnc ) ,
where the NCONT-th order descriptor system (Ac-lambda*Ec,Bc,Cc)
is a finite and/or infinite controllable. The pencil
Anc - lambda*Enc is regular of order N-NCONT and contains the
uncontrollable finite and/or infinite eigenvalues of the pencil
A-lambda*E.
For JOBCON = 'C' or 'I', the pencil ( Bc Ec-lambda*Ac ) has full
row rank NCONT for all finite lambda and is in a staircase form
with
_ _ _ _
( E1,0 E1,1 ... E1,k-1 E1,k )
( _ _ _ )
( Bc Ec ) = ( 0 E2,1 ... E2,k-1 E2,k ) , (1)
( ... _ _ )
( 0 0 ... Ek,k-1 Ek,k )
_ _ _
( A1,1 ... A1,k-1 A1,k )
( _ _ )
Ac = ( 0 ... A2,k-1 A2,k ) , (2)
( ... _ )
( 0 ... 0 Ak,k )
_
where Ei,i-1 is an rtau(i)-by-rtau(i-1) full row rank matrix
_
(with rtau(0) = M) and Ai,i is an rtau(i)-by-rtau(i)
upper triangular matrix.
For JOBCON = 'F', the pencil ( Bc Ac-lambda*Ec ) has full
row rank NCONT for all finite lambda and is in a staircase form
with
_ _ _ _
( A1,0 A1,1 ... A1,k-1 A1,k )
( _ _ _ )
( Bc Ac ) = ( 0 A2,1 ... A2,k-1 A2,k ) , (3)
( ... _ _ )
( 0 0 ... Ak,k-1 Ak,k )
_ _ _
( E1,1 ... E1,k-1 E1,k )
( _ _ )
Ec = ( 0 ... E2,k-1 E2,k ) , (4)
( ... _ )
( 0 ... 0 Ek,k )
_
where Ai,i-1 is an rtau(i)-by-rtau(i-1) full row rank matrix
_
(with rtau(0) = M) and Ei,i is an rtau(i)-by-rtau(i)
upper triangular matrix.
For JOBCON = 'C', the (N-NCONT)-by-(N-NCONT) regular pencil
Anc - lambda*Enc has the form
( Ainc - lambda*Einc * )
Anc - lambda*Enc = ( ) ,
( 0 Afnc - lambda*Efnc )
where:
1) the NIUCON-by-NIUCON regular pencil Ainc - lambda*Einc,
with Ainc upper triangular and nonsingular, contains the
uncontrollable infinite eigenvalues of A - lambda*E;
2) the (N-NCONT-NIUCON)-by-(N-NCONT-NIUCON) regular pencil
Afnc - lambda*Efnc, with Efnc upper triangular and
nonsingular, contains the uncontrollable finite
eigenvalues of A - lambda*E.
Note: The significance of the two diagonal blocks can be
interchanged by calling the routine with the
arguments A and E interchanged. In this case,
Ainc - lambda*Einc contains the uncontrollable zero
eigenvalues of A - lambda*E, while Afnc - lambda*Efnc
contains the uncontrollable nonzero finite and infinite
eigenvalues of A - lambda*E.
For JOBCON = 'F', the pencil Anc - lambda*Enc has the form
Anc - lambda*Enc = Afnc - lambda*Efnc ,
where the regular pencil Afnc - lambda*Efnc, with Efnc
upper triangular and nonsingular, contains the uncontrollable
finite eigenvalues of A - lambda*E.
For JOBCON = 'I', the pencil Anc - lambda*Enc has the form
Anc - lambda*Enc = Ainc - lambda*Einc ,
where the regular pencil Ainc - lambda*Einc, with Ainc
upper triangular and nonsingular, contains the uncontrollable
nonzero finite and infinite eigenvalues of A - lambda*E.
The left and/or right orthogonal transformations Q and Z
performed to reduce the system matrices can be optionally
accumulated.
The reduced order descriptor system (Ac-lambda*Ec,Bc,Cc) has
the same transfer-function matrix as the original system
(A-lambda*E,B,C).
Specification
SUBROUTINE TG01HD( JOBCON, COMPQ, COMPZ, N, M, P, A, LDA, E, LDE,
$ B, LDB, C, LDC, Q, LDQ, Z, LDZ, NCONT, NIUCON,
$ NRBLCK, RTAU, TOL, IWORK, DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, COMPZ, JOBCON
INTEGER INFO, LDA, LDB, LDC, LDE, LDQ, LDZ,
$ M, N, NCONT, NIUCON, NRBLCK, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK( * ), RTAU( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ DWORK( * ), E( LDE, * ), Q( LDQ, * ),
$ Z( LDZ, * )
Arguments
Mode Parameters
JOBCON CHARACTER*1
= 'C': separate both finite and infinite uncontrollable
eigenvalues;
= 'F': separate only finite uncontrollable eigenvalues:
= 'I': separate only nonzero finite and infinite
uncontrollable eigenvalues.
COMPQ CHARACTER*1
= 'N': do not compute Q;
= 'I': Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= 'U': Q must contain an orthogonal matrix Q1 on entry,
and the product Q1*Q is returned.
COMPZ CHARACTER*1
= 'N': do not compute Z;
= 'I': Z is initialized to the unit matrix, and the
orthogonal matrix Z is returned;
= 'U': Z must contain an orthogonal matrix Z1 on entry,
and the product Z1*Z is returned.
Input/Output Parameters
N (input) INTEGER
The dimension of the descriptor state vector; also the
order of square matrices A and E, the number of rows of
matrix B, and the number of columns of matrix C. N >= 0.
M (input) INTEGER
The dimension of descriptor system input vector; also the
number of columns of matrix B. M >= 0.
P (input) INTEGER
The dimension of descriptor system output vector; also the
number of rows of matrix C. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the N-by-N state matrix A.
On exit, the leading N-by-N part of this array contains
the transformed state matrix Q'*A*Z,
( Ac * )
Q'*A*Z = ( ) ,
( 0 Anc )
where Ac is NCONT-by-NCONT and Anc is
(N-NCONT)-by-(N-NCONT).
If JOBCON = 'F', the matrix ( Bc Ac ) is in the
controllability staircase form (3).
If JOBCON = 'C' or 'I', the submatrix Ac is upper
triangular.
If JOBCON = 'C', the Anc matrix has the form
( Ainc * )
Anc = ( ) ,
( 0 Afnc )
where the NIUCON-by-NIUCON matrix Ainc is nonsingular and
upper triangular.
If JOBCON = 'I', Anc is nonsingular and upper triangular.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain the N-by-N descriptor matrix E.
On exit, the leading N-by-N part of this array contains
the transformed descriptor matrix Q'*E*Z,
( Ec * )
Q'*E*Z = ( ) ,
( 0 Enc )
where Ec is NCONT-by-NCONT and Enc is
(N-NCONT)-by-(N-NCONT).
If JOBCON = 'C' or 'I', the matrix ( Bc Ec ) is in the
controllability staircase form (1).
If JOBCON = 'F', the submatrix Ec is upper triangular.
If JOBCON = 'C', the Enc matrix has the form
( Einc * )
Enc = ( ) ,
( 0 Efnc )
where the NIUCON-by-NIUCON matrix Einc is nilpotent
and the (N-NCONT-NIUCON)-by-(N-NCONT-NIUCON) matrix Efnc
is nonsingular and upper triangular.
If JOBCON = 'F', Enc is nonsingular and upper triangular.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the N-by-M input matrix B.
On exit, the leading N-by-M part of this array contains
the transformed input matrix
( Bc )
Q'*B = ( ) ,
( 0 )
where Bc is NCONT-by-M.
For JOBCON = 'C' or 'I', the matrix ( Bc Ec ) is in the
controllability staircase form (1).
For JOBCON = 'F', the matrix ( Bc Ac ) is in the
controllability staircase form (3).
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed matrix C*Z.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
If COMPQ = 'N': Q is not referenced.
If COMPQ = 'I': on entry, Q need not be set;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix Q,
where Q' is the product of transformations
which are applied to A, E, and B on
the left.
If COMPQ = 'U': on entry, the leading N-by-N part of this
array must contain an orthogonal matrix
Qc;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix
Qc*Q.
LDQ INTEGER
The leading dimension of array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1,N), if COMPQ = 'U' or 'I'.
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
If COMPZ = 'N': Z is not referenced.
If COMPZ = 'I': on entry, Z need not be set;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix Z,
which is the product of transformations
applied to A, E, and C on the right.
If COMPZ = 'U': on entry, the leading N-by-N part of this
array must contain an orthogonal matrix
Zc;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix
Zc*Z.
LDZ INTEGER
The leading dimension of array Z.
LDZ >= 1, if COMPZ = 'N';
LDZ >= MAX(1,N), if COMPZ = 'U' or 'I'.
NCONT (output) INTEGER
The order of the reduced matrices Ac and Ec, and the
number of rows of reduced matrix Bc; also the order of
the controllable part of the pair (A-lambda*E,B).
NIUCON (output) INTEGER
For JOBCON = 'C', the order of the reduced matrices
Ainc and Einc; also the number of uncontrollable
infinite eigenvalues of the pencil A - lambda*E.
For JOBCON = 'F' or 'I', NIUCON has no significance
and is set to zero.
NRBLCK (output) INTEGER
For JOBCON = 'C' or 'I', the number k, of full row rank
_
blocks Ei,i in the staircase form of the pencil
(Bc Ec-lambda*Ac) (see (1) and (2)).
For JOBCON = 'F', the number k, of full row rank blocks
_
Ai,i in the staircase form of the pencil (Bc Ac-lambda*Ec)
(see (3) and (4)).
RTAU (output) INTEGER array, dimension (N)
RTAU(i), for i = 1, ..., NRBLCK, is the row dimension of
_ _
the full row rank block Ei,i-1 or Ai,i-1 in the staircase
form (1) or (3) for JOBCON = 'C' or 'I', or
for JOBCON = 'F', respectively.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in rank determinations when
transforming (A-lambda*E, B). If the user sets TOL > 0,
then the given value of TOL is used as a lower bound for
reciprocal condition numbers in rank determinations; a
(sub)matrix whose estimated condition number is less than
1/TOL is considered to be of full rank. If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
defined by TOLDEF = N*N*EPS, is used instead, where EPS
is the machine precision (see LAPACK Library routine
DLAMCH). TOL < 1.
Workspace
IWORK INTEGER array, dimension (M) DWORK DOUBLE PRECISION array, dimension (MAX(N,2*M))Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The subroutine is based on the reduction algorithms of [1].References
[1] A. Varga
Computation of Irreducible Generalized State-Space
Realizations.
Kybernetika, vol. 26, pp. 89-106, 1990.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( N**3 ) floating point operations.Further Comments
If the system matrices A, E and B are badly scaled, it is generally recommendable to scale them with the SLICOT routine TG01AD, before calling TG01HD.Example
Program Text
* TG01HD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER LMAX, NMAX, MMAX, PMAX
PARAMETER ( LMAX = 20, NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDE, LDQ, LDZ
PARAMETER ( LDA = LMAX, LDB = LMAX, LDC = PMAX,
$ LDE = LMAX, LDQ = LMAX, LDZ = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( 1, NMAX, 2*MMAX ) )
* .. Local Scalars ..
CHARACTER*1 COMPQ, COMPZ, JOBCO
INTEGER I, INFO, J, M, N, NCONT, NIUCON, NRBLCK, P
DOUBLE PRECISION TOL
* .. Local Arrays ..
INTEGER IWORK(MMAX), RTAU(NMAX)
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,LMAX),
$ Z(LDZ,NMAX)
* .. External Subroutines ..
EXTERNAL TG01HD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL, JOBCO
COMPQ = 'I'
COMPZ = 'I'
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find the transformed descriptor system (A-lambda E,B,C).
CALL TG01HD( JOBCO, COMPQ, COMPZ, N, M, P, A, LDA,
$ E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ,
$ NCONT, NIUCON, NRBLCK, RTAU, TOL, IWORK,
$ DWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) NCONT, NIUCON
WRITE ( NOUT, FMT = 99985 )
WRITE ( NOUT, FMT = 99984 ) ( RTAU(I), I = 1,NRBLCK )
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 30 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
30 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 40 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
40 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 50 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,N )
50 CONTINUE
WRITE ( NOUT, FMT = 99990 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
60 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01HD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01HD = ',I2)
99997 FORMAT (/' The transformed state dynamics matrix Q''*A*Z is ')
99996 FORMAT (/' The transformed descriptor matrix Q''*E*Z is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (' Dimension of controllable part =', I5/
$ ' Number of uncontrollable infinite eigenvalues =', I5)
99993 FORMAT (/' The transformed input/state matrix Q''*B is ')
99992 FORMAT (/' The transformed state/output matrix C*Z is ')
99991 FORMAT (/' The left transformation matrix Q is ')
99990 FORMAT (/' The right transformation matrix Z is ')
99989 FORMAT (/' L is out of range.',/' L = ',I5)
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
99986 FORMAT (/' P is out of range.',/' P = ',I5)
99985 FORMAT (/' The staircase form row dimensions are ' )
99984 FORMAT (10I5)
END
Program Data
TG01HD EXAMPLE PROGRAM DATA
7 3 2 0.0 C
2 0 2 0 -1 3 1
0 1 0 0 1 0 0
0 0 0 1 0 0 1
0 0 2 0 -1 3 1
0 0 0 1 0 0 1
0 1 0 0 1 0 0
0 0 0 1 0 0 1
0 0 1 0 0 0 0
0 0 0 0 0 1 0
0 0 0 0 0 0 1
0 0 0 0 0 0 1
0 0 0 1 0 0 0
0 0 1 0 -1 0 0
1 3 0 2 0 0 0
2 1 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
1 2 3
1 0 0 1 0 0 1
0 -1 1 0 -1 1 0
Program Results
TG01HD EXAMPLE PROGRAM RESULTS
Dimension of controllable part = 3
Number of uncontrollable infinite eigenvalues = 1
The staircase form row dimensions are
2 1
The transformed state dynamics matrix Q'*A*Z is
0.0000 0.0000 0.0000 0.0000 -1.2627 0.4334 0.4666
0.0000 2.0000 0.0000 -3.7417 -0.8520 0.2924 -0.4342
0.0000 0.0000 1.7862 0.3780 -0.2651 -0.7723 0.0000
0.0000 0.0000 0.0000 3.7417 0.8520 -0.2924 0.4342
0.0000 0.0000 0.0000 0.0000 -1.5540 0.5334 0.5742
0.0000 0.0000 0.0000 0.0000 -0.6533 0.2242 0.2414
0.0000 0.0000 0.0000 0.0000 -0.5892 0.2022 0.2177
The transformed descriptor matrix Q'*E*Z is
-1.8325 1.0000 2.3752 0.0000 -0.8214 0.2819 1.8016
0.4887 0.0000 0.3770 -0.5345 0.1874 0.5461 0.0000
-0.1728 0.0000 -0.1333 -1.1339 0.1325 0.3861 0.0000
0.0000 0.0000 0.0000 0.0000 0.8520 -0.2924 0.4342
0.0000 0.0000 0.0000 0.0000 -1.0260 -0.1496 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 1.1937 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000
The transformed input/state matrix Q'*B is
1.0000 2.0000 3.0000
2.0000 1.0000 0.0000
0.0000 0.0000 0.0000
0.0000 0.0000 0.0000
0.0000 0.0000 0.0000
0.0000 0.0000 0.0000
0.0000 0.0000 0.0000
The transformed state/output matrix C*Z is
0.0000 1.0000 0.0000 0.0000 -1.2627 0.4334 0.4666
0.3665 0.0000 -0.9803 -1.6036 0.1874 0.5461 0.0000
The left transformation matrix Q is
0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.7071 0.0000 0.2740 -0.6519 0.0000
0.0000 0.0000 0.0000 0.0000 0.8304 0.3491 -0.4342
0.0000 0.0000 0.0000 -1.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.4003 0.1683 0.9008
0.0000 0.0000 0.7071 0.0000 -0.2740 0.6519 0.0000
1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
The right transformation matrix Z is
0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000
-0.6108 0.0000 0.7917 0.0000 0.0000 0.0000 0.0000
0.4887 0.0000 0.3770 -0.5345 0.1874 0.5461 0.0000
0.0000 0.0000 0.0000 0.0000 -0.4107 0.1410 0.9008
0.6108 0.0000 0.4713 0.2673 -0.1874 -0.5461 0.0000
-0.1222 0.0000 -0.0943 -0.8018 -0.1874 -0.5461 0.0000
0.0000 0.0000 0.0000 0.0000 -0.8520 0.2924 -0.4342
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TG01HU.html 0000664 0000000 0000000 00000033160 14560147231 0020365 0 ustar 00root root 0000000 0000000
Purpose
Given the descriptor system (A-lambda*E,B,C) with the system
matrices A, E and B of the form
( A1 X1 ) ( E1 Y1 ) ( B1 B2 )
A = ( ) , E = ( ) , B = ( ) ,
( 0 X2 ) ( 0 Y2 ) ( 0 0 )
where
- B is an L-by-(M1+M2) matrix,
with B1 an N1-by-M1 submatrix, B2 an N1-by-M2 submatrix,
- A is an L-by-N matrix, with A1 an N1-by-N1 submatrix,
- E is an L-by-N matrix, with E1 an N1-by-N1 submatrix
with LBE nonzero sub-diagonals,
this routine reduces the pair (A1-lambda*E1,[B1 B2]) to the form
Qc'*[B1 B2 A1-lambda*E1 ]*diag(I,Zc) =
( Bc1 Bc2 Ac-lambda*Ec * )
( ) ,
( 0 0 0 Anc-lambda*Enc )
where:
1) the pencil ( Bc1 Bc2 Ac-lambda*Ec ) has full row rank NR for
all finite lambda and is in a staircase form with
[ A11 A12 . . . A1,p-2 A1,p-1 A1p ]
[ A21 A22 . . . A2,p-2 A2,p-1 A2p ]
[ A31 A32 . . . A3,p-2 A3,p-1 A3p ]
[ 0 A42 . . . A4,p-2 A4,p-1 A4p ]
Ac = [ . . . . . . . . ], (1)
[ . . . . . . . ]
[ . . . . . . ]
[ 0 0 . . . Ap,p-2 Ap,p-1 App ]
[ A1,-1 A1,0 ]
[ 0 A2,0 ]
[ 0 0 ] ( E11 E12 ... E1p )
[ 0 0 ] ( 0 E22 ... E2p )
[Bc1 Bc2] = [ . . ], Ec = ( . . . . ),
[ . . ] ( . . . . )
[ . . ] ( 0 0 ... Epp )
[ 0 0 ]
where the block Ai,i-2 is an rtau(i)-by-rtau(i-2) full row
rank matrix (with rtau(-1) = M1, rtau(0) = M2) and Ei,i is an
rtau(i)-by-rtau(i) upper triangular matrix.
2) the pencil Anc-lambda*Enc is regular of order N1-NR with Enc
upper triangular; this pencil contains the uncontrollable
finite eigenvalues of the pencil (A1-lambda*E1).
The transformations are applied to the whole matrices A, E, B
and C. The left and/or right orthogonal transformations Qc and Zc,
performed to reduce the pencil, can be optionally accumulated in
the matrices Q and Z, respectively.
The reduced order descriptor system (Ac-lambda*Ec,Bc,Cc) has no
uncontrollable finite eigenvalues and has the same transfer-
function matrix as the original system (A-lambda*E,B,C).
Specification
SUBROUTINE TG01HU( COMPQ, COMPZ, L, N, M1, M2, P, N1, LBE, A, LDA,
$ E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ, NR,
$ NRBLCK, RTAU, TOL, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, COMPZ
INTEGER INFO, L, LBE, LDA, LDB, LDC, LDE, LDQ, LDWORK,
$ LDZ, M1, M2, N, N1, NR, NRBLCK, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK( * ), RTAU( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ DWORK( * ), E( LDE, * ), Q( LDQ, * ),
$ Z( LDZ, * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
= 'N': do not compute Q;
= 'I': Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= 'U': Q must contain an orthogonal matrix Q1 on entry,
and the product Q1*Q is returned.
COMPZ CHARACTER*1
= 'N': do not compute Z;
= 'I': Z is initialized to the unit matrix, and the
orthogonal matrix Z is returned;
= 'U': Z must contain an orthogonal matrix Z1 on entry,
and the product Z1*Z is returned.
Input/Output Parameters
L (input) INTEGER
The number of descriptor state equations; also the number
of rows of the matrices A, E and B. L >= 0.
N (input) INTEGER
The dimension of the descriptor state vector; also the
number of columns of the matrices A, E and C. N >= 0.
M1 (input) INTEGER
The number of system inputs in U1, or of columns of B1.
M1 >= 0.
M2 (input) INTEGER
The number of system inputs in U2, or of columns of B2.
M2 >= 0.
P (input) INTEGER
The dimension of descriptor system output; also the
number of rows of the matrix C. P >= 0.
N1 (input) INTEGER
The order of the subsystem (A1-lambda*E1,B1,C1) to be
reduced. MIN(L,N) >= N1 >= 0.
LBE (input) INTEGER
The number of nonzero sub-diagonals of the submatrix E1.
MAX(0,N1-1) >= LBE >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading L-by-N part of this array must
contain the L-by-N state matrix A in the partitioned form
( A1 X1 )
A = ( ) ,
( 0 X2 )
where A1 is N1-by-N1.
On exit, the leading L-by-N part of this array contains
the transformed state matrix,
( Ac * * )
Qc'*A*diag(Zc,I) = ( 0 Anc * ) ,
( 0 0 * )
where Ac is NR-by-NR and Anc is (N1-NR)-by-(N1-NR).
The matrix ( Bc Ac ) is in the controllability staircase
form (1).
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,L).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading L-by-N part of this array must
contain the L-by-N descriptor matrix E in the partitioned
form
( E1 Y1 )
E = ( ) ,
( 0 Y2 )
where E1 is an N1-by-N1 matrix with LBE nonzero
sub-diagonals.
On exit, the leading L-by-N part of this array contains
the transformed descriptor matrix
( Ec * * )
Qc'*E*diag(Zc,I) = ( 0 Enc * ) ,
( 0 0 * )
where Ec is NR-by-NR and Enc is (N1-NR)-by-(N1-NR).
Both Ec and Enc are upper triangular and Enc is
nonsingular.
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,L).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
with M = M1 + M2.
On entry, the leading L-by-M part of this array must
contain the L-by-M input matrix B in the partitioned form
( Bi )
B = ( ) ,
( 0 )
where Bi is N1-by-M.
On exit, the leading L-by-M part of this array contains
the transformed input matrix
( Bc )
Qc'*B = ( ) ,
( 0 )
where Bc is NR-by-M.
The matrix ( Bc Ac ) is in the controllability staircase
form (1).
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,L).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed matrix C*Zc.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,P).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,L)
If COMPQ = 'N': Q is not referenced.
If COMPQ = 'I': on entry, Q need not be set;
on exit, the leading L-by-L part of this
array contains the orthogonal matrix Qc,
where Qc' is the product of the
transformations applied to A, E, and B on
the left.
If COMPQ = 'U': on entry, the leading L-by-L part of this
array must contain an orthogonal matrix Q;
on exit, the leading L-by-L part of this
array contains the orthogonal matrix
Q*Qc.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1,L), if COMPQ = 'I' or 'U'.
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
If COMPZ = 'N': Z is not referenced.
If COMPZ = 'I': on entry, Z need not be set;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix Zc,
i.e., the product of the transformations
applied to A, E, and C on the right.
If COMPZ = 'U': on entry, the leading N-by-N part of this
array must contain an orthogonal matrix Z;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix
Z*Zc.
LDZ INTEGER
The leading dimension of the array Z.
LDZ >= 1, if COMPZ = 'N';
LDZ >= MAX(1,N), if COMPZ = 'I' or 'U'.
NR (output) INTEGER
The order of the reduced matrices Ac and Ec, and the
number of rows of the reduced matrix Bc; also the order of
the controllable part of the pair (B, A-lambda*E).
NRBLCK (output) INTEGER
The number p, of full row rank blocks Ai,i-2 in the
staircase form of the pencil (Bc1 Bc2 Ac-lambda*Ec).
RTAU (output) INTEGER array, dimension (2*N1)
The leading NRBLCK elements of this array contain the
orders of the diagonal blocks of Ac. NRBLCK is always
an even number, and the NRBLCK/2 odd and even components
of RTAU have decreasing values, respectively.
Note that some elements of RTAU can be zero.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in rank determinations when
transforming (A-lambda*E, B). If the user sets TOL > 0,
then the given value of TOL is used as a lower bound for
reciprocal condition numbers in rank determinations; a
(sub)matrix whose estimated condition number is less than
1/TOL is considered to be of full rank. If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
defined by TOLDEF = L*N*EPS, is used instead, where
EPS is the machine precision (see LAPACK Library routine
DLAMCH). TOL < 1.
Workspace
IWORK INTEGER array, dimension (M)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 1, if MIN(N1,M) = 0; otherwise,
LDWORK >= MAX(N1+MAX(L,N,M),2*M), if LBE > 0 and N1 > 2;
LDWORK >= MAX(1,L,N,2*M), if LBE = 0 or N1 <= 2.
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The implemented algorithm [1] represents a specialization of the controllability staircase algorithm of [2] to the special structure of the input matrix B = [B1,B2].References
[1] Varga, A.
Reliable algorithms for computing minimal dynamic covers for
descriptor systems.
Proc. of MTNS'04, Leuven, Belgium, 2004.
[2] Varga, A.
Computation of Irreducible Generalized State-Space
Realizations.
Kybernetika, vol. 26, pp. 89-106, 1990.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( N*N1**2 ) floating point operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
Given the descriptor system (A-lambda*E,B,C) with the system
matrices A, E and B of the form
( A1 X1 ) ( E1 Y1 ) ( B1 )
A = ( ) , E = ( ) , B = ( ) ,
( 0 X2 ) ( 0 Y2 ) ( 0 )
where
- B is an L-by-M matrix, with B1 an N1-by-M submatrix
- A is an L-by-N matrix, with A1 an N1-by-N1 submatrix
- E is an L-by-N matrix, with E1 an N1-by-N1 submatrix
with LBE nonzero sub-diagonals,
this routine reduces the pair (A1-lambda*E1,B1) to the form
Qc'*[B1 A1-lambda*E1]*diag(I,Zc) =
( Bc Ac-lambda*Ec * )
( ) ,
( 0 0 Anc-lambda*Enc )
where:
1) the pencil ( Bc Ac-lambda*Ec ) has full row rank NR for
all finite lambda and is in a staircase form with
_ _ _ _
( A1,0 A1,1 ... A1,k-1 A1,k )
( _ _ _ )
( Bc Ac ) = ( 0 A2,1 ... A2,k-1 A2,k ) , (1)
( ... _ _ )
( 0 0 ... Ak,k-1 Ak,k )
_ _ _
( E1,1 ... E1,k-1 E1,k )
( _ _ )
Ec = ( 0 ... E2,k-1 E2,k ) , (2)
( ... _ )
( 0 ... 0 Ek,k )
_
where Ai,i-1 is an rtau(i)-by-rtau(i-1) full row rank
_
matrix (with rtau(0) = M) and Ei,i is an rtau(i)-by-rtau(i)
upper triangular matrix.
2) the pencil Anc-lambda*Enc is regular of order N1-NR with Enc
upper triangular; this pencil contains the uncontrollable
finite eigenvalues of the pencil (A1-lambda*E1).
The transformations are applied to the whole matrices A, E, B
and C. The left and/or right orthogonal transformations Qc and Zc
performed to reduce the pencil can be optionally accumulated
in the matrices Q and Z, respectively.
The reduced order descriptor system (Ac-lambda*Ec,Bc,Cc) has no
uncontrollable finite eigenvalues and has the same
transfer-function matrix as the original system (A-lambda*E,B,C).
Specification
SUBROUTINE TG01HX( COMPQ, COMPZ, L, N, M, P, N1, LBE, A, LDA,
$ E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ, NR,
$ NRBLCK, RTAU, TOL, IWORK, DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, COMPZ
INTEGER INFO, L, LBE, LDA, LDB, LDC, LDE, LDQ, LDZ, M,
$ N, N1, NR, NRBLCK, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK( * ), RTAU( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ DWORK( * ), E( LDE, * ), Q( LDQ, * ),
$ Z( LDZ, * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
= 'N': do not compute Q;
= 'I': Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= 'U': Q must contain an orthogonal matrix Q1 on entry,
and the product Q1*Q is returned.
COMPZ CHARACTER*1
= 'N': do not compute Z;
= 'I': Z is initialized to the unit matrix, and the
orthogonal matrix Z is returned;
= 'U': Z must contain an orthogonal matrix Z1 on entry,
and the product Z1*Z is returned.
Input/Output Parameters
L (input) INTEGER
The number of descriptor state equations; also the number
of rows of matrices A, E and B. L >= 0.
N (input) INTEGER
The dimension of the descriptor state vector; also the
number of columns of matrices A, E and C. N >= 0.
M (input) INTEGER
The dimension of descriptor system input vector; also the
number of columns of matrix B. M >= 0.
P (input) INTEGER
The dimension of descriptor system output; also the
number of rows of matrix C. P >= 0.
N1 (input) INTEGER
The order of subsystem (A1-lambda*E1,B1,C1) to be reduced.
MIN(L,N) >= N1 >= 0.
LBE (input) INTEGER
The number of nonzero sub-diagonals of submatrix E1.
MAX(0,N1-1) >= LBE >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading L-by-N part of this array must
contain the L-by-N state matrix A in the partitioned
form
( A1 X1 )
A = ( ) ,
( 0 X2 )
where A1 is N1-by-N1.
On exit, the leading L-by-N part of this array contains
the transformed state matrix,
( Ac * * )
Qc'*A*Zc = ( 0 Anc * ) ,
( 0 0 * )
where Ac is NR-by-NR and Anc is (N1-NR)-by-(N1-NR).
The matrix ( Bc Ac ) is in the controllability
staircase form (1).
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,L).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading L-by-N part of this array must
contain the L-by-N descriptor matrix E in the partitioned
form
( E1 Y1 )
E = ( ) ,
( 0 Y2 )
where E1 is N1-by-N1 matrix with LBE nonzero
sub-diagonals.
On exit, the leading L-by-N part of this array contains
the transformed descriptor matrix
( Ec * * )
Qc'*E*Zc = ( 0 Enc * ) ,
( 0 0 * )
where Ec is NR-by-NR and Enc is (N1-NR)-by-(N1-NR).
Both Ec and Enc are upper triangular and Enc is
nonsingular.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,L).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading L-by-M part of this array must
contain the L-by-M input matrix B in the partitioned
form
( B1 )
B = ( ) ,
( 0 )
where B1 is N1-by-M.
On exit, the leading L-by-M part of this array contains
the transformed input matrix
( Bc )
Qc'*B = ( ) ,
( 0 )
where Bc is NR-by-M.
The matrix ( Bc Ac ) is in the controllability
staircase form (1).
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,L).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed matrix C*Zc.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,L)
If COMPQ = 'N': Q is not referenced.
If COMPQ = 'I': on entry, Q need not be set;
on exit, the leading L-by-L part of this
array contains the orthogonal matrix Qc,
where Qc' is the product of transformations
which are applied to A, E, and B on
the left.
If COMPQ = 'U': on entry, the leading L-by-L part of this
array must contain an orthogonal matrix Q;
on exit, the leading L-by-L part of this
array contains the orthogonal matrix
Q*Qc.
LDQ INTEGER
The leading dimension of array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1,L), if COMPQ = 'U' or 'I'.
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
If COMPZ = 'N': Z is not referenced.
If COMPZ = 'I': on entry, Z need not be set;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix Zc,
which is the product of transformations
applied to A, E, and C on the right.
If COMPZ = 'U': on entry, the leading N-by-N part of this
array must contain an orthogonal matrix Z;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix
Z*Zc.
LDZ INTEGER
The leading dimension of array Z.
LDZ >= 1, if COMPZ = 'N';
LDZ >= MAX(1,N), if COMPZ = 'U' or 'I'.
NR (output) INTEGER
The order of the reduced matrices Ac and Ec, and the
number of rows of the reduced matrix Bc; also the order of
the controllable part of the pair (B, A-lambda*E).
NRBLCK (output) INTEGER _
The number k, of full row rank blocks Ai,i in the
staircase form of the pencil (Bc Ac-lambda*Ec) (see (1)
and (2)).
RTAU (output) INTEGER array, dimension (N1)
RTAU(i), for i = 1, ..., NRBLCK, is the row dimension of
_
the full row rank block Ai,i-1 in the staircase form (1).
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in rank determinations when
transforming (A-lambda*E, B). If the user sets TOL > 0,
then the given value of TOL is used as a lower bound for
reciprocal condition numbers in rank determinations; a
(sub)matrix whose estimated condition number is less than
1/TOL is considered to be of full rank. If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
defined by TOLDEF = L*N*EPS, is used instead, where
EPS is the machine precision (see LAPACK Library routine
DLAMCH). TOL < 1.
Workspace
IWORK INTEGER array, dimension (M) DWORK DOUBLE PRECISION array, dimension (MAX(N,L,2*M))Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The subroutine is based on the reduction algorithm of [1].References
[1] Varga, A.
Computation of Irreducible Generalized State-Space
Realizations.
Kybernetika, vol. 26, pp. 89-106, 1990.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( N*N1**2 ) floating point operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
Given the descriptor system (A-lambda*E,B,C) with the system
matrices A, E and B of the form
( A1 X1 ) ( E1 Y1 ) ( B1 )
A = ( ) , E = ( ) , B = ( ) ,
( 0 X2 ) ( 0 Y2 ) ( 0 )
where
- B is an L-by-M matrix, with B1 an N1-by-M submatrix,
- A is an L-by-N matrix, with A1 an N1-by-N1 submatrix,
- E is an L-by-N matrix, with E1 an N1-by-N1 submatrix
with LBE nonzero sub-diagonals,
this routine reduces the pair (A1-lambda*E1,B1) to the form
Qc'*[ B1 A1-lambda*E1 ]*diag(I,Zc) =
( Bc Ac-lambda*Ec * )
( ) ,
( 0 0 Anc-lambda*Enc )
where:
1) the pencil ( Bc Ac-lambda*Ec ) has full row rank NR for
all finite lambda and is in a staircase form with
_ _ _ _
( A1,0 A1,1 ... A1,k-1 A1,k )
( _ _ _ )
( Bc Ac ) = ( 0 A2,1 ... A2,k-1 A2,k ) , (1)
( ... _ _ )
( 0 0 ... Ak,k-1 Ak,k )
_ _ _
( E1,1 ... E1,k-1 E1,k )
( _ _ )
Ec = ( 0 ... E2,k-1 E2,k ) , (2)
( ... _ )
( 0 ... 0 Ek,k )
_
where Ai,i-1 is an rtau(i)-by-rtau(i-1) full row rank
_
matrix (with rtau(0) = M) and Ei,i is an rtau(i)-by-rtau(i)
upper triangular matrix.
2) the pencil Anc-lambda*Enc is regular of order N1-NR with Enc
upper triangular; this pencil contains the uncontrollable
finite eigenvalues of the pencil (A1-lambda*E1).
The transformations are applied to the whole matrices A, E, B
and C. The left and/or right orthogonal transformations Qc and Zc,
performed to reduce the pencil, can be optionally accumulated in
the matrices Q and Z, respectively.
The reduced order descriptor system (Ac-lambda*Ec,Bc,Cc) has no
uncontrollable finite eigenvalues and has the same transfer-
function matrix as the original system (A-lambda*E,B,C).
Specification
SUBROUTINE TG01HY( COMPQ, COMPZ, L, N, M, P, N1, LBE, A, LDA,
$ E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ, NR,
$ NRBLCK, RTAU, TOL, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, COMPZ
INTEGER INFO, L, LBE, LDA, LDB, LDC, LDE, LDQ, LDWORK,
$ LDZ, M, N, N1, NR, NRBLCK, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK( * ), RTAU( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ DWORK( * ), E( LDE, * ), Q( LDQ, * ),
$ Z( LDZ, * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
= 'N': do not compute Q;
= 'I': Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= 'U': Q must contain an orthogonal matrix Q1 on entry,
and the product Q1*Q is returned.
COMPZ CHARACTER*1
= 'N': do not compute Z;
= 'I': Z is initialized to the unit matrix, and the
orthogonal matrix Z is returned;
= 'U': Z must contain an orthogonal matrix Z1 on entry,
and the product Z1*Z is returned.
Input/Output Parameters
L (input) INTEGER
The number of descriptor state equations; also the number
of rows of the matrices A, E and B. L >= 0.
N (input) INTEGER
The dimension of the descriptor state vector; also the
number of columns of the matrices A, E and C. N >= 0.
M (input) INTEGER
The dimension of descriptor system input vector; also the
number of columns of the matrix B. M >= 0.
P (input) INTEGER
The dimension of descriptor system output; also the
number of rows of the matrix C. P >= 0.
N1 (input) INTEGER
The order of the subsystem (A1-lambda*E1,B1,C1) to be
reduced. MIN(L,N) >= N1 >= 0.
LBE (input) INTEGER
The number of nonzero sub-diagonals of the submatrix E1.
MAX(0,N1-1) >= LBE >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading L-by-N part of this array must
contain the L-by-N state matrix A in the partitioned form
( A1 X1 )
A = ( ) ,
( 0 X2 )
where A1 is N1-by-N1.
On exit, the leading L-by-N part of this array contains
the transformed state matrix,
( Ac * * )
Qc'*A*diag(Zc,I) = ( 0 Anc * ) ,
( 0 0 * )
where Ac is NR-by-NR and Anc is (N1-NR)-by-(N1-NR).
The matrix ( Bc Ac ) is in the controllability staircase
form (1).
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,L).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading L-by-N part of this array must
contain the L-by-N descriptor matrix E in the partitioned
form
( E1 Y1 )
E = ( ) ,
( 0 Y2 )
where E1 is an N1-by-N1 matrix with LBE nonzero
sub-diagonals.
On exit, the leading L-by-N part of this array contains
the transformed descriptor matrix
( Ec * * )
Qc'*E*diag(Zc,I) = ( 0 Enc * ) ,
( 0 0 * )
where Ec is NR-by-NR and Enc is (N1-NR)-by-(N1-NR).
Both Ec and Enc are upper triangular.
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,L).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading L-by-M part of this array must
contain the L-by-M input matrix B in the partitioned form
( B1 )
B = ( ) ,
( 0 )
where B1 is N1-by-M.
On exit, the leading L-by-M part of this array contains
the transformed input matrix
( Bc )
Qc'*B = ( ) ,
( 0 )
where Bc is NR-by-M.
The matrix ( Bc Ac ) is in the controllability staircase
form (1).
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,L).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed matrix C*Zc.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,P).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,L)
If COMPQ = 'N': Q is not referenced.
If COMPQ = 'I': on entry, Q need not be set;
on exit, the leading L-by-L part of this
array contains the orthogonal matrix Qc,
where Qc' is the product of the
transformations applied to A, E, and B on
the left.
If COMPQ = 'U': on entry, the leading L-by-L part of this
array must contain an orthogonal matrix Q;
on exit, the leading L-by-L part of this
array contains the orthogonal matrix
Q*Qc.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1,L), if COMPQ = 'U' or 'I'.
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
If COMPZ = 'N': Z is not referenced.
If COMPZ = 'I': on entry, Z need not be set;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix Zc,
i.e., the product of the transformations
applied to A, E, and C on the right.
If COMPZ = 'U': on entry, the leading N-by-N part of this
array must contain an orthogonal matrix Z;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix
Z*Zc.
LDZ INTEGER
The leading dimension of the array Z.
LDZ >= 1, if COMPZ = 'N';
LDZ >= MAX(1,N), if COMPZ = 'U' or 'I'.
NR (output) INTEGER
The order of the reduced matrices Ac and Ec, and the
number of rows of the reduced matrix Bc; also the order of
the controllable part of the pair (B, A-lambda*E).
NRBLCK (output) INTEGER _
The number k, of full row rank blocks Ai,i in the
staircase form of the pencil (Bc Ac-lambda*Ec) (see (1)
and (2)).
RTAU (output) INTEGER array, dimension (N1)
RTAU(i), for i = 1, ..., NRBLCK, is the row dimension of
_
the full row rank block Ai,i-1 in the staircase form (1).
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in rank determinations when
transforming (A-lambda*E, B). If the user sets TOL > 0,
then the given value of TOL is used as a lower bound for
reciprocal condition numbers in rank determinations; a
(sub)matrix whose estimated condition number is less than
1/TOL is considered to be of full rank. If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
defined by TOLDEF = L*N*EPS, is used instead, where
EPS is the machine precision (see LAPACK Library routine
DLAMCH). TOL < 1.
Workspace
IWORK INTEGER array, dimension (M)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,N,L,2*(M+N1-1))
For good performance, LDWORK should be generally larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The subroutine is based on the reduction algorithm of [1]. If suitable, it uses block algorithms for the reduction of the matrix E and for the corresponding updates of the matrices A, B, and Q. Moreover, for large systems, the row transformations are applied on panels of columns of the matrices A, B, and E.References
[1] Varga, A.
Computation of Irreducible Generalized State-Space
Realizations.
Kybernetika, vol. 26, pp. 89-106, 1990.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( N*N1**2 ) floating point operations.Further Comments
If INFO > 0 on entry, that value is used as block size for the block algorithms. Otherwise, the block size is chosen internally.Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute orthogonal transformation matrices Q and Z which
reduce the N-th order descriptor system (A-lambda*E,B,C)
to the form
( Ano * ) ( Eno * ) ( Bno )
Q'*A*Z = ( ) , Q'*E*Z = ( ) , Q'*B = ( ) ,
( 0 Ao ) ( 0 Eo ) ( Bo )
C*Z = ( 0 Co ) ,
where the NOBSV-th order descriptor system (Ao-lambda*Eo,Bo,Co)
is a finite and/or infinite observable. The pencil
Ano - lambda*Eno is regular of order N-NOBSV and contains the
unobservable finite and/or infinite eigenvalues of the pencil
A-lambda*E.
For JOBOBS = 'O' or 'I', the pencil ( Eo-lambda*Ao ) has full
( Co )
column rank NOBSV for all finite lambda and is in a staircase form
with
_ _ _ _
( Ek,k Ek,k-1 ... Ek,2 Ek,1 )
( _ _ _ _ )
( Eo ) = ( Ek-1,k Ek-1,k-1 ... Ek-1,2 Ek-1,1 ) , (1)
( Co ) ( ... ... _ _ )
( 0 0 ... E1,2 E1,1 )
( _ )
( 0 0 ... 0 E0,1 )
_ _ _
( Ak,k ... Ak,2 Ak,1 )
( ... _ _ )
Ao = ( 0 ... A2,2 A2,1 ) , (2)
( _ )
( 0 ... 0 A1,1 )
_
where Ei-1,i is a CTAU(i-1)-by-CTAU(i) full column rank matrix
_
(with CTAU(0) = P) and Ai,i is a CTAU(i)-by-CTAU(i)
upper triangular matrix.
For JOBOBS = 'F', the pencil ( Ao-lambda*Eo ) has full
( Co )
column rank NOBSV for all finite lambda and is in a staircase form
with
_ _ _ _
( Ak,k Ak,k-1 ... Ak,2 Ak,1 )
( _ _ _ _ )
( Ao ) = ( Ak-1,k Ak-1,k-1 ... Ak-1,2 Ak-1,1 ) , (3)
( Co ) ( ... ... _ _ )
( 0 0 ... A1,2 A1,1 )
( _ )
( 0 0 ... 0 A0,1 )
_ _ _
( Ek,k ... Ek,2 Ek,1 )
( ... _ _ )
Eo = ( 0 ... E2,2 E2,1 ) , (4)
( _ )
( 0 ... 0 E1,1 )
_
where Ai-1,i is a CTAU(i-1)-by-CTAU(i) full column rank matrix
_
(with CTAU(0) = P) and Ei,i is a CTAU(i)-by-CTAU(i)
upper triangular matrix.
For JOBOBS = 'O', the (N-NOBSV)-by-(N-NOBSV) regular pencil
Ano - lambda*Eno has the form
( Afno - lambda*Efno * )
Ano - lambda*Eno = ( ) ,
( 0 Aino - lambda*Eino )
where:
1) the NIUOBS-by-NIUOBS regular pencil Aino - lambda*Eino,
with Aino upper triangular and nonsingular, contains the
unobservable infinite eigenvalues of A - lambda*E;
2) the (N-NOBSV-NIUOBS)-by-(N-NOBSV-NIUOBS) regular pencil
Afno - lambda*Efno, with Efno upper triangular and
nonsingular, contains the unobservable finite
eigenvalues of A - lambda*E.
Note: The significance of the two diagonal blocks can be
interchanged by calling the routine with the
arguments A and E interchanged. In this case,
Aino - lambda*Eino contains the unobservable zero
eigenvalues of A - lambda*E, while Afno - lambda*Efno
contains the unobservable nonzero finite and infinite
eigenvalues of A - lambda*E.
For JOBOBS = 'F', the pencil Ano - lambda*Eno has the form
Ano - lambda*Eno = Afno - lambda*Efno ,
where the regular pencil Afno - lambda*Efno, with Efno
upper triangular and nonsingular, contains the unobservable
finite eigenvalues of A - lambda*E.
For JOBOBS = 'I', the pencil Ano - lambda*Eno has the form
Ano - lambda*Eno = Aino - lambda*Eino ,
where the regular pencil Aino - lambda*Eino, with Aino
upper triangular and nonsingular, contains the unobservable
nonzero finite and infinite eigenvalues of A - lambda*E.
The left and/or right orthogonal transformations Q and Z
performed to reduce the system matrices can be optionally
accumulated.
The reduced order descriptor system (Ao-lambda*Eo,Bo,Co) has
the same transfer-function matrix as the original system
(A-lambda*E,B,C).
Specification
SUBROUTINE TG01ID( JOBOBS, COMPQ, COMPZ, N, M, P, A, LDA, E, LDE,
$ B, LDB, C, LDC, Q, LDQ, Z, LDZ, NOBSV, NIUOBS,
$ NLBLCK, CTAU, TOL, IWORK, DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, COMPZ, JOBOBS
INTEGER INFO, LDA, LDB, LDC, LDE, LDQ, LDZ,
$ M, N, NIUOBS, NLBLCK, NOBSV, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER CTAU( * ), IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ DWORK( * ), E( LDE, * ), Q( LDQ, * ),
$ Z( LDZ, * )
Arguments
Mode Parameters
JOBOBS CHARACTER*1
= 'O': separate both finite and infinite unobservable
eigenvalues;
= 'F': separate only finite unobservable eigenvalues;
= 'I': separate only nonzero finite and infinite
unobservable eigenvalues.
COMPQ CHARACTER*1
= 'N': do not compute Q;
= 'I': Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= 'U': Q must contain an orthogonal matrix Q1 on entry,
and the product Q1*Q is returned.
COMPZ CHARACTER*1
= 'N': do not compute Z;
= 'I': Z is initialized to the unit matrix, and the
orthogonal matrix Z is returned;
= 'U': Z must contain an orthogonal matrix Z1 on entry,
and the product Z1*Z is returned.
Input/Output Parameters
N (input) INTEGER
The dimension of the descriptor state vector; also the
order of square matrices A and E, the number of rows of
matrix B, and the number of columns of matrix C. N >= 0.
M (input) INTEGER
The dimension of descriptor system input vector; also the
number of columns of matrix B. M >= 0.
P (input) INTEGER
The dimension of descriptor system output vector; also the
number of rows of matrix C. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the N-by-N state matrix A.
On exit, the leading N-by-N part of this array contains
the transformed state matrix Q'*A*Z,
( Ano * )
Q'*A*Z = ( ) ,
( 0 Ao )
where Ao is NOBSV-by-NOBSV and Ano is
(N-NOBSV)-by-(N-NOBSV).
If JOBOBS = 'F', the matrix ( Ao ) is in the observability
( Co )
staircase form (3).
If JOBOBS = 'O' or 'I', the submatrix Ao is upper
triangular.
If JOBOBS = 'O', the submatrix Ano has the form
( Afno * )
Ano = ( ) ,
( 0 Aino )
where the NIUOBS-by-NIUOBS matrix Aino is nonsingular and
upper triangular.
If JOBOBS = 'I', Ano is nonsingular and upper triangular.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain the N-by-N descriptor matrix E.
On exit, the leading N-by-N part of this array contains
the transformed state matrix Q'*E*Z,
( Eno * )
Q'*E*Z = ( ) ,
( 0 Eo )
where Eo is NOBSV-by-NOBSV and Eno is
(N-NOBSV)-by-(N-NOBSV).
If JOBOBS = 'O' or 'I', the matrix ( Eo ) is in the
( Co )
observability staircase form (1).
If JOBOBS = 'F', the submatrix Eo is upper triangular.
If JOBOBS = 'O', the Eno matrix has the form
( Efno * )
Eno = ( ) ,
( 0 Eino )
where the NIUOBS-by-NIUOBS matrix Eino is nilpotent
and the (N-NOBSV-NIUOBS)-by-(N-NOBSV-NIUOBS) matrix Efno
is nonsingular and upper triangular.
If JOBOBS = 'F', Eno is nonsingular and upper triangular.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension
(LDB,MAX(M,P))
On entry, the leading N-by-M part of this array must
contain the N-by-M input matrix B.
On exit, the leading N-by-M part of this array contains
the transformed input matrix Q'*B.
LDB INTEGER
The leading dimension of array B.
LDB >= MAX(1,N) if M > 0 or LDB >= 1 if M = 0.
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed matrix
C*Z = ( 0 Co ) ,
where Co is P-by-NOBSV.
If JOBOBS = 'O' or 'I', the matrix ( Eo ) is in the
( Co )
observability staircase form (1).
If JOBOBS = 'F', the matrix ( Ao ) is in the observability
( Co )
staircase form (3).
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,M,P).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
If COMPQ = 'N': Q is not referenced.
If COMPQ = 'I': on entry, Q need not be set;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix Q,
where Q' is the product of transformations
which are applied to A, E, and B on
the left.
If COMPQ = 'U': on entry, the leading N-by-N part of this
array must contain an orthogonal matrix
Qc;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix
Qc*Q.
LDQ INTEGER
The leading dimension of array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1,N), if COMPQ = 'U' or 'I'.
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
If COMPZ = 'N': Z is not referenced.
If COMPZ = 'I': on entry, Z need not be set;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix Z,
which is the product of transformations
applied to A, E, and C on the right.
If COMPZ = 'U': on entry, the leading N-by-N part of this
array must contain an orthogonal matrix
Zc;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix
Zc*Z.
LDZ INTEGER
The leading dimension of array Z.
LDZ >= 1, if COMPZ = 'N';
LDZ >= MAX(1,N), if COMPZ = 'U' or 'I'.
NOBSV (output) INTEGER
The order of the reduced matrices Ao and Eo, and the
number of columns of reduced matrix Co; also the order of
observable part of the pair (C, A-lambda*E).
NIUOBS (output) INTEGER
For JOBOBS = 'O', the order of the reduced matrices
Aino and Eino; also the number of unobservable
infinite eigenvalues of the pencil A - lambda*E.
For JOBOBS = 'F' or 'I', NIUOBS has no significance
and is set to zero.
NLBLCK (output) INTEGER
For JOBOBS = 'O' or 'I', the number k, of full column rank
_
blocks Ei-1,i in the staircase form of the pencil
(Eo-lambda*Ao) (see (1) and (2)).
( Co )
For JOBOBS = 'F', the number k, of full column rank blocks
_
Ai-1,i in the staircase form of the pencil (Ao-lambda*Eo)
( Co )
(see (3) and (4)).
CTAU (output) INTEGER array, dimension (N)
CTAU(i), for i = 1, ..., NLBLCK, is the column dimension
_ _
of the full column rank block Ei-1,i or Ai-1,i in the
staircase form (1) or (3) for JOBOBS = 'O' or 'I', or
for JOBOBS = 'F', respectively.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in rank determinations when
transforming (A'-lambda*E',C')'. If the user sets TOL > 0,
then the given value of TOL is used as a lower bound for
reciprocal condition numbers in rank determinations; a
(sub)matrix whose estimated condition number is less than
1/TOL is considered to be of full rank. If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
defined by TOLDEF = N*N*EPS, is used instead, where EPS
is the machine precision (see LAPACK Library routine
DLAMCH). TOL < 1.
Workspace
IWORK INTEGER array, dimension (P) DWORK DOUBLE PRECISION array, dimension (MAX(N,2*P))Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The subroutine is based on the dual of the reduction algorithms of [1].References
[1] A. Varga
Computation of Irreducible Generalized State-Space
Realizations.
Kybernetika, vol. 26, pp. 89-106, 1990.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( N**3 ) floating point operations.Further Comments
If the system matrices A, E and C are badly scaled, it is generally recommendable to scale them with the SLICOT routine TG01AD, before calling TG01ID.Example
Program Text
* TG01ID EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER LMAX, NMAX, MMAX, PMAX
PARAMETER ( LMAX = 20, NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER MPMX
PARAMETER ( MPMX = MAX( MMAX, PMAX ) )
INTEGER LDA, LDB, LDC, LDE, LDQ, LDZ
PARAMETER ( LDA = LMAX, LDB = LMAX, LDC = MAX(MMAX,PMAX),
$ LDE = LMAX, LDQ = LMAX, LDZ = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( 1, NMAX, 2*PMAX ) )
* .. Local Scalars ..
CHARACTER*1 COMPQ, COMPZ, JOBOBS
INTEGER I, INFO, J, M, N, NOBSV, NIUOBS, NLBLCK, P
DOUBLE PRECISION TOL
* .. Local Arrays ..
INTEGER IWORK(MMAX), CTAU(NMAX)
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MPMX), C(LDC,NMAX),
$ DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,LMAX),
$ Z(LDZ,NMAX)
* .. External Subroutines ..
EXTERNAL TG01ID
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL, JOBOBS
COMPQ = 'I'
COMPZ = 'I'
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find the transformed descriptor system (A-lambda E,B,C).
CALL TG01ID( JOBOBS, COMPQ, COMPZ, N, M, P, A, LDA,
$ E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ,
$ NOBSV, NIUOBS, NLBLCK, CTAU, TOL, IWORK,
$ DWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) NOBSV, NIUOBS
WRITE ( NOUT, FMT = 99985 )
WRITE ( NOUT, FMT = 99984 ) ( CTAU(I), I = 1,NLBLCK )
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 30 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
30 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 40 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
40 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 50 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,N )
50 CONTINUE
WRITE ( NOUT, FMT = 99990 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
60 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01ID EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01ID = ',I2)
99997 FORMAT (/' The transformed state dynamics matrix Q''*A*Z is ')
99996 FORMAT (/' The transformed descriptor matrix Q''*E*Z is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (' Dimension of observable part =', I5/
$ ' Number of unobservable infinite eigenvalues =', I5)
99993 FORMAT (/' The transformed input/state matrix Q''*B is ')
99992 FORMAT (/' The transformed state/output matrix C*Z is ')
99991 FORMAT (/' The left transformation matrix Q is ')
99990 FORMAT (/' The right transformation matrix Z is ')
99989 FORMAT (/' L is out of range.',/' L = ',I5)
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
99986 FORMAT (/' P is out of range.',/' P = ',I5)
99985 FORMAT (/' The staircase form column dimensions are ' )
99984 FORMAT (10I5)
END
Program Data
TG01ID EXAMPLE PROGRAM DATA
7 2 3 0.0 O
2 0 0 0 0 0 0
0 1 0 0 0 1 0
2 0 0 2 0 0 0
0 0 1 0 1 0 1
-1 1 0 -1 0 1 0
3 0 0 3 0 0 0
1 0 1 1 1 0 1
0 0 0 0 0 0 1
0 0 0 0 0 0 3
1 0 0 0 0 1 0
0 0 0 0 1 0 2
0 0 0 0 0 -1 0
0 1 0 0 0 0 0
0 0 1 1 0 0 0
1 0
0 -1
0 1
1 0
0 -1
0 1
1 0
2 0 0 0 0 0 1
1 0 0 0 0 0 2
0 0 0 0 0 0 3
Program Results
TG01ID EXAMPLE PROGRAM RESULTS
Dimension of observable part = 3
Number of unobservable infinite eigenvalues = 1
The staircase form column dimensions are
2 1
The transformed state dynamics matrix Q'*A*Z is
0.2177 0.2414 0.5742 0.4342 0.0000 -0.4342 0.4666
0.2022 0.2242 0.5334 -0.2924 -0.7723 0.2924 0.4334
-0.5892 -0.6533 -1.5540 0.8520 -0.2651 -0.8520 -1.2627
0.0000 0.0000 0.0000 3.7417 0.3780 -3.7417 0.0000
0.0000 0.0000 0.0000 0.0000 1.7862 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 2.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
The transformed descriptor matrix Q'*E*Z is
1.0000 0.0000 0.0000 0.4342 0.0000 0.0000 1.8016
0.0000 1.1937 -0.1496 -0.2924 0.3861 0.5461 0.2819
0.0000 0.0000 -1.0260 0.8520 0.1325 0.1874 -0.8214
0.0000 0.0000 0.0000 0.0000 -1.1339 -0.5345 0.0000
0.0000 0.0000 0.0000 0.0000 -0.1333 0.3770 2.3752
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000
0.0000 0.0000 0.0000 0.0000 -0.1728 0.4887 -1.8325
The transformed input/state matrix Q'*B is
0.4666 0.0000
0.4334 0.5461
-1.2627 0.1874
0.0000 -1.6036
0.0000 -0.9803
1.0000 0.0000
0.0000 0.3665
The transformed state/output matrix C*Z is
0.0000 0.0000 0.0000 0.0000 0.0000 2.0000 1.0000
0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 2.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.0000
The left transformation matrix Q is
0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.7917 0.0000 -0.6108
0.0000 0.5461 0.1874 -0.5345 0.3770 0.0000 0.4887
0.9008 0.1410 -0.4107 0.0000 0.0000 0.0000 0.0000
0.0000 -0.5461 -0.1874 0.2673 0.4713 0.0000 0.6108
0.0000 -0.5461 -0.1874 -0.8018 -0.0943 0.0000 -0.1222
-0.4342 0.2924 -0.8520 0.0000 0.0000 0.0000 0.0000
The right transformation matrix Z is
0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000
0.0000 -0.6519 0.2740 0.0000 0.7071 0.0000 0.0000
-0.4342 0.3491 0.8304 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 -1.0000 0.0000 0.0000 0.0000
0.9008 0.1683 0.4003 0.0000 0.0000 0.0000 0.0000
0.0000 0.6519 -0.2740 0.0000 0.7071 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TG01JD.html 0000664 0000000 0000000 00000045304 14560147231 0020351 0 ustar 00root root 0000000 0000000
Purpose
To find a reduced (controllable, observable, or irreducible) descriptor representation (Ar-lambda*Er,Br,Cr) for an original descriptor representation (A-lambda*E,B,C). The pencil Ar-lambda*Er is in an upper block Hessenberg form, with either Ar or Er upper triangular.Specification
SUBROUTINE TG01JD( JOB, SYSTYP, EQUIL, N, M, P, A, LDA, E, LDE,
$ B, LDB, C, LDC, NR, INFRED, TOL, IWORK, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER EQUIL, JOB, SYSTYP
INTEGER INFO, LDA, LDB, LDC, LDE, LDWORK, M, N, NR, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER INFRED(*), IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), E(LDE,*)
Arguments
Mode Parameters
JOB CHARACTER*1
Indicates whether the user wishes to remove the
uncontrollable and/or unobservable parts as follows:
= 'I': Remove both the uncontrollable and unobservable
parts to get an irreducible descriptor
representation;
= 'C': Remove the uncontrollable part only to get a
controllable descriptor representation;
= 'O': Remove the unobservable part only to get an
observable descriptor representation.
SYSTYP CHARACTER*1
Indicates the type of descriptor system algorithm
to be applied according to the assumed
transfer-function matrix as follows:
= 'R': Rational transfer-function matrix;
= 'S': Proper (standard) transfer-function matrix;
= 'P': Polynomial transfer-function matrix.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily scale
the system (A-lambda*E,B,C) as follows:
= 'S': Perform scaling;
= 'N': Do not perform scaling.
Input/Output Parameters
N (input) INTEGER
The dimension of the descriptor state vector; also the
order of square matrices A and E, the number of rows of
matrix B, and the number of columns of matrix C. N >= 0.
M (input) INTEGER
The dimension of descriptor system input vector; also the
number of columns of matrix B. M >= 0.
P (input) INTEGER
The dimension of descriptor system output vector; also the
number of rows of matrix C. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state matrix A.
On exit, the leading NR-by-NR part of this array contains
the reduced order state matrix Ar of an irreducible,
controllable, or observable realization for the original
system, depending on the value of JOB, JOB = 'I',
JOB = 'C', or JOB = 'O', respectively.
The matrix Ar is upper triangular if SYSTYP = 'R' or 'P'.
If SYSTYP = 'S' and JOB = 'C', the matrix [Br Ar]
is in a controllable staircase form (see TG01HD).
If SYSTYP = 'S' and JOB = 'I' or 'O', the matrix ( Ar )
( Cr )
is in an observable staircase form (see TG01HD).
The block structure of staircase forms is contained
in the leading INFRED(7) elements of IWORK.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain the original descriptor matrix E.
On exit, the leading NR-by-NR part of this array contains
the reduced order descriptor matrix Er of an irreducible,
controllable, or observable realization for the original
system, depending on the value of JOB, JOB = 'I',
JOB = 'C', or JOB = 'O', respectively.
The resulting Er has INFRED(6) nonzero sub-diagonals.
If at least for one k = 1,...,4, INFRED(k) >= 0, then the
resulting Er is structured being either upper triangular
or block Hessenberg, in accordance to the last
performed order reduction phase (see METHOD).
The block structure of staircase forms is contained
in the leading INFRED(7) elements of IWORK.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M),
if JOB = 'C', or (LDB,MAX(M,P)), otherwise.
On entry, the leading N-by-M part of this array must
contain the original input matrix B; if JOB = 'I',
or JOB = 'O', the remainder of the leading N-by-MAX(M,P)
part is used as internal workspace.
On exit, the leading NR-by-M part of this array contains
the reduced input matrix Br of an irreducible,
controllable, or observable realization for the original
system, depending on the value of JOB, JOB = 'I',
JOB = 'C', or JOB = 'O', respectively.
If JOB = 'C', only the first IWORK(1) rows of B are
nonzero.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original output matrix C; if JOB = 'I',
or JOB = 'O', the remainder of the leading MAX(M,P)-by-N
part is used as internal workspace.
On exit, the leading P-by-NR part of this array contains
the transformed state/output matrix Cr of an irreducible,
controllable, or observable realization for the original
system, depending on the value of JOB, JOB = 'I',
JOB = 'C', or JOB = 'O', respectively.
If JOB = 'I', or JOB = 'O', only the last IWORK(1) columns
(in the first NR columns) of C are nonzero.
LDC INTEGER
The leading dimension of array C.
LDC >= MAX(1,M,P) if N > 0.
LDC >= 1 if N = 0.
NR (output) INTEGER
The order of the reduced descriptor representation
(Ar-lambda*Er,Br,Cr) of an irreducible, controllable,
or observable realization for the original system,
depending on JOB = 'I', JOB = 'C', or JOB = 'O',
respectively.
INFRED (output) INTEGER array, dimension 7
This array contains information on performed reduction
and on structure of resulting system matrices as follows:
INFRED(k) >= 0 (k = 1, 2, 3, or 4) if Phase k of reduction
(see METHOD) has been performed. In this
case, INFRED(k) is the achieved order
reduction in Phase k.
INFRED(k) < 0 (k = 1, 2, 3, or 4) if Phase k was not
performed.
INFRED(5) - the number of nonzero sub-diagonals of A.
INFRED(6) - the number of nonzero sub-diagonals of E.
INFRED(7) - the number of blocks in the resulting
staircase form at last performed reduction
phase. The block dimensions are contained
in the first INFRED(7) elements of IWORK.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in rank determinations when
transforming (A-lambda*E,B,C). If the user sets TOL > 0,
then the given value of TOL is used as a lower bound for
reciprocal condition numbers in rank determinations; a
(sub)matrix whose estimated condition number is less than
1/TOL is considered to be of full rank. If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
defined by TOLDEF = N*N*EPS, is used instead, where
EPS is the machine precision (see LAPACK Library routine
DLAMCH). TOL < 1.
Workspace
IWORK INTEGER array, dimension ((c*N+MAX(M,P)), where
c = 2, if JOB = 'I' or SYSTYP = 'R', and c = 1, otherwise.
On exit, if INFO = 0, the leading INFRED(7) elements of
IWORK contain the orders of the diagonal blocks of
Ar-lambda*Er.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(8*N,2*M,2*P), if EQUIL = 'S';
LDWORK >= MAX(N,2*M,2*P), if EQUIL = 'N'.
If LDWORK >= 2*N*N+N*M+N*P+MAX(N,2*M,2*P) then more
accurate results are to be expected by performing only
those reductions phases (see METHOD), where effective
order reduction occurs. This is achieved by saving the
system matrices before each phase and restoring them if no
order reduction took place.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The subroutine is based on the reduction algorithms of [1].
The order reduction is performed in 4 phases:
Phase 1: Eliminate all finite uncontrollable eigenvalues.
The resulting matrix ( Br Ar ) is in a controllable
staircase form (see SLICOT Library routine TG01HD), and
Er is upper triangular.
This phase is performed if JOB = 'I' or 'C' and
SYSTYP = 'R' or 'S'.
Phase 2: Eliminate all infinite and finite nonzero uncontrollable
eigenvalues. The resulting matrix ( Br Er ) is in a
controllable staircase form (see TG01HD), and Ar is
upper triangular.
This phase is performed if JOB = 'I' or 'C' and
SYSTYP = 'R' or 'P'.
Phase 3: Eliminate all finite unobservable eigenvalues.
The resulting matrix ( Ar ) is in an observable
( Cr )
staircase form (see SLICOT Library routine TG01ID), and
Er is upper triangular.
This phase is performed if JOB = 'I' or 'O' and
SYSTYP = 'R' or 'S'.
Phase 4: Eliminate all infinite and finite nonzero unobservable
eigenvalues. The resulting matrix ( Er ) is in an
( Cr )
observable staircase form (see TG01ID), and Ar is
upper triangular.
This phase is performed if JOB = 'I' or 'O' and
SYSTYP = 'R' or 'P'.
References
[1] A. Varga
Computation of Irreducible Generalized State-Space
Realizations.
Kybernetika, vol. 26, pp. 89-106, 1990.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( N**3 ) floating point operations.Further Comments
If the pencil (A-lambda*E) has no zero eigenvalues, then an irreducible realization can be computed skipping Phases 1 and 3 by using the setting: JOB = 'I' and SYSTYP = 'P'.Example
Program Text
* TG01JD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDE
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDE = NMAX )
INTEGER LDWORK, LIWORK
PARAMETER ( LDWORK = MAX( 8*NMAX,2*MMAX,2*PMAX ),
$ LIWORK = 2*NMAX + MAX( MMAX, PMAX ) )
* .. Local Scalars ..
CHARACTER EQUIL, JOB, SYSTYP
INTEGER I, INFO, J, M, N, NR, P
DOUBLE PRECISION TOL
* .. Local Arrays ..
INTEGER INFRED(7), IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ DWORK(LDWORK), E(LDE,NMAX)
* .. External Subroutines ..
EXTERNAL TG01JD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL, JOB, SYSTYP, EQUIL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find the irreducible descriptor system (Ar-lambda Er,Br,Cr).
CALL TG01JD( JOB, SYSTYP, EQUIL, N, M, P, A, LDA, E, LDE,
$ B, LDB, C, LDC, NR, INFRED, TOL, IWORK,
$ DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) NR
WRITE ( NOUT, FMT = 99991 )
DO 10 I = 1, 4
IF( INFRED(I).GE.0 )
$ WRITE ( NOUT, FMT = 99990 ) I, INFRED(I)
10 CONTINUE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
20 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 30 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,NR )
30 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 50 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
50 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01JD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01JD = ',I2)
99997 FORMAT (/' The reduced state dynamics matrix Ar is ')
99996 FORMAT (/' The reduced descriptor matrix Er is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (' Order of reduced system =', I5 )
99993 FORMAT (/' The reduced input/state matrix Br is ')
99992 FORMAT (/' The reduced state/output matrix Cr is ')
99991 FORMAT (/' Achieved order reductions in different phases')
99990 FORMAT (' Phase',I2,':', I3, ' elliminated eigenvalue(s)' )
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
99986 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
TG01JD EXAMPLE PROGRAM DATA
9 2 2 0.0 I R N
-2 -3 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0
0 0 -2 -3 0 0 0 0 0
0 0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1 0
1 0
0 0
0 1
0 0
-1 0
0 0
0 -1
0 0
0 0
1 0 1 -3 0 1 0 2 0
0 1 1 3 0 1 0 0 1
Program Results
TG01JD EXAMPLE PROGRAM RESULTS Order of reduced system = 7 Achieved order reductions in different phases Phase 1: 0 elliminated eigenvalue(s) Phase 2: 0 elliminated eigenvalue(s) Phase 3: 2 elliminated eigenvalue(s) Phase 4: 0 elliminated eigenvalue(s) The reduced state dynamics matrix Ar is 1.0000 -0.0393 -0.0980 -0.1066 0.0781 -0.2330 0.0777 0.0000 1.0312 0.2717 0.2609 -0.1533 0.6758 -0.3553 0.0000 0.0000 1.3887 0.6699 -0.4281 1.6389 -0.7615 0.0000 0.0000 0.0000 -1.2147 0.2423 -0.9792 0.4788 0.0000 0.0000 0.0000 0.0000 -1.0545 0.5035 -0.2788 0.0000 0.0000 0.0000 0.0000 0.0000 1.6355 -0.4323 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 The reduced descriptor matrix Er is 0.4100 0.2590 0.5080 -0.3109 0.0705 0.1429 -0.1477 -0.7629 -0.3464 0.0992 -0.3007 0.0619 0.2483 -0.0152 0.1120 -0.2124 -0.4184 -0.1288 0.0569 -0.4213 -0.6182 0.0000 0.1122 -0.0039 0.2771 -0.0758 0.0975 0.3923 0.0000 0.0000 0.3708 -0.4290 0.1006 0.1402 -0.2699 0.0000 0.0000 0.0000 0.0000 0.9458 -0.2211 0.2378 0.0000 0.0000 0.0000 0.5711 0.2648 0.5948 -0.5000 The reduced input/state matrix Br is -0.5597 0.2363 -0.4843 -0.0498 -0.4727 -0.1491 0.1802 1.1574 0.5995 0.1556 -0.1729 -0.3999 0.0000 0.2500 The reduced state/output matrix Cr is 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 4.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.1524 -1.7500
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TG01JY.html 0000664 0000000 0000000 00000054050 14560147231 0020374 0 ustar 00root root 0000000 0000000
Purpose
To find a reduced (controllable, observable, or irreducible) descriptor representation (Ar-lambda*Er,Br,Cr) for an original descriptor representation (A-lambda*E,B,C). The pencil Ar-lambda*Er is in an upper block Hessenberg form, with either Ar or Er upper triangular.Specification
SUBROUTINE TG01JY( JOB, SYSTYP, EQUIL, CKSING, RESTOR, N, M, P, A,
$ LDA, E, LDE, B, LDB, C, LDC, NR, INFRED, TOL,
$ IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER CKSING, EQUIL, JOB, RESTOR, SYSTYP
INTEGER INFO, LDA, LDB, LDC, LDE, LDWORK, M, N, NR, P
C .. Array Arguments ..
INTEGER INFRED(*), IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
$ E(LDE,*), TOL(*)
Arguments
Mode Parameters
JOB CHARACTER*1
Indicates whether the user wishes to remove the
uncontrollable and/or unobservable parts as follows:
= 'I': Remove both the uncontrollable and unobservable
parts to get an irreducible descriptor
representation;
= 'C': Remove the uncontrollable part only to get a
controllable descriptor representation;
= 'O': Remove the unobservable part only to get an
observable descriptor representation.
SYSTYP CHARACTER*1
Indicates the type of descriptor system algorithm
to be applied according to the assumed
transfer-function matrix as follows:
= 'R': Rational transfer-function matrix;
= 'S': Proper (standard) transfer-function matrix;
= 'P': Polynomial transfer-function matrix.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily scale
the system (A-lambda*E,B,C) as follows:
= 'S': Perform scaling;
= 'N': Do not perform scaling.
CKSING CHARACTER*1
Specifies whether the user wishes to check if the pencil
(A-lambda*E) is singular as follows:
= 'C': Check singularity;
= 'N': Do not check singularity.
If the pencil is singular, the reduced system computed for
CKSING = 'N' can be wrong.
RESTOR CHARACTER*1
Specifies whether the user wishes to save the system
matrices before each phase and restore them if no order
reduction took place as follows:
= 'R': Save and restore;
= 'N': Do not save the matrices.
Input/Output Parameters
N (input) INTEGER
The dimension of the descriptor state vector; also the
order of square matrices A and E, the number of rows of
matrix B, and the number of columns of matrix C. N >= 0.
M (input) INTEGER
The dimension of descriptor system input vector; also the
number of columns of matrix B. M >= 0.
P (input) INTEGER
The dimension of descriptor system output vector; also the
number of rows of matrix C. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state matrix A.
On exit, the leading NR-by-NR part of this array contains
the reduced order state matrix Ar of an irreducible,
controllable, or observable realization for the original
system, depending on the value of JOB, JOB = 'I',
JOB = 'C', or JOB = 'O', respectively.
The matrix Ar is upper triangular if SYSTYP = 'P'.
If SYSTYP = 'S' and JOB = 'C', the matrix [Br Ar]
is in a controllable staircase form (see SLICOT Library
routine TG01HD).
If SYSTYP = 'S' and JOB = 'I' or 'O', the matrix ( Ar )
( Cr )
is in an observable staircase form (see TG01HD).
The resulting Ar has INFRED(5) nonzero sub-diagonals.
The block structure of staircase forms is contained
in the leading INFRED(7) elements of IWORK.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain the original descriptor matrix E.
On exit, the leading NR-by-NR part of this array contains
the reduced order descriptor matrix Er of an irreducible,
controllable, or observable realization for the original
system, depending on the value of JOB, JOB = 'I',
JOB = 'C', or JOB = 'O', respectively.
The resulting Er has INFRED(6) nonzero sub-diagonals.
If at least for one k = 1,...,4, INFRED(k) >= 0, then the
resulting Er is structured being either upper triangular
or block Hessenberg, in accordance to the last
performed order reduction phase (see METHOD).
The block structure of staircase forms is contained
in the leading INFRED(7) elements of IWORK.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M),
if JOB = 'C', or (LDB,MAX(M,P)), otherwise.
On entry, the leading N-by-M part of this array must
contain the original input matrix B; if JOB = 'I',
or JOB = 'O', the remainder of the leading N-by-MAX(M,P)
part is used as internal workspace.
On exit, the leading NR-by-M part of this array contains
the reduced input matrix Br of an irreducible,
controllable, or observable realization for the original
system, depending on the value of JOB, JOB = 'I',
JOB = 'C', or JOB = 'O', respectively.
If JOB = 'C', only the first IWORK(1) rows of B are
nonzero.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original output matrix C; if JOB = 'I',
or JOB = 'O', the remainder of the leading MAX(M,P)-by-N
part is used as internal workspace.
On exit, the leading P-by-NR part of this array contains
the transformed state/output matrix Cr of an irreducible,
controllable, or observable realization for the original
system, depending on the value of JOB, JOB = 'I',
JOB = 'C', or JOB = 'O', respectively.
If JOB = 'I', or JOB = 'O', only the last IWORK(1) columns
(in the first NR columns) of C are nonzero.
LDC INTEGER
The leading dimension of array C.
LDC >= MAX(1,M,P) if N > 0.
LDC >= 1 if N = 0.
NR (output) INTEGER
The order of the reduced descriptor representation
(Ar-lambda*Er,Br,Cr) of an irreducible, controllable,
or observable realization for the original system,
depending on JOB = 'I', JOB = 'C', or JOB = 'O',
respectively.
INFRED (output) INTEGER array, dimension 7
This array contains information on performed reduction
and on structure of resulting system matrices as follows:
INFRED(k) >= 0 (k = 1, 2, 3, or 4) if Phase k of reduction
(see METHOD) has been performed. In this
case, INFRED(k) is the achieved order
reduction in Phase k.
INFRED(k) < 0 (k = 1, 2, 3, or 4) if Phase k was not
performed.
INFRED(5) - the number of nonzero sub-diagonals of A.
INFRED(6) - the number of nonzero sub-diagonals of E.
INFRED(7) - the number of blocks in the resulting
staircase form at last performed reduction
phase. The block dimensions are contained
in the first INFRED(7) elements of IWORK.
Tolerances
TOL DOUBLE PRECISION array, dimension 3
TOL(1) is the tolerance to be used in rank determinations
when transforming (A-lambda*E,B,C). If the user sets
TOL(1) > 0, then the given value of TOL(1) is used as a
lower bound for reciprocal condition numbers in rank
determinations; a (sub)matrix whose estimated condition
number is less than 1/TOL(1) is considered to be of full
rank. If the user sets TOL(1) <= 0, then an implicitly
computed, default tolerance, defined by TOLDEF1 = N*N*EPS,
is used instead, where EPS is the machine precision (see
LAPACK Library routine DLAMCH). TOL(1) < 1.
TOL(2) is the tolerance to be used for checking pencil
singularity when CKSING = 'C', or singularity of the
matrices A and E when CKSING = 'N'. If the user sets
TOL(2) > 0, then the given value of TOL(2) is used.
If the user sets TOL(2) <= 0, then an implicitly
computed, default tolerance, defined by TOLDEF2 = 10*EPS,
is used instead. TOL(2) < 1.
TOL(3) is the threshold value for magnitude of the matrix
elements, if EQUIL = 'S': elements with magnitude less
than or equal to TOL(3) are ignored for scaling. If the
user sets TOL(3) >= 0, then the given value of TOL(3) is
used. If the user sets TOL(3) < 0, then an implicitly
computed, default threshold, defined by THRESH = c*EPS,
where c = MAX(norm_1(A,E,B,C)) is used instead.
TOL(3) = 0 is not always a good choice. TOL(3) < 1.
TOL(3) is not used if EQUIL = 'N'.
Workspace
IWORK INTEGER array, dimension (2*N+MAX(M,P))
On exit, if INFO = 0, the leading INFRED(7) elements of
IWORK contain the orders of the diagonal blocks of
Ar-lambda*Er.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,x,y,8*N), if EQUIL = 'S',
LDWORK >= MAX(1,x,y), if EQUIL = 'N',
where x = MAX(2*(z+MAX(M,P)+N-1),N*N+4*N), if RESTOR = 'R'
x = MAX( 2*(MAX(M,P)+N-1),N*N+4*N), if RESTOR = 'N'
y = 2*N*N+10*N+MAX(N,23), if CKSING = 'C',
y = 0, if CKSING = 'N',
z = 2*N*N+N*M+N*P, if JOB = 'I',
z = 0, if JOB <> 'I'.
For good performance, LDWORK should be generally larger.
If RESTOR = 'R', or
LDWORK >= MAX(1,2*N*N+N*M+N*P+2*(MAX(M,P)+N-1),
more accurate results are to be expected by considering
only those reductions phases (see METHOD), where effective
order reduction occurs. This is achieved by saving the
system matrices before each phase and restoring them if no
order reduction took place. Actually, if JOB = 'I' and
RESTOR = 'N', then the saved matrices are those obtained
after orthogonally triangularizing the matrix A (if
SYSTYP = 'R' or 'P'), or the matrix E (if SYSTYP = 'R'
or 'S').
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA. The optimal workspace includes the
extra space for improving the accuracy.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the given pencil A - lambda*E is numerically
singular and the reduced system is not computed.
This error can be returned only if CKSING = 'C'.
Method
The subroutine is based on the reduction algorithms of [1], but
with a different ordering of the phases.
The order reduction is performed in 4 phases:
Phase 1: Eliminate all infinite and finite nonzero uncontrollable
eigenvalues. The resulting matrix ( Br Er ) is in a
controllable staircase form (see TG01HD), and Ar is
upper triangular.
This phase is performed if JOB = 'I' or 'C' and
SYSTYP = 'R' or 'P'.
Phase 2: Eliminate all infinite and finite nonzero unobservable
eigenvalues. The resulting matrix ( Er ) is in an
( Cr )
observable staircase form (see SLICOT Library routine
TG01ID), and Ar is upper triangular.
This phase is performed if JOB = 'I' or 'O' and
SYSTYP = 'R' or 'P'.
Phase 3: Eliminate all finite uncontrollable eigenvalues.
The resulting matrix ( Br Ar ) is in a controllable
staircase form (see TG01HD), and Er is upper triangular.
This phase is performed if JOB = 'I' or 'C' and
SYSTYP = 'R' or 'S'.
Phase 4: Eliminate all finite unobservable eigenvalues.
The resulting matrix ( Ar ) is in an observable
( Cr )
staircase form (see TG01ID), and Er is upper triangular.
This phase is performed if JOB = 'I' or 'O' and
SYSTYP = 'R' or 'S'.
The routine checks the singularity of the matrices A and/or E
(depending on JOB and SYSTYP) and skips the unnecessary phases.
See FURTHER COMMENTS.
References
[1] A. Varga
Computation of Irreducible Generalized State-Space
Realizations.
Kybernetika, vol. 26, pp. 89-106, 1990.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( N**3 ) floating point operations.Further Comments
If the pencil A-lambda*E has no zero eigenvalues, then an irreducible realization is computed skipping Phases 3 and 4 (equivalent to setting: JOB = 'I' and SYSTYP = 'P').Example
Program Text
* TG01JY EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDE
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDE = NMAX )
INTEGER LDWORK, LIWORK
PARAMETER ( LDWORK = 2*NMAX*NMAX +
$ MAX( 2*( NMAX*( NMAX + MMAX + PMAX ) +
$ MAX( MMAX, PMAX ) + NMAX - 1 ),
$ 10*NMAX + MAX( NMAX, 23 ) ),
$ LIWORK = 2*NMAX + MAX( MMAX, PMAX ) )
* .. Local Scalars ..
CHARACTER CKSING, EQUIL, JOB, RESTOR, SYSTYP
INTEGER I, INFO, J, M, N, NR, P
* .. Local Arrays ..
INTEGER INFRED(7), IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ DWORK(LDWORK), E(LDE,NMAX), TOL(3)
* .. External Subroutines ..
EXTERNAL TG01JY
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL(1), TOL(2), TOL(3), JOB,
$ SYSTYP, EQUIL, CKSING, RESTOR
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find the irreducible descriptor system (Ar-lambda Er,Br,Cr).
CALL TG01JY( JOB, SYSTYP, EQUIL, CKSING, RESTOR, N, M, P,
$ A, LDA, E, LDE, B, LDB, C, LDC, NR, INFRED,
$ TOL, IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) NR
WRITE ( NOUT, FMT = 99991 )
DO 10 I = 1, 4
IF( INFRED(I).GE.0 )
$ WRITE ( NOUT, FMT = 99990 ) I, INFRED(I)
10 CONTINUE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
20 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 30 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,NR )
30 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 50 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
50 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01JY EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01JY = ',I2)
99997 FORMAT (/' The reduced state dynamics matrix Ar is ')
99996 FORMAT (/' The reduced descriptor matrix Er is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (' Order of reduced system =', I5 )
99993 FORMAT (/' The reduced input/state matrix Br is ')
99992 FORMAT (/' The reduced state/output matrix Cr is ')
99991 FORMAT (/' Achieved order reductions in different phases')
99990 FORMAT (' Phase',I2,':', I3, ' elliminated eigenvalue(s)' )
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
99986 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
TG01JY EXAMPLE PROGRAM DATA
9 2 2 0.0 0.0 0.0 I R N N N
-2 -3 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0
0 0 -2 -3 0 0 0 0 0
0 0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1 0
1 0
0 0
0 1
0 0
-1 0
0 0
0 -1
0 0
0 0
1 0 1 -3 0 1 0 2 0
0 1 1 3 0 1 0 0 1
Program Results
TG01JY EXAMPLE PROGRAM RESULTS Order of reduced system = 7 Achieved order reductions in different phases Phase 1: 0 elliminated eigenvalue(s) Phase 2: 2 elliminated eigenvalue(s) The reduced state dynamics matrix Ar is 1.0000 -0.0393 -0.0980 0.1066 -0.0781 0.2330 -0.0777 0.0000 1.0312 0.2717 -0.2609 0.1533 -0.6758 0.3553 0.0000 0.0000 1.3887 -0.6699 0.4281 -1.6389 0.7615 0.0000 0.0000 0.0000 1.2147 -0.2423 0.9792 -0.4788 0.0000 0.0000 0.0000 0.0000 1.0545 -0.5035 0.2788 0.0000 0.0000 0.0000 0.0000 0.0000 1.6355 -0.4323 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 The reduced descriptor matrix Er is 0.4100 0.2590 0.5080 0.3109 -0.0705 -0.1429 0.1477 -0.7629 -0.3464 0.0992 0.3007 -0.0619 -0.2483 0.0152 0.1120 -0.2124 -0.4184 0.1288 -0.0569 0.4213 0.6182 0.0000 0.1122 -0.0039 -0.2771 0.0758 -0.0975 -0.3923 0.0000 0.0000 0.3708 0.4290 -0.1006 -0.1402 0.2699 0.0000 0.0000 0.0000 0.0000 0.9458 -0.2211 0.2378 0.0000 0.0000 0.0000 0.5711 0.2648 0.5948 -0.5000 The reduced input/state matrix Br is 0.5597 -0.2363 0.4843 0.0498 0.4727 0.1491 -0.1802 -1.1574 -0.5995 -0.1556 -0.1729 -0.3999 0.0000 0.2500 The reduced state/output matrix Cr is 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 4.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.1524 -1.7500
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TG01KD.html 0000664 0000000 0000000 00000016242 14560147231 0020351 0 ustar 00root root 0000000 0000000
Purpose
To compute for a single-input single-output descriptor system, (A, E, B, C), with E upper triangular, a transformed system, (Q'*A*Z, Q'*E*Z, Q'*B, C*Z), via an orthogonal equivalence transformation, so that Q'*B has only the first element nonzero and Q'*E*Z remains upper triangular.Specification
SUBROUTINE TG01KD( JOBE, COMPC, COMPQ, COMPZ, N, A, LDA, E, LDE,
$ B, C, INCC, Q, LDQ, Z, LDZ, INFO )
C .. Scalar Arguments ..
CHARACTER COMPC, COMPQ, COMPZ, JOBE
INTEGER INCC, INFO, LDA, LDE, LDQ, LDZ, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(*), C(*), E(LDE,*), Q(LDQ,*),
$ Z(LDZ,*)
Arguments
Mode Parameters
JOBE CHARACTER*1
Specifies whether E is an upper triangular or an identity
matrix, as follows:
= 'U': The matrix E is an upper triangular matrix;
= 'I': The matrix E is assumed identity and is not given.
COMPC CHARACTER*1
Indicates whether the user wishes to transform the system
output matrix C, as follows:
= 'C': Transform the system output matrix C;
= 'N': Do not transform the system output matrix C.
COMPQ CHARACTER*1
Indicates whether the user wishes to accumulate in a
matrix Q the orthogonal row transformations, as follows:
= 'N': Do not form Q;
= 'I': Q is initialized to the unit matrix and the
orthogonal transformation matrix Q is returned;
= 'U': The given matrix Q is updated by the orthogonal
transformations used.
COMPZ CHARACTER*1
Indicates whether the user wishes to accumulate in a
matrix Z the orthogonal column transformations, as
follows:
= 'N': Do not form Z;
= 'I': Z is initialized to the unit matrix and the
orthogonal transformation matrix Z is returned;
= 'U': The given matrix Z is updated by the orthogonal
transformations used.
Input/Output Parameters
N (input) INTEGER
The dimension of the descriptor state vector; also the
order of square matrices A and E, the number of rows of
matrix B, and the number of columns of matrix C. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state matrix A.
On exit, the leading N-by-N part of this array contains
the transformed state matrix Q'*A*Z.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
E (input/output) DOUBLE PRECISION array, dimension (LDE,*)
On entry, if JOBE = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the descriptor matrix E. The lower
triangular part under the first subdiagonal is not
referenced.
On exit, if JOBE = 'U', the leading N-by-N upper
triangular part of this array contains the upper
triangular part of the transformed descriptor matrix,
Q'*E*Z.
If JOBE = 'I', this array is not referenced.
LDE INTEGER
The leading dimension of the array E.
LDE >= MAX(1,N), if JOBE = 'U';
LDE >= 1, if JOBE = 'I'.
B (input/output) DOUBLE PRECISION array, dimension (N)
On entry, the leading N part of this array must contain
the original input matrix B.
On exit, the leading N part of this array contains the
transformed input matrix Q'*B with all elements but the
first set to zero.
C (input/output) DOUBLE PRECISION array, dimension
((N-1)*INCC+1)
On entry, if COMPC = 'C', the elements 1, INCC+1, ...,
(N-1)*INCC+1 of this array must contain the original
output vector C.
On exit, if COMPC = 'C', the elements 1, INCC+1, ...,
(N-1)*INCC+1 of this array contain the transformed output
vector C*Z.
If COMPC = 'N', this array is not referenced.
INCC INTEGER
If COMPC = 'C', the increment between successive values
of C. INCC > 0.
If COMPC = 'N', INCC is not used.
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,*)
On entry, if COMPQ = 'U', the leading N-by-N part of this
array must contain the given matrix Q1. Otherwise, this
array need not be set on input.
On exit, if COMPU <> 'N', the leading N-by-N part of this
array contains the orthogonal transformation matrix used
(Q1*Q if COMPQ = 'U').
If COMPQ = 'N', this array is not referenced.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= max(1,N), if COMPQ <> 'N'.
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,*)
On entry, if COMPZ = 'U', the leading N-by-N part of this
array must contain the given matrix Z1. Otherwise, this
array need not be set on input.
On exit, if COMPZ <> 'N', the leading N-by-N part of this
array contains the orthogonal transformation matrix used
(Z1*Z if COMPZ = 'U').
If COMPZ = 'N', this array is not referenced.
LDZ INTEGER
The leading dimension of the array Z.
LDZ >= 1, if COMPZ = 'N';
LDZ >= max(1,N), if COMPZ <> 'N'.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Givens rotations are used to annihilate the last N-1 elements of B in reverse order, but preserve the form of E.Numerical Aspects
The algorithm is numerically backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute for a single-input single-output descriptor system, (A, E, B, C), with E upper triangular, a transformed system, (Q'*A*Z, Q'*E*Z, Q'*B, C*Z), via an orthogonal equivalence transformation, so that Q'*B has only the first element nonzero and Q'*E*Z remains upper triangular.Specification
SUBROUTINE TG01KZ( JOBE, COMPC, COMPQ, COMPZ, N, A, LDA, E, LDE,
$ B, C, INCC, Q, LDQ, Z, LDZ, INFO )
C .. Scalar Arguments ..
CHARACTER COMPC, COMPQ, COMPZ, JOBE
INTEGER INCC, INFO, LDA, LDE, LDQ, LDZ, N
C .. Array Arguments ..
COMPLEX*16 A(LDA,*), B(*), C(*), E(LDE,*), Q(LDQ,*),
$ Z(LDZ,*)
Arguments
Mode Parameters
JOBE CHARACTER*1
Specifies whether E is an upper triangular or an identity
matrix, as follows:
= 'U': The matrix E is an upper triangular matrix;
= 'I': The matrix E is assumed identity and is not given.
COMPC CHARACTER*1
Indicates whether the user wishes to transform the system
output matrix C, as follows:
= 'C': Transform the system output matrix C;
= 'N': Do not transform the system output matrix C.
COMPQ CHARACTER*1
Indicates whether the user wishes to accumulate in a
matrix Q the orthogonal row transformations, as follows:
= 'N': Do not form Q;
= 'I': Q is initialized to the unit matrix and the
orthogonal transformation matrix Q is returned;
= 'U': The given matrix Q is updated by the orthogonal
transformations used.
COMPZ CHARACTER*1
Indicates whether the user wishes to accumulate in a
matrix Z the orthogonal column transformations, as
follows:
= 'N': Do not form Z;
= 'I': Z is initialized to the unit matrix and the
orthogonal transformation matrix Z is returned;
= 'U': The given matrix Z is updated by the orthogonal
transformations used.
Input/Output Parameters
N (input) INTEGER
The dimension of the descriptor state vector; also the
order of square matrices A and E, the number of rows of
matrix B, and the number of columns of matrix C. N >= 0.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state matrix A.
On exit, the leading N-by-N part of this array contains
the transformed state matrix Q'*A*Z.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
E (input/output) COMPLEX*16 array, dimension (LDE,*)
On entry, if JOBE = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the descriptor matrix E. The lower
triangular part under the first subdiagonal is not
referenced.
On exit, if JOBE = 'U', the leading N-by-N upper
triangular part of this array contains the upper
triangular part of the transformed descriptor matrix,
Q'*E*Z.
If JOBE = 'I', this array is not referenced.
LDE INTEGER
The leading dimension of the array E.
LDE >= MAX(1,N), if JOBE = 'U';
LDE >= 1, if JOBE = 'I'.
B (input/output) COMPLEX*16 array, dimension (N)
On entry, the leading N part of this array must contain
the original input matrix B.
On exit, the leading N part of this array contains the
transformed input matrix Q'*B with all elements but the
first set to zero.
C (input/output) COMPLEX*16 array, dimension
((N-1)*INCC+1)
On entry, if COMPC = 'C', the elements 1, INCC+1, ...,
(N-1)*INCC+1 of this array must contain the original
output vector C.
On exit, if COMPC = 'C', the elements 1, INCC+1, ...,
(N-1)*INCC+1 of this array contain the transformed output
vector C*Z.
If COMPC = 'N', this array is not referenced.
INCC INTEGER
If COMPC = 'C', the increment between successive values
of C. INCC > 0.
If COMPC = 'N', INCC is not used.
Q (input/output) COMPLEX*16 array, dimension (LDQ,*)
On entry, if COMPQ = 'U', the leading N-by-N part of this
array must contain the given matrix Q1. Otherwise, this
array need not be set on input.
On exit, if COMPU <> 'N', the leading N-by-N part of this
array contains the orthogonal transformation matrix used
(Q1*Q if COMPQ = 'U').
If COMPQ = 'N', this array is not referenced.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= max(1,N), if COMPQ <> 'N'.
Z (input/output) COMPLEX*16 array, dimension (LDZ,*)
On entry, if COMPZ = 'U', the leading N-by-N part of this
array must contain the given matrix Z1. Otherwise, this
array need not be set on input.
On exit, if COMPZ <> 'N', the leading N-by-N part of this
array contains the orthogonal transformation matrix used
(Z1*Z if COMPZ = 'U').
If COMPZ = 'N', this array is not referenced.
LDZ INTEGER
The leading dimension of the array Z.
LDZ >= 1, if COMPZ = 'N';
LDZ >= max(1,N), if COMPZ <> 'N'.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Givens rotations are used to annihilate the last N-1 elements of B in reverse order, but preserve the form of E.Numerical Aspects
The algorithm is numerically backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute orthogonal transformation matrices Q and Z which
reduce the regular pole pencil A-lambda*E of the descriptor system
(A-lambda*E,B,C) to the form (if JOB = 'F')
( Af * ) ( Ef * )
Q'*A*Z = ( ) , Q'*E*Z = ( ) , (1)
( 0 Ai ) ( 0 Ei )
or to the form (if JOB = 'I')
( Ai * ) ( Ei * )
Q'*A*Z = ( ) , Q'*E*Z = ( ) , (2)
( 0 Af ) ( 0 Ef )
where the subpencil Af-lambda*Ef, with Ef nonsingular and upper
triangular, contains the finite eigenvalues, and the subpencil
Ai-lambda*Ei, with Ai nonsingular and upper triangular, contains
the infinite eigenvalues. The subpencil Ai-lambda*Ei is in a
staircase form (see METHOD). If JOBA = 'H', the submatrix Af
is further reduced to an upper Hessenberg form.
Specification
SUBROUTINE TG01LD( JOB, JOBA, COMPQ, COMPZ, N, M, P, A, LDA,
$ E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ, NF, ND,
$ NIBLCK, IBLCK, TOL, IWORK, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, COMPZ, JOB, JOBA
INTEGER INFO, LDA, LDB, LDC, LDE, LDQ, LDWORK, LDZ, M,
$ N, ND, NF, NIBLCK, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IBLCK( * ), IWORK(*)
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ DWORK( * ), E( LDE, * ), Q( LDQ, * ),
$ Z( LDZ, * )
Arguments
Mode Parameters
JOB CHARACTER*1
= 'F': perform the finite-infinite separation;
= 'I': perform the infinite-finite separation.
JOBA CHARACTER*1
= 'H': reduce Af further to an upper Hessenberg form;
= 'N': keep Af unreduced.
COMPQ CHARACTER*1
= 'N': do not compute Q;
= 'I': Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= 'U': Q must contain an orthogonal matrix Q1 on entry,
and the product Q1*Q is returned.
COMPZ CHARACTER*1
= 'N': do not compute Z;
= 'I': Z is initialized to the unit matrix, and the
orthogonal matrix Z is returned;
= 'U': Z must contain an orthogonal matrix Z1 on entry,
and the product Z1*Z is returned.
Input/Output Parameters
N (input) INTEGER
The number of rows of the matrix B, the number of columns
of the matrix C and the order of the square matrices A
and E. N >= 0.
M (input) INTEGER
The number of columns of the matrix B. M >= 0.
P (input) INTEGER
The number of rows of the matrix C. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the N-by-N state matrix A.
On exit, the leading N-by-N part of this array contains
the transformed state matrix Q'*A*Z,
( Af * ) ( Ai * )
Q'*A*Z = ( ) , or Q'*A*Z = ( ) ,
( 0 Ai ) ( 0 Af )
depending on JOB, with Af an NF-by-NF matrix, and Ai an
(N-NF)-by-(N-NF) nonsingular and upper triangular matrix.
If JOBA = 'H', Af is in an upper Hessenberg form.
Otherwise, Af is unreduced.
Ai has a block structure as in (3) or (4), where A0,0 is
ND-by-ND and Ai,i , for i = 1, ..., NIBLCK, is
IBLCK(i)-by-IBLCK(i).
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain the N-by-N descriptor matrix E.
On exit, the leading N-by-N part of this array contains
the transformed descriptor matrix Q'*E*Z,
( Ef * ) ( Ei * )
Q'*E*Z = ( ) , or Q'*E*Z = ( ) ,
( 0 Ei ) ( 0 Ef )
depending on JOB, with Ef an NF-by-NF nonsingular matrix,
and Ei an (N-NF)-by-(N-NF) nilpotent matrix in an upper
block triangular form, as in (3) or (4).
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,K),
where K = M if JOB = 'F', and K = MAX(M,P) if JOB = 'I'.
On entry, the leading N-by-M part of this array must
contain the N-by-M input matrix B.
On exit, the leading N-by-M part of this array contains
the transformed input matrix Q'*B.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed matrix C*Z.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,K),
where K = P if JOB = 'F', and K = MAX(M,P) if JOB = 'I'.
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
If COMPQ = 'N': Q is not referenced.
If COMPQ = 'I': on entry, Q need not be set;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix Q,
where Q' is the product of Householder
transformations applied to A, E, and B on
the left.
If COMPQ = 'U': on entry, the leading N-by-N part of this
array must contain an orthogonal matrix
Q1;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix
Q1*Q.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1,N), if COMPQ = 'I' or 'U'.
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
If COMPZ = 'N': Z is not referenced.
If COMPZ = 'I': on entry, Z need not be set;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix Z,
which is the product of Householder
transformations applied to A, E, and C on
the right.
If COMPZ = 'U': on entry, the leading N-by-N part of this
array must contain an orthogonal matrix
Z1;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix
Z1*Z.
LDZ INTEGER
The leading dimension of the array Z.
LDZ >= 1, if COMPZ = 'N';
LDZ >= MAX(1,N), if COMPZ = 'I' or 'U'.
NF (output) INTEGER.
The order of the reduced matrices Af and Ef; also, the
number of finite generalized eigenvalues of the pencil
A-lambda*E.
ND (output) INTEGER.
The number of non-dynamic infinite eigenvalues of the
pair (A,E). Note: N-ND is the rank of the matrix E.
NIBLCK (output) INTEGER
If ND > 0, the number of infinite blocks minus one.
If ND = 0, then NIBLCK = 0.
IBLCK (output) INTEGER array, dimension (N)
IBLCK(i) contains the dimension of the i-th block in the
staircase form (3) or (4), with i = 1,2, ..., NIBLCK.
Tolerances
TOL DOUBLE PRECISION
A tolerance used in rank decisions to determine the
effective rank, which is defined as the order of the
largest leading (or trailing) triangular submatrix in the
QR factorization with column pivoting whose estimated
condition number is less than 1/TOL. If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
TOLDEF = N**2*EPS, is used instead, where EPS is the
machine precision (see LAPACK Library routine DLAMCH).
TOL < 1.
Workspace
IWORK INTEGER array, dimension (N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= 1, and if N > 0,
LDWORK >= N + MAX(3*N,M,P).
If LDWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message related to LDWORK is issued by
XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the pencil A-lambda*E is not regular.
Method
The subroutine is based on the reduction algorithm of [1].
If JOB = 'F', the matrices Ai and Ei have the form
( A0,0 A0,k ... A0,1 ) ( 0 E0,k ... E0,1 )
Ai = ( 0 Ak,k ... Ak,1 ) , Ei = ( 0 0 ... Ek,1 ) ; (3)
( : : . : ) ( : : . : )
( 0 0 ... A1,1 ) ( 0 0 ... 0 )
if JOB = 'I', the matrices Ai and Ei have the form
( A1,1 ... A1,k A1,0 ) ( 0 ... E1,k E1,0 )
Ai = ( : . : : ) , Ei = ( : . : : ) , (4)
( : ... Ak,k Ak,0 ) ( : ... 0 Ek,0 )
( 0 ... 0 A0,0 ) ( 0 ... 0 0 )
where Ai,i , for i = 0, 1, ..., k, are nonsingular upper
triangular matrices. A0,0 corresponds to the non-dynamic infinite
modes of the system.
References
[1] Misra, P., Van Dooren, P., and Varga, A.
Computation of structural invariants of generalized
state-space systems.
Automatica, 30, pp. 1921-1936, 1994.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( N**3 ) floating point operations.Further Comments
The number of infinite poles is computed as
NIBLCK
NINFP = Sum IBLCK(i) = N - ND - NF.
i=1
The multiplicities of infinite poles can be computed as follows:
there are IBLCK(k)-IBLCK(k+1) infinite poles of multiplicity
k, for k = 1, ..., NIBLCK, where IBLCK(NIBLCK+1) = 0.
Note that each infinite pole of multiplicity k corresponds to
an infinite eigenvalue of multiplicity k+1.
Example
Program Text
* TG01LD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDE, LDQ, LDZ
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDE = NMAX, LDQ = NMAX, LDZ = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX+MAX( 3*NMAX, MMAX, PMAX ) )
* .. Local Scalars ..
CHARACTER*1 COMPQ, COMPZ, JOB, JOBA
INTEGER I, INFO, J, M, N, ND, NF, NIBLCK, P
DOUBLE PRECISION TOL
* .. Local Arrays ..
INTEGER IBLCK(NMAX), IWORK(NMAX)
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,NMAX),
$ Z(LDZ,NMAX)
* .. External Subroutines ..
EXTERNAL TG01LD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, JOB, JOBA, TOL
COMPQ = 'I'
COMPZ = 'I'
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find the reduced descriptor system
* (A-lambda E,B,C).
CALL TG01LD( JOB, JOBA, COMPQ, COMPZ, N, M, P, A, LDA,
$ E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ, NF,
$ ND, NIBLCK, IBLCK, TOL, IWORK, DWORK,
$ LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) NF, ND
WRITE ( NOUT, FMT = 99989 ) NIBLCK + 1
IF ( NIBLCK.GT.0 ) THEN
WRITE ( NOUT, FMT = 99985 )
$ ( IBLCK(I), I = 1, NIBLCK )
END IF
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 30 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
30 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 40 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
40 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 50 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,N )
50 CONTINUE
WRITE ( NOUT, FMT = 99990 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
60 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01LD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01LD = ',I2)
99997 FORMAT (/' The reduced state dynamics matrix Q''*A*Z is ')
99996 FORMAT (/' The reduced descriptor matrix Q''*E*Z is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (' Order of reduced system =', I5/
$ ' Number of non-dynamic infinite eigenvalues =', I5)
99993 FORMAT (/' The reduced input/state matrix Q''*B is ')
99992 FORMAT (/' The reduced state/output matrix C*Z is ')
99991 FORMAT (/' The left transformation matrix Q is ')
99990 FORMAT (/' The right transformation matrix Z is ')
99989 FORMAT ( ' Number of infinite blocks = ',I5)
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
99986 FORMAT (/' P is out of range.',/' P = ',I5)
99985 FORMAT ( ' Dimension of the blocks'/20I5)
END
Program Data
TG01LD EXAMPLE PROGRAM DATA
4 2 2 F N 0.0
-1 0 0 3
0 0 1 2
1 1 0 4
0 0 0 0
1 2 0 0
0 1 0 1
3 9 6 3
0 0 2 0
1 0
0 0
0 1
1 1
-1 0 1 0
0 1 -1 1
Program Results
TG01LD EXAMPLE PROGRAM RESULTS Order of reduced system = 3 Number of non-dynamic infinite eigenvalues = 1 Number of infinite blocks = 1 The reduced state dynamics matrix Q'*A*Z is 2.4497 -1.3995 0.2397 -4.0023 -0.0680 -0.0030 0.1739 -1.6225 0.3707 0.0161 -0.9482 0.1049 0.0000 0.0000 0.0000 2.2913 The reduced descriptor matrix Q'*E*Z is 9.9139 4.7725 -3.4725 -2.3836 0.0000 -1.2024 2.0137 0.7926 0.0000 0.0000 0.2929 -0.9914 0.0000 0.0000 0.0000 0.0000 The reduced input/state matrix Q'*B is -0.2157 -0.9705 0.3015 0.9516 0.7595 0.0991 1.1339 0.3780 The reduced state/output matrix C*Z is 0.5345 -1.1134 0.3758 0.5774 -1.0690 0.2784 -1.2026 0.5774 The left transformation matrix Q is -0.2157 -0.5088 0.6109 0.5669 -0.1078 -0.2544 -0.7760 0.5669 -0.9705 0.1413 -0.0495 -0.1890 0.0000 0.8102 0.1486 0.5669 The right transformation matrix Z is -0.5345 0.6263 0.4617 -0.3299 -0.8018 -0.5219 -0.2792 -0.0825 0.0000 -0.4871 0.8375 0.2474 -0.2673 0.3132 -0.0859 0.9073
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TG01LY.html 0000664 0000000 0000000 00000024710 14560147231 0020376 0 ustar 00root root 0000000 0000000
Purpose
To compute orthogonal transformation matrices Q and Z which reduce
the regular pole pencil A-lambda*E of the descriptor system
(A-lambda*E,B,C), with the A and E matrices in the form
( A11 A12 A13 ) ( E11 0 0 )
A = ( A21 A22 A23 ) , E = ( 0 0 0 ) , (1)
( A31 0 0 ) ( 0 0 0 )
where E11 and A22 are nonsingular and upper triangular matrices,
to the form
( Af * ) ( Ef * )
Q'*A*Z = ( ) , Q'*E*Z = ( ) ,
( 0 Ai ) ( 0 Ei )
where the subpencil Af-lambda*Ef contains the finite eigenvalues
and the subpencil Ai-lambda*Ei contains the infinite eigenvalues.
The subpencil Ai-lambda*Ei is in a staircase form with the
matrices Ai and Ei of form
( A0,0 A0,k ... A0,1 ) ( 0 E0,k ... E0,1 )
Ai = ( 0 Ak,k ... Ak,1 ) , Ei = ( 0 0 ... Ek,1 ) , (2)
( : : ... : ) ( : : ... : )
( 0 0 ... A1,1 ) ( 0 0 ... 0 )
where Ai,i, for i = 0, 1, ..., k, are nonsingular upper triangular
matrices.
Specification
SUBROUTINE TG01LY( COMPQ, COMPZ, N, M, P, RANKE, RNKA22, A, LDA,
$ E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ, NF,
$ NIBLCK, IBLCK, TOL, IWORK, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
LOGICAL COMPQ, COMPZ
INTEGER INFO, LDA, LDB, LDC, LDE, LDQ, LDWORK, LDZ, M,
$ N, NF, NIBLCK, P, RANKE, RNKA22
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IBLCK( * ), IWORK(*)
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ DWORK( * ), E( LDE, * ), Q( LDQ, * ),
$ Z( LDZ, * )
Arguments
Mode Parameters
COMPQ LOGICAL
Specify the option to accumulate or not the performed
left transformations:
COMPQ = .FALSE. : do not accumulate the transformations;
COMPQ = .TRUE. : accumulate the transformations; in this
case, Q must contain an orthogonal matrix Q1
on entry, and the product Q1*Q is returned.
COMPZ LOGICAL
Specify the option to accumulate or not the performed
right transformations:
COMPZ = .FALSE. : do not accumulate the transformations;
COMPZ = .TRUE. : accumulate the transformations; in this
case, Z must contain an orthogonal matrix Z1
on entry, and the product Z1*Z is returned.
Input/Output Parameters
N (input) INTEGER
The number of rows of the matrix B, the number of columns
of the matrix C and the order of the square matrices A
and E. N >= 0.
M (input) INTEGER
The number of columns of the matrix B. M >= 0.
P (input) INTEGER
The number of rows of the matrix C. P >= 0.
RANKE (input) INTEGER
The rank of the matrix E; also, the order of the upper
triangular matrix E11. 0 <= RANKE <= N.
RNKA22 (input) DOUBLE PRECISION
The order of the nonsingular submatrix A22 of A.
0 <= RNKA22 <= N - RANKE.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the N-by-N state matrix A in the form (1).
On exit, the leading N-by-N part of this array contains
the transformed state matrix Q'*A*Z,
( Af * )
Q'*A*Z = ( ) ,
( 0 Ai )
where Af is NF-by-NF and Ai is (N-NF)-by-(N-NF).
The submatrix Ai is in the staircase form (2), where A0,0
is (N-RANKE)-by-(N-RANKE), and Ai,i , for i = 1, ...,
NIBLCK is IBLCK(i)-by-IBLCK(i).
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain the N-by-N descriptor matrix E in the form (1).
On exit, the leading N-by-N part of this array contains
the transformed descriptor matrix Q'*E*Z,
( Ef * )
Q'*E*Z = ( ) ,
( 0 Ei )
where Ef is an NF-by-NF nonsingular matrix and Ei is an
(N-NF)-by-(N-NF) nilpotent matrix in the staircase
form (2).
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the N-by-M input matrix B.
On exit, the leading N-by-M part of this array contains
the transformed input matrix Q'*B.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed matrix C*Z.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,P).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
If COMPQ = .FALSE., Q is not referenced.
If COMPQ = .TRUE., on entry, the leading N-by-N part of
this array must contain an orthogonal matrix
Q1; on exit, the leading N-by-N part of this
array contains the orthogonal matrix Q1*Q.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = .FALSE.;
LDQ >= MAX(1,N), if COMPQ = .TRUE. .
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
If COMPZ = .FALSE., Z is not referenced.
If COMPZ = .TRUE., on entry, the leading N-by-N part of
this array must contain an orthogonal matrix
Z1; on exit, the leading N-by-N part of this
array contains the orthogonal matrix Z1*Z.
LDZ INTEGER
The leading dimension of the array Z.
LDZ >= 1, if COMPZ = .FALSE.;
LDZ >= MAX(1,N), if COMPZ = .TRUE. .
NF (output) INTEGER
The order of the reduced matrices Af and Ef; also, the
number of finite generalized eigenvalues of the pencil
A-lambda*E.
NIBLCK (output) INTEGER
If RANKE < N, the number of infinite blocks minus one.
If RANKE = N, NIBLCK = 0.
IBLCK (output) INTEGER array, dimension (N)
IBLCK(i) contains the dimension of the i-th block in the
staircase form (2), where i = 1, 2, ..., NIBLCK.
Tolerances
TOL DOUBLE PRECISION
A tolerance used in rank decisions to determine the
effective rank, which is defined as the order of the
largest leading (or trailing) triangular submatrix in the
QR factorization with column pivoting whose estimated
condition number is less than 1/TOL. If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
TOLDEF = N**2*EPS, is used instead, where EPS is the
machine precision (see LAPACK Library routine DLAMCH).
TOL < 1.
Workspace
IWORK INTEGER array, dimension (N-RANKE)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 1, if RANKE = N; otherwise,
LDWORK >= MAX(4*(N-RANKE)-1, N-RANKE-RNKA22+MAX(N,M)).
For optimal performance, LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the pencil A-lambda*E is not regular.
Method
The subroutine is based on the reduction algorithm of [1].References
[1] Misra, P., Van Dooren, P., and Varga, A.
Computation of structural invariants of generalized
state-space systems.
Automatica, 30, pp. 1921-1936, 1994.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( N**3 ) floating point operations.Further Comments
The number of infinite poles is computed as
NIBLCK
Sum IBLCK(i) = RANKE - NF.
i=1
The multiplicities of infinite poles can be computed as follows:
there are IBLCK(k)-IBLCK(k+1) infinite poles of multiplicity
k, for k = 1, ..., NIBLCK, where IBLCK(NIBLCK+1) = 0.
Note that each infinite pole of multiplicity k corresponds to
an infinite eigenvalue of multiplicity k+1.
Example
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute orthogonal transformation matrices Q and Z which
reduce the regular pole pencil A-lambda*E of the descriptor system
(A-lambda*E,B,C) to the form (if JOB = 'F')
( Af * ) ( Ef * )
Q'*A*Z = ( ) , Q'*E*Z = ( ) , (1)
( 0 Ai ) ( 0 Ei )
or to the form (if JOB = 'I')
( Ai * ) ( Ei * )
Q'*A*Z = ( ) , Q'*E*Z = ( ) , (2)
( 0 Af ) ( 0 Ef )
where the pair (Af,Ef) is in a generalized real Schur form, with
Ef nonsingular and upper triangular and Af in real Schur form.
The subpencil Af-lambda*Ef contains the finite eigenvalues.
The pair (Ai,Ei) is in a generalized real Schur form with
both Ai and Ei upper triangular. The subpencil Ai-lambda*Ei,
with Ai nonsingular and Ei nilpotent contains the infinite
eigenvalues and is in a block staircase form (see METHOD).
Specification
SUBROUTINE TG01MD( JOB, N, M, P, A, LDA, E, LDE, B, LDB, C, LDC,
$ ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, NF, ND,
$ NIBLCK, IBLCK, TOL, IWORK, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER JOB
INTEGER INFO, LDA, LDB, LDC, LDE, LDQ, LDWORK, LDZ, M,
$ N, ND, NF, NIBLCK, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IBLCK( * ), IWORK(*)
DOUBLE PRECISION A(LDA,*), ALPHAI(*), ALPHAR(*), B(LDB,*),
$ BETA(*), C(LDC,*), DWORK(*), E(LDE,*),
$ Q(LDQ,*), Z(LDZ,*)
Arguments
Mode Parameters
JOB CHARACTER*1
= 'F': perform the finite-infinite separation;
= 'I': perform the infinite-finite separation.
Input/Output Parameters
N (input) INTEGER
The number of rows of the matrix B, the number of columns
of the matrix C and the order of the square matrices A
and E. N >= 0.
M (input) INTEGER
The number of columns of the matrix B. M >= 0.
P (input) INTEGER
The number of rows of the matrix C. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the N-by-N state matrix A.
On exit, the leading N-by-N part of this array contains
the transformed state matrix Q'*A*Z in the form
( Af * ) ( Ai * )
( ) for JOB = 'F', or ( ) for JOB = 'I',
( 0 Ai ) ( 0 Af )
where Af is an NF-by-NF matrix in real Schur form, and Ai
is an (N-NF)-by-(N-NF) nonsingular and upper triangular
matrix. Ai has a block structure as in (3) or (4), where
A0,0 is ND-by-ND and Ai,i , for i = 1, ..., NIBLCK, is
IBLCK(i)-by-IBLCK(i). (See METHOD.)
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain the N-by-N descriptor matrix E.
On exit, the leading N-by-N part of this array contains
the transformed descriptor matrix Q'*E*Z in the form
( Ef * ) ( Ei * )
( ) for JOB = 'F', or ( ) for JOB = 'I',
( 0 Ei ) ( 0 Ef )
where Ef is an NF-by-NF nonsingular and upper triangular
matrix, and Ei is an (N-NF)-by-(N-NF) nilpotent matrix in
an upper triangular block form as in (3) or (4).
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the N-by-M input matrix B.
On exit, the leading N-by-M part of this array contains
the transformed input matrix Q'*B.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed matrix C*Z.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,P).
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAR(1:NF) will be set to the real parts of the diagonal
elements of Af that would result from reducing A and E to
the Schur form, and then further reducing both of them to
triangular form using unitary transformations, subject to
having the diagonal of E positive real. Thus, if Af(j,j)
is in a 1-by-1 block (i.e., Af(j+1,j) = Af(j,j+1) = 0),
then ALPHAR(j) = Af(j,j). Note that the (real or complex)
values (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,NF, are
the finite generalized eigenvalues of the matrix pencil
A - lambda*E.
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
ALPHAI(1:NF) will be set to the imaginary parts of the
diagonal elements of Af that would result from reducing A
and E to Schur form, and then further reducing both of
them to triangular form using unitary transformations,
subject to having the diagonal of E positive real. Thus,
if Af(j,j) is in a 1-by-1 block (see above), then
ALPHAI(j) = 0. Note that the (real or complex) values
(ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,NF, are the
finite generalized eigenvalues of the matrix pencil
A - lambda*E.
BETA (output) DOUBLE PRECISION array, dimension (N)
BETA(1:NF) will be set to the (real) diagonal elements of
Ef that would result from reducing A and E to Schur form,
and then further reducing both of them to triangular form
using unitary transformations, subject to having the
diagonal of E positive real. Thus, if Af(j,j) is in a
1-by-1 block (see above), then BETA(j) = Ef(j,j).
Note that the (real or complex) values
(ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,NF, are the
finite generalized eigenvalues of the matrix pencil
A - lambda*E.
Q (output) DOUBLE PRECISION array, dimension (LDQ,N)
The leading N-by-N part of this array contains the
orthogonal matrix Q, which is the accumulated product of
the transformations applied to A, E, and B on the left.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= MAX(1,N).
Z (output) DOUBLE PRECISION array, dimension (LDZ,N)
The leading N-by-N part of this array contains the
orthogonal matrix Z, which is the accumulated product of
the transformations applied to A, E, and C on the right.
LDZ INTEGER
The leading dimension of the array Z. LDZ >= MAX(1,N).
NF (output) INTEGER
The order of the reduced matrices Af and Ef; also, the
number of finite generalized eigenvalues of the pencil
A-lambda*E.
ND (output) INTEGER
The number of non-dynamic infinite eigenvalues of the
matrix pair (A,E). Note: N-ND is the rank of the matrix E.
NIBLCK (output) INTEGER
If ND > 0, the number of infinite blocks minus one.
If ND = 0, then NIBLCK = 0.
IBLCK (output) INTEGER array, dimension (N)
IBLCK(i) contains the dimension of the i-th block in the
staircase form (3), where i = 1,2,...,NIBLCK.
Tolerances
TOL DOUBLE PRECISION
A tolerance used in rank decisions to determine the
effective rank, which is defined as the order of the
largest leading (or trailing) triangular submatrix in the
QR factorization with column pivoting whose estimated
condition number is less than 1/TOL. If the user sets
TOL <= 0, then an implicitly computed, default tolerance
TOLDEF = N**2*EPS, is used instead, where EPS is the
machine precision (see LAPACK Library routine DLAMCH).
TOL < 1.
Workspace
IWORK INTEGER array, dimension (N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= 1, and if N > 0,
LDWORK >= 4*N.
If LDWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message related to LDWORK is issued by
XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the pencil A-lambda*E is not regular;
= 2: the QZ iteration did not converge.
Method
For the separation of infinite structure, the reduction algorithm
of [1] is employed.
If JOB = 'F', the matrices Ai and Ei have the form
( A0,0 A0,k ... A0,1 ) ( 0 E0,k ... E0,1 )
Ai = ( 0 Ak,k ... Ak,1 ) , Ei = ( 0 0 ... Ek,1 ) ; (3)
( : : . : ) ( : : . : )
( 0 0 ... A1,1 ) ( 0 0 ... 0 )
if JOB = 'I' the matrices Ai and Ei have the form
( A1,1 ... A1,k A1,0 ) ( 0 ... E1,k E1,0 )
Ai = ( : . : : ) , Ei = ( : . : : ) , (4)
( : ... Ak,k Ak,0 ) ( : ... 0 Ek,0 )
( 0 ... 0 A0,0 ) ( 0 ... 0 0 )
where Ai,i, for i = 0, 1, ..., k, are nonsingular upper triangular
matrices. A0,0 corresponds to the non-dynamic infinite modes of
the system.
References
[1] Misra, P., Van Dooren, P., and Varga, A.
Computation of structural invariants of generalized
state-space systems.
Automatica, 30, pp. 1921-1936, 1994.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( N**3 ) floating point operations.Further Comments
The number of infinite poles is computed as
NIBLCK
NINFP = Sum IBLCK(i) = N - ND - NF.
i=1
The multiplicities of infinite poles can be computed as follows:
there are IBLCK(k)-IBLCK(k+1) infinite poles of multiplicity
k, for k = 1, ..., NIBLCK, where IBLCK(NIBLCK+1) = 0.
Note that each infinite pole of multiplicity k corresponds to
an infinite eigenvalue of multiplicity k+1.
Example
Program Text
* TG01MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDE, LDQ, LDZ
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDE = NMAX, LDQ = NMAX, LDZ = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 4*NMAX )
* .. Local Scalars ..
CHARACTER*1 JOB
INTEGER I, INFO, J, M, N, ND, NF, NIBLCK, P
DOUBLE PRECISION TOL
* .. Local Arrays ..
INTEGER IBLCK(NMAX), IWORK(NMAX)
DOUBLE PRECISION A(LDA,NMAX), ALPHAI(NMAX), ALPHAR(NMAX),
$ B(LDB,MMAX), BETA(NMAX), C(LDC,NMAX),
$ DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,NMAX),
$ Z(LDZ,NMAX)
* .. External Subroutines ..
EXTERNAL TG01MD
* .. Intrinsic Functions ..
INTRINSIC DCMPLX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, JOB, TOL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find the reduced descriptor system
* (A-lambda E,B,C).
CALL TG01MD( JOB, N, M, P, A, LDA, E, LDE, B, LDB, C,
$ LDC, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ,
$ NF, ND, NIBLCK, IBLCK, TOL, IWORK, DWORK,
$ LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) NF, ND
WRITE ( NOUT, FMT = 99989 ) NIBLCK + 1
IF ( NIBLCK.GT.0 ) THEN
WRITE ( NOUT, FMT = 99985 )
$ ( IBLCK(I), I = 1, NIBLCK )
END IF
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 30 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
30 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 40 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
40 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 50 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,N )
50 CONTINUE
WRITE ( NOUT, FMT = 99990 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
60 CONTINUE
WRITE ( NOUT, FMT = 99985 )
DO 70 I = 1, NF
WRITE ( NOUT, FMT = 99984 )
$ DCMPLX( ALPHAR(I), ALPHAI(I) )/BETA(I)
70 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01MD = ',I2)
99997 FORMAT (/' The transformed state dynamics matrix Q''*A*Z is ')
99996 FORMAT (/' The transformed descriptor matrix Q''*E*Z is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (' Order of reduced system =', I5/
$ ' Number of non-dynamic infinite eigenvalues =', I5)
99993 FORMAT (/' The transformed input/state matrix Q''*B is ')
99992 FORMAT (/' The transformed state/output matrix C*Z is ')
99991 FORMAT (/' The left transformation matrix Q is ')
99990 FORMAT (/' The right transformation matrix Z is ')
99989 FORMAT ( ' Number of infinite blocks = ',I5)
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
99986 FORMAT (/' P is out of range.',/' P = ',I5)
99985 FORMAT (/' The finite generalized eigenvalues are '/
$ ' real part imag part ')
99984 FORMAT (1X,F9.4,SP,F9.4,S,'i ')
END
Program Data
TG01MD EXAMPLE PROGRAM DATA
4 2 2 F 0.0
-1 0 0 3
0 0 1 2
1 1 0 4
0 0 0 0
1 2 0 0
0 1 0 1
3 9 6 3
0 0 2 0
1 0
0 0
0 1
1 1
-1 0 1 0
0 1 -1 1
Program Results
TG01MD EXAMPLE PROGRAM RESULTS
Order of reduced system = 3
Number of non-dynamic infinite eigenvalues = 1
Number of infinite blocks = 1
The transformed state dynamics matrix Q'*A*Z is
1.2803 -2.3613 -0.9025 -3.9982
0.0000 -0.5796 0.8504 0.4350
0.0000 0.0000 0.0000 1.5770
0.0000 0.0000 0.0000 2.2913
The transformed descriptor matrix Q'*E*Z is
9.3142 -4.1463 5.4026 -2.3944
0.0000 0.1594 0.1212 -1.0948
0.0000 0.0000 2.3524 -0.6008
0.0000 0.0000 0.0000 0.0000
The transformed input/state matrix Q'*B is
-0.2089 -0.9712
0.6948 -0.0647
-0.4336 -0.9538
1.1339 0.3780
The transformed state/output matrix C*Z is
-0.0469 -0.9391 -0.8847 0.5774
-1.0697 0.3620 1.1795 0.5774
The left transformation matrix Q is
-0.2089 0.6948 0.3902 0.5669
-0.1148 -0.7163 0.3902 0.5669
-0.9712 -0.0647 -0.1301 -0.1890
0.0000 0.0000 -0.8238 0.5669
The right transformation matrix Z is
0.0469 0.9391 -0.0843 -0.3299
-0.9962 0.0189 -0.0211 -0.0825
0.0000 -0.0000 -0.9689 0.2474
-0.0735 0.3432 0.2317 0.9073
The finite generalized eigenvalues are
real part imag part
0.1375 +0.0000i
-3.6375 +0.0000i
0.0000 +0.0000i
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TG01ND.html 0000664 0000000 0000000 00000050303 14560147231 0020350 0 ustar 00root root 0000000 0000000
Purpose
To compute equivalence transformation matrices Q and Z which
reduce the regular pole pencil A-lambda*E of the descriptor system
(A-lambda*E,B,C) to the form (if JOB = 'F')
( Af 0 ) ( Ef 0 )
Q*A*Z = ( ) , Q*E*Z = ( ) , (1)
( 0 Ai ) ( 0 Ei )
or to the form (if JOB = 'I')
( Ai 0 ) ( Ei 0 )
Q*A*Z = ( ) , Q*E*Z = ( ) , (2)
( 0 Af ) ( 0 Ef )
where the pair (Af,Ef) is in a generalized real Schur form, with
Ef nonsingular and upper triangular and Af in real Schur form.
The subpencil Af-lambda*Ef contains the finite eigenvalues.
The pair (Ai,Ei) is in a generalized real Schur form with
both Ai and Ei upper triangular. The subpencil Ai-lambda*Ei,
with Ai nonsingular and Ei nilpotent contains the infinite
eigenvalues and is in a block staircase form (see METHOD).
This decomposition corresponds to an additive decomposition of
the transfer-function matrix of the descriptor system as the
sum of a proper term and a polynomial term.
Specification
SUBROUTINE TG01ND( JOB, JOBT, N, M, P, A, LDA, E, LDE, B, LDB,
$ C, LDC, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ,
$ NF, ND, NIBLCK, IBLCK, TOL, IWORK, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOB, JOBT
INTEGER INFO, LDA, LDB, LDC, LDE, LDQ, LDWORK, LDZ, M,
$ N, ND, NF, NIBLCK, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IBLCK( * ), IWORK(*)
DOUBLE PRECISION A(LDA,*), ALPHAR(*), ALPHAI(*), B(LDB,*),
$ BETA(*), C(LDC,*), DWORK(*), E(LDE,*),
$ Q(LDQ,*), Z(LDZ,*)
Arguments
Mode Parameters
JOB CHARACTER*1
= 'F': perform the finite-infinite separation;
= 'I': perform the infinite-finite separation.
JOBT CHARACTER*1
= 'D': compute the direct transformation matrices;
= 'I': compute the inverse transformation matrices
inv(Q) and inv(Z).
Input/Output Parameters
N (input) INTEGER
The number of rows of the matrix B, the number of columns
of the matrix C and the order of the square matrices A
and E. N >= 0.
M (input) INTEGER
The number of columns of the matrix B. M >= 0.
P (input) INTEGER
The number of rows of the matrix C. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the N-by-N state matrix A.
On exit, the leading N-by-N part of this array contains
the transformed state matrix Q*A*Z (if JOBT = 'D') or
inv(Q)*A*inv(Z) (if JOBT = 'I') in the form
( Af 0 ) ( Ai 0 )
( ) for JOB = 'F', or ( ) for JOB = 'I',
( 0 Ai ) ( 0 Af )
where Af is an NF-by-NF matrix in real Schur form, and Ai
is an (N-NF)-by-(N-NF) nonsingular and upper triangular
matrix. Ai has a block structure as in (3) or (4), where
A0,0 is ND-by-ND and Ai,i , for i = 1, ..., NIBLCK, is
IBLCK(i)-by-IBLCK(i). (See METHOD.)
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain the N-by-N descriptor matrix E.
On exit, the leading N-by-N part of this array contains
the transformed descriptor matrix Q*E*Z (if JOBT = 'D') or
inv(Q)*E*inv(Z) (if JOBT = 'I') in the form
( Ef 0 ) ( Ei 0 )
( ) for JOB = 'F', or ( ) for JOB = 'I',
( 0 Ei ) ( 0 Ef )
where Ef is an NF-by-NF nonsingular and upper triangular
matrix, and Ei is an (N-NF)-by-(N-NF) nilpotent matrix in
an upper triangular block form as in (3) or (4).
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the N-by-M input matrix B.
On exit, the leading N-by-M part of this array contains
the transformed input matrix Q*B (if JOBT = 'D') or
inv(Q)*B (if JOBT = 'I').
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed matrix C*Z (if JOBT = 'D') or C*inv(Z)
(if JOBT = 'I').
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,P).
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAR(1:NF) will be set to the real parts of the diagonal
elements of Af that would result from reducing A and E to
the Schur form, and then further reducing both of them to
triangular form using unitary transformations, subject to
having the diagonal of E positive real. Thus, if Af(j,j)
is in a 1-by-1 block (i.e., Af(j+1,j) = Af(j,j+1) = 0),
then ALPHAR(j) = Af(j,j). Note that the (real or complex)
values (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,NF, are
the finite generalized eigenvalues of the matrix pencil
A - lambda*E.
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
ALPHAI(1:NF) will be set to the imaginary parts of the
diagonal elements of Af that would result from reducing A
and E to Schur form, and then further reducing both of
them to triangular form using unitary transformations,
subject to having the diagonal of E positive real. Thus,
if Af(j,j) is in a 1-by-1 block (see above), then
ALPHAI(j) = 0. Note that the (real or complex) values
(ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,NF, are the
finite generalized eigenvalues of the matrix pencil
A - lambda*E.
BETA (output) DOUBLE PRECISION array, dimension (N)
BETA(1:NF) will be set to the (real) diagonal elements of
Ef that would result from reducing A and E to Schur form,
and then further reducing both of them to triangular form
using unitary transformations, subject to having the
diagonal of E positive real. Thus, if Af(j,j) is in a
1-by-1 block (see above), then BETA(j) = Ef(j,j).
Note that the (real or complex) values
(ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,NF, are the
finite generalized eigenvalues of the matrix pencil
A - lambda*E.
Q (output) DOUBLE PRECISION array, dimension (LDQ,N)
The leading N-by-N part of this array contains the
left transformation matrix Q, if JOBT = 'D', or its
inverse inv(Q), if JOBT = 'I'.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= MAX(1,N).
Z (output) DOUBLE PRECISION array, dimension (LDZ,N)
The leading N-by-N part of this array contains the
right transformation matrix Z, if JOBT = 'D', or its
inverse inv(Z), if JOBT = 'I'.
LDZ INTEGER
The leading dimension of the array Z. LDZ >= MAX(1,N).
NF (output) INTEGER
The order of the reduced matrices Af and Ef; also, the
number of finite generalized eigenvalues of the pencil
A-lambda*E.
ND (output) INTEGER
The number of non-dynamic infinite eigenvalues of the
matrix pair (A,E). Note: N-ND is the rank of the matrix E.
NIBLCK (output) INTEGER
If ND > 0, the number of infinite blocks minus one.
If ND = 0, then NIBLCK = 0.
IBLCK (output) INTEGER array, dimension (N)
IBLCK(i) contains the dimension of the i-th block in the
staircase form (3), where i = 1,2,...,NIBLCK.
Tolerances
TOL DOUBLE PRECISION
A tolerance used in rank decisions to determine the
effective rank, which is defined as the order of the
largest leading (or trailing) triangular submatrix in the
QR factorization with column pivoting whose estimated
condition number is less than 1/TOL. If the user sets
TOL <= 0, then an implicitly computed, default tolerance
TOLDEF = N**2*EPS, is used instead, where EPS is the
machine precision (see LAPACK Library routine DLAMCH).
TOL < 1.
Workspace
IWORK INTEGER array, dimension (N+6)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= 1, and if N > 0,
LDWORK >= 4*N.
If LDWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message related to LDWORK is issued by
XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the pencil A-lambda*E is not regular;
= 2: the QZ iteration did not converge;
= 3: (Af,Ef) and (Ai,Ei) have too close generalized
eigenvalues.
Method
For the separation of infinite structure, the reduction algorithm
of [1] is employed. This separation is achieved by computing
orthogonal matrices Q1 and Z1 such that Q1*A*Z1 and Q1*E*Z1
have the form (if JOB = 'F')
( Af Ao ) ( Ef Eo )
Q1*A*Z1 = ( ) , Q1*E*Z1 = ( ) ,
( 0 Ai ) ( 0 Ei )
or to the form (if JOB = 'I')
( Ai Ao ) ( Ei Eo )
Q1*A*Z1 = ( ) , Q1*E*Z1 = ( ) .
( 0 Af ) ( 0 Ef )
If JOB = 'F', the matrices Ai and Ei have the form
( A0,0 A0,k ... A0,1 ) ( 0 E0,k ... E0,1 )
Ai = ( 0 Ak,k ... Ak,1 ) , Ei = ( 0 0 ... Ek,1 ) ; (3)
( : : . : ) ( : : . : )
( 0 0 ... A1,1 ) ( 0 0 ... 0 )
if JOB = 'I' the matrices Ai and Ei have the form
( A1,1 ... A1,k A1,0 ) ( 0 ... E1,k E1,0 )
Ai = ( : . : : ) , Ei = ( : . : : ) , (4)
( : ... Ak,k Ak,0 ) ( : ... 0 Ek,0 )
( 0 ... 0 A0,0 ) ( 0 ... 0 0 )
where Ai,i, for i = 0, 1, ..., k, are nonsingular upper triangular
matrices. A0,0 corresponds to the non-dynamic infinite modes of
the system.
In a second step, the transformation matrices Q2 and Z2 are
determined, of the form
( I -X ) ( I Y )
Q2 = ( ) , Z2 = ( )
( 0 I ) ( 0 I )
such that with Q = Q2*Q1 and Z = Z1*Z2, Q*A*Z and Q*E*Z are
block diagonal as in (1) (if JOB = 'F') or in (2) (if JOB = 'I').
X and Y are computed by solving generalized Sylvester equations.
If we partition Q*B and C*Z according to (1) or (2) in the form
( Bf ) and ( Cf Ci ), if JOB = 'F', or ( Bi ) and ( Ci Cf ), if
( Bi ) ( Bf )
JOB = 'I', then (Af-lambda*Ef,Bf,Cf) is the strictly proper part
of the original descriptor system and (Ai-lambda*Ei,Bi,Ci) is its
polynomial part.
References
[1] Misra, P., Van Dooren, P., and Varga, A.
Computation of structural invariants of generalized
state-space systems.
Automatica, 30, pp. 1921-1936, 1994.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( N**3 ) floating point operations.Further Comments
The number of infinite poles is computed as
NIBLCK
NINFP = Sum IBLCK(i) = N - ND - NF.
i=1
The multiplicities of infinite poles can be computed as follows:
there are IBLCK(k)-IBLCK(k+1) infinite poles of multiplicity
k, for k = 1, ..., NIBLCK, where IBLCK(NIBLCK+1) = 0.
Note that each infinite pole of multiplicity k corresponds to
an infinite eigenvalue of multiplicity k+1.
Example
Program Text
* TG01ND EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDE, LDQ, LDZ
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDE = NMAX, LDQ = NMAX, LDZ = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 4*NMAX )
* .. Local Scalars ..
CHARACTER*1 JOB, JOBT
INTEGER I, INFO, J, M, N, ND, NF, NIBLCK, P
DOUBLE PRECISION TOL
* .. Local Arrays ..
INTEGER IBLCK(NMAX), IWORK(NMAX+6)
DOUBLE PRECISION A(LDA,NMAX), ALPHAI(NMAX), ALPHAR(NMAX),
$ B(LDB,MMAX), BETA(NMAX), C(LDC,NMAX),
$ DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,NMAX),
$ Z(LDZ,NMAX)
* .. External Subroutines ..
EXTERNAL TG01ND
* .. Intrinsic Functions ..
INTRINSIC DCMPLX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, JOB, JOBT, TOL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find the reduced descriptor system
* (A-lambda E,B,C).
CALL TG01ND( JOB, JOBT, N, M, P, A, LDA, E, LDE, B, LDB,
$ C, LDC, ALPHAR, ALPHAI, BETA, Q, LDQ, Z,
$ LDZ, NF, ND, NIBLCK, IBLCK, TOL, IWORK,
$ DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) NF, ND
WRITE ( NOUT, FMT = 99989 ) NIBLCK + 1
IF ( NIBLCK.GT.0 ) THEN
WRITE ( NOUT, FMT = 99985 )
$ ( IBLCK(I), I = 1, NIBLCK )
END IF
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 30 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
30 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 40 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
40 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 50 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,N )
50 CONTINUE
WRITE ( NOUT, FMT = 99990 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
60 CONTINUE
WRITE ( NOUT, FMT = 99985 )
DO 70 I = 1, NF
WRITE ( NOUT, FMT = 99984 )
$ DCMPLX( ALPHAR(I), ALPHAI(I) )/BETA(I)
70 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01ND EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01ND = ',I2)
99997 FORMAT (/' The transformed state dynamics matrix Q*A*Z is ')
99996 FORMAT (/' The transformed descriptor matrix Q*E*Z is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (' Order of reduced system =', I5/
$ ' Number of non-dynamic infinite eigenvalues =', I5)
99993 FORMAT (/' The transformed input/state matrix Q*B is ')
99992 FORMAT (/' The transformed state/output matrix C*Z is ')
99991 FORMAT (/' The left transformation matrix Q is ')
99990 FORMAT (/' The right transformation matrix Z is ')
99989 FORMAT ( ' Number of infinite blocks = ',I5)
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
99986 FORMAT (/' P is out of range.',/' P = ',I5)
99985 FORMAT (/' The finite generalized eigenvalues are '/
$ ' real part imag part ')
99984 FORMAT (1X,F9.4,SP,F9.4,S,'i ')
END
Program Data
TG01ND EXAMPLE PROGRAM DATA
4 2 2 F D 0.0
-1 0 0 3
0 0 1 2
1 1 0 4
0 0 0 0
1 2 0 0
0 1 0 1
3 9 6 3
0 0 2 0
1 0
0 0
0 1
1 1
-1 0 1 0
0 1 -1 1
Program Results
TG01ND EXAMPLE PROGRAM RESULTS
Order of reduced system = 3
Number of non-dynamic infinite eigenvalues = 1
Number of infinite blocks = 1
The transformed state dynamics matrix Q*A*Z is
1.2803 -2.3613 -0.9025 0.0000
0.0000 -0.5796 0.8504 0.0000
0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 2.2913
The transformed descriptor matrix Q*E*Z is
9.3142 -4.1463 5.4026 0.0000
0.0000 0.1594 0.1212 0.0000
0.0000 0.0000 2.3524 0.0000
0.0000 0.0000 0.0000 0.0000
The transformed input/state matrix Q*B is
7.7328 1.6760
2.2870 0.4660
-1.2140 -1.2140
1.1339 0.3780
The transformed state/output matrix C*Z is
-0.0469 -0.9391 -0.8847 -6.0622
-1.0697 0.3620 1.1795 -0.0000
The left transformation matrix Q is
3.7620 3.8560 -2.2948 3.9708
1.4909 0.0798 -0.3301 0.7961
-0.0000 -0.0000 0.0000 -1.2140
0.5669 0.5669 -0.1890 0.5669
The right transformation matrix Z is
0.0469 0.9391 -0.0843 6.0622
-0.9962 0.0189 -0.0211 -3.0311
0.0000 -0.0000 -0.9689 -0.0000
-0.0735 0.3432 0.2317 3.0311
The finite generalized eigenvalues are
real part imag part
0.1375 +0.0000i
-3.6375 +0.0000i
0.0000 +0.0000i
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TG01NX.html 0000664 0000000 0000000 00000016721 14560147231 0020402 0 ustar 00root root 0000000 0000000
Purpose
To compute equivalence transformation matrices Q and Z which
reduce the regular pole pencil A-lambda*E of the descriptor system
(A-lambda*E,B,C), with (A,E) in a generalized real Schur form, to
the block-diagonal form
( A1 0 ) ( E1 0 )
Q*A*Z = ( ) , Q*E*Z = ( ) , (1)
( 0 A2 ) ( 0 E2 )
where the pair (Q*A*Z,Q*E*Z) is in a generalized real Schur form,
with (A1,E1) and (A2,E2) having no common generalized eigenvalues.
This decomposition corresponds to an additive spectral
decomposition of the transfer-function matrix of the descriptor
system as the sum of two terms containing the generalized
eigenvalues of (A1,E1) and (A2,E2), respectively.
Specification
SUBROUTINE TG01NX( JOBT, N, M, P, NDIM, A, LDA, E, LDE, B, LDB,
$ C, LDC, Q, LDQ, Z, LDZ, IWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBT
INTEGER INFO, LDA, LDB, LDC, LDE, LDQ, LDZ, M, N, NDIM,
$ P
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), E(LDE,*),
$ Q(LDQ,*), Z(LDZ,*)
Arguments
Mode Parameters
JOBT CHARACTER*1
= 'D': compute the direct transformation matrices;
= 'I': compute the inverse transformation matrices
inv(Q) and inv(Z).
Input/Output Parameters
N (input) INTEGER
The number of rows of the matrix B, the number of columns
of the matrix C and the order of the square matrices A
and E. N >= 0.
M (input) INTEGER
The number of columns of the matrix B. M >= 0.
P (input) INTEGER
The number of rows of the matrix C. P >= 0.
NDIM (input) INTEGER
The dimension of the leading diagonal blocks of (A,E)
having generalized eigenvalues distinct from those of the
trailing diagonal block. 0 <= NDIM <= N.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the N-by-N state matrix A in a real Schur form.
On exit, the leading N-by-N part of this array contains
the transformed state matrix Q*A*Z (if JOBT = 'D') or
inv(Q)*A*inv(Z) (if JOBT = 'I'), in the form (1), where
A1 is a NDIM-by-NDIM matrix.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain the N-by-N descriptor matrix E in upper triangular
form.
On exit, the leading N-by-N part of this array contains
the transformed descriptor matrix Q*E*Z (if JOBT = 'D') or
inv(Q)*E*inv(Z) (if JOBT = 'I'), in the form (1), where
E1 is an NDIM-by-NDIM matrix.
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the N-by-M input matrix B.
On exit, the leading N-by-M part of this array contains
the transformed input matrix Q*B (if JOBT = 'D') or
inv(Q)*B (if JOBT = 'I').
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed matrix C*Z (if JOBT = 'D') or C*inv(Z)
(if JOBT = 'I').
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,P).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, the leading N-by-N part of this array contains
Q1, the orthogonal left transformation matrix Q used to
reduce the pair (A,E) to the generalized real Schur form.
On exit, the leading N-by-N part of this array contains
the left transformation matrix Q = Q2*Q1, if JOBT = 'D',
or its inverse inv(Q), if JOBT = 'I'.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= MAX(1,N).
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
On entry, the leading N-by-N part of this array contains
the orthogonal right transformation matrix Z1 used to
reduce the pair (A,E) to the generalized real Schur form.
On exit, the leading N-by-N part of this array contains
the right transformation matrix Z = Z1*Z2, if JOBT = 'D',
or its inverse inv(Z), if JOBT = 'I'.
LDZ INTEGER
The leading dimension of the array Z. LDZ >= MAX(1,N).
Workspace
IWORK INTEGER array, dimension (N+6)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the separation of the two diagonal blocks failed
because of very close eigenvalues.
Method
For the separation, transformation matrices Q2 and Z2 of the form
( I -X ) ( I Y )
Q2 = ( ) , Z2 = ( )
( 0 I ) ( 0 I )
are determined, such that Q2*A*Z2 and Q2*E*Z2 are block diagonal
as in (1). X and Y are computed by solving generalized Sylvester
equations.
If we partition Q2*B and C*Z2 according to (1) in the form ( B1 )
( B2 )
and ( C1 C2 ), then (A1-lambda*E1,B1,C1) and (A2-lambda*E2,B2,C2)
represent an additive spectral decomposition of the system
transfer-function matrix.
References
[1] Kagstrom, B. and Van Dooren, P.
Additive decomposition of a transfer function with respect
to a specified region.
Proc. MTNS Symp., Brussels, 1989.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( N**3 ) floating point operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute for a single-input single-output descriptor system,
given by the system matrix
[ D C ]
[ B A - s*E ],
with E upper triangular, a transformed system, (Q'*A*Z, Q'*E*Z,
Q'*B, C*Z), via an orthogonal equivalence transformation, so that
Q'*B has only the first element nonzero and Q'*E*Z remains upper
triangular.
Specification
SUBROUTINE TG01OA( JOBE, N, DCBA, LDDCBA, E, LDE, INFO )
C .. Scalar Arguments ..
CHARACTER JOBE
INTEGER INFO, LDDCBA, LDE, N
C .. Array Arguments ..
DOUBLE PRECISION DCBA(LDDCBA,*), E(LDE,*)
Arguments
Mode Parameters
JOBE CHARACTER*1
Specifies whether E is an upper triangular or an identity
matrix, as follows:
= 'U': The matrix E is an upper triangular matrix;
= 'I': The matrix E is assumed identity and is not given.
Input/Output Parameters
N (input) INTEGER
The dimension of the descriptor state vector; also the
order of square matrices A and E, the number of rows of
matrix B, and the number of columns of matrix C. N >= 0.
DCBA (input/output) DOUBLE PRECISION array, dimension
(LDDCBA,N+1)
On entry, the leading (N+1)-by-(N+1) part of this array
must contain the original system matrices A, B, C, and D,
stored as follows
[ D C ]
[ B A ].
On exit, the leading (N+1)-by-(N+1) part of this array
contains the transformed matrices C*Z, Q'*B, and Q'*A*Z,
replacing C, B, and A. The scalar D is unchanged.
LDDCBA INTEGER
The leading dimension of the array DCBA.
LDDCBA >= N+1.
E (input/output) DOUBLE PRECISION array, dimension (LDE,*)
On entry, if JOBE = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the descriptor matrix E. The lower
triangular part under the first subdiagonal is not
referenced.
On exit, if JOBE = 'U', the leading N-by-N upper
triangular part of this array contains the upper
triangular part of the transformed descriptor matrix,
Q'*E*Z.
If JOBE = 'I', this array is not referenced.
LDE INTEGER
The leading dimension of the array E.
LDE >= MAX(1,N), if JOBE = 'U';
LDE >= 1, if JOBE = 'I'.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Givens rotations are used to annihilate the last N-1 elements of B in reverse order, but preserve the form of E.Numerical Aspects
The algorithm is numerically backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute for a single-input single-output descriptor system,
given by the system matrix
[ D C ]
[ B A - s*E ],
with E upper triangular, a transformed system, (Q'*A*Z, Q'*E*Z,
Q'*B, C*Z), via an orthogonal equivalence transformation, so that
Q'*B has only the first element nonzero and Q'*E*Z remains upper
triangular. The matrices have complex elements.
Specification
SUBROUTINE TG01OB( JOBE, N, DCBA, LDDCBA, E, LDE, INFO )
C .. Scalar Arguments ..
CHARACTER JOBE
INTEGER INFO, LDDCBA, LDE, N
C .. Array Arguments ..
COMPLEX*16 DCBA(LDDCBA,*), E(LDE,*)
Arguments
Mode Parameters
JOBE CHARACTER*1
Specifies whether E is an upper triangular or an identity
matrix, as follows:
= 'U': The matrix E is an upper triangular matrix;
= 'I': The matrix E is assumed identity and is not given.
Input/Output Parameters
N (input) INTEGER
The dimension of the descriptor state vector; also the
order of square matrices A and E, the number of rows of
matrix B, and the number of columns of matrix C. N >= 0.
DCBA (input/output) COMPLEX*16 array, dimension (LDDCBA,N+1)
On entry, the leading (N+1)-by-(N+1) part of this array
must contain the original system matrices A, B, C, and D,
stored as follows
[ D C ]
[ B A ].
On exit, the leading (N+1)-by-(N+1) part of this array
contains the transformed matrices C*Z, Q'*B, and Q'*A*Z,
replacing C, B, and A. The scalar D is unchanged.
LDDCBA INTEGER
The leading dimension of the array DCBA.
LDDCBA >= N+1.
E (input/output) COMPLEX*16 array, dimension (LDE,*)
On entry, if JOBE = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the descriptor matrix E. The lower
triangular part under the first subdiagonal is not
referenced.
On exit, if JOBE = 'U', the leading N-by-N upper
triangular part of this array contains the upper
triangular part of the transformed descriptor matrix,
Q'*E*Z.
If JOBE = 'I', this array is not referenced.
LDE INTEGER
The leading dimension of the array E.
LDE >= MAX(1,N), if JOBE = 'U';
LDE >= 1, if JOBE = 'I'.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Givens rotations are used to annihilate the last N-1 elements of B in reverse order, but preserve the form of E.Numerical Aspects
The algorithm is numerically backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Purpose
To compute for a single-input single-output descriptor system,
given by the system matrix
[ D C ]
[ B A - s*E ],
with E nonsingular, a reduced system matrix,
[ d c ]
[ b a - s*e ],
such that d has a "sufficiently" large magnitude.
Specification
SUBROUTINE TG01OD( JOBE, N, DCBA, LDDCBA, E, LDE, NZ, G, TOL,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBE
INTEGER INFO, LDDCBA, LDWORK, LDE, N, NZ
DOUBLE PRECISION G, TOL
C .. Array Arguments ..
DOUBLE PRECISION DCBA(LDDCBA,*), DWORK(*), E(LDE,*)
Arguments
Mode Parameters
JOBE CHARACTER*1
Specifies whether E is a general or an identity matrix,
as follows:
= 'G': The matrix E is a general matrix;
= 'I': The matrix E is assumed identity and is not given.
Input/Output Parameters
N (input) INTEGER
The dimension of the descriptor state vector; also the
order of square matrices A and E, the number of rows of
matrix B, and the number of columns of matrix C. N >= 0.
DCBA (input/output) DOUBLE PRECISION array, dimension
(LDDCBA,N+1)
On entry, the leading (N+1)-by-(N+1) part of this array
must contain the original system matrices A, B, C, and D,
stored as follows
[ D C ]
[ B A ].
On exit, the leading (NZ+1)-by-(NZ+1) part of this array
contains the reduced system matrices a, b, c, and d.
LDDCBA INTEGER
The leading dimension of the array DCBA. LDDCBA >= N+1.
E (input/output) DOUBLE PRECISION array, dimension (LDE,*)
On entry, if JOBE = 'G', the leading N-by-N part of this
array must contain the nonsingular descriptor matrix E.
On exit, if JOBE = 'G', the leading NZ-by-NZ part of this
array contains the reduced descriptor matrix e.
If JOBE = 'I', this array is not referenced.
LDE INTEGER
The leading dimension of the array E.
LDE >= MAX(1,N), if JOBE = 'G';
LDE >= 1, if JOBE = 'I'.
NZ (output) INTEGER
The order of the reduced system.
G (output) DOUBLE PRECISION
The gain of the reduced system.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in determining if the transformed
d has a "sufficiently" large magnitude. If the user sets
TOL > 0, then the given value of TOL is used. If the user
sets TOL <= 0, then an implicitly computed, default
tolerance, defined by TOLDEF = EPS**(3/4), is used
instead, where EPS is the machine precision (see LAPACK
Library routine DLAMCH).
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
On exit, if INFO = -11, DWORK(1) returns the minimum value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 2*N+1, if JOBE = 'G';
LDWORK >= N+1, if JOBE = 'I'.
For good performance when JOBE = 'G', LDWORK should be
larger. Specifically,
LDWORK >= MAX( N*NB(DGEQRF), (N+1)*NB(DORMQR) ),
where NB(X) is the optimal block sizes for the LAPACK
Library routine X.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Householder transformations and Givens rotations are used to process the matrices. If E is a general matrix, it is first triangularized using the QR decomposition, and the triangular form is preserved during the remaining computations.Numerical Aspects
The algorithm is numerically backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TG01OZ.html 0000664 0000000 0000000 00000012701 14560147231 0020377 0 ustar 00root root 0000000 0000000
Purpose
To compute for a single-input single-output descriptor system,
given by the system matrix with complex elements
[ D C ]
[ B A - s*E ],
with E nonsingular, a reduced system matrix,
[ d c ]
[ b a - s*e ],
such that d has a "sufficiently" large magnitude.
Specification
SUBROUTINE TG01OZ( JOBE, N, DCBA, LDDCBA, E, LDE, NZ, G, TOL,
$ ZWORK, LZWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBE
INTEGER INFO, LDDCBA, LDE, LZWORK, N, NZ
DOUBLE PRECISION TOL
COMPLEX*16 G
C .. Array Arguments ..
COMPLEX*16 DCBA(LDDCBA,*), E(LDE,*), ZWORK(*)
Arguments
Mode Parameters
JOBE CHARACTER*1
Specifies whether E is a general or an identity matrix,
as follows:
= 'G': The matrix E is a general matrix;
= 'I': The matrix E is assumed identity and is not given.
Input/Output Parameters
N (input) INTEGER
The dimension of the descriptor state vector; also the
order of square matrices A and E, the number of rows of
matrix B, and the number of columns of matrix C. N >= 0.
DCBA (input/output) COMPLEX*16 array, dimension (LDDCBA,N+1)
On entry, the leading (N+1)-by-(N+1) part of this array
must contain the original system matrices A, B, C, and D,
stored as follows
[ D C ]
[ B A ].
On exit, the leading (NZ+1)-by-(NZ+1) part of this array
contains the reduced system matrices a, b, c, and d.
LDDCBA INTEGER
The leading dimension of the array DCBA. LDDCBA >= N+1.
E (input/output) COMPLEX*16 array, dimension (LDE,*)
On entry, if JOBE = 'G', the leading N-by-N part of this
array must contain the nonsingular descriptor matrix E.
On exit, if JOBE = 'G', the leading NZ-by-NZ part of this
array contains the reduced descriptor matrix e.
If JOBE = 'I', this array is not referenced.
LDE INTEGER
The leading dimension of the array E.
LDE >= MAX(1,N), if JOBE = 'G';
LDE >= 1, if JOBE = 'I'.
NZ (output) INTEGER
The order of the reduced system.
G (output) COMPLEX*16
The gain of the reduced system.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in determining if the transformed
d has a "sufficiently" large magnitude. If the user sets
TOL > 0, then the given value of TOL is used. If the user
sets TOL <= 0, then an implicitly computed, default
tolerance, defined by TOLDEF = EPS**(3/4), is used
instead, where EPS is the machine precision (see LAPACK
Library routine DLAMCH).
Workspace
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) returns the optimal value
of LZWORK.
On exit, if INFO = -11, ZWORK(1) returns the minimum value
of LZWORK.
LZWORK INTEGER
The length of the array ZWORK.
LZWORK >= 2*N+1, if JOBE = 'G';
LZWORK >= N+1, if JOBE = 'I'.
For good performance when JOBE = 'G', LZWORK should be
larger. Specifically,
LZWORK >= MAX( N*NB(ZGEQRF), (N+1)*NB(ZUNMQR) ),
where NB(X) is the optimal block sizes for the LAPACK
Library routine X.
If LZWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
ZWORK array, returns this value as the first entry of
the ZWORK array, and no error message related to LZWORK
is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Householder transformations and Givens rotations are used to process the matrices. If E is a general matrix, it is first triangularized using the QR decomposition, and the triangular form is preserved during the remaining computations.Numerical Aspects
The algorithm is numerically backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TG01PD.html 0000664 0000000 0000000 00000047112 14560147231 0020356 0 ustar 00root root 0000000 0000000
Purpose
To compute orthogonal transformation matrices Q and Z which
reduce the regular pole pencil A-lambda*E of the descriptor system
(A-lambda*E,B,C) to the generalized real Schur form with ordered
generalized eigenvalues. The pair (A,E) is reduced to the form
( * * * * ) ( * * * * )
( ) ( )
( 0 A1 * * ) ( 0 E1 * * )
Q'*A*Z = ( ) , Q'*E*Z = ( ) ,
( 0 0 A2 * ) ( 0 0 E2 * )
( ) ( )
( 0 0 0 * ) ( 0 0 0 * )
where the subpencil A1-lambda*E1 contains the eigenvalues which
belong to a suitably defined domain of interest and the subpencil
A2-lambda*E2 contains the eigenvalues which are outside of the
domain of interest.
If JOBAE = 'S', the pair (A,E) is assumed to be already in a
generalized real Schur form and the reduction is performed only
on the subpencil A12 - lambda*E12 defined by rows and columns
NLOW to NSUP of A - lambda*E.
Specification
SUBROUTINE TG01PD( DICO, STDOM, JOBAE, COMPQ, COMPZ, N, M, P,
$ NLOW, NSUP, ALPHA, A, LDA, E, LDE, B, LDB,
$ C, LDC, Q, LDQ, Z, LDZ, NDIM, ALPHAR, ALPHAI,
$ BETA, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, COMPZ, DICO, JOBAE, STDOM
INTEGER INFO, LDA, LDB, LDC, LDE, LDQ, LDWORK, LDZ, M, N,
$ NDIM, NLOW, NSUP, P
DOUBLE PRECISION ALPHA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), ALPHAI(*), ALPHAR(*), B(LDB,*),
$ BETA(*), C(LDC,*), DWORK(*), E(LDE,*),
$ Q(LDQ,*), Z(LDZ,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the descriptor system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
STDOM CHARACTER*1
Specifies whether the domain of interest is of stability
type (left part of complex plane or inside of a circle)
or of instability type (right part of complex plane or
outside of a circle) as follows:
= 'S': stability type domain;
= 'U': instability type domain.
JOBAE CHARACTER*1
Specifies the shape of the matrix pair (A,E) on entry
as follows:
= 'S': (A,E) is in a generalized real Schur form;
= 'G': A and E are general square dense matrices.
COMPQ CHARACTER*1
= 'I': Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= 'U': Q must contain an orthogonal matrix Q1 on entry,
and the product Q1*Q is returned.
This option can not be used when JOBAE = 'G'.
COMPZ CHARACTER*1
= 'I': Z is initialized to the unit matrix, and the
orthogonal matrix Z is returned;
= 'U': Z must contain an orthogonal matrix Z1 on entry,
and the product Z1*Z is returned.
This option can not be used when JOBAE = 'G'.
Input/Output Parameters
N (input) INTEGER
The number of rows of the matrix B, the number of columns
of the matrix C, and the order of the square matrices A
and E. N >= 0.
M (input) INTEGER
The number of columns of the matrix B. M >= 0.
P (input) INTEGER
The number of rows of the matrix C. P >= 0.
NLOW, (input) INTEGER
NSUP (input) INTEGER
NLOW and NSUP specify the boundary indices for the rows
and columns of the principal subpencil of A - lambda*E
whose diagonal blocks are to be reordered.
0 <= NLOW <= NSUP <= N, if JOBAE = 'S'.
NLOW = MIN( 1, N ), NSUP = N, if JOBAE = 'G'.
ALPHA (input) DOUBLE PRECISION
The boundary of the domain of interest for the generalized
eigenvalues of the pair (A,E). For a continuous-time
system (DICO = 'C'), ALPHA is the boundary value for the
real parts of the generalized eigenvalues, while for a
discrete-time system (DICO = 'D'), ALPHA >= 0 represents
the boundary value for the moduli of the generalized
eigenvalues.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A.
If JOBAE = 'S' then A must be a matrix in real Schur form.
On exit, the leading N-by-N part of this array contains
the matrix Q'*A*Z in real Schur form, with the elements
below the first subdiagonal set to zero.
The leading NDIM-by-NDIM part of the principal subpencil
A12 - lambda*E12, defined by A12 := A(NLOW:NSUP,NLOW:NSUP)
and E12 := E(NLOW:NSUP,NLOW:NSUP), has generalized
eigenvalues in the domain of interest, and the trailing
part of this subpencil has generalized eigenvalues outside
the domain of interest.
The domain of interest for eig(A12,E12), the generalized
eigenvalues of the pair (A12,E12), is defined by the
parameters ALPHA, DICO and STDOM as follows:
For DICO = 'C':
Real(eig(A12,E12)) < ALPHA if STDOM = 'S';
Real(eig(A12,E12)) > ALPHA if STDOM = 'U'.
For DICO = 'D':
Abs(eig(A12,E12)) < ALPHA if STDOM = 'S';
Abs(eig(A12,E12)) > ALPHA if STDOM = 'U'.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain the descriptor matrix E.
If JOBAE = 'S', then E must be an upper triangular matrix.
On exit, the leading N-by-N part of this array contains an
upper triangular matrix Q'*E*Z, with the elements below
the diagonal set to zero.
The leading NDIM-by-NDIM part of the principal subpencil
A12 - lambda*E12 (see description of A) has generalized
eigenvalues in the domain of interest, and the trailing
part of this subpencil has generalized eigenvalues outside
the domain of interest.
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input matrix B.
On exit, the leading N-by-M part of this array contains
the transformed input matrix Q'*B.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed output matrix C*Z.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,P).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
If COMPQ = 'I': on entry, Q need not be set;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix Q,
where Q' is the product of orthogonal
transformations which are applied to A,
E, and B on the left.
If COMPQ = 'U': on entry, the leading N-by-N part of this
array must contain an orthogonal matrix
Q1;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix
Q1*Q.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= MAX(1,N).
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
If COMPZ = 'I': on entry, Z need not be set;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix Z,
which is the product of orthogonal
transformations applied to A, E, and C
on the right.
If COMPZ = 'U': on entry, the leading N-by-N part of this
array must contain an orthogonal matrix
Z1;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix
Z1*Z.
LDZ INTEGER
The leading dimension of the array Z. LDZ >= MAX(1,N).
NDIM (output) INTEGER
The number of generalized eigenvalues of the principal
subpencil A12 - lambda*E12 (see description of A) lying
inside the domain of interest for eigenvalues.
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
BETA (output) DOUBLE PRECISION array, dimension (N)
On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N,
are the generalized eigenvalues.
ALPHAR(j) + ALPHAI(j)*i, and BETA(j), j = 1,...,N, are the
diagonals of the complex Schur form (S,T) that would
result if the 2-by-2 diagonal blocks of the real Schur
form of (A,B) were further reduced to triangular form
using 2-by-2 complex unitary transformations.
If ALPHAI(j) is zero, then the j-th eigenvalue is real;
if positive, then the j-th and (j+1)-st eigenvalues are a
complex conjugate pair, with ALPHAI(j+1) negative.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 8*N+16, if JOBAE = 'G';
LDWORK >= 4*N+16, if JOBAE = 'S'.
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message related to LDWORK is issued by
XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the QZ algorithm failed to compute all generalized
eigenvalues of the pair (A,E);
= 2: a failure occured during the ordering of the
generalized real Schur form of the pair (A,E).
Method
If JOBAE = 'G', the pair (A,E) is reduced to an ordered generalized real Schur form using an orthogonal equivalence transformation A <-- Q'*A*Z and E <-- Q'*E*Z. This transformation is determined so that the leading diagonal blocks of the resulting pair (A,E) have generalized eigenvalues in a suitably defined domain of interest. Then, the transformations are applied to the matrices B and C: B <-- Q'*B and C <-- C*Z. If JOBAE = 'S', then the diagonal blocks of the subpencil A12 - lambda*E12, defined by A12 := A(NLOW:NSUP,NLOW:NSUP) and E12 := E(NLOW:NSUP,NLOW:NSUP), are reordered using orthogonal equivalence transformations, such that the leading blocks have generalized eigenvalues in a suitably defined domain of interest.Numerical Aspects
3 The algorithm requires about 25N floating point operations.Further Comments
NoneExample
Program Text
* TG01PD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDE, LDQ, LDZ
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDE = NMAX, LDQ = NMAX, LDZ = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 8*NMAX+16 )
* .. Local Scalars ..
CHARACTER*1 COMPQ, COMPZ, DICO, JOBAE, STDOM
INTEGER I, INFO, J, M, N, NDIM, NLOW, NSUP, P
DOUBLE PRECISION ALPHA, TOL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), ALPHAI(NMAX), ALPHAR(NMAX),
$ B(LDB,MMAX), BETA(NMAX), C(LDC,NMAX),
$ DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,NMAX),
$ Z(LDZ,NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL TG01PD
* .. Intrinsic Functions ..
INTRINSIC DCMPLX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, DICO, STDOM, JOBAE, COMPQ, COMPZ,
$ NLOW, NSUP, ALPHA, TOL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
IF ( LSAME( COMPQ, 'U' ) )
$ READ ( NIN, FMT = * ) ( ( Q(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( COMPZ, 'U' ) )
$ READ ( NIN, FMT = * ) ( ( Z(I,J), J = 1,N ), I = 1,N )
* Find the reduced descriptor system
* (A-lambda E,B,C).
CALL TG01PD( DICO, STDOM, JOBAE, COMPQ, COMPZ, N, M, P,
$ NLOW, NSUP, ALPHA, A, LDA, E, LDE, B, LDB,
$ C, LDC, Q, LDQ, Z, LDZ, NDIM, ALPHAR,
$ ALPHAI, BETA, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) NDIM
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 30 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
30 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 40 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
40 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 50 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,N )
50 CONTINUE
WRITE ( NOUT, FMT = 99990 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
60 CONTINUE
WRITE ( NOUT, FMT = 99985 )
DO 70 I = 1, N
IF ( BETA(I).EQ.ZERO .OR. ALPHAI(I).EQ.ZERO ) THEN
WRITE ( NOUT, FMT = 99984 )
$ ALPHAR(I)/BETA(I)
ELSE
WRITE ( NOUT, FMT = 99984 )
$ DCMPLX( ALPHAR(I), ALPHAI(I) )/BETA(I)
END IF
70 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01PD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01PD = ',I2)
99997 FORMAT (/' The transformed state dynamics matrix Q''*A*Z is ')
99996 FORMAT (/' The transformed descriptor matrix Q''*E*Z is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (' Number of eigenvalues in the domain =', I5)
99993 FORMAT (/' The transformed input/state matrix Q''*B is ')
99992 FORMAT (/' The transformed state/output matrix C*Z is ')
99991 FORMAT (/' The left transformation matrix Q is ')
99990 FORMAT (/' The right transformation matrix Z is ')
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
99986 FORMAT (/' P is out of range.',/' P = ',I5)
99985 FORMAT (/' The finite generalized eigenvalues are '/
$ ' real part imag part ')
99984 FORMAT (1X,F9.4,SP,F9.4,S,'i ')
END
Program Data
TG01PD EXAMPLE PROGRAM DATA
4 2 2 C S G I I 1 4 -1.E-7 0.0
-1 0 0 3
0 0 1 2
1 1 0 4
0 0 0 0
1 2 0 0
0 1 0 1
3 9 6 3
0 0 2 0
1 0
0 0
0 1
1 1
-1 0 1 0
0 1 -1 1
Program Results
TG01PD EXAMPLE PROGRAM RESULTS
Number of eigenvalues in the domain = 1
The transformed state dynamics matrix Q'*A*Z is
-1.6311 2.1641 -3.6829 -0.3369
0.0000 0.4550 -1.9033 0.6425
0.0000 0.0000 2.6950 0.6882
0.0000 0.0000 0.0000 0.0000
The transformed descriptor matrix Q'*E*Z is
0.4484 9.6340 -1.2601 -5.6475
0.0000 3.3099 0.6641 -1.4869
0.0000 0.0000 0.0000 -1.3765
0.0000 0.0000 0.0000 2.0000
The transformed input/state matrix Q'*B is
0.0232 -0.9413
-0.7251 -0.2478
0.6882 -0.2294
1.0000 1.0000
The transformed state/output matrix C*Z is
-0.8621 0.3754 0.3405 1.0000
-0.1511 -1.1192 0.8513 -1.0000
The left transformation matrix Q is
0.0232 -0.7251 0.6882 0.0000
-0.3369 0.6425 0.6882 0.0000
-0.9413 -0.2478 -0.2294 0.0000
0.0000 0.0000 0.0000 1.0000
The right transformation matrix Z is
0.8621 -0.3754 -0.3405 0.0000
-0.4258 -0.9008 -0.0851 0.0000
0.0000 0.0000 0.0000 1.0000
0.2748 -0.2184 0.9364 0.0000
The finite generalized eigenvalues are
real part imag part
-3.6375
0.1375
Infinity
0.0000
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TG01QD.html 0000664 0000000 0000000 00000050325 14560147231 0020357 0 ustar 00root root 0000000 0000000
Purpose
To compute orthogonal transformation matrices Q and Z which
reduce the regular pole pencil A-lambda*E of the descriptor system
(A-lambda*E,B,C) to the generalized real Schur form with ordered
generalized eigenvalues. The pair (A,E) is reduced to the form
( A1 * * ) ( E1 * * )
Q'*A*Z = ( 0 A2 * ) , Q'*E*Z = ( 0 E2 * ) , (1)
( 0 0 A3 ) ( 0 0 E2 )
where the subpencils Ak-lambda*Ek, for k = 1, 2, 3, contain the
generalized eigenvalues which belong to certain domains of
interest.
Specification
SUBROUTINE TG01QD( DICO, STDOM, JOBFI, N, M, P, ALPHA, A, LDA,
$ E, LDE, B, LDB, C, LDC, N1, N2, N3, ND, NIBLCK,
$ IBLCK, Q, LDQ, Z, LDZ, ALPHAR, ALPHAI, BETA,
$ TOL, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, JOBFI, STDOM
INTEGER INFO, LDA, LDB, LDC, LDE, LDQ, LDWORK, LDZ, M, N,
$ N1, N2, N3, ND, NIBLCK, P
DOUBLE PRECISION ALPHA, TOL
C .. Array Arguments ..
INTEGER IBLCK( * ), IWORK(*)
DOUBLE PRECISION A(LDA,*), ALPHAI(*), ALPHAR(*), B(LDB,*),
$ BETA(*), C(LDC,*), DWORK(*), E(LDE,*), Q(LDQ,*),
$ Z(LDZ,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the descriptor system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
STDOM CHARACTER*1
Specifies the type of the domain of interest for the
generalized eigenvalues, as follows:
= 'S': stability type domain (i.e., left part of complex
plane or inside of a circle);
= 'U': instability type domain (i.e., right part of complex
plane or outside of a circle);
= 'N': whole complex domain, excepting infinity.
JOBFI CHARACTER*1
Specifies the type of generalized eigenvalues in the
leading diagonal block(s) as follows:
= 'F': finite generalized eigenvalues are in the
leading diagonal blocks (Af,Ef), and the resulting
transformed pair has the form
( Af * ) ( Ef * )
Q'*A*Z = ( ) , Q'*E*Z = ( ) ;
( 0 Ai ) ( 0 Ei )
= 'I': infinite generalized eigenvalues are in the
leading diagonal blocks (Ai,Ei), and the resulting
transformed pair has the form
( Ai * ) ( Ei * )
Q'*A*Z = ( ) , Q'*E*Z = ( ) .
( 0 Af ) ( 0 Ef )
Input/Output Parameters
N (input) INTEGER
The number of rows of the matrix B, the number of columns
of the matrix C and the order of the square matrices A
and E. N >= 0.
M (input) INTEGER
The number of columns of the matrix B. M >= 0.
P (input) INTEGER
The number of rows of the matrix C. P >= 0.
ALPHA (input) DOUBLE PRECISION
The boundary of the domain of interest for the finite
generalized eigenvalues of the pair (A,E). For a
continuous-time system (DICO = 'C'), ALPHA is the boundary
value for the real parts of the generalized eigenvalues,
while for a discrete-time system (DICO = 'D'), ALPHA >= 0
represents the boundary value for the moduli of the
generalized eigenvalues.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A.
On exit, the leading N-by-N part of this array contains
the matrix Q'*A*Z in real Schur form, with the elements
below the first subdiagonal set to zero.
If JOBFI = 'I', the N1-by-N1 pair (A1,E1) contains the
infinite spectrum, the N2-by-N2 pair (A2,E2) contains the
finite spectrum in the domain of interest, and the
N3-by-N3 pair (A3,E3) contains the finite spectrum ouside
of the domain of interest.
If JOBFI = 'F', the N1-by-N1 pair (A1,E1) contains the
finite spectrum in the domain of interest, the N2-by-N2
pair (A2,E2) contains the finite spectrum ouside of the
domain of interest, and the N3-by-N3 pair (A3,E3) contains
the infinite spectrum.
Ai has a block structure as in (2), where A0,0 is ND-by-ND
and Ai,i is IBLCK(i)-by-IBLCK(i), for i = 1, ..., NIBLCK.
The domain of interest for the pair (Af,Ef), containing
the finite generalized eigenvalues, is defined by the
parameters ALPHA, DICO and STDOM as follows:
For DICO = 'C':
Real(eig(Af,Ef)) < ALPHA if STDOM = 'S';
Real(eig(Af,Ef)) > ALPHA if STDOM = 'U'.
For DICO = 'D':
Abs(eig(Af,Ef)) < ALPHA if STDOM = 'S';
Abs(eig(Af,Ef)) > ALPHA if STDOM = 'U'.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain the descriptor matrix E.
On exit, the leading N-by-N part of this array contains
the matrix Q'*E*Z in upper triangular form, with the
elements below the diagonal set to zero. Its structure
corresponds to the block structure of the matrix Q'*A*Z
(see description of A).
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input matrix B.
On exit, the leading N-by-M part of this array contains
the transformed input matrix Q'*B.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed output matrix C*Z.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,P).
N1 (output) INTEGER
N2 (output) INTEGER
N3 (output) INTEGER
The number of the generalized eigenvalues of the pairs
(A1,E1), (A2,E2) and (A3,E3), respectively.
ND (output) INTEGER.
The number of non-dynamic infinite eigenvalues of the
pair (A,E). Note: N-ND is the rank of the matrix E.
NIBLCK (output) INTEGER
If ND > 0, the number of infinite blocks minus one.
If ND = 0, then NIBLCK = 0.
IBLCK (output) INTEGER array, dimension (N)
IBLCK(i) contains the dimension of the i-th block in the
staircase form (2), where i = 1,2,...,NIBLCK.
Q (output) DOUBLE PRECISION array, dimension (LDQ,N)
The leading N-by-N part of this array contains the
orthogonal matrix Q, where Q' is the product of orthogonal
transformations applied to A, E, and B on the left.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= MAX(1,N).
Z (output) DOUBLE PRECISION array, dimension (LDZ,N)
The leading N-by-N part of this array contains the
orthogonal matrix Z, which is the product of orthogonal
transformations applied to A, E, and C on the right.
LDZ INTEGER
The leading dimension of the array Z. LDZ >= MAX(1,N).
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
BETA (output) DOUBLE PRECISION array, dimension (N)
On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j = 1, ..., N,
are the generalized eigenvalues.
ALPHAR(j) + ALPHAI(j)*i, and BETA(j), j = 1, ..., N, are
the diagonals of the complex Schur form (S,T) that would
result if the 2-by-2 diagonal blocks of the real Schur
form of (A,E) were further reduced to triangular form
using 2-by-2 complex unitary transformations.
If ALPHAI(j) is zero, then the j-th eigenvalue is real;
if positive, then the j-th and (j+1)-st eigenvalues are a
complex conjugate pair, with ALPHAI(j+1) negative.
Tolerances
TOL DOUBLE PRECISION
A tolerance used in rank decisions to determine the
effective rank, which is defined as the order of the
largest leading (or trailing) triangular submatrix in the
QR factorization with column pivoting whose estimated
condition number is less than 1/TOL. If the user sets
TOL <= 0, then an implicitly computed, default tolerance
TOLDEF = N**2*EPS, is used instead, where EPS is the
machine precision (see LAPACK Library routine DLAMCH).
TOL < 1.
Workspace
IWORK INTEGER array, dimension (N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= 1, and if N = 0,
LDWORK >= 4*N, if STDOM = 'N';
LDWORK >= 4*N+16, if STDOM = 'S' or 'U'.
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message related to LDWORK is issued by
XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the pencil A-lambda*E is not regular;
= 2: the QZ algorithm failed to compute all generalized
eigenvalues of the pair (A,E);
= 3: a failure occured during the ordering of the
generalized real Schur form of the pair (A,E).
Method
The separation of the finite and infinite parts is based on the
reduction algorithm of [1].
If JOBFI = 'F', the matrices of the pair (Ai,Ei), containing the
infinite generalized eigenvalues, have the form
( A0,0 A0,k ... A0,1 ) ( 0 E0,k ... E0,1 )
Ai = ( 0 Ak,k ... Ak,1 ) , Ei = ( 0 0 ... Ek,1 ) ; (2)
( : : . : ) ( : : . : )
( 0 0 ... A1,1 ) ( 0 0 ... 0 )
if JOBFI = 'I', the matrices Ai and Ei have the form
( A1,1 ... A1,k A1,0 ) ( 0 ... E1,k E1,0 )
Ai = ( : . : : ) , Ei = ( : . : : ) , (3)
( : ... Ak,k Ak,0 ) ( : ... 0 Ek,0 )
( 0 ... 0 A0,0 ) ( 0 ... 0 0 )
where Ai,i , for i = 0, 1, ..., k, are nonsingular upper
triangular matrices, and A0,0 corresponds to the non-dynamic
infinite modes of the system.
References
[1] Misra, P., Van Dooren, P., and Varga, A.
Computation of structural invariants of generalized
state-space systems.
Automatica, 30, pp. 1921-1936, 1994.
Numerical Aspects
3 The algorithm requires about 25N floating point operations.Further Comments
The number of infinite poles is computed as
NIBLCK
NINFP = Sum IBLCK(i) = N - ND - NF,
i=1
where NF is the number of finite generalized eigenvalues.
The multiplicities of infinite poles can be computed as follows:
there are IBLCK(k)-IBLCK(k+1) infinite poles of multiplicity k,
for k = 1, ..., NIBLCK, where IBLCK(NIBLCK+1) = 0.
Note that each infinite pole of multiplicity k corresponds to an
infinite eigenvalue of multiplicity k+1.
Example
Program Text
* TG01QD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDE, LDQ, LDZ
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDE = NMAX, LDQ = NMAX, LDZ = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 4*NMAX+16 )
* .. Local Scalars ..
CHARACTER*1 DICO, JOBFI, STDOM
INTEGER I, INFO, J, M, N, N1, N2, N3, ND, NIBLCK, P
DOUBLE PRECISION ALPHA, TOL
* .. Local Arrays ..
INTEGER IBLCK(NMAX), IWORK(NMAX)
DOUBLE PRECISION A(LDA,NMAX), ALPHAI(NMAX), ALPHAR(NMAX),
$ B(LDB,MMAX), BETA(NMAX), C(LDC,NMAX),
$ DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,NMAX),
$ Z(LDZ,NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL TG01QD
* .. Intrinsic Functions ..
INTRINSIC DCMPLX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, DICO, STDOM, JOBFI, ALPHA, TOL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find the reduced descriptor system
* (A-lambda E,B,C).
CALL TG01QD( DICO, STDOM, JOBFI, N, M, P, ALPHA, A, LDA,
$ E, LDE, B, LDB, C, LDC, N1, N2, N3, ND,
$ NIBLCK, IBLCK, Q, LDQ, Z, LDZ, ALPHAR,
$ ALPHAI, BETA, TOL, IWORK, DWORK, LDWORK,
$ INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) N1, N2, N3
WRITE ( NOUT, FMT = 99983 ) ND
WRITE ( NOUT, FMT = 99989 ) NIBLCK + 1
IF ( NIBLCK.GT.0 ) THEN
WRITE ( NOUT, FMT = 99985 )
$ ( IBLCK(I), I = 1, NIBLCK )
END IF
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 30 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
30 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 40 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
40 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 50 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,N )
50 CONTINUE
WRITE ( NOUT, FMT = 99990 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
60 CONTINUE
WRITE ( NOUT, FMT = 99985 )
DO 70 I = 1, N
IF ( BETA(I).EQ.ZERO .OR. ALPHAI(I).EQ.ZERO ) THEN
WRITE ( NOUT, FMT = 99984 )
$ ALPHAR(I)/BETA(I)
ELSE
WRITE ( NOUT, FMT = 99984 )
$ DCMPLX( ALPHAR(I), ALPHAI(I) )/BETA(I)
END IF
70 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01QD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01QD = ',I2)
99997 FORMAT (/' The transformed state dynamics matrix Q''*A*Z is ')
99996 FORMAT (/' The transformed descriptor matrix Q''*E*Z is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (' Number of eigenvalues of the three blocks =', 3I5)
99993 FORMAT (/' The transformed input/state matrix Q''*B is ')
99992 FORMAT (/' The transformed state/output matrix C*Z is ')
99991 FORMAT (/' The left transformation matrix Q is ')
99990 FORMAT (/' The right transformation matrix Z is ')
99989 FORMAT ( ' Number of infinite blocks = ',I5)
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
99986 FORMAT (/' P is out of range.',/' P = ',I5)
99985 FORMAT (/' The finite generalized eigenvalues are '/
$ ' real part imag part ')
99984 FORMAT (1X,F9.4,SP,F9.4,S,'i ')
99983 FORMAT ( ' Number of non-dynamic infinite eigenvalues = ',I5)
END
Program Data
TG01QD EXAMPLE PROGRAM DATA
4 2 2 C S F -1.E-7 0.0
-1 0 0 3
0 0 1 2
1 1 0 4
0 0 0 0
1 2 0 0
0 1 0 1
3 9 6 3
0 0 2 0
1 0
0 0
0 1
1 1
-1 0 1 0
0 1 -1 1
Program Results
TG01QD EXAMPLE PROGRAM RESULTS
Number of eigenvalues of the three blocks = 1 2 1
Number of non-dynamic infinite eigenvalues = 1
Number of infinite blocks = 1
The transformed state dynamics matrix Q'*A*Z is
1.6311 2.1641 -0.5848 -3.6517
0.0000 -0.4550 1.0935 1.6851
0.0000 0.0000 0.0000 1.5770
0.0000 0.0000 0.0000 2.2913
The transformed descriptor matrix Q'*E*Z is
-0.4484 9.6340 5.1601 -2.6183
0.0000 -3.3099 -1.6050 -0.2756
0.0000 0.0000 2.3524 -0.6008
0.0000 0.0000 0.0000 0.0000
The transformed input/state matrix Q'*B is
0.0232 -0.9413
0.7251 0.2478
-0.4336 -0.9538
1.1339 0.3780
The transformed state/output matrix C*Z is
0.8621 0.3754 -0.8847 0.5774
0.1511 -1.1192 1.1795 0.5774
The left transformation matrix Q is
0.0232 0.7251 0.3902 0.5669
-0.3369 -0.6425 0.3902 0.5669
-0.9413 0.2478 -0.1301 -0.1890
0.0000 0.0000 -0.8238 0.5669
The right transformation matrix Z is
-0.8621 -0.3754 -0.0843 -0.3299
0.4258 -0.9008 -0.0211 -0.0825
0.0000 0.0000 -0.9689 0.2474
-0.2748 -0.2184 0.2317 0.9073
The finite generalized eigenvalues are
real part imag part
-3.6375
0.1375
0.0000
Infinity
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/TG01WD.html 0000664 0000000 0000000 00000015027 14560147231 0020365 0 ustar 00root root 0000000 0000000
Purpose
To reduce the pair (A,E) to a real generalized Schur form by using an orthogonal equivalence transformation (A,E) <-- (Q'*A*Z,Q'*E*Z) and to apply the transformation to the matrices B and C: B <-- Q'*B and C <-- C*Z.Specification
SUBROUTINE TG01WD( N, M, P, A, LDA, E, LDE, B, LDB, C, LDC,
$ Q, LDQ, Z, LDZ, ALPHAR, ALPHAI, BETA, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDE, LDQ, LDWORK, LDZ,
$ M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), ALPHAI(*), ALPHAR(*), B(LDB,*),
$ BETA(*), C(LDC,*), DWORK(*), E(LDE,*),
$ Q(LDQ,*), Z(LDZ,*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation,
i.e., the order of the matrices A and E. N >= 0.
M (input) INTEGER
The number of system inputs, or of columns of B. M >= 0.
P (input) INTEGER
The number of system outputs, or of rows of C. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state dynamics matrix A.
On exit, the leading N-by-N part of this array contains
the matrix Q' * A * Z in an upper quasi-triangular form.
The elements below the first subdiagonal are set to zero.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain the original descriptor matrix E.
On exit, the leading N-by-N part of this array contains
the matrix Q' * E * Z in an upper triangular form.
The elements below the diagonal are set to zero.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input matrix B.
On exit, the leading N-by-M part of this array contains
the transformed input matrix Q' * B.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed output matrix C * Z.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
Q (output) DOUBLE PRECISION array, dimension (LDQ,N)
The leading N-by-N part of this array contains the left
orthogonal transformation matrix used to reduce (A,E) to
the real generalized Schur form.
The columns of Q are the left generalized Schur vectors
of the pair (A,E).
LDQ INTEGER
The leading dimension of array Q. LDQ >= max(1,N).
Z (output) DOUBLE PRECISION array, dimension (LDZ,N)
The leading N-by-N part of this array contains the right
orthogonal transformation matrix used to reduce (A,E) to
the real generalized Schur form.
The columns of Z are the right generalized Schur vectors
of the pair (A,E).
LDZ INTEGER
The leading dimension of array Z. LDZ >= max(1,N).
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
BETA (output) DOUBLE PRECISION array, dimension (N)
On exit, if INFO = 0, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j),
j=1,...,N, will be the generalized eigenvalues.
ALPHAR(j) + ALPHAI(j)*i, and BETA(j), j=1,...,N, are the
diagonals of the complex Schur form that would result if
the 2-by-2 diagonal blocks of the real Schur form of
(A,E) were further reduced to triangular form using
2-by-2 complex unitary transformations.
If ALPHAI(j) is zero, then the j-th eigenvalue is real;
if positive, then the j-th and (j+1)-st eigenvalues are a
complex conjugate pair, with ALPHAI(j+1) negative.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The dimension of working array DWORK. LDWORK >= 8*N+16.
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, the QZ algorithm failed to compute
the generalized real Schur form; elements i+1:N of
ALPHAR, ALPHAI, and BETA should be correct.
Method
The pair (A,E) is reduced to a real generalized Schur form using an orthogonal equivalence transformation (A,E) <-- (Q'*A*Z,Q'*E*Z) and the transformation is applied to the matrices B and C: B <-- Q'*B and C <-- C*Z.Numerical Aspects
3 The algorithm requires about 25N floating point operations.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/UD01BD.html 0000664 0000000 0000000 00000014224 14560147231 0020334 0 ustar 00root root 0000000 0000000
Purpose
To read the coefficients of a matrix polynomial
dp-1 dp
P(s) = P(0) + P(1) * s + . . . + P(dp-1) * s + P(dp) * s .
Specification
SUBROUTINE UD01BD( MP, NP, DP, NIN, P, LDP1, LDP2, INFO )
C .. Scalar Arguments ..
INTEGER DP, INFO, LDP1, LDP2, MP, NP, NIN
C .. Array Arguments ..
DOUBLE PRECISION P(LDP1,LDP2,*)
Arguments
Input/Output Parameters
MP (input) INTEGER
The number of rows of the matrix polynomial P(s).
MP >= 1.
NP (input) INTEGER
The number of columns of the matrix polynomial P(s).
NP >= 1.
DP (input) INTEGER
The degree of the matrix polynomial P(s). DP >= 0.
NIN (input) INTEGER
The input channel from which the elements of P(s) are
read. NIN >= 0.
P (output) DOUBLE PRECISION array, dimension
(LDP1,LDP2,DP+1)
The leading MP-by-NP-by-(DP+1) part of this array contains
the coefficients of the matrix polynomial P(s).
Specifically, P(i,j,k) contains the coefficient of
s**(k-1) of the polynomial which is the (i,j)-th element
of P(s), where i = 1,2,...,MP, j = 1,2,...,NP and
k = 1,2,...,DP+1.
LDP1 INTEGER
The leading dimension of array P. LDP1 >= MP.
LDP2 INTEGER
The second dimension of array P. LDP2 >= NP.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The coefficients P(i), i = 0, ..., DP, which are MP-by-NP matrices, are read from the input file NIN row by row. Each P(i) must be preceded by a text line. This text line can be used to indicate the coefficient matrices.References
None.Numerical Aspects
None.Further Comments
NoneExample
Program Text
* UD01BD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MPMAX, NPMAX, DPMAX
PARAMETER ( MPMAX = 10, NPMAX = 10, DPMAX = 5 )
INTEGER LDP1, LDP2
PARAMETER ( LDP1 = MPMAX, LDP2 = NPMAX )
* .. Local Scalars ..
INTEGER DP, INFO, L, MP, NP
* .. Local Arrays ..
DOUBLE PRECISION P(LDP1,LDP2,DPMAX)
* .. External Subroutines ..
EXTERNAL UD01BD, UD01ND
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) MP, NP, DP
IF ( MP.LE.0 .OR. MP.GT.MPMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) MP
ELSE IF ( NP.LE.0 .OR. NP.GT.NPMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) NP
ELSE IF ( DP.LT.0 .OR. DP.GT.DPMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) DP
ELSE
* Read the coefficients of the matrix polynomial P(s).
CALL UD01BD( MP, NP, DP, NIN, P, LDP1, LDP2, INFO )
IF ( INFO.EQ.0 ) THEN
WRITE ( NOUT, 99996 ) MP, NP, DP
* Write the coefficients of the matrix polynomial P(s).
L = 5
CALL UD01ND( MP, NP, DP, L, NOUT, P, LDP1, LDP2, ' P',
$ INFO )
IF ( INFO.NE.0 )
$ WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99998 ) INFO
END IF
END IF
STOP
*
99999 FORMAT (' UD01BD EXAMPLE PROGRAM RESULTS', /1X)
99998 FORMAT (' INFO on exit from UD01BD = ',I2)
99997 FORMAT (' INFO on exit from UD01ND = ',I2)
99996 FORMAT (' MP =', I2, 2X, ' NP =', I2, 3X, 'DP =', I2)
99995 FORMAT (/' NP is out of range.',/' NP = ',I5)
99994 FORMAT (/' MP is out of range.',/' MP = ',I5)
99993 FORMAT (/' DP is out of range.',/' DP = ',I5)
END
Program Data
UD01BD EXAMPLE PROGRAM DATA 4 3 2 P0 1.0D-00 0.0D-00 0.0D-00 0.0D-00 2.0D-00 4.0D-00 0.0D-00 4.0D-00 8.0D-00 0.0D-00 6.0D-00 1.2D+01 P1 0.0D-00 1.0D-00 2.0D-00 1.0D-00 0.0D-00 0.0D-00 2.0D-00 0.0D-00 0.0D-00 3.0D-00 0.0D-00 0.0D-00 P2 1.0D-00 0.0D-00 0.0D-00 0.0D-00 0.0D-00 0.0D-00 0.0D-00 0.0D-00 0.0D-00 0.0D-00 0.0D-00 0.0D-00Program Results
UD01BD EXAMPLE PROGRAM RESULTS
MP = 4 NP = 3 DP = 2
P( 0) ( 4X 3)
1 2 3
1 0.1000000D+01 0.0000000D+00 0.0000000D+00
2 0.0000000D+00 0.2000000D+01 0.4000000D+01
3 0.0000000D+00 0.4000000D+01 0.8000000D+01
4 0.0000000D+00 0.6000000D+01 0.1200000D+02
P( 1) ( 4X 3)
1 2 3
1 0.0000000D+00 0.1000000D+01 0.2000000D+01
2 0.1000000D+01 0.0000000D+00 0.0000000D+00
3 0.2000000D+01 0.0000000D+00 0.0000000D+00
4 0.3000000D+01 0.0000000D+00 0.0000000D+00
P( 2) ( 4X 3)
1 2 3
1 0.1000000D+01 0.0000000D+00 0.0000000D+00
2 0.0000000D+00 0.0000000D+00 0.0000000D+00
3 0.0000000D+00 0.0000000D+00 0.0000000D+00
4 0.0000000D+00 0.0000000D+00 0.0000000D+00
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/UD01CD.html 0000664 0000000 0000000 00000015721 14560147231 0020340 0 ustar 00root root 0000000 0000000
Purpose
To read the elements of a sparse matrix polynomial
dp-1 dp
P(s) = P(0) + P(1) * s + . . . + P(dp-1) * s + P(dp) * s .
Specification
SUBROUTINE UD01CD( MP, NP, DP, NIN, P, LDP1, LDP2, INFO )
C .. Scalar Arguments ..
INTEGER DP, INFO, LDP1, LDP2, MP, NP, NIN
C .. Array Arguments ..
DOUBLE PRECISION P(LDP1,LDP2,*)
Arguments
Input/Output Parameters
MP (input) INTEGER
The number of rows of the matrix polynomial P(s).
MP >= 1.
NP (input) INTEGER
The number of columns of the matrix polynomial P(s).
NP >= 1.
DP (input) INTEGER
The degree of the matrix polynomial P(s). DP >= 0.
NIN (input) INTEGER
The input channel from which the elements of P(s) are
read. NIN >= 0.
P (output) DOUBLE PRECISION array, dimension
(LDP1,LDP2,DP+1)
The leading MP-by-NP-by-(DP+1) part of this array contains
the coefficients of the matrix polynomial P(s).
Specifically, P(i,j,k) contains the coefficient of
s**(k-1) of the polynomial which is the (i,j)-th element
of P(s), where i = 1,2,...,MP, j = 1,2,...,NP and
k = 1,2,...,DP+1.
The not assigned elements are set to zero.
LDP1 INTEGER
The leading dimension of array P. LDP1 >= MP.
LDP2 INTEGER
The second dimension of array P. LDP2 >= NP.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1 : if a row index i is read with i < 1 or i > MP or
a column index j is read with j < 1 or j > NP or
a coefficient degree d is read with d < 0 or
d > DP + 1. This is a warning.
Method
First, the elements P(i,j,k) with 1 <= i <= MP, 1 <= j <= NP and
1 <= k <= DP + 1 are set to zero. Next the nonzero (polynomial)
elements are read from the input file NIN. Each nonzero element is
given by the values i, j, d, P(i,j,k), k = 1, ..., d+1, where d is
the degree and P(i,j,k) is the coefficient of s**(k-1) in the
(i,j)-th element of P(s), i.e., let
d
P (s) = P (0) + P (1) * s + . . . + P (d) * s
i,j i,j i,j i,j
be the nonzero (i,j)-th element of the matrix polynomial P(s).
Then P(i,j,k) corresponds to coefficient P (k-1), k = 1,...,d+1.
i,j
For each nonzero element, the values i, j, and d are read as one
record of the file NIN, and the values P(i,j,k), k = 1,...,d+1,
are read as the following record.
The routine terminates after the last line has been read.
References
None.Numerical Aspects
None.Further Comments
NoneExample
Program Text
* UD01CD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MPMAX, NPMAX, DPMAX
PARAMETER ( MPMAX = 10, NPMAX = 10, DPMAX = 5 )
INTEGER LDP1, LDP2
PARAMETER ( LDP1 = MPMAX, LDP2 = NPMAX )
* .. Local Scalars ..
INTEGER DP, INFO, INFO1, L, MP, NP
* .. Local Arrays ..
DOUBLE PRECISION P(LDP1,LDP2,DPMAX)
* .. External Subroutines ..
EXTERNAL UD01CD, UD01ND
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) MP, NP, DP
IF ( MP.LE.0 .OR. MP.GT.MPMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) MP
ELSE IF ( NP.LE.0 .OR. NP.GT.NPMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) NP
ELSE IF ( DP.LT.0 .OR. DP.GT.DPMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) DP
ELSE
* Read the coefficients of the matrix polynomial P(s).
CALL UD01CD( MP, NP, DP, NIN, P, LDP1, LDP2, INFO )
IF ( INFO.GE.0 ) THEN
WRITE ( NOUT, 99996 ) MP, NP, DP
* Write the coefficients of the matrix polynomial P(s).
L = 5
CALL UD01ND( MP, NP, DP, L, NOUT, P, LDP1, LDP2, ' P',
$ INFO1 )
IF ( INFO1.NE.0 )
$ WRITE ( NOUT, FMT = 99997 ) INFO1
END IF
IF ( INFO.NE.0 )
$ WRITE ( NOUT, FMT = 99998 ) INFO
END IF
STOP
*
99999 FORMAT (' UD01CD EXAMPLE PROGRAM RESULTS', /1X)
99998 FORMAT (' INFO on exit from UD01CD = ',I2)
99997 FORMAT (' INFO on exit from UD01ND = ',I2)
99996 FORMAT (' MP =', I2, 2X, ' NP =', I2, 3X, 'DP =', I2)
99995 FORMAT (/' NP is out of range.',/' NP = ',I5)
99994 FORMAT (/' MP is out of range.',/' MP = ',I5)
99993 FORMAT (/' DP is out of range.',/' DP = ',I5)
END
Program Data
UD01CD EXAMPLE PROGRAM DATA 4 3 2 1 1 1 1.0 1.0 2 2 2 2.0 0.0 1.0 3 3 2 0.0 3.0 1.0 4 1 0 4.0Program Results
UD01CD EXAMPLE PROGRAM RESULTS
MP = 4 NP = 3 DP = 2
P( 0) ( 4X 3)
1 2 3
1 0.1000000D+01 0.0000000D+00 0.0000000D+00
2 0.0000000D+00 0.2000000D+01 0.0000000D+00
3 0.0000000D+00 0.0000000D+00 0.0000000D+00
4 0.4000000D+01 0.0000000D+00 0.0000000D+00
P( 1) ( 4X 3)
1 2 3
1 0.1000000D+01 0.0000000D+00 0.0000000D+00
2 0.0000000D+00 0.0000000D+00 0.0000000D+00
3 0.0000000D+00 0.0000000D+00 0.3000000D+01
4 0.0000000D+00 0.0000000D+00 0.0000000D+00
P( 2) ( 4X 3)
1 2 3
1 0.0000000D+00 0.0000000D+00 0.0000000D+00
2 0.0000000D+00 0.1000000D+01 0.0000000D+00
3 0.0000000D+00 0.0000000D+00 0.1000000D+01
4 0.0000000D+00 0.0000000D+00 0.0000000D+00
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/UD01DD.html 0000664 0000000 0000000 00000011672 14560147231 0020342 0 ustar 00root root 0000000 0000000
Purpose
To read the elements of a sparse matrix.Specification
SUBROUTINE UD01DD( M, N, NIN, A, LDA, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, M, N, NIN
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*)
Arguments
Input/Output Parameters
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
NIN (input) INTEGER
The input channel from which the elements of A are read.
NIN >= 0.
A (output) DOUBLE PRECISION array, dimension (LDA,N)
The leading M-by-N part of this array contains the sparse
matrix A. The not assigned elements are set to zero.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,M).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1 : if a row index i is read with i < 1 or i > M or
a column index j is read with j < 1 or j > N.
This is a warning.
Method
First, the elements A(i,j) with 1 <= i <= M and 1 <= j <= N are set to zero. Next the nonzero elements are read from the input file NIN. Each line of NIN must contain consecutively the values i, j, A(i,j). The routine terminates after the last line has been read.References
None.Numerical Aspects
None.Further Comments
NoneExample
Program Text
* UD01DD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, NMAX
PARAMETER ( MMAX = 10, NMAX = 10 )
INTEGER LDA
PARAMETER ( LDA = NMAX )
* .. Local Scalars ..
INTEGER INFO, INFO1, M, N
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX)
* .. External Subroutines ..
EXTERNAL UD01DD, UD01MD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, N
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) M
ELSE IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) N
ELSE
* Read the coefficients of the matrix polynomial P(s).
CALL UD01DD( M, N, NIN, A, LDA, INFO )
IF ( INFO.GE.0 ) THEN
* Write the matrix A.
CALL UD01MD( M, N, 5, NOUT, A, LDA, ' Matrix A', INFO1 )
IF ( INFO1.NE.0 )
$ WRITE ( NOUT, FMT = 99998 ) INFO1
END IF
IF ( INFO.NE.0 )
$ WRITE ( NOUT, FMT = 99997 ) INFO
END IF
STOP
*
99999 FORMAT (' UD01DD EXAMPLE PROGRAM RESULTS', /1X)
99998 FORMAT (' INFO on exit from UD01MD = ',I2)
99997 FORMAT (' INFO on exit from UD01DD = ',I2)
99995 FORMAT (/' N is out of range.',/' N = ',I5)
99994 FORMAT (/' M is out of range.',/' M = ',I5)
END
Program Data
UD01DD EXAMPLE PROGRAM DATA 6 5 1 1 -1.1 6 1 1.5 2 2 -2.2 6 2 2.5 3 3 -3.3 6 3 3.5 4 4 -4.4 6 4 4.5 5 5 -5.5 6 5 5.5Program Results
UD01DD EXAMPLE PROGRAM RESULTS
Matrix A ( 6X 5)
1 2 3 4 5
1 -0.1100000D+01 0.0000000D+00 0.0000000D+00 0.0000000D+00 0.0000000D+00
2 0.0000000D+00 -0.2200000D+01 0.0000000D+00 0.0000000D+00 0.0000000D+00
3 0.0000000D+00 0.0000000D+00 -0.3300000D+01 0.0000000D+00 0.0000000D+00
4 0.0000000D+00 0.0000000D+00 0.0000000D+00 -0.4400000D+01 0.0000000D+00
5 0.0000000D+00 0.0000000D+00 0.0000000D+00 0.0000000D+00 -0.5500000D+01
6 0.1500000D+01 0.2500000D+01 0.3500000D+01 0.4500000D+01 0.5500000D+01
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/UD01MD.html 0000664 0000000 0000000 00000012003 14560147231 0020340 0 ustar 00root root 0000000 0000000
Purpose
To print an M-by-N real matrix A row by row. The elements of A are output to 7 significant figures.Specification
SUBROUTINE UD01MD( M, N, L, NOUT, A, LDA, TEXT, INFO )
C .. Scalar Arguments ..
INTEGER INFO, L, LDA, M, N, NOUT
CHARACTER*(*) TEXT
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*)
Arguments
Input/Output Parameters
M (input) INTEGER
The number of rows of matrix A to be printed. M >= 1.
N (input) INTEGER
The number of columns of matrix A to be printed. N >= 1.
L (input) INTEGER
The number of elements of matrix A to be printed per line.
1 <= L <= 5.
NOUT (input) INTEGER
The output channel to which the results are sent.
NOUT >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading M-by-N part of this array must contain the
matrix to be printed.
LDA INTEGER
The leading dimension of array A. LDA >= M.
TEXT (input) CHARACTER*72.
Title caption of the matrix to be printed (up to a
maximum of 72 characters). For example, TEXT = 'Matrix A'.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine first prints the contents of TEXT as a title, followed
by the elements of the matrix A such that
(i) if N <= L, the leading M-by-N part is printed;
(ii) if N = k*L + p (where k,p > 0), then k M-by-L blocks of
consecutive columns of A are printed one after another
followed by one M-by-p block containing the last p columns
of A.
Row numbers are printed on the left of each row and a column
number appears on top of each column.
The routine uses 2 + (k + 1)*(m + 1) lines and 8 + 15*c positions
per line where c is the actual number of columns, (i.e. c = L
or c = p).
References
None.Numerical Aspects
None.Further Comments
NoneExample
Program Text
* UD01MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, NMAX
PARAMETER ( MMAX = 20, NMAX = 20 )
INTEGER LDA
PARAMETER ( LDA = MMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, L, M, N
CHARACTER*72 TEXT
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX)
* .. External Subroutines ..
EXTERNAL UD01MD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, N, L, TEXT
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99996 ) M
ELSE IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99997 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,M )
* Print out the matrix A.
CALL UD01MD( M, N, L, NOUT, A, LDA, TEXT, INFO )
IF ( INFO.NE.0 ) WRITE ( NOUT, FMT = 99998 ) INFO
END IF
STOP
*
99999 FORMAT (' UD01MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from UD01MD = ',I2)
99997 FORMAT (/' N is out of range.',/' N = ',I5)
99996 FORMAT (/' M is out of range.',/' M = ',I5)
END
Program Data
UD01MD EXAMPLE PROGRAM DATA 4 4 4 'Matrix A' 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0Program Results
UD01MD EXAMPLE PROGRAM RESULTS
Matrix A ( 4X 4)
1 2 3 4
1 0.1000000D+01 0.2000000D+01 0.3000000D+01 0.4000000D+01
2 0.5000000D+01 0.6000000D+01 0.7000000D+01 0.8000000D+01
3 0.9000000D+01 0.1000000D+02 0.1100000D+02 0.1200000D+02
4 0.1300000D+02 0.1400000D+02 0.1500000D+02 0.1600000D+02
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/UD01MZ.html 0000664 0000000 0000000 00000006333 14560147231 0020377 0 ustar 00root root 0000000 0000000
Purpose
To print an M-by-N real matrix A row by row. The elements of A are output to 7 significant figures.Specification
SUBROUTINE UD01MZ( M, N, L, NOUT, A, LDA, TEXT, INFO )
C .. Scalar Arguments ..
INTEGER INFO, L, LDA, M, N, NOUT
CHARACTER*(*) TEXT
C .. Array Arguments ..
COMPLEX*16 A(LDA,*)
Arguments
Input/Output Parameters
M (input) INTEGER
The number of rows of matrix A to be printed. M >= 1.
N (input) INTEGER
The number of columns of matrix A to be printed. N >= 1.
L (input) INTEGER
The number of elements of matrix A to be printed per line.
1 <= L <= 3.
NOUT (input) INTEGER
The output channel to which the results are sent.
NOUT >= 0.
A (input) COMPLEX*16 array, dimension (LDA,N)
The leading M-by-N part of this array must contain the
matrix to be printed.
LDA INTEGER
The leading dimension of array A. LDA >= M.
TEXT (input) CHARACTER*72.
Title caption of the matrix to be printed (up to a
maximum of 72 characters). For example, TEXT = 'Matrix A'.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine first prints the contents of TEXT as a title, followed
by the elements of the matrix A such that
(i) if N <= L, the leading M-by-N part is printed;
(ii) if N = k*L + p (where k,p > 0), then k M-by-L blocks of
consecutive columns of A are printed one after another
followed by one M-by-p block containing the last p columns
of A.
Row numbers are printed on the left of each row and a column
number appears on top of each complex column.
The routine uses 2 + (k + 1)*(m + 1) lines and 7 + 32*c positions
per line where c is the actual number of columns, (i.e. c = L
or c = p).
References
None.Numerical Aspects
None.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/UD01ND.html 0000664 0000000 0000000 00000016061 14560147231 0020351 0 ustar 00root root 0000000 0000000
Purpose
To print the MP-by-NP coefficient matrices of a matrix polynomial
dp-1 dp
P(s) = P(0) + P(1) * s + . . . + P(dp-1) * s + P(dp) * s .
The elements of the matrices are output to 7 significant figures.
Specification
SUBROUTINE UD01ND( MP, NP, DP, L, NOUT, P, LDP1, LDP2, TEXT,
$ INFO )
C .. Scalar Arguments ..
INTEGER DP, INFO, L, LDP1, LDP2, MP, NP, NOUT
CHARACTER*(*) TEXT
C .. Array Arguments ..
DOUBLE PRECISION P(LDP1,LDP2,*)
Arguments
Input/Output Parameters
MP (input) INTEGER
The number of rows of the matrix polynomial P(s).
MP >= 1.
NP (input) INTEGER
The number of columns of the matrix polynomial P(s).
NP >= 1.
DP (input) INTEGER
The degree of the matrix polynomial P(s). DP >= 0.
L (input) INTEGER
The number of elements of the coefficient matrices to be
printed per line. 1 <= L <= 5.
NOUT (input) INTEGER
The output channel to which the results are sent.
NOUT >= 0.
P (input) DOUBLE PRECISION array, dimension (LDP1,LDP2,DP+1)
The leading MP-by-NP-by-(DP+1) part of this array must
contain the coefficients of the matrix polynomial P(s).
Specifically, P(i,j,k) must contain the coefficient of
s**(k-1) of the polynomial which is the (i,j)-th element
of P(s), where i = 1,2,...,MP, j = 1,2,...,NP and
k = 1,2,...,DP+1.
LDP1 INTEGER
The leading dimension of array P. LDP1 >= MP.
LDP2 INTEGER
The second dimension of array P. LDP2 >= NP.
TEXT (input) CHARACTER*72
Title caption of the coefficient matrices to be printed.
TEXT is followed by the degree of the coefficient matrix,
within brackets. If TEXT = ' ', then the coefficient
matrices are separated by an empty line.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
For i = 1, 2, ..., DP + 1 the routine first prints the contents of
TEXT followed by (i-1) as a title, followed by the elements of the
MP-by-NP coefficient matrix P(i) such that
(i) if NP < L, then the leading MP-by-NP part is printed;
(ii) if NP = k*L + p (where k, p > 0), then k MP-by-L blocks of
consecutive columns of P(i) are printed one after another
followed by one MP-by-p block containing the last p columns
of P(i).
Row numbers are printed on the left of each row and a column
number on top of each column.
References
None.Numerical Aspects
None.Further Comments
NoneExample
Program Text
* UD01ND EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MPMAX, NPMAX, DPMAX
PARAMETER ( MPMAX = 10, NPMAX = 10, DPMAX = 5 )
INTEGER LDP1, LDP2
PARAMETER ( LDP1 = MPMAX, LDP2 = NPMAX )
* .. Local Scalars ..
INTEGER DP, INFO, L, MP, NP
CHARACTER*72 TEXT
* .. Local Arrays ..
DOUBLE PRECISION P(LDP1,LDP2,DPMAX)
* .. External Subroutines ..
EXTERNAL UD01BD, UD01ND
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) MP, NP, DP, L, TEXT
IF ( MP.LE.0 .OR. MP.GT.MPMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) MP
ELSE IF ( NP.LE.0 .OR. NP.GT.NPMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) NP
ELSE IF ( DP.LT.0 .OR. DP.GT.DPMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) DP
ELSE
* Read the coefficients of the matrix polynomial P(s).
CALL UD01BD( MP, NP, DP, NIN, P, LDP1, LDP2, INFO )
IF ( INFO.EQ.0 ) THEN
WRITE ( NOUT, 99996 ) MP, NP, DP
* Write the coefficients of the matrix polynomial P(s).
CALL UD01ND( MP, NP, DP, L, NOUT, P, LDP1, LDP2, TEXT,
$ INFO )
IF ( INFO.NE.0 )
$ WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) INFO
END IF
END IF
STOP
*
99999 FORMAT (' UD01ND EXAMPLE PROGRAM RESULTS', /1X)
99998 FORMAT (' INFO on exit from UD01ND = ',I2)
99997 FORMAT (' INFO on exit from UD01BD = ',I2)
99996 FORMAT (' MP =', I2, 2X, ' NP =', I2, 3X, 'DP =', I2)
99995 FORMAT (/' NP is out of range.',/' NP = ',I5)
99994 FORMAT (/' MP is out of range.',/' MP = ',I5)
99993 FORMAT (/' DP is out of range.',/' DP = ',I5)
END
Program Data
UD01ND EXAMPLE PROGRAM DATA 4 3 2 5 P P0 1.0D-00 0.0D-00 0.0D-00 0.0D-00 2.0D-00 4.0D-00 0.0D-00 4.0D-00 8.0D-00 0.0D-00 6.0D-00 1.2D+01 P1 0.0D-00 1.0D-00 2.0D-00 1.0D-00 0.0D-00 0.0D-00 2.0D-00 0.0D-00 0.0D-00 3.0D-00 0.0D-00 0.0D-00 P2 1.0D-00 0.0D-00 0.0D-00 0.0D-00 0.0D-00 0.0D-00 0.0D-00 0.0D-00 0.0D-00 0.0D-00 0.0D-00 0.0D-00Program Results
UD01ND EXAMPLE PROGRAM RESULTS
MP = 4 NP = 3 DP = 2
P( 0) ( 4X 3)
1 2 3
1 0.1000000D+01 0.0000000D+00 0.0000000D+00
2 0.0000000D+00 0.2000000D+01 0.4000000D+01
3 0.0000000D+00 0.4000000D+01 0.8000000D+01
4 0.0000000D+00 0.6000000D+01 0.1200000D+02
P( 1) ( 4X 3)
1 2 3
1 0.0000000D+00 0.1000000D+01 0.2000000D+01
2 0.1000000D+01 0.0000000D+00 0.0000000D+00
3 0.2000000D+01 0.0000000D+00 0.0000000D+00
4 0.3000000D+01 0.0000000D+00 0.0000000D+00
P( 2) ( 4X 3)
1 2 3
1 0.1000000D+01 0.0000000D+00 0.0000000D+00
2 0.0000000D+00 0.0000000D+00 0.0000000D+00
3 0.0000000D+00 0.0000000D+00 0.0000000D+00
4 0.0000000D+00 0.0000000D+00 0.0000000D+00
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/UE01MD.html 0000664 0000000 0000000 00000005746 14560147231 0020361 0 ustar 00root root 0000000 0000000
Purpose
To provide an extension of the LAPACK routine ILAENV to machine-specific parameters for SLICOT routines. The default values in this version aim to give good performance on a wide range of computers. For optimal performance, however, the user is advised to modify this routine. Note that an optimized BLAS is a crucial prerequisite for any speed gains. For further details, see ILAENV.Specification
INTEGER FUNCTION UE01MD( ISPEC, NAME, OPTS, N1, N2, N3 )
C .. Scalar Arguments ..
CHARACTER*( * ) NAME, OPTS
INTEGER ISPEC, N1, N2, N3
Function Value
UE01MD INTEGER
The function value set according to ISPEC.
Arguments
Input/Output Parameters
ISPEC (input) INTEGER
Specifies the parameter to be returned as the value of
UE01MD, as follows:
= 1: the optimal blocksize; if the returned value is 1, an
unblocked algorithm will give the best performance;
= 2: the minimum block size for which the block routine
should be used; if the usable block size is less than
this value, an unblocked routine should be used;
= 3: the crossover point (in a block routine, for N less
than this value, an unblocked routine should be used)
= 4: the number of shifts, used in the product eigenvalue
routine;
= 8: the crossover point for the multishift QR method for
product eigenvalue problems.
NAME (input) CHARACTER*(*)
The name of the calling subroutine, in either upper case
or lower case.
OPTS (input) CHARACTER*(*)
The character options to the subroutine NAME, concatenated
into a single character string.
N1 (input) INTEGER
N2 (input) INTEGER
N3 (input) INTEGER
Problem dimensions for the subroutine NAME; these may not
all be required.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
Return to index SLICOT-SLICOT-Reference-a037f7e/doc/readme 0000664 0000000 0000000 00000001722 14560147231 0017746 0 ustar 00root root 0000000 0000000 SLICOT Library Subdirectory doc ------------------------------- SLICOT Library Subdirectory doc contains documentation files (*.html) for almost all SLICOT Library routines. Omitted are few auxiliary routines, called by other SLICOT routines, but which seem to have no general use. The documentation conforms to the SLICOT standards. Besides function, arguments, and algorithm description, most documents for user-callable rotines include example programs, which could be easily modified, by copying, pasting, etc., to solve related, specific problems. The documentation files could be viewed individually, or via the provided index files: the main SLICOT Library index (libindex.html), stored in the SLICOT root directory, or the auxiliary, supporting routines index (support.html), stored in the subdirectory doc. The second file is also accessible from the first one. These index files are organized according to the SLICOT Library chapters, sections, and subsections. SLICOT-SLICOT-Reference-a037f7e/doc/support.html 0000664 0000000 0000000 00000125033 14560147231 0021172 0 ustar 00root root 0000000 0000000
AB08NX Construction of a reduced system with input/output matrix Dr of full row rank, preserving transmission zeros AB08NY Construction of a reduced system with input/output matrix Dr of full row rank, preserving transmission zeros (extended variant) AB8NXZ Construction of a reduced system with input/output matrix Dr of full row rank, preserving transmission zeros (complex case)
AB09AX Balance & Truncate model reduction with state matrix in real Schur form AB09BX Singular perturbation approximation based model reduction with state matrix in real Schur form AB09CX Hankel norm approximation based model reduction with state matrix in real Schur form AB09HX Stochastic balancing model reduction of stable systems AB09HY Cholesky factors of the controllability and observability Grammians AB09IX Accuracy enhanced balancing related model reduction AB09IY Cholesky factors of the frequency-weighted controllability and observability Grammians AB09JV State-space representation of a projection of a left weighted transfer-function matrix AB09JW State-space representation of a projection of a right weighted transfer-function matrix AB09JX Check stability/antistability of finite eigenvalues AB09KX Stable projection of V*G*W or conj(V)*G*conj(W)
AB13AX Hankel-norm of a stable system with state matrix in real Schur form AB13DX Maximum singular value of a transfer-function matrix
AG08BY Construction of a reduced system with input/output matrix Dr of full row rank, preserving the finite Smith zeros AG8BYZ Construction of a reduced system with input/output matrix Dr of full row rank, preserving the finite Smith zeros (complex case)
IB01MD Upper triangular factor in QR factorization of a block-Hankel-block matrix IB01MY Upper triangular factor in fast QR factorization of a block-Hankel-block matrix IB01ND Singular value decomposition giving the system order IB01OD Estimating the system order IB01OY User's confirmation of the system order IB01PD Estimating the system matrices and covariances IB01PX Estimating the matrices B and D of a system using Kronecker products IB01PY Estimating the matrices B and D of a system exploiting the structure IB01QD Estimating the initial state and the matrices B and D of a system IB01RD Estimating the initial state of a system
MA01AD Complex square root of a complex number in real arithmetic MA01BD Safely computing the general product of K real scalars MA01BZ Safely computing the general product of K complex scalars MA01CD Safely computing the sign of a sum of two real numbers represented using integer powers of a base MA01DD Approximate symmetric chordal metric for two finite complex numbers MA01DZ Approximate symmetric chordal metric for two, possibly infinite, complex numbers
MA02AD Transpose of a matrix MA02AZ (Conjugate) transpose of a complex matrix MA02BD Reversing the order of rows and/or columns of a matrix MA02BZ Reversing the order of rows and/or columns of a matrix (complex case) MA02CD Pertranspose of the central band of a square matrix MA02CZ Pertranspose of the central band of a square matrix (complex case) MA02DD Pack/unpack the upper or lower triangle of a symmetric matrix MA02ED Construct a triangle of a symmetric matrix, given the other triangle MA02ES Construct a triangle of a skew-symmetric real matrix, given the other triangle MA02EZ Construct a triangle of a (skew-)symmetric/Hermitian complex matrix, given the other triangle MA02FD Hyperbolic plane rotation MA02GD Column interchanges on a real matrix MA02GZ Column interchanges on a complex matrix MA02HD Check if a matrix is a scalar multiple of an identity-like matrix MA02HZ Check if a complex matrix is a scalar multiple of an identity-like matrix MA02ID Matrix 1-, Frobenius, or infinity norms of a skew-Hamiltonian matrix MA02IZ Matrix 1-, Frobenius, or infinity norms of a complex skew-Hamiltonian matrix MA02JD Test if a matrix is an orthogonal symplectic matrix MA02JZ Test if a matrix is a unitary symplectic matrix MA02MD Norms of a real skew-symmetric matrix MA02MZ Norms of a complex skew-symmetric matrix MA02NZ Two rows and columns permutation of a (skew-)symmetric/Hermitian complex matrix MA02OD Number of zero rows of a real (skew-)Hamiltonian matrix MA02OZ Number of zero rows of a complex (skew-)Hamiltonian matrix MA02PD Number of zero rows and columns of a real matrix MA02PZ Number of zero rows and columns of a complex matrix MA02RD Sorting a real vector and rearranging another vector MA02SD Smallest nonzero absolute value of the elements of a real matrix MB01KD Rank 2k operation alpha*A*trans(B) - alpha*B*trans(A) + beta*C, with A and C skew-symmetric matrices MB01LD Computation of matrix expression alpha*R + beta*A*X*trans(A) with skew-symmetric matrices R and X MB01MD Matrix-vector operation alpha*A*x + beta*y, with A a skew-symmetric matrix MB01ND Rank 2 operation alpha*x*trans(y) - alpha*y*trans(x) + A, with A a skew-symmetric matrix MB01SD Rows and/or columns scaling of a matrix MB01SS Symmetric scaling of a symmetric matrix
MB01OC Computation of matrix expression alpha R + beta ( op(H) X + X op(H)' ) with R, X symmetric and H upper Hessenberg MB01OD Computation of matrix expression alpha R + beta ( op(H) X op(E)' + op(E) X op(H)' ) with R, X symmetric, H upper Hessenberg, and E upper triangular MB01OE Computation of matrix expression alpha R + beta ( op(H) op(E)' + op(E) op(H)' ) with R symmetric, H upper Hessenberg, and E upper triangular MB01OH Computation of matrix expression alpha R + beta ( op(H) op(A)' + op(A) op(H)' ) with R symmetric, and A, H upper Hessenberg MB01OO Computation of P or P' with P = op(H) X op(E)' with X symmetric, H upper Hessenberg, and E upper triangular MB01OS Computation of matrix expression P = H X or P = X H, with X symmetric and H upper Hessenberg MB01OT Computation of matrix expression alpha R + beta ( op(E) op(T)' + op(T) op(E)' ) with R symmetric and E, T upper triangular MB01RH Computation of matrix expression alpha R + beta op(H) X op(H)' with R, X symmetric and H upper Hessenberg MB01RT Computation of matrix expression alpha R + beta op(E) X op(E)' with R, X symmetric and E upper triangular MB01RU Computation of matrix expression alpha*R + beta*A*X*trans(A) (MB01RD variant) MB01RW Computation of matrix expression alpha*A*X*trans(A), X symmetric (BLAS 2) MB01RX Computing a triangle of the matrix expressions alpha*R + beta*A*B or alpha*R + beta*B*A MB01RY Computing a triangle of the matrix expressions alpha*R + beta*H*B or alpha*R + beta*B*H, with H an upper Hessenberg matrix MB01UW Computation of matrix expressions alpha*H*A or alpha*A*H, overwritting A, with H an upper Hessenberg matrix MB01VD Kronecker product of two matrices MB01XY Computation of the product U'*U or L*L', with U and L upper and lower triangular matrices (unblock algorithm) SB03OV Construction of a complex plane rotation to annihilate a real number, modifying a complex number SG03BY Computing a complex plane rotation in real arithmetic SG03BR Computing a complex plane rotation in real arithmetic (SG03BY version - adaptation of LAPACK ZLARTG)
MB02CU Bringing the first blocks of a generator in proper form (extended version of MB02CX) MB02CV Applying the MB02CU transformations on other columns / rows of the generator MB02CX Bringing the first blocks of a generator in proper form MB02CY Applying the MB02CX transformations on other columns / rows of the generator MB02NY Separation of a zero singular value of a bidiagonal submatrix MB02QY Minimum-norm least squares solution, given a rank-revealing QR factorization MB02UU Solution of linear equations using LU factorization with complete pivoting MB02UV LU factorization with complete pivoting MB02UW Solution of linear equations of order at most 2 with possible scaling and perturbation of system matrix MB02WD Solution of a positive definite linear system A*x = b, or f(A, x) = b, using conjugate gradient algorithm MB02XD Solution of a set of positive definite linear systems, A'*A*X = B, or f(A)*X = B, using Gaussian elimination MB02YD Solution of the linear system A*x = b, D*x = 0, D diagonal
MB03AD Reducing the first column of a real Wilkinson shift polynomial for a product of matrices to the first unit vector MB03AB Reducing the first column of a real Wilkinson shift polynomial for a product of matrices to the first unit vector (variant with explicit shifts) MB03AE Reducing the first column of a real Wilkinson shift polynomial for a product of matrices to the first unit vector (variant with partial evaluation, Hessenberg factor is the first one) MB03AF Reducing the first column of a real Wilkinson shift polynomial for a product of matrices to the first unit vector (variant, Hessenberg factor is the last one) MB03AG Reducing the first column of a real Wilkinson shift polynomial for a product of matrices to the first unit vector (variant with evaluation, Hessenberg factor is the first one) MB03AH Reducing the first column of a real Wilkinson shift polynomial for a product of matrices to the first unit vector (variant with partial evaluation, Hessenberg factor is the last one) MB03AI Reducing the first column of a real Wilkinson shift polynomial for a product of matrices to the first unit vector (variant with evaluation, Hessenberg factor is the last one) MB03BA Computing maps for Hessenberg index and signature array MB03BB Eigenvalues of a 2-by-2 matrix product via a complex single shifted periodic QZ algorithm MB03BC Product singular value decomposition of K-1 triangular factors of order 2 MB03BD Finding eigenvalues of a generalized matrix product in Hessenberg-triangular form MB03BE Applying iterations of a real single shifted periodic QZ algorithm to a 2-by-2 matrix product MB03BF Applying iterations of a real single shifted periodic QZ algorithm to a 2-by-2 matrix product, with Hessenberg factor the last one MB03BZ Finding eigenvalues of a complex generalized matrix product in Hessenberg-triangular form MB03CD Exchanging eigenvalues of a real 2-by-2, 3-by-3 or 4-by-4 block upper triangular pencil (factored version) MB03CZ Exchanging eigenvalues of a complex 2-by-2 upper triangular pencil (factored version) MB03DD Exchanging eigenvalues of a real 2-by-2, 3-by-3 or 4-by-4 block upper triangular pencil MB03DZ Exchanging eigenvalues of a complex 2-by-2 upper triangular pencil MB03ED Reducing a real 2-by-2 or 4-by-4 block (anti-)diagonal skew-Hamiltonian/Hamiltonian pencil to generalized Schur form and moving eigenvalues with negative real parts to the top (factored version) MB03FD Reducing a real 2-by-2 or 4-by-4 block (anti-)diagonal skew-Hamiltonian/Hamiltonian pencil to generalized Schur form and moving eigenvalues with negative real parts to the top MB03GD Exchanging eigenvalues of a real 2-by-2 or 4-by-4 block upper triangular skew-Hamiltonian/Hamiltonian pencil (factored version) MB03GZ Exchanging eigenvalues of a complex 2-by-2 skew-Hamiltonian/ Hamiltonian pencil in structured Schur form (factored version) MB03HD Exchanging eigenvalues of a real 2-by-2 or 4-by-4 skew-Hamiltonian/ Hamiltonian pencil in structured Schur form MB03HZ Exchanging eigenvalues of a complex 2-by-2 skew-Hamiltonian/ Hamiltonian pencil in structured Schur form MB03ID Moving eigenvalues with negative real parts of a real skew-Hamiltonian/Hamiltonian pencil in structured Schur form to the leading subpencil (factored version) MB03IZ Moving eigenvalues with negative real parts of a complex skew-Hamiltonian/Hamiltonian pencil in structured Schur form to the leading subpencil (factored version) MB03JD Moving eigenvalues with negative real parts of a real skew-Hamiltonian/Hamiltonian pencil in structured Schur form to the leading subpencil MB03JP Moving eigenvalues with negative real parts of a real skew-Hamiltonian/Hamiltonian pencil in structured Schur form to the leading subpencil (applying transformations on panels of columns) MB03JZ Moving eigenvalues with negative real parts of a complex skew-Hamiltonian/Hamiltonian pencil in structured Schur form to the leading subpencil MB3JZP Moving eigenvalues with negative real parts of a complex skew-Hamiltonian/Hamiltonian pencil in structured Schur form to the leading subpencil (applying transformations on panels of columns) MB03KA Moving diagonal blocks at a specified position in a formal matrix product to another position MB03KB Swapping pairs of adjacent diagonal blocks of sizes 1 and/or 2 in a formal matrix product MB03KC Reducing a 2-by-2 formal matrix product to periodic Hessenberg-triangular form MB03KD Reordering the diagonal blocks of a formal matrix product using periodic QZ algorithm MB03KE Solving periodic Sylvester-like equations with matrices of order at most 2 MB03NY The smallest singular value of A - jwI MB03OY Matrix rank determination by incremental condition estimation, during the pivoted QR factorization process MB3OYZ Matrix rank determination by incremental condition estimation, during the pivoted QR factorization process (complex case) MB03PY Matrix rank determination by incremental condition estimation, during the pivoted RQ factorization process (row pivoting) MB3PYZ Matrix rank determination by incremental condition estimation, during the pivoted RQ factorization process (row pivoting, complex case) MB03QV Eigenvalues of an upper quasi-triangular matrix pencil MB03QW Standardization and eigenvalues of a 2-by-2 diagonal block pair of an upper quasi-triangular matrix pencil MB03QX Eigenvalues of an upper quasi-triangular matrix MB03QY Transformation to Schur canonical form of a selected 2-by-2 diagonal block of an upper quasi-triangular matrix MB03RX Reordering the diagonal blocks of a principal submatrix of a real Schur form matrix MB03RY Tentative solution of Sylvester equation -AX + XB = C (A, B in real Schur form) MB03RW Tentative solution of Sylvester equation -AX + XB = C (A, B in complex Schur form) MB03TS Swapping two diagonal blocks of a matrix in (skew-)Hamiltonian canonical Schur form MB03VY Generating orthogonal matrices for reduction to periodic Hessenberg form of a product of matrices MB03WA Swapping two adjacent diagonal blocks in a periodic real Schur canonical form MB03WX Eigenvalues of a product of matrices, T = T_1*T_2*...*T_p, with T_1 upper quasi-triangular and T_2, ..., T_p upper triangular MB03XS Eigenvalues and real skew-Hamiltonian Schur form of a skew-Hamiltonian matrix MB03XU Panel reduction of columns and rows of a real (k+2n)-by-(k+2n) matrix by orthogonal symplectic transformations MB03YA Annihilation of one or two entries on the subdiagonal of a Hessenberg matrix corresponding to zero elements on the diagonal of a triangular matrix MB03YT Periodic Schur factorization of a real 2-by-2 matrix pair (A,B) with B upper triangular MB03ZA Reordering a selected cluster of eigenvalues of a given matrix pair in periodic Schur form MB05MY Computing an orthogonal matrix reducing a matrix to real Schur form T, the eigenvalues, and the upper triangular matrix of right eigenvectors of T MB05OY Restoring a matrix after balancing transformations
MB04CD Reducing a special real block (anti-)diagonal skew-Hamiltonian/ Hamiltonian pencil in factored form to generalized Schur form MB04DB Applying the inverse of a balancing transformation for a real skew-Hamiltonian/Hamiltonian matrix pencil MB4DBZ Applying the inverse of a balancing transformation for a complex skew-Hamiltonian/Hamiltonian matrix pencil MB04DD Balancing a real Hamiltonian matrix MB04DZ Balancing a complex Hamiltonian matrix MB04DI Applying the inverse of a balancing transformation for a real Hamiltonian matrix MB04DS Balancing a real skew-Hamiltonian matrix MB04DY Symplectic scaling of a Hamiltonian matrix MB04HD Reducing a special real block (anti-)diagonal skew-Hamiltonian/ Hamiltonian pencil to generalized Schur form MB04IY Applying the product of elementary reflectors used for QR factorization of a matrix having a lower left zero triangle MB04NY Applying an elementary reflector to a matrix C = ( A B ), from the right, where A has one column MB04OY Applying an elementary reflector to a matrix C = ( A' B' )', from the left, where A has one row MB04OW Rank-one update of a Cholesky factorization for a 2-by-2 block matrix MB04OX Rank-one update of a Cholesky factorization MB04PA Special reduction of a (skew-)Hamiltonian like matrix MB04PU Computation of the Paige/Van Loan (PVL) form of a Hamiltonian matrix (unblocked algorithm) MB04PY Applying an elementary reflector to a matrix from the left or right MB04QB Applying a product of symplectic reflectors and Givens rotations to two general real matrices MB04QC Premultiplying a real matrix with an orthogonal symplectic block reflector MB04QF Forming the triangular block factors of a symplectic block reflector MB04QS Multiplication with a product of symplectic reflectors and Givens rotations MB04QU Applying a product of symplectic reflectors and Givens rotations to two general real matrices (unblocked algorithm) MB04RB Reduction of a skew-Hamiltonian matrix to Paige/Van Loan (PVL) form (blocked version) MB04RU Reduction of a skew-Hamiltonian matrix to Paige/Van Loan (PVL) form (unblocked version) MB04RS Solution of a generalized real Sylvester equation with matrix pairs in generalized real Schur form MB04RV Solution of a generalized complex Sylvester equation with matrix pairs in generalized complex Schur form MB04SU Symplectic QR decomposition of a real 2M-by-N matrix MB04TS Symplectic URV decomposition of a real 2N-by-2N matrix (unblocked version) MB04TU Applying a row-permuted Givens transformation to two row vectors MB04WD Generating an orthogonal basis spanning an isotropic subspace MB04WP Generating an orthogonal symplectic matrix which performed the reduction in MB04PU MB04WR Generating orthogonal symplectic matrices defined as products of symplectic reflectors and Givens rotations MB04WU Generating an orthogonal basis spanning an isotropic subspace (unblocked version) MB04XY Applying Householder transformations for bidiagonalization (stored in factored form) to one or two matrices, from the left MB04YW One QR or QL iteration step onto an unreduced bidiagonal submatrix of a bidiagonal matrix
MC01PY Coefficients of a real polynomial, stored in decreasing order, given its zeros
MC03NX Construction of a pencil sE-A related to a given polynomial matrix
MD03BX QR factorization with column pivoting and error vector transformation MD03BY Finding the Levenberg-Marquardt parameter
NF01AD Computing the output of a Wiener system NF01AY Computing the output of a set of neural networks NF01BD Computing the Jacobian of a Wiener system NF01BP Finding the Levenberg-Marquardt parameter NF01BQ Solution of the linear system J*x = b, D*x = 0, D diagonal NF01BR Solution of the linear system op(R)*x = b, R block upper triangular stored in a compressed form NF01BS QR factorization of a structured Jacobian matrix NF01BU Computing J'*J + c*I, for the Jacobian J given in a compressed form NF01BV Computing J'*J + c*I, for a full Jacobian J (one output variable) NF01BW Matrix-vector product x <-- (J'*J + c*I)*x, for J in a compressed form NF01BX Matrix-vector product x <-- (A'*A + c*I)*x, for a full matrix A NF01BY Computing the Jacobian of the error function for a neural network (for one output variable)
SB01BX Choosing the closest real (complex conjugate) eigenvalue(s) to a given real (complex) value SB01BY Pole placement for systems of order 1 or 2 SB01FY Inner denominator of a right-coprime factorization of an unstable system of order 1 or 2
SB02MU Constructing the 2n-by-2n Hamiltonian or symplectic matrix for linear-quadratic optimization problems SB02RU Constructing the 2n-by-2n Hamiltonian or symplectic matrix for linear-quadratic optimization problems (efficient and accurate version of SB02MU) SB02OY Constructing and compressing the extended Hamiltonian or symplectic matrix pairs for linear-quadratic optimization problems
SB03MV Solving a discrete-time Lyapunov equation for a 2-by-2 matrix SB03MW Solving a continuous-time Lyapunov equation for a 2-by-2 matrix SB03MX Solving a discrete-time Lyapunov equation with matrix A quasi-triangular SB03MY Solving a continuous-time Lyapunov equation with matrix A quasi-triangular SB03OT Solving (for Cholesky factor) stable continuous- or discrete-time Lyapunov equations, with A quasi-triangular and R triangular SB03OS Solving (for Cholesky factor) stable continuous- or discrete-time complex Lyapunov equations, with matrices S and R triangular SB03OU Solving (for Cholesky factor) stable continuous- or discrete-time Lyapunov equations, with A in real Schur form and B rectangular SB03OY Solving (for Cholesky factor) stable 2-by-2 continuous- or discrete-time Lyapunov equations, with matrix A having complex conjugate eigenvalues SB03QX Forward error bound for continuous-time Lyapunov equations SB03QY Separation and Theta norm for continuous-time Lyapunov equations SB03SX Forward error bound for discrete-time Lyapunov equations SB03SY Separation and Theta norm for discrete-time Lyapunov equations
SB03MU Solving a discrete-time Sylvester equation for an m-by-n matrix X, 1 <= m,n <= 2 SB03OR Solving quasi-triangular continuous- or discrete-time Sylvester equations, for an n-by-m matrix X, 1 <= m <= 2 SB04MR Solving a linear algebraic system whose coefficient matrix (stored compactly) has zeros below the second subdiagonal SB04MU Constructing and solving a linear algebraic system whose coefficient matrix (stored compactly) has zeros below the second subdiagonal SB04MW Solving a linear algebraic system whose coefficient matrix (stored compactly) has zeros below the first subdiagonal SB04MY Constructing and solving a linear algebraic system whose coefficient matrix (stored compactly) has zeros below the first subdiagonal SB04NV Constructing right-hand sides for a system of equations in Hessenberg form solved via SB04NX SB04NW Constructing the right-hand side for a system of equations in Hessenberg form solved via SB04NY SB04NX Solving a system of equations in Hessenberg form with two consecutive offdiagonals and two right-hand sides SB04NY Solving a system of equations in Hessenberg form with one offdiagonal and one right-hand side SB04OW Solving a periodic Sylvester equation with matrices in periodic Schur form SB04PX Solving a discrete-time Sylvester equation for matrices of order <= 2 SB04PY Solving a discrete-time Sylvester equation with matrices in Schur form SB04QR Solving a linear algebraic system whose coefficient matrix (stored compactly) has zeros below the third subdiagonal SB04QU Constructing and solving a linear algebraic system whose coefficient matrix (stored compactly) has zeros below the third subdiagonal SB04QY Constructing and solving a linear algebraic system whose coefficient matrix (stored compactly) has zeros below the first subdiagonal (discrete-time case) SB04RV Constructing right-hand sides for a system of equations in Hessenberg form solved via SB04RX SB04RW Constructing the right-hand side for a system of equations in Hessenberg form solved via SB04RY SB04RX Solving a system of equations in Hessenberg form with two consecutive offdiagonals and two right-hand sides (discrete-time case) SB04RY Solving a system of equations in Hessenberg form with one offdiagonal and one right-hand side (discrete-time case)
SB10JD Conversion of a descriptor state-space system into regular state-space form SB10LD Closed-loop system matrices for a system with robust controller SB10PD Normalization of a system for H-infinity controller design SB10QD State feedback and output injection matrices for an H-infinity (sub)optimal state controller (continuous-time) SB10RD H-infinity (sub)optimal controller matrices using state feedback and output injection matrices (continuous-time) SB10SD H2 optimal controller matrices for a normalized discrete-time system SB10TD H2 optimal controller matrices for a discrete-time system SB10UD Normalization of a system for H2 controller design SB10VD State feedback and output injection matrices for an H2 optimal state controller (continuous-time) SB10WD H2 optimal controller matrices using state feedback and output injection matrices (continuous-time) SB10YD Fitting frequency response data with a stable, minimum phase SISO system SB10ZP Transforming a SISO system into a stable and minimum phase one
SB16AY Cholesky factors of the frequency-weighted controllability and observability Grammians for controller reduction SB16CY Cholesky factors of controllability and observability Grammians of coprime factors of a state-feedback controller
SG02CV Computation of residual matrix for a continuous-time or discrete-time reduced Lyapunov equation SG03AX Solving a generalized discrete-time Lyapunov equation with A quasi-triangular and E upper triangular SG03AY Solving a generalized continuous-time Lyapunov equation with A quasi-triangular and E upper triangular SG03BU Solving (for Cholesky factor) stable generalized discrete-time Lyapunov equations with A quasi-triangular, and E, B upper triangular SG03BS Solving (for Cholesky factor) stable generalized discrete-time complex Lyapunov equations with A, E, and B upper triangular SG03BV Solving (for Cholesky factor) stable generalized continuous-time Lyapunov equations with A quasi-triangular, and E, B upper triangular SG03BT Solving (for Cholesky factor) stable generalized continuous-time complex Lyapunov equations with A, E, and B upper triangular SG03BX Solving (for Cholesky factor) stable generalized 2-by-2 Lyapunov equations
SG03BW Solving a generalized Sylvester equation with A quasi-triangular and E upper triangular, for X m-by-n, n = 1 or 2
TB01KX Additive spectral decomposition of the transfer-function matrix of a standard system TB01UX Observable-unobservable decomposition of a standard system TB01VD Conversion of a discrete-time system to output normal form TB01VY Conversion of the output normal form of a discrete-time system to a state-space representation TB01XD Special similarity transformation of the dual state-space system TB01XZ Special similarity transformation of the dual state-space system (complex case) TB01YD Special similarity transformation of a state-space system
TB04BV Strictly proper part of a proper transfer function matrix TB04BW Sum of a rational matrix and a real matrix TB04BX Gain of a SISO linear system, given (A,b,c,d), its poles and zeros
TF01MX Output response of a linear discrete-time system, given a general system matrix (each output is a column of the result) TF01MY Output response of a linear discrete-time system, given the system matrices (each output is a column of the result)
TG01HU Staircase controllability representation of a multi-input descriptor system TG01HX Orthogonal reduction of a descriptor system to a system with the same transfer-function matrix and without uncontrollable finite eigenvalues TG01HY Orthogonal reduction of a descriptor system to a system with the same transfer-function matrix and without uncontrollable finite eigenvalues (blocked version) TG01KD Orthogonal equivalence transformation of a SISO descriptor system with E upper triangular (TG01OA version with more complex interface) TG01KZ Unitary equivalence transformation of a complex SISO descriptor system with E upper triangular (TG01OB version with more complex interface) TG01LY Finite-infinite decomposition of a structured descriptor system TG01NX Block-diagonal decomposition of a descriptor system in generalized real Schur form TG01OA Orthogonal equivalence transformation of a SISO descriptor system with E upper triangular, so that B becomes parallel to the first unit vector and E keeps its structure TG01OB Unitary equivalence transformation of a complex SISO descriptor system with E upper triangular, so that B becomes parallel to the first unit vector and E keeps its structure