scalapack-doc-1.5/0040755000056400000620000000000007151422255013543 5ustar pfrauenfstaffscalapack-doc-1.5/html/0040755000056400000620000000000007151420741014505 5ustar pfrauenfstaffscalapack-doc-1.5/html/faq.html0100644000056400000620000007226607141341617016157 0ustar pfrauenfstaff ScaLAPACK FAQ

ScaLAPACK Frequently Asked Questions (FAQ)

scalapack@cs.utk.edu

Many thanks to the netlib_maintainers@netlib.org from whose FAQ list I have patterned this list for ScaLAPACK.

Table of Contents

ScaLAPACK
1.1) What is ScaLAPACK?
1.2) How do I reference ScaLAPACK in a scientific publication?
1.3) Are there vendor-specific versions of ScaLAPACK?
1.4) What is the difference between the vendor and Netlib version of ScaLAPACK and which should I use?
1.5) Are there legal restrictions on the use of ScaLAPACK software?
1.6) What is two-dimensional block cyclic data distribution?
1.7) Where can I find out more information about ScaLAPACK?
1.8) What and where are the PBLAS?
1.9) Are example programs available?
1.10) How do I run an example program?
1.11) How do I install ScaLAPACK?
1.12) How do I install ScaLAPACK using MPIch-G and Globus?
1.13) How do I achieve high performance using ScaLAPACK?
1.14) Are prebuilt ScaLAPACK libraries available?
1.15) How do I find a particular routine?
1.16) I can't get a program to work. What should I do?
1.17) How can I unpack scalapack.tgz?
1.18) What technical support for ScaLAPACK is available?
1.19) How do I submit a bug report?
1.20) How do I gather a distributed vector back to one processor?

BLACS
2.1) What and where are the BLACS?
2.2) Is there a Quick Reference Guide to the BLACS available?
2.3) How do I install the BLACS?
2.4) Are prebuilt BLACS libraries available?
2.5) Are example BLACS programs available?

BLAS
3.1) What are the BLAS?
3.2) Publications/references for the BLAS?
3.3) Is there a Quick Reference Guide to the BLAS avail able?
3.4) Are optimized BLAS libraries available?
3.5) What is ATLAS?
3.6) Where can I find vendor supplied BLAS?
3.7) Where can I find the Intel BLAS for Linux?
3.8) Where can I find Java BLAS?
3.9) Are prebuilt Fortran77 ref implementation BLAS lib raries available from Netlib?

1) ScaLAPACK

1.1) What is ScaLAPACK?

The ScaLAPACK (or Scalable LAPACK) library includes a subset of LAPACK routines redesigned for distributed memory MIMD parallel computers. It is currently written in a Single-Program-Multiple-Data style using explicit message passing for interprocessor communication. It assumes matrices are laid out in a two-dimensional block cyclic decomposition.

Like LAPACK, the ScaLAPACK routines are based on block-partitioned algorithms in order to minimize the frequency of data movement between different levels of the memory hierarchy. (For such machines, the memory hierarchy includes the off-processor memory of other processors, in addition to the hierarchy of registers, cache, and local memory on each processor.) The fundamental building blocks of the ScaLAPACK library are distributed memory versions (PBLAS) of the Level 1, 2 and 3 BLAS, and a set of Basic Linear Algebra Communication Subprograms (BLACS) for communication tasks that arise frequently in parallel linear algebra computations. In the ScaLAPACK routines, all interprocessor communication occurs within the PBLAS and the BLACS. One of the design goals of ScaLAPACK was to have the ScaLAPACK routines resemble their LAPACK equivalents as much as possible.

For detailed information on ScaLAPACK, please refer to the ScaLAPACK Users' Guide.

1.2) How do I reference ScaLAPACK in a scientific publication?

We ask that you cite the ScaLAPACK Users' Guide. 

@BOOK{slug,
      AUTHOR = {Blackford, L. S. and Choi, J. and Cleary, A. and
                D'Azevedo, E. and Demmel, J. and Dhillon, I. and
                Dongarra, J. and Hammarling, S. and Henry, G. and
                Petitet, A. and Stanley, K. and Walker, D. and
                Whaley, R. C.},
      TITLE = {{ScaLAPACK} Users' Guide},
      PUBLISHER = {Society for Industrial and Applied Mathematics},
      YEAR = {1997},
      ADDRESS = {Philadelphia, PA},
      ISBN = {0-89871-397-8 (paperback)} }

1.3) Are there vendor-specific versions of ScaLAPACK?

Yes.

ScaLAPACK has been incorporated into several commercial packages, including the Sun Scalable Scientific Subroutine Library (Sun S3L), NAG Parallel Library, IBM Parallel ESSL, and Cray LIBSCI, and is being integrated into the VNI IMSL Numerical Library, as well as software libraries for Fujitsu, Hewlett-Packard/Convex, Hitachi, NEC, and SGI.

1.4) What is the difference between the vendor and Netlib version of ScaLAPACK and which should I use?

The publically available version of ScaLAPACK (on netlib) is designed to be portable and efficient across a wide range of computers. It is not hand-tuned for a specific computer architecture.

The vendor-specific versions of ScaLAPACK have been optimized for a specific architecture. Therefore, for best performance, we recommend using a vendor-optimized version of ScaLAPACK if it is available.

However, as new ScaLAPACK routines are introduced with each release, the vendor-specific versions of ScaLAPACK may only contain a subset of the existing routines.

If a user suspects an error in a vendor-specific ScaLAPACK routine, he is recommended to download the ScaLAPACK Test Suite from netlib.

1.5) Are there legal restrictions on the use of ScaLAPACK software?

ScaLAPACK (like LINPACK, EISPACK, LAPACK, etc) is a freely-available software package. It is available from netlib via anonymous ftp and the World Wide Web. It can, and is, being included in commercial packages (e.g., Sun's S3L, IBM's Parallel ESSL, NAG Numerical PVM and MPI Library). We only ask that proper credit be given to the authors.

Like all software, it is copyrighted. It is not trademarked, but we do ask the following:

If you modify the source for these routines we ask that you change the name of the routine and comment the changes made to the original.

We will gladly answer any questions regarding the software. If a modification is done, however, it is the responsibility of the person who modified the routine to provide support.

1.6) What is two-dimensional block cyclic data distribution?
two-dimensional block cyclic decomposition

1.7) Where can I find more information about ScaLAPACK?

A variety of working notes related to the ScaLAPACK library were published as LAPACK Working Notes and are available in postscript and pdf format at:

http://www.netlib.org/lapack/lawns/ and
http://www.netlib.org/lapack/lawnspdf/

1.8) What and where are the PBLAS?

The Parallel Basic Linear Algebra Subprograms (PBLAS) are distributed memory versions of the Level 1, 2 and 3 BLAS. A Quick Reference Guide to the PBLAS is available. The software is available as part of the ScaLAPACK distribution tar file (scalapack.tgz).

There is also a new prototype version of the PBLAS (version 2.0), which is alignment-restriction free and uses logical algorithmic blocking techniques. For details, please refer to the scalapack/prototype/readme.pblas.

1.9) Are example ScaLAPACK programs available?

Yes, example ScaLAPACK programs are available. Refer to

http://www.netlib.org/scalapack/examples/
for a list of available example programs.

A detailed description of how to run a ScaLAPACK example program is discussed in Chapter 2 of the ScaLAPACK Users' Guide.

1.10) How do I run an example program?

A detailed description of how to run a ScaLAPACK example program is discussed in Chapter 2 of the ScaLAPACK Users' Guide.

1.11) How do I install ScaLAPACK?

A comprehensive Installation Guide for ScaLAPACK is provided. In short, a user only needs to modify one file, SLmake.inc, to specify his compiler, compiler flags, location of his MPI library, BLACS library, and BLAS library. And then type make lib to build the ScaLAPACK library, and make exe to build the testing/timing executables. Example SLmake.inc files for various architectures are supplied in the SCALAPACK/INSTALL subdirectory in the distribution.

When you install ScaLAPACK, the installation assumes that the user has available a low-level message passing layer (like MPI, PVM, or a native message-passing library), a BLACS library (MPIBLACS or PVMBLACS, etc), and a BLAS library. If any of these required components is not available, then the user must build the needed component before proceeding with the ScaLAPACK installation.

If a vendor-optimized BLAS library is not available, ATLAS can be used to automatically generate an optimized BLAS library for your architecture. Only as a last resort should the user use the reference implementation Fortran77 BLAS contained on the BLAS webpage.

1.12) How do I install ScaLAPACK using MPIch-G and Globus?

A detailed explanation of how to run a ScaLAPACK program using MPIch-G and Globus can be found at: http://www.cs.utk.edu/~petitet/grads/.

See Question 1.11 for general installation instructions.

1.13) How do I achieve high performance using ScaLAPACK?

ScaLAPACK performance relies on an efficient low-level message-passing layer and high speed interconnection network for communication, and an optimized BLAS library for local computation.

For a detailed description of performance-related issues, please refer to Chapter 5 of the ScaLAPACK Users' Guide.

1.14) Are prebuilt ScaLAPACK libraries available?

Yes, prebuilt ScaLAPACK libraries are available for a variety of architectures. Refer to

http://www.netlib.org/scalapack/archives/
for a complete list of available prebuilt libraries.

1.15) How do I find a particular routine?

Indexes of individual ScaLAPACK driver and computational routines are available. These indexes contain brief descriptions of each routine.

ScaLAPACK routines are available in four types: single precision real, double precision real, single precision complex, and double precision complex. At the present time, the nonsymmetric eigenproblem is only available in single and double precision real.

1.16) I can't get a program to work. What should I do?

Technical questions should be directed to the authors at

scalapack@cs.utk.edu.

Please tell us the type of machine on which the tests were run, the compiler and compiler options that were used, details of the BLACS library that was used, as well as the BLAS library, and a copy of the input file if appropriate.

Be prepared to answer the following questions:

  1. Have you run the BLAS, BLACS, PBLAS and ScaLAPACK test suites?
  2. Have you checked the appropriate errata lists on netlib?
  3. Have you attempted to replicate this error using the appropriate ScaLAPACK test code and/or one of the ScaLAPACK example routines?
  4. If you are using an optimized BLAS or BLACS library, have you tried using the reference implementations from netlib?

1.17) How can I unpack scalapack.tgz?
 

   gunzip scalapack.tgz
   tar xvf scalapack.tar

The compression program gzip (and gunzip) is Gnu software. If it is not already available on your machine, you can download it via anonymous ftp:

   ncftp prep.ai.mit.edu
   cd pub/gnu/
   get gzip-1.2.4.tar

See Question 1.11 for installation instructions.

1.18) What technical support for ScaLAPACK is available?

Technical questions and comments should be directed to the authors at

scalapack@cs.utk.edu.

See Question 1.16

1.19) How do I submit a bug report?

Technical questions should be directed to the authors at

scalapack@cs.utk.edu.

Be prepared to answer the questions as outlined in Question 1.15. Those are the first questions that we will ask!

1.20) How do I gather a distributed vector back to one processor?

There are several ways to accomplish this task.

  1. You can create a local array of the global size and each process will write his pieces of the matrix in the appropriate locations, and then you can do a call to the BLACS routine DGSUM2D to add all of them together and then leave the answer on one process or on all processes.
  2. You can modify SCALAPACK/TOOLS/pdlaprnt.f to write to an array instead of writing to a file.
  3. You can modify the routine pdlawrite.f from the example program http://www.netlib.org/scalapack/examples/scaex.tgz.
  4. You can create a second "context" containing only one process, and then call the redistribution routines in SCALAPACK/REDIST/SRC/ to redistribute the matrix to that process grid.

2) BLACS

2.1) What and where are the BLACS?

The BLACS (Basic Linear Algebra Communication Subprograms) project is an ongoing investigation whose purpose is to create a linear algebra oriented message passing interface that may be implemented efficiently and uniformly across a large range of distributed memory platforms.

The length of time required to implement efficient distributed memory algorithms makes it impractical to rewrite programs for every new parallel machine. The BLACS exist in order to make linear algebra applications both easier to program and more portable. It is for this reason that the BLACS are used as the communication layer of ScaLAPACK.

For further information on the BLACS, please refer to the blacs directory on netlib, as well as the BLACS Homepage.

2.2) Is there a Quick Reference Guide to the BLACS available?

Yes, there is a postscript version of the Quick Reference Guide to the BLACS available.

2.3) How do I install the BLACS?

First, you must choose which underlying message-passing layer that the BLACS will use (MPI, PVM, NX, MPL, etc). Once this decision has been made, you download the respective gzip tar file.

An Installation Guide for the BLACS is provided, as well as a comprehensive BLACS Test Suite. In short, a user only needs to modify one file, Bmake.inc, to specify his compiler, compiler flags, and location of his MPI library. And then type make mpi to build the MPI BLACS library, for example. Example Bmake.inc files for various architectures are supplied in the BLACS/BMAKES subdirectory in the distribution. There are also scripts in BLACS/INSTALL which can be run to help the user to determine some of the settings in the Bmake.inc file.

It is highly recommended that the user run the BLACS Tester to ensure that his installation is correct, and that no bugs have been detected in the low-level message-passing layer. If you suspect an error, please consult the

file on netlib.

2.4) Are prebuilt BLACS libraries available?

Yes, prebuilt BLACS libraries are available for a variety of architectures and message-passing interfaces. Refer to

http://www.netlib.org/blacs/archives/
for a complete list of available prebuilt libraries.

2.5) Are example BLACS programs available?

Yes, example BLACS programs are available. Refer to

http://www.netlib.org/scalapack/examples/
for a list of available example programs.

3) BLAS

3.1) What and where are the BLAS?

The BLAS (Basic Linear Algebra Subprograms) are high quality "building block" routines for performing basic vector and matrix operations. Level 1 BLAS do vector-vector operations, Level 2 BLAS do matrix-vector operations, and Level 3 BLAS do matrix-matrix operations. Because the BLAS are efficient, portable, and widely available, they're commonly used in the development of high quality linear algebra software, LINPACK and LAPACK for example.

A Fortran77 reference implementation of the BLAS is located in the blas directory of Netlib.

3.2) Publications/references for the BLAS?

  1. C. L. Lawson, R. J. Hanson, D. Kincaid, and F. T. Krogh, Basic Linear Algebra Subprograms for FORTRAN usage, ACM Trans. Math. Soft., 5 (1979), pp. 308--323.

  2. J. J. Dongarra, J. Du Croz, S. Hammarling, and R. J. Hanson, An extended set of FORTRAN Basic Linear Algebra Subprograms, ACM Trans. Math. Soft., 14 (1988), pp. 1--17.

  3. J. J. Dongarra, J. Du Croz, S. Hammarling, and R. J. Hanson, Algorithm 656: An extended set of FORTRAN Basic Linear Algebra Subprograms, ACM Trans. Math. Soft., 14 (1988), pp. 18--32.

  4. J. J. Dongarra, J. Du Croz, I. S. Duff, and S. Hammarling, A set of Level 3 Basic Linear Algebra Subprograms, ACM Trans. Math. Soft., 16 (1990), pp. 1--17.

  5. J. J. Dongarra, J. Du Croz, I. S. Duff, and S. Hammarling, Algorithm 679: A set of Level 3 Basic Linear Algebra Subprograms, ACM Trans. Math. Soft., 16 (1990), pp. 18--28.

3.3) Is there a Quick Reference Guide to the BLAS available?

Yes, there is a postscript version of the Quick Reference Guide to the BLAS available.

3.4) Are optimized BLAS libraries available?

YES! Machine-specific optimized BLAS libraries are available for a variety of computer architectures. These optimized BLAS libraries are provided by the computer vendor or by an independent software vendor (ISV). For further details, please contact your local vendor representative.

Alternatively, the user can download ATLAS to automatically generate an optimized BLAS library for his architecture.

If all else fails, the user can download a Fortran77 reference implementation of the BLAS from netlib. However, keep in mind that this is a reference implementation and is not optimized.

3.5) What is ATLAS?

ATLAS is an approach for the automatic generation and optimization of numerical software for processors with deep memory hierarchies and pipelined functional units. The production of such software for machines ranging from desktop workstations to embedded processors can be a tedious and time consuming task. ATLAS has been designed to automate much of this process. We concentrate our efforts on the widely used linear algebra kernels called the Basic Linear Algebra Subroutines (BLAS).

For further information, refer to the ATL AS webpage.

3.6) Where can I find vendor supplied BLAS?

BLAS Vendor List
Last updated: November 13, 1998

Vendor

URL

Cray http://www.sgi.com/Products/appsdirectory.dir/Applications/Math_Physics_Other_ Sciences/
DEC http://www.digital.com/info/hpc/software/dxml.html
HP http://www.hp.com/esy/systems_networking/tech_servers/products/library.html
IBM http://www.rs6000.ibm.com/software/Apps/essl.html
http://www.rs6000.ibm.com/software/sp_products/esslpara.html
Intel http://developer.intel.com/vtune/perflibst/mkl/mklperf.htm (NT)
SGI http://www.sgi.com/software/scsl.html
SUN http://www.sun.com/workshop/fortran/

3.7) Where can I find the Intel BLAS for Linux?

Yes, the Intel BLAS for Linux are now available! Refer to the following URL: Intel BLAS for Linux.

3.8) Where can I find Java BLAS?

Yes, Java BLAS are available. Refer to the following URLs: Java LAPACK and JavaNumerics The JavaNumerics webpage provides a focal point for information on numerical computing in Java.

3.9) Are prebuilt Fortran77 ref implementation BLAS libraries available?

Yes. However, it is assumed that you have a machine-specific optimized BLAS library already available on the architecture to which you are installing LAPACK. If this is not the case, you can download a prebuilt Fortran77 reference implementation BLAS library or compile the Fortran77 reference implementation source code of the BLAS from netlib.

Although a model implementation of the BLAS in available from netlib in the blas directory, it is not expected to perform as well as a specially tuned implementation on most high-performance computers -- on some machines it may give much worse performance -- but it allows users to run LAPACK software on machines that do not offer any other implementation of the BLAS.

scalapack@cs.utk.edu
 scalapack-doc-1.5/html/slug/0040755000056400000620000000000007151420274015460 5ustar pfrauenfstaffscalapack-doc-1.5/html/slug/copyright.gif0100644000056400000620000000011706336145701020156 0ustar pfrauenfstaffGIF89a!,& Ǒړ 5EU#v覙X9Ӯ;scalapack-doc-1.5/html/slug/footnode.html0100644000056400000620000002153606336057275020202 0ustar pfrauenfstaff Footnotes
...output
This example program computes the relative machine precision which causes, on some systems, the IEEE floating-point exception flags to be raised. This may result in the printing of a warning message. This is normal. 
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...output
This example program computes the relative machine precision which causes, on some systems, the IEEE floating-point exception flags to be raised. This may result in the printing of a warning message. This is normal. 
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...output),
(Local input or local output) means that the argument may be either a local input argument or a local output argument, depending on the values of other arguments; for example, in the PxyySVX driver routines, some arguments are used either as local output arguments to return details of a factorization, or as local input arguments to supply details of a previously computed factorization. 
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...input),
(local or global input) is used to describe the length of the workspace arguments, e.g., LWORK, where the value can be local input specifying the size of the local WORK array, or global input LWORK=-1 specifying a global query for the amount of workspace required. 
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...input)
(local or global input) is used to describe the length of the workspace arguments, e.g., LWORK, where the value can be local input specifying the size of the local WORK array, or global input LWORK=-1 specifying a global query for the amount of workspace required. 
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...size
The block size must be large enough that the local matrix multiply is efficient. A block size of 64 suffices for most computers that have only one processor per node. Computers that have multiple shared-memory processors on each node may require a larger block size. 
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...(DSSL)
Dakota Scientific Software, Inc., 501 East Saint Joseph Street, Rapid City, SD 57701-3995 USA, (605) 394-2471 
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...(DSSL)
Dakota Scientific Software, Inc., 501 East Saint Joseph Street, Rapid City, SD 57701-3995 USA, (605) 394-2471 
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...PSLAHQR/PDLAHQR.
Strictly speaking, PSLAHQR/PDLAHQR is an auxiliary routine for computing the eigenvalues and optionally the corresponding eigenvectors of the more general case of nonsymmetric matrices. 
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...computer.
The ScaLAPACK sample timer page (contained within the ScaLAPACK examples directory on netlib and on the CD-ROM) has a BLACS port of the message-passing program and instructions for building the BLAS timing program. 
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...PxSYEVX.
See section 3.1.3 for explanation of the naming convention used for ScaLAPACK routines. 
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...'E'))
ICTXT refers to the BLACS CONTEXT parameter. Refer to section 4.1.2 for further details. 
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...digit,
This is the case on the Cray C90 and its predecessors and emulators. 
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...PCs,
Important machines that do not implement the IEEE standard include the CRAY X-MP, CRAY Y-MP, CRAY 2, CRAY C90, IBM 370, DEC Vax, and their emulators. 
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...[#lawn112##1#].
Running either machine in non-default mode to avoid this problem, either so that the IBM RS/6000 flushes denormalized numbers to zero or so that the DEC Alpha handles denormalized numbers correctly by doing gradual underflow, slows down the machine significantly [42]. 
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Sometimes our algorithms satisfy only 535#535 where both 532#532 and 536#536 are small. This does not significantly change the following analysis. 
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...small.
More generally, we need only Lipschitz continuity of f and may use the Lipschitz constant in place of f' in deriving error bounds. 
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...exist).
This is a different use of the term ill-posed from that used in other contexts. For example, to be well-posed (not ill-posed) in the sense of Hadamard, it is sufficient for f to be continuous, whereas we require Lipschitz continuity. 
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...described
There are some caveats to this statement. When computing the inverse of a matrix, the backward error E is small, taking the columns of the computed inverse one at a time, with a different E for each column [62]. The same is true when computing the eigenvectors of a nonsymmetric matrix. When computing the eigenvalues and eigenvectors of 550#550, 551#551 or 552#552, with A symmetric and B symmetric and positive definite (using PxSYGVX or PxHEGVX), the method may not be backward normwise stable if      B has a large condition number 553#553, although it has useful error bounds in this case too (see section 6.9). Solving the Sylvester equation  AX+XB=C for the matrix X may not be backward stable, although there are again useful error bounds for X [83]. 
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...error.
For other algorithms, the answers (and computed error bounds) are as accurate as though the algorithms were componentwise relatively backward stable, even though they are not. These algorithms are called componentwise relatively forward stable. 
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...bound
As discussed in section 6.3, this approximate error bound may underestimate the true error by a factor p(n), which is a modestly growing function of the problem dimension n. Often 569#569. 
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...vectors
These bounds are special cases of those in section 6.7 since the singular values and vectors of A are simply related to the eigenvalues and eigenvectors of the Hermitian matrix 700#700 [71, p. 427,]. 
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...magnitude:
This bound is guaranteed only if the Level 3 BLAS are implemented in a conventional way, not in a fast way. 
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...large.
Another interpretation of chordal distance is as half the usual Euclidean distance between the projections of 714#714 and 637#637 on the Riemann sphere, i.e., half the length of the chord connecting the projections. 
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Susan Blackford
Tue May 13 09:21:01 EDT 1997
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LWORK tex2html_wrap_inline14966 WORK(1)

 

In some ScaLAPACK eigenvalue routines, such as the symmetric eigenproblems (PxSYEV and PxSYEVX/PxHEEVX) and the generalized symmetric eigenproblem (PxSYGVX/PxHEGVX), a larger value of LWORK can guarantee the orthogonality of the returned eigenvectors at the risk of potentially degraded performance of the algorithm. The minimum amount of workspace required is returned in the first element of the work array, but a larger amount of workspace can allow for additional orthogonalization if desired by the user. Refer to section 5.3.6 and the leading comments of the source code for complete details.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Error Handling and the Diagnostic Argument INFO

 

All  driver and computational routines  have a diagnostic argument INFO   that indicates the success or failure of the computation. It is recommended that the user always check the value of INFO on exit from calling a ScaLAPACK routine. The value of INFO is defined as follows:

  • INFO = 0: successful exit
  • INFO < 0: illegal value of one or more arguments -- no computation performed
  • INFO > 0: failure in the course of computation 

The value of (INFO<0)  is calculated as follows: if the error is detected in the jth entry of a descriptor array, which is the ith argument in the parameter list, the number passed to the error-handling routine PXERBLA()   has been arbitrarily chosen to be tex2html_wrap_inline16134. This allows the user to distinguish an error on a descriptor entry from an error on a scalar argument.

The standard version of PXERBLA() only issues an error message and does not halt execution of the program. The main reason for this behavior is that some ``errors'' are deemed recoverable and we wanted to allow the user the flexibility to continue program execution if certain values were corrected. If user wish to change this behavior and additionally halt execution of the program, they may add a call to BLACS_ABORT() to their version of PXERBLA().

If an input error (INFO<0) is detected at a high-level routine (ScaLAPACK driver or computational routine), it is possible for the user to recover from such an error and proceed with the computation. An error message is printed by PXERBLA(), a RETURN is issued, and the program execution continues. However, if an error is detected in a low-level ScaLAPACK routine, this error is considered unrecoverable, a message is printed by PXERBLA(), and program execution is terminated by a call to BLACS_ABORT() .

Likewise, if an error is detected at a low-level routine, such as a PBLAS or BLACS routine, this error is deemed fatal. An error message is printed, and the program execution is terminated by the specific error-handling routine.

All ScaLAPACK driver and computational routines perform global and local input error-checking. In general, no input error-checking is performed on the auxiliary routines. The exception to this rule is for the auxiliary routines which are Level 2 versions of computational routines (e.g., PxGETF2, PxGEQR2, PxORMR2, PxORM2R, etc.). For efficiency purposes, these specialized low-level routines perform only a local validity check of their argument list. If an error is detected in at least one process of the current context, the program execution is stopped.

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Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Alignment Restrictions

    

Most routines in the present ScaLAPACK library have alignment restrictions. Alignment restrictions are constraints in the type of distributions and the indexing into the matrix that the user may utilize when calling a particular routine. For example, some routines will not accept submatrices whose starting index is not a multiple of the physical blocking factor.

More commonly, routines require that their various operand matrices have certain alignment commonalities. For instance, the solver routines generally require that row i of the matrix A be distributed across the same process row as row i of the right hand side matrix B.

Because of their idiosyncratic nature, it is almost impossible to give a full description of the alignment restrictions inherit in the present library without doing so on a routine-specific basis. All ScaLAPACK routines provide a description of the assumed alignment restrictions in the leading comments to the routine, and at this time the user must consult the actual code to find out what restrictions exist, if any.

We are working to remove the alignment restrictions, (with the exceptions noted below) so that the user will not have to worry about alignment, save as a performance issue.

Certain fundamental restrictions about data distributions are not currently being removed the library. Examples include the fact that the operands should be block-cyclicly distributed for the dense codes and one-dimensional block distributed for the banded codes. Also included here is the restriction that all operands be distributed across the same context (process grid).

Note that the ScaLAPACK library includes a redistribution/copy routine which allows the user to explicitly move matrices across contexts. Similar routines could be provided for distributions that do not match the ones presently employed in ScaLAPACK.

Finally, we note that the current descriptor structure does not accommodate the definition of replicated vectors. A replicated vector is a vector that is distributed across a row or column within the process grid and duplicated across subsequent process rows or columns, respectively. Such vectors occur, for example, as the IPIV  vector in the LU factorization and the TAU  vector in QR factorizations.

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Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Extensions

Extensions to the library are under way and will remove the majority of the alignment restrictions in the PBLAS. The ScaLAPACK library and the PBLAS are also being modified to allow the possibility of a partial first block, as well as the incorporation of aggregate (algorithmic) blocking. The partial first block extension makes ScaLAPACK fully compatible with HPF and necessitates the establishment of a new matrix descriptor.

The incorporation of aggregate (algorithmic) blocking at the top-level ScaLAPACK routines, as well as in the PBLAS, removes the restriction of a user's performance being tied to his physical matrix distribution. Instead, the algorithms will perform at an optimal block size predetermined inside the PBLAS.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Performance of ScaLAPACK

  

This chapter presents performance numbers for ScaLAPACK routines. The numbers are provided for illustration only and should not be regarded as a definitive up-to-date statement of performance. They have been selected from performance numbers obtained in 1996-1997 during the development of version 1.4 of ScaLAPACK. To obtain up-to-date performance figures, users should use the timing programs provided with ScaLAPACK.



Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Achieving High Performance with ScaLAPACK

   

ScaLAPACK achieves high performance on distributed memory computers, such as the SP2, T3D, T3E, and Paragon. ScaLAPACK can also achieve high performance on some networks of workstations    .

Distributed memory computers   are intended to be used primarily to run parallel programs. They typically include an efficient message-passing system, a one-to-one mapping of processes to processors, a gang scheduler  and a well-connected communications network. Networks of workstations may be designed primarily for use as individual workstations and may not have all of these important features.



Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Achieving High Performance on a Distributed Memory Computer

      

Assuming that the ScaLAPACK installation was done correctly, the users need only make sure that they are using an appropriate number of processors and that their matrices are efficiently distributed. Here is a checklist to get started.

  • Use the right number of processors.
    • Rule of thumb: tex2html_wrap_inline16164 for an tex2html_wrap_inline15127 matrix. This provides a local matrix of size approximately 1000 by 1000. 
    • Do not try to solve a small problem on too many processors.
    • Do not exceed physical memory.
  • Use an efficient data distribution.
    • Block sizegif (i.e., MB,NB) = 64. 
    • Square processor grid, tex2html_wrap_inline16172. 
  • Use efficient machine-specific BLAS (not the Fortran 77 reference implementation BLAS) and BLACS (nondebug, BLACSDBGLVL=0 in Bmake.inc)

If the performance is still below that expected, see section 5.3. For guidelines on tuning for higher performance, see section 5.4.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node107.html0100644000056400000620000000637006336113732017530 0ustar pfrauenfstaff Achieving High Performance on a Network of Workstations next up previous contents index
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Achieving High Performance on a Network of Workstations

 

If a network meets the following guidelines, ScaLAPACK will perform well on it (see section 5.1.1). If a network of workstations does not meet one or more of these guidelines, read the rest of this chapter for more information.

  • The bandwidth per node, if measured in Megabytes per second per node, should be no less than one tenth of the peak floating-point rate as measured in megaflops/second/node.
  • The underlying network must allow simultaneous messages, that is, not standard ethernet and not FDDI  .
  • Message latency should be no more than 500 microseconds.
  • All processors should be similar in architecture and performance. ScaLAPACK will be limited by the slowest processor. Data format conversion significantly reduces communication performance.
  • No other jobs should be allowed to execute on the processors that are being used. If the processors are gang scheduled and there is enough physical memory for all jobs on all processors, this requirement may be relaxed, but we do not recommend doing so without careful study.
  • No more than one process should be executed per processor.

Vendor specifications and actual performance often differ considerably, especially in communication latency and bandwidth. Users should make sure that they are using the most efficient BLAS and BLACS available on their system.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Performance, Portability and Scalability

 

How can we provide portable  software for dense linear algebra computations that is efficient on a wide range of modern distributed-memory computers? Answering this question -- and providing the appropriate software -- has been an objective of the ScaLAPACK project.

The ScaLAPACK software has been designed specifically to achieve high efficiency for a wide range of modern distributed-memory computers. Examples of such computers include the Cray T3D and T3E computers, the IBM Scalable POWERparallel SP series, the Intel iPSC and Paragon computers, the nCube-2/3 computer, networks and clusters of workstations (NoWs  and CoWs ), and ``piles'' of PCs (PoPCs) .



Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node109.html0100644000056400000620000002140306336113733017525 0ustar pfrauenfstaff The BLAS as the Key to (Trans)portable Efficiency next up previous contents index
Next: Two-Dimensional Block Cyclic Data Up: PerformancePortability and Scalability Previous: PerformancePortability and Scalability

The BLAS as the Key to (Trans)portable Efficiency

  The total number of floating-point operations performed by most of the ScaLAPACK driver routines for dense matrices can be approximated by the quantity tex2html_wrap_inline12066, where tex2html_wrap_inline16191 is a constant and N is the order of the largest matrix operand. For solving linear equations or linear least squares, tex2html_wrap_inline16191  is a constant depending solely on the selected algorithm. The algorithms used to find eigenvalues and singular values are iterative; hence, for these operations the constant tex2html_wrap_inline16191 truly depends on the input data as well. It is, however, customary or ``standard'' to consider the values of the constants tex2html_wrap_inline16191 for a fixed number of iterations. The ``standard'' constants tex2html_wrap_inline16191 range from 1/3 to approximately 18, as shown in Table 5.8.

The performance of the ScaLAPACK drivers is thus bounded above by the performance of a computation that could be partitioned into P independent chunks of tex2html_wrap_inline16211 floating-point operations each. This upper bound, referred to hereafter as the peak performance , can be computed as the product of tex2html_wrap_inline16211 and the highest reachable local node flop rate. Hence, for a given problem size N and assuming a uniform distribution of the computational tasks, the most important factors determining the overall performance are the number P of nodes involved in the computation and the local node flop rate.

In a serial computational environment, transportable efficiency   is the essential motivation for developing blocking strategies and block-partitioned algorithms [2, 3, 35, 90]  . The linear algebra package (LAPACK) [3]  is the archetype of such a strategy. The LAPACK software is constructed as much as possible out of calls to the BLAS. These kernels confine the impact of the computer architecture differences to a small number of routines. The efficiency and portability of the LAPACK software are then achieved by combining native and efficient BLAS implementations with portable high-level components.

The BLAS  are subdivided into three levels, each of which offers increased scope for exploiting parallelism. This subdivision corresponds to three different types of basic linear algebra operations:

  • Level 1 BLAS [93] : for vector operations, such as tex2html_wrap_inline16219,
  • Level 2 BLAS [59, 58] : for matrix-vector operations, such as tex2html_wrap_inline16221,
  • Level 3 BLAS [57, 56] : for matrix-matrix operations, such as tex2html_wrap_inline16223.
Here, A, B, and C are matrices, x and y are vectors, and tex2html_wrap_inline16235 and tex2html_wrap_inline14473 are scalars.

The performance potential of the three levels of BLAS is strongly related to the ratio of floating-point operations to memory references, as well as to the reuse of data when it is stored in the higher levels of the memory hierarchy. Consequently, the Level 1 BLAS cannot achieve high efficiency on most modern supercomputers. The Level 2 BLAS can achieve near-peak performance on many vector processors; on RISC microprocessors, however, their performance is limited by the memory access bandwidth bottleneck. The greatest scope for exploiting the highest levels of the memory hierarchy as well as other forms of parallelism is offered by the Level 3 BLAS [3].

The previous reasoning applies to distributed-memory computational environments in two ways. First, in order to achieve overall high performance, it is necessary to express the bulk of the computation local to each node in terms of Level 3 BLAS operations. Second, designing and developing a set of parallel BLAS (PBLAS)  for distributed-memory computers should lead to an efficient and straightforward port of the LAPACK software. This is the path followed by the ScaLAPACK initiative [25, 53] as well as others [1, 21, 30, 63]. As part of the ScaLAPACK project, a set of PBLAS has been early designed and developed [29, 26].

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Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node10.html0100644000056400000620000000770306336113641017441 0ustar pfrauenfstaff Structure and Functionality next up previous contents index
Next: Software Components Up: Essentials Previous: ScaLAPACK

Structure and Functionality

  

ScaLAPACK can solve systems of linear equations, linear least squares problems, eigenvalue problems, and singular value problems. ScaLAPACK can also handle many associated computations such as matrix factorizations  or estimating condition numbers.

Like LAPACK, the ScaLAPACK routines are based on block-partitioned algorithms   in order to minimize the frequency of data movement between different levels of the memory hierarchy  . The fundamental building blocks of the ScaLAPACK library are distributed-memory  versions of the Level 1, Level 2, and Level 3 BLAS, called the Parallel BLAS or PBLAS [26, 104], and a set of Basic Linear Algebra Communication Subprograms (BLACS) [54] for communication tasks that arise frequently in parallel linear algebra computations. In the ScaLAPACK routines, the majority of interprocessor communication occurs within the PBLAS, so the source code of the top software layer of ScaLAPACK looks similar to that of LAPACK.

ScaLAPACK contains driver routines  for solving standard types of problems, computational routines  to perform a distinct computational task, and auxiliary routines  to perform a certain subtask or common low-level computation. Each driver routine typically calls a sequence of computational routines. Taken as a whole, the computational routines can perform a wider range of tasks than are covered by the driver routines. Many of the auxiliary routines may be of use to numerical analysts or software developers, so we have documented the Fortran source for these routines with the same level of detail used for the ScaLAPACK computational routines and driver routines.

Dense and band matrices are provided for, but not general sparse matrices. Similar functionality is provided for real and complex matrices. See Chapter 3 for a complete summary of the contents.

Not all the facilities of LAPACK are covered by Release 1.5 of ScaLAPACK.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node110.html0100644000056400000620000000733506336113733017525 0ustar pfrauenfstaff Two-Dimensional Block Cyclic Data Distribution as a Key to Load Balancing and Software Reuse next up previous contents index
Next: BLACS as an Efficient Up: PerformancePortability and Scalability Previous: The BLAS as the

Two-Dimensional Block Cyclic Data Distribution as a Key to Load Balancing and Software Reuse

 

The way the data is distributed over the memory hierarchy of a computer is of fundamental importance to load balancing and software reuse. The block cyclic data distribution allows a reduction of the overhead due to load imbalance and data movement. Block-partitioned algorithms   are used to maximize the local node performance.

Since the data decomposition largely determines the performance and scalability of a concurrent algorithm, a great deal of research [27, 65, 69, 78] has focused on different data decompositions [10, 20, 85]. In particular, the two-dimensional block cyclic distribution [92]  has been suggested as a possible general-purpose basic decomposition for parallel dense linear algebra libraries [31, 76, 97, 17] such as ScaLAPACK.

Block cyclic distribution is beneficial because of its scalability [51] , load balance,  and communication [76] properties. The block-partitioned computation then proceeds in consecutive order just as a conventional serial algorithm does. This essential property of the block cyclic data distribution explains why the ScaLAPACK design has been able to reuse the numerical and software expertise of the sequential LAPACK library.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node111.html0100644000056400000620000001655006336113734017526 0ustar pfrauenfstaff BLACS as an Efficient, Portable and Adequate Message-Passing Interface next up previous contents index
Next: Parallel Efficiency Up: PerformancePortability and Scalability Previous: Two-Dimensional Block Cyclic Data

BLACS as an Efficient, Portable and Adequate Message-Passing Interface

 

The total volume of data communicated by most of the ScaLAPACK driver routines for dense matrices can be approximated by the quantity tex2html_wrap_inline16256 , where N is the order of the largest matrix operand. The number of messages, however, is proportional to N and can be approximated by the quantity tex2html_wrap_inline12070 , where NB is the logical blocking factor used in the computation. Similar to the situation described above, the ``standard'' constants tex2html_wrap_inline16270 for the communication volume depend upon the performed computation and are of the same order as the floating-point operation constants tex2html_wrap_inline16191 shown in Table 5.8. The values of the ``standard'' constants tex2html_wrap_inline16270 for a few selected ScaLAPACK drivers are presented in Table 5.8. As a result, a significant percentage of the ScaLAPACK software aims at exchanging messages between processes.

Developing an adequate message-passing interface specialized for linear algebra operations has been one of the first achievements of the ScaLAPACK project. The Basic Linear Algebra Communications Subprograms (BLACS)  [50, 54] were thus specifically designed to facilitate the expression of the relevant communication operations. The simplicity of the BLACS interface, as well as the rigor of their specification, allows for an easy port of the entire ScaLAPACK software. Currently, the BLACS have been efficiently ported on machine-specific message-passing libraries such as the IBM (MPL)   and Intel (NX)   message-passing libraries, as well as more generic interfaces such as PVM   and MPI  . The BLACS  overhead has been shown to be negligible [54].

The BLACS interface provides the user and library designer with an appropriate level of notation. Indeed, the BLACS operate on typed two-dimensional arrays. The computational model consists of a one- or two-dimensional grid of processes, where each process stores matrices and vectors. The BLACS include synchronous send/receive routines to send a matrix or submatrix from one process to another, to broadcast submatrices, or to perform global reductions (sums, maxima and minima). Other routines establish, change, or query the process grid. The BLACS provide an adequate interface level for linear algebra communication operations.

For ease of use and flexibility, the BLACS send operation is locally blocking;  that is, the return from the send operation indicates that the resources may be reused. However, since this depends only on local information, it is unknown whether the receive operation has been called. Buffering is necessary on the sending or the receiving process. The BLACS receive operation is globally blocking . The return from the receive operation indicates that the message has been (sent and) received. On a system natively supporting globally blocking sends such as the IBM SP2 computer, nonblocking sends coupled with buffering are used to simulate locally blocking sends. This extra buffering operation may cause a slight performance degradation on those systems.

The BLACS broadcast and combine operations feature the ability of selecting different virtual network topologies. This easy-to-use built-in facility allows for the expression of various message scheduling approaches, such as a communication pipeline. This unique and distinctive BLACS characteristic is necessary for achieving the highest performance levels on distributed-memory platforms.


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Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node112.html0100644000056400000620000002235506336113735017530 0ustar pfrauenfstaff Parallel Efficiency next up previous contents index
Next: ScaLAPACK Performance Up: BLACS as an Efficient Previous: BLACS as an Efficient

Parallel Efficiency

   

An important performance metric is parallel efficiency. Parallel efficiency, E(N, P),  for a problem of size N on P nodes is defined in the usual way [65, 92] by
displaymath16290
where T(N,P)  is the runtime of the parallel algorithm, and tex2html_wrap_inline16300  is the runtime of the best sequential algorithm. For dense matrix computations, an implementation is said to be scalable  if the parallel efficiency is an increasing function of tex2html_wrap_inline16304, the problem size per node. The algorithms implemented in the ScaLAPACK library are scalable in this sense.

Figure 5.1 shows the scalability of the ScaLAPACK implementation of the LU factorization on the Intel XP/S Paragon computer. The nodes of the Intel XP/S Paragon computer are general-purpose (GP) or multiprocessor (MP) nodes, based on the Intel i860 XP RISC processors. Each Intel i860 processor is capable of a peak performance of 50 Mflop/s. On such a processor, however, the vendor-supplied BLAS matrix-matrix multiply routine DGEMM can achieve only approximately 45 Mflop/s. The computer used for obtaining the performance results presented in this chapter consisted of MP nodes configured as follows: each MP node had three Intel i860 XP processors -- two to execute application code and a third used exclusively as a message coprocessor  . On such a node, the vendor-supplied BLAS matrix-matrix multiply routine DGEMM can achieve approximately 90 Mflop/s.

  figure3673
Figure 5.1: LU Performance per Intel XP/S MP Paragon node

Figure 5.1 shows the speed in Mflop/s per node of the ScaLAPACK LU factorization routine PDGETRF for different computer configurations. This figure illustrates that when the number of nodes is scaled by a constant factor, the same efficiency or speed per node is achieved for equidistant problem sizes on a logarithmic scale. In other words, maintaining a constant memory use per node allows efficiency to be maintained. (This scalability behavior is also referred to as isoefficiency,  or isogranularity .) In practice, however, a slight degradation is acceptable. The ScaLAPACK driver routines, in general, feature the same scalability behavior up to a constant factor that depends on the exact number of floating-point operations and the total volume of data exchanged during the computation.

In large dense linear algebra computations, the computation cost dominates the communication cost. In the following, the time to execute one floating-point operation by one node is denoted by tex2html_wrap_inline12202 . The time to communicate a message between two nodes is approximated by a linear function of the number of items communicated. The function is the sum of the time to prepare the message for transmission (tex2html_wrap_inline12208)  and the time taken by the message to traverse the network to its destination, that is, the product of its length by the time to transfer one data item (tex2html_wrap_inline12228) . Alternatively, tex2html_wrap_inline12208 is also called the latency,   since it is the time to communicate a message of zero length. On most modern interconnection networks, the order of magnitude of the latency varies between a microsecond and a millisecond.

The bandwidth   of the network is also referred to as its throughput.  It is proportional to the reciprocal of tex2html_wrap_inline12228. On modern networks, the order of magnitude of the bandwidth is the megabyte per second. For a scalable algorithm with tex2html_wrap_inline16304 held constant, one expects the performance to be proportional to P. The algorithms implemented in ScaLAPACK are scalable in this sense. Table 5.1 summarizes the relevant constants used in our scalability analysis.

  table3694
Table 5.1: Variable definitions

Using the notation presented in table 5.1, the execution time of the ScaLAPACK drivers can be approximated by
 equation3728

The corresponding parallel efficiency  can then be approximated by
 equation3739
Equation 5.2 illustrates, in particular, that the communication versus computation performance ratio of a distributed-memory computer significantly affects parallel efficiency. The ratio of the latency to the time per flop tex2html_wrap_inline16374 greatly affects the parallel efficiency of small problems. The ratio of the network throughput to the flop rate tex2html_wrap_inline16376 significantly affects the parallel efficiency of medium-sized problems. For large problems, the node flop rate tex2html_wrap_inline16378 is the dominant factor contributing to the parallel efficiency of the parallel algorithms implemented in ScaLAPACK.

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Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node113.html0100644000056400000620000001630006336113735017522 0ustar pfrauenfstaff ScaLAPACK Performance next up previous contents index
Next: Performance of Selected BLACS Up: PerformancePortability and Scalability Previous: Parallel Efficiency

ScaLAPACK Performance

 

In this section, we present performance data for Version 1.4 of ScaLAPACK on four distributed memory computers and two networks of workstations. The four distributed memory computers are the Cray T3E computer, the IBM Scalable POWERparallel 2 computer, the Intel XP/S MP Paragon computer, and the Intel ASCI Option Red Supercomputer. One of the networks of workstations consists of Sun Ultra Enterprise 2 (Model 2170s) connected via switched ATM. The other network of workstations, the Berkeley NOW [34] , consists of 100+ Sun UltraSPARC-1 workstations and 40+ Myricom crossbar switches and LANai 4.1 network interface cards. ScaLAPACK on the NOW uses MPI BLACS, where the MPI is a port of the freely-available MPICH reference code. MPI uses Active Messages  as its underlying communications layer. Active Messages [98] provide ultra-lightweight remote-procedure calls for processes on the NOW. The system currently uses AM-II , a generalized active message layer that supports more than SPMD parallel programs, e.g., client-server programs and distributed filesystems. It retains the simple request/response paradigm common to all previous active message implementations as well as its high-performance. These six computers are a collection of processing nodes interconnected via a network. Each node has local memory and one or more processors. Tables 5.2, 5.3, and 5.4 describe the characteristics of these six computers.

  table3764
Table 5.2: Characteristics of the Cray T3E and IBM SP2 computers timed

  table3773
Table 5.3: Characteristics of the Intel computers timed

  table3782
Table 5.4: Characteristics of the networks of workstations timed

As noted in Tables 5.2, 5.3, and 5.4, a machine-specific optimized BLAS implementation was used for all the performance numbers reported in this chapter. For the IBM Scalable POWERparallel 2 (SP2) computer, the IBM Engineering and Scientific Subroutine Library (ESSL) was used [88]. On the Intel XP/S MP Paragon computer, the Intel Basic Math Library Software (Release 5.0) [89] was used. The Intel ASCI Option Red Supercomputer was tested using a pre-alpha version of the Cougar operating system and using an unoptimized functional version of the dual processor Basic Math Library from Kuck and Associates, Inc. The communication performance and library performance was still being enhanced. On the Sun Ultra Enterprise 2 workstation, the Dakota Scientific Software Library (DSSL)gif was used. The DSSL BLAS implementation used only one processor per node. On the Berkeley NOW, the Sun Performance Library, version 1.2, was used. It should also be noted that for the IBM Scalable POWERparallel 2 (SP2) the communication layer used was the IBM Parallel Operating Environment (POE), which is a combination of MPI and MPL libraries.

Several data distributions were tried for N=2000. The fastest data distribution for N=2000 was used for all problem sizes, although this data distribution may not be optimal for all problem sizes. Whenever applicable, only the options UPLO=`U' and TRANS=`N' were timed. The test matrices were generated with randomly distributed entries. All runtimes are reported in seconds. Block size is denoted by NB.

This section first reports performance data for a relevant selection of BLAS and BLACS routines. Then, timing results obtained for some PBLAS routines are presented. Finally, performance numbers for selected ScaLAPACK driver routines are shown.

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Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: Performance of Selected PBLAS Up: PerformancePortability and Scalability Previous: ScaLAPACK Performance

Performance of Selected BLACS and Level 3 BLAS Routines

 

The efficiency  of the ScaLAPACK software depends on efficient implementations of the BLAS and the BLACS being provided by computer vendors (or others) for their computers. The BLAS and the BLACS form a low-level interface between ScaLAPACK software and different computer architectures. Table 5.5 presents performance numbers indicating how well the BLACS and Level 3 BLAS perform on different distributed-memory computers. For each computer this table shows the flop rate achieved by the matrix-matrix multiply Level 3 BLAS routine SGEMM/DGEMM (tex2html_wrap_inline12088)  on a node versus the theoretical peak performance of that node, the underlying message-passing library called by the BLACS, and the approximated values of the latency (tex2html_wrap_inline12208)  and the bandwidth (tex2html_wrap_inline12064)  achieved by the BLACS versus the underlying message-passing software for the machine.

  table3805
Table 5.5: BLACS and Level 3 BLAS performance indicators

The values for latency in table 5.5 were obtained by timing the cost of a 0-byte message. The bandwidth numbers table 5.5 were obtained by increasing message length until message bandwidth was saturated. We used the same timing mechanism for both the BLACS and the underlying message-passing library.

These numbers are actual timing numbers, not values based on hardware peaks, for instance. Therefore, they should be considered as approximate values or indicators of the observed performance between two nodes, as opposed to precise evaluations of the interconnection network capabilities. On the CRAY, the numbers reported are for MPI and the MPIBLACS, instead of the more optimal shmem library with CRAY's native BLACS.

For all four computers, a machine-specific optimized BLAS implementation was used for all the performance numbers reported in this chapter. For the IBM Scalable POWERparallel 2 (SP2) computer, the IBM Engineering and Scientific Subroutine Library (ESSL) was used [88]. On the Intel XP/S MP Paragon computer, the Intel Basic Math Library Software (Release 5.0) [89] was used. On the Sun Ultra Enterprise 2 workstation, the Dakota Scientific Software Library (DSSL)gif was used. The DSSL BLAS implementation used only one processor per node. The speed of the BLAS matrix-matrix multiply routine shown in Table 5.5 has been obtained for the following operation tex2html_wrap_inline16435, where A, B, and C are square matrices of order 500.


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Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node115.html0100644000056400000620000000706006336113737017531 0ustar pfrauenfstaff Performance of Selected PBLAS routines next up previous contents index
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Performance of Selected PBLAS routines

 

The performance of Level 2 PBLAS routines is dependent on the performance of Level 2 BLAS routines which is dependent on the bulk transfer rate from main memory.

  table3850
Table 5.6: Speed in Mflop/s for the PBLAS matrix-vector multiply routine PSGEMV/PDGEMV

Table 5.6  shows execution rates for the 64-bit matrix-vector multiply PBLAS routine PSGEMV /PDGEMV . The rates listed are for a matrix-vector product tex2html_wrap_inline16483, where A is a square matrix of order N and x and y are vectors that are both distributed over a process column.

The Level 3 PBLAS are not necessarily limited by memory bandwidth because they perform many flops for each word involved. The flop rate is correspondingly higher. Table 5.7 

  table3877
Table 5.7: Speed in Mflop/s for the PBLAS matrix-matrix multiply routine PSGEMM/PDGEMM

shows the performance results obtained by the general matrix-matrix multiply PBLAS routine PSGEMM /PDGEMM . These results have been obtained for the matrix-matrix multiply operation tex2html_wrap_inline16435, where A, B, and C are square matrices of order N.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node116.html0100644000056400000620000001472306336113737017536 0ustar pfrauenfstaff Solution of Common Numerical Linear Algebra Problems next up previous contents index
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Solution of Common Numerical Linear Algebra Problems

 

This section contains performance numbers for selected driver routines. These routines provide complete solutions for common linear algebra problems.

  • Solve a general N-by-N system of linear equations with one right-hand side using the routine PSGESV/PDGESV.
  • Solve a symmetric positive definite N-by-N system of linear equations with one right-hand side, using PSPOSV/PDPOSV.
  • Solve an N-by-N linear least squares problem with one right-hand side using the routine PSGELS/PDGELS.
  • Find the eigenvalues and optionally the corresponding eigenvectors of an N-by-N symmetric matrix, using the routine PSSYEVX/PDSYEVX.
  • Find the eigenvalues and optionally the corresponding eigenvectors of an N-by-N symmetric matrix, using the routine PSSYEV/PDSYEV.
  • Find the singular values and optionally the corresponding right and left singular vectors of an N-by-N matrix, using PSGESVD/PDGESVD.
  • Find the eigenvalues and optionally the corresponding right eigenvectors of an N-by-N Hessenberg matrix, using the routine PSLAHQR/PDLAHQR.gif
Table 5.8 presents ``standard'' floating-point operation costs  (tex2html_wrap_inline12066) for selected ScaLAPACK drivers for matrices of order N. Approximate values of the constants tex2html_wrap_inline16577 and tex2html_wrap_inline16270 defined in section 5.2.3 are also provided.

  table3908
Table 5.8: ``Standard'' floating-point operation (tex2html_wrap_inline16191) and communication costs (tex2html_wrap_inline16270, tex2html_wrap_inline16577) for selected ScaLAPACK drivers

The operation counts given for the eigenvalue and SVD drivers are incomplete. They do not include any of the tex2html_wrap_inline16659 computation costs (i.e., the entire tridiagonal eigendecomposition is ignored in PxxxEVX). Furthermore, the reductions involved require matrix-vector multiplies, which are less efficient than the matrix-matrix multiplies required by the other drivers listed here. Hence this table greatly underestimates the execution time of the eigenvalue and SVD drivers, especially the expert symmetric eigensolver drivers. For PxLAHQR, when only eigenvalues are computed, tex2html_wrap_inline16577 and tex2html_wrap_inline16270 look the same as the full Schur form case, in terms of ``order of magnitude''. There is actually tex2html_wrap_inline16665 to tex2html_wrap_inline16667 the number of messages/volume depending on the circumstances.

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Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Solving Linear Systems of Equations

 

Table 5.9  illustrates the speed of the ScaLAPACK driver routine PSGESV /PDGESV  for solving a square linear system of order N by LU factorization with partial row pivoting of a real matrix. For all timings, 64-bit floating-point arithmetic was used. Thus, single-precision timings are reported for the Cray T3E, and double precision timings are reported on all other computers. The distribution block size  is also used as the partitioning unit  for the computation and communication phases.

Table 5.10  illustrates the speed of the ScaLAPACK routine PSPOSV /PDPOSV  for solving a symmetric positive definite linear system of order N via the Cholesky factorization.

Right-looking variants of the LU and Cholesky factorizations were chosen for ScaLAPACK because they minimize total communication volume, that is, the aggregated

  table3947
Table 5.9: Speed in Mflop/s of PSGESV/PDGESV for square matrices of order N

amount of data transferred between processes during the operation.

  table3969
Table 5.10: Speed in Mflop/s of PSPOSV/PDPOSV for matrices of order N with UPLO=`U'



Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node118.html0100644000056400000620000000771406336113740017534 0ustar pfrauenfstaff Solving Linear Least Squares Problems next up previous contents index
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Solving Linear Least Squares Problems

 

Table 5.11  summarizes performance results obtained for the ScaLAPACK routine PSGELS /PDGELS  that solves full-rank linear least squares problems. Solving such problems of the form tex2html_wrap_inline16768, where x and b are vectors and A is a rectangular matrix having full rank is traditionally achieved via the computation of the QR factorization of the matrix A. In ScaLAPACK, the QR factorization   is based on the use of elementary Householder   matrices of the general form
displaymath16782
where v is a column vector and tex2html_wrap_inline14435 is a scalar. This leads to an algorithm with excellent vector performance, especially if coded to use Level 2 PBLAS.

The key to developing a distributed block form of this algorithm is to represent a product of K elementary Householder matrices of order N as a block form of a Householder matrix.   This can be done in various ways. ScaLAPACK uses the form [108]
displaymath16792
where V is an N-by-K matrix whose columns are the individual vectors tex2html_wrap_inline16800 associated with the Householder matrices tex2html_wrap_inline16802 and T is an upper triangular matrix of order K. Extra work is required to compute the elements of T, but this is compensated for by the greater speed of applying the block form.

  table4001
Table 5.11: Speed in Mflop/s of PSGELS/PDGELS for square matrices of order N


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Eigenvalue Problems

 

ScaLAPACK includes block algorithms for solving symmetric  and nonsymmetric eigenvalue problems as well as for computing the singular value decomposition.

The first step in solving many types of eigenvalue problems is to reduce the original matrix to a ``condensed form'' by orthogonal transformations.     In the reduction to condensed forms, the unblocked algorithms all use elementary Householder matrices and have good vector performance. Block forms of these algorithms have been developed [28], but all require additional operations, and a significant proportion of the work must still be performed by the Level 2 PBLAS. Thus, there is less possibility of compensating for the extra operations.

The algorithms concerned are listed below:

  • Reduction of a symmetric matrix to tridiagonal form  to solve a symmetric eigenvalue problem: ScaLAPACK routine PSSYTRD /PDSYTRD  applies a symmetric block update of the form
    displaymath16854
    using the Level 3 PBLAS routine PSSYR2K /PDSYR2K ; Level 3 PBLAS account for at most half the work.
  • Reduction of a rectangular matrix to bidiagonal form  to compute a singular value decomposition: ScaLAPACK routine PSGEBRD  /PDGEBRD  applies a block update of the form
    displaymath16855
    using two calls to the Level 3 PBLAS routine PSGEMM/PDGEMM; Level 3 PBLAS account for at most half the work.
  • Reduction of a nonsymmetric matrix to Hessenberg form   to solve a nonsymmetric eigenvalue problem: ScaLAPACK routine PSGEHRD /PDGEHRD  applies a block update of the form
    displaymath16856
    Level 3 PBLAS account for at most three-quarters of the work.

Extra work must be performed to compute the N-by-K matrices X and Y that are required for the block updates (K is the block size), and extra workspace is needed to store them.

Following the reduction of a dense symmetric matrix to tridiagonal form T, one must compute the eigenvalues and (optionally) eigenvectors of T. The current version of ScaLAPACK includes two different routines PSSYEVX /PDSYEVX  and PSSYEV /PDSYEV  for solving symmetric eigenproblems. PSSYEVX/PDSYEVX uses bisection and inverse iteration. PSSYEV/PDSYEV uses the QR algorithm. Table 5.12  and Table 5.13  show the execution time in seconds of the routines PSSYEVX/PDSYEVX and PSSYEV /PDSYEV , respectively, for computing the eigenvalues and eigenvectors of symmetric matrices of order N. The performance of PSSYEVX /PDSYEVX  deteriorates in the face of large clusters of eigenvalues. ScaLAPACK uses a nonscalable definition of clusters (because we chose to remain consistent with LAPACK). Hence, matrices larger than N=1000 tend to have at least one very large cluster (see section 5.3.6). This needs further study. More detailed information concerning the performance of these routines may be found in [40]. Table 5.14  shows the execution time in seconds of the routines PSGESVD /PDGESVD  for computing the singular values and the corresponding right and left singular vectors of a general matrix of order N.

  table4060
Table 5.12: Execution time in seconds of PSSYEVX/PDSYEVX for square matrices of order N

For computing the eigenvalues and eigenvectors of a Hessenberg matrix--or rather, for computing its Schur factorization-- two flavors of block algorithms have been developed. The first algorithm implemented in the routine PSLAHQR /PDLAHQR  results from the parallelization of the QR algorithm. The key idea is to generate many shifts at once rather than two at a time, thereby allowing all bulges to carry out up-to-date shifts. The second algorithm that is currently implemented as a prototype code  is based on the computation of the matrix sign function [14, 13, 12]. In this section, however, only performance results of the first approach are reported.

  table4085
Table 5.13: Execution time in seconds of PSSYEV/PDSYEV for square matrices of order N

  table4106
Table 5.14: Execution time in seconds of PSGESVD/PDGESVD for square matrices of order N

Table 5.15  summarizes performance results obtained for the ScaLAPACK routine PDLAHQR doing a full Schur decomposition of an order N upper Hessenberg matrix. The supercomputers the table gives timings for are the Intel XP/S MP Paragon supercomputer and technology from the Intel ASCI Option Red Supercomputer. For both machines, we assume only one CPU is being used for computation on this code. The Schur decomposition is based on iteratively applying orthogonal similarity transformations on a Hessenberg matrix H such as
displaymath16857
until T becomes pseudo-upper triangular (i.e., in the real case, having one by one or two by two subdiagonal blocks.) The serial performance (assuming roughly tex2html_wrap_inline16994 flops) of the LAPACK routine DLAHQR for computing a complex Schur decomposition is around 8.5 Mflops on the Intel MP Paragon supercomputer. The enhanced performance shown in Table 5.15 is slightly faster, a bit above 9 Mflops, and ends up peaking around 10 Mflops because of the block application of Householder transforms found in the ScaLAPACK serial auxiliary routine DLAREF. For the technology behind the Intel ASCI Option Red Supercomputer, it peaks at several times the speed of the Paragon, and has a slightly faster drop off in efficiency. For further details and timings, please see [79].

  table4130
Table 5.15: Execution time in seconds of PDLAHQR for square matrices of order N

A more detailed performance analysis of the eigensolvers included in the ScaLAPACK software library can be found in [48, 79]. Finally, we note that research into parallel algorithms for symmetric and nonsymmetric eigenproblems continues [11, 86, 45], and future versions of ScaLAPACK will be updated to contain the best algorithms available.

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Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: LAPACK Up: Essentials Previous: Structure and Functionality

Software Components

 

Figure 1.1 describes the ScaLAPACK software hierarchy . The components below the line, labeled Local , are called on a single processor, with arguments stored on single processors only. The components above the line, labeled Global , are synchronous parallel routines, whose arguments include matrices and vectors distributed across multiple processors. We describe each component in turn.

  figure398
Figure 1.1: ScaLAPACK software hierarchy



Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: Obtaining High Performance with Up: Performance of ScaLAPACK Previous: Eigenvalue Problems

Performance Evaluation

   



Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node121.html0100644000056400000620000000666406336113742017533 0ustar pfrauenfstaff Obtaining High Performance with ScaLAPACK Codes next up previous contents index
Next: Checking the BLAS and Up: Performance Evaluation Previous: Performance Evaluation

Obtaining High Performance with ScaLAPACK Codes

We suggest the following approach to obtain high performance with ScaLAPACK codes:

  • Use the best BLAS and BLACS libraries available.
  • Start with a standard data distribution.
    • A square processor grid (tex2html_wrap_inline17051) if tex2html_wrap_inline17053 
    • A one dimensional processor grid (Ptex2html_wrap_inline12112=1, Ptex2html_wrap_inline12114=P) if P < 9
    • Block size = 64 
  • Determine whether reasonable performance is being achieved.
  • Identify the performance bottleneck(s), if any,
  • Tune the distribution or routine parameters to improve performance further.

The standard data distribution will typically achieve 25-50% of the peak performance possible (depending in part on how many processors are ignored, i.e., the difference between tex2html_wrap_inline17061 and tex2html_wrap_inline17063). We do not recommend experimenting with different data distributions until performance that is acceptable (or nearly so) has been achieved. If each individual node requires a block size larger than 64 to achieve near-peak performance on local matrix-matrix multiply, the block size may have to be increased. This step is unlikely, however, unless the computer has a shared-memory multiprocessor with more than four processors on each node.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node122.html0100644000056400000620000000570706336113743017532 0ustar pfrauenfstaff Checking the BLAS and BLACS Libraries next up previous contents index
Next: Estimate Execution Time Up: Performance Evaluation Previous: Obtaining High Performance with

Checking the BLAS and BLACS Libraries

 

The best way to determine whether one is using efficient BLAS and BLACS libraries is to time them. ScaLAPACK provides a rudimentary BLAS and BLACS timer in the examples/timers directory on netlib and on the CD-ROM. This directory also contains pointers and instructions for more complete timers, such as the LAPACK BLAS timer and the message-passing benchmark program [46]. We encourage users to use these timers to measure the performance of the BLAS.

This ScaLAPACK examples/timers directory also contains pointers to some benchmark results and some pointers on interpreting the results of the timers.

To determine which BLAS and BLACS libraries are being linked in, users should check the output of the linker. If the Makefile includes SLmake.inc, the BLAS library is given by the macro BLASLIB in SLmake.inc while the BLACS library is given by the macro BLACSLIB. If the BLAS library name is of the form blas_LINUX.a, this is probably the (slow) reference implementation BLAS. If the BLAS library name is -lblas, -lessl, -ldxml or the like, this may be an optimized BLAS library.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node123.html0100644000056400000620000000765106336113743017533 0ustar pfrauenfstaff Estimate Execution Time next up previous contents index
Next: Determine Whether Reasonable Performance Up: Performance Evaluation Previous: Checking the BLAS and

Estimate Execution Time

 

This section describes how one can estimate the execution time of a ScaLAPACK routine on a given platform, using Equation 5.1 and the values provided in table 5.5 and table 5.8. By comparing this estimate with experimental data, the user can determine whether reasonable performance has been achieved and can (possibly) identify the performance bottlenecks, if any.

For linear system solvers, the estimate typically is accurate to within 50% for moderate-sized problems (i.e., 160,000 or more matrix elements per node). For eigensolvers, the estimate may be low by a factor of 2 for moderate-sized problems and by more than that for smaller problems. The eigensolvers take longer because they involve matrix-vector flops, as well as matrix-matrix flops, and involve substantial numbers of o(tex2html_wrap_inline17067) flops that are not included in the approximation. The accuracy of performance estimates increases with the problem size. Unfortunately, because ScaLAPACK eigensolvers require more memory than the other ScaLAPACK drivers, large problems cannot be solved; hence, execution times for small and medium-sized problems (rather than medium-sized and large problems) are reported.

  table4184
Table 5.16: Estimated (Est) versus obtained (Obt) Mflop/s rates of PDGESV and PDPOSV on P nodes of the IBM SP2 computer for matrices of order N and a block size (NB) equal to 50

Table 5.16 shows the estimated versus obtained Mflop/s rates for two ScaLAPACK driver routines solving linear systems of equations on the IBM Scalable POWERparallel 2 computer. The results show that for these drivers the estimated execution times are within approximately 35 % of the experimental data on the SP2. (The estimated times for the symmetric eigensolvers and SVD codes would not be as accurate.)


Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node124.html0100644000056400000620000000642506336113744017533 0ustar pfrauenfstaff Determine Whether Reasonable Performance Is Achieved next up previous contents index
Next: Identify Performance Bottlenecks Up: Performance Evaluation Previous: Estimate Execution Time

Determine Whether Reasonable Performance Is Achieved

This chapter contains performance results for some ScaLAPACK drivers on a range of different platforms. We recommend that users compare their performance results against the ones presented in the previous tables. For those users whose computer is not listed, the BLAS timing program [3] and the message-passing benchmark program [46] can be used to estimate the values of tex2html_wrap_inline12088, tex2html_wrap_inline12208 and tex2html_wrap_inline12228 for a specific computer. gif The execution time can then be estimated by using the material presented in the previous section.

If this chapter does not contain performance data for the ScaLAPACK routine a user is using, we recommend verifying the performance with one of the drivers whose performance is presented in this chapter. The publicly available ScaLAPACK distribution contains timing programs for each driver. This performance sanity check should convince the user not only that the library has been correctly installed, but also that it is being used properly. Users may also send questions, suggestions, or comments regarding ScaLAPACK performance issues to scalapack@cs.utk.edu.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: Performance Bottlenecks in the Up: Performance Evaluation Previous: Determine Whether Reasonable Performance

Identify Performance Bottlenecks

    

The formulas mentioned in section 5.3.3, in addition to providing an estimate of performance, can help one identify whether the performance is limited by computation, by the number of messages, or by the volume of communication. Even if the estimate is far from correct, the user may get some information about the performance bottleneck by studying the computation and communication estimates provided by those formulas.

Comparing the execution times of a problem of size N and one of size N/2 may also provide insight into the performance of the ScaLAPACK routine being used. Let tex2html_wrap_inline12212 and tex2html_wrap_inline12220 be the time required for a problem of size N and size N/2, respectively, on P processors.

  • If tex2html_wrap_inline17099, the physical memory of each node may be exceeded.
  • If tex2html_wrap_inline17101, the performance may be limited by the rate at which flops are performed. If the flop rate is significantly less than expected, the user should check the data distribution (try the standard data distribution suggested in section 5.1.1) and the underlying BLAS.
  • If tex2html_wrap_inline17103, the major performance factor may be bandwidth (tex2html_wrap_inline12064). This is what one should obtain for medium values of N.
  • If tex2html_wrap_inline17109, the major performance factor may be latency (tex2html_wrap_inline12208). This is what one should obtain for small values of N.
This performance analysis suggests which computer characteristic is most likely limiting the performance. It cannot say whether one is getting good performance.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node126.html0100644000056400000620000000720306336113745017531 0ustar pfrauenfstaff Performance Bottlenecks in the Expert Symmetric Eigenproblem Drivers next up previous contents index
Next: Performance Improvement Up: Performance Evaluation Previous: Identify Performance Bottlenecks

Performance Bottlenecks in the Expert Symmetric Eigenproblem Drivers

 

Large clusters of eigenvalues in the input matrix may cause poor performance in the expert symmetric eigenproblem drivers, PSSYEVX /PDSYEVX . If the execution time observed for the ScaLAPACK drivers PxSYGVX and PxSYEVX is more than double the estimate in section 5.3.3 and more than the minimum LWORK is provided, we recommend that this part of the code be retimed after relaxing the orthogonalization requirements. This can be achieved either by setting the value of the formal parameter LWORK to the minimum allowed by the driver as specified in the leading comments of the source code, or by calling the driver with the value of the parameter ORTOL set to the machine epsilon multiplied by the norm of the matrix. These last two values may be obtained by calling respectively the ScaLAPACK routines PxLAMCH and PxLANSY. If the execution time obtained for the driver after relaxing the orthogonalization requirements is substantially reduced, it is likely that the spectrum of the matrix or matrix pencil has a large cluster of eigenvalues that the driver attempts to reorthogonalize. Otherwise, it is likely that the performance bottleneck is caused by other factors as mentioned in section 5.3.5. If the matrix or matrix pencil has a large cluster of eigenvalues, we recommend using the corresponding simple driver PxSYEV, instead. If the application can tolerate loss of orthogonality, the drivers PxSYGVX and PxSYEVX may achieve good performance by relaxing the orthogonalization requirements using the method suggested above. Please check the value of the INFO parameter returned by these and all ScaLAPACK drivers.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node127.html0100644000056400000620000000473606336113746017543 0ustar pfrauenfstaff Performance Improvement next up previous contents index
Next: Choosing a Faster BLACS Up: Performance of ScaLAPACK Previous: Performance Bottlenecks in the

Performance Improvement

   

Before experimenting with different data layouts, users should make sure that they are using the fastest BLACS and BLAS libraries.

Three major factors influence the performance of a ScaLAPACK routine: the flop rate achieved by the BLAS on each node, the computational load balance, and the communication costs.



Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node128.html0100644000056400000620000000645006336113746017537 0ustar pfrauenfstaff Choosing a Faster BLACS Library next up previous contents index
Next: Choosing a Faster BLAS Up: Performance Improvement Previous: Performance Improvement

Choosing a Faster BLACS Library

Users should choose vendor-supplied BLACS optimized for their computer; these BLAS will be the fastest BLACS implementation. If no vendor-supplied BLACS exists, users will have to choose among the publicly available BLACS libraries.

Many distributed-memory computers offer several communication libraries. The SP2, for example, offers MPI, PVM and MPL communication libraries. Since implementations of the BLACS exist on each of several communication libraries, one may have a choice of several different BLACS implementations. On the SP2, for example, the user can run the BLACS MPI, BLACS MPL, or BLACS PVM version.

Unfortunately, no hard rule exists as to which BLACS implementation will be fastest. However, since the BLACS cannot be faster than the communication library upon which it is built, and since the BLACS typically add little overhead, it is usually best to choose the BLACS implementation that is based on the fastest communication library.

Identifying the fastest communication library may not be trivial. The speed of communication libraries may be reported in different ways. Moreover, although the speed of blocking sends is reported because they are faster than nonblocking sends, the BLACS must use the nonblocking sends or provide its own buffering. Those who are using one of the computers listed in this chapter should refer to Tables 5.2 and 5.3 to see which library we used for timing. Our experience is that the fastest communication library was the library that is native to that particular computer.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node129.html0100644000056400000620000000542706336113747017544 0ustar pfrauenfstaff Choosing a Faster BLAS Library next up previous contents index
Next: Tuning the Distribution Parameters Up: Performance Improvement Previous: Choosing a Faster BLACS

Choosing a Faster BLAS Library

Highly efficient machine-specific implementations of the BLAS are available for many modern high-performance computers. Users who cannot obtain an efficient BLAS for your architecture may be able to create one from by using a set of BLAS that requires only an efficient implementation of the matrix-matrix multiply BLAS routine xGEMM [35, 90], combined with an automatically generated machine-specific and efficient implementation of xGEMM [16].

Users who are using one of the computers listed in this chapter should refer to Tables 5.2 and 5.3 to see which library we used for timing. Otherwise, the computer vendor may be able to provide information about optimized BLAS for a specific computer.

A reference Fortran 77 implementation of the BLAS is available from the blas directory on netlib.

http://www.netlib.org/blas/blas.shar

Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: BLAS Up: Software Components Previous: Software Components

LAPACK

  As mentioned before, LAPACK, or Linear Algebra PACKage [3], is a collection of routines for solving linear systems, least squares problems, eigenproblems, and singular problems. High performance is attained by using algorithms that do most of their work in calls to the BLAS, with an emphasis on matrix-matrix multiplication. Each routine has one or more performance tuning parameters, such as the sizes of the blocks operated on by the BLAS. These parameters are machine dependent and are obtained from a table defined when the package is installed and referenced at runtime.

The LAPACK routines are written as a single thread of execution. LAPACK can accommodate shared-memory machines, provided parallel BLAS are available (in other words, the only parallelism is implicit in calls to BLAS). Extensive performance results for LAPACK can be found in the LAPACK Users' Guide [3].


Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node130.html0100644000056400000620000002063106336113747017526 0ustar pfrauenfstaff Tuning the Distribution Parameters for Better Performance next up previous contents index
Next: Performance of Banded and Up: Performance Improvement Previous: Choosing a Faster BLAS

Tuning the Distribution Parameters for Better Performance

    

By adjusting the data distribution of the matrices, users may be able to achieve 10-50 % greater performance than by using the standard data distribution suggested in section 5.1.1.

The performance attained using the standard data distribution is usually fairly close to optimal; hence, if one is getting poor performance, it is unlikely that modifying the data distribution will solve the performance problem.

An optimal data distribution depends upon several factors including the performance characteristics of the hardware, the ScaLAPACK routine invoked, and (to a certain extent) the problem size. The algorithms currently implemented in ScaLAPACK fall into two main classes.

The first class of algorithms is distinguished by the fact that at each step a block of rows or columns is replicated in all process rows or columns. Furthermore, the process row or column source of this broadcast operation is the one immediately following -- or preceding depending on the algorithm -- the process row or column source of the broadcast operation performed at the previous step of the algorithm. The QR factorization and the right looking variant of the LU factorization are typical examples of such algorithms, where it is thus possible to establish and maintain a communication pipeline in order to overlap computation and communication. The direction of the pipeline determines the best possible shapes of the process grid. For instance, the LU, QR, and QL factorizations perform better for ``flat'' process grids (tex2html_wrap_inline17130). These factorizations perform a reduction operation for each matrix column for pivoting in the LU factorization and for computing the Householder transformation in the QR and QL decompositions. Moreover, after this reduction has been performed, it is important to update the next block of columns as fast as possible. This update is done by broadcasting the current block of columns using a ring topology, that is, feeding the ongoing communication pipe. Similarly, the performance of the LQ and RQ factorizations take advantage of ``tall'' grids (tex2html_wrap_inline17142) for the same, but transposed, reasons.

The second group of algorithms is characterized by the physical transposition of a block of rows and/or columns at each step. Square or near square grids are more adequate from a performance point of view for these transposition operations. Examples of such algorithms implemented in ScaLAPACK include the right-looking variant of the Cholesky factorization, the matrix inversion algorithm, and the reductions to bidiagonal form (PxGEBRD), to Hessenberg form (PxGEHRD), and to tridiagonal form (PxSYTRD). It is interesting to note that if square grids are more efficient for these matrix reduction operations, the corresponding eigensolver usually prefers flatter grids.

Table 5.17 summarizes this paragraph and provides suggestions for selecting the most appropriate shape of the logical tex2html_wrap_inline12182 process grid from a performance point of view. The results presented in this table may need to be refined depending on the physical characteristics of the physical interconnection network.

  table4271
Table 5.17: Process grid suggestions for some ScaLAPACK drivers

Assume that at most P nodes are available. A natural question is: Could we decide which tex2html_wrap_inline17180 process grid should be used? Similarly, depending on the value of P, it is not always possible to factor tex2html_wrap_inline17184 to create an appropriate grid shape. For example, if the number of nodes available is a prime number and a square grid is suitable with respect to performance, it may be beneficial to let some nodes remain idle so that the remaining nodes can be arranged in a ``squarer'' grid.

If the BLACS implementation or the interconnection network features high latency, a one-dimensional data distribution will improve the performance for small and medium problem sizes. The number of messages significantly impacts the performance achieved for small problem sizes, whereas the total message volume becomes a dominant factor for medium-sized problems. The performance cost due to floating-point operations dominates for large problem sizes. One-dimensional data distributions reduce the total number of messages exchanged on the interconnection network but increase the total volume of message traffic. Therefore, one-dimensional data distributions are better for small problem sizes but are worse for large problem sizes, especially when one is using eight or more processors.

Determining optimal, or near-optimal, distribution block sizes with respect to performance   for a given platform is a difficult task. However, it is empirically true that as soon as a good block size or even a set of good distribution parameters is found, the performance is not highly sensitive to small changes of the values of these parameters.

next up previous contents index
Next: Performance of Banded and Up: Performance Improvement Previous: Choosing a Faster BLAS

Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node131.html0100644000056400000620000000502406336113750017520 0ustar pfrauenfstaff Performance of Banded and Out-of-Core Drivers next up previous contents index
Next: Accuracy and Stability Up: Performance of ScaLAPACK Previous: Tuning the Distribution Parameters

Performance of Banded and Out-of-Core Drivers

ScaLAPACK provides LU and Cholesky factorizations for band matrices. For small bandwidth, divide and conquer algorithms have been chosen even though they require more floating-point operations. A more detailed performance analysis can be found in [18].

ScaLAPACK also provides prototype out-of-core linear system solvers. Information on these particular routines as well as the algorithms that have been selected can be found in [47, 55, 18]. In particular, it is shown in [55] that these out-of-core solvers incur approximately a 20% overhead over the corresponding in-core ScaLAPACK solvers.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node132.html0100644000056400000620000001746006336113751017531 0ustar pfrauenfstaff Accuracy and Stability next up previous contents index
Next: Sources of Error in Up: Guide Previous: Performance of Banded and

Accuracy and Stability

 

In this chapter we explain our overall approach to obtaining error bounds and provide enough information to use the software. The comments at the beginning of the individual routines should be consulted for more details. It is beyond the scope of this chapter to justify all the bounds we present. Instead, we give references to the literature. For example, standard material on error analysis can be found in [71, 114, 84, 38].

To make this chapter easy to read, we have labeled parts not essential for a first reading as Further Details. The sections not labeled as Further Details should provide all the information needed to understand and use the main error bounds computed by ScaLAPACK. The Further Details sections provide mathematical background, references, and tighter but more expensive error bounds, and may be read later.

Since ScaLAPACK uses the same overall algorithmic approach as LAPACK, its error  bounds  are essentially the same as those for LAPACK. Therefore, this chapter is largely analogous to Chapter 4 of the LAPACK Users' Guide [3]. Significant differences between LAPACK and ScaLAPACK include the following:

  • Section 6.1 discusses how machine constants in a heterogeneous network of machines with differing floating-point arithmetics must be redefined. ScaLAPACK can also exploit arithmetic with tex2html_wrap_inline17191, which is available in IEEE standard floating-point arithmetic.  
  • Section 6.2 discusses reliability problems that can arise on heterogeneous networks of machines and how to guarantee reliability on a homogeneous network.
  • Section 6.5 discusses some routines that do Gaussian elimination on band matrices with the pivot order chosen for parallelism rather than numerical stability. These routines are numerically stable only when applied to matrices that are do not require partial pivoting for stability (such as diagonally dominant and symmetric positive definite matrices).
  • Section 6.7 discusses PxSYEVX.gif In contrast to its LAPACK analogue, xSYEVX, PxSYEVX allows the user to trade off orthogonality of computed eigenvectors and runtime.

In section 6.1 we discuss the sources of numerical error, in particular roundoff error. We also briefly discuss IEEE arithmetic. Section 6.2 discusses the new sources of numerical error specific to parallel libraries, and the restrictions they impose on the reliable use of ScaLAPACK. Section 6.3 discusses how to measure errors, as well as some standard notation. Section 6.4 discusses further details of how error bounds are derived. Sections 6.5 through 6.9 present error bounds for linear equations, linear least squares problems, the symmetric eigenproblem, the singular value decomposition, and the generalized symmetric definite eigenproblem, respectively.


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Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Sources of Error in Numerical Calculations

 

          The effects of two sources of error can be measured by the bounds in this chapter: roundoff error and input error. Roundoff error arises from rounding results of floating-point operations during the algorithm. Input error is error in the input to the algorithm from prior calculations or measurements. We describe roundoff error first, and then input error.

Almost all the error bounds ScaLAPACK provides are multiples of relative machine precision,     which we abbreviate by tex2html_wrap_inline17202. Relative machine precision (epsilon) bounds the roundoff in individual floating-point operations. It may be loosely defined as the largest relative error    in any floating-point operation that neither overflows nor underflows. (Overflow means the result is too large to represent accurately, and underflow means the result is too small to represent accurately.) Relative machine precision (epsilon) is available either by the function call   PSLAMCH(ICTXT, 'Epsilon') (or simply PSLAMCH(ICTXT, 'E'))gif in single precision, or by the function call PDLAMCH(ICTXT, 'Epsilon') (or PDLAMCH(ICTXT, 'E')) in double precision. See section 6.1 and Table 6.1 for a discussion of common values of machine epsilon.      

PDLAMCH(ICTXT,`E') returns a single value for the selected machine parameter `E' on all processes within the context ICTXT. If these processes are running on a network of heterogeneous processors , with different floating-point arithmetics, then a ``safe'' common value is returned, the maximum value of machine epsilon for all the processors.

In case of overflow,   there are two common system responses: stopping with an error message, or returning tex2html_wrap_inline17191 and continuing to compute. The latter is the default response of IEEE standard floating-point arithmetic [7, 8]  , the most commonly used arithmetic. It is possible to change this default to abort with an error message, which is often useful for debugging.

In contrast to LAPACK, ScaLAPACK can take advantage of arithmetic with tex2html_wrap_inline17191   to accelerate the routines that compute eigenvalues of symmetric matrices using PxLAIECT (the drivers PxSYEVX and PxSYGVX, and their complex counterparts).           PxLAIECT comes in two different versions, one in which arithmetic with tex2html_wrap_inline17191 is available (the default) and one in which it is not. When tex2html_wrap_inline17191 is available, the inner loop of PxLAIECT is accelerated by removing a branch to test for and avoid division by zero. This speed advantage is realized only when arithmetic with tex2html_wrap_inline17191 is as fast as arithmetic with normalized floating-point numbers; this is usually but not always the case [42]. The compile time flag NO_IEEE can be used during installation to run without using tex2html_wrap_inline17191; see the ScaLAPACK Installation Guide for details [24].

Since underflow is almost always less significant than roundoff, we will not consider it further in this section (but see section 6.1).

Bounds on input errors may be easily incorporated into most ScaLAPACK error bounds. Suppose the input data is accurate to, say, five decimal digits (we discuss exactly what this means in section 6.3). Then one simply replaces tex2html_wrap_inline17202 by tex2html_wrap_inline17220 in the error bounds.

Further Details: Floating-Point Arithmetic 

    Roundoff error is bounded in terms of the relative machine precision tex2html_wrap_inline17202,   which is the smallest value satisfying
displaymath17196
where a and b are floating-point numbers , tex2html_wrap_inline17228 is any one of the four operations +, -, tex2html_wrap_inline12420, and tex2html_wrap_inline17236, and tex2html_wrap_inline17238 is the floating-point result of tex2html_wrap_inline17240. Relative machine precision, tex2html_wrap_inline17202, is the smallest value for which this inequality is true for all tex2html_wrap_inline17228, and for all a and b such that tex2html_wrap_inline17240 is neither too large (magnitude exceeds the overflow threshold)   nor too small (is nonzero with magnitude less than the underflow threshold)   to be represented accurately in the machine. We also assume tex2html_wrap_inline17202 bounds the relative error in unary   operations such as square root:
displaymath17197

A precise characterization of tex2html_wrap_inline17202 depends on the details of the machine arithmetic and sometimes even of the compiler. For example, if addition and subtraction are implemented without a guard digit,gif we must redefine tex2html_wrap_inline17202 to be the smallest number such that
displaymath17198

In order to assure portability , machine parameters such as relative machine precision (epsilon), the overflow threshold and underflow threshold are computed at runtime by the auxiliary    routine PxLAMCH. The alternative, keeping a fixed table of machine parameter values, would degrade portability because the table would have to be changed when moving from one machine or combination of machines, or even one compiler, to another.

Most machines (but not yet all) do have the same machine parameters because they implement IEEE Standard Floating Point Arithmetic   [7, 8], which exactly specifies floating-point number representations and operations. For these machines, including all modern workstations and PCs,gif the values of these parameters are given in Table 6.1.

Unfortunately, machines claiming to implement IEEE arithmetic may still compute different results from the same program and input. Here are some examples. Intel processors have 80-bit floating-point registers, and the fastest way to use them is to evaluate all results to 80-bit accuracy until they are stored back to memory in 32-bit or 64-bit format. The IBM RS/6000 has a fused multiply-add instruction that evaluates a+b*c with one rounding error instead of two. The DEC Alpha's default (fast) mode is to flush underflowed values to zero instead of returning subnormal numbers, which is the default demanded by the IEEE standard;     in this mode the DEC Alpha aborts if it encounters a subnormal number. In all these cases machines may be made to operate absolutely identically, for example, by rounding all intermediate results back to single or double on an Intel machine, or by doing subnormal arithmetic carefully and slowly on a DEC Alpha. These heterogeneities lead to errors encountered only in parallel computing; see section 6.2 for further discussion.

  table4445
Table 6.1: Values of machine parameters in IEEE floating-point arithmetic

As stated above, we will ignore underflow in discussing error bounds. Reference [37] discusses extending error bounds to include underflow and shows that for many common computations, when underflow occurs, it is less significant than roundoff.

Overflow historically resulted in an error message and stopped execution, in which case our error bounds would not apply. But with IEEE floating-point arithmetic,   the default is that overflow returns tex2html_wrap_inline17191 and execution continues. Indeed, with IEEE arithmetic machines can continue to compute past overflows, even division by zero, square roots of negative numbers, etc., by producing tex2html_wrap_inline17191 and NaN (``Not a Number'') symbols according to special rules of arithmetic. The default on many systems is to continue computing with these symbols. Routine PxLAIECT exploits this arithmetic to accelerate the computations of eigenvalues, as discussed above. It is also possible to stop with an error message when overflow occurs, a feature that is often useful for debugging. The user should consult the system manual to see how to turn error messages on or off.                            

Most of our error bounds will simply be proportional to relative machine precision (epsilon). This means, for example, that if the same problem in solved in double precision and single precision, the error bound in double precision will be smaller than the error bound in single precision by a factor of tex2html_wrap_inline17278. In IEEE arithmetic, this ratio is tex2html_wrap_inline17280, meaning that one expects the double-precision answer to have approximately nine more decimal digits correct than the single-precision answer.

Like their counterparts in LAPACK. ScaLAPACK routines are generally insensitive to the details of rounding, provided all processes perform arithmetic identically. The one exception is PxLAIECT, as mentioned above. The next section discusses what can happen when processes do not perform arithmetic identically, that is, are heterogeneous.

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Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node134.html0100644000056400000620000002426406336113753017535 0ustar pfrauenfstaff New Sources of Error in Parallel Numerical Computations next up previous contents index
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New Sources of Error in Parallel Numerical Computations

 

An important difference between ScaLAPACK and LAPACK is that a parallel computing environment, possibly consisting of a heterogeneous collection of processors, introduces new sources of possible errors not found in the serial environment in which LAPACK runs. These errors could indeed afflict any parallel algorithm that uses floating-point arithmetic. For example, consider the following pseudocode, executed in parallel by several processors: 

 
   		 s= global_sum(x)  ...  each processor receives the sum s of global array x


   		 if s < thresh then


   		  		 return my part of answer 1


   		  else


   		  		 do more computations


   		  		 return my part of answer 2


   		 end if

It is possible for the value of s to differ from processor to processor; we call this incoherence.  This can happen if the floating-point arithmetic varies from processor to processor (we call this heterogeneity),  since processors may not even share the same set of floating-point numbers. The value of s can also vary if global_sum accumulates the sum in different orders on different processors, since floating-point addition is not associative. In either case, the test s< thresh may be true on one processor but not another, so that the program may inconsistently return answer 1 on some processors and answer 2 on others. If the ``more computations'' include communication with synchronization, even deadlock could result.   

Deadlock can also result if the floating-point numbers communicated from one processor to another cause fatal floating-point errors on the receiving processor. For example, if an IBM RS/6000, running in its default mode, sends a message containing a denormalized number [7, 8]   to a DEC Alpha running in its default mode, then the DEC Alpha aborts [19].gif

It is also possible for global_sum to compute the same s on all processors but compute a different s from run to run of the program, for example, if global_sum computes the sum in a nondeterministic order on one processor and broadcasts the result to all processors. We call this nonrepeatability.  If this happens, debugging the overall code can be more difficult.

Coherence   and repeatability   are independent properties of an algorithm. It is possible in principle for an algorithm running on a particular platform to be incoherent and repeatable, coherent and nonrepeatable, or any other combination. On a different platform, the same algorithm may have different properties.

Reference [19] contains a more extensive discussion of these possible errors.

One run of a ScaLAPACK routine is designed to be as reliable  as LAPACK, so that errors due to incoherence cannot occur as long as ScaLAPACK is executed on a homogeneous network  of processors. The following conditions apply:

  • The processors are completely identical. This also means that relevant flags, like those controlling the way overflow and underflow are handled in IEEE floating-point arithmetic,   must be identical.
  • The communication library used by the BLACS may only ``copy bits'' and not modify any floating-point numbers (by translation to a different internal floating-point format, as XDR [111] may do).  
  • The identical ScaLAPACK object code must be executed by each processor.

The above conditions guarantee that a single ScaLAPACK call is as reliable  as its LAPACK counterpart. If, in addition, identical answers from one run to another are desired (i.e., repeatability),  this can be guaranteed at runtime by calling BLACS_SET  to enforce repeatability  of the BLACS, and the ScaLAPACK routines that use them, by using an appropriate topology (see the BLACS users guide [54] for details).

Maintaining coherence  on a heterogeneous network  is harder, and not always possible. If floating-point formats differ (say, on a Cray C90 and IBM RS/6000, which uses IEEE arithmetic), there is no cost-effective way to guarantee coherence. If floating-point formats are the same, however, operations such as global sums can accumulate the result on one processor and broadcast it to guarantee coherence (except for the problem of DEC Alphas and denormalized numbers mentioned above). The BLACS do this, except when using the ``bidirectional exchange'' topology. One can avoid using ``bidirectional exchange'' and so guarantee coherence whenever possible, by calling BLACS_SET  to enforce coherence  (see the BLACS users guide [54] for details).

Still other ScaLAPACK routines are guaranteed to work only on homogeneous networks (PxGESVD and PxSYEV). These routines do large numbers of redundant calculations on all processors and depend on the results of these calculations being the same. There are too many of these calculations to cost-effectively compute them all on one processor and broadcast the results.

The user may wonder why ScaLAPACK and the BLACS are not designed to guarantee coherence and repeatability in the most general possible situations, so that calling BLACS_SET would not be necessary. The reason is that the possible bugs described above are quite rare, and so ScaLAPACK and the BLACS were designed to maximize performance instead. Provided the mere sending of floating-point numbers does not cause a fatal error, these bugs cannot occur at all in most ScaLAPACK routines, because branches depending on a supposedly identical floating-point value like s do not occur. For most other ScaLAPACK routines where such branches do occur, we have not seen these bugs despite extensive testing, including attempts to cause them to occur. Complete understanding and cost-effective elimination of such possible bugs are future work.

In the meantime, to get repeatability when running on a homogeneous network, we recommend calling BLACS_SET  as described above when using the following ScaLAPACK drivers: PxGESVX, PxPOSVX, PxSYEV, PxSYEVX, PxGESVD, and PxSYGVX.                          

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Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node135.html0100644000056400000620000007021306336113754017532 0ustar pfrauenfstaff How to Measure Errors next up previous contents index
Next: Further Details: How Error Up: Accuracy and Stability Previous: New Sources of Error

How to Measure Errors

 

  ScaLAPACK routines return four types of floating-point output arguments:

     
  • Scalar, such as an eigenvalue of a matrix,  
  • Vector, such as the solution x of a linear system Ax=b,  
  • Matrix, such as a matrix inverse tex2html_wrap_inline13063, and  
  • Subspace, such as the space spanned by one or more eigenvectors of a matrix.
This section provides measures for errors in these quantities, which we need in order to express error bounds.

  First, consider scalars. Let the scalar tex2html_wrap_inline17436 be an approximation of the true answer tex2html_wrap_inline16235. We can measure the difference between tex2html_wrap_inline16235 and tex2html_wrap_inline17436 either by the absolute error tex2html_wrap_inline17444, or, if tex2html_wrap_inline16235 is nonzero, by the relative error tex2html_wrap_inline17448. Alternatively, it is sometimes more convenient to use tex2html_wrap_inline17450 instead of the standard expression for relative error. If the relative error of tex2html_wrap_inline17436 is, say, tex2html_wrap_inline17454, we say that tex2html_wrap_inline17436 is accurate to 5 decimal digits.      

  To measure the error in vectors, we need to measure the size or norm of a vector x . A popular norm is the magnitude of the largest component, tex2html_wrap_inline17460, which we denote by tex2html_wrap_inline17462. This is read the infinity norm of x. See Table 6.2 for a summary of norms.

  table4627
Table 6.2: Vector and matrix norms

If tex2html_wrap_inline17482 is an approximation to the exact vector x, we will refer to tex2html_wrap_inline17486 as the absolute error in tex2html_wrap_inline17482 (where p is one of the values in Table 6.2)       and refer to tex2html_wrap_inline17492 as the relative error in tex2html_wrap_inline17482 (assuming tex2html_wrap_inline17496). As with scalars, we will sometimes use tex2html_wrap_inline17498 for the relative error. As above, if the relative error of tex2html_wrap_inline17482 is, say tex2html_wrap_inline17454, we say that tex2html_wrap_inline17482 is accurate to 5 decimal digits. The following example illustrates these ideas.


displaymath17410

eqnarray4677
Thus, we would say that tex2html_wrap_inline17482 approximates x to 2 decimal digits.

  Errors in matrices may also be measured with norms . The most obvious generalization of tex2html_wrap_inline17462 to matrices would appear to be tex2html_wrap_inline17512, but this does not have certain important mathematical properties that make deriving error bounds convenient. Instead, we will use tex2html_wrap_inline17514, where A is an m-by-n matrix, or tex2html_wrap_inline17522; see Table 6.2 for other matrix norms. As before, tex2html_wrap_inline17524 is the absolute error   in tex2html_wrap_inline17526, tex2html_wrap_inline17528 is the relative error   in tex2html_wrap_inline17526, and a relative error in tex2html_wrap_inline17526 of tex2html_wrap_inline17454 means tex2html_wrap_inline17526 is accurate to 5 decimal digits. The following example illustrates these ideas.
displaymath17411

eqnarray4739
so tex2html_wrap_inline17526 is accurate to 1 decimal digit.

We now introduce some related notation we will use in our error bounds. The condition number of a matrix A is defined as   tex2html_wrap_inline17542, where A is square and invertible, and p is tex2html_wrap_inline17548 or one of the other possibilities in Table 6.2. The condition number measures how sensitive tex2html_wrap_inline13063 is to changes in A; the larger the condition number, the more sensitive is tex2html_wrap_inline13063. For example, for the same A as in the last example,
displaymath17412
ScaLAPACK error estimation routines typically compute a variable called RCOND , which is the reciprocal of the condition number (or an approximation of the reciprocal). The reciprocal of the condition number is used instead of the condition number itself in order to avoid the possibility of overflow when the condition number is very large.     Also, some of our error bounds will use the vector of absolute values of x, |x| (tex2html_wrap_inline17562), or similarly |A| (tex2html_wrap_inline17566).

  Now we consider errors in subspaces. Subspaces are the outputs of routines that compute eigenvectors and invariant subspaces of matrices. We need a careful definition of error in these cases for the following reason. The nonzero vector x is called a (right) eigenvector of the matrix A with eigenvalue tex2html_wrap_inline12778 if tex2html_wrap_inline17574. From this definition, we see that -x, 2x, or any other nonzero multiple tex2html_wrap_inline17580 of x is also an eigenvector. In other words, eigenvectors are not unique. This means we cannot measure the difference between two supposed eigenvectors tex2html_wrap_inline17482 and x by computing tex2html_wrap_inline17588, because this may be large while tex2html_wrap_inline17590 is small or even zero for some tex2html_wrap_inline17592. This is true even if we normalize x so that tex2html_wrap_inline17596, since both x and -x can be normalized simultaneously. Hence, to define error in a useful way, we need instead to consider the set tex2html_wrap_inline17602 of all scalar multiples tex2html_wrap_inline17604 of x. The set tex2html_wrap_inline17602 is called the subspace spanned by x and is uniquely determined by any nonzero member of tex2html_wrap_inline17602. We will measure the difference between two such sets by the acute angle between them. Suppose tex2html_wrap_inline17614 is spanned by tex2html_wrap_inline17616 and tex2html_wrap_inline17602 is spanned by tex2html_wrap_inline17620. Then the acute angle between tex2html_wrap_inline17614 and tex2html_wrap_inline17602 is defined as    
displaymath17413
One can show that tex2html_wrap_inline17626 does not change when either tex2html_wrap_inline17482 or x is multiplied by any nonzero scalar. For example, if
displaymath17414
as above, then tex2html_wrap_inline17632 for any nonzero scalars tex2html_wrap_inline14473 and tex2html_wrap_inline17636.

Let us consider another way to interpret the angle tex2html_wrap_inline17638 between tex2html_wrap_inline17614 and tex2html_wrap_inline17602.     Suppose tex2html_wrap_inline17482 is a unit vector (tex2html_wrap_inline17646). Then there is a scalar tex2html_wrap_inline14473 such that
displaymath17415
The approximation tex2html_wrap_inline17650 holds when tex2html_wrap_inline17638 is much less than 1 (less than .1 will do nicely). If tex2html_wrap_inline17482 is an approximate eigenvector with error bound tex2html_wrap_inline17656, where x is a true eigenvector, there is another true eigenvector tex2html_wrap_inline17580 satisfying tex2html_wrap_inline17662. For example, if
displaymath17416
then tex2html_wrap_inline17664 for tex2html_wrap_inline17666.

Finally, many of our error bounds will contain a factor p(n) (or p(m,n)), which grows as a function of matrix dimension n (or dimensions m and n). It represents a potentially different function for each problem. In practice, the true errors usually grow at most linearly; using p(n)=1 in the error bound formulas will often give a reasonable estimate; p(n)=n is more conservative. Therefore, we will refer to p(n) as a ``modestly growing'' function of n; however. it can occasionally be much larger. For simplicity, the error bounds computed by the code fragments in the following sections will use p(n)=1. This means these computed error bounds may occasionally slightly underestimate the true error. For this reason we refer to these computed error bounds as ``approximate error bounds.''

Further Details: How to Measure Errors 

   The relative error tex2html_wrap_inline17448 in the approximation tex2html_wrap_inline17436 of the true solution tex2html_wrap_inline16235 has a drawback: it often cannot be computed directly, because it depends on the unknown quantity tex2html_wrap_inline17694. However, we can often instead estimate tex2html_wrap_inline17450, since tex2html_wrap_inline17436 is known (it is the output of our algorithm). Fortunately, these two quantities are necessarily close together, provided either one is small, which is the only time they provide a useful bound anyway. For example, tex2html_wrap_inline17700 implies
displaymath17417
so they can be used interchangeably.

Table 6.2 contains a variety of norms we will use to measure errors. These norms have the properties that tex2html_wrap_inline17702, and tex2html_wrap_inline17704, where p is one of 1, 2, tex2html_wrap_inline17548, and F. These properties are useful for deriving error bounds.

An error bound that uses a given norm may be changed into an error bound that uses another norm. This is accomplished by multiplying the first error bound by an appropriate function of the problem dimension. Table 6.3 gives the factors tex2html_wrap_inline17716 such that tex2html_wrap_inline17718, where n is the dimension of x.

  table4862
Table 6.3: Bounding one vector norm in terms of another

Table 6.4 gives the factors tex2html_wrap_inline17746 such that tex2html_wrap_inline17748, where A is m-by-n.

  table4892
Table 6.4: Bounding one matrix norm in terms of another

The two-norm of A, tex2html_wrap_inline17802, is also called the spectral norm of A and is equal to the largest singular value tex2html_wrap_inline17806 of A. We shall also need to refer to the smallest singular value tex2html_wrap_inline17810 of A; its value can be defined in a similar way to the definition of the two-norm in Table 6.2, namely, as tex2html_wrap_inline17814 when A has at least as many rows as columns, and defined as tex2html_wrap_inline17818 when A has more columns than rows. The two-norm, Frobenius norm  , and singular values of a matrix do not change if the matrix is multiplied by a real orthogonal (or complex unitary) matrix.

Now we define subspaces spanned by more than one vector, and angles between subspaces.     Given a set of k n-dimensional vectors tex2html_wrap_inline17826, they determine a subspace tex2html_wrap_inline17602 consisting of all their possible linear combinations tex2html_wrap_inline17830, tex2html_wrap_inline17832 scalars tex2html_wrap_inline17834. We also say that tex2html_wrap_inline17826 spans tex2html_wrap_inline17602. The difficulty in measuring the difference between subspaces is that the sets of vectors spanning them are not unique. For example, tex2html_wrap_inline17620, tex2html_wrap_inline17842 and tex2html_wrap_inline17844 all determine the same subspace. Therefore, we cannot simply compare the subspaces spanned by tex2html_wrap_inline17846 and tex2html_wrap_inline17826 by comparing each tex2html_wrap_inline17850 to tex2html_wrap_inline17852. Instead, we will measure the angle between the subspaces, which is independent of the spanning set of vectors. Suppose subspace tex2html_wrap_inline17614 is spanned by tex2html_wrap_inline17846 and that subspace tex2html_wrap_inline17602 is spanned by tex2html_wrap_inline17826. If k=1, we instead write more simply tex2html_wrap_inline17616 and tex2html_wrap_inline17620. When k=1, we define the angle tex2html_wrap_inline17870 between tex2html_wrap_inline17614 and tex2html_wrap_inline17602 as the acute angle between tex2html_wrap_inline17482 and x. When k>1, we define the acute angle between tex2html_wrap_inline17614 and tex2html_wrap_inline17602 as the largest acute angle between any vector tex2html_wrap_inline17482 in tex2html_wrap_inline17614 and the closest vector x in tex2html_wrap_inline17602 to tex2html_wrap_inline17482:


eqnarray4963
ScaLAPACK routines that compute subspaces return vectors tex2html_wrap_inline17846 spanning a subspace tex2html_wrap_inline17614 that are orthonormal. This means the n-by-k matrix tex2html_wrap_inline17904 satisfies tex2html_wrap_inline17906. Suppose also that the vectors tex2html_wrap_inline17826 spanning tex2html_wrap_inline17602 are orthonormal, so tex2html_wrap_inline17912 also satisfies tex2html_wrap_inline17914. Then there is a simple expression for the angle between tex2html_wrap_inline17614 and tex2html_wrap_inline17602:    
displaymath17418
For example, if
displaymath17419
then tex2html_wrap_inline17920.

As stated above, all our bounds will contain a factor p(n) (or p(m,n)), which measures how roundoff errors can grow as a function of matrix dimension n (or m and m). In practice, the true error usually grows just linearly with n, or even slower, but we can generally prove only much weaker bounds of the form tex2html_wrap_inline17934. This is because we cannot rule out the extremely unlikely possibility of rounding errors all adding together instead of canceling on average. Using tex2html_wrap_inline17934 would give pessimistic and unrealistic bounds, especially for large n, so we content ourselves with describing p(n) as a ``modestly growing'' polynomial function of n. Using p(n)=1 in the error bound formulas will often give a reasonable error estimate. For detailed derivations of various p(n), see [71, 114, 84, 38].

There is also one situation where p(n) can grow as large as tex2html_wrap_inline17950: Gaussian elimination. This typically occurs only on specially constructed matrices presented in numerical analysis courses [114, p. 212,]. However, the expert driver for solving linear systems, PxGESVX    , provides error bounds incorporating p(n), and so this rare possibility can be detected.

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Next: Further Details: How Error Up: Accuracy and Stability Previous: New Sources of Error

Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node136.html0100644000056400000620000000403206336113755017530 0ustar pfrauenfstaff Further Details: How Error Bounds Are Derived next up previous contents index
Next: Standard Error Analysis Up: Accuracy and Stability Previous: How to Measure Errors

Further Details: How Error Bounds Are Derived

 



Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node137.html0100644000056400000620000002300306336113756017531 0ustar pfrauenfstaff Standard Error Analysis next up previous contents index
Next: Improved Error Bounds Up: Further Details: How Error Previous: Further Details: How Error

Standard Error Analysis

 

  We illustrate standard error analysis with the simple example of evaluating the scalar function y=f(z). Let the output of the subroutine that implements f(z) be denoted tex2html_wrap_inline18005; this includes the effects of roundoff. If tex2html_wrap_inline18007, where tex2html_wrap_inline18009 is small, we say that tex2html_wrap_inline18011 is a backward stable     algorithm for f, or that the backward error tex2html_wrap_inline18009 is small.     In other words, tex2html_wrap_inline18005 is the exact value of f at a slightly perturbed input tex2html_wrap_inline18021.gif

Suppose now that f is a smooth function, so that we may approximate it near z by a straight line: tex2html_wrap_inline18033. Then we have the simple error estimate
displaymath17997
Thus, if tex2html_wrap_inline18009 is small and the derivative f'(z) is moderate, the error tex2html_wrap_inline18039 will be small.gif This is often written in the similar form
displaymath17998
This approximately bounds the relative error    tex2html_wrap_inline18045 by the product of the condition number of f at z, tex2html_wrap_inline18051, and the relative backward error tex2html_wrap_inline18053.     Thus we get an error bound by multiplying a condition  number and a backward error (or bounds for these quantities). We call a problem ill-conditioned  if its condition number is large, and ill-posed  if its condition number is infinite (or does not exist).gif

If f and z are vector quantities, then f'(z) is a matrix (the Jacobian). Hence, instead of using absolute values as before, we now measure tex2html_wrap_inline18009 by a vector norm tex2html_wrap_inline18065 and f'(z) by a matrix norm tex2html_wrap_inline18069. The conventional (and coarsest) error analysis uses a norm such as the infinity norm. We therefore call this normwise backward stability.     For example, a normwise stable method for solving a system of linear equations Ax=b will produce a solution tex2html_wrap_inline17482 satisfying tex2html_wrap_inline18075, where tex2html_wrap_inline18077 and tex2html_wrap_inline18079 are both small (close to machine epsilon). In this case the condition number is tex2html_wrap_inline18081 (see section 6.5).  

Almost all of the algorithms in ScaLAPACK (as well as LAPACK) are stable in the sense just describedgif: when applied to a matrix A they produce the exact result for a slightly different matrix A+E, where tex2html_wrap_inline18077 is of order tex2html_wrap_inline17202.

Condition numbers may be expensive to compute exactly. For example, it costs about tex2html_wrap_inline18115 operations to solve Ax=b for a general matrix A, and computing tex2html_wrap_inline18121 exactly costs an additional tex2html_wrap_inline18123 operations, or twice as much. But tex2html_wrap_inline18121 can be estimated in only tex2html_wrap_inline18127 operations beyond those tex2html_wrap_inline18115 necessary for solution, a tiny extra cost. Therefore, most of ScaLAPACK's condition numbers and error bounds are based on estimated condition numbers , using the method of [72, 80, 81].

The price one pays for using an estimated rather than an exact condition number is occasional (but very rare) underestimates of the true error; years of experience attest to the reliability of our estimators, although examples where they badly underestimate the error can be constructed [82]. Note that once a condition estimate is large enough (usually tex2html_wrap_inline18131), it confirms that the computed answer may be completely inaccurate, and so the exact magnitude of the condition estimate conveys little information.

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Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: Error Bounds for Linear Up: Further Details: How Error Previous: Standard Error Analysis

Improved Error Bounds

 

The standard error analysis just outlined has a drawback: by using the infinity norm tex2html_wrap_inline18154 to measure the backward error, entries of equal magnitude in tex2html_wrap_inline18009 contribute equally to the final error bound tex2html_wrap_inline18158. This means that if z is sparse or has some tiny entries, a normwise backward stable algorithm may make large changes in these entries compared wit their original values. If these tiny values are known accurately by the user, these errors may be unacceptable, or the error bounds may be unacceptably large.

For example, consider solving a diagonal system of linear equations Ax=b. Each component of the solution is computed accurately by Gaussian elimination: tex2html_wrap_inline18164. The usual error bound is approximately tex2html_wrap_inline18166, which can arbitrarily overestimate the true error, tex2html_wrap_inline17202, if at least one tex2html_wrap_inline18170 is tiny and another one is large.

LAPACK addresses this inadequacy by providing some algorithms whose backward error tex2html_wrap_inline18009 is a tiny relative change in each component of z: tex2html_wrap_inline18176. This backward error retains both the sparsity structure of z as well as the information in tiny entries. These algorithms are therefore called componentwise relatively backward stable. Furthermore, computed error bounds reflect this stronger form of backward error.gif          

If the input data has independent uncertainty in each component, each component must have at least a small relative uncertainty, since each is a floating-point number. In this case, the extra uncertainty contributed by the algorithm is not much worse than the uncertainty in the input data, so one could say the answer provided by a componentwise relatively backward stable algorithm is as accurate as the data warrants [4].

When solving Ax=b using expert driver PxyySVX or computational routine PxyyRFS, for example, we almost always compute tex2html_wrap_inline17482 satisfying tex2html_wrap_inline18075, where tex2html_wrap_inline18186 is a small relative change in tex2html_wrap_inline18188 and tex2html_wrap_inline18190 is a small relative change in tex2html_wrap_inline18192. In particular, if A is diagonal, the corresponding error bound is always tiny, as one would expect (see the next section).

ScaLAPACK can achieve this accuracy   for linear equation solving, the bidiagonal singular value decomposition, and the symmetric tridiagonal eigenproblem and provides facilities for achieving this accuracy for least squares problems.

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Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node139.html0100644000056400000620000004176006336113760017540 0ustar pfrauenfstaff Error Bounds for Linear Equation Solving next up previous contents index
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Error Bounds for Linear Equation Solving

 

Let Ax=b be the system to be solved, and tex2html_wrap_inline17482 the computed solution. Let n be the dimension of A. An approximate error bound  for tex2html_wrap_inline17482 may be obtained in one of the following two ways, depending on whether the solution is computed by a simple driver or an expert driver:

  1. Suppose that Ax=b is solved using the simple driver PSGESV    (section 3.2.1). Then the approximate error boundgif


    displaymath18202
    can be computed by the following code fragment. 

          EPSMCH = PSLAMCH( ICTXT, 'E' )
*     Get infinity-norm of A
      ANORM = PSLANGE( 'I', N, N, A, IA, JA, DESCA, WORK )
*     Solve system; The solution X overwrites B
      CALL PSGESV( N, 1, A, IA, JA, DESCA, IPIV, B, IB, JB, DESCB, INFO )
      IF( INFO.GT.0 ) THEN
         PRINT *,'Singular Matrix'
      ELSE IF( N.GT.0 ) THEN
*        Get reciprocal condition number RCOND of A
         CALL PSGECON( 'I', N, A, IA, JA, DESCA, ANORM, RCOND, WORK,
     $                 LWORK, IWORK, LIWORK, INFO )
         RCOND = MAX( RCOND, EPSMCH )
         ERRBD = EPSMCH / RCOND
      END IF
     

    For example, suppose

    tex2html_wrap_inline18242,
    displaymath18203
    Then (to 4 decimal places)
    displaymath18204
    tex2html_wrap_inline18244, tex2html_wrap_inline18246, the true reciprocal condition number tex2html_wrap_inline18248, tex2html_wrap_inline18250, and the true error tex2html_wrap_inline18252.  

  2. Suppose that Ax=b is solved using the expert driver PSGESVX (section 3.2.1).    This routine provides an explicit error bound FERR, measured with the infinity-norm:  
    displaymath18205
    For example, the following code fragment solves Ax=b and computes an approximate error bound FERR: 

          CALL PSGESVX( 'E', 'N', N, 1, A, IA, JA, DESCA, AF, IAF, JAF,
     $              DESCAF, IPIV, EQUED, R, C, B, IB, JB, DESCB, X, IX,
     $              JX, DESCX, RCOND, FERR, BERR, WORK, LWORK, IWORK,
     $              LIWORK, INFO )
      IF( INFO.GT.0 ) PRINT *,'(Nearly) Singular Matrix'

    For the same A and b as above, tex2html_wrap_inline18258, tex2html_wrap_inline18260, and the actual error is tex2html_wrap_inline18262.

This example illustrates that the expert driver provides an error bound with less programming effort than the simple driver, and also that it may produce a significantly more accurate answer.

Similar code fragments, with obvious adaptations, may be used with all the driver routines PxPOSV and PxPOSVX      in Table 3.2. For example, if a symmetric positive definite or Hermitian positive definite system is solved by using the simple driver PxPOSV,      then PxLANSY or PxLANHE, respectively, must be used to compute ANORM, and PxPOCON     must be used to compute RCOND.

The drivers PxGBSV      (for solving general band matrices with partial pivoting), PxPBSV      (for solving positive definite band matrices) and PxPTSV      (for solving positive definite tridiagonal matrices), do not yet have the corresponding routines needed to compute error bounds, namely, PxLAnHE to compute ANORM and PxyyCON to compute RCOND.

The drivers PxDBSV      (for solving general band matrices) and PxDTSV      (for solving general tridiagonal matrices) do not pivot for numerical stability, and so may be faster but less accurate than their pivoting counterparts above. These routines may be used safely when any diagonal pivot sequence leads to a stable factorization; diagonally dominant matrices and symmetric positive definite matrices [71] have this property, for example.

Further Details: Error Bounds for Linear Equation Solving 

The conventional error analysis of linear equation  solving goes as follows. Let Ax=b be the system to be solved. Let tex2html_wrap_inline17482 be the solution computed by ScaLAPACK (or LAPACK) using any of their linear equation solvers. Let r be the residual tex2html_wrap_inline18270. In the absence of rounding error, r would be zero and tex2html_wrap_inline17482 would equal x; with rounding error, one can only say the following:

The normwise backward error of the computed solution tex2html_wrap_inline17482,     with respect to the infinity norm, is the pair E,f, which minimizes
displaymath18206
subject to the constraint tex2html_wrap_inline18075. The minimal value of tex2html_wrap_inline18284 is given by
displaymath18207
One can show that the computed solution tex2html_wrap_inline17482 satisfies tex2html_wrap_inline18288, where p(n) is a modestly growing function of n. The corresponding condition number is tex2html_wrap_inline18294.   The error tex2html_wrap_inline18296 is bounded by
displaymath18208
In the first code fragment in the preceding section, tex2html_wrap_inline18300, which is tex2html_wrap_inline18302 in the numerical example, is approximated by tex2html_wrap_inline18304. Approximations  of tex2html_wrap_inline18121 -- or, strictly speaking, its reciprocal RCOND -- are returned by computational routines PxyyCON (section 3.3.1) or driver routines                PxyySVX (section 3.2.1). The code fragment makes sure RCOND is at least tex2html_wrap_inline18308 EPSMCH to avoid overflow in computing ERRBD.   This limits ERRBD to a maximum of 1, which is no loss of generality because a relative error of 1 or more indicates the same thing:    a complete loss of accuracy.   Note that the value of RCOND returned by PxyySVX may apply to a linear system obtained from Ax=b by equilibration, namely, scaling the rows and columns of A in order to make the condition number smaller. This is the case in the second code fragment in the preceding section, where the program chose to scale the rows by the factors returned in tex2html_wrap_inline18314, resulting in tex2html_wrap_inline18316.

As stated in section 6.4.2, this approach does not respect the presence of zero or tiny entries in A. In contrast, the ScaLAPACK computational routines PxyyRFS (section 3.3.1) or driver routines PxyySVX (section 3.2.1) will (except in rare cases) compute a solution tex2html_wrap_inline17482 with the following properties:

The componentwise backward error of the computed solution tex2html_wrap_inline17482 is the pair E,f which minimizes    
displaymath18209
(where we interpret 0/0 as 0) subject to the constraint tex2html_wrap_inline18075. The minimal value of tex2html_wrap_inline18330 is given by
displaymath18210
One can show that for most problems the tex2html_wrap_inline17482 computed by PxyySVX satisfies tex2html_wrap_inline18334, where p(n) is a modestly growing function of n. In other words, tex2html_wrap_inline17482 is the exact solution of the perturbed problem tex2html_wrap_inline18075, where E and f are small relative perturbations in each entry of A and b, respectively. The corresponding condition number is tex2html_wrap_inline18352.   The error tex2html_wrap_inline18296 is bounded by
displaymath18211

The routines PxyyRFS and PxyySVX return tex2html_wrap_inline18356,                              which is called BERR  (for Backward ERRor), and a bound on the the actual error tex2html_wrap_inline18358, called FERR   (for Forward ERRor), as in the second code fragment in the last section. FERR is actually calculated by the following formula, which can be smaller than the bound tex2html_wrap_inline18360 given above:
displaymath18212
Here, tex2html_wrap_inline18362 is the computed value of the residual tex2html_wrap_inline18364, and the norm in the numerator is estimated by using the same estimation subroutine used for RCOND.

The value of BERR for the example in the preceding section is tex2html_wrap_inline18366.

Even in the rare cases where PxyyRFS fails to make BERR close to its minimum tex2html_wrap_inline17202, the error bound FERR may remain small. See [9] for details.

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Susan Blackford
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Next: PBLAS Up: Software Components Previous: LAPACK

BLAS

  The BLAS (Basic Linear Algebra Subprograms) [57, 59, 93] include subroutines for common linear algebra computations such as dot-products, matrix-vector multiplication, and matrix-matrix multiplication. As is well known, using matrix-matrix operations (in particular, matrix multiplication) tuned for a particular architecture can mask the effects of the memory hierarchy   (cache misses, TLB misses, etc.) and permit floating-point operations to be performed near peak speed of the machine. An important aim of the BLAS is to provide a portability layer for computation.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Error Bounds for Linear Least Squares Problems

 

The linear least squares problem is to find x that minimizes tex2html_wrap_inline18475. We discuss error bounds for the most common case where A is m-by-n with m > n, and A has full rank ; this is called an overdetermined least squares problem   (the following code fragments deal with m=n as well).

Let tex2html_wrap_inline17482 be the solution computed by the driver routine PxGELS (see section 3.2.2). An approximate error bound     
displaymath18463
may be computed in the following way: 

      EPSMCH = PSLAMCH( ICTXT, 'E' )
*     Get the 2-norm of the right hand side B
      BNORM = PSLANGE( 'F', M, 1, B, IB, JB, DESCB, WORK )
*     Solve the least squares problem; the solution X overwrites B
      CALL PSGELS( 'N', M, N, 1, A, IA, JA, DESCA, B, IB, JB, DESCB,
     $             WORK, LWORK, INFO )
      IF( MIN( M, N ).GT.0 ) THEN
*        Get the 2-norm of the residual A*X-B
         RNORM = PSLANGE( 'F', M-N, 1, B, IB+N, JB, DESCB, WORK ) 
*        Get the reciprocal condition number RCOND of A
         CALL PSTRCON( 'I', 'U', 'N', N, A, IA, JA, DESCA, RCOND, WORK,
     $                 LWORK, IWORK, LIWORK, INFO )
         RCOND = MAX( RCOND, EPSMCH )
         IF( BNORM.GT.0.0 ) THEN
            SINT = RNORM / BNORM
         ELSE
            SINT = 0.0
         END IF
         COST = MAX( SQRT( ( 1.0E0 - SINT )*( 1.0E0 + SINT ) ), EPSMCH )
         TANT = SINT / COST
         ERRBD = EPSMCH*( 2.0E0 / ( RCOND*COST ) + TANT / RCOND**2 )
      END IF
 

For example, if tex2html_wrap_inline18242,
displaymath18464
then, to four decimal places,
displaymath18465
tex2html_wrap_inline18495, tex2html_wrap_inline18497, tex2html_wrap_inline18499, tex2html_wrap_inline18501, and the true error is tex2html_wrap_inline18503.

Note that in the preceding code fragment, the routine PSLANGE was used to compute the two-norm of the right hand side matrix B and the residual tex2html_wrap_inline18507. This routine was chosen because the result of the computation (BNORM or RNORM, respectively) is automatically known on all process columns within the process grid. The routine PSNRM2 could have also been used to perform this calculation; however, the use of PSNRM2 in this example would have required an additional communication broadcast, because the resulting value of BNORM or RNORM, respectively, is known only within the process column owning B(:,JB).

Further Details: Error Bounds for Linear Least Squares Problems 

The conventional error analysis of linear least squares problems goes as follows . As above, let tex2html_wrap_inline17482 be the solution to minimizing tex2html_wrap_inline18475 computed by ScaLAPACK using the least squares driver PxGELS (see section 3.2.2). We discuss the most common case, where A is overdetermined  (i.e., has more rows than columns) and has full rank [71]:     

The computed solution tex2html_wrap_inline17482 has a small normwise backward error. In other words, tex2html_wrap_inline17482 minimizes tex2html_wrap_inline18519, where E and f satisfy    
displaymath18466
and p(n) is a modestly growing function of n. We take p(n)=1 in the code fragments above. Let tex2html_wrap_inline18531 (approximated by 1/RCOND in the above code fragments), tex2html_wrap_inline18533 (= RNORM above), and tex2html_wrap_inline18535 (SINT = RNORM / BNORM above). Here, tex2html_wrap_inline17638 is the acute angle between the vectors tex2html_wrap_inline18539 and b.     Then when tex2html_wrap_inline18543 is small, the error tex2html_wrap_inline18545 is bounded by
displaymath18467
where tex2html_wrap_inline18549 = COST and tex2html_wrap_inline18551 = TANT in the code fragments above.

We avoid overflow by making sure RCOND and COST are both at least tex2html_wrap_inline18308 EPSMCH, and by handling the case of a zero B matrix separately (BNORM = 0).    

tex2html_wrap_inline18531 may be computed directly from the singular values of A returned by PxGESVD. It may also be approximated by using PxTRCON following calls to PxGELS. PxTRCON estimates tex2html_wrap_inline18559 or tex2html_wrap_inline18561 instead of tex2html_wrap_inline18563, but these can differ from tex2html_wrap_inline18563 by at most a factor of n.          

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Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Error Bounds for the Symmetric Eigenproblem

 

The eigendecomposition  of an n-by-n real symmetric matrix is the factorization tex2html_wrap_inline13904 (tex2html_wrap_inline18618 in the complex Hermitian case), where Z is orthogonal (unitary) and tex2html_wrap_inline18622 is real and diagonal, with tex2html_wrap_inline18624. The tex2html_wrap_inline18626 are the eigenvalues  of A, and the columns tex2html_wrap_inline18630 of Z are the eigenvectors . This is also often written tex2html_wrap_inline18634. The eigendecomposition of a symmetric matrix is computed by the driver routines PxSYEV and PxSYEVX. The complex counterparts of these routines, which compute the eigendecomposition of complex Hermitian matrices, are the driver routines PxHEEV and PxHEEVX (see section 3.2.3).         

The approximate error bounds       for the computed eigenvalues tex2html_wrap_inline18636 are
displaymath18596
The approximate error bounds for the computed eigenvectors tex2html_wrap_inline18638, which bound the acute angles between the computed eigenvectors and true eigenvectors tex2html_wrap_inline18630, are    
displaymath18597
These bounds can be computed by the following code fragment: 

      EPSMCH = PSLAMCH( ICTXT, 'E' )
*     Compute eigenvalues and eigenvectors of A
*     The eigenvalues are returned in W
*     The eigenvector matrix Z overwrites A
      CALL PSSYEV( 'V', UPLO, N, A, IA, JA, DESCA, W, Z, IZ, JZ,
     $             DESCZ, WORK, LWORK, INFO )
      IF( INFO.GT.0 ) THEN
         PRINT *,'PSSYEV did not converge'
      ELSE IF( N.GT.0 ) THEN
*        Compute the norm of A
         ANORM = MAX( ABS( W( 1 ) ), ABS( W( N ) ) )
         EERRBD = EPSMCH * ANORM
*        Compute reciprocal condition numbers for eigenvectors
         CALL SDISNA( 'Eigenvectors', N, N, W, RCONDZ, INFO )
         DO 10 I = 1, N
            ZERRBD( I ) = EPSMCH * ( ANORM / RCONDZ( I ) )
10       CONTINUE
      END IF

For example, if tex2html_wrap_inline18242 and


displaymath18598
then the eigenvalues, approximate error bounds, and true errors are as follows.


tabular5436

Further Details: Error Bounds for the Symmetric Eigenproblem 

The usual error analysis of the symmetric  eigenproblem using ScaLAPACK driver PSSYEV   (see subsection 3.2.3) is as follows [101]:

The computed eigendecomposition tex2html_wrap_inline18680 is nearly the exact eigendecomposition of A+E, namely, tex2html_wrap_inline18684 is a true eigendecomposition so that tex2html_wrap_inline18686 is orthogonal, where tex2html_wrap_inline18688 and tex2html_wrap_inline18690. Here p(n) is a modestly growing function of n. We take p(n)=1 in the above code fragment. Each computed eigenvalue tex2html_wrap_inline18698 differs from a true tex2html_wrap_inline18626 by at most
displaymath18599
Thus, large eigenvalues (those near tex2html_wrap_inline18702) are computed to high relative accuracy,   while small ones may not be.  

The angular difference between the computed unit eigenvector tex2html_wrap_inline18704 and a true unit eigenvector tex2html_wrap_inline18630 satisfies the approximate bound  
displaymath18600
if tex2html_wrap_inline18543 is small enough. Here, tex2html_wrap_inline18712 is the absolute gap   between tex2html_wrap_inline18626 and the nearest other eigenvalue. Thus, if tex2html_wrap_inline18626 is close to other eigenvalues, its corresponding eigenvector tex2html_wrap_inline18630 may be inaccurate. The gaps may be easily computed from the array of computed eigenvalues by using subroutine SDISNA  . The gaps computed by SDISNA are ensured not to be so small as to cause overflow when used as divisors.    

Let tex2html_wrap_inline17614 be the invariant subspace spanned by a collection of eigenvectors tex2html_wrap_inline18722, where tex2html_wrap_inline18724 is a subset of the integers from 1 to n. Let tex2html_wrap_inline17602 be the corresponding true subspace. Then
displaymath18601
  where
displaymath18602
is the absolute gap between the eigenvalues in tex2html_wrap_inline18724 and the nearest other eigenvalue. Thus, a cluster  of close eigenvalues that is far away from any other eigenvalue may have a well-determined invariant subspace tex2html_wrap_inline17614 even if its individual eigenvectors are ill-conditioned.

A small possibility exists that PSSYEV will fail to achieve the above error bounds on a heterogeneous network of processors   for reasons discussed in section 6.2. On a homogeneous network, PSSYEV is as robust as the corresponding LAPACK routine SSYEV. A future release will attempt to detect heterogeneity and warn the user to use an alternative algorithm.

In contrast to LAPACK, where the same error analysis applied to the simple and expert drivers, the expert driver PSSYEVX      satisfies slightly weaker error bounds than PSSYEV. The bounds tex2html_wrap_inline18736 and tex2html_wrap_inline18738 continue to hold, but the computed eigenvectors tex2html_wrap_inline18638 are no longer guaranteed to be orthogonal to one another. The corresponding LAPACK routine SSYEVX tries to guarantee orthogonality by reorthogonalizing computed eigenvectors against one another provided their corresponding computed eigenvalues are close enough together: tex2html_wrap_inline18742, where the threshold tex2html_wrap_inline18744. If m eigenvalues lie in a cluster satisfying this closeness criterion, the SSYEVX requires tex2html_wrap_inline18748 serial time to execute. When m is a large fraction of n, this serial bottleneck is expensive and does not always improve orthogonality.

ScaLAPACK addresses this problem in two ways. First, it lets the user use more or less time and space to perform reorthogonalization, rather than have a fixed criterion. In particular, the user can set the threshold ORTOL used above, decreasing it to make reorthogonalization less frequent or increasing it to reorthogonalize more. Furthermore, since each processor computes a subset of the eigenvectors, ScaLAPACK permits reorthogonalization only with the local eigenvectors; that is, no communication is allowed. Hence, if a cluster of eigenvalues is small enough for the corresponding eigenvectors to fit on one processor, the same reorthogonalization will be done as in LAPACK. The user can supply more or less workspace to limit the size of a cluster on one processor. Hence, at one extreme, with a large cluster and lots of workspace, the algorithm will be essentially equivalent to SSYEVX. At the other extreme, with all small clusters or little workspace supplied, the algorithm will be perfectly load balanced and perform minimal communication to compute the eigenvectors.

The second way ScaLAPACK will deal with reorthogonalization is to introduce a new algorithm [103, 102, 44] that requires nearly no reorthogonalization to compute orthogonal eigenvectors in a fully parallel way. This algorithm will be introduced in future ScaLAPACK and LAPACK releases.

In the special case of a real symmetric tridiagonal matrix T, the eigenvalues can sometimes be computed much more accurately. PxSYEV (and the other symmetric eigenproblem drivers) computes the eigenvalues and eigenvectors of a dense symmetric matrix by first reducing it to tridiagonal form  T and then finding the eigenvalues and eigenvectors of T. Reduction of a dense matrix to tridiagonal form  T can introduce additional errors, so the following bounds for the tridiagonal case do not apply to the dense case.

The eigenvalues of the symmetric tridiagonal matrix T may be computed with small componentwise relative backward error     (tex2html_wrap_inline18764) by using subroutine PxSTEBZ (section 3.3.4).    To compute tighter error bounds for the computed eigenvalues tex2html_wrap_inline18698 we must make some assumptions about T. The bounds discussed here are from [15]. Suppose T is positive definite, and write T=DHD where tex2html_wrap_inline18774 and tex2html_wrap_inline18776. Then the computed eigenvalues tex2html_wrap_inline18698 can differ from true eigenvalues tex2html_wrap_inline18626 by
displaymath18603
where p(n) is a modestly growing function of n. Thus, if tex2html_wrap_inline18786 is moderate, each eigenvalue will be computed to high relative accuracy,   no matter how tiny it is.

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Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Error Bounds for the Singular Value Decomposition

 

The singular  value decomposition (SVD) of a real m-by-n matrix A is defined as follows. Let tex2html_wrap_inline14127. The SVD of A is tex2html_wrap_inline14119 (tex2html_wrap_inline14133 in the complex case), where U and V are orthogonal (unitary) matrices and tex2html_wrap_inline18852 is diagonal, with tex2html_wrap_inline18854. The tex2html_wrap_inline12826 are the singular values of A and the leading r columns tex2html_wrap_inline12840 of U and tex2html_wrap_inline12842 of V the left and right singular vectors, respectively. The SVD of a general matrix is computed by PxGESVD      (see subsection 3.2.3).

The approximate error bounds for the computed singular values tex2html_wrap_inline18870 are
displaymath18830
The approximate error bounds for the computed singular vectors tex2html_wrap_inline18872 and tex2html_wrap_inline18874, which bound the acute angles between the computed singular vectors and true singular vectors tex2html_wrap_inline12842 and tex2html_wrap_inline12840, are    
eqnarray5899
These bounds can be computing by the following code fragment:     

      EPSMCH = PSLAMCH( ICTXT, 'E' )
*     Compute singular value decomposition of A
*     The singular values are returned in S
*     The left singular vectors are returned in U
*     The transposed right singular vectors are returned in VT
      CALL PSGESVD( 'V', 'V', M, N, A, IA, JA, DESCA, S, U, IU, JU,
     $              DESCU, VT, IVT, JVT, DESCVT, WORK, LWORK, INFO )
      IF( INFO.GT.0 ) THEN
         PRINT *,'PSGESVD did not converge'
      ELSE IF( MIN( M, N ).GT.0 ) THEN
         SERRBD  = EPSMCH * S( 1 )
*        Compute reciprocal condition numbers for singular vectors
         CALL SDISNA( 'Left', M, N, S, RCONDU, INFO )
         CALL SDISNA( 'Right', M, N, S, RCONDV, INFO )
         DO 10 I = 1, MIN( M, N )
            VERRBD( I ) = EPSMCH*( S( 1 ) / RCONDV( I ) )
            UERRBD( I ) = EPSMCH*( S( 1 ) / RCONDU( I ) )
10       CONTINUE
      END IF

For example, if tex2html_wrap_inline18242 and
displaymath18831
then the singular values, approximate error bounds, and true errors are given below.


tabular5913


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Further Details: Error Bounds for the Singular Value Decomposition

 

The usual error analysis of the SVD algorithm  PxGESVD in ScaLAPACK (see subsection 3.2.3)is as follows [71]:

The SVD algorithm is backward stable.     This means that the computed SVD, tex2html_wrap_inline18949, is nearly the exact SVD of A+E where tex2html_wrap_inline18953, and p(m,n) is a modestly growing function of m and n. This means tex2html_wrap_inline18961 is the true SVD, so that tex2html_wrap_inline18963 and tex2html_wrap_inline18965 are both orthogonal, where tex2html_wrap_inline18967, and tex2html_wrap_inline18969. Each computed singular value tex2html_wrap_inline18971 differs from true tex2html_wrap_inline12826 by at most
displaymath18941
(we take p(m,n)=1 in the above code fragment). Thus, large singular values (those near tex2html_wrap_inline18977) are computed to high relative accuracy   and small ones may not be.    

The angular difference between the computed left singular vector tex2html_wrap_inline18979 and a true tex2html_wrap_inline12840 satisfies the approximate bound
displaymath18942
where tex2html_wrap_inline18985 is the absolute gap   between tex2html_wrap_inline12826 and the nearest other singular value (we take p(m,n)=1 in the above code fragment). Thus, if tex2html_wrap_inline12826 is close to other singular values, its corresponding singular vector tex2html_wrap_inline12840 may be inaccurate. When n < m, then tex2html_wrap_inline18997 must be redefined as tex2html_wrap_inline18999. The gaps may be easily computed from the array of computed singular values using function    SDISNA. The gaps computed by SDISNA are ensured not to be so small as to cause overflow when used as divisors.     The same bound applies to the computed right singular vector tex2html_wrap_inline19001 and a true vector tex2html_wrap_inline12842.

Let tex2html_wrap_inline17614 be the space spanned by a collection of computed left singular vectors tex2html_wrap_inline19007, where tex2html_wrap_inline18724 is a subset of the integers from 1 to n. Let tex2html_wrap_inline17602 be the corresponding true space. Then
displaymath18943
where
displaymath18944
is the absolute gap between the singular values in tex2html_wrap_inline18724 and the nearest other singular value. Thus, a cluster  of close singular values which is far away from any other singular value may have a well determined space tex2html_wrap_inline17614 even if its individual singular vectors are ill-conditioned. The same bound applies to a set of right singular vectors tex2html_wrap_inline19021gif.

There is a small possibility that PxGESVD will fail to achieve the above error bounds on a heterogeneous network of processors   for reasons discussed in section 6.2. On a homogeneous network, PxGESVD is as robust as the corresponding LAPACK routine xGESVD. A future release will attempt to detect heterogeneity and warn the user to use an alternative algorithm.

In the special case of bidiagonal matrices, the singular values and singular vectors may be computed much more accurately. A bidiagonal matrix B has nonzero entries only on the main diagonal and the diagonal immediately above it (or immediately below it). PxGESVD computes the SVD of a general      matrix by first reducing it to bidiagonal form B, and then calling xBDSQR (subsection 3.3.6) to compute the SVD of B. Reduction of a dense matrix to bidiagonal form B can introduce additional errors, so the following bounds for the bidiagonal case do not apply to the dense case. For the error analysis of xBDSQR, see the LAPACK manual.

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Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: Troubleshooting Up: Accuracy and Stability Previous: Further Details: Error Bounds

Error Bounds for the Generalized Symmetric Definite Eigenproblem

 

Three types of problems must be considered.   In all cases A and B are real symmetric (or complex Hermitian) and B is positive definite. These decompositions are computed for real symmetric matrices by the driver routines PxSYGVX (see section 3.2.4). These decompositions are computed for complex Hermitian matrices by the driver routines PxHEGVX (see subsection 3.2.4). In each of the following three decompositions, tex2html_wrap_inline12784 is real and diagonal with diagonal entries tex2html_wrap_inline19079, and the columns tex2html_wrap_inline18630 of Z are linearly independent vectors. The tex2html_wrap_inline18626 are called eigenvalues and the tex2html_wrap_inline18630 are eigenvectors.     

  1. tex2html_wrap_inline18087. The eigendecomposition may be written tex2html_wrap_inline12895 and tex2html_wrap_inline12903 (or tex2html_wrap_inline12913 and tex2html_wrap_inline12921 if A and B are complex). This may also be written tex2html_wrap_inline19103.
  2. tex2html_wrap_inline18089. The eigendecomposition may be written tex2html_wrap_inline19107 and tex2html_wrap_inline12903 (tex2html_wrap_inline19111 and tex2html_wrap_inline12921 if A and B are complex). This may also be written tex2html_wrap_inline19119.
  3. tex2html_wrap_inline18091. The eigendecomposition may be written tex2html_wrap_inline12895 and tex2html_wrap_inline12905 (tex2html_wrap_inline12913 and tex2html_wrap_inline12923 if A and B are complex). This may also be written tex2html_wrap_inline19135.

The approximate error bounds for the computed eigenvalues tex2html_wrap_inline18636 are
displaymath19053
The approximate error bounds   for the computed eigenvectors tex2html_wrap_inline18638, which bound the acute angles between the computed eigenvectors and true eigenvectors tex2html_wrap_inline18630, are    
displaymath18597
These bounds are computed differently, depending on which of the above three problems are to be solved. The following code fragments show how.

  1. First we consider error bounds for problem 1. 

          EPSMCH = PSLAMCH( ICTXT, 'E' )
      UNFL = PSLAMCH( ICTXT, 'U' )
*     Solve the eigenproblem A - lambda B (ITYPE = 1)
      ITYPE = 1
*     Compute the norms of A and B
      ANORM = PSLANSY( '1', UPLO, N, A, IA, JA, DESCA, WORK )
      BNORM = PSLANSY( '1', UPLO, N, B, IB, JB, DESCB, WORK )
*     The eigenvalues are returned in W
*     The eigenvectors are returned in A
      SUBROUTINE PSSYGVX( ITYPE, 'V', 'A', UPLO, N, A, IA, JA,
     $                    DESCA, B, IB, JB, DESCB, VL, VU, IL, IU,
     $                    UNFL, M, NZ, W, -1.0, Z, IZ, JZ, DESCZ,
     $                    WORK, LWORK, IWORK, LIWORK, IFAIL, ICLUSTR,
     $                    GAP, INFO )
      IF( INFO.GT.0 ) THEN
         PRINT *,'PSSYGVX did not converge, or B not positive definite'
      ELSE IF( N.GT.0 ) THEN
*        Get reciprocal condition number RCONDB of Cholesky factor of B
         CALL PSTRCON( '1', UPLO, 'N', N, B, IB, JB, DESCB, RCONDB,
     $                 WORK, LWORK, IWORK, LIWORK, INFO )
         RCONDB = MAX( RCONDB, EPSMCH )
         CALL SDISNA( 'Eigenvectors', N, N, W, RCONDZ, INFO )
         DO 10 I = 1, N
            EERRBD( I ) = ( EPSMCH / RCONDB**2 ) * ( ANORM / BNORM +
     $                      ABS( W( I ) ) )
            ZERRBD( I ) = ( EPSMCH / RCONDB**3 ) * ( ( ANORM / BNORM )
     $                     / RCONDZ( I ) + ( ABS( W( I ) ) /
     $                     RCONDZ( I ) ) * RCONDB )
10       CONTINUE
      END IF
     

    For example, if tex2html_wrap_inline18242,
    displaymath19055
    then tex2html_wrap_inline19145, tex2html_wrap_inline19147, and tex2html_wrap_inline19149, and the approximate eigenvalues, approximate error bounds, and true errors are shown below.


    tabular6044

  2. Problem types 2 and 3 have the same error bounds. We illustrate only type 2.     

          EPSMCH = PSLAMCH( ICTXT, 'E' )
*     Solve the eigenproblem A*B - lambda I (ITYPE = 2)
      ITYPE = 2
*     Compute the norms of A and B
      ANORM = PSLANSY( '1', UPLO, N, A, IA, JA, DESCA, WORK )
      BNORM = PSLANSY( '1', UPLO, N, B, IB, JB, DESCB, WORK )
*     The eigenvalues are returned in W
*     The eigenvectors are returned in A
      SUBROUTINE PSSYGVX( ITYPE, 'V', 'A', UPLO, N, A, IA, JA,
     $                    DESCA, B, IB, JB, DESCB, VL, VU, IL, IU,
     $                    UNFL, M, NZ, W, -1.0, Z, IZ, JZ, DESCZ,
     $                    WORK, LWORK, IWORK, LIWORK, IFAIL, ICLUSTR,
     $                    GAP, INFO )
      IF( INFO.GT.0 .AND. INFO.LE.N ) THEN
         PRINT *,'PSSYGVX did not converge'
      ELSE IF( INFO.GT.N ) THEN
         PRINT *,'B not positive definite'
      ELSE IF( N.GT.0 ) THEN
*        Get reciprocal condition number RCONDB of Cholesky factor of B
         CALL PSTRCON( '1', UPLO, 'N', N, B, IB, JB, DESCB, RCONDB,
     $                 WORK, LWORK, IWORK, LIWORK, INFO )
         RCONDB = MAX( RCONDB, EPSMCH )
         CALL SDISNA( 'Eigenvectors', N, N, W, RCONDZ, INFO )
         DO 10 I = 1, N
            EERRBD(I) = ( ANORM * BNORM ) * EPSMCH + 
     $                  ( EPSMCH / RCONDB**2 ) * ABS( W( I ) )
            ZERRBD(I) = ( EPSMCH / RCONDB ) * ( ( ANORM * BNORM ) / 
     $                    RCONDZ( I ) + 1.0 / RCONDB )
10       CONTINUE
      END IF
     

    For the same A and B as above, the approximate eigenvalues, approximate error bounds, and true errors are shown below.


    tabular6068

Further Details: Error Bounds for the Generalized Symmetric Definite Eigenproblem 

The error analysis of the driver routine PxSYGVX, or PxHEGVX in the complex case      (see section 3.2.4), goes as follows. In all cases tex2html_wrap_inline18712 is the absolute gap   between tex2html_wrap_inline18626 and the nearest other eigenvalue.  

  1. tex2html_wrap_inline18087. The computed eigenvalues tex2html_wrap_inline19245 can differ from true eigenvalues tex2html_wrap_inline18626 by at most about
    displaymath19056

        The angular difference between the computed eigenvector tex2html_wrap_inline18638 and a true eigenvector tex2html_wrap_inline18630 is
    displaymath19057

  2. tex2html_wrap_inline18089 or tex2html_wrap_inline18091. The computed eigenvalues tex2html_wrap_inline19245 can differ from true eigenvalues tex2html_wrap_inline18626 by at most about
    displaymath19058

       

    The angular difference between the computed eigenvector tex2html_wrap_inline18638 and a true eigenvector tex2html_wrap_inline18630 is
    displaymath19059

The code fragments above replace p(n) by 1 and make sure neither RCONDB nor RCONDZ is so small as to cause overflow when used as divisors in the expressions for error bounds.

These error bounds are large when B is ill-conditioned with respect to inversion (tex2html_wrap_inline19277 is large). Often, the eigenvalues and eigenvectors are much better conditioned than indicated here. We mention two ways to get tighter bounds. The first way is effective when the diagonal entries of B differ widely in magnitude:gif

  1. tex2html_wrap_inline18087. Let tex2html_wrap_inline19283 be a diagonal matrix. Then replace B by DBD and A by DAD in the above bounds.
  2. tex2html_wrap_inline18089 or tex2html_wrap_inline18091. Let tex2html_wrap_inline19283 be a diagonal matrix. Then replace B by DBD and A by tex2html_wrap_inline19305 in the above bounds.

The second way to get tighter bounds does not actually supply guaranteed bounds, but its estimates are often better in practice. It is not guaranteed because it assumes the algorithm is backward stable, which is not necessarily true when B is ill-conditioned.     It estimates the chordal distance between a true eigenvalue tex2html_wrap_inline18626 and a computed eigenvalue tex2html_wrap_inline19245:  
displaymath19060
To interpret this measure, we write tex2html_wrap_inline19313 and tex2html_wrap_inline19315. Then tex2html_wrap_inline19317. In other words, if tex2html_wrap_inline19245 represents the one-dimensional subspace tex2html_wrap_inline17614 consisting of the line through the origin with slope tex2html_wrap_inline19245, and tex2html_wrap_inline18626 represents the analogous subspace tex2html_wrap_inline17602, then tex2html_wrap_inline19329 is the sine of the acute angle tex2html_wrap_inline17870 between these subspaces.     Thus, tex2html_wrap_inline19333 is bounded by one and is small when both arguments are large.gif It applies only to the first problem, tex2html_wrap_inline18087:

Suppose a computed eigenvalue tex2html_wrap_inline19245 of tex2html_wrap_inline18087 is the exact eigenvalue of a perturbed problem tex2html_wrap_inline19345. Let tex2html_wrap_inline17852 be the unit eigenvector (tex2html_wrap_inline19349) for the exact eigenvalue tex2html_wrap_inline18626. Then if tex2html_wrap_inline19353 is small compared with tex2html_wrap_inline19355, and if tex2html_wrap_inline19357 is small compared with tex2html_wrap_inline19359, we have
displaymath19061
Thus tex2html_wrap_inline19363 is a condition number for eigenvalue tex2html_wrap_inline18626.  
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Susan Blackford
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Troubleshooting

  

       Successful installation, testing, and use of ScaLAPACK rely heavily on the proper installation of its building blocks (PVM or MPI, BLACS, BLAS, and PBLAS). Frequently Asked Questions (FAQ) lists are maintained in the directories on netlib to answer some of the most common user questions. For the user's convenience, prebuilt ScaLAPACK and BLACS libraries are provided for a variety of computer architectures in the following URLs:

http://www.netlib.org/scalapack/archives/
http://www.netlib.org/blacs/archives/

Test suites are provided for PVM, the BLACS, the BLAS, and the PBLAS. It is highly recommended that each of these respective test suites be run prior to the execution of the ScaLAPACK test suite. Installation Guides  are also provided for the BLACS and ScaLAPACK. Refer to the appropriate directory on netlib for further information.

We begin this chapter by discussing a set of first-step debugging hints to pinpoint where the problem is occurring. Following these debugging hints, we discuss the types of error messages that can be encountered during the execution of a ScaLAPACK routine: ScaLAPACK error messages, PBLAS error messages, BLACS error messages, and system-dependent error messages or failures.

If these suggestions do not help evaluate specific difficulties, we suggest that the user review the following ``bug report checklist'' and then feel free to contact the authors at scalapack@cs.utk.edu   or blacs@cs.utk.edu, respectively. The user should tell us the type of machine on which the tests were run, the compiler and compiler options that were used, details of the BLACS library and message-passing library that were used and the BLAS library; also, the user should send a copy of the input file, if appropriate.



Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Bug Report Checklist

  

When the user sends e-mail to our mailing alias, some of the first questions we will ask are the following:

  1. Have you run the BLAS, BLACS, PBLAS and ScaLAPACK test suites?
  2. Have you checked the errata lists (errata.scalapack and errata.blacs) on netlib?
    http://www.netlib.org/scalapack/errata.scalapack
    http://www.netlib.org/blacs/errata.blacs
  3. If you are using an optimized BLAS or BLACS library, have you tried using the reference implementations from netlib?
    http://www.netlib.org/blas/
    http://www.netlib.org/blacs/
  4. If you are using an optimized MPI or PVM library, have you tried using the reference implementations from netlib?
    http://www.netlib.org/pvm3/
    http://www.netlib.org/mpi/
  5. Have you attempted to replicate this error using the appropriate ScaLAPACK test code and/or one of the ScaLAPACK example routines?

Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Installation Debugging Hints

 

If the user encounters difficulty in the installation process, we suggest the following:

  • Obtain prebuilt ScaLAPACK and BLACS libraries on netlib for a variety of architectures.
    http://www.netlib.org/scalapack/archives/
    http://www.netlib.org/blacs/archives/
  • Obtain sample SLmake.inc files for a variety of architectures in the SCALAPACK/INSTALL directory in the scalapack distribution tar file. Sample Bmake.inc files are included in the BLACS/BMAKES directory in the blacs distribution file.
  • Consult the ScaLAPACK FAQ list on netlib.
  • Consult the errata.scalapack file in the scalapack directory on netlib, and/or the errata.blacs file in the blacs directory on netlib. These files contain a list of known difficulties that have been diagnosed and corrected (or will be corrected in the next release), or reported to the vendor as in the case of message-passing libraries or optimized BLAS  .
  • Always run the BLACS, BLAS, and PBLAS test suites to ensure that these libraries have been properly installed. (If PVM is the underlying message-passing layer, please also run the PVM test suite.) If a problem is detected in the BLAS or BLACS libraries, try linking to the reference implementations to be found in the respective blas or blacs directory on netlib.
  • If ScaLAPACK is being tested on a heterogeneous cluster of computers, please ensure that all executables are linked with the same debug level of the BLACS. Otherwise, unpredictable results will occur because the debug level 1 BLACS (specified by BLACSDBGLVL=1 in Bmake.inc) perform error-checking and thus send more messages than the performance debug level 0 BLACS (specified by BLACSDBGLVL=0 in Bmake.inc).

Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: Common Errors in Calling Up: Troubleshooting Previous: Installation Debugging Hints

Application Debugging Hints

 

We highly recommend the following as a list of debugging hints (and tools) for writing parallel application programs that call ScaLAPACK:

  • Look at the ScaLAPACK example programs as a good starting point.
    http://www.netlib.org/scalapack/examples/
  • Always check the value of INFO on exit from a ScaLAPACK routine.
  • All routines in ScaLAPACK that require workspace also require the length of that workspace to be specified in the calling sequence. If in doubt about the amount of workspace to supply to a ScaLAPACK routine, supply LWORK=-1, and use the returned value in WORK(1) as the correct value for LWORK. Refer to section 4.6.5 for further details on determining workspace requirements  .
  • If you are calling a ScaLAPACK routine that has an LAPACK equivalent, write a serial code calling LAPACK first. Code can be converted in pieces from LAPACK to ScaLAPACK by debugging on a one-process grid. When all of the LAPACK codes have been removed and the code has been fully parallelized, execute it on a multiple process grid.
  • When writing a parallel program, first debug the code to work on one process, and then expand to more processes.
  • When writing a parallel program, debug with small matrices.
  • Use the TOOLS routine PxLAPRNT  to print out each process's portion of a distributed matrix. A variety of utility routines   are provided in the TOOLS directory and are commonly used as debugging aids in the development of the ScaLAPACK library.
  • Sprinkle synchronization points via BLACS_BARRIER   near suspected error.
  • Link to the debug level 1 BLACS (specified by BLACSDBGLVL=1 in Bmake.inc)  until the program is completely debugged.
  • Specify the``Repeatability'' flag in BLACS_SET.
  • If running a heterogeneous application, please ensure that all executables are linked with the same debug level of the BLACS. Otherwise, unpredictable results will occur because the debug level 1 BLACS perform error-checking and thus send more messages than the performance debug level 0 BLACS.
  • Always run the BLACS, BLAS, and PBLAS test suites to ensure that these libraries have been properly installed. (If PVM is the underlying message-passing layer, please also run the PVM test suite.) If a problem is detected in the BLAS or BLACS libraries, try linking to the reference implementations in the respective blas or blacs directory on netlib.
  • Consult the errata.scalapack file in the scalapack directory on netlib, and/or the errata.blacs file in the blacs directory on netlib. These files contain a list of known difficulties that have been diagnosed and corrected (or will be corrected in the next release), or reported to the vendor as in the case of message-passing libraries or optimized BLAS  .
  • Refer to section 4.6.7 and the leading comments of the source code for the alignment restrictions currently needed in some of the ScaLAPACK routines.
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Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Common Errors in Calling ScaLAPACK Routines

 

The user must read the leading comments of a ScaLAPACK routine before invoking the routine. The wording of the leading comments is explained in Chapter 4. Basic terminology is explained in the Glossary and List of Notation.

For the benefit of less experienced programmers, we provide a list of common programming errors in calling a ScaLAPACK routine. These errors may cause the ScaLAPACK routine to report a failure, as described in section 7.4 ; they may cause an error to be reported by the system; or they may lead to wrong results -- see also section 7.5.

  • Wrong number of arguments
  • Arguments in the wrong order
  • Argument of the wrong type (especially real and complex arguments of the wrong precision)
  • Wrong dimensions for an array argument
  • Insufficient space in a workspace argument
  • Failure to assign a value to an input argument
  • Routine designed for homogeneous computers was executed on a heterogeneous system (see section 6.2)

Some modern compilation systems, as well as software tools such as the Fortran 77 syntax checker ftnchek (freely available on netlib) and the portability checker in Toolpack [105], can check that arguments agree in number and type; and many compilation systems offer run-time detection of errors such as an array element out-of-bounds or use of an unassigned variable.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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PBLAS

 

To simplify the design of ScaLAPACK, and because the BLAS have proven to be useful tools outside LAPACK, we chose to build a parallel set of BLAS, called PBLAS [26, 104], which perform message-passing and whose interface is as similar to the BLAS as possible. This decision has permitted the ScaLAPACK code to be quite similar, and sometimes nearly identical, to the analogous LAPACK code.

We hope that the PBLAS will provide a distributed memory standard  , just as the BLAS have provided a shared memory standard. This would simplify and encourage the development of high performance and portable parallel numerical software, as well as providing manufacturers with a small set of routines to be optimized. Further details of the PBLAS can be found in [26], [104], and Appendix D.2.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Failures Detected by ScaLAPACK Routines

   

A ScaLAPACK routine has two ways to report a failure to complete a computation successfully.



Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Invalid Arguments and PXERBLA

    If an illegal value is supplied for one of the input arguments to a ScaLAPACK routine, it will call the error handler PXERBLA to write a message to the standard output unit of the form: 

 ** On entry to PSGESV  parameter number  4 had an illegal value
This particular message could be caused by passing to PSGESV  a value of NRHS that was less than zero, for example. The arguments are checked in order, beginning with the first. As mentioned in Chapter 4, if an error is detected in the tex2html_wrap_inline19432 entry of a descriptor array, which is the tex2html_wrap_inline19434 argument in the parameter list, the number passed to PXERBLA has been arbitrarily chosen to be 100*i+j. This allows the user to distinguish an error on a descriptor entry from an error on a scalar argument. Invalid arguments are often caused by the kind of error listed in section 7.3.

In the model implementation of PXERBLA  that is supplied with ScaLAPACK, the only action that is performed is the printing of an error message to standard output. Program execution is not terminated. For the ScaLAPACK driver and computational routines, a RETURN statement is issued following the call to PXERBLA. Control returns to the higher-level calling routine, and it is left to the user to determine how the program should proceed. However, in the specialized low-level ScaLAPACK routines (auxiliary routines that are Level 2 equivalents of computational routines), the call to PXERBLA() is immediately followed by a call to BLACS_ABORT()  to terminate program execution since recovery from an error at this level in the computation is not possible.

It is always good practice to check for a nonzero value of INFO on return from a ScaLAPACK routine.  


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Computational Failures and INFO > 0

   A positive value of INFO on return from a ScaLAPACK routine indicates a failure in the course of the algorithm. Common causes are

  • a matrix is singular (to working precision),
  • a symmetric matrix is not positive definite, or
  • an iterative algorithm for computing eigenvalues or eigenvectors fails to converge in the permitted number of iterations.
For example, if PSGESVX  is called to solve a system of equations with a coefficient matrix  that is approximately singular, it may detect exact singularity at the ith stage of the LU factorization, in which case it returns INFO = i; or (more probably) it may compute an estimate of the reciprocal condition number that is less than relative machine precision, in which case it returns INFO = n+1. Again, the documentation in Part ii should be consulted for a description of the error.

When a failure with INFO > 0 occurs, control is always returned to the calling program; PXERBLA() is not called, and no error message is written. Thus, it is always good practice to check for a nonzero value of INFO on return from a ScaLAPACK routine.

A failure with INFO > 0 may indicate any of the following:

  • An inappropriate routine was used:. For example, if a routine fails because a symmetric matrix turns out not to be positive definite, consider using a routine for symmetric indefinite matrices.
  • A single-precision routine was used when double precision was needed. For example, if PSGESVX  reports approximate singularity (as illustrated above), the corresponding double precision routine PDGESVX may be able to solve the problem (but nevertheless the problem is ill-conditioned).
  • A programming error occurred in generating the data supplied to a routine. For example, even though theoretically a matrix should be well-conditioned and positive-definite, a programming error in generating the matrix could easily destroy either of those properties.
  • A programming error occurred in calling the routine, of the kind listed in section 7.3.

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Wrong Results

  

Wrong results from ScaLAPACK routines are most often caused by incorrect usage. It is also possible that wrong results are caused by a bug outside of ScaLAPACK, in the compiler or in one of the library routines, such as the BLAS, the BLACS, or the underlying message-passing layer, that are linked with ScaLAPACK. Test suites are available for ScaLAPACK, the PBLAS, the BLACS, and the BLAS. The ScaLAPACK installation guide [24] or the BLACS installation guide should be consulted for descriptions of the tests and for advice on resolving problems.

A list of known problems, compiler errors, and bugs in ScaLAPACK routines is maintained on netlib; see Chapter 1.

Users who suspect they have found a new bug in a ScaLAPACK routine are encouraged to report it promptly to the developers as directed in Chapter 1. The bug report should include a test case, a description of the problem and expected results, and the actions, if any, that the user has already taken to fix the bug.


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Error Handling in the PBLAS

 

If a PBLAS routine is called with an invalid value for any of its arguments, it must report the fact and terminate the execution of the program. In the model implementation, each routine, on detecting an error, calls a common error-handling PBLAS routine  , passing to it the current BLACS context, the name of the routine and the number of the first argument that is in error. If an error is detected in the tex2html_wrap_inline19432 entry of a descriptor array, which is the tex2html_wrap_inline19434 argument in the parameter list, the number passed to the PBLAS error-handler routine has been arbitrarily chosen to be tex2html_wrap_inline19471. This allows the user to distinguish an error on a descriptor entry from an error on a scalar argument. For efficiency purposes, the PBLAS routines performs only a local validity check of their argument list. If an error is detected in at least one process of the current context, the program execution is stopped.

A global validity check of the input arguments passed to a PBLAS routine must be performed in the higher-level calling procedure. To understand the need and cost of global checking, as well as the reason why this type of checking is not performed in the PBLAS, consider the following example. The value of a global input argument is legal but differs from one process to another. The results are unpredictable. In order to detect this kind of error situation, a synchronization point would be necessary, which may result in a significant performance degradation. Since every process must call the same routine to perform the desired operation successfully, it is natural and safe to restrict somewhat the amount of checking operations performed in the PBLAS routines.

Specialized implementations may call system-specific exception-handling facilities, either via an auxiliary routine for error handling or directly from the routine. In addition, the testing programs can take advantage of this exception handling mechanism by simulating specific erroneous input argument lists and then verifying that particular errors are correctly detected.

For complete details on the specification of all routines, please refer to [26]. Appendix D.2 contains the Quick Reference Guide to the PBLAS. An html version of this Quick Reference Guide, along with the leading comments from each of the routines, is available on the ScaLAPACK homepage.

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Error Handling in the BLACS

 

This section describes the BLACS error-handling features. The BLACS error-handling behavior may be changed at compile time by using the C preprocessor macro BlacsDebugLvl. a call to BLACS_GET (see  [54] for details) will help determine what debug level the BLACS are using.

If the BLACS are compiled with a BLACS debug level of 0, little error checking is performed. A few critical items will be checked (for instance, BLACS_GRIDINIT will still not allow the user to allocate a process grid with more processes than there are available), but for performance reasons, the BLACS will not check most of the parameters.

It is therefore highly recommended that users link their code to a BLACS library compiled with debug level 1 while debugging their code. BLACS debug level 1 mainly does parameter checking. A few other services are also provided. For instance, users will be warned if a process sends a message to itself. Having a process send to itself is legal, but it displays poor performance and requires enough buffer space that it can occasionally cause hangs for large messages. The BLACS therefore issue a warning when this behavior is detected.

Many times, the debug level 0 code will simply hang, leaving the developer without any clue as to what has gone wrong. This may be caused, for instance, by trying to receive from a process that is not in the current context. The debug level 1 BLACS can detect this type of user error, and issue a (we hope helpful) message.

The BLACS issue three types of messages:

  1. BLACS warning: BLACS detect risky behavior, but attempt to correct or ignore. Warning message is printed, and execution proceeds.
  2. BLACS error: BLACS detects an error, prints an error message, and kills the machine via a call to BLACS_ABORT.
  3. System error: The BLACS receive an error message from the underlying system, which is then passed on to the user, and the BLACS kills the machine.

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BLACS Warning and Error Messages

All BLACS warning messages are printed by the internal routine BlacsWarn, and all BLACS error messages are printed by the internal routine BlacsErr. The only real difference between BlacsWarn and BlacsErr is that BlacsErr calls BLACS_ABORT after the message is printed.

With these central routines handling BLACS error messages, it should be relatively easy for the programmer to modify error handling if the default routines are not adequate for his needs. One particularly annoying problem is that on many systems a print to the screen takes a long time to finish. BlacsErr may then kill the machine before the print reaches the screen, and the error message is lost. In this case, the user may wish to make BlacsErr wait before killing, or not kill at all, for instance.

BLACS warning messages have the following form: 

BLACS WARNING '<explanation string>'
from {<p>,<q>}, pnum=<pnum>, Contxt=<ictxt>, on line <#> of file '<fname>'.

BLACS error messages have the following form: 

BLACS ERROR '<explanation string>'
from {<p>,<q>}, pnum=<pnum>, Contxt=<ictxt>, on line <#> of file '<fname>'.

The meaning of these parameters are as follows:

  • explanation string The message that should help the user determine what is wrong. For example, on an incorrect call to BLACS_GRIDINIT, the user might get:
    Process 0 had 2 x 4 grid; correct is 1 x 4.
  • {p, q}: The row and column process grid coordinates of the process issuing the warning/error.
  • pnum: The process number returned in the first argument of BLACS_PINFO.
  • ictxt: The integer context handle. Please note that this value is not the same across all processes. For instance, process {0, 0} may have ictxt = 0 and process {0, 1} have ictxt = 1 for the same context. However, the pnum and ictxt together provide an unambiguous process/context identifier.
  • #: The line number within the file fname that issued the warning.
  • fname: The file name where the routine that issued the warning/error is located.

Not all of this information may be available at the time an error or warning is issued. For instance, if the error occurs before the creation of the grid, the process grid coordinates will be unavailable. For any value that the BLACS cannot figure out, a -1 is printed to indicate that the value is unknown.

Examples of these BLACS error messages can be found on the BLACS homepage
(http://www.netlib.org/blacs/index.html) or in ``A User's Guide to the BLACS'' [54].

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System Error Messages

Occasionally, ScaLAPACK will receive an error message from the underlying system. At this time, the BLACS will print the system error message, and exit. Since these error messages come from the underlying system, their form will necessarily vary depending on which BLACS version is being used. The user may need to obtain vendor documentation or on-line manpages describing system error messages in order to understand the message. For example, if the PVM BLACS are being used, a PVM error number will be returned. The PVM quick reference guide, for example, could then be consulted to translate the error number into an understandable error message.

Examples of system error messages can be found on the BLACS homepage
(http://www.netlib.org/blacs/index.html) or in ``A User's Guide to the BLACS'' [54].


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Poor Performance

ScaLAPACK ultimately relies on an efficient implementation of the BLAS  and the data distribution for load balance. Refer to Chapter 5.

To avoid poor performance  from ScaLAPACK routines, note the following recommendations :

BLAS:
One should use machine-specific optimized BLAS if they are available. Many manufacturers and research institutions have developed, or are developing, efficient versions of the BLAS for particular machines. The BLAS enable LAPACK and ScaLAPACK routines to achieve high performance with transportable software. Users are urged to determine whether such an implementation of the BLAS exists for their platform. When such an optimized implementation of the BLAS is available, it should be used to ensure optimal performance. If such a machine-specific implementation of the BLAS does not exist for a particular platform, one should consider installing a publicly available set of BLAS that requires only an efficient implementation of the matrix-matrix multiply BLAS routine xGEMM. Examples of such implementations are [35, 90]. A machine-specific and efficient implementation of the routine GEMM can be automatically generated by publicly available software such as [16]. Although a reference implementation of the Fortran77 BLAS is available from the blas directory on netlib, these routines are not expected to perform as well as a specially tuned implementation on most high-performance computers - on some machines it may give much worse performance - but it allows users to run LAPACK and ScaLAPACK software on machines that do not offer any other implementation of the BLAS.

BLACS:
With the few exceptions mentioned in section 5.2.3, the performance achieved by the BLACS should be close to the one of the underlying message-passing library it is calling. Since publicly available implementations of the BLACS exist for a range of native message-passing libraries such as NX for the Intel supercomputers and MPL for the IBM SP series, as well as more generic interfaces such as PVM and MPI, users should select the BLACS implementation that is based on the most efficient message-passing library available. Some vendors, such as Cray and IBM, supply an optimized implementation of the BLACS for their systems. Users are urged to rely on these BLACS libraries whenever possible.

LWORK tex2html_wrap_inline14966 WORK(1):
In some ScaLAPACK eigenvalue routines, such as the symmetric eigenproblems (PxSYEV and PxSYEVX/PxHEEVX) and the generalized symmetric eigenproblem (PxSYGVX/PxHEGVX), a larger value of LWORK can guarantee the orthogonality of the returned eigenvectors at the risk of potentially degraded performance of the algorithm. The minimum amount of workspace required is returned in the first element of the work array, but a larger amount of workspace can allow for additional orthogonalization if desired by the user. Refer to section 5.3.6 and the leading comments of the source code for complete details.

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Index of ScaLAPACK Routines

 

A separate index is provided for each of the following classifications of routines: driver and computational routines, auxiliary routines, and matrix redistribution/copy routines.



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BLACS

 

The BLACS (Basic Linear Algebra Communication Subprograms) [50, 54] are a message-passing library designed for linear algebra. The computational model consists of a one- or two-dimensional process grid, where each process stores pieces of the matrices and vectors. The BLACS include synchronous send/receive routines to communicate a matrix or submatrix from one process to another, to broadcast submatrices to many processes, or to compute global reductions (sums, maxima and minima). There are also routines to construct, change, or query the process grid. Since several ScaLAPACK algorithms require broadcasts or reductions among different subsets of processes, the BLACS permit a process to be a member of several overlapping or disjoint process grids, each one labeled by a context. Some message-passing systems, such as MPI [64, 110], also include this context  concept; MPI calls this a communicator . The BLACS provide facilities for safe inter-operation of system contexts and BLACS contexts. Further details of the BLACS can be found in [54]. An important aim of the BLACS is to provide a portable, linear algebra specific layer for communication.


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Index of Driver and Computational Routines

 

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Notes

  1. This index   lists related pairs of real and complex routines together, for example, PSGETRF and PCGETRF.
  2. Driver routines are listed in bold type, for example, PSGESV and PCGESV.
  3. Routines are listed in alphanumeric order of the real (single precision) routine name (which always begins with PS-). (See section 3.1.3 for details of the ScaLAPACK naming scheme.)
  4. Double precision routines are not listed here; they have names beginning with PD- instead of PS-, or PZ- instead of PC-.
  5. This index gives only a brief description of the purpose of each routine. For a precise description, consult the Specifications in Part ii, where the routines appear in the same order as here.
  6. The text of the descriptions applies to both real and complex routines, except where alternative words or phrases are indicated, for example, ``symmetric/Hermitian'', ``orthogonal/unitary'', or ``quasi-triangular/triangular''. For the real routines tex2html_wrap_inline12724 is equivalent to tex2html_wrap_inline12722. (The same convention is used in Part ii.)
  7. A few routines for real matrices have no complex equivalent (for example, PSSTEBZ).


tabular6406

tabular6460

tabular6529

tabular6587


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Index of Auxiliary Routines

 


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Notes

  1. This index  lists related pairs of real and complex routines together.
  2. Routines are listed in alphanumeric order of the real (single precision) routine name (which always begins with PS-). (See section 3.1.3 for details of the ScaLAPACK naming scheme.)
  3. A few complex routines have no real equivalents, and they are listed first.
  4. Double-precision routines are not listed here; they have names beginning with PD- instead of PS-, or PZ- instead of PC-. The only exceptions to this simple rule are that the double-precision versions of PCMAX1, PSCSUM1, and PCSRSCL are named PZMAX1, PDZSUM1, and PZDRSCL.
  5. A few routines in the list have names that are independent of data type: PXERBLA.
  6. This index gives only a brief description of the purpose of each routine. For a precise description, consult the leading comments in the code, which have been written in the same style as for the driver and computational routines.


tabular6639

tabular6653

tabular6673

tabular6694

tabular6720


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Matrix Redistribution/Copy Routines

 

ScaLAPACK provides two matrix redistribution/copy routines for each data type [107, 49, 106]. These routines provide a truly general copy from any block cyclicly distributed (sub)matrix to any other block cyclicly distributed (sub)matrix. These routines are the only ones in the entire ScaLAPACK library which provide inter-context operations. By this we mean that they can take a (sub)matrix in context A (distributed over process grid A) and copy it to a (sub)matrix in context B.

There need be no relation between the two operand (sub)matrices other than their global size and the fact that they are both legal block cyclicly distributed (sub)matrices. This means that they may be distributed across different process grids, have varying block sizes, and differing matrix starting points, be contained in different size distributed matrices, etc.

Because of the generality of these routines, they may be used for many operations not usually associated with copy routines. For instance, they may be used to a take a matrix on one process and distribute it across a process grid, or the reverse. If a supercomputer is grouped into a virtual parallel machine with a workstation, for instance, this routine can be used to move the matrix from the workstation to the supercomputer and back. In ScaLAPACK, these routines are called to copy matrices from a two-dimensional process grid to a one-dimensional process grid. They can be used to redistribute matrices so that distributions providing maximal performance can be used by various component libraries, as well. This list of uses is hardly exhaustive, but it gives an idea of the power of a general copy in parallel computing.

The two routine classifications are as follows:

  • P_GEMR2D copies between general, rectangular matrices.
  • P_TRMR2D copies between trapezoidal matrices.
                 

All routines are available in integer, single precision real, double precision real, single precision complex, and double precision complex. In the following sections, we describe only the singe precision routines for each data type. Double precision routines are the same as their single precision counterparts, but they have names beginning with PD- instead of PS-, or PZ- instead of PC-.

Note that these routines require an array descriptor of type DESC_(DTYPE_)=1.


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Fortran Interface


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PxGEMR2D

      SUBROUTINE PSGEMR2D( M, N, A, IA, JA, DESCA, B, IB, JB, DESCB,
     $                     ICTXT )
      INTEGER            IA, IB, ICTXT, JA, JB, M, N
      INTEGER            DESCA( * ), DESCB( * )
      REAL               A( * ), B( * )

      SUBROUTINE PCGEMR2D( M, N, A, IA, JA, DESCA, B, IB, JB, DESCB,
     $                     ICTXT )
      INTEGER            IA, IB, ICTXT, JA, JB, M, N
      INTEGER            DESCA( * ), DESCB( * )
      COMPLEX            A( * ), B( * )

      SUBROUTINE PIGEMR2D( M, N, A, IA, JA, DESCA, B, IB, JB, DESCB,
     $                     ICTXT )
      INTEGER            IA, IB, ICTXT, JA, JB, M, N
      INTEGER            DESCA( * ), DESCB( * )
      INTEGER            A( * ), B( * )


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Purpose

PxGEMR2D copies the indicated (sub)matrix of A to the indicated (sub)matrix of B. A and B can have arbitrary block-cyclic distributions: they can be distributed across different process grids, have different blocking factors, etc.

Particular care must be taken when the process grid over which matrix A is distributed (call this context A) is disjoint from the process grid over which matrix B is distributed (call this context B). The general rules for which parameters need to be set are:

  • All calling processes must have the correct M and N.
  • Processes in context A must correctly define all parameters describing A.
  • Processes in context B must correctly define all parameters describing B.
  • Processes which are not members of context A must pass DESCA(CTXT_) = -1 and need not set other parameters describing A.
  • Processes which are not members of context B must pass DESCB(CTXT_) = -1 and need not set other parameters describing B.

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Arguments

M
(global input) INTEGER
On entry, M specifies the number of rows of the (sub)matrix A to be copied. M tex2html_wrap_inline14966 0.

N
(global input) INTEGER
On entry, N specifies the number of columns of the (sub)matrix A to be copied. N tex2html_wrap_inline14966 zero.

A
(local input) REAL/COMPLEX/INTEGER array, dimension (LLD_A,LOCc(JA+N-1))
On entry, the source matrix.

IA
(global input) INTEGER
On entry,the global row index of the beginning of the (sub)matrix of A to copy.
1 tex2html_wrap_inline14970 IA tex2html_wrap_inline14970 M_A - M + 1.

JA
(global input) INTEGER
On entry,the global column index of the beginning of the (sub)matrix of A to copy.
1 tex2html_wrap_inline14970 JA tex2html_wrap_inline14970 N_A - N + 1.

DESCA
(global and local input) INTEGER array, dimension (DLEN_)
The array descriptor for the distributed matrix A.
Only DESCA(DTYPE_)=1 is supported, and thus DLEN_ = 9.
If the calling process is not part of the context of A, then DESCA(CTXT_) must be equal to -1.

B
(local output) REAL/COMPLEX/INTEGER array, dimension (LLD_B,LOCc(JB+N-1))
On exit, the defined (sub)matrix is overwritten by the indicated (sub)matrix from A.

IB
(global input) INTEGER
On entry, the global row index of the beginning of the (sub)matrix of B that will be overwritten.
1 tex2html_wrap_inline14970 IB tex2html_wrap_inline14970 M_B - M + 1.

JB
(global input) INTEGER
On entry, the global column index of the beginning of the submatrix of B that will be overwritten.
1 tex2html_wrap_inline14970 JB tex2html_wrap_inline14970 N_B - N + 1.

DESCB
(global and local input) INTEGER array, dimension (DLEN_)
The array descriptor for the distributed matrix B.
Only DESCB(DTYPE_)=1 is supported, and thus DLEN_ = 9.
If the calling process is not part of the context of B, then DESCB(CTXT_) must be equal to -1.

ICTXT
(global input) INTEGER
The context encompassing at least the union of all processes in context A and context B. All processes in the context ICTXT must call this routine, even if they do not own a piece of either matrix.

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PxTRMR2D

      SUBROUTINE PSTRMR2D( UPLO, DIAG, M, N, A, IA, JA, DESCA, B, IB,
     $                     JB, DESCB, ICTXT )
      CHARACTER          DIAG, UPLO
      INTEGER            IA, IB, ICTXT, JA, JB, M, N
      INTEGER            DESCA( * ), DESCB( * )
      REAL               A( * ), B( * )

      SUBROUTINE PCTRMR2D( UPLO, DIAG, M, N, A, IA, JA, DESCA, B, IB,
     $                     JB, DESCB, ICTXT )
      CHARACTER          DIAG, UPLO
      INTEGER            IA, IB, ICTXT, JA, JB, M, N
      INTEGER            DESCA( * ), DESCB( * )
      COMPLEX            A( * ), B( * )

      SUBROUTINE PITRMR2D( UPLO, DIAG, M, N, A, IA, JA, DESCA, B, IB,
     $                     JB, DESCB, ICTXT )
      CHARACTER          DIAG, UPLO
      INTEGER            IA, IB, ICTXT, JA, JB, M, N
      INTEGER            DESCA( * ), DESCB( * )
      INTEGER            A( * ), B( * )


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Efficiency and Portability

  

ScaLAPACK is designed to give high efficiency  on MIMD    distributed memory concurrent supercomputers, such as the Intel Paragon, IBM SP series, and the Cray T3 series. In addition, the software is designed so that it can be used with clusters of workstations through a networked environment and with a heterogeneous  computing environment via PVM or MPI. Indeed, ScaLAPACK can run on any machine that supports either PVM  or MPI . See Chapter 5 for some examples of the performance achieved by ScaLAPACK routines.

The ScaLAPACK strategy for combining efficiency with portability  is to construct the software so that as much as possible of the computation is performed by calls to the Parallel Basic Linear Algebra Subprograms (PBLAS). The PBLAS [26, 104] perform global computation by relying on the Basic Linear Algebra Subprograms (BLAS) [93, 59, 57]  for local computation and the Basic Linear Algebra Communication Subprograms (BLACS) [54, 113] for communication.

The efficiency of ScaLAPACK software depends on the use of block-partitioned algorithms   and on efficient implementations of the BLAS  and the BLACS  being provided by computer vendors (and others) for their machines. Thus, the BLAS and the BLACS form a low-level interface between ScaLAPACK software and different machine architectures. Above this level, all of the ScaLAPACK software is portable.

The BLAS, PBLAS, and the BLACS are not, strictly speaking, part of ScaLAPACK. C code for the PBLAS is included in the ScaLAPACK distribution. Since the performance of the package depends upon the BLAS and the BLACS being implemented efficiently, we have not included this software with the ScaLAPACK distribution. A machine-specific implementation of the BLAS and the BLACS should be used. If a machine-optimized version of the BLAS is not available, a Fortran 77 reference implementation of the BLAS is available from netlib (see section 1.5). This code constitutes the ``model implementation'' [58, 56]. The model implementation of the BLAS is not expected to perform as well as a specially tuned implementation on most high-performance computers -- on some machines it may give much worse performance -- but it allows users to run ScaLAPACK codes on machines that do not offer any other implementation of the BLAS.

If a vendor-optimized version of the BLACS is not available for a specific architecture, efficiently ported versions of the BLACS are available on netlib. Currently, the BLACS have been efficiently ported on machine-specific message-passing libraries such as the IBM (MPL)   and Intel (NX)   message-passing libraries, as well as more generic interfaces such as PVM   and MPI  . The BLACS  overhead has been shown to be negligible in [54]. Refer to the URL for the blacs directory on netlib for more details:

http://www.netlib.org/blacs/index.html
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Next: Availability Up: Essentials Previous: BLACS

Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: Arguments Up: PxTRMR2D Previous: PxTRMR2D

Purpose

PxTRMR2D copies the indicated (sub)matrix of A to the indicated (sub)matrix of B. A and B can have arbitrary block-cyclic distributions: they can be distributed across different process grids, have different blocking factors, etc.

The (sub)matrix to be copied is assumed to be trapezoidal. So only the upper or the lower part will be copied. The other part is unchanged.

Particular care must be taken when the process grid over which matrix A is distributed (call this context A) is disjoint from the process grid over which matrix B is distributed (call this context B). The general rules for which parameters need to be set are as follows:

  • All calling processes must have the correct M and N.
  • Processes in context A must correctly define all parameters describing A.
  • Processes in context B must correctly define all parameters describing B.
  • Processes that are not members of context A must pass DESCA(CTXT_) = -1 and need not set other parameters describing A.
  • Processes that are not members of context B must pass DESCB(CTXT_) = -1 and need not set other parameters describing B.

Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: C Interface Up: PxTRMR2D Previous: Purpose

Arguments

UPLO
(global input) CHARACTERtex2html_wrap_inline152631
On entry, UPLO specifies whether we should copy the upper part or the lower part of the indicated (sub)matrix:
= 'U':
Copy the upper triangular part.
= 'L':
Copy the lower triangular part.

DIAG
(global input) CHARACTERtex2html_wrap_inline152631
On entry, DIAG specifies whether we should copy the diagonal.
= 'U':
Do NOT copy the diagonal of the (sub)matrix.
= 'N':
Do copy the diagonal of the (sub)matrix.

M
(global input) INTEGER
On entry, M specifies the number of rows of the (sub)matrix to be copied. M tex2html_wrap_inline14966 0.

N
(global input) INTEGER
On entry, N specifies the number of columns of the (sub)matrix to be copied. N tex2html_wrap_inline14966 0.

A
(local input) REAL/COMPLEX/INTEGER array, dimension (LLD_A,LOCc(JA+N-1))
On entry, the source matrix.

IA
(global input) INTEGER
On entry,the global row index of the beginning of the (sub)matrix of A to copy.
1 tex2html_wrap_inline14970 IA tex2html_wrap_inline14970 M_A - M + 1.

JA
(global input) INTEGER
On entry,the global column index of the beginning of the (sub)matrix of A to copy.
1 tex2html_wrap_inline14970 JA tex2html_wrap_inline14970 N_A - N + 1.

DESCA
(global and local input) INTEGER array, dimension (DLEN_)
The array descriptor for the distributed matrix A.
Only DESCA(DTYPE_)=1 is supported, and thus DLEN_ = 9.
If the current process is not part of the context of A, then DESCA(CTXT_) must be equal to -1.

B
(local output) REAL/COMPLEX/INTEGER array, dimension (LLD_B,LOCc(JB+N-1))
On exit, the defined (sub)matrix is overwritten by the indicated (sub)matrix from A.

IB
(global input) INTEGER
On entry, the global row index of the beginning of the (sub)matrix of B that will be overwritten.
1 tex2html_wrap_inline14970 IB tex2html_wrap_inline14970 M_B - M + 1.

JB
(global input) INTEGER
On entry, the global column index of the beginning of the submatrix of B that will be overwritten.
1 tex2html_wrap_inline14970 JB tex2html_wrap_inline14970 N_B - N + 1.

DESCB
(global and local input) INTEGER array, dimension (DLEN_)
The array descriptor for the distributed matrix B.
Only DESCB(DTYPE_)=1 is supported, and thus DLEN_ = 9.
If the calling process is not part of the context of B, then DESCB(CTXT_) must be equal to -1.

ICTXT
(global input) INTEGER
The context encompassing at least the union of all processes in context A and context B. All processes in the context ICTXT must call this routine, even if they do not own a piece of either matrix.

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Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: Code Fragment Calling C Up: Matrix Redistribution/Copy Routines Previous: Arguments

C Interface

void Cp_gemr2d(int m, int n, TYPE *A, int IA, int JA, int *descA, 
               TYPE *B, int IB, int JB, int *descB, int gcontext);

void Cp_trmr2d(char *uplo, char *diag, int m, int n, TYPE *A, int IA,
               int JA, int *descA, TYPE *B, int IB, int JB, int *descB,
               int gcontext);
where _ and TYPE are as defined below:


tabular6773

Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: Call Conversion: LAPACK to Up: Matrix Redistribution/Copy Routines Previous: C Interface

Code Fragment Calling C Interface Cpdgemr2d

      /* scatter of the matrix A from 1 processor to a P*Q grid */       
         Cpdgemr2d(m, n,
                   Aseq, ia, ja, &descA_1x1,
                   Apar, ib, jb, &descA_PxQ, gcontext);     
       
      /* computation of the system solution */
         Cpdgesv( m, n, 
                  Apar , 1, 1, &descA_PxQ, ipiv , 
                  Cpar, 1, 1, &descC_PxQ, &info);
     
      /* gather of the solution matrix C on 1 processor */
         Cpdgemr2d(m, n,
                   Cpar, ia, ja, &descC_PxQ,
                   Cseq, ib, jb, &descC_1x1, gcontext);

Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: Translating BLAS-based programs to Up: Guide Previous: Code Fragment Calling C

Call Conversion: LAPACK to ScaLAPACK and BLAS to PBLAS

 

This section   is designed to assist people in converting serial programs based on calls to the BLAS and LAPACK to parallel programs using the PBLAS and ScaLAPACK.



Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: Translating LAPACK-based programs to Up: Call Conversion: LAPACK to Previous: Call Conversion: LAPACK to

Translating BLAS-based programs to the PBLAS

   

With a concrete understanding of array descriptors (Chapter 4), it is relatively simple to translate the serial version of a BLAS call into its parallel equivalent. Translating BLAS calls to PBLAS calls primarily consists of the following steps:

  • a `P' has to be inserted in front of the routine name,
  • the leading dimensions should be replaced by the global array descriptors, and
  • the global indices into the distributed matrices should be inserted as separate parameters in the calling sequence.
An example of translating a DGEMM call to a PDGEMM call is given below. 
      CALL DGEMM( 'No transpose', 'No transpose', M-J-JB+1, N-J-JB+1,
     $            JB, -ONE, A( J+JB, J ), LDA, A( J, J+JB ), LDA, ONE,
     $            A( J+JB, J+JB ), LDA )
tex2html_wrap_inline19869 
      CALL PDGEMM( 'No transpose', 'No transpose', M-J-JB+JA, N-J-JB+JA,
     $             JB, -ONE, A, J+JB, J, DESCA, A, J, J+JB, DESCA, ONE,
     $             A, J+JB, J+JB, DESCA )

This simple translation process considerably simplifies the implementation phase of linear algebra codes built on top of the BLAS.

The steps necessary to write a program to call a PBLAS routine are analogous to the steps presented in section 2.4.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Translating LAPACK-based programs to ScaLAPACK

 

This section  demonstrates how sequential LAPACK-based programs are parallelized and converted to ScaLAPACK.

As with the BLAS conversion, it is relatively simple to translate the serial version of an LAPACK call into its parallel equivalent. Translating LAPACK calls to ScaLAPACK calls primarily consists of the following steps:

  • a `P' has to be inserted in front of the routine name,
  • the leading dimensions should be replaced by the global array descriptors, and
  • the global indices into the distributed matrices should be inserted as separate parameters in the calling sequence.

As an example of this translation process, let us consider the parallelization of the LAPACK driver routine DGESV, which solves a general system of linear equations. The calling sequence comparison for DGESV versus its ScaLAPACK equivalent, PDGESV, is presented below. 

      CALL DGESV( N, NRHS, A( I, J ), LDA, IPIV, B( I, 1 ), LDB, INFO )
tex2html_wrap_inline19869 
      CALL PDGESV( N, NRHS, A, I, J, DESCA, IPIV, B, I, 1, DESCB, INFO )

For a more complete example, let us consider parallelizing a serial LU factorization code, as demonstrated in sections B.2.1 and B.2.2. Note that the parallel routine assumes the existence of the auxiliary routines PDGETF2 (unblocked LU) and PDLASWP (parallel swap routine) in addition to the PBLAS. With this in mind, the serial and parallel versions are very similar since most of the details of the parallel implementation such as communication and synchronization are hidden at lower levels of the software.



Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: Parallel LU Factorization Up: Translating LAPACK-based programs to Previous: Translating LAPACK-based programs to

Sequential LU Factorization

  

      SUBROUTINE DGETRF( M, N, A, LDA, IPIV, INFO )
*
*  LU factorization of a M-by-N matrix A using partial pivoting with
*  row interchanges.
*
      INTEGER            INFO, LDA, M, N, IPIV( * )
      DOUBLE PRECISION   A( LDA, * )
*
      INTEGER            I, IINFO, J, JB, NB
      PARAMETER          ( NB = 64 )
      EXTERNAL           DGEMM, DGETF2, DLASWP, DTRSM
      INTRINSIC          MIN
*
      DO 20 J = 1, MIN(M,N), NB
         JB = MIN( MIN(M,N)-J+1, NB )
*
*        Factor diagonal block and test for exact singularity.
*
         CALL DGETF2( M-J+1, JB, A(J,J), LDA, IPIV(J), IINFO )
*
*        Adjust INFO and the pivot indices.
*
         IF( INFO.EQ.0 .AND. IINFO.GT.0 ) INFO = IINFO + J - 1
         DO 10 I = J, MIN(M,J+JB-1)
            IPIV(I) = J - 1 + IPIV(I)
   10    CONTINUE
*
*        Apply interchanges to columns 1:J-1 and J+JB:N.
*
         CALL DLASWP( J-1, A, LDA, J, J+JB-1, IPIV, 1 )
         IF( J+JB.LE.N ) THEN
            CALL DLASWP( N-J-JB+1, A(1,J+JB), LDA, J, J+JB-1, IPIV, 1 )
*
*           Compute block row of U and update trailing submatrix.
*
            CALL DTRSM( 'Left', 'Lower', 'No transpose', 'Unit', JB,
     $                  N-J-JB+1, 1.0D+0, A(J,J), LDA, A(J,J+JB), LDA )
            IF( J+JB.LE.M )
     $         CALL DGEMM( 'No transpose', 'No transpose', M-J-JB+1,
     $                     N-J-JB+1, JB, -1.0D+0, A(J+JB,J), LDA,
     $                     A(J,J+JB), LDA, 1.0D+0, A(J+JB,J+JB), LDA )
         END IF
   20 CONTINUE
      RETURN
*
      END


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Parallel LU Factorization

  

      SUBROUTINE PDGETRF( M, N, A, IA, JA, DESCA, IPIV, INFO )
*
      INTEGER            IA, INFO, JA, M, N, DESCA( * ), IPIV( * )
      DOUBLE PRECISION   A( * )
*
*  LU factorization of a M-by-N distributed matrix A(IA:IA+M-1,JA:JA+N-1)
*  using partial pivoting with row interchanges.
*
      INTEGER            I, IINFO, J, JB
      EXTERNAL           IGAMN2D, PDGEMM, PDGETF2, PDLASWP, PDTRSM
      INTRINSIC          MIN
*
      DO 10 J = JA, JA+MIN(M,N)-1, DESCA( NB_ )
         JB = MIN( MIN(M,N)-J+JA, DESCA( NB_ ) )
         I = IA + J - JA
*
*        Factor diagonal block and test for exact singularity.
*
         CALL PDGETF2( M-J+JA, JB, A, I, J, DESCA, IPIV, IINFO )
         IF( INFO.EQ.0 .AND. IINFO.GT.0 ) INFO = IINFO + J - JA
*
*        Apply interchanges to columns JA:J-JA and J+JB:JA+N-1.
*
         CALL PDLASWP( 'Forward', 'Rows', J-JA, A, IA, JA, DESCA,
     $                 I, I+JB-1, IPIV )
         IF( J-JA+JB+1.LE.N ) THEN
            CALL PDLASWP( 'Forward', 'Rows', N-J-JB+JA, A, IA, J+JB,
     $                    DESCA, I, I+JB-1, IPIV )
*
*           Compute block row of U and update trailing submatrix.
*
            CALL PDTRSM( 'Left', 'Lower', 'No transpose', 'Unit', JB,
     $                   N-J-JB+JA, 1.0D+0, A, I, J, DESCA, A, I, J+JB,
     $                   DESCA )
            IF( J-JA+JB+1.LE.M ) THEN
     $         CALL PDGEMM( 'No transpose', 'No transpose', M-J-JB+JA,
     $                      N-J-JB+JA, JB, -1.0D+0, A, I+JB, J, DESCA, A,
     $                      I, J+JB, DESCA, 1.0D+0, A, I+JB, J+JB, DESCA )
         END IF
   10 CONTINUE
      IF( INFO.EQ.0 ) INFO = MIN(M,N) + 1
      CALL IGAMN2D( ICTXT, 'Row', ' ', 1, 1, INFO, 1, I, J, -1, -1, MYCOL )
      IF( INFO.EQ.MIN(M,N)+1 ) INFO = 0
*
      RETURN
*
      END

The required steps to call a ScaLAPACK routine from a parallel program are demonstrated in Example Program #1 in Chapter 2 and explained in section 2.3.


Susan Blackford
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Example Programs

 

This Appendix provides additional example programs. Section C.1 presents a more memory-efficient program calling the ScaLAPACK driver routine PDGESV. Section C.2 presents an HPF program calling the HPF interface to ScaLAPACK. These example programs are available from the respective URLs:

http://www.netlib.org/scalapack/examples/scaex.shar
http://www.netlib.org/scalapack/examples/sample_hpf_gesv.f


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: Commercial Use Up: Essentials Previous: Efficiency and Portability

Availability

  

The complete ScaLAPACK package is freely available on netlib [60, 22, 23]  and can be obtained via the World Wide Web or anonymous ftp.

The ScaLAPACK homepage  can be accessed on the World Wide Web via the URL address:

http://www.netlib.org/scalapack/index.html

Prebuilt ScaLAPACK and BLACS libraries     are available on netlib for a variety of architectures. Refer to the following URLs:

http://www.netlib.org/scalapack/archives/index.html
http://www.netlib.org/blacs/archives/index.html

At the time of this writing, the e-mail addresses for netlib [22, 23] are

netlib@www.netlib.org
netlib@research.bell-labs.com
Both repositories provide electronic mail and anonymous ftp service (the netlib@www.netlib.org site is available via anonymous ftp to ftp.netlib.org). The URL for netlib is http://www.netlib.org/  .

The following sites are mirror repositories :

tabular506

General information about ScaLAPACK (and the PBLAS) can be obtained by contacting any of the URLs listed above. If additional information is desired, feel free to contact the authors at scalapack@cs.utk.edu.

The complete ScaLAPACK package, including test code and timing programs in four different data types, constitutes some 500,000 lines of Fortran and C source and comments.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: HPF Interface to ScaLAPACK Up: Example Programs Previous: Example Programs

Example Program #2

   

In Chapter 2, we presented an example program using ScaLAPACK. Here we present a second example--a more flexible and memory efficient program to solve a system of linear equations using the ScaLAPACK driver routine PDGESV. In this example we will read the input matrices from a file, distribute these matrices to the processes in the grid. After calling the ScaLAPACK routine, we will write the output solution matrix to a file. The input data files for the program are SCAEX.dat, SCAEXMAT.dat, and SCAEXRHS.dat.

This program is also available in the scalapack directory on netlib
(http://www.netlib.org/scalapack/examples/scaex.shar).

SCAEX.dat is: 

'ScaLAPACK Example Program 2'
'May 1997'
'SCAEX.out'             output file name (if any)
6                       device out
6                       value of N
1                       value of NRHS
2                       values of NB
2                       values of NPROW
2                       values of NPCOL

SCAEXMAT.dat is: 

6 6
 6.0000D+0
 3.0000D+0
 0.0000D+0
 0.0000D+0
 3.0000D+0
 0.0000D+0
 0.0000D+0
-3.0000D+0
-1.0000D+0
 1.0000D+0
 1.0000D+0
 0.0000D+0
-1.0000D+0
 0.0000D+0
11.0000D+0
 0.0000D+0
 0.0000D+0
10.0000D+0
 0.0000D+0
 0.0000D+0
 0.0000D+0
-11.0000D+0
 0.0000D+0
 0.0000D+0
 0.0000D+0
 0.0000D+0
 0.0000D+0
 2.0000D+0
-4.0000D+0
 0.0000D+0
 0.0000D+0
 0.0000D+0
 0.0000D+0
 8.0000D+0
 0.0000D+0
-10.0000D+0

SCAEXRHS.dat is: 

   6  1
     72.000000000000000000D+00
      0.000000000000000000D+00
    160.000000000000000000D+00
      0.000000000000000000D+00
      0.000000000000000000D+00
      0.000000000000000000D+00

      PROGRAM PDSCAEX
*
*  -- ScaLAPACK example code --
*     University of Tennessee, Knoxville, Oak Ridge National Laboratory,
*     and University of California, Berkeley.
*
*     Written by Antoine Petitet, (petitet@cs.utk.edu)
*
*     This program solves a linear system by calling the ScaLAPACK 
*     routine PDGESV. The input matrix and right-and-sides are
*     read from a file. The solution is written to a file.
*
*     .. Parameters ..
      INTEGER            DBLESZ, INTGSZ, MEMSIZ, TOTMEM
      PARAMETER          ( DBLESZ = 8, INTGSZ = 4, TOTMEM = 2000000,
     $                     MEMSIZ = TOTMEM / DBLESZ )
      INTEGER            BLOCK_CYCLIC_2D, CSRC_, CTXT_, DLEN_, DTYPE_,
     $                   LLD_, MB_, M_, NB_, N_, RSRC_
      PARAMETER          ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1,
     $                     CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6,
     $                     RSRC_ = 7, CSRC_ = 8, LLD_ = 9 )
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      CHARACTER*80       OUTFILE
      INTEGER            IAM, ICTXT, INFO, IPA, IPACPY, IPB, IPPIV, IPX,
     $                   IPW, LIPIV, MYCOL, MYROW, N, NB, NOUT, NPCOL,
     $                   NPROCS, NPROW, NP, NQ, NQRHS, NRHS, WORKSIZ
      DOUBLE PRECISION   ANORM, BNORM, EPS, XNORM, RESID
*     ..
*     .. Local Arrays ..
      INTEGER            DESCA( DLEN_ ), DESCB( DLEN_ ), DESCX( DLEN_ )
      DOUBLE PRECISION   MEM( MEMSIZ )
*     ..
*     .. External Subroutines ..
      EXTERNAL           BLACS_EXIT, BLACS_GET, BLACS_GRIDEXIT,
     $                   BLACS_GRIDINFO, BLACS_GRIDINIT, BLACS_PINFO,
     $                   DESCINIT, IGSUM2D, PDSCAEXINFO, PDGESV,
     $                   PDGEMM, PDLACPY, PDLAPRNT, PDLAREAD, PDLAWRITE
*     ..
*     .. External Functions ..
      INTEGER            ICEIL, NUMROC
      DOUBLE PRECISION   PDLAMCH, PDLANGE
      EXTERNAL           ICEIL, NUMROC, PDLAMCH, PDLANGE
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DBLE, MAX
*     ..
*     .. Executable Statements ..
*
*     Get starting information
*
      CALL BLACS_PINFO( IAM, NPROCS )
      CALL PDSCAEXINFO( OUTFILE, NOUT, N, NRHS, NB, NPROW, NPCOL, MEM,
     $                  IAM, NPROCS )
*
*     Define process grid
*
      CALL BLACS_GET( -1, 0, ICTXT )
      CALL BLACS_GRIDINIT( ICTXT, 'Row-major', NPROW, NPCOL )
      CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL )
*
*     Go to bottom of process grid loop if this case doesn't use my
*     process
*
      IF( MYROW.GE.NPROW .OR. MYCOL.GE.NPCOL )
     $   GO TO 20
*
      NP    = NUMROC( N, NB, MYROW, 0, NPROW )
      NQ    = NUMROC( N, NB, MYCOL, 0, NPCOL )
      NQRHS = NUMROC( NRHS, NB, MYCOL, 0, NPCOL )
*
*     Initialize the array descriptor for the matrix A and B
*
      CALL DESCINIT( DESCA, N, N, NB, NB, 0, 0, ICTXT, MAX( 1, NP ),
     $               INFO )
      CALL DESCINIT( DESCB, N, NRHS, NB, NB, 0, 0, ICTXT, MAX( 1, NP ),
     $               INFO )
      CALL DESCINIT( DESCX, N, NRHS, NB, NB, 0, 0, ICTXT, MAX( 1, NP ),
     $               INFO )
*
*     Assign pointers into MEM for SCALAPACK arrays, A is
*     allocated starting at position MEM( 1 )
*
      IPA = 1
      IPACPY = IPA + DESCA( LLD_ )*NQ
      IPB = IPACPY + DESCA( LLD_ )*NQ
      IPX = IPB + DESCB( LLD_ )*NQRHS
      IPPIV = IPX + DESCB( LLD_ )*NQRHS
      LIPIV = ICEIL( INTGSZ*( NP+NB ), DBLESZ )
      IPW = IPPIV + MAX( NP, LIPIV )
*
      WORKSIZ = MAX( NB, NP )
*
*     Check for adequate memory for problem size
*
      INFO = 0
      IF( IPW+WORKSIZ.GT.MEMSIZ ) THEN
         IF( IAM.EQ.0 )
     $      WRITE( NOUT, FMT = 9998 ) 'test', ( IPW+WORKSIZ )*DBLESZ
         INFO = 1
      END IF
*
*     Check all processes for an error
*
      CALL IGSUM2D( ICTXT, 'All', ' ', 1, 1, INFO, 1, -1, 0 )
      IF( INFO.GT.0 ) THEN
         IF( IAM.EQ.0 )
     $      WRITE( NOUT, FMT = 9999 ) 'MEMORY'
         GO TO 10
      END IF
*
*     Read from file and distribute matrices A and B
*
      CALL PDLAREAD( 'SCAEXMAT.dat', MEM( IPA ), DESCA, 0, 0,
     $               MEM( IPW ) )
      CALL PDLAREAD( 'SCAEXRHS.dat', MEM( IPB ), DESCB, 0, 0,
     $               MEM( IPW ) )
*
*     Make a copy of A and the rhs for checking purposes
*
      CALL PDLACPY( 'All', N, N, MEM( IPA ), 1, 1, DESCA,
     $              MEM( IPACPY ), 1, 1, DESCA )
      CALL PDLACPY( 'All', N, NRHS, MEM( IPB ), 1, 1, DESCB,
     $              MEM( IPX ), 1, 1, DESCX )
*
**********************************************************************
*     Call ScaLAPACK PDGESV routine
**********************************************************************
*
      IF( IAM.EQ.0 ) THEN
         WRITE( NOUT, FMT = * )
         WRITE( NOUT, FMT = * )
     $         '***********************************************'
         WRITE( NOUT, FMT = * )
     $         'Example of ScaLAPACK routine call: (PDGESV)'
         WRITE( NOUT, FMT = * )
     $         '***********************************************'
         WRITE( NOUT, FMT = * )
         WRITE( NOUT, FMT = * ) 'A * X = B, Matrix A:'
         WRITE( NOUT, FMT = * )
      END IF
      CALL PDLAPRNT( N, N, MEM( IPA ), 1, 1, DESCA, 0, 0,
     $               'A', NOUT, MEM( IPW ) )
      IF( IAM.EQ.0 ) THEN
         WRITE( NOUT, FMT = * )
         WRITE( NOUT, FMT = * ) 'Matrix B:'
         WRITE( NOUT, FMT = * )
      END IF
      CALL PDLAPRNT( N, NRHS, MEM( IPB ), 1, 1, DESCB, 0, 0,
     $               'B', NOUT, MEM( IPW ) )
*
      CALL PDGESV( N, NRHS, MEM( IPA ), 1, 1, DESCA, MEM( IPPIV ),
     $             MEM( IPB ), 1, 1, DESCB, INFO )
*
      IF( MYROW.EQ.0 .AND. MYCOL.EQ.0 ) THEN
         WRITE( NOUT, FMT = * )
         WRITE( NOUT, FMT = * ) 'INFO code returned by PDGESV = ', INFO
         WRITE( NOUT, FMT = * )
         WRITE( NOUT, FMT = * ) 'Matrix X = A^{-1} * B'
         WRITE( NOUT, FMT = * )
      END IF
      CALL PDLAPRNT( N, NRHS, MEM( IPB ), 1, 1, DESCB, 0, 0, 'X', NOUT,
     $               MEM( IPW ) )
      CALL PDLAWRITE( 'SCAEXSOL.dat', N, NRHS, MEM( IPB ), 1, 1, DESCB,
     $                0, 0, MEM( IPW ) )
*
*     Compute residual ||A * X  - B|| / ( ||X|| * ||A|| * eps * N )
*
      EPS = PDLAMCH( ICTXT, 'Epsilon' )
      ANORM = PDLANGE( 'I', N, N, MEM( IPA ), 1, 1, DESCA, MEM( IPW ) )
      BNORM = PDLANGE( 'I', N, NRHS, MEM( IPB ), 1, 1, DESCB,
     $                 MEM( IPW ) )
      CALL PDGEMM( 'No transpose', 'No transpose', N, NRHS, N, ONE,
     $             MEM( IPACPY ), 1, 1, DESCA, MEM( IPB ), 1, 1, DESCB,
     $             -ONE, MEM( IPX ), 1, 1, DESCX )
      XNORM = PDLANGE( 'I', N, NRHS, MEM( IPX ), 1, 1, DESCX,
     $                 MEM( IPW ) )
      RESID = XNORM / ( ANORM * BNORM * EPS * DBLE( N ) )
*
      IF( MYROW.EQ.0 .AND. MYCOL.EQ.0 ) THEN
         WRITE( NOUT, FMT = * )
         WRITE( NOUT, FMT = * )
     $     '||A * X  - B|| / ( ||X|| * ||A|| * eps * N ) = ', RESID
         WRITE( NOUT, FMT = * )
         IF( RESID.LT.10.0D+0 ) THEN
            WRITE( NOUT, FMT = * ) 'The answer is correct.'
         ELSE
            WRITE( NOUT, FMT = * ) 'The answer is suspicious.'
         END IF
      END IF
*
   10 CONTINUE
*
      CALL BLACS_GRIDEXIT( ICTXT )
*
   20 CONTINUE
*
*     Print ending messages and close output file
*
      IF( IAM.EQ.0 ) THEN
         WRITE( NOUT, FMT = * )
         WRITE( NOUT, FMT = * )
         WRITE( NOUT, FMT = 9997 )
         WRITE( NOUT, FMT = * )
         IF( NOUT.NE.6 .AND. NOUT.NE.0 )
     $      CLOSE ( NOUT )
      END IF
*
      CALL BLACS_EXIT( 0 )
*
 9999 FORMAT( 'Bad ', A6, ' parameters: going on to next test case.' )
 9998 FORMAT( 'Unable to perform ', A, ': need TOTMEM of at least',
     $        I11 )
 9997 FORMAT( 'END OF TESTS.' )
*
      STOP
*
*     End of PDSCAEX
*
      END

      SUBROUTINE PDSCAEXINFO( SUMMRY, NOUT, N, NRHS, NB, NPROW, NPCOL,
     $                        WORK, IAM, NPROCS )
*
*  -- ScaLAPACK example code --
*     University of Tennessee, Knoxville, Oak Ridge National Laboratory,
*     and University of California, Berkeley.
*
*     Written by Antoine Petitet, (petitet@cs.utk.edu)
*
*     This program solves a linear system by calling the ScaLAPACK 
*     routine PDGESV. The input matrix and right-and-sides are
*     read from a file. The solution is written to a file.
*
*     .. Scalar Arguments ..
      CHARACTER*( * )    SUMMRY
      INTEGER            IAM, N, NRHS, NB, NOUT, NPCOL, NPROCS, NPROW
*     ..
*     .. Array Arguments ..
      INTEGER            WORK( * )
*     ..
*
* ======================================================================
*
*     .. Parameters ..
      INTEGER            NIN
      PARAMETER          ( NIN = 11 )
*     ..
*     .. Local Scalars ..
      CHARACTER*79       USRINFO
      INTEGER            ICTXT
*     ..
*     .. External Subroutines ..
      EXTERNAL           BLACS_ABORT, BLACS_GET, BLACS_GRIDEXIT,
     $                   BLACS_GRIDINIT, BLACS_SETUP, IGEBR2D, IGEBS2D
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Process 0 reads the input data, broadcasts to other processes and
*     writes needed information to NOUT
*
      IF( IAM.EQ.0 ) THEN
*
*        Open file and skip data file header
*
         OPEN( NIN, FILE='SCAEX.dat', STATUS='OLD' )
         READ( NIN, FMT = * ) SUMMRY
         SUMMRY = ' '
*
*        Read in user-supplied info about machine type, compiler, etc.
*
         READ( NIN, FMT = 9999 ) USRINFO
*
*        Read name and unit number for summary output file
*
         READ( NIN, FMT = * ) SUMMRY
         READ( NIN, FMT = * ) NOUT
         IF( NOUT.NE.0 .AND. NOUT.NE.6 )
     $      OPEN( NOUT, FILE = SUMMRY, STATUS = 'UNKNOWN' )
*
*        Read and check the parameter values for the tests.
*
*        Get matrix dimensions
*
         READ( NIN, FMT = * ) N
         READ( NIN, FMT = * ) NRHS
*
*        Get value of NB
*
         READ( NIN, FMT = * ) NB
*
*        Get grid shape
*
         READ( NIN, FMT = * ) NPROW
         READ( NIN, FMT = * ) NPCOL
*
*        Close input file
*
         CLOSE( NIN )
*
*        If underlying system needs additional set up, do it now
*
         IF( NPROCS.LT.1 ) THEN
            NPROCS = NPROW * NPCOL
            CALL BLACS_SETUP( IAM, NPROCS )
         END IF
*
*        Temporarily define blacs grid to include all processes so
*        information can be broadcast to all processes
*
         CALL BLACS_GET( -1, 0, ICTXT )
         CALL BLACS_GRIDINIT( ICTXT, 'Row-major', 1, NPROCS )
*
*        Pack information arrays and broadcast
*
         WORK( 1 ) = N
         WORK( 2 ) = NRHS
         WORK( 3 ) = NB
         WORK( 4 ) = NPROW
         WORK( 5 ) = NPCOL
         CALL IGEBS2D( ICTXT, 'All', ' ', 5, 1, WORK, 5 )
*
*        regurgitate input
*
         WRITE( NOUT, FMT = 9999 )
     $               'SCALAPACK example driver.'
         WRITE( NOUT, FMT = 9999 ) USRINFO
         WRITE( NOUT, FMT = * )
         WRITE( NOUT, FMT = 9999 )
     $               'The matrices A and B are read from '//
     $               'a file.'
         WRITE( NOUT, FMT = * )
         WRITE( NOUT, FMT = 9999 )
     $               'An explanation of the input/output '//
     $               'parameters follows:'
*
         WRITE( NOUT, FMT = 9999 )
     $               'N       : The order of the matrix A.'
         WRITE( NOUT, FMT = 9999 )
     $               'NRHS    : The number of right and sides.'
         WRITE( NOUT, FMT = 9999 )
     $               'NB      : The size of the square blocks the'//
     $               ' matrices A and B are split into.'
         WRITE( NOUT, FMT = 9999 )
     $               'P       : The number of process rows.'
         WRITE( NOUT, FMT = 9999 )
     $               'Q       : The number of process columns.'
         WRITE( NOUT, FMT = * )
         WRITE( NOUT, FMT = 9999 )
     $               'The following parameter values will be used:'
         WRITE( NOUT, FMT = 9998 ) 'N    ', N
         WRITE( NOUT, FMT = 9998 ) 'NRHS ', NRHS
         WRITE( NOUT, FMT = 9998 ) 'NB   ', NB
         WRITE( NOUT, FMT = 9998 ) 'P    ', NPROW
         WRITE( NOUT, FMT = 9998 ) 'Q    ', NPCOL
         WRITE( NOUT, FMT = * )
*
      ELSE
*
*        If underlying system needs additional set up, do it now
*
         IF( NPROCS.LT.1 )
     $      CALL BLACS_SETUP( IAM, NPROCS )
*
*        Temporarily define blacs grid to include all processes so
*        information can be broadcast to all processes
*
         CALL BLACS_GET( -1, 0, ICTXT )
         CALL BLACS_GRIDINIT( ICTXT, 'Row-major', 1, NPROCS )
*
         CALL IGEBR2D( ICTXT, 'All', ' ', 5, 1, WORK, 5, 0, 0 )
         N     = WORK( 1 )
         NRHS  = WORK( 2 )
         NB    = WORK( 3 )
         NPROW = WORK( 4 )
         NPCOL = WORK( 5 )
*
      END IF
*
      CALL BLACS_GRIDEXIT( ICTXT )
*
      RETURN
*
   20 WRITE( NOUT, FMT = 9997 )
      CLOSE( NIN )
      IF( NOUT.NE.6 .AND. NOUT.NE.0 )
     $   CLOSE( NOUT )
      CALL BLACS_ABORT( ICTXT, 1 )
*
      STOP
*
 9999 FORMAT( A )
 9998 FORMAT( 2X, A5, '   :        ', I6 )
 9997 FORMAT( ' Illegal input in file ',40A,'.  Aborting run.' )
*
*     End of PDSCAEXINFO
*
      END

      SUBROUTINE PDLAREAD( FILNAM, A, DESCA, IRREAD, ICREAD, WORK )
*
*  -- ScaLAPACK example --
*     University of Tennessee, Knoxville, Oak Ridge National Laboratory,
*     and University of California, Berkeley.
* 
*     written by Antoine Petitet, (petitet@cs.utk.edu)
*
*     .. Scalar Arguments ..
      INTEGER            ICREAD, IRREAD
*     ..
*     .. Array Arguments ..
      CHARACTER*(*)      FILNAM
      INTEGER            DESCA( * )
      DOUBLE PRECISION   A( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  PDLAREAD reads from a file named FILNAM a matrix and distribute
*  it to the process grid.
*
*  Only the process of coordinates {IRREAD, ICREAD} read the file.
*
*  WORK must be of size >= MB_ = DESCA( MB_ ).
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            NIN
      PARAMETER          ( NIN = 11 )
      INTEGER            BLOCK_CYCLIC_2D, CSRC_, CTXT_, DLEN_, DTYPE_,
     $                   LLD_, MB_, M_, NB_, N_, RSRC_
      PARAMETER          ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1,
     $                     CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6,
     $                     RSRC_ = 7, CSRC_ = 8, LLD_ = 9 )
*     ..
*     .. Local Scalars ..
      INTEGER            H, I, IB, ICTXT, ICURCOL, ICURROW, II, J, JB,
     $                   JJ, K, LDA, M, MYCOL, MYROW, N, NPCOL, NPROW
*     ..
*     .. Local Arrays ..
      INTEGER            IWORK( 2 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           BLACS_GRIDINFO, INFOG2L, DGERV2D, DGESD2D,
     $                   IGEBS2D, IGEBR2D
*     ..
*     .. External Functions ..
      INTEGER            ICEIL
      EXTERNAL           ICEIL
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MIN
*     ..
*     .. Executable Statements ..
*
*     Get grid parameters
*
      ICTXT = DESCA( CTXT_ )
      CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL )
*
      IF( MYROW.EQ.IRREAD .AND. MYCOL.EQ.ICREAD ) THEN
         OPEN( NIN, FILE=FILNAM, STATUS='OLD' )
         READ( NIN, FMT = * ) ( IWORK( I ), I = 1, 2 )
         CALL IGEBS2D( ICTXT, 'All', ' ', 2, 1, IWORK, 2 )
      ELSE
         CALL IGEBR2D( ICTXT, 'All', ' ', 2, 1, IWORK, 2, IRREAD,
     $                 ICREAD )
      END IF
      M = IWORK( 1 )
      N = IWORK( 2 )
*
      IF( M.LE.0 .OR. N.LE.0 )
     $   RETURN
*
      IF( M.GT.DESCA( M_ ).OR. N.GT.DESCA( N_ ) ) THEN
         IF( MYROW.EQ.0 .AND. MYCOL.EQ.0 ) THEN
            WRITE( *, FMT = * ) 'PDLAREAD: Matrix too big to fit in'
            WRITE( *, FMT = * ) 'Abort ...'
         END IF
         CALL BLACS_ABORT( ICTXT, 0 )
      END IF
*
      II = 1
      JJ = 1
      ICURROW = DESCA( RSRC_ )
      ICURCOL = DESCA( CSRC_ )
      LDA = DESCA( LLD_ )
*
*     Loop over column blocks
*
      DO 50 J = 1, N, DESCA( NB_ )
         JB = MIN(  DESCA( NB_ ), N-J+1 )
         DO 40 H = 0, JB-1
*
*           Loop over block of rows
*
            DO 30 I = 1, M, DESCA( MB_ )
               IB = MIN( DESCA( MB_ ), M-I+1 )
               IF( ICURROW.EQ.IRREAD .AND. ICURCOL.EQ.ICREAD ) THEN
                  IF( MYROW.EQ.IRREAD .AND. MYCOL.EQ.ICREAD ) THEN
                     DO 10 K = 0, IB-1
                        READ( NIN, FMT = * ) A( II+K+(JJ+H-1)*LDA )
   10                CONTINUE
                  END IF
               ELSE
                  IF( MYROW.EQ.ICURROW .AND. MYCOL.EQ.ICURCOL ) THEN
                     CALL DGERV2D( ICTXT, IB, 1, A( II+(JJ+H-1)*LDA ),
     $                             LDA, IRREAD, ICREAD )
                   ELSE IF( MYROW.EQ.IRREAD .AND. MYCOL.EQ.ICREAD ) THEN
                     DO 20 K = 1, IB
                        READ( NIN, FMT = * ) WORK( K )
   20                CONTINUE
                     CALL DGESD2D( ICTXT, IB, 1, WORK, DESCA( MB_ ),
     $                             ICURROW, ICURCOL )
                  END IF
               END IF
               IF( MYROW.EQ.ICURROW )
     $            II = II + IB
               ICURROW = MOD( ICURROW+1, NPROW )
   30       CONTINUE
*
            II = 1
            ICURROW = DESCA( RSRC_ )
   40    CONTINUE
*
         IF( MYCOL.EQ.ICURCOL )
     $      JJ = JJ + JB
         ICURCOL = MOD( ICURCOL+1, NPCOL )
*
   50 CONTINUE
*
      IF( MYROW.EQ.IRREAD .AND. MYCOL.EQ.ICREAD ) THEN
         CLOSE( NIN )
      END IF
*
      RETURN
*
*     End of PDLAREAD
*
      END

      SUBROUTINE PDLAWRITE( FILNAM, M, N, A, IA, JA, DESCA, IRWRIT,
     $                      ICWRIT, WORK )
*
*  -- ScaLAPACK example --
*     University of Tennessee, Knoxville, Oak Ridge National Laboratory,
*     and University of California, Berkeley.
*
*     written by Antoine Petitet, (petitet@cs.utk.edu)
*
*     .. Scalar Arguments ..
      INTEGER            IA, ICWRIT, IRWRIT, JA, M, N
*     ..
*     .. Array Arguments ..
      CHARACTER*(*)      FILNAM
      INTEGER            DESCA( * )
      DOUBLE PRECISION   A( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  PDLAWRITE writes to a file named FILNAMa distributed matrix sub( A )
*  denoting A(IA:IA+M-1,JA:JA+N-1). The local pieces are sent to and
*  written by the process of coordinates (IRWWRITE, ICWRIT).
*
*  WORK must be of size >= MB_ = DESCA( MB_ ).
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            NOUT
      PARAMETER          ( NOUT = 13 )
      INTEGER            BLOCK_CYCLIC_2D, CSRC_, CTXT_, DLEN_, DTYPE_,
     $                   LLD_, MB_, M_, NB_, N_, RSRC_
      PARAMETER          ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1,
     $                     CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6,
     $                     RSRC_ = 7, CSRC_ = 8, LLD_ = 9 )
*     ..
*     .. Local Scalars ..
      INTEGER            H, I, IACOL, IAROW, IB, ICTXT, ICURCOL,
     $                   ICURROW, II, IIA, IN, J, JB, JJ, JJA, JN, K,
     $                   LDA, MYCOL, MYROW, NPCOL, NPROW
*     ..
*     .. External Subroutines ..
      EXTERNAL           BLACS_BARRIER, BLACS_GRIDINFO, INFOG2L,
     $                   DGERV2D, DGESD2D
*     ..
*     .. External Functions ..
      INTEGER            ICEIL
      EXTERNAL           ICEIL
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MIN
*     ..
*     .. Executable Statements ..
*
*     Get grid parameters
*
      ICTXT = DESCA( CTXT_ )
      CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL )
*
      IF( MYROW.EQ.IRWRIT .AND. MYCOL.EQ.ICWRIT ) THEN
         OPEN( NOUT, FILE=FILNAM, STATUS='UNKNOWN' )
         WRITE( NOUT, FMT = * ) M, N
      END IF
*
      CALL INFOG2L( IA, JA, DESCA, NPROW, NPCOL, MYROW, MYCOL,
     $              IIA, JJA, IAROW, IACOL )
      ICURROW = IAROW
      ICURCOL = IACOL
      II = IIA
      JJ = JJA
      LDA = DESCA( LLD_ )
*
*     Handle the first block of column separately
*
      JN = MIN( ICEIL( JA, DESCA( NB_ ) ) * DESCA( NB_ ), JA+N-1 )
      JB = JN-JA+1
      DO 60 H = 0, JB-1
         IN = MIN( ICEIL( IA, DESCA( MB_ ) ) * DESCA( MB_ ), IA+M-1 )
         IB = IN-IA+1
         IF( ICURROW.EQ.IRWRIT .AND. ICURCOL.EQ.ICWRIT ) THEN
            IF( MYROW.EQ.IRWRIT .AND. MYCOL.EQ.ICWRIT ) THEN
               DO 10 K = 0, IB-1
                  WRITE( NOUT, FMT = 9999 ) A( II+K+(JJ+H-1)*LDA )
   10          CONTINUE
            END IF
         ELSE
            IF( MYROW.EQ.ICURROW .AND. MYCOL.EQ.ICURCOL ) THEN
               CALL DGESD2D( ICTXT, IB, 1, A( II+(JJ+H-1)*LDA ), LDA,
     $                       IRWRIT, ICWRIT )
            ELSE IF( MYROW.EQ.IRWRIT .AND. MYCOL.EQ.ICWRIT ) THEN
               CALL DGERV2D( ICTXT, IB, 1, WORK, DESCA( MB_ ),
     $                       ICURROW, ICURCOL )
               DO 20 K = 1, IB
                  WRITE( NOUT, FMT = 9999 ) WORK( K )
   20          CONTINUE
            END IF
         END IF
         IF( MYROW.EQ.ICURROW )
     $      II = II + IB
         ICURROW = MOD( ICURROW+1, NPROW )
         CALL BLACS_BARRIER( ICTXT, 'All' )
*
*        Loop over remaining block of rows
*
         DO 50 I = IN+1, IA+M-1, DESCA( MB_ )
            IB = MIN( DESCA( MB_ ), IA+M-I )
            IF( ICURROW.EQ.IRWRIT .AND. ICURCOL.EQ.ICWRIT ) THEN
               IF( MYROW.EQ.IRWRIT .AND. MYCOL.EQ.ICWRIT ) THEN
                  DO 30 K = 0, IB-1
                     WRITE( NOUT, FMT = 9999 ) A( II+K+(JJ+H-1)*LDA )
   30             CONTINUE
               END IF
            ELSE
               IF( MYROW.EQ.ICURROW .AND. MYCOL.EQ.ICURCOL ) THEN
                  CALL DGESD2D( ICTXT, IB, 1, A( II+(JJ+H-1)*LDA ),
     $                          LDA, IRWRIT, ICWRIT )
               ELSE IF( MYROW.EQ.IRWRIT .AND. MYCOL.EQ.ICWRIT ) THEN
                  CALL DGERV2D( ICTXT, IB, 1, WORK, DESCA( MB_ ),
     $                          ICURROW, ICURCOL )
                  DO 40 K = 1, IB
                     WRITE( NOUT, FMT = 9999 ) WORK( K )
   40             CONTINUE
               END IF
            END IF
            IF( MYROW.EQ.ICURROW )
     $         II = II + IB
            ICURROW = MOD( ICURROW+1, NPROW )
            CALL BLACS_BARRIER( ICTXT, 'All' )
   50    CONTINUE
*
        II = IIA
        ICURROW = IAROW
   60 CONTINUE
*
      IF( MYCOL.EQ.ICURCOL )
     $   JJ = JJ + JB
      ICURCOL = MOD( ICURCOL+1, NPCOL )
      CALL BLACS_BARRIER( ICTXT, 'All' )
*
*     Loop over remaining column blocks
*
      DO 130 J = JN+1, JA+N-1, DESCA( NB_ )
         JB = MIN(  DESCA( NB_ ), JA+N-J )
         DO 120 H = 0, JB-1
            IN = MIN( ICEIL( IA, DESCA( MB_ ) ) * DESCA( MB_ ), IA+M-1 )
            IB = IN-IA+1
            IF( ICURROW.EQ.IRWRIT .AND. ICURCOL.EQ.ICWRIT ) THEN
               IF( MYROW.EQ.IRWRIT .AND. MYCOL.EQ.ICWRIT ) THEN
                  DO 70 K = 0, IB-1
                     WRITE( NOUT, FMT = 9999 ) A( II+K+(JJ+H-1)*LDA )
   70             CONTINUE
               END IF
            ELSE
               IF( MYROW.EQ.ICURROW .AND. MYCOL.EQ.ICURCOL ) THEN
                  CALL DGESD2D( ICTXT, IB, 1, A( II+(JJ+H-1)*LDA ),
     $                          LDA, IRWRIT, ICWRIT )
               ELSE IF( MYROW.EQ.IRWRIT .AND. MYCOL.EQ.ICWRIT ) THEN
                  CALL DGERV2D( ICTXT, IB, 1, WORK, DESCA( MB_ ),
     $                          ICURROW, ICURCOL )
                  DO 80 K = 1, IB
                     WRITE( NOUT, FMT = 9999 ) WORK( K )
   80             CONTINUE
               END IF
            END IF
            IF( MYROW.EQ.ICURROW )
     $         II = II + IB
            ICURROW = MOD( ICURROW+1, NPROW )
            CALL BLACS_BARRIER( ICTXT, 'All' )
*
*           Loop over remaining block of rows
*
            DO 110 I = IN+1, IA+M-1, DESCA( MB_ )
               IB = MIN( DESCA( MB_ ), IA+M-I )
               IF( ICURROW.EQ.IRWRIT .AND. ICURCOL.EQ.ICWRIT ) THEN
                  IF( MYROW.EQ.IRWRIT .AND. MYCOL.EQ.ICWRIT ) THEN
                     DO 90 K = 0, IB-1
                        WRITE( NOUT, FMT = 9999 ) A( II+K+(JJ+H-1)*LDA )
   90                CONTINUE
                  END IF
               ELSE
                  IF( MYROW.EQ.ICURROW .AND. MYCOL.EQ.ICURCOL ) THEN
                     CALL DGESD2D( ICTXT, IB, 1, A( II+(JJ+H-1)*LDA ),
     $                             LDA, IRWRIT, ICWRIT )
                   ELSE IF( MYROW.EQ.IRWRIT .AND. MYCOL.EQ.ICWRIT ) THEN
                     CALL DGERV2D( ICTXT, IB, 1, WORK, DESCA( MB_ ),
     $                             ICURROW, ICURCOL )
                     DO 100 K = 1, IB
                        WRITE( NOUT, FMT = 9999 ) WORK( K )
  100                CONTINUE
                  END IF
               END IF
               IF( MYROW.EQ.ICURROW )
     $            II = II + IB
               ICURROW = MOD( ICURROW+1, NPROW )
               CALL BLACS_BARRIER( ICTXT, 'All' )
  110       CONTINUE
*
            II = IIA
            ICURROW = IAROW
  120    CONTINUE
*
         IF( MYCOL.EQ.ICURCOL )
     $      JJ = JJ + JB
         ICURCOL = MOD( ICURCOL+1, NPCOL )
         CALL BLACS_BARRIER( ICTXT, 'All' )
*
  130 CONTINUE
*
      IF( MYROW.EQ.IRWRIT .AND. MYCOL.EQ.ICWRIT ) THEN
         CLOSE( NOUT )
      END IF
*
 9999 FORMAT( D30.18 )
*
      RETURN
*
*     End of PDLAWRITE
*
      END


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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HPF Interface to ScaLAPACK

        

We are investigating issues related to interfacing ScaLAPACK with High Performance Fortran (HPF) [91]. As a part of this effort, we have provided prototype interfaces to some of the ScaLAPACK routines. We are collecting user feedback on these codes, as well as allowing additional time for compiler maturation, before providing a more complete interface.

Initially, interfaces are provided for the following ScaLAPACK routines: the general and symmetric positive definite linear equation solvers (PxGESV and PxPOSV), the linear least squares solver (PxGELS), and the PBLAS matrix multiply routine (PxGEMM).             

      LA_GESV(A, B, IPIV, INFO)
         TYPE, intent(inout), dimension(:,:) :: A, B
         integer, optional, intent(out) :: IPIV(:), INFO
 
      LA_POSV(A, B, UPLO, INFO)
         TYPE, intent(inout), dimension(:,:) :: A, B
         character(LEN=1), optional, intent(in) :: UPLO
         integer, optional, intent(out) :: INFO
 
      LA_GELS(A, B, TRANS, INFO)
         TYPE, intent(inout), dimension(:,:) :: A, B
         character(LEN=1), optional, intent(in) :: TRANS
         integer, optional, intent(out) :: INFO
 
      LA_GEMM(A, B, C, transA, transB, alpha, beta)
         TYPE, intent(in), dimension(:,:) :: A, B
         TYPE, intent(inout), dimension(:,:) :: C
         character(LEN=1), optional, intent(in) :: transA, transB
         TYPE, optional, intent(in) :: alpha, beta

With this interface, all matrices are inherited, and query functions are used to determine the distribution of the matrices. Only when ScaLAPACK cannot handle the user's distribution are the matrices redistributed. In such a case, it is done transparently to the user, and only performance will show that it has occurred.

The prototype interfaces can be downloaded from netlib at the following URL:

http://www.netlib.org/scalapack/prototypes/slhpf.tar.gz

Questions or comments on these routines may be mailed to scalapack@cs.utk.edu.

The following example code is a complete HPF code calling and testing the ScaLAPACK LU factorization/solve in HPF.

This program is also available in the scalapack directory on netlib
(http://www.netlib.org/scalapack/examples/sample_hpf_gesv.f). 

      program simplegesv
      use HPF_LAPACK
      integer, parameter :: N=500, NRHS=20, NB=64, NBRHS=64, P=1, Q=3
      integer, parameter :: DP=kind(0.0D0)
      integer :: IPIV(N)
      real(DP) :: A(N, N), X(N, NRHS), B(N, NRHS)
!HPF$ PROCESSORS PROC(P,Q)
!HPF$ DISTRIBUTE A(cyclic(NB), cyclic(NB)) ONTO PROC
!HPF$ DISTRIBUTE (cyclic(NB), cyclic(NBRHS)) ONTO PROC :: B, X

!
!     Randomly generate the coefficient matrix A and the solution
!     matrix X.  Set the right hand side matrix B such that B = A * X.
!
      call random_number(A)
      call random_number(X)
      B = matmul(A, X)
!
!     Solve the linear system; the computed solution overwrites B
!
      call la_gesv(A, B, IPIV)
!
!     As a simple test, print the largest difference (in absolute value)
!     between the computed solution (B) and the generated solution (X).
!
      print*,'MAX( ABS(X~ - X) ) = ',maxval( abs(B - X) )
!
!     Shutdown the ScaLAPACK system, I'm done
!
      call SLhpf_exit()

      stop
      end


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Quick Reference Guides

   

Quick References Guides are provided for ScaLAPACK, the PBLAS, and the BLACS. In addition, quick reference guides are also available for LAPACK and the BLAS.

http://www.netlib.org/scalapack/scalapackqref.ps
http://www.netlib.org/scalapack/pblasqref.ps
http://www.netlib.org/blacs/cblacsqref.ps
http://www.netlib.org/blacs/f77blacsqref.ps
http://www.netlib.org/blas/blasqr.ps


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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ScaLAPACK Quick Reference Guide

 

A postscript version of this Quick Reference Guide is available on the ScaLAPACK homepage.

http://www.netlib.org/scalapack/scalapackqref.ps


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Expert Drivers


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Meaning of prefixes
Routines beginning with ``PS'' are available in:

PS - REAL
PD - DOUBLE PRECISION

Routines beginning with ``PC'' are available in:

PC - COMPLEX
PZ - COMPLEX*16

Note: COMPLEX*16 may not be supported by all machines


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Quick Reference Guide to the PBLAS

 

An html version of this Quick Reference Guide , along with the leading comments from each of the routines, is available via the ScaLAPACK homepage.

http://www.netlib.org/scalapack/index.html

At the lowest level, the efficiency of the PBLAS is determined by the local performance of the BLAS and the BLACS. In addition, depending on the shape of its input and output distributed matrices, the PBLAS select the best algorithm in terms of data transfer across the process grid. Transparent to the user, this relatively simple selection process ensures high efficiency independent of the actual computation performed.

Level 1 PBLAS 

         dim scalar     vector                  vector
P_SWAP ( N,             X, IX, JX, DESCX, INCX, Y, IY, JY, DESCY, INCY )
P_SCAL ( N, ALPHA,      X, IX, JX, DESCX, INCX )
P_COPY ( N,             X, IX, JX, DESCX, INCX, Y, IY, JY, DESCY, INCY )
P_AXPY ( N, ALPHA,      X, IX, JX, DESCX, INCX, Y, IY, JY, DESCY, INCY )
P_DOT  ( N, DOT,        X, IX, JX, DESCX, INCX, Y, IY, JY, DESCY, INCY )
P_DOTU ( N, DOTU,       X, IX, JX, DESCX, INCX, Y, IY, JY, DESCY, INCY )
P_DOTC ( N, DOTC,       X, IX, JX, DESCX, INCX, Y, IY, JY, DESCY, INCY )
P_NRM2 ( N, NORM2,      X, IX, JX, DESCX, INCX )
P_ASUM ( N, ASUM,       X, IX, JX, DESCX, INCX )
P_AMAX ( N, AMAX, INDX, X, IX, JX, DESCX, INCX )

Level 2 PBLAS 

         options            dim   scalar matrix            vector                 scalar vector
P_GEMV (       TRANS,       M, N, ALPHA, A, IA, JA, DESCA, X, IX, JX, DESCX, INCX, BETA, Y, IY, JY, DESCY, INCY )
P_HEMV ( UPLO,                 N, ALPHA, A, IA, JA, DESCA, X, IX, JX, DESCX, INCX, BETA, Y, IY, JY, DESCY, INCY )
P_SYMV ( UPLO,                 N, ALPHA, A, IA, JA, DESCA, X, IX, JX, DESCX, INCX, BETA, Y, IY, JY, DESCY, INCY )
P_TRMV ( UPLO, TRANS, DIAG,    N,        A, IA, JA, DESCA, X, IX, JX, DESCX, INCX )
P_TRSV ( UPLO, TRANS, DIAG,    N,        A, IA, JA, DESCA, X, IX, JX, DESCX, INCX )

         options            dim   scalar vector                  vector                  matrix
P_GER  (                    M, N, ALPHA, X, IX, JX, DESCX, INCX, Y, IY, JY, DESCY, INCY, A, IA, JA, DESCA )
P_GERU (                    M, N, ALPHA, X, IX, JX, DESCX, INCX, Y, IY, JY, DESCY, INCY, A, IA, JA, DESCA )
P_GERC (                    M, N, ALPHA, X, IX, JX, DESCX, INCX, Y, IY, JY, DESCY, INCY, A, IA, JA, DESCA )
P_HER  ( UPLO,                 N, ALPHA, X, IX, JX, DESCX, INCX,                         A, IA, JA, DESCA )
P_HER2 ( UPLO,                 N, ALPHA, X, IX, JX, DESCX, INCX, Y, IY, JY, DESCY, INCY, A, IA, JA, DESCA )
P_SYR  ( UPLO,                 N, ALPHA, X, IX, JX, DESCX, INCX,                         A, IA, JA, DESCA )
P_SYR2 ( UPLO,                 N, ALPHA, X, IX, JX, DESCX, INCX, Y, IY, JY, DESCY, INCY, A, IA, JA, DESCA )

Level 3 PBLAS 

         options                           dim      scalar matrix            matrix            scalar matrix
P_GEMM (             TRANSA, TRANSB,       M, N, K, ALPHA, A, IA, JA, DESCA, B, IB, JB, DESCB, BETA, C, IC, JC, DESCC )
P_SYMM ( SIDE, UPLO,                       M, N,    ALPHA, A, IA, JA, DESCA, B, IB, JB, DESCB, BETA, C, IC, JC, DESCC )
P_HEMM ( SIDE, UPLO,                       M, N,    ALPHA, A, IA, JA, DESCA, B, IB, JB, DESCB, BETA, C, IC, JC, DESCC )
P_SYRK (       UPLO, TRANS,                   N, K, ALPHA, A, IA, JA, DESCA,                   BETA, C, IC, JC, DESCC )
P_HERK (       UPLO, TRANS,                   N, K, ALPHA, A, IA, JA, DESCA,                   BETA, C, IC, JC, DESCC )
P_SYR2K(       UPLO, TRANS,                   N, K, ALPHA, A, IA, JA, DESCA, B, IB, JB, DESCB, BETA, C, IC, JC, DESCC )
P_HER2K(       UPLO, TRANS,                   N, K, ALPHA, A, IA, JA, DESCA, B, IB, JB, DESCB, BETA, C, IC, JC, DESCC )
P_TRAN (                                   M, N,    ALPHA, A, IA, JA, DESCA,                   BETA, C, IC, JC, DESCC )
P_TRANU(                                   M, N,    ALPHA, A, IA, JA, DESCA,                   BETA, C, IC, JC, DESCC )
P_TRANC(                                   M, N,    ALPHA, A, IA, JA, DESCA,                   BETA, C, IC, JC, DESCC )
P_TRMM ( SIDE, UPLO, TRANSA,         DIAG, M, N,    ALPHA, A, IA, JA, DESCA, B, IB, JB, DESCB )
P_TRSM ( SIDE, UPLO, TRANSA,         DIAG, M, N,    ALPHA, A, IA, JA, DESCA, B, IB, JB, DESCB )


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Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Quick Reference Guide to the BLACS

     

An html version of this Quick Reference Guide, along with the leading comments from each of the routines, is available on the BLACS homepage.

http://www.netlib.org/blacs/index.html


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Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Glossary

 

The following is a glossary of terms and notation used throughout this users guide and the leading comments of the source code. The first time notation from this glossary appears in the text, it will be italicized.

  • Array descriptor: Contains the information required to establish the mapping between a global matrix entry and its corresponding process and memory location  .

    The notations x_ used in the entries of the array descriptor denote the attributes of a global matrix. For example, M_ denotes the number of rows, and M_A specifically denotes the number of rows in global matrix A. See sections 4.2,  4.3.3,  4.4.5,  4.4.6, and  4.5.1 for complete details.

  • BLACS : Basic Linear Algebra Communication Subprograms, a message-passing library designed for linear algebra. They provide a portability layer for communication between ScaLAPACK and message-passing systems such as MPI and PVM, as well as native message-passing libraries such as NX and MPL. See section 1.3.4.
  • BLAS : Basic Linear Algebra Subprograms [57, 59, 93], a standard for subroutines for common linear algebra computations such as dot-products, matrix-vector multiplication, and matrix-matrix multiplication. They provide a portability layer for computation. See section 1.3.2.
  • Block size: The number of contiguous rows or columns of a global matrix to be distributed consecutively to each of the processes in the process grid. The block size is quantified by the notation tex2html_wrap_inline14817, where MB is the row block size and NB is the column block size.

    The distribution block size can be square, MB=NB, or rectangular, tex2html_wrap_inline20032. Block size  is also referred to as the partitioning unit  or blocking factor .

  • Distributed memory computer: A term used in two senses:
    • A computer marketed as a distributed memory computer (such as the Cray T3 computers, the IBM SP computers, or the Intel Paragon), including one or more message-passing libraries.
    • A distributed shared-memory computer (e.g., the Origin 2000) or network of workstations (e.g., the Berkeley NOW) with message passing.
    ScaLAPACK delivers high performance on these computers provided that they include certain key features such as an efficient message-passing system, a one-to-one mapping of processes to processors, a gang scheduler and a well-connected communication network.
  • Distribution : Method by which the entries of a global matrix are allocated among the processes, also commonly referred to as decomposition   or data layout . Examples of distributions used by ScaLAPACK include block and block-cyclic distributions and these will be illustrated and explained in detail later.

    Data distribution in ScaLAPACK is controlled primarily by the process grid and the block size.

  • Global: A term ``global''   used in two ways:
    • To define the mathematical matrix , e.g. the global matrix A.
    • To identify arguments that must have the same value on all processes .
  • tex2html_wrap_inline12124(K_) : Number of columns that a process receives if tex2html_wrap_inline12126 columns of a matrix are distributed over c columns of its process row.

    To be consistent in notation, we have used a ``modifying character'' subscript on LOC to denote the dimension of the process grid to which we are referring. The subscript ``r'' indicates ``row'' whenever it is appended to LOC; likewise, the subscript ``c'' indicates ``column'' when it is appended to LOC.

    The value of tex2html_wrap_inline12124() may differ from process to process within the process grid. For example, in figure 4.6 (section 4.3.4), we can see that for process (0,0) tex2html_wrap_inline12124(N_)= 4; however, for process (0,1) tex2html_wrap_inline12124(N_) = 3.

  • tex2html_wrap_inline12132(K_) : Number of rows that a process would receive if tex2html_wrap_inline12126 rows of a matrix are distributed over r rows of its process column.

    To be consistent in notation, we have used a ``modifying character'' subscript on LOC to denote the dimension of the process grid to which we are referring. The subscript ``r'' indicates ``row'' whenever it is appended to LOC; likewise, the subscript ``c'' indicates ``column'' when it is appended to LOC.

    The value of tex2html_wrap_inline12132() may differ from process to process within the process grid. For example, in figure 4.6 (section 4.3.4), we can see that for process (0,0) tex2html_wrap_inline12132(M_)= 5; however, for process (1,0) tex2html_wrap_inline12132(M_) = 4.

  • Local: A term    used in two ways:
    • To express the array elements or blocks stored on each process, e.g., the local part of the global matrix A, also referred to as the local array . The size of the local array may differ from process to process. See section 2.3 for further details.
    • To identify arguments that may have different values on different processes. 

  • Local leading dimension   of a local array: Specification of entry size for local array. When a global array is distributed among the processes in the process grid, locally the entries are stored in a two-dimensional array, the size of which may vary from process to process. Thus, a leading dimension needs to be specified for each local array. For example, in Figure 2.2 in section 2.3, we can see that for process (0,0) the local leading dimension of the local array A (denoted tex2html_wrap_inline12116) is 5, whereas for process (1,0) the local leading dimension of local array A is 4.
  • MYCOL : The calling process's column coordinate in the process grid. Each process within the process grid is uniquely identified by its process coordinates (MYROW, MYCOL).
  • MYROW : The calling process's row coordinate in the process grid. Each process within the process grid is uniquely identified by its process coordinates (MYROW, MYCOL).
  • P : The total number of processes in the process grid, i.e., tex2html_wrap_inline17184.

    In terms of notation for process grids, we have used a ``modifying character'' subscript on P to denote the dimension of the process grid to which we are referring. The subscript ``r'' indicates ``row'' whenever it is appended to P, and thus tex2html_wrap_inline12172 is the number of process rows in the process grid. Likewise, the subscript ``c'' indicates ``column'' when it is appended to P, and thus tex2html_wrap_inline12162 is the number of process columns in the process grid.

  • tex2html_wrap_inline12162 : The number of process columns in the process grid (i.e., the second dimension of the two-dimensional process grid).
  • tex2html_wrap_inline12172 : The number of process rows in the process grid (i.e., the first dimension of the two-dimensional process grid).
  • PBLAS : A distributed-memory version of the BLAS (Basic Linear Algebra Subprograms), also referred to as the Parallel BLAS or Parallel Basic Linear Algebra Subprograms. Refer to section 1.3.3 for further details.
  • Process: Basic unit or thread of execution  that minimally includes a stack, registers, and memory. Multiple processes may share a physical processor. The term processor  refers to the actual hardware.

    In ScaLAPACK, each process is treated as if it were a processor: the process must exist for the lifetime of the ScaLAPACK run, and its execution should affect other processes' execution only through the use of message-passing calls. With this in mind, we use the term process in all sections of this users guide except those dealing with timings. When discussing timings, we specify processors as our unit of execution, since speedup will be determined largely by actual hardware resources.

    In ScaLAPACK, algorithms are presented in terms of processes, rather than physical processors. In general there may be several processes on a processor, in which case we assume that the runtime system handles the scheduling of processes. In the absence of such a runtime system, ScaLAPACK assumes one process per processor.

  • Process column : A specific column of processes within the two-dimensional process grid. For further details, consult the definition of process grid.
  • Process grid  : The way we logically view a parallel machine as a one- or two-dimensional rectangular grid of processes.

    For two-dimensional process grids, the variable tex2html_wrap_inline12172  is used to indicate the number of rows in the process grid (i.e., the first dimension of the two-dimensional process grid). The variable tex2html_wrap_inline12162  is used to indicate the number of columns in the process grid (i.e., the second dimension of the two-dimensional process grid). The collection of processes need not physically be connected in the two-dimensional process grid.

    For example, the following figure shows six processes mapped to a tex2html_wrap_inline14922 grid, where tex2html_wrap_inline12422 and tex2html_wrap_inline12424.

    figure7281

    A user may perform an operation within a process row or process column of the process grid. A process row refers to a specific row of processes within the process grid, and a process column refers to a specific column of processes with the process grid. In the example, process row 0 contains the processes with natural ordering 0, 1, and 2, and process column 0 contains the processes with natural ordering 0 and 3.

    For further details, please refer to section 4.1.1.

  • Process row : A specific row of processes within the two-dimensional process grid. For further details, consult the definition of process grid.
  • Scope : A term used in two ways:
    • The portion of the process grid within which an operation is defined. For example, in the Level 1 PBLAS, the resultant output array or scalar will be global or local within a process column or row of the process grid, and undefined elsewhere .

      Equivalently, in Appendix D.3, scope indicates the processes that participate in the broadcast or global combine operations. Scope can equal ``all'', ``row'', or ``column'' .

    • The portion of the parallel program within which the definition of an argument remains unchanged. When the scope of an argument is defined as global , the argument must have the same value on all processes. When the scope of an argument is defined as local , the argument may have different values on different processes .
    Refer to section 4.1.3 for further details.

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Specifications of Routines

 


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Notes

  1. The specifications that follow give the calling sequence, purpose, and descriptions of the arguments of each ScaLAPACK driver and computational routine (but not of auxiliary routines).
  2. Specifications of pairs of real and complex routines have been merged (for example PSGETRF/PCGETRF).
  3. Specifications are given only for single-precision routines. To adapt them for the double precision version of the software, simply interpret REAL as DOUBLE PRECISION, COMPLEX as COMPLEX*16 (or DOUBLE COMPLEX), and the initial letters PS- and PC- of ScaLAPACK routine names as PD- and PZ-.
  4. Specifications are arranged in alphabetical order of the real routine name.
  5. The text of the specifications has been derived from the leading comments in the source-text of the routines. It makes only limited use of mathematical typesetting facilities. To eliminate redundancy, tex2html_wrap_inline12724 has been used throughout the specifications. Thus, the reader should note that tex2html_wrap_inline12724 is equivalent to tex2html_wrap_inline12722 in the real case.
  6. If there is a discrepancy between the specifications listed in this section and the actual source code, the source code should be regarded as the most up to date.

Included in the leading comments of each subroutine (immediately preceding the Argument section) is a brief note describing the array descriptor and some commonly used expressions in calculating workspace. For brevity, we have listed this information below and not included it in the specifications of the routines. 

*  Notes
*  =====
*
*  Each global data object is described by an associated description
*  vector.  This vector stores the information required to establish
*  the mapping between an object element and its corresponding process
*  and memory location.
*
*  Let A be a generic term for any 2D block cyclicly distributed array.
*  Such a global array has an associated description vector DESCA.
*  In the following comments, the character _ should be read as
*  "of the global array".
*
*  NOTATION        STORED IN      EXPLANATION
*  --------------- -------------- --------------------------------------
*  DTYPE_A(global) DESCA( DTYPE_ )The descriptor type.  In this case,
*                                 DTYPE_A = 1.
*  CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
*                                 the BLACS process grid A is distribu-
*                                 ted over. The context itself is glo-
*                                 bal, but the handle (the integer
*                                 value) may vary.
*  M_A    (global) DESCA( M_ )    The number of rows in the global
*                                 array A.
*  N_A    (global) DESCA( N_ )    The number of columns in the global
*                                 array A.
*  MB_A   (global) DESCA( MB_ )   The blocking factor used to distribute
*                                 the rows of the array.
*  NB_A   (global) DESCA( NB_ )   The blocking factor used to distribute
*                                 the columns of the array.
*  RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
*                                 row of the array A is distributed.
*  CSRC_A (global) DESCA( CSRC_ ) The process column over which the
*                                 first column of the array A is
*                                 distributed.
*  LLD_A  (local)  DESCA( LLD_ )  The leading dimension of the local
*                                 array.  LLD_A >= MAX(1,LOCr(M_A)).
*
*  Let K be the number of rows or columns of a distributed matrix,
*  and assume that its process grid has dimension p x q.
*  LOCr( K ) denotes the number of elements of K that a process
*  would receive if K were distributed over the p processes of its
*  process column.
*  Similarly, LOCc( K ) denotes the number of elements of K that a
*  process would receive if K were distributed over the q processes of
*  its process row.
*  The values of LOCr() and LOCc() may be determined via a call to the
*  ScaLAPACK tool function, NUMROC:
*          LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
*          LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).
*  An upper bound for these quantities may be computed by:
*          LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
*          LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A





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Susan Blackford
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References

1
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3
E. ANDERSON, Z. BAI, C. BISCHOF, J. DEMMEL, J. DONGARRA, J. DU CROZ, A. GREENBAUM, S. HAMMARLING, A. MCKENNEY, S. OSTROUCHOV, AND D. SORENSEN, LAPACK Users' Guide, Society for Industrial and Applied Mathematics, Philadelphia, PA, second ed., 1995.

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5
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6
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13
Z. BAI, J. DEMMEL, J. DONGARRA, A. PETITET, H. ROBINSON, AND K. STANLEY, The spectral decomposition of nonsymmetric matrices on distributed memory computers, Computer Science Dept. Technical Report CS-95-273, University of Tennessee, Knoxville, TN, 1995. (Also LAPACK Working Note No. 91), To appear in SIAM J. Sci. Stat. Comput.

14
Z. BAI AND J. W. DEMMEL, Design of a parallel nonsymmetric eigenroutine toolbox, Part I, in Proceedings of the Sixth SIAM Conference on Parallel Processing for Scientific Computing, R. F. et al. Sincovec, ed., Philadelphia, PA, 1993, Society for Industrial and Applied Mathematics, pp. 391-398. Long version available as Computer Science Report CSD-92-718, University of California, Berkeley, 1992.

15
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16
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17
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18
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19
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20
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21
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22
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23
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24
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26
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27
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28
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29
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30
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31
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32
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33
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36
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37
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40
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41
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42
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43
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45
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46
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47
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48
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49
J. DONGARRA, C. RANDRIAMARO, L. PRYLLI, AND B. TOURANCHEAU, Array redistribution in ScaLAPACK using PVM, in EuroPVM users' group, Hermes, 1995.

50
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51
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52
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53
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54
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55
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56
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59
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61
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62
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63
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68
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78
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79
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80
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94
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96
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Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: Installation Up: Essentials Previous: Availability

Commercial Use

  

LAPACK and ScaLAPACK are freely available software packages provided on the World Wide Web via netlib, anonymous ftp, and http access. Thus they can be included in commercial packages (and have been). We ask only that proper credit be given to the authors.

Like all software, these packages are copyrighted. They are not trademarked; however, if modifications are made that affect the interface, functionality, or accuracy of the resulting software, the name of the routine should be changed. Any modification to our software should be noted in the modifier's documentation.

We will gladly answer questions regarding our software. If modifications are made to the software, however, it is the responsibility of the individuals/company who modified the routine to provide support.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: About this document Up: ScaLAPACK Users' Guide Previous: References

Index

absolute error , gif , gif
absolute gap , gif , gif
accuracy , gif
accuracy
high , gif , gif , gif
Active Messages
algorithmic
coherence
reliability , gif , gif
repeatability
algorithms, block-partitioned , gif
alignment restrictions
AM-II, see Active Messages
angle between vectors and subspaces , gif , gif , gif , gif , gif , gif , gif , gif
arguments
BERR
description conventions , gif , gif , gif
descriptor , gif , gif , gif , gif , gif
DIAG , gif
dimensions
FERR , gif
global
global column index , gif , gif , gif
global input
global output
global row index , gif , gif , gif
IA and JA , gif
INFO , gif , gif
local
local input
local output
LWORK
options
order of
RCOND
SIDE
TRANS
UPLO
work space
ARPACK
array argument descriptions
block cyclic arrays
block-column and block-row arrays
array descriptor , gif , gif , gif , gif , gif , gif
array descriptor
block , gif , gif
block cyclic
CTXT_
definition of
DLEN_
DTYPE_ , gif
example
LLD_
M_ and N_ , gif
MB_ and NB_ , gif
out-of-core
RSRC_ and CSRC_
auxiliary routines , gif
auxiliary routines, index of: see Appendix A
avoiding poor performance
backward error , gif , gif , gif , gif , gif , gif
backward stability , gif , gif
backward stability
componentwise
normwise
band matrix, distribution of
bandwidth
Berkeley NOW
bidiagonal form , gif , gif
bidiagonal matrix, distribution of
BLACS , gif
BLACS
BLACS_ABORT , gif
BLACS_BARRIER
BLACS_EXIT
BLACS_GRIDEXIT , gif
BLACS_GRIDINFO
BLACS_GRIDINIT
BLACS_SET , gif , gif
C interface
context
context handle
efficiency , gif
Fortran 77 interface
prebuilt libraries, availability of
quick reference guide: see Appendix D
BLAS , gif , gif , gif
BLAS
Level 1
Level 2
Level 3
block-column distribution , gif , gif
block-partitioned algorithms , gif , gif , gif
block-row distribution , gif , gif
block cyclic distribution , gif
blocking factor, see block size
block size , gif
block size
size of , gif
bug reports
bug reports
checklist
bug reports
mailing alias
Ctex2html_wrap_inline12068
Ctex2html_wrap_inline12072
Ctex2html_wrap_inline12076
Ctex2html_wrap_inline12068
Ctex2html_wrap_inline12076
Ctex2html_wrap_inline12072
CAPSS
checklist, high performance
chordal distance
cluster
eigenvalues
singular values
clusters of workstations
coefficient band matrix
coefficient matrix , gif , gif , gif , gif
coefficient matrix
distribution of , gif , gif
coefficient tridiagonal matrix , gif
coherence , gif , gif
column-major order
column pivoting (QR)
commercial use
communication bandwidth
communication deadlock
communication latency
communicator, MPI
complete orthogonal factorization
computational routines , gif , gif
Computational Routines, index of: see Appendix A
computers, see distributed memory computers
condensed form, reduction to
condition number , gif , gif , gif , gif , gif , gif , gif , gif , gif , gif
condition number
estimate , gif
context, BLACS
context handle
coprocessor
CoWs , gif
CSRC_, see array descriptor
CTXT_, see array descriptor
data distribution
data distribution
block
block cyclic , gif
block-column , gif , gif
block-row , gif , gif
general band matrix , gif
general tridiagonal matrix , gif
symmetric band matrix
symmetric positive definite band matrix
symmetric positive definite tridiagonal matrix
symmetric tridiagonal matrix
DDISNA , gif
deadlock
debugging
alignment restrictions
errata.blacs , gif
errata.scalapack , gif
INFO > 0
INFO < 0
INFO < 0
INFO > 0
synchronization points
TOOLS
workspace requirements
debugging hints
application
installation
debug level BLACS
decomposition, see data distribution
DESC_, see array descriptor
descriptor , gif , gif , gif , gif , gif
descriptor, see array descriptor
determining the size of local arrays , gif , gif
diagonally dominant-like , gif , gif , gif
distributed across process columns
distributed across process rows
distributed matrix
distributed matrix, partitioning of
distributed memory , gif , gif , gif , gif , gif
distributed memory computers , gif
distributed memory standards
distributed vector
distribution, see data distribution
DLEN_, see array descriptor
documentation
documentation
installation guide , gif
documentation, structure
downloading instructions
driver routine
generalized symmetric definite eigenvalue problem
linear equations
linear least squares
SVD
driver routine, symmetric eigenvalue problem
driver routines , gif , gif
driver routines
expert , gif
simple , gif
driver routines, index of: see Appendix A
DSTEQR2
DTYPE_, see array descriptor
E() , gif
effective rank
efficiency
efficiency
BLACS
BLAS
block-partitioned algorithms
efficiency, transportable
eigendecomposition
blocked form
symmetric
eigenvalue , gif , gif , gif , gif , gif
eigenvalue
error bound , gif , gif , gif , gif , gif
GSEP
SEP
eigenvalue problem
generalized symmetric
generalized symmetric/Hermitian
eigenvalue problem,symmetric/Hermitian
eigenvalue problems
large scale
large sparse
nonsymmetric
eigenvalue problems, symmetric/Hermitian
eigenvector , gif , gif
eigenvector
error bound , gif , gif , gif , gif , gif
GSEP , gif
left
right
SEP
elementary Householder matrix, see Householder matrix , gif , gif , gif , gif , gif
elementary reflector, see Householder matrix , gif , gif , gif , gif
equilibration , gif
errata
BLACS
ScaLAPACK and PBLAS
error
absolute , gif , gif
analysis
backward , gif , gif , gif , gif , gif , gif
measurement of
measurement of
matrix , gif
scalar , gif
subspace , gif
vector , gif
relative , gif , gif , gif , gif , gif , gif , gif
error bounds
generalized symmetric definite eigenproblem , gif
linear equations , gif
linear least squares , gif
singular value decomposition , gif
symmetric eigenproblem , gif
error handler, BLACS
error handler, PBLAS , gif
error handler, PXERBLA , gif
example program
how to execute
HPF
output, using MPICH
output, using PVM
ScaLAPACK (PDGESV) , gif
expert driver
expert driver, see driver routines, expert , gif
External Data Representation (XDR)
Ftex2html_wrap_inline12090
Ftex2html_wrap_inline12090
factorizations, see matrix factorizations
failures
failures
common causes
error handling
INFO
FDDI
Fiber Distributed Data Interface (FDDI)
floating-point arithmetic
floating-point arithmetic
tex2html_wrap_inline17191
gradual underflow
infinity
NaN
Not-a-Number
overflow , gif , gif , gif , gif , gif , gif , gif , gif , gif
relative machine precision , gif
roundoff error
subnormal numbers
underflow , gif , gif , gif
floating-point arithmetic, IEEE standard , gif , gif , gif , gif , gif
flop counts, for ScaLAPACK drivers
forward error
forward stability , gif
functionality
gang scheduler,
gap , gif , gif
generalized eigenproblem, symmetric definite
generalized orthogonal factorization
global
global
matrix
software components
variables , gif
global column index (JA)
global column index (JA) , gif
global input
globally blocking receive
global output
global row index (IA) , gif , gif
glossary
GQR , gif , gif , gif
granularity, of computation , gif , gif , gif
grid, see process grid
GRQ , gif
GSEP
guidelines, for high performance
Hermitian eigenvalue problem, see eigenvalue problems
Hessenberg form
Hessenberg form
reduction to
upper
heterogeneity
heterogeneous environment
heterogeneous network
heterogeneous software issues
algorithmic integrity , gif
machine parameters (PxLAMCH)
hierarchical memory , gif , gif
homepage
BLACS
MPI
netlib
PBLAS
PVM
ScaLAPACK , gif
homogeneous network
Householder matrix , gif
Householder matrix
complex
Householder transformation - blocked form
Householder vector
HPF
DISTRIBUTE , gif , gif , gif , gif , gif , gif , gif , gif , gif
example program
interface LA_GELS
interface LA_GEMM
interface LA_GESV
interface LA_POSV
interface to ScaLAPACK , gif
LA_GELS
LA_GEMM
LA_GESV
LA_POSV
PROCESSORS
ICTXT
ICTXT, see BLACS
ill-conditioned
ill-posed
incoherence
infinity
INFO
input error
installation
installation
PxLAMCH
insufficient workspace
invariant subspaces
error bound
inverse iteration
inverting matrix, using factorization
isoefficiency
isogranularity
iterative refinement
LAPACK
latency , gif
layout, see data distribution
lcm(Ptex2html_wrap_inline12112,Ptex2html_wrap_inline12114)
LDLtex2html_wrap_inline13041 factorization
leading dimension, see local array
linear equations
linear least squares problem
linear least squares problem
linear least squares problem , gif , gif
linear least squares problem
full rank
overdetermined , gif
rank-deficient , gif , gif
underdetermined
linear systems, solution of , gif
LLD_, see array descriptor
LLS , gif
load balance
LOCtex2html_wrap_inline12114()
LOCtex2html_wrap_inline12112()
LOCtex2html_wrap_inline12112()
LOCtex2html_wrap_inline12114()
LOCtex2html_wrap_inline12114()
LOCtex2html_wrap_inline12112()
local
local
software components
variables , gif
local array
local array
determining the size of , gif , gif
local leading dimension
local input
locally blocking send
local output
LQ factorization
LU factorization
LU factorization
matrix types
LWORK
LWORK tex2html_wrap_inline14966 WORK(1)
LWORK query
M_, see array descriptor
machine precision , gif
matrix factorizations
Cholesky , gif
complete orthogonal
generalized orthogonal
generalized QR , gif
generalized RQ
LDLtex2html_wrap_inline13041
LQ , gif
LU
orthogonal , gif
QL
QR , gif , gif
RQ
matrix inversion , gif
matrix of right-hand-side vectors
matrix of solution vectors
matrix partitioning
matrix sign function
MB_, see array descriptor
memory hierarchy , gif , gif
message-passing libraries
MPI , gif
MPL , gif
NX , gif
PVM , gif
message coprocessor
message passing, see distributed memory
MIMD , gif
minimum norm least squares solution
minimum norm solution
minimum norm solution , gif , gif
MPI , gif , gif , gif , gif
MPI
communicator
homepage
reference implementation (MPICH)
MPICH
MPL , gif
Multiple Instruction Multiple Data (MIMD)
Multiple Instruction Multiple Data (MIMD)
MYCOL , gif , gif
MYROW , gif , gif
N_, see array descriptor
naming scheme
naming scheme
auxiliary
driver and computational
NaN
NB_, see array descriptor
NBRHS
netlib
netlib
homepage
mirror repositories
networks of workstations
nonrepeatability
nonsymmetric eigenproblem
norm
Frobenius norm
matrix
two norm
vector
Not-a-Number
notation, table of
NoWs , gif
NPCOL, see Ptex2html_wrap_inline12114
NPROCS, see P
NPROW, see Ptex2html_wrap_inline12112
NRHS
numerical error, sources of
numerical error, sources of
input error
roundoff error , gif
NX , gif
optimal block size, determination of
orthogonal (unitary) factorizations , gif
orthogonal (unitary) transformation , gif
orthogonal factorization, generalized
orthogonal matrix , gif
out-of-core linear solvers
overdetermined system , gif , gif
overflow , gif , gif , gif , gif , gif , gif , gif , gif
P , gif , gif
Ptex2html_wrap_inline12114
Ptex2html_wrap_inline12112
Ptex2html_wrap_inline12114
Ptex2html_wrap_inline12112
P_ARPACK
parallel efficiency , gif
parallelizing BLAS-based programs: see Appendix B
parallelizing LAPACK-based programs: see Appendix B
ParPre
partial pivoting
partitioning unit, see block size , gif
PBLAS , gif
PBLAS
converting from the BLAS to: see Appendix B
error handling
PDGEMM
PDGEMV
PSGEMM
PSGEMV
quick reference guide: see Appendix D
PCDBSV
PCDBTRF , gif , gif
PCDBTRS , gif
PCDTSV
PCDTTRF , gif , gif
PCDTTRS , gif
PCGBSV
PCGBTRF , gif , gif
PCGBTRS , gif
PCGEBRD
PCGECON , gif
PCGEEQU
PCGEHRD
PCGELQF , gif
PCGELS , gif , gif
PCGEMR2D
PCGEQLF
PCGEQPF
PCGEQRF , gif , gif
PCGERFS , gif
PCGERQF
PCGESVD , gif , gif , gif
PCGESVX , gif , gif
PCGETRF
PCGETRI
PCGETRS
PCGGQRF
PCGGRQF
PCHEEVX , gif , gif , gif
PCHEGST
PCHEGVX , gif , gif , gif , gif
PCHETRD
PCHETRF
PCHETRS
PCPBSV
PCPBTRF , gif , gif
PCPBTRS , gif
PCPOCON , gif , gif
PCPOEQU
PCPORFS , gif
PCPOSV , gif
PCPOSVX , gif
PCPOTRF
PCPOTRI
PCPOTRS
PCPTSV
PCPTTRF
PCPTTRS
PCSTEIN , gif
PCTRCON , gif , gif
PCTRMR2D
PCTRRFS , gif
PCTRTRI
PCTRTRS , gif , gif
PCTZRZF
PCUNGLQ
PCUNGQR , gif
PCUNMBR
PCUNMHR
PCUNMLQ
PCUNMQR , gif , gif , gif
PCUNMRZ
PCUNMTR
PDDBSV
PDDBTRF , gif , gif
PDDBTRS , gif
PDDTSV
PDDTTRF , gif , gif
PDDTTRS , gif
PDGBSV
PDGBTRF , gif , gif
PDGBTRS , gif
PDGEBRD , gif
PDGECON , gif
PDGEEQU
PDGEHRD , gif
PDGELQF , gif
PDGELS , gif , gif , gif
PDGEMR2D
PDGEQLF
PDGEQPF
PDGEQRF , gif , gif
PDGERFS , gif
PDGERQF
PDGESV , gif
PDGESVD , gif , gif , gif , gif , gif , gif
PDGESVX , gif , gif , gif
PDGETRF
PDGETRI
PDGETRS
PDGGQRF
PDGGRQF
PDLAHQR , gif
PDLAMCH , gif
PDORGLQ
PDORGQR , gif
PDORMBR , gif
PDORMHR
PDORMLQ , gif
PDORMQR , gif , gif , gif
PDORMRZ
PDORMTR
PDPBSV
PDPBTRF , gif , gif
PDPBTRS , gif
PDPOCON , gif , gif
PDPOEQU
PDPORFS , gif
PDPOSV , gif , gif
PDPOSVX , gif
PDPOTRF
PDPOTRI
PDPOTRS
PDPTSV
PDPTTRF
PDPTTRS
PDSTEBZ , gif
PDSTEIN , gif
PDSYEV , gif , gif , gif , gif
PDSYEVX , gif , gif , gif , gif , gif , gif
PDSYGST
PDSYGVX , gif , gif , gif , gif
PDSYR2K
PDSYTRD , gif
PDTRCON , gif , gif
PDTRMR2D
PDTRRFS , gif
PDTRTRI
PDTRTRS , gif , gif
PDTZRZF
peak performance
performance , gif , gif
performance
diagnosing poor
evaluation of
improvement guidelines
PDLAHQR
PSGELS/PDGELS
PSGEMM/PDGEMM
PSGEMV/PDGEMV
PSGESV/PDGESV
PSGESVD/PDGESVD
PSPOSV/PDPOSV
PSSYEV/PDSYEV
PSSYEVX/PDSYEVX
recommendations
tuning
performance guidelines
PETSc
PIGEMR2D
PITRMR2D
pivoting
column (QR)
partial
poor performance
PoPCs
portability , gif , gif , gif , gif
prebuilt libraries
blacs
scalapack
process
column coordinate (MYCOL)
row coordinate (MYROW)
process column
process grid , gif
process grid
initialization , gif
mapping (column-major) , gif
mapping (row-major) , gif
mapping (user-defined) , gif
obtaining BLACS context
Ptex2html_wrap_inline12112
Ptex2html_wrap_inline12114
shape , gif
process mesh, see process grid
processor
process row
prototype codes
prototype codes
HPF interface to ScaLAPACK
prototype codes
matrix sign function , gif
out-of-core linear solvers
SuperLU
PSDBSV
PSDBTRF , gif , gif
PSDBTRS , gif
PSDTSV
PSDTTRF , gif , gif
PSDTTRS , gif
PSGBSV
PSGBTRF , gif , gif
PSGBTRS , gif
PSGEBRD , gif
PSGECON , gif
PSGEEQU
PSGEHRD , gif
PSGELQF , gif
PSGELS , gif , gif , gif
PSGEMR2D
PSGEQLF
PSGEQPF
PSGEQRF , gif , gif
PSGERFS , gif
PSGERQF
PSGESV , gif , gif
PSGESVD , gif , gif , gif , gif , gif , gif
PSGESVX , gif , gif , gif , gif , gif
PSGETRF
PSGETRI
PSGETRS
PSGGQRF
PSGGRQF
PSLAHQR , gif
PSLAMCH , gif
PSORGLQ
PSORGQR , gif
PSORMBR , gif
PSORMHR
PSORMLQ , gif
PSORMQR , gif , gif , gif
PSORMRZ
PSORMTR
PSPBSV
PSPBTRF , gif , gif
PSPBTRS , gif
PSPOCON , gif , gif
PSPOEQU
PSPORFS , gif
PSPOSV , gif , gif
PSPOSVX , gif
PSPOTRF
PSPOTRI
PSPOTRS
PSPTSV
PSPTTRF
PSPTTRS
PSSTEBZ , gif
PSSTEIN , gif
PSSYEV , gif , gif , gif , gif
PSSYEVX , gif , gif , gif , gif , gif , gif
PSSYGST
PSSYGVX , gif , gif , gif , gif
PSSYR2K
PSSYTRD , gif
PSTRCON , gif , gif
PSTRMR2D
PSTRRFS , gif
PSTRTRI
PSTRTRS , gif , gif
PSTZRZF
PVM , gif , gif , gif
PVM
homepage
PXERBLA , gif , gif
PZDBSV
PZDBTRF , gif , gif
PZDBTRS , gif
PZDTSV
PZDTTRF , gif , gif
PZDTTRS , gif
PZGBSV
PZGBTRF , gif , gif
PZGBTRS , gif
PZGEBRD
PZGECON , gif
PZGEEQU
PZGEHRD
PZGELQF , gif
PZGELS , gif , gif
PZGEMR2D
PZGEQLF
PZGEQPF
PZGEQRF , gif , gif
PZGERFS , gif
PZGERQF
PZGESVD , gif , gif , gif
PZGESVX , gif , gif
PZGETRF
PZGETRI
PZGETRS
PZGGQRF
PZGGRQF
PZHEEVX , gif , gif , gif
PZHEGST
PZHEGVX , gif , gif , gif , gif
PZHETRD
PZHETRF
PZHETRS
PZPBSV
PZPBTRF , gif , gif
PZPBTRS , gif
PZPOCON , gif , gif
PZPOEQU
PZPORFS , gif
PZPOSV , gif
PZPOSVX , gif
PZPOTRF
PZPOTRI
PZPOTRS
PZPTSV
PZPTTRF
PZPTTRS
PZSTEIN , gif
PZTRCON , gif , gif
PZTRMR2D
PZTRRFS , gif
PZTRTRI
PZTRTRS , gif , gif
PZTZRZF
PZUNGLQ
PZUNGQR , gif
PZUNMBR
PZUNMHR
PZUNMLQ
PZUNMQR , gif , gif , gif
PZUNMRZ
PZUNMTR
QL algorithm
implicit
QL factorization
QR algorithm
implicit
QR factorization , gif
QR factorization
blocked form
generalized (GQR) , gif
with pivoting , gif
query
quick reference guides: see Appendix D
rank
rank-deficient, see linear least squares problem
redistribution routines
PxGEMR2D
PxTRMR2D
reduction
bidiagonal
standard form
tridiagonal
upper Hessenberg
relative error , gif , gif , gif , gif , gif , gif , gif
relative machine precision
reliability, see test suites , gif
repeatability , gif
repeatability
enforcement of (BLACS_SET)
replicated across process columns
replicated across process rows
replicated and distributed
replicated vector , gif
reporting bugs, see bug reports , gif
right-hand-side matrix
roundoff error , gif
RQ factorization
RQ factorization
generalized (GRQ) , gif
RSRC_, see array descriptor
rule of thumb
scalability , gif , gif , gif
ScaLAPACK
commercial use of
converting from LAPACK to: see Appendix B
prebuilt libraries, availability of
quick reference guide: see Appendix D
scaling
Schur factorization
Schur form
Schur vectors
scope , gif
scope
groupings of processes
in communication
of a variable
within a process grid
scoped operation
SDISNA , gif
SEP , gif
simple driver, see driver routines, simple , gif
Single Program Multiple Data (SPMD)
singular value
singular value decomposition (SVD)
singular value decomposition (SVD)
singular value
error bound , gif
singular vector
error bound , gif
singular vectors
left , gif
right , gif
software hierarchy
software standards
software standards
distributed memory
solution matrix
spectral factorization
SPMD
SSTEQR2
stability , gif
stability
backward , gif , gif , gif , gif
forward
standard form
storage scheme
general tridiagonal
Hermitian , gif
orthogonal or unitary matrices
symmetric
symmetric tridiagonal
triangular
submatrix access
subspaces
angle between
subspaces, angle between , gif , gif , gif , gif , gif , gif , gif
suggestions for reading
SuperLU, see prototype codes
support
SVD
Sylvester equation
symmetric eigenproblem
T() , gif
Ttex2html_wrap_inline12200()
ttex2html_wrap_inline12068
ttex2html_wrap_inline12072
ttex2html_wrap_inline12218
ttex2html_wrap_inline12226
ttex2html_wrap_inline12076
Ttex2html_wrap_inline16302
ttex2html_wrap_inline12068
ttex2html_wrap_inline12072, see latency
ttex2html_wrap_inline12076
ttex2html_wrap_inline12072
ttex2html_wrap_inline12076
terminology
test suites , gif , gif , gif , gif , gif , gif , gif , gif , gif
thread of execution, see process
throughput, see bandwidth
TOOLS
DESCINIT
NUMROC
PxLAPRNT
SL_INIT
transportable efficiency
tridiagonal form , gif , gif , gif , gif , gif
tridiagonal matrix, distribution of
troubleshooting
tuning
underdetermined system , gif , gif
underflow , gif , gif , gif
unitary matrix , gif
upper Hessenberg form
utility routines
workspace query
World Wide Web
wrong results
WWW
XDR


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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About this document ...

ScaLAPACK Users' Guide

This document was generated using the LaTeX2HTML translator Version 96.1-h (September 30, 1996) Copyright © 1993, 1994, 1995, 1996, Nikos Drakos, Computer Based Learning Unit, University of Leeds.

The command line arguments were:
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Susan Blackford
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Next: Documentation Up: Essentials Previous: Commercial Use

Installation

   

To ease the installation process, prebuilt ScaLAPACK libraries are available on netlib for a variety of architectures.

http://www.netlib.org/scalapack/archives/
Included with each prebuilt library archive is the make include file SLmake.inc detailing the compiler options, and so on, used to compile the library. If a prebuilt library is not available for the specific architecture, the user will need to download the source code from netlib
http://www.netlib.org/scalapack/scalapack.tar.gz
and build the library as instructed in the ScaLAPACK Installation Guide [24]. Sample SLmake.inc files for various architectures are included in the distribution tar file and will require only limited modifications to customize for a specific architecture.

A comprehensive ScaLAPACK Installation Guide (LAPACK Working Note 93) [24]  is distributed with the complete package and contains descriptions of the testing programs, as well as detailed installation instructions.

A BLAS library and BLACS library must have been installed or be available on the architecture on which the user is planning to run ScaLAPACK. Users who plan to run ScaLAPACK on top of PVM [68] or MPI [64, 110] must also have PVM and/or MPI available.

If a vendor-optimized version of the BLAS is not available, one can obtain a Fortran77 reference implementation from the blas directory on netlib. If a BLACS library is not available, prebuilt BLACS libraries are available in the blacs/archives directory on netlib for a variety of architecture and message-passing library combinations. Otherwise, BLACS implementations for the Intel series, IBM SP series, PVM, and MPI are available from the blacs directory on netlib. PVM is available from the pvm3 directory on netlib, and a reference implementation of MPI is also available. Refer to the following URLs    :

http://www.netlib.org/blas/index.html
http://www.netlib.org/blacs/index.html
http://www.netlib.org/blacs/archives/
http://www.netlib.org/pvm3/index.html
http://www.netlib.org/mpi/index.html

Comprehensive test suites       for the BLAS, BLACS, and PVM are provided on netlib, and it is highly recommended that these test suites be run to ensure proper installation of the packages.

If the user will be using PVM, it is important to note that only PVM version 3.3 or later is supported with the BLACS [113, 52]. Because of major changes in PVM and the resulting changes required in the BLACS, earlier versions of PVM are not supported. User who have a previous release of PVM must obtain version 3.3 or later to run the PVM BLACS and thus ScaLAPACK.

next up previous contents index
Next: Documentation Up: Essentials Previous: Commercial Use

Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: List of Figures Up: ScaLAPACK Users' Guide Previous: ScaLAPACK Users' Guide

Contents


Susan Blackford
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Documentation

  

This users guide provides an informal introduction to the design of the package, a detailed description of its contents, and a reference manual for the leading comments of the source code. A brief discussion of the contents of each chapter, as well as guidance for novice and expert users, can be found in the Suggestions for Reading at the beginning of this book. A List of Notation and Glossary are also provided.

On-line manpages (troff files) for ScaLAPACK routines, as well as for LAPACK and the BLAS, are available on netlib. These files are automatically generated at the time of each release. For more information, see the manpages.tar.gz entry on the scalapack index on netlib. A comprehensive Installation Guide for ScaLAPACK [24] is also available; refer to section 1.7 for further details.

Using a World Wide Web   browser such as Netscape, one can access the ScaLAPACK homepage  via the URL:

http://www.netlib.org/scalapack/index.html
This homepage contains hyperlinks for additional documentation as well as the ability to view individual ScaLAPACK driver and computational routines.


Susan Blackford
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Support

 

ScaLAPACK has been thoroughly tested before release, on many different types of computers and configurations. The ScaLAPACK project supports the package in the sense that reports of errors or poor performance will gain immediate attention from the developers  . Refer to section 7 for a list of questions asked when the user submits a bug report. Such reports -- and also descriptions of interesting applications and other comments -- should be sent to

ScaLAPACK Project
c/o J. J. Dongarra
Computer Science Department
University of Tennessee
Knoxville, TN 37996-1301
USA
E-mail: scalapack@cs.utk.edu

Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Errata

A list of known problems, bugs, and compiler errors  for ScaLAPACK and the PBLAS, as well as an errata list for this guide, is maintained on netlib. For a copy of this report, refer to the URL

http://www.netlib.org/scalapack/errata.scalapack

Similarly, an errata file  for the BLACS can be obtained by the request:

http://www.netlib.org/blacs/errata.blacs

A ScaLAPACK FAQ (Frequently Asked Questions) file is also maintained via the URL

http://www.netlib.org/scalapack/faq.html

Susan Blackford
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Related Projects

As mentioned in the Preface, the ScaLAPACK library discussed in this Users' Guide is only one facet of the ScaLAPACK project. A variety of other software is also available in the scalapack directory on netlib.

http://www.netlib.org/scalapack/index.html

P_ARPACK (Parallel ARPACK)  is an extension of the ARPACK  software package used for solving large-scale eigenvalue problems   on distributed-memory parallel architectures. The message-passing layers currently supported are BLACS and MPI. Serial ARPACK must be retrieved and installed prior to installing P_ARPACK. All core ARPACK routines are available in single-precision real, double-precision real, single-precision complex, and double-precision complex. An extensive set of driver routines is available for ARPACK, and a subset of these is available for parallel computation with P_ARPACK. These may be used as templates that are easily modified to construct a problem specific parallel interface to P_ARPACK.

CAPSS  is a fully parallel package to solve a sparse linear system of the form Ax=b on a message passing multiprocessor; the matrix A is assumed to be symmetric positive definite and associated with a mesh in two or three dimensions. This version has been tested on the Intel Paragon and makes possible efficient parallel solution for several right-hand-side vectors.

ParPre  is a package of parallel preconditioners for general sparse matrices. It includes classical point/block relaxation methods, generalized block SSOR preconditioners (this includes ILU), and domain decomposition methods (additive and multiplicative Schwarz, Schur complement). The communication protocol is MPI, and low-level routines from the PETSc [109]  library are used, but installing the complete PETSc library is not necessary.

Prototype codes are provided for out-of-core   solvers [55] for LU, Cholesky, and QR, the matrix sign function for eigenproblems [14, 13, 12]  , an HPF interface   to a subset of ScaLAPACK routines, and SuperLU [96, 39, 41]  .

http://www.netlib.org/scalapack/prototype/
These software contributions are classified as prototype codes  because they are still under development and their calling sequences may change. They have been tested only on a limited number of architectures and have not been rigorously tested on all of the architectures to which the ScaLAPACK library is portable.

Refer to Appendix C.2 for a brief description of the HPF interface to ScaLAPACK, as well as an example program.

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Contents of the CD-ROM

Each Users' Guide includes a CD-ROM containing

  • the HTML version of the ScaLAPACK Users' Guide,
  • the source code for the ScaLAPACK, PBLAS, BLACS, and LAPACK packages, including testing and timing programs,
  • prebuilt ScaLAPACK, BLACS, and LAPACK libraries for a variety of architectures,
  • example programs, and
  • the full set of LAPACK Working Notes in postscript and pdf format.

Instructions for reading and traversing the directory structure on the CD-ROM are provided in the booklet packaged with the CD-ROM. A readme file is provided in each directory on the CD-ROM. The directory structure on the CD-ROM mimics the scalapack, blacs, and lapack directory contents on netlib.


Susan Blackford
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Getting Started with ScaLAPACK

 

This chapter provides the background information to enable users to call ScaLAPACK software, together with a simple example program. The chapter begins by presenting a set of instructions to execute a ScaLAPACK example program, followed by the source code of Example Program #1. A detailed explanation of the example program is given, and finally the necessary steps to call a ScaLAPACK routine are described by referencing the example program.

For an explanation of the terminology used within this chapter, please refer to the List of Notation and/or Glossary at the beginning of this book.



Susan Blackford
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How to Run an Example Program Using MPI

 

This section presents the instructions for installing ScaLAPACK and running a simple example program in parallel. The example assumes that the underlying system is a Sun Solaris system; the problem is run on one physical processor, using six processes that do message passing. The example uses MPI as the message-passing layer. The version of MPI used in this example is MPICH (version 1.0.13), and we assume the user has this version installed. MPICH   is a freely available, portable implementation of MPI. If MPICH is not installed, refer to http://www.netlib.org/mpi/. If this is run on a different architecture, the user will have to make a number of changes. In particular, the prebuilt libraries will have to be changed. If prebuilt libraries do not exist for the specific architecture, the user will need to download the source
(http://www.netlib.org/scalapack/scalapack.tar.gz) and build them.

To use ScaLAPACK for the first time (on a network of workstations using MPI), one should

  1. Make a directory for this testing. 
          mkdir SCALAPACK
      cd SCALAPACK
  2. Download the ScaLAPACK example program (about 7 KB) into directory SCALAPACK.
    http://www.netlib.org/scalapack/examples/example1.f
  3. Download the prebuilt ScaLAPACK library (about 3MB) for the specific architecture into directory SCALAPACK (e.g., SUN4SOL2) and uncompress.
    http://www.netlib.org/scalapack/archives/scalapack_SUN4SOL2.tar.gz 
          gunzip scalapack_SUN4SOL2.tar.gz
      tar xvf scalapack_SUN4SOL2.tar
      rm scalapack_SUN4SOL2.tar
    (Note that this tar file contains the library archive and the SLmake.inc used to build the library. Details of compiler flags, etc. can be found in this make include file.)
  4. Download the prebuilt BLACS library (about 60 KB) for the architecture (e.g., SUN4SOL2) and message-passing layer (e.g., MPICH), and uncompress into directory SCALAPACK.
    http://www.netlib.org/blacs/archives/blacs_MPI-SUN4SOL2-0.tar.gz 
          gunzip blacs_MPI-SUN4SOL2-0.tar.gz
      tar xvf blacs_MPI-SUN4SOL2-0.tar
      rm blacs_MPI-SUN4SOL2-0.tar
    (Note that this tar file contains the library archive(s) and the Bmake.inc used to build the library. Details of compiler flags, etc. can be found in this make include file.)
  5. Find the optimized BLAS library on the specific architecture.

    If not available, download reference implementation (about 1 MB) into directory SCALAPACK/BLAS, compile, and build the library. 

            mkdir BLAS
        cd BLAS
    Download http://www.netlib.org/blas/blas.shar. 
            sh blas.shar
        f77 -O -f -c *.f
        ar cr ../blas_SUN4SOL2.a *.o
        cd ..
    (Note that this reference implementation of the BLAS will not deliver high performance.)
  6. Compile and link to prebuilt libraries. 

        sun4sol2> f77 -f -o example1 example1.f scalapack_SUN4SOL2.a \
              blacsF77init_MPI-SUN4SOL2-0.a blacs_MPI-SUN4SOL2-0.a \
              blacsF77init_MPI-SUN4SOL2-0.a blas_SUN4SOL2.a \
              $MPI_ROOT/lib/solaris/ch_p4/libmpi.a -lnsl -lsocket
    example1.f:
     MAIN example1:
           matinit:
    Note that the -lnsl -lsocket libraries are machine specific to Solaris. Refer to the SLmake.inc for details. MPICH can be found in the directory $MPI_ROOT. On our system we did 
        sun4sol2> setenv MPI_ROOT /src/icl/MPI/mpich
  7. Run the ScaLAPACK example program.

    To run an MPI program with MPICH, one will need to add $MPI_ROOT/bin to the path. On our system we did 

        sun4sol2> set path = ($path $MPI_ROOT/bin)

    To run the example: 

        sun4sol2> mpirun -np 6 example1

    The example runs on six processes and prints out a statement that ``the solution is correct'' or ``the solution is incorrect''.

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Source Code for Example Program #1

This program is also available in the scalapack directory on netlib
(http://www.netlib.org/scalapack/examples/example1.f). 

      PROGRAM EXAMPLE1
*
*     Example Program solving Ax=b via ScaLAPACK routine PDGESV
*
*     .. Parameters ..
      INTEGER            DLEN_, IA, JA, IB, JB, M, N, MB, NB, RSRC,
     $                   CSRC, MXLLDA, MXLLDB, NRHS, NBRHS, NOUT,
     $                   MXLOCR, MXLOCC, MXRHSC
      PARAMETER          ( DLEN_ = 9, IA = 1, JA = 1, IB = 1, JB = 1,
     $                   M = 9, N = 9, MB = 2, NB = 2, RSRC = 0,
     $                   CSRC = 0, MXLLDA = 5, MXLLDB = 5, NRHS = 1,
     $                   NBRHS = 1, NOUT = 6, MXLOCR = 5, MXLOCC = 4,
     $                   MXRHSC = 1 )
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            ICTXT, INFO, MYCOL, MYROW, NPCOL, NPROW
      DOUBLE PRECISION   ANORM, BNORM, EPS, RESID, XNORM
*     ..
*     .. Local Arrays ..
      INTEGER            DESCA( DLEN_ ), DESCB( DLEN_ ),
     $                   IPIV( MXLOCR+NB )
      DOUBLE PRECISION   A( MXLLDA, MXLOCC ), A0( MXLLDA, MXLOCC ),
     $                   B( MXLLDB, MXRHSC ), B0( MXLLDB, MXRHSC ),
     $                   WORK( MXLOCR )
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   PDLAMCH, PDLANGE
      EXTERNAL           PDLAMCH, PDLANGE
*     ..
*     .. External Subroutines ..
      EXTERNAL           BLACS_EXIT, BLACS_GRIDEXIT, BLACS_GRIDINFO,
     $                   DESCINIT, MATINIT, PDGEMM, PDGESV, PDLACPY,
     $                   SL_INIT
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DBLE
*     ..
*     .. Data statements ..
      DATA               NPROW / 2 / , NPCOL / 3 /
*     ..
*     .. Executable Statements ..
*
*     INITIALIZE THE PROCESS GRID
*
      CALL SL_INIT( ICTXT, NPROW, NPCOL )
      CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL )
*
*     If I'm not in the process grid, go to the end of the program
*
      IF( MYROW.EQ.-1 )
     $   GO TO 10
*
*     DISTRIBUTE THE MATRIX ON THE PROCESS GRID
*     Initialize the array descriptors for the matrices A and B
*
      CALL DESCINIT( DESCA, M, N, MB, NB, RSRC, CSRC, ICTXT, MXLLDA,
     $               INFO )
      CALL DESCINIT( DESCB, N, NRHS, NB, NBRHS, RSRC, CSRC, ICTXT,
     $               MXLLDB, INFO )
*
*     Generate matrices A and B and distribute to the process grid
*
      CALL MATINIT( A, DESCA, B, DESCB )
*
*     Make a copy of A and B for checking purposes
*
      CALL PDLACPY( 'All', N, N, A, 1, 1, DESCA, A0, 1, 1, DESCA )
      CALL PDLACPY( 'All', N, NRHS, B, 1, 1, DESCB, B0, 1, 1, DESCB )
*
*     CALL THE SCALAPACK ROUTINE
*     Solve the linear system A * X = B
*
      CALL PDGESV( N, NRHS, A, IA, JA, DESCA, IPIV, B, IB, JB, DESCB,
     $             INFO )
*
      IF( MYROW.EQ.0 .AND. MYCOL.EQ.0 ) THEN
         WRITE( NOUT, FMT = 9999 )
         WRITE( NOUT, FMT = 9998 )M, N, NB
         WRITE( NOUT, FMT = 9997 )NPROW*NPCOL, NPROW, NPCOL
         WRITE( NOUT, FMT = 9996 )INFO
      END IF
*
*     Compute residual ||A * X  - B|| / ( ||X|| * ||A|| * eps * N )
*
      EPS = PDLAMCH( ICTXT, 'Epsilon' )
      ANORM = PDLANGE( 'I', N, N, A, 1, 1, DESCA, WORK )
      BNORM = PDLANGE( 'I', N, NRHS, B, 1, 1, DESCB, WORK )
      CALL PDGEMM( 'N', 'N', N, NRHS, N, ONE, A0, 1, 1, DESCA, B, 1, 1,
     $             DESCB, -ONE, B0, 1, 1, DESCB )
      XNORM = PDLANGE( 'I', N, NRHS, B0, 1, 1, DESCB, WORK )
      RESID = XNORM / ( ANORM*BNORM*EPS*DBLE( N ) )
*
      IF( MYROW.EQ.0 .AND. MYCOL.EQ.0 ) THEN
         IF( RESID.LT.10.0D+0 ) THEN
            WRITE( NOUT, FMT = 9995 )
            WRITE( NOUT, FMT = 9993 )RESID
         ELSE
            WRITE( NOUT, FMT = 9994 )
            WRITE( NOUT, FMT = 9993 )RESID
         END IF
      END IF
*
*     RELEASE THE PROCESS GRID
*     Free the BLACS context
*
      CALL BLACS_GRIDEXIT( ICTXT )
   10 CONTINUE
*
*     Exit the BLACS
*
      CALL BLACS_EXIT( 0 )
*
 9999 FORMAT( / 'ScaLAPACK Example Program #1 -- May 1, 1997' )
 9998 FORMAT( / 'Solving Ax=b where A is a ', I3, ' by ', I3,
     $      ' matrix with a block size of ', I3 )
 9997 FORMAT( 'Running on ', I3, ' processes, where the process grid',
     $      ' is ', I3, ' by ', I3 )
 9996 FORMAT( / 'INFO code returned by PDGESV = ', I3 )
 9995 FORMAT( /
     $   'According to the normalized residual the solution is correct.'
     $       )
 9994 FORMAT( /
     $ 'According to the normalized residual the solution is incorrect.'
     $       )
 9993 FORMAT( / '||A*x - b|| / ( ||x||*||A||*eps*N ) = ', 1P, E16.8 )
      STOP
      END
      SUBROUTINE MATINIT( AA, DESCA, B, DESCB )
*
*     MATINIT generates and distributes matrices A and B (depicted in
*     figures 2.5 and 2.6) to a 2 x 3 process grid
*
*     .. Array Arguments ..
      INTEGER            DESCA( * ), DESCB( * )
      DOUBLE PRECISION   AA( * ), B( * )
*     ..
*     .. Parameters ..
      INTEGER            CTXT_, LLD_
      PARAMETER          ( CTXT_ = 2, LLD_ = 9 )
*     ..
*     .. Local Scalars ..
      INTEGER            ICTXT, MXLLDA, MYCOL, MYROW, NPCOL, NPROW
      DOUBLE PRECISION   A, C, K, L, P, S
*     ..
*     .. External Subroutines ..
      EXTERNAL           BLACS_GRIDINFO
*     ..
*     .. Executable Statements ..
*
      ICTXT = DESCA( CTXT_ )
      CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL )
*
      S = 19.0D0
      C = 3.0D0
      A = 1.0D0
      L = 12.0D0
      P = 16.0D0
      K = 11.0D0
*
      MXLLDA = DESCA( LLD_ )
*
      IF( MYROW.EQ.0 .AND. MYCOL.EQ.0 ) THEN
         AA( 1 ) = S
         AA( 2 ) = -S
         AA( 3 ) = -S
         AA( 4 ) = -S
         AA( 5 ) = -S
         AA( 1+MXLLDA ) = C
         AA( 2+MXLLDA ) = C
         AA( 3+MXLLDA ) = -C
         AA( 4+MXLLDA ) = -C
         AA( 5+MXLLDA ) = -C
         AA( 1+2*MXLLDA ) = A
         AA( 2+2*MXLLDA ) = A
         AA( 3+2*MXLLDA ) = A
         AA( 4+2*MXLLDA ) = A
         AA( 5+2*MXLLDA ) = -A
         AA( 1+3*MXLLDA ) = C
         AA( 2+3*MXLLDA ) = C
         AA( 3+3*MXLLDA ) = C
         AA( 4+3*MXLLDA ) = C
         AA( 5+3*MXLLDA ) = -C
         B( 1 ) = 0.0D0
         B( 2 ) = 0.0D0
         B( 3 ) = 0.0D0
         B( 4 ) = 0.0D0
         B( 5 ) = 0.0D0
      ELSE IF( MYROW.EQ.0 .AND. MYCOL.EQ.1 ) THEN
         AA( 1 ) = A
         AA( 2 ) = A
         AA( 3 ) = -A
         AA( 4 ) = -A
         AA( 5 ) = -A
         AA( 1+MXLLDA ) = L
         AA( 2+MXLLDA ) = L
         AA( 3+MXLLDA ) = -L
         AA( 4+MXLLDA ) = -L
         AA( 5+MXLLDA ) = -L
         AA( 1+2*MXLLDA ) = K
         AA( 2+2*MXLLDA ) = K
         AA( 3+2*MXLLDA ) = K
         AA( 4+2*MXLLDA ) = K
         AA( 5+2*MXLLDA ) = K
      ELSE IF( MYROW.EQ.0 .AND. MYCOL.EQ.2 ) THEN
         AA( 1 ) = A
         AA( 2 ) = A
         AA( 3 ) = A
         AA( 4 ) = -A
         AA( 5 ) = -A
         AA( 1+MXLLDA ) = P
         AA( 2+MXLLDA ) = P
         AA( 3+MXLLDA ) = P
         AA( 4+MXLLDA ) = P
         AA( 5+MXLLDA ) = -P
      ELSE IF( MYROW.EQ.1 .AND. MYCOL.EQ.0 ) THEN
         AA( 1 ) = -S
         AA( 2 ) = -S
         AA( 3 ) = -S
         AA( 4 ) = -S
         AA( 1+MXLLDA ) = -C
         AA( 2+MXLLDA ) = -C
         AA( 3+MXLLDA ) = -C
         AA( 4+MXLLDA ) = C
         AA( 1+2*MXLLDA ) = A
         AA( 2+2*MXLLDA ) = A
         AA( 3+2*MXLLDA ) = A
         AA( 4+2*MXLLDA ) = -A
         AA( 1+3*MXLLDA ) = C
         AA( 2+3*MXLLDA ) = C
         AA( 3+3*MXLLDA ) = C
         AA( 4+3*MXLLDA ) = C
         B( 1 ) = 1.0D0
         B( 2 ) = 0.0D0
         B( 3 ) = 0.0D0
         B( 4 ) = 0.0D0
      ELSE IF( MYROW.EQ.1 .AND. MYCOL.EQ.1 ) THEN
         AA( 1 ) = A
         AA( 2 ) = -A
         AA( 3 ) = -A
         AA( 4 ) = -A
         AA( 1+MXLLDA ) = L
         AA( 2+MXLLDA ) = L
         AA( 3+MXLLDA ) = -L
         AA( 4+MXLLDA ) = -L
         AA( 1+2*MXLLDA ) = K
         AA( 2+2*MXLLDA ) = K
         AA( 3+2*MXLLDA ) = K
         AA( 4+2*MXLLDA ) = K
      ELSE IF( MYROW.EQ.1 .AND. MYCOL.EQ.2 ) THEN
         AA( 1 ) = A
         AA( 2 ) = A
         AA( 3 ) = -A
         AA( 4 ) = -A
         AA( 1+MXLLDA ) = P
         AA( 2+MXLLDA ) = P
         AA( 3+MXLLDA ) = -P
         AA( 4+MXLLDA ) = -P
      END IF
      RETURN
      END


Susan Blackford
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Details of Example Program #1

   

This example program demonstrates the basic requirements to call a ScaLAPACK routine -- initializing the process grid, assigning the matrix to the processes, calling the ScaLAPACK routine, and releasing the process grid. For further details on each of these steps, please refer to section 2.4.

This example program solves the tex2html_wrap_inline12402 system of linear equations given by
displaymath12392

using the ScaLAPACK driver routine PDGESV. The ScaLAPACK routine PDGESV solves a system of linear equations tex2html_wrap_inline12404, where the coefficient matrix (denoted by A) and the right-hand-side matrix (denoted by B) are real, general distributed matrices. The coefficient matrix A is distributed as depicted below, and for simplicity, we shall solve the system for one right-hand side (NRHS=1); that is, the matrix B is a vector. The third element of the matrix B is equal to 1, and all other elements are equal to 0. After solving this system of equations, the solution vector X is given by
displaymath12393

Let us assume that the matrix A is partitioned and distributed as denoted in figure 4.6; that is, we have chosen the row and column block sizes as MB=NB=2, and the matrix is distributed on a 2 tex2html_wrap_inline12420 3 process grid (tex2html_wrap_inline12422,tex2html_wrap_inline12424). The partitioning and distribution of our example matrix A is represented in figures 2.1 and 2.2, where, to aid visualization, we use the notation s=19, c=3, a=1, l=12, p=16, and k=11.

  figure705
Figure 2.1: Partitioning of global matrix A (s=19;c=3;a=1;l=12;p=16;k=11)

  figure710
Figure 2.2: Mapping of matrix A onto process grid (tex2html_wrap_inline12422,tex2html_wrap_inline12424). For example, note that process (0,0) contains a local array of size A(5,4).

The partitioning and distribution of our example matrix B are demonstrated in figure 2.3.

  figure717
Figure 2.3: Mapping of matrix B onto process grid (tex2html_wrap_inline12422,tex2html_wrap_inline12424)

Note that matrix B is distributed only in column 0 of the process grid. All other columns in the process grid possess an empty local portion of the matrix B.

On exit from PDGESV, process (0,0) contains (in the global view) the global vector X and (in the local view) the local array B given by
displaymath12394
and process (1,0) contains (in the global view) the global vector X and (in the local view) local array B given by
displaymath12395

The normalized residual check
displaymath12396
is performed on the solution to verify the accuracy of the results.

For more information on the BLACS routines called in this program, please refer to section 2.4, Appendix D.3, [54], and the BLACS homepage (http://www.netlib.org/blacs/index.html). Further details of the matrix distribution and storage scheme can be found in section 2.3.2, figure 4.6, and table 4.8. Complete details on matrix distribution can be found in Chapter 4 and details of the array descriptors can be found in section 4.3.2. For a more flexible and memory efficient example program, please refer to Appendix C.1.


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Simplifying Assumptions Used in Example Program

Several simplifying assumptions and/or restrictions have been made in this example program in order to present the most basic example for the user:

  • We have chosen a small block size, MB=NB=2; however, this should not be regarded as a typical choice of block size in a user's application. For best performance, a choice of MB=NB=32 or MB=NB=64 is more suitable. Refer to Chapter 5 for further details.
  • A simplistic subroutine MATINIT is used to assign matrices A and B to the process grid. Note that this subroutine hardcodes the local arrays on each process and does not perform communication. It is not a ScaLAPACK routine and is provided only for the purposes of this example program.
  • We assume RSRC=CSRC=0 , and thus both matrices A and B are distributed across the process grid starting with process (0,0). In general, however, any process in the current process grid can be assigned to receive the first element of the distributed matrix.
  • We have set the local leading dimension of local array A and the local leading dimension of local array B to be the same over all process rows in the process grid. The variable MXLLDA is equal to the maximum local leading dimension for array A (denoted tex2html_wrap_inline12116) over all process rows. Likewise, variable MXLLDB is the maximum local leading dimension for array B (denoted tex2html_wrap_inline12120) over all process rows. In general, however, the local leading dimension of the local array can differ from process to process in the process grid.
  • The system is solved by using the entire matrix A, as opposed to a submatrix of A, so the global indices, denoted by IA, JA, IB, and JB     , into the matrix are equal to 1. Refer to figure 4.7 in section 4.3.5 for more information on the representation of global addressing into a distributed submatrix.
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List of Figures


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Notation Used in Example Program

 

The following is a list of notational variables and definitions specific to Example Program #1. A complete List of Notation can be found at the beginning of this book. 

 
Variable 		 Definition


CSRC 		 (global) Process column over which the first column of the matrix is distributed.


DESCA 		 (global and local) Array descriptor for matrix A.


DESCB 		 (global and local) Array descriptor for matrix B.


ICTXT 		 (global) BLACS context associated with a process grid.


M 		 (global) Number of rows in the global matrix A.


MB 		 (global) Row block size for the matrix A.


MXLLDA 		 (global) Maximum local leading dimension of the array A.


MXLLDB 		 (global) Maximum local leading dimension of the array B.


MXLOCC  		 (global) Maximum number of columns of the matrix A owned by any


        		  process column.


MXLOCR  		 (global) Maximum number of rows of the matrix A owned by any


         		  process row.


MXRHSC  		 (global) Maximum number of columns of the matrix B owned by any


        		  process column.


MYCOL 		 (local) Calling process's column coordinate in the process grid.


MYROW 		 (local) Calling process's row coordinate in the process grid.


N 		 (global) Number of columns in the global matrix A,


     		 and the number of rows of the global solution matrix B.


NB 		 (global) Column block size for the matrix A,


     		 and the row block size for the matrix B.


NBRHS		 (global) Column block size for the global solution matrix B.


NPCOL 		 (global) Number of columns in the process grid.


NPROW 		 (global) Number of rows in the process grid.


NRHS 		 (global) Number of columns in the global solution matrix B.


RSRC 		 (global) Process row over which the first row of the matrix is distributed.

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Output of Example Program #1 Using MPI

When this example program is executed on a Sun Solaris architecture using MPICH (version 1.0.13) and the MPI BLACS, the following outputgif is received:  

sun4sol2> f77 -o example1 example1.f scalapack_SUN4SOL2.a \
          blacsF77init_MPI-SUN4SOL2-0.a blacs_MPI-SUN4SOL2-0.a \
          blacsF77init_MPI-SUN4SOL2-0.a blas_SUN4SOL2.a \
          $MPI_ROOT/lib/solaris/ch_p4/libmpi.a -lnsl -lsocket
example1.f:
 MAIN example1:
        matinit:
sun4sol2> mpirun -np 6 example1
 
ScaLAPACK Example Program #1 -- May 1, 1997
 
Solving Ax=b where A is a   9 by   9 matrix with a block size of   2
Running on   6 processes, where the process grid is   2 by   3
 
INFO code returned by PDGESV =   0
 
According to the normalized residual the solution is correct.
 
||A*x - b|| / ( ||x||*||A||*eps*N ) =   0.00000000E+00


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Output of Example Program #1 Using PVM

When this example program is executed on a SUN4 architecture using PVM (version 3.3.11) and the PVM BLACS, the following outputgif is received:  

sun4> f77 -o example1 example1.f scalapack_SUN4.a \
      blacs_PVM-SUN4-0.a blas_SUN4.a $PVM_ROOT/lib/SUN4/libpvm3.a
example1.f:
 MAIN example1:
        matinit:
sun4> cp example1 $HOME/pvm3/bin/SUN4/
sun4> cd $HOME/pvm3/bin/SUN4/
sun4> pvm
pvm> quit
 
pvmd still running.
sun4> example1
File 'blacs_setup.dat' not found.  Spawning processes to current
configuration.
Enter the name of the executable to run: example1
Spawning 5 more copies of example1
Spawning process 'example1' to host sun4
[t40003] BEGIN
Spawning process 'example1' to host sun4
[t40004] BEGIN
Spawning process 'example1' to host sun4
[t40005] BEGIN
Spawning process 'example1' to host sun4
[t40006] BEGIN
Spawning process 'example1' to host sun4
[t40007] BEGIN
 
ScaLAPACK Example Program #1 -- May 1, 1997
 
Solving Ax=b where A is a   9 by   9 matrix with a block size of   2
Running on   6 processes, where the process grid is   2 by   3
 
INFO code returned by PDGESV =   0
 
According to the normalized residual the solution is correct.
 
||A*x - b|| / ( ||x||*||A||*eps*N ) =   0.00000000E+00
sun4> pvm
pvmd already running.
pvm> halt
sun4>


Susan Blackford
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Four Basic Steps Required to Call a ScaLAPACK Routine

 

Four basic steps are required to call a ScaLAPACK routine.

  1. Initialize the process grid
  2. Distribute the matrix on the process grid
  3. Call ScaLAPACK routine
  4. Release the process grid

Each of these steps is detailed below. The example program in section 2.3 illustrates these basic requirements. Refer to section 2.3.2 for an explanation of notational variables.

For more information on the BLACS routines called in this program, and more specifically their calling sequences, please refer to Appendix D.3, [54], and the BLACS homepage 
(http://www.netlib.org/blacs/index.html). Further details of the matrix distribution and storage scheme can be found in Chapter 4 and section 4.3.2.



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Initialize the Process Grid

 

A call to the ScaLAPACK TOOLS routine SL_INIT initializes the process grid. This routine initializes a tex2html_wrap_inline12182 (denoted tex2html_wrap_inline12614 in the source code) process grid by using a row-major ordering of the processes, and obtains a default system context . For more information on contexts, refer to section 4.1.2 or [54].

The user can then query the process grid to identify each process's coordinates (MYROW,MYCOL)  )   via a call to BLACS_GRIDINFO .

A typical code fragment (as obtained from the example program in section 2.3) to accomplish this task would be the following: 

      CALL SL_INIT( ICTXT, NPROW, NPCOL )
      CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL )

where details of the calling sequence for SL_INIT are provided below. Detailed descriptions of calling sequences for each BLACS routine can be found in [54] and Appendix D.3. Note that underlined arguments in the calling sequence denote output arguments.

  • SL_INIT( , NPROW, NPCOL )
     
    ICTXT
    (global output) INTEGER
    ICTXT specifies the BLACS context identifying the created process grid.
    NPROW
    (global input) INTEGER
    NPROW specifies the number of process rows in the process grid to be created.
    NPCOL
    (global input) INTEGER
    NPCOL specifies the number of process columns in the process grid to be created.

For a description of these variables names, please refer to the example program notation in section 2.3 and [54].


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Distribute the Matrix on the Process Grid

All global matrices must be distributed on the process grid prior to the invocation of a ScaLAPACK routine. It is the user's responsibility to perform this data distribution. For further information on the appropriate data distribution, please refer to Chapter 4.

Each global matrix that is to be distributed across the process grid must be assigned an array descriptor   . Details of the entries in the array descriptor can be found in section 4.3.3. This array descriptor is most easily initialized with a call to a ScaLAPACK TOOLS routine called DESCINIT  and must be set prior to the invocation of a ScaLAPACK routine.

As an example, the array descriptors  for the matrices in figures 2.2 and 2.3 are assigned with the following code excerpt from the example program in section 2.3. 

      CALL DESCINIT( DESCA, M, N, MB, NB, RSRC, CSRC, ICTXT, MXLLDA,
     $               INFO )
      CALL DESCINIT( DESCB, N, NRHS, NB, NBRHS, RSRC, CSRC, ICTXT,
     $               MXLLDB, INFO )
These two calls to DESCINIT are equivalent to the assignment statements: 
      DESCA( 1 ) = 1
      DESCA( 2 ) = ICTXT
      DESCA( 3 ) = M
      DESCA( 4 ) = N
      DESCA( 5 ) = MB
      DESCA( 6 ) = NB
      DESCA( 7 ) = RSRC
      DESCA( 8 ) = CSRC
      DESCA( 9 ) = MXLLDA
*
      DESCB( 1 ) = 1
      DESCB( 2 ) = ICTXT
      DESCB( 3 ) = N
      DESCB( 4 ) = NRHS
      DESCB( 5 ) = NB
      DESCB( 6 ) = NBRHS
      DESCB( 7 ) = RSRC
      DESCB( 8 ) = CSRC
      DESCB( 9 ) = MXLLDB

Details of the entries in the array descriptor can be found in section 4.3.3.

A simplistic mapping of the global matrix in figure 2.2 to a process grid is accomplished in the example program in section 2.3 via a call to the subroutine MATINIT. Please note that the routine MATINIT is not a ScaLAPACK routine and is used in this example program for demonstrative purposes only.

Appendix C.1 provides a more detailed example program, which reads a matrix from a file, distributes it onto a process grid, and then writes the solution to a file.


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Call the ScaLAPACK Routine

All ScaLAPACK routines assume that the data has been distributed on the process grid prior to the invocation of the routine. Detailed descriptions of the appropriate calling sequences for each of the ScaLAPACK routines can be found in the leading comments of the source code or in Part ii of this users guide. The required data distribution for the ScaLAPACK routine, as well as the amount of input error checking to be performed, is described in Chapter 4. For debugging hints, the user should refer to Chapter 7.


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Release the Process Grid

After the desired computation on a process grid has been completed, it is advisable to release the process grid via a call to BLACS_GRIDEXIT . When all computations have been completed, the program is exited with a call to BLACS_EXIT.

A typical code fragment to accomplish these steps would be 

      CALL BLACS_GRIDEXIT( ICTXT )
      CALL BLACS_EXIT( 0 )

A detailed explanation of the BLACS calling sequences can be found in Appendix D and [54].


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Contents of ScaLAPACK

 



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Structure of ScaLAPACK

 



Susan Blackford
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List of Tables


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Levels of Routines

 

The routines in ScaLAPACK are classified into three broad categories:

  • Driver routines , each of which solves a complete problem, for example, solving a system of linear equations or computing the eigenvalues of a real symmetric matrix. Users are recommended to use a driver routine if one meets their requirements. Driver routines are described in section 3.2, and a complete list of routine names can be found in Appendix A.1. Global and local input error-checking are performed where possible for these routines.
  • Computational routines , each of which performs a distinct computational task, for example an LU factorization or the reduction of a real symmetric matrix to tridiagonal form. Each driver routine calls a sequence of computational routines. Users (especially software developers) may need to call computational routines directly to perform tasks, or sequences of tasks, that cannot conveniently be performed by the driver routines. Computational routines are described in section 3.3 and a complete list of routine names can be found in Appendix A.1. Global and local input error-checking are performed for these routines.
  • Auxiliary routines , which in turn can be classified as follows:

    • routines that perform subtasks of block-partitioned algorithms  -- in particular, routines that implement unblocked versions of the algorithms; and
    • routines that perform some commonly required low-level computations, for example, scaling a matrix, computing a matrix-norm, or generating an elementary Householder matrix; some of these may be of interest to numerical analysts or software developers and could be considered for future additions to the PBLAS.

In general, no input error-checking is performed in the auxiliary routines. The exception to this rule is for the auxiliary routines that are Level 2 equivalents of computational routines (e.g., PxGETF2, PxGEQR2, PxORMR2, PxORM2R). For these routines, local input error-checking is performed.

Both driver routines and computational routines are fully described in this users guide, but not the auxiliary routines. A list of the auxiliary routines, with brief descriptions of their functions, is given in Appendix A.2. LAPACK auxiliary routines are also used whenever possible for local computation. Refer to the LAPACK Users' Guide [3] for details.

The PBLAS, BLAS, BLACS, and LAPACK are strictly-speaking not part of the ScaLAPACK routines. However, the ScaLAPACK routines make frequent calls to these packages.

ScaLAPACK also provides two matrix redistribution/copy routines for each data type [107, 49, 106]. These routines provide a truly general copy from any block cyclicly distributed (sub)matrix to any other block cyclicly distributed (sub)matrix. These routines are the only ones in the entire ScaLAPACK library which provide inter-context operations. Because of the generality of these routines, they may be used for many operations not usually associated with copy routines. For instance, they may be used to a take a matrix on one process and distribute it across a process grid, or the reverse. If a supercomputer is grouped into a virtual parallel machine with a workstation, for instance, this routine can be used to move the matrix from the workstation to the supercomputer and back. In ScaLAPACK, these routines are called to copy matrices from a two-dimensional process grid to a one-dimensional process grid. They can be used to redistribute matrices so that distributions providing maximal performance can be used by various component libraries, as well. For further details on these routines, refer to Appendix A.3.

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Data Types and Precision

ScaLAPACK provides the same range of functionality for real and complex data, with a few exceptions. The complex Hermitian eigensolver (PCHEEV) and complex singular value decomposition (PCGESVD) are still under development. They may be available in a future release of ScaLAPACK.

Matching routines for real and complex data have been coded to maintain a close correspondence between the two, wherever possible. However, there are cases where the corresponding complex version calling sequence has more arguments than the real version.

All routines in ScaLAPACK are provided in both single- and double-precision versions.

Double-precision routines for complex matrices require the nonstandard Fortran 77 data type COMPLEX*16, which is available on most machines where double precision computation is usual.


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Naming Scheme

  

Each subroutine name in ScaLAPACK, which has an LAPACK equivalent, is simply the LAPACK name prepended by a P. Thus, we have relaxed (violated) the Fortran 77 standard by allowing subroutine names to be greater than 6-characters in length and allowing an underscore _ in the names of certain TOOLS routines.

All driver and computational routines  have names of the form PXYYZZZ, where for some driver routines the seventh character is blank.

The second letter, X, indicates the data type as follows:


tabular885

When we wish to refer to a ScaLAPACK routine generically, regardless of data type, we replace the second letter by ``x''. Thus PxGESV refers to any or all of the routines PSGESV, PCGESV, PDGESV, and PZGESV.

The next two letters, YY, indicate the type of matrix (or of the most significant matrix). Most of these two-letter codes apply to both real and complex matrices; a few apply specifically to one or the other, as indicated in table 3.1.

  table891
Table 3.1: Matrix types in the ScaLAPACK naming scheme

A diagonally dominant-like  matrix is one for which it is known a priori that pivoting for stability is NOT required in the LU factorization of the matrix. Diagonally dominant matrices themselves are examples of diagonally dominant-like matrices.

When we wish to refer to a class of routines that performs the same function on different types of matrices, we replace the second, third, and fourth letters by ``xyy''. Thus, PxyySVX refers to all the expert driver routines for systems of linear equations that are listed in table 3.2.

The last three letters ZZZ indicate the computation performed. Their meanings will be explained in section 3.3. For example, PSGEBRD is a single-precision routine that performs a bidiagonal reduction (BRD) of a real general matrix.

The names of auxiliary routines  follow a similar scheme except that the third and fourth characters YY are usually LA (for example, PSLASCL or PCLARFG). There are two kinds of exception. Auxiliary routines that implement an unblocked version of a block-partitioned algorithm  have similar names to the routines that perform the block-partitioned algorithm, with the seventh character being ``2'' (for example, PSGETF2 is the unblocked version of PSGETRF). A few routines that may be regarded as extensions to the BLAS are named similar to the BLAS naming schemes (for example, PCMAX1, PSCSUM1).

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Driver Routines

 

This section describes the driver routines  in ScaLAPACK. Further details on the terminology and the numerical operations they perform are given in section 3.3, which describes the computational routines. If the parallel algorithm or implementation differs significantly from the serial LAPACK equivalent, this fact will be noted and the user directed to consult the appropriate LAPACK working note.



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Linear Equations

 

Two types of driver routines are provided for solving systems of linear equations :

  • a simple driver (name ending -SV)  , which solves the system AX = B by factorizing A and overwriting B with the solution X;
  • an expert driver (name ending -SVX)  , which can also perform some or all of the following functions (some of them optionally):

    • solve tex2html_wrap_inline12654 or tex2html_wrap_inline12656 (unless A is symmetric or Hermitian);
    • estimate the condition number of A, check for near-singularity, and check for pivot growth;
    • refine the solution and compute forward and backward error bounds;
    • equilibrate  the system if A is poorly scaled.

    The expert driver requires roughly twice as much storage as the simple driver in order to perform these extra functions.

Both types of driver routines can handle multiple right-hand sides (the columns of B).

Different driver routines are provided to take advantage of special properties or storage schemes of the matrix A, as shown in table 3.2.

These driver routines cover all the functionality of the computational routines for linear systems , except matrix inversion . It is seldom necessary to compute the inverse of a matrix explicitly, and such computation is certainly not recommended as a means of solving linear systems.

At present, only simple drivers (name ending -SV) are provided for systems involving band and tridiagonal matrices. It is important to note that in the banded and tridiagonal factorizations (PxDBTRF, PxDTTRF, PxGBTRF, PxPBTRF, and PxPTTRF) used within these drivers, the resulting factorization is not the same factorization as returned from LAPACK. Additional permutations are performed on the matrix for the sake of parallelism. Further details of the algorithmic implementations can be found in [32].

  table931
Table 3.2: Driver routines for linear equations

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Linear Least Squares Problems

   

The linear least squares (LLS) problem  is:
 equation988
where A is an m-by-n matrix, b is a given m element vector and x is the n element solution vector.

In the most usual case, tex2html_wrap_inline12690 and tex2html_wrap_inline12692. In this case the solution to problem (3.1) is unique. The problem is also referred to as finding a least squares solution to an overdetermined  system of linear equations.

When m < n and tex2html_wrap_inline12696, there are an infinite number of solutions x that exactly satisfy b-Ax=0. In this case it is often useful to find the unique solution x that minimizes tex2html_wrap_inline12704, and the problem is referred to as finding a minimum norm solution  to an underdetermined  system of linear equations.

The driver routine PxGELS  solves problem (3.1) on the assumption that tex2html_wrap_inline12706 -- in other words, A has full rank -- finding a least squares solution of an overdetermined   system when m > n, and a minimum norm solution of an underdetermined  system when m < n. PxGELS     uses a QR or LQ factorization   of A and also allows A to be replaced by tex2html_wrap_inline12722 in the statement of the problem (or by tex2html_wrap_inline12724 if A is complex).

In the general case when we may have tex2html_wrap_inline12728 -- in other words, A may be rank-deficient  -- we seek the minimum norm least squares solution  x that minimizes both tex2html_wrap_inline12704 and tex2html_wrap_inline12736.

The LLS  driver routines are listed in table 3.3.

All routines allow several right-hand-side vectors b and corresponding solutions x to be handled in a single call, storing these vectors as columns of matrices B and X, respectively. Note, however, that equation 3.1 is solved for each right-hand-side vector independently; this is not the same as finding a matrix X that minimizes tex2html_wrap_inline12748.

  table1027
Table 3.3: Driver routines for linear least squares problems

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Standard Eigenvalue and Singular Value Problems

 


Susan Blackford
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Symmetric Eigenproblems

   

The symmetric eigenvalue problem (SEP) is to find the eigenvalues  , tex2html_wrap_inline12778, and corresponding eigenvectors , tex2html_wrap_inline12780, such that
displaymath12772
For the Hermitian eigenvalue problem  we have
displaymath12773
For both problems the eigenvalues tex2html_wrap_inline12778 are real.

When all eigenvalues and eigenvectors have been computed, we write
displaymath12774
where tex2html_wrap_inline12784 is a diagonal matrix whose diagonal elements are the eigenvalues , and Z is an orthogonal (or unitary) matrix whose columns are the eigenvectors. This is the classical spectral factorization   of A.

Two types of driver routines  are provided for symmetric or Hermitian eigenproblems:

  • a simple driver (name ending -EV)  , which computes all the eigenvalues and (optionally) the eigenvectors of a symmetric or Hermitian matrix A;
  • an expert driver (name ending -EVX)  , which can compute either all or a selected subset of the eigenvalues, and (optionally) the corresponding eigenvectors.

The driver routines are shown in table 3.4. Currently the only simple drivers provided are PSSYEV and PDSYEV.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Singular Value Decomposition

The singular value decomposition (SVD) of an m-by-n matrix A is given by   
displaymath12804
where U and V are orthogonal (unitary) and tex2html_wrap_inline12820 is an m-by-n diagonal matrix with real diagonal elements, tex2html_wrap_inline12826, such that
displaymath12805
The tex2html_wrap_inline12826 are the singular values of A and the first min(m,n) columns of U and V are the left and right singular vectors of A.   

The singular values and singular vectors satisfy
displaymath12806
where tex2html_wrap_inline12840 and tex2html_wrap_inline12842 are the ith columns of U and V, respectively.

A single driver  routine, PxGESVD  , computes the ``economy size'' or ``thin'' singular value decomposition of a general nonsymmetric matrix (see table 3.4). Thus, if A is m-by-n with m>n, then only the first n columns of U are computed and tex2html_wrap_inline12820 is an n-by-n matrix. For a detailed discussion of the ``thin'' singular value decomposition, refer to [71, p. 72,].

Currently, only PSGESVD and PDGESVD are provided.

  table1093
Table 3.4: Driver routines for standard eigenvalue and singular value problems


Susan Blackford
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Generalized Symmetric Definite Eigenproblems (GSEP)

  

   An expert driver  is provided to compute all the eigenvalues and (optionally) the eigenvectors of the following types of problems:

  1. tex2html_wrap_inline12875
  2. tex2html_wrap_inline12877
  3. tex2html_wrap_inline12879

where A and B are symmetric or Hermitian and B is positive definite. For all these problems the eigenvalues  tex2html_wrap_inline12778 are real. When A and B are symmetric, the matrices Z of computed eigenvectors  satisfy tex2html_wrap_inline12895 (problem types 1 and 3) or tex2html_wrap_inline12897 (problem type 2), where tex2html_wrap_inline12784 is a diagonal matrix with the eigenvalues on the diagonal. Z also satisfies tex2html_wrap_inline12903 (problem types 1 and 2) or tex2html_wrap_inline12905 (problem type 3). When A and B are Hermitian, the matrices Z of computed eigenvectors  satisfy tex2html_wrap_inline12913 (problem types 1 and 3) or tex2html_wrap_inline12915 (problem type 2), where tex2html_wrap_inline12784 is a diagonal matrix with the eigenvalues on the diagonal. Z also satisfies tex2html_wrap_inline12921 (problem types 1 and 2) or tex2html_wrap_inline12923 (problem type 3).

The routine is listed in table 3.5.

  table1135
Table 3.5: Driver routine for the generalized symmetric definite eigenvalue problems


Susan Blackford
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Preface

Following the initial release of LAPACK and the emerging importance of distributed memory computing , work began on adapting LAPACK to distributed-memory architectures. Since porting software efficiently from one distributed-memory architecture to another is a challenging task, this work is an effort to establish standards for library development in the varied world of distributed-memory computing.

ScaLAPACK is an acronym for Scalable Linear Algebra PACKage, or Scalable LAPACK. As in LAPACK, the ScaLAPACK routines are based on block-partitioned algorithms in order to minimize the frequency of data movement between different levels of the memory hierarchy. (For distributed-memory machines, the memory hierarchy includes the off-processor memory of other processors, in addition to the hierarchy of registers, cache, and local memory on each processor.) The fundamental building block of the ScaLAPACK library is a distributed-memory version of the Level 1, 2, and 3 BLAS, called the PBLAS (Parallel BLAS). The PBLAS are in turn built on the BLAS for computation on single nodes and on a set of Basic Linear Algebra Communication Subprograms (BLACS) for communication tasks that arise frequently in parallel linear algebra computations. For optimal performance, it is necessary, first, that the BLAS be implemented efficiently on the target machine, and second, that an efficient version of the BLACS be available.

Versions of the BLACS exist for both MPI and PVM, as well as versions for the Intel series (NX), IBM SP series (MPL), and Thinking Machines CM-5 (CMMD). A vendor-optimized version of the BLACS is available for the Cray T3 series. Thus, ScaLAPACK is portable on any computer or network of computers that supports MPI or PVM (as well as the aforementioned native message-passing protocols).

Most of the ScaLAPACK code is written in standard Fortran 77; the PBLAS and the BLACS are written in C, but with Fortran 77 interfaces.

The first ScaLAPACK software was written in 1989-1990, and the appearance of the code has undergone many changes since then in our pursuit to resemble and enable code reuse from LAPACK.

The first public release (version 1.0) of ScaLAPACK occurred on February 28, 1995, and subsequent releases occurred in 1996.

The ScaLAPACK library is only one facet of the ``ScaLAPACK Project,'' which is a collaborative effort involving several institutions:

  • Oak Ridge National Laboratory
  • Rice University
  • University of California, Berkeley
  • University of California, Los Angeles
  • University of Illinois, Champaign-Urbana
  • University of Tennessee, Knoxville
and comprises four components:
  • dense and band matrix software (ScaLAPACK)
  • large sparse eigenvalue software (P_ARPACK)
  • sparse direct systems software (CAPSS)
  • preconditioners for large sparse iterative solvers (ParPre)

For further information on any of the related ScaLAPACK projects, please refer to the scalapack index on netlib:

http://www.netlib.org/scalapack/index.html

This users guide describes version 1.5 of the dense and band matrix software package (ScaLAPACK).

The University of Tennessee, Knoxville, provided the routines for the solution of dense, band, and tridiagonal linear systems of equations, condition estimation and iterative refinement, for LU and Cholesky factorization, matrix inversion, full-rank linear least squares problems, orthogonal and generalized orthogonal factorizations, orthogonal transformation routines, reductions to upper Hessenberg, bidiagonal and tridiagonal form, and reduction of a symmetric-definite generalized eigenproblem to standard form. And finally, the BLACS, the PBLAS, and the HPF wrappers were also written at the University of Tennessee, Knoxville.

The University of California, Berkeley, provided the routines for the symmetric and generalized symmetric eigenproblem and the singular value decomposition.

Greg Henry at Intel Corporation provided the routines for the nonsymmetric eigenproblem.

Oak Ridge National Laboratory provided the out-of-core linear solvers for LU, Cholesky, and QR factorizations.

ScaLAPACK has been incorporated into several commercial packages, including the NAG Parallel Library, IBM Parallel ESSL, and Cray LIBSCI, and is being integrated into the VNI IMSL Numerical Library, as well as software libraries for Fujitsu, Hewlett-Packard/Convex, Hitachi, and NEC. Additional information can be found on the respective Web pages:

http://www.nag.co.uk:80/numeric/FM.html
http://www.rs6000.ibm.com/software/sp_products/esslpara.html
http://www.cray.com/PUBLIC/product-info/sw/PE/LibSci.html
http://www.sgi.com/Products/hardware/Power/ch_complib.html
http://www.vni.com/products/imsl/index.html

A number of technical reports have been written during the development of ScaLAPACK and published as LAPACK Working Notes by the University of Tennessee. Refer to the following URL for a complete set of working notes:

http://www.netlib.org/lapack/lawns/index.html
Many of these reports subsequently appeared as journal articles. The Bibliography gives the most recent published reference.

As the distributed-memory  version of LAPACK, ScaLAPACK has drawn heavily on the software and documentation standards set by LAPACK. The test and timing software for the Level 2 and 3 BLAS was used as a model for the PBLAS test and timing software, and the ScaLAPACK test suite was patterned after the LAPACK test suite. Because of the large amount of software, all BLACS, PBLAS, and ScaLAPACK routines are maintained in basefiles whereby the codes can be re-extracted as needed. Final formatting of the software was done using Toolpack/1 [105].

We have tried to be consistent with our documentation and coding style throughout ScaLAPACK in the hope that it will serve as a model for other distributed-memory software development efforts. ScaLAPACK has been designed as a source of building blocks for larger parallel applications.

The development of ScaLAPACK was supported in part by National Science Foundation Grant ASC-9005933; by the Defense Advanced Research Projects Agency under contract DAAH04-95-1-0077, administered by the Army Research Office; by the Division of Mathematical, Information, and Computational Sciences, of the U.S. Department of Energy, under Contract DE-AC05-96OR22464; and by the National Science Foundation Science and Technology Center Cooperative Agreement CCR-8809615.

The performance results presented in this book were obtained using computer resources at various sites:

  • Cray T3E, located at Lawrence Berkeley National Laboratory, National Energy Research Scientific Computing Center (NERSC), supported by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy under contract number 76SF00098.
  • IBM SP-2, located at the Cornell Theory Center, which receives major funding from the National Science Foundation (NSF) and New York State, with additional support from the Defense Advanced Research Projects Agency (DARPA), the National Center for Research Resources at the National Institutes of Health (NIH), IBM Corporation, and other members of the center's Corporate Partnership Program.
  • Intel MP Paragon XPS/35, located at Intel Corporation, Portland, Oregon.
  • Intel ASCI Option Red Supercomputer Technology located in Beaverton, Oregon.
  • Network of Sun Ultra Enterprise 2 (Model 2170s) workstations, located in the Department of Computer Science at the University of Tennessee, funded by National Science Foundation Grant CDA-9529459, the Center of Excellence - Science Alliance, UT Networking Services, and the UT Computer Science Department.
  • Network of Sun UltraSPARC-1 workstations, located in the Department of Computer Science at the University of California, Berkeley supported by DARPA Grant F30602-95-C-0014, NSF Grants CCR-9257974 and PFF-CCR-9253705, as well as California MICRO Grants. Corporate sponsors are: the AT&T Foundation, Digital Equipment Corporation, Exabyte Corporation, Hewlett-Packard Company, Informix Software Inc, Intel Corporation, International Business Machines, Internet Archive, Microsoft Corporation, Mitsubishi Electric Research Laboratories, Myricom Inc, Siemens Corporation, Sun Microsystems, Synoptics Corporation, Tandem Corporation, and TIBCO Inc.

The cover of this book was designed by Andy Cleary at Lawrence Livermore National Laboratory.

We acknowledge with gratitude the support that we have received from the following organizations, and the help of individual members of their staff: Cornell Theory Center, Cray Research, a Silicon Graphics Company, IBM (Parallel ESSL Development and Research), Lawrence Berkeley National Laboratory, National Energy Research Scientific Computing Center (NERSC), Maui High Performance Computer Center, Minnesota Supercomputing Center, NAG Ltd., and Oak Ridge National Laboratory Center for Computational Sciences (CCS).

We also thank the many, many people who have contributed code, criticism, ideas and encouragement. We especially acknowledge the contributions of Mark Adams, Peter Arbenz, Scott Betts, Shirley Browne, Henri Casanova, Soumen Chakrabarti, Mishi Derakhshan, Frederic Desprez, Brett Ellis, Ray Fellers, Markus Hegland, Nick Higham, Adolfy Hoisie, Velvel Kahan, Xiaoye Li, Bill Magro, Osni Marques, Paul McMahan, Caroline Papadopoulos, Beresford Parlett, Loic Prylli, Yves Robert, Howard Robinson, Tom Rowan, Shilpa Singhal, Françoise Tisseur, Bernard Tourancheau, Anne Trefethen, Robert van de Geijn, and Andrey Zege.

We express appreciation to all those who helped in the preparation of this work, in particular to Gail Pieper for her tireless efforts in proofreading the draft and improving the quality of the presentation.

Finally, we thank all the test sites that received several test releases of the ScaLAPACK software and that ran an extensive series of test programs for us.

tex2html_wrap12059

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Computational Routines

  

As previously stated, if the parallel algorithm or implementation differs significantly from the serial LAPACK equivalent, this fact will be noted and the user directed to consult the appropriate LAPACK Working Note.



Susan Blackford
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Linear Equations

 

We use the standard notation for a system of simultaneous linear  equations :
 equation1162
where A is the coefficient matrix , b is the right-hand side  , and x is the solution  . In (3.2) A is assumed to be a square matrix of order n, but some of the individual routines allow A to be rectangular. If there are several right-hand sides we write
 equation1174
where the columns of B are the individual right-hand sides, and the columns of X are the corresponding solutions. The basic task is to compute X, given A and B.

If A is upper or lower triangular, (3.2) can be solved by a straightforward process of backward or forward substitution. Otherwise, the solution is obtained after first factorizing A as a product of triangular matrices (and possibly also a diagonal matrix or permutation matrix).

The form of the factorization depends on the properties of the matrix A. ScaLAPACK provides routines for the following types of matrices, based on the stated  factorizations:

  • general matrices   (LU factorization with partial pivoting)  :
    displaymath12933
    where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n).
  • symmetric and Hermitian positive definite matrices (Cholesky factorization) :
    displaymath12934

    displaymath12935
    where U is an upper triangular matrix and L is lower triangular.
  • general band matrices (LU factorization with partial pivoting):

    If A is m-by-n with bwl subdiagonals and bwu superdiagonals, the factorization is
    displaymath12936
    where P and Q are permutation matrices and L and U are banded lower and upper triangular matrices, respectively.

  • general diagonally dominant-like band matrices  including general tridiagonal matrices (LU factorization without pivoting):

    A diagonally dominant-like  matrix is one for which it is known a priori that pivoting for stability is NOT required in the LU factorization of the matrix. Diagonally dominant matrices themselves are examples of diagonally dominant-like matrices.

    If A is m-by-n with bwl subdiagonals and bwu superdiagonals, the factorization is
    displaymath12937
    where P is a permutation matrix and L and U are banded lower and upper triangular matrices respectively.

  • symmetric and Hermitian positive definite band matrices (Cholesky factorization) :
    displaymath12938

    displaymath12939
    where P is a permutation matrix and U and L are banded upper and lower triangular matrices, respectively.
  • symmetric and Hermitian positive definite tridiagonal matrices (tex2html_wrap_inline13039 factorization):  
    displaymath12940

    displaymath12941
    where P is a permutation matrix and U and L are bidiagonal upper and lower triangular matrices respectively.

Note: In the banded and tridiagonal factorizations (PxDBTRF, PxDTTRF, PxGBTRF, PxPBTRF, and PxPTTRF), the resulting factorization is not the same factorization as returned from LAPACK. Additional permutations are performed on the matrix for the sake of parallelism. Further details of the algorithmic implementations can be found in [32].

The factorization for a general diagonally dominant-like  tridiagonal matrix is like that for a general diagonally dominant-like band matrix with bwl = 1 and bwu = 1. Band matrices use the band storage scheme described in section 4.4.3.

While the primary use of a matrix factorization is to solve a system of equations, other related tasks are provided as well. Wherever possible, ScaLAPACK provides routines to perform each of these tasks for each type of matrix and storage scheme (see table 3.6). The following list relates the tasks to the last three characters of the name of the corresponding computational routine:

PxyyTRF:
factorize (obviously not needed for triangular matrices);                                 

PxyyTRS:
use the factorization (or the matrix A itself if it is triangular) to solve (3.3) by forward or backward substitution;                                      

PxyyCON:
estimate the reciprocal of the condition number tex2html_wrap_inline13057; Higham's modification [81] of Hager's method [72] is used to estimate tex2html_wrap_inline13059 (not provided for band or tridiagonal matrices);               

PxyyRFS:
compute bounds on the error in the computed solution (returned by the PxyyTRS routine), and refine the solution to reduce the backward error (see below) (not provided for band or tridiagonal matrices);               

PxyyTRI:
use the factorization (or the matrix A itself if it is triangular) to compute tex2html_wrap_inline13063 (not provided for band matrices, because the inverse does not in general preserve bandedness);                   

PxyyEQU:
compute scaling factors to equilibrate  A (not provided for band, tridiagonal, or triangular matrices). These routines do not actually scale the matrices: auxiliary routines PxLAQyy may be used for that purpose -- see the code of the driver routines PxyySVX for sample usage .          

Note that some of the above routines depend on the output of others:

PxyyTRF:
may work on an equilibrated matrix produced by PxyyEQU and PxLAQyy, if yy is one of {GE, PO};

PxyyTRS:
requires the factorization returned by PxyyTRF;

PxyyCON:
requires the norm of the original matrix A and the factorization returned by PxyyTRF;

PxyyRFS:
requires the original matrices A and B, the factorization returned by PxyyTRF, and the solution X returned by PxyyTRS;

PxyyTRI:
requires the factorization returned by PxyyTRF.

The RFS (``refine solution'') routines perform iterative refinement  and compute backward and forward error  bounds for the solution. Iterative refinement is done in the same precision as the input data. In particular, the residual is not computed with extra precision, as has been traditionally done. The benefit of this procedure is discussed in section 6.5.

  table1358
Table 3.6: Computational routines for linear equations

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Orthogonal Factorizations and Linear Least Squares Problems

 

ScaLAPACK provides a number of routines for factorizing a general rectangular m-by-n matrix A, as the product of an orthogonal matrix (unitary if complex) and a triangular (or possibly trapezoidal) matrix.  

A real matrix Q is orthogonal if tex2html_wrap_inline13210 ; a complex matrix Q is unitary if tex2html_wrap_inline13214 . Orthogonal or unitary matrices   have the important property that they leave the two-norm of a vector invariant:
displaymath13200
As a result, they help to maintain numerical stability because they do not   amplify rounding errors.

Orthogonal factorizations    are used in the solution of linear least squares problems . They may also be used to perform preliminary steps in the solution of eigenvalue or singular value problems.

Table 3.7 lists all routines provided by ScaLAPACK to perform orthogonal factorizations and the generation or pre- or post-multiplication of the matrix Q for each matrix type and storage scheme.



Susan Blackford
Tue May 13 09:21:01 EDT 1997
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QR Factorization

 

The most common, and best known, of the factorizations is the QR factorization  given by
displaymath13230
where R is an n-by-n upper triangular matrix and Q is an m-by-m orthogonal (or unitary) matrix. If A is of full rank n, then R is nonsingular. It is sometimes convenient to write the factorization as
displaymath13231
which reduces to
displaymath13232
where tex2html_wrap_inline13270 consists of the first n columns of Q, and tex2html_wrap_inline13276 the remaining m-n columns.

If m < n, R is trapezoidal, and the factorization can be written
displaymath13233
where tex2html_wrap_inline13286 is upper triangular and tex2html_wrap_inline13288 is rectangular.

The routine PxGEQRF     computes the QR factorization  . The matrix Q is not formed explicitly, but is represented as a product of elementary reflectors,     as described in section 3.4. Users need not be aware of the details of this representation, because associated routines are provided to work with Q: PxORGQR   (or PxUNGQR   in the complex case) can generate all or part of Q, while PxORMQR   (or PxUNMQR)   can pre- or post-multiply a given matrix by Q or tex2html_wrap_inline13300 (tex2html_wrap_inline13302 if complex).

The QR factorization can be used to solve the linear least squares problem (3.1)   when tex2html_wrap_inline13306 and A is of full rank, since
displaymath13234
c can be computed by PxORMQR   (or PxUNMQR  ), and tex2html_wrap_inline13312 consists of its first n elements. Then x is the solution of the upper triangular system
displaymath13235
which can be computed by PxTRTRS    . The residual vector r is given by
displaymath13236
and may be computed using PxORMQR   (or PxUNMQR  ). The residual sum of squares tex2html_wrap_inline13320 may be computed without forming r explicitly, since
displaymath13237

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Susan Blackford
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LQ Factorization

The LQ factorization   is given by
displaymath13356
where L is m-by-m lower triangular, Q is n-by-n orthogonal (or unitary), tex2html_wrap_inline13270 consists of the first m rows of Q, and tex2html_wrap_inline13276 consists of the remaining n-m rows.

This factorization is computed by the routine PxGELQF, and again Q is      represented as a product of elementary reflectors; PxORGLQ      (or PxUNGLQ   in the complex case) can generate all or part of Q, and PxORMLQ   (or PxUNMLQ  ) can pre- or post-multiply a given matrix by Q or tex2html_wrap_inline13300 (tex2html_wrap_inline13302 if Q is complex).

The LQ factorization of A is essentially the same as the QR factorization of tex2html_wrap_inline12722 (tex2html_wrap_inline12724 if A is complex), since
displaymath13357

The LQ factorization may be used to find a minimum norm solution  of an underdetermined   system of linear equations A x = b, where A is m-by-n with m < n and has rank m. The solution is given by
displaymath13358
and may be computed by calls to PxTRTRS and PxORMLQ.        


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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QR Factorization with Column Pivoting

To solve a linear least squares problem (3.1)   when A is not of full rank, or the rank of A is in doubt, we can perform either a QR factorization with column pivoting     or a singular value decomposition (see subsection 3.3.6).

The QR factorization with column pivoting is given by
displaymath13452
where Q and R are as before and P is a permutation matrix, chosen (in general) so that
displaymath13453
and moreover, for each k,
displaymath13454
In exact arithmetic, if tex2html_wrap_inline13482, then the whole of the submatrix tex2html_wrap_inline13484 in rows and columns k+1 to n would be zero. In numerical computation, the aim must be to determine an index k such that the leading submatrix tex2html_wrap_inline13492 in the first k rows and columns is well conditioned and tex2html_wrap_inline13484 is negligible:
displaymath13455
Then k is the effective rank of A. See Golub and Van Loan [71] for a further discussion of numerical rank determination.   

The so-called basic solution to the linear least squares problem (3.1)  can be obtained from this factorization as
displaymath13456
where tex2html_wrap_inline13502 consists of just the first k elements of tex2html_wrap_inline13506.

The routine PxGEQPF     computes the QR factorization with column pivoting  but does not attempt to determine the rank of A. The matrix Q is represented in exactly the same way as after a call of PxGEQRF    , and so the routines PxORGQR and PxORMQR can be used to work with Q (PxUNGQR and PxUNMQR if Q is complex).          


Susan Blackford
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Complete Orthogonal Factorization

The QR factorization with column pivoting does not enable us to compute a minimum norm solution to a rank-deficient linear least squares problem   unless tex2html_wrap_inline13551. However, by applying further orthogonal (or unitary) transformations  from the right to the upper trapezoidal matrix tex2html_wrap_inline13553, using the routine PxTZRZF, tex2html_wrap_inline13555 can be eliminated:     
displaymath13543
This gives the complete orthogonal factorization  
displaymath13544
from which the minimum norm solution  can be obtained as
displaymath13545

The matrix Z is not formed explicitly but is represented as a product of elementary reflectors,     as described in section 3.4. Users need not be aware of the details of this representation, because associated routines are provided to work with Z: PxORMRZ   (or PxUNMRZ  ) can pre- or post-multiply a given matrix by Z or tex2html_wrap_inline13563 (tex2html_wrap_inline13565 if complex).


Susan Blackford
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Other Factorizations

The QL and RQ factorizations      are given by
displaymath13582
and
displaymath13583
These factorizations are computed by PxGEQLF and PxGERQF, respectively; they are         less commonly used than either the QR or LQ factorizations described above, but have applications in, for example, the computation of generalized QR factorizations [5].    

All the factorization routines discussed here (except PxTZRZF) allow arbitrary m and n, so that in some cases the matrices R or L are trapezoidal rather than triangular. A routine that performs pivoting is provided only for the QR factorization.

  table1747
Table 3.7: Computational routines for orthogonal factorizations


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Generalized Orthogonal Factorizations

      



Susan Blackford
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Generalized QR Factorization

 

   The generalized QR (GQR) factorization of an n-by-m matrix A and an n-by-p matrix B is given by the pair of factorizations
displaymath13656
where Q and Z are respectively n-by-n and p-by-p orthogonal matrices (or unitary matrices if A and B are complex). R has the form
displaymath13657
or
displaymath13658
where tex2html_wrap_inline13492 is upper triangular. T has the form
displaymath13659
or
displaymath13660
where tex2html_wrap_inline13706 or tex2html_wrap_inline13708 is upper triangular.

Note that if B is square and nonsingular, the GQR factorization of A and B implicitly gives the QR factorization of the matrix tex2html_wrap_inline13718:
displaymath13661
without explicitly computing the matrix inverse tex2html_wrap_inline13720 or the product tex2html_wrap_inline13718.

The routine PxGGQRF computes the GQR  factorization by     computing first the QR factorization of A and then the RQ factorization of tex2html_wrap_inline13730. The orthogonal (or unitary) matrices Q and Z can be formed explicitly or can be used just to multiply another given matrix in the same way as the orthogonal (or unitary) matrix in the QR factorization (see section 3.3.2).

The GQR factorization was introduced in [73, 100]. The implementation of the GQR factorization here follows that in [5]. Further generalizations of the GQR  factorization can be found in [36].


Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node5.html0100644000056400000620000001743606336113633017372 0ustar pfrauenfstaff Suggestions for Reading next up previous contents index
Next: List of Notation Up: List of Tables Previous: Preface

Suggestions for Reading

 

This users guide is divided into two parts. Part I: Guide contains chapters and appendices providing a thorough explanation of the design and functionality of the ScaLAPACK library. These chapters should be read in the order in which they are presented. Part II: Specifications of Routines is a reference manual of the leading comments of each routine in alphabetical order by routine name. A Bibliography is also provided, as well as two indexes- Index by Keyword and Index by Routine Name.

This book assumes a basic knowledge of distributed-memory parallel programming, and is written for an audience of both novice and expert users. Users intimately familiar with specific concepts discussed in this book may choose to not read certain chapters or sections within this book. Some of the chapters can be regarded as stand-alone and read independently. Novice users are directed to focus their attention on special introductory chapters, sections, and example programs, as detailed below.

All users are encouraged to frequently refer to the List of Notation and the Glossary. The first time notation from the glossary appears in the text, it will be italicized. If the user is unfamiliar with any of the concepts defined, a number of books provide background information in parallel programming [6, 33, 65, 66, 67, 70, 75, 87, 92, 95, 99, 112].

  • Chapter 1: Essentials provides a brief overview of the components of the library, downloading instructions, and details of support for the package. Users who are familiar with the design of the BLAS and LAPACK and acquainted with the existing Web pages may wish to skip this chapter.
  • Chapter 2: Getting Started with ScaLAPACK presents the basic requirements to enable users to call ScaLAPACK software, together with a very simple example program. Users who are well versed in using ScaLAPACK software may choose to skip this chapter.
  • Chapter 3: Contents of ScaLAPACK outlines the functionality provided by the package. This is a stand-alone chapter and important for both expert and novice users.
  • Chapter 4: Data Distributions and Software Conventions discusses process grid layout, contexts, block and block-cyclic data distributions, and documentation and software conventions. This chapter is essential reading for any user who is not familiar with data distributions, array descriptors, and the calling sequences of ScaLAPACK routines.
  • Chapter 5: Performance of ScaLAPACK provides guidelines to achieve high performance by using ScaLAPACK and presents performance results for a subset of the ScaLAPACK routines on a variety of distributed-memory MIMD computers and networks of workstations. This is a stand-alone chapter.
  • Chapter 6: Accuracy and Stability discusses the accuracy and stability of the algorithms used in ScaLAPACK, as well as issues of heterogeneous computing. This chapter provides varying degrees of detail, catering to novice as well as expert users.
  • Chapter 7: Troubleshooting provides a set of installation and application debugging hints for first-time ScaLAPACK users.
  • Appendix A provides a list of routine names for all driver, computational, and auxiliary routines in ScaLAPACK, as well as the matrix redistribution/copy routines.
  • Appendix B provides a brief tutorial on how to convert programs using the BLAS to the PBLAS and LAPACK to ScaLAPACK.
  • Appendix C provides two additional example programs. Section C.1 contains a more memory-efficient and practical example program, which reads a matrix from a file, distributes this matrix to the process grid, calls the desired ScaLAPACK routine, and writes the solution matrix to a file. Section C.2 provides a brief description of the HPF interface to ScaLAPACK, as well as an example program.
  • Appendix D contains Quick Reference Guides for ScaLAPACK, the PBLAS, and the BLACS.
  • Part ii: Specifications of Routines is a reference manual of the leading comments from the source code of all driver and computational routines. This manual can be read selectively as needed.

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Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node60.html0100644000056400000620000001023506336113674017446 0ustar pfrauenfstaff Generalized RQ factorization next up previous contents index
Next: Symmetric Eigenproblems Up: Generalized Orthogonal Factorizations Previous: Generalized QR Factorization

Generalized RQ factorization

    The generalized RQ (GRQ) factorization of an m-by-n matrix A and a p-by-n matrix B is given by the pair of factorizations
displaymath13747
where Q and Z are respectively n-by-n and p-by-p orthogonal matrices (or unitary matrices if A and B are complex). R has the form
displaymath13748
or
displaymath13749
where tex2html_wrap_inline13555 or tex2html_wrap_inline13795 is upper triangular. T has the form
displaymath13750
or
displaymath13751
where tex2html_wrap_inline13799 is upper triangular.

Note that if B is square and nonsingular, the GRQ factorization of A and B implicitly gives the RQ factorization of the matrix tex2html_wrap_inline13809:
displaymath13752
without explicitly computing the matrix inverse tex2html_wrap_inline13720 or the product tex2html_wrap_inline13809.

The routine PxGGRQF computes the GRQ factorization       by computing first the RQ factorization of A and then the QR factorization of tex2html_wrap_inline13821. The orthogonal (or unitary) matrices Q and Z can be formed explicitly or can be used just to multiply another given matrix in the same way as the orthogonal (or unitary) matrix in the RQ factorization (see section 3.3.2).


Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node61.html0100644000056400000620000002131706336113675017453 0ustar pfrauenfstaff Symmetric Eigenproblems next up previous contents index
Next: Nonsymmetric Eigenproblems Up: Computational Routines Previous: Generalized RQ factorization

Symmetric Eigenproblems

   

Let A be a real symmetric   or complex Hermitian n-by-n matrix. A scalar tex2html_wrap_inline12778 is called an eigenvalue  and a nonzero column vector z the corresponding eigenvector  if tex2html_wrap_inline13848. tex2html_wrap_inline12778 is always real when A is real symmetric or complex Hermitian.

The basic task of the symmetric eigenproblem routines is to compute values of tex2html_wrap_inline12778 and, optionally, corresponding vectors z for a given matrix A.

This computation proceeds in the following stages:

  1. The real symmetric or complex Hermitian matrix A is reduced to real tridiagonal form   T. If A is real symmetric, the decomposition is tex2html_wrap_inline13866 with Q orthogonal and T symmetric tridiagonal. If A is complex Hermitian, the decomposition is tex2html_wrap_inline13874 with Q unitary and T, as before, real symmetric tridiagonal .
  2. Eigenvalues and eigenvectors of the real symmetric tridiagonal matrix T are computed. If all eigenvalues and eigenvectors are computed, this process is equivalent to factorizing T as tex2html_wrap_inline13884, where S is orthogonal and tex2html_wrap_inline12784 is diagonal. The diagonal entries of tex2html_wrap_inline12784 are the eigenvalues of T, which are also the eigenvalues of A, and the columns of S are the eigenvectors of T; the eigenvectors of A are the columns of Z=QS, so that tex2html_wrap_inline13904 (tex2html_wrap_inline13906 when A is complex Hermitian).

In the real case, the decomposition tex2html_wrap_inline13866 is computed by the routine PxSYTRD   (see table 3.8). The complex analogue of this routine is called PxHETRD.    The routine PxSYTRD (or PxHETRD) represents the matrix Q as a product of elementary reflectors, as described in section 3.4. The routine PxORMTR   (or in the complex case PxUNMTR)    is provided to multiply another matrix by Q without forming Q explicitly; this can be used to transform eigenvectors of T, computed by PxSTEIN, back to eigenvectors of A.     

The following routines compute eigenvalues  and eigenvectors  of T.

xSTEQR2
   This routine [77] is a modified version of LAPACK routine xSTEQR. It has fewer iterations than the LAPACK routine due to a modification in the look-ahead technique described in [77]. Some additional modifications allow each process to perform partial updates to matrix Q. This routine computes all eigenvalues and, optionally, eigenvectors of a symmetric tridiagonal matrix using the implicit QL or QR method.   
PxSTEBZ
   This routine uses bisection to compute some or all of the eigenvalues. Options provide for computing all the eigenvalues in a real interval or all the eigenvalues from the ith to the jth largest. It can be highly accurate but may be adjusted to run faster if lower accuracy is acceptable.
PxSTEIN
     Given accurate eigenvalues, this routine uses inverse iteration  to compute some or all of the eigenvectors.

Without any reorthogonalization, inverse iteration may produce vectors that have large dot products. To cure this, most implementations of inverse iteration such as LAPACK's xSTEIN reorthogonalize when eigenvalues differ by less than tex2html_wrap_inline13934. As a result, the eigenvectors computed by xSTEIN are almost always orthogonal, but the increase in cost can result in tex2html_wrap_inline13936 work. On some rare examples, xSTEIN may still fail to deliver accurate answers; see [43, 44]. The orthogonalization done by PxSTEIN is limited by the amount of workspace provided; whenever it performs less reorthogonalization than xSTEIN, there is a danger that the dot products may not be satisfactory.

  table1946
Table 3.8: Computational routines for the symmetric eigenproblem

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Next: Nonsymmetric Eigenproblems Up: Computational Routines Previous: Generalized RQ factorization

Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node62.html0100644000056400000620000000364606336113676017462 0ustar pfrauenfstaff Nonsymmetric Eigenproblems next up previous contents index
Next: EigenvaluesEigenvectors, and Schur Up: Computational Routines Previous: Symmetric Eigenproblems

Nonsymmetric Eigenproblems

   



Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node63.html0100644000056400000620000001646406336113676017465 0ustar pfrauenfstaff Eigenvalues, Eigenvectors, and Schur Factorization next up previous contents index
Next: Singular Value Decomposition Up: Nonsymmetric Eigenproblems Previous: Nonsymmetric Eigenproblems

Eigenvalues, Eigenvectors, and Schur Factorization

  Let A be a square n-by-n matrix. A scalar tex2html_wrap_inline12778 is called an eigenvalue  and a nonzero column vector v the corresponding right eigenvector  if tex2html_wrap_inline13982. A nonzero column vector u satisfying tex2html_wrap_inline13986 is called the left eigenvector . The first basic task of the routines described in this section is to compute, for a given matrix A, all n values of tex2html_wrap_inline12778 and, if desired, their associated right eigenvectors v and/or left eigenvectors u.

A second basic task is to compute the Schur factorization of a matrix A.   If A is complex, then its Schur factorization is tex2html_wrap_inline14002, where Z is unitary and T is upper triangular. If A is real, its Schur factorization is tex2html_wrap_inline14010, where Z is orthogonal, and T is upper quasi-triangular (1-by-1 and 2-by-2 blocks on its diagonal). The columns of Z are called the Schur vectors of A.   The eigenvalues of A appear on the diagonal of T; complex conjugate eigenvalues of a real A correspond to 2-by-2 blocks on the diagonal of T.

These two basic tasks can be performed in the following stages:

  1. A general matrix A is reduced to upper Hessenberg form  H     which is zero below the first subdiagonal. The reduction may be written tex2html_wrap_inline14044 with Q orthogonal if A is real, or tex2html_wrap_inline14050 with Q unitary if A is complex. The reduction is performed by subroutine PxGEHRD, which represents      Q in a factored form, as described in section 3.4. The routine PxORMHR   (or in the complex case PxUNMHR) is provided to   multiply another matrix by Q without forming Q explicitly.
  2. The upper Hessenberg matrix H is reduced to Schur form T,   giving the Schur factorization tex2html_wrap_inline14066 (for H real) or tex2html_wrap_inline14070 (for H complex). The matrix S (the Schur vectors of H) may optionally be computed as well. Alternatively S may be postmultiplied into the matrix Q determined in stage 1, to give the matrix Z = Q S, the Schur vectors of A. The eigenvalues  are obtained from the diagonal of T. All this is done by subroutine PxLAHQR.   

The algorithm used in PxLAHQR is similar to the LAPACK routine xLAHQR. Unlike xLAHQR, however, instead of sending one double shift through the largest unreduced submatrix, this algorithm sends multiple double shifts and spaces them apart so that there can be parallelism across several processor row/columns. Another critical difference is that this algorithm applies multiple double shifts in a block fashion, as opposed to xLAHQR which applies one double shift at a time, and xHSEQR from LAPACK which attempts to achieve a blocked code by combining the double shifts into one single large multi-shift. For complete details, please refer to [79].

See table 3.9 for a complete list of the routines.

  table2014
Table 3.9: Computational routines for the nonsymmetric eigenproblem

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Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: Generalized Symmetric Definite Eigenproblems Up: Computational Routines Previous: EigenvaluesEigenvectors, and Schur

Singular Value Decomposition

 

Let A be a general real m-by-n matrix. The singular value decomposition (SVD) of A is the factorization  tex2html_wrap_inline14119, where U and V are orthogonal and tex2html_wrap_inline14125, tex2html_wrap_inline14127, with tex2html_wrap_inline14129. If A is complex, its SVD is tex2html_wrap_inline14133, where U and V are unitary and tex2html_wrap_inline12820 is as before with real diagonal elements. The tex2html_wrap_inline12826 are called the singular values , the first r columns of V the right singular vectors , and the first r columns of U the left singular vectors .

The routines described in this section, and listed in table 3.10, are used to compute this decomposition. The computation proceeds in the following stages:

  1. The matrix A is reduced to bidiagonal  form: tex2html_wrap_inline14153 if A is real (tex2html_wrap_inline14157 if A is complex), where tex2html_wrap_inline14161 and tex2html_wrap_inline14163 are orthogonal (unitary if A is complex), and B is real and upper-bidiagonal when tex2html_wrap_inline13306 and lower bidiagonal when m < n, so that B is nonzero only on the main diagonal and either on the first superdiagonal (if tex2html_wrap_inline13306) or the first subdiagonal (if m<n).
  2. The SVD of the bidiagonal matrix B is computed: tex2html_wrap_inline14181, where tex2html_wrap_inline14183 and tex2html_wrap_inline14185 are orthogonal and tex2html_wrap_inline12820 is diagonal as described above. The singular vectors of A are then tex2html_wrap_inline14191 and tex2html_wrap_inline14193.

The reduction to bidiagonal form is performed by the subroutine PxGEBRD and the SVD of B is computed using the LAPACK routine xBDSQR.       

The routine PxGEBRD represents tex2html_wrap_inline14161 and tex2html_wrap_inline14163 in factored form as products of elementary reflectors,   as described in section 3.4. If A is real, the matrices tex2html_wrap_inline14161 and tex2html_wrap_inline14163 may be multiplied by other matrices without forming tex2html_wrap_inline14161 and tex2html_wrap_inline14163 using routine PxORMBR  . If A is complex, one instead uses PxUNMBR  .

If tex2html_wrap_inline14213, it may be more efficient to first perform a QR factorization of A, using the routine PxGEQRF    , and then to compute the SVD of the n-by-n matrix R, since if A = QR and tex2html_wrap_inline14227, then the SVD of A is given by tex2html_wrap_inline14231. Similarly, if tex2html_wrap_inline14233, it may be more efficient to first perform an LQ factorization of A, using xGELQF. These preliminary QR and LQ      factorizations are performed by the driver PxGESVD.     

The SVD may be used to find a minimum norm solution  to a (possibly) rank-deficient linear least squares   problem (3.1). The effective rank, k, of A can be determined as the number of singular values which exceed a suitable threshold. Let tex2html_wrap_inline14247 be the leading k-by-k submatrix of tex2html_wrap_inline12820, and tex2html_wrap_inline14255 be the matrix consisting of the first k columns of V. Then the solution is given by
displaymath14109
where tex2html_wrap_inline13502 consists of the first k elements of tex2html_wrap_inline14265. tex2html_wrap_inline14267 can be computed by using PxORMBR.   

  table2093
Table 3.10: Computational routines for the singular value decomposition

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Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node65.html0100644000056400000620000001250606336113700017444 0ustar pfrauenfstaff Generalized Symmetric Definite Eigenproblems next up previous contents index
Next: Orthogonal or Unitary Matrices Up: Computational Routines Previous: Singular Value Decomposition

Generalized Symmetric Definite Eigenproblems

  

This section is concerned with the solution of the generalized eigenvalue problems tex2html_wrap_inline12875, tex2html_wrap_inline12877, and tex2html_wrap_inline12879, where A and B are real symmetric or complex Hermitian and B is positive definite. Each of these problems can be reduced to a standard symmetric eigenvalue problem, using a Cholesky factorization of B as either tex2html_wrap_inline14317 or tex2html_wrap_inline14319 (tex2html_wrap_inline14321 or tex2html_wrap_inline14323 in the Hermitian case).

With tex2html_wrap_inline14317, we have
displaymath14301
Hence the eigenvalues of tex2html_wrap_inline12875 are those of tex2html_wrap_inline14329, where C is the symmetric matrix tex2html_wrap_inline14333 and tex2html_wrap_inline14335. In the complex case C is Hermitian with tex2html_wrap_inline14339 and tex2html_wrap_inline14341.

Table 3.11 summarizes how each of the three types of problem may be reduced to standard form   tex2html_wrap_inline14329, and how the eigenvectors z of the original problem may be recovered from the eigenvectors y of the reduced problem. The table applies to real problems; for complex problems, transposed matrices must be replaced by conjugate transposes.

  table2129
Table 3.11: Reduction of generalized symmetric definite eigenproblems to standard problems

Given A and a Cholesky factorization of B, the routines PxyyGST overwrite A      with the matrix C of the corresponding standard problem tex2html_wrap_inline14329 (see table 3.12). This may then be solved by using the routines described in subsection 3.3.4. No special routines are needed to recover the eigenvectors z of the generalized problem from the eigenvectors y of the standard problem, because these computations are simple applications of Level 2 or Level 3 BLAS.

  table2167
Table 3.12: Computational routines for the generalized symmetric definite eigenproblem


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Orthogonal or Unitary Matrices

 

A real orthogonal or complex unitary matrix (usually denoted Q) is often represented  in ScaLAPACK as a product of elementary reflectors -- also referred to as     elementary Householder matrices (usually denoted tex2html_wrap_inline14421). For example,
displaymath14415
Most users need not be aware of the details, because ScaLAPACK routines are provided to work with this representation:

  • routines whose names begin PSORG- (real) or PCUNG- (complex) can generate all or part of Q explicitly;
  • routines whose name begin PSORM- (real) or PCUNM- (complex) can multiply a given matrix by Q or tex2html_wrap_inline13302 without forming Q explicitly.

The following details may occasionally be useful.

An elementary reflector (or elementary Householder matrix) H of order n is a unitary matrix  of the form    
 equation2204
where tex2html_wrap_inline14435 is a scalar and v is an n-vector, with tex2html_wrap_inline14441); v is often referred to as the Householder vector.  Often v has several leading or trailing zero elements, but for the purpose of this discussion assume that H has no such special structure.

Some redundancy in the representation (3.4) exists, which can be removed in various ways. Like LAPACK, the representation used in ScaLAPACK (which differs from that used in LINPACK or EISPACK) sets tex2html_wrap_inline14449; hence tex2html_wrap_inline14451 need not be stored. In real arithmetic, tex2html_wrap_inline14453, except that tex2html_wrap_inline14455 implies H = I.

In complex arithmetic , tex2html_wrap_inline14435 may be complex and satisfies tex2html_wrap_inline14461 and tex2html_wrap_inline14463. Thus a complex H is not Hermitian (as it is in other representations), but it is unitary, which is the important property. The advantage of allowing tex2html_wrap_inline14435 to be complex is that, given an arbitrary complex vector x, H can be computed so that
displaymath14416
with real tex2html_wrap_inline14473. This is useful, for example, when reducing a complex Hermitian matrix to real symmetric tridiagonal form  or a complex rectangular matrix to real bidiagonal form .

For further details, see Lehoucq [94].

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Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: Data Distributions and Software Up: Contents of ScaLAPACK Previous: Orthogonal or Unitary Matrices

Algorithmic Differences between LAPACK and ScaLAPACK

 

The following ScaLAPACK routines use different algorithms from their LAPACK counterparts. Refer to the relevant LAPACK working notes or the leading comments of the source code for details.

  • PxDBSV, PxDBTRF, PxDBTRS: No LAPACK equivalent; refer to [32].
  • PxGBSV, PxGBTRF, PxGBTRS: Refer to [32].
  • PxLAHQR: Refer to [79].
  • PxPBSV, PxPBTRF, PxPBTRS: Refer to [32].
  • PxPTSV, PxPTTRF, PxPTTRS: Refer to [32].
  • PxSYEV
  • PxSYEVX/PxHEEVX: Refer to [40].
  • PxSYGVX/PxHEGVX.

Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: Basics Up: Guide Previous: Algorithmic Differences between LAPACK

Data Distributions and Software Conventions

 

The ScaLAPACK software library provides routines that operate on three types of matrices: in-core dense matrices, in-core narrow band matrices and out-of-core dense matrices. On entry, these routines assume that the data has been distributed on the processors according to a specific data decomposition scheme. Conventional arrays are used to store locally the data when it resides in the processors' memory. The data layout information as well as the local storage scheme for these different matrix operands is conveyed to the routines via a simple array of integers called an array descriptor. The first entry of this array identifies the type of the descriptor, i.e., the data distribution scheme it describes. This chapter first presents the fundamental concepts of process grids, communication contexts and array descriptors. Then, for each of the three possible matrix operand mentioned above, the data distribution scheme and the corresponding descriptor array used by ScaLAPACK are discussed in detail. Finally, the software conventions common to all ScaLAPACK routines are presented.



Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Basics

 

ScaLAPACK requires that all global data (vectors or matrices) be distributed across the processes prior to invoking the ScaLAPACK routines. The storage schemes of global data structures in ScaLAPACK are conceptually the same as for LAPACK.

Global data is mapped to the local memories of processes assuming specific data distributions. The local data on each process is referred to as the local array.

The layout of an application's data within the hierarchical memory   of a concurrent computer is critical in determining the performance and scalability  of the parallel code. On shared-memory concurrent computers (or multiprocessors) LAPACK seeks to make efficient use of the hierarchical memory by maximizing data reuse (e.g., on a cache-based computer LAPACK avoids having to reload the cache too frequently). Specifically, LAPACK casts linear algebra computations in terms of block-oriented, matrix-matrix operations through the use of the Level 3 BLAS whenever possible. This approach generally results in maximizing the ratio of floating-point operations to memory references and enables data reuse as much as possible while it is stored in the highest levels of the memory hierarchy (e.g., vector registers or high-speed cache).

An analogous approach has been followed in the design of ScaLAPACK for distributed-memory  machines. By using block-partitioned algorithms  we seek to reduce the frequency with which data must be transferred between processes, thereby reducing the fixed startup cost (or latency)  incurred each time a message is communicated.

The ScaLAPACK routines for solving dense linear systems and eigenvalue problems assume that all global data has been distributed to the processes with a one-dimensional or two-dimensional block-cyclic data distribution. This distribution is a natural expression of the block-partitioned algorithms present in ScaLAPACK. The ScaLAPACK routines for solving band linear systems and tridiagonal systems assume that all global data has been distributed to the processes with a one-dimensional block data distribution. Each of these distributions is supported in the High Performance Fortran standard [91]. Explanations for each distribution will be presented and accompanied by the appropriate HPF directives.

Our implementation of ScaLAPACK emphasizes the mathematical view of a matrix over its storage. In fact, it is even possible to reuse our interface for a different block data distribution that would not fit in the block-cyclic scheme.


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Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Next: Guide Up: List of Tables Previous: Suggestions for Reading

List of Notation

   

 
1D 		 One-dimensional (as in 1D data distribution)


tex2html_wrap_inline12064 		 Bandwidth (or throughput) for the network


2D 		 Two-dimensional (as in 2D data distribution)


tex2html_wrap_inline12066 		 Total number of floating-point operations 


tex2html_wrap_inline12070 		 Total number of messages 


CSRC_ Entry in DESC_ indicating the process column over which


        		 the first column of the array is distributed 


CTXT_ Entry in DESC_ indicating the BLACS context associated with the


        		 global array  


tex2html_wrap_inline12074 		 Total number of data items communicated 


DESC_ Array descriptor for a global array 


DESCA  		 Array descriptor for global array A


DESCB  		 Array descriptor for global array B


DLEN_ Length of the descriptor DESC_ 


DTYPE_ First entry of the descriptor DESC_, identifying


        		 the descriptor type  


E()   		 Estimated parallel efficiency 


tex2html_wrap_inline12088 		 Time per floating point operation in matrix-matrix multiply 


Gflop/s 		 Gigaflops (tex2html_wrap_inline12092 floating point operations) per second


IA     		 Global row index in the global array A indicating the


       		 first row of tex2html_wrap_inline12098 


ICTXT 		 BLACS context associated with a process grid 


INFO 		 Output integer argument of driver and computational routines


     		 indicating the success or failure of the routine


JA     		 Global column index in the global array A indicating the


       		 first column of tex2html_wrap_inline12098 


tex2html_wrap_inline12108 		 Least common multiple of (tex2html_wrap_inline12110) 


LLD_ Entry in DESC_ indicating the local leading dimension of the


      		 local array 


tex2html_wrap_inline12116 		 Local leading dimension of the local array A


tex2html_wrap_inline12120 		 Local leading dimension of the local array B


tex2html_wrap_inline12124(K_) 		 Number of columns that a process receives if tex2html_wrap_inline12126 columns


               		 of a matrix are distributed over c columns of its process row. 


tex2html_wrap_inline12132(K_) 		 Number of rows that a process would receive if tex2html_wrap_inline12126 rows


               		 of a matrix are distributed over r rows of its process column. 


M     		 Global number of rows of the distributed submatrix tex2html_wrap_inline12098


M_ Entry of DESC_ indicating the number of rows in the global


      		 array 


MB    		 Global row block size for partitioning the global matrix


MB_ Entry in DESC_ indicating the block size used to distribute


      		 the rows of the global array 


MB/s 		 Megabyte per second


Mflop/s 		 Megaflops (tex2html_wrap_inline12146 floating point operations) per second


MYCOL  		 The calling process's column coordinate in the process grid. 


MYROW  		 The calling process's row coordinate in the process grid 


N     		 Global number of columns of the distributed submatrix tex2html_wrap_inline12098


N_ Entry of DESC_ indicating the number of columns in the


      		 global array 


NB    		 Global column block size for partitioning the global matrix


NB_ Entry in DESC_ indicating the block size used to distribute


      		 the columns of the global array 


NBRHS 		 Global column block size for the solution matrix 


NPCOL  		 Number of process columns in the process grid (equivalent to tex2html_wrap_inline12162) 


NPROCS 		 Total number of processes in the process grid (equivalent to P) 


NPROW  		 Number of process rows in the process grid (equivalent to tex2html_wrap_inline12172) 


NRHS  		 Global number of columns in the global solution matrix B 


NUMROC 		 TOOLS routine used to calculate the number of rows or columns


       		 in a local array 


P     		 Total number of processes in the process grid, i.e., tex2html_wrap_inline12182 


tex2html_wrap_inline12162 		 Number of process columns in the process grid 


tex2html_wrap_inline12172 		 Number of process rows in the process grid 


RSRC_ Entry in DESC_ indicating the process row over which the


        		 first row of the array is distributed 


tex2html_wrap_inline12098 		 Distributed submatrix A(IA:IA+M-1,JA:JA+N-1)


T()   		 Estimated parallel execution time 


tex2html_wrap_inline12198() 		 Estimated serial execution time 


tex2html_wrap_inline12202 		 Time per floating-point operation (typically tex2html_wrap_inline12204) 


tex2html_wrap_inline12208 		 Time per message (latency) 


tex2html_wrap_inline12212 		 Time required to solve a problem of size N on P processors 


tex2html_wrap_inline12220 		 Time required to solve a problem of size N/2 on P processors 


tex2html_wrap_inline12228 		 Time per data item communicated 



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Susan Blackford
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Next: Contexts Up: Basics Previous: Basics

Process Grid

   

The P  processes of an abstract parallel computer are often represented as a one-dimensional linear array of processes labeled tex2html_wrap_inline14493. For reasons described below, it is often more convenient to map this one-dimensional array of processes into a two-dimensional rectangular grid, or process grid. (A process grid is also referred to as a process mesh.)  This grid will have tex2html_wrap_inline12172 process rows and tex2html_wrap_inline12162 process columns, where tex2html_wrap_inline14499. A process can now be referenced by its row and column coordinates, (tex2html_wrap_inline14501), within the grid, where tex2html_wrap_inline14503, and tex2html_wrap_inline14505). An example of such a mapping is shown in figure 4.1, where tex2html_wrap_inline12422 and tex2html_wrap_inline14509.

  figure2256
Figure 4.1: Eight processes mapped to a tex2html_wrap_inline14511 process grid

In figure 4.1, the processes are mapped to the process grid by using row-major order ; in other words, the numbering of the processes increases sequentially across each row. Similarly, the processes can be mapped in a column-major order  whereby the numbering of the processes proceeds down each column of the process grid. The BLACS routine BLACS_GRIDINIT  performs this task of mapping the processes to the process grid. By default, BLACS_GRIDINIT assumes a row-major ordering, although a column-major ordering can also be specified. The companion routine BLACS_GRIDMAP is a more general form of BLACS_GRIDINIT and allows the user to define the mapping of the processes . Refer to the Appendix D.3 for further details.

All ScaLAPACK routines, with the exception of the band linear system routines, allow the processes to be viewed as a one-dimensional or two-dimensional process grid. The band routines support only one-dimensional process grids.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Contexts

 

In ScaLAPACK, and thus the BLACS, each process grid is enclosed in a context  . Similarly, a context is associated with every global matrix in ScaLAPACK. The use of a context provides the ability to have separate ``universes'' of message passing. This means that a process grid can safely communicate even if other (possibly overlapping) process grids are also communicating. Thus, a context is a powerful mechanism for avoiding unintentional nondeterminism in message passing and provides support for the design of safe, modular software libraries. In MPI  , this concept is referred to as a communicator.

A context partitions the communication space. A message sent from one context cannot be received in another context. The use of separate communication contexts by distinct libraries (or distinct library routine invocations) insulates communication internal to a specific library routine from external communication that may be going on within the user's program.

In most respects, we can use the terms process grid and context interchangeably. For example, we may say we perform an operation ``in context X'' or ``in process grid X''. The slight difference here is that the user may define two identical process grids (say, two tex2html_wrap_inline14534 process grids, both of which use processes 0, 1, and 2), but each will be enclosed in its own context, so that they are distinct in operation, even though they are indistinguishable from a process grid standpoint.

Another example of the use of context might be to define a normal two-dimensional process grid within which most computation takes place. However, in certain portions of the code it may be more convenient to access the processes as a one-dimensional process grid, whereas at other times we may wish, for instance, to share information among nearest neighbors. In such cases, we will want each process to have access to three contexts: the two-dimensional process grid, the one-dimensional process grid, and a small process grid that contains the process and its nearest neighbors.

Therefore, we see that context allows us to

  • create arbitrary groups of processes,
  • create an indeterminate number of overlapping and/or disjoint process grids, and
  • isolate the process grid so that they do not interfere with each other.

The BLACS has two process grid creation  routines, BLACS_GRIDINIT and BLACS_GRIDMAP,    that create a process grid and its enclosing context. These routines return context handles  , which are simple integers, assigned by the BLACS to identify the context. Subsequent BLACS routines will be passed these handles, which allow the BLACS to determine from which context/process grid a routine is being called. The user should never alter or change these handles; they are opaque data objects that are only meaningful for the BLACS routines.

A defined context consumes resources. It is therefore advisable to release contexts when they are no longer needed. This release is done via the routine BLACS_GRIDEXIT . When the entire BLACS system is shut down (via a call to BLACS_EXIT ), all outstanding contexts are automatically freed. Further details about these routines can be found in Appendix D.3.

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Susan Blackford
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Scoped Operations

   

An operation that involves more than just a sender and a receiver is called a scoped operation . All processes that participate in a scoped operation are said to be within the operation's scope.

On a system using a linear array of processes, the only natural scope is all processes. Using a two-dimensional rectangular process grid, we have three natural scopes , as shown in table 4.1. Refer to [24] for further details.

 table2322
Table 4.1: Scopes provided by a two-dimensional process grid 

These groupings of processes are of particular interest to the linear algebra programmer, since distributed data decompositions of a two-dimensional array (a linear algebra matrix) tend to follow this process mapping. For instance, all of a distributed matrix row can be found on a process row.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Array Descriptors

   

An array descriptor    is associated with each global array. This array stores the information required to establish the mapping between each global array entry and its corresponding process and memory location. The notations x_ used in the entries of the array descriptor denote the attributes of a global array. For example, M_ denotes the number of rows and M_A specifically denotes the number of rows in global matrix A. These descriptors assume different storage for the global data. The length of the array descriptor  is specified by DLEN_ and varies according to the descriptor type DTYPE_.

Array descriptors are provided for

  • dense matrices,
  • band and tridiagonal matrices, and
  • out-of-core matrices
and are differentiated by the DTYPE_ entry in the descriptor. At the present time the following values of DESC_(DTYPE_) are valid.  

 table2340
Table 4.2: Valid values of DESC_(DTYPE_)


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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In-core Dense Matrices

 

The choice of an appropriate data distribution heavily depends on the characteristics or flow of the computation in the algorithm. For dense matrix computations, ScaLAPACK assumes the data to be distributed according to the two-dimensional block-cyclic data layout scheme. This section presents this distribution and demonstrates how the ScaLAPACK software encodes this essential information as well as the related software conventions.

Dense matrix computations feature a large amount of parallelism, so that a wide variety of distribution schemes have the potential for achieving high performance. The block-cyclic data layout has been selected for the dense algorithms implemented in ScaLAPACK principally because of its scalability [51], load balance, and communication [76] properties. The block-partitioned computation proceeds in consecutive order just like a conventional serial algorithm. This essential property of the block cyclic data layout explains why the ScaLAPACK design has been able to reuse the numerical and software expertise of the sequential LAPACK library.



Susan Blackford
Tue May 13 09:21:01 EDT 1997
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The Two-dimensional Block-Cyclic Distribution

    

In this section, we consider the data layout of dense matrices on distributed-memory machines, with the goal of making dense matrix computations as efficient as possible. We shall discuss a sequence of data layouts, starting with the most simple, obvious, and inefficient one and working up to the complicated but efficient ScaLAPACK ultimately uses. Even though our justification is based on Gaussian elimination, analysis of many other algorithms has led to the same set of layouts. As a result, these layouts have been standardized as part of the High Performance Fortran standard [91], with corresponding data declarations as part of that language.

The two main issues in choosing a data layout for dense matrix computations are

  • load balance, or splitting the work reasonably evenly among the processors throughout the algorithm, and
  • use of the Level 3 BLAS during computations on a single processor, to account for the memory hierarchy on each processor.

It will help to remember the pictorial representation of Gaussian elimination below. As the algorithm proceeds, it works on successively smaller square southeast corners of the matrix. In addition, there is extra Level 2 BLAS work to factorize the submatrix tex2html_wrap_inline14557.

  figure2362
Figure 4.2: Gaussian elimination using Level 3 BLAS

For convenience we will number the processes from 0 to P-1, and matrix columns (or rows) from 1 to N. The following two figures shows a sequence of data layouts we will consider. In all cases, each submatrix is labeled with the number of the process (from 0 to 3) that contains it. Process 0 owns the shaded submatrices.

Consider the layout illustrated on the left of figure 4.3, the one-dimensional block column distribution. This distribution

  figure2369
Figure 4.3: The one-dimensional block and cyclic column distributions

assigns a block of contiguous columns of a matrix to successive processes. Each process receives only one block of columns of the matrix. Column k is stored on process tex2html_wrap_inline14569 where tex2html_wrap_inline14571 is the maximum number of columns stored per process. In the figure N=16 and P=4. This layout does not permit good load balancing for the above Gaussian elimination algorithm because as soon as the first tc columns are complete, process 0 is idle for the rest of the computation. The transpose of this layout, the one-dimensional block row distribution, has a similar shortfall for dense computations.

The second layout illustrated on the right of figure 4.3, the one-dimensional cyclic column distribution, addressed this problem by assigning column k to process (k-1) mod P. In the figure, N=16 and P=4. With this layout, each process owns approximately tex2html_wrap_inline14591 of the square southeast corner of the matrix, so the load balance is good. However, since single columns (rather than blocks) are stored, we cannot use the Level 2 BLAS to factorize tex2html_wrap_inline14557 and may not be able to use the Level 3 BLAS to update tex2html_wrap_inline14595. The transpose of this layout, the one-dimensional cyclic row distribution, has a similar shortfall.

The third layout shown on the left of figure 4.4, the one-dimensional block-cyclic column distribution, is a compromise between the distribution schemes shown in figure 4.3. We choose a block size NB, divide the columns into groups of size NB, and distribute these groups in a cyclic manner. This means column k is stored in process tex2html_wrap_inline14601. In fact, this layout includes the first two as the special cases, tex2html_wrap_inline14603 and NB=1, respectively. In the figure N=16, P=4 and NB=2. For NB larger than 1, this has a slightly worse balance than the one-dimensional cyclic column distribution, but can use the Level 2 BLAS and Level 3 BLAS for the local computations. For NB less than tc, it has a better load balance than the one-dimensional block column distribution, but can call the BLAS only on smaller subproblems. Hence, it takes less advantage of the local memory hierarchy. Moreover, this layout has the disadvantage that the factorization of tex2html_wrap_inline14557 will take place on one process (in the natural situation where column blocks in the layout correspond to column blocks in Gaussian elimination), thereby representing a serial bottleneck.

  figure2387
Figure 4.4: The one-dimensional block-cyclic column- and the two-dimensional block-cyclic distributions

This serial bottleneck is eased by the fourth layout shown on the right of figure 4.4, the two-dimensional block cyclic distribution. Here, we think of our P processes arranged in a tex2html_wrap_inline12172 tex2html_wrap_inline12420 tex2html_wrap_inline12162 rectangular array of processes, indexed in a two-dimensional fashion by tex2html_wrap_inline14629, with tex2html_wrap_inline14631 and tex2html_wrap_inline14633. All the processes tex2html_wrap_inline14629 with a fixed tex2html_wrap_inline14637 are referred to as process column tex2html_wrap_inline14637. All the processes tex2html_wrap_inline14629 with a fixed tex2html_wrap_inline14643 are referred to as process row tex2html_wrap_inline14643. Thus, this layout includes all the previous layouts, and their transposes, as special cases. In the figure, N=16, P=4, tex2html_wrap_inline14651, and MB=NB=2. This layout permits tex2html_wrap_inline12162-fold parallelism in any column, and calls to the Level 2 BLAS and Level 3 BLAS on local subarrays. Finally, this layout also features good scalability properties as shown in [61].

The two-dimensional block cyclic distribution scheme is the data layout that is used in the ScaLAPACK library for dense matrix computations.

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Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Local Storage Scheme and Block-Cyclic Mapping

 

The block-cyclic distribution scheme is a mapping of a set of blocks onto the processes. The previous section informally described this mapping as well as some of its properties. To be complete, we must now explain how the blocks that are mapped to the same process are arranged and stored in the local process memory. In other words, we shall describe the precise mapping that associates to a matrix entry identified by its global indexes the coordinates of the process that owns it and its local position within that process's memory.

Suppose we have an array of length N to be stored on P processes. By convention, the array entries are numbered 1 through N and the processes are numbered 0 through P-1. First, the array is divided into contiguous blocks of size NB. When NB does not divide N evenly, the last block of array elements will only contain tex2html_wrap_inline14699 entries instead of NB. By convention, these blocks are numbered starting from zero and dealt out to the processes like a deck of cards. In other words, if we assume that the process 0 receives the first block, the tex2html_wrap_inline14705 block is assigned to the process of coordinate tex2html_wrap_inline14707. The blocks assigned to the same process are stored contiguously in memory. The mapping of an array entry globally indexed by I is defined by the following analytical equation:
displaymath14669
where I is a global index in the array, l is the local block coordinate into which this entry resides, p is the coordinate of the process owning that block, and finally x is the coordinate within that block where the global array entry of index I is to be found. It is then fairly easy to establish the analytical relationship between these variables. One obtains:
 equation2409
These equations allow to determine the local information, i.e. the local index tex2html_wrap_inline14721 as well as the process coordinate p corresponding to a global entry identified by its global index I and conversely. Table 4.3 illustrates this mapping for the block layout when P=2 and N=16, i.e., NB=8. At most one block is assigned to each process.

  table2416
Table 4.3: One-dimensional block mapping example for P=2 and N=16

This example of the one-dimensional block distribution mapping can be expressed in HPF  by using the following statements: 

      REAL :: X( N )
!HPF$ PROCESSORS PROC( P )
!HPF$ DISTRIBUTE X( BLOCK( NB ) ) ONTO PROC

Table 4.4 illustrates Equation 4.1 for the cyclic layout, i.e., NB=1 when P=2 and N=16.

  table2428
Table 4.4: One-dimensional cyclic mapping example for P=2 and N=16

This example of the one-dimensional cyclic distribution mapping can be expressed in HPF  by using the following statements: 

      REAL :: X( N )
!HPF$ PROCESSORS PROC( P )
!HPF$ DISTRIBUTE X( CYCLIC ) ONTO PROC

Table 4.5 illustrates Equation 4.1 for the block-cyclic layout when P=2, NB=3 and N=16.

  table2440
Table 4.5: One-dimensional block-cyclic mapping example for P=2, NB=3 and N=16

This example of the one-dimensional cyclic distribution mapping can be expressed in HPF  by using the following statements: 

      REAL :: X( N )
!HPF$ PROCESSORS PROC( P )
!HPF$ DISTRIBUTE X( CYCLIC( NB ) ) ONTO PROC

There is in fact no real reason to always deal out the blocks starting with the process 0. In fact, it is sometimes useful to start the data distribution with the process of arbitrary coordinate SRC, in which case Equation 4.1 becomes:
 equation2451

Table 4.6 illustrates Equation 4.2 for the block-cyclic layout when tex2html_wrap_inline14791, tex2html_wrap_inline14793, tex2html_wrap_inline14795 and tex2html_wrap_inline14797.

  table2461
Table 4.6: One-dimensional block-cyclic mapping example for P=2, SRC=1, NB=3 and N=16

This example of the one-dimensional block-cyclic distribution mapping can be expressed in HPF  by using the following statements: 

      REAL :: X( N )
!HPF$ PROCESSORS PROC( P )
!HPF$ TEMPLATE T( N + P*NB )
!HPF$ DISTRIBUTE T( CYCLIC( NB ) ) ONTO PROC
!HPF$ ALIGN X( I ) WITH T( SRC*NB + I )

In the two-dimensional case, assuming the matrix is partitioned in tex2html_wrap_inline14817 blocks and that the first block is given to the process of coordinates (RSRC, CSRC), the analytical formula given above for the one-dimensional case are simply reused independently in each dimension of the tex2html_wrap_inline12182 process grid. For example, the matrix entry (I,J) is thus to be found in the process of coordinates tex2html_wrap_inline14629 within the local (l,m) block at the position (x,y) given by:
displaymath14670

These formula specify how an tex2html_wrap_inline14831 by tex2html_wrap_inline14833  matrix A is mapped and stored on the process grid. It is first decomposed into tex2html_wrap_inline14837 by tex2html_wrap_inline14839  blocks starting at its upper left corner. These blocks are then uniformly distributed across the process grid in a cyclic manner.

Every process owns a collection of blocks, which are contiguously stored by column in a two-dimensional ``column major'' array.

This local storage convention allows the ScaLAPACK software to use efficiently the local memory hierarchy by calling the BLAS on subarrays that may be larger than a single tex2html_wrap_inline14837 by tex2html_wrap_inline14839 block. We present in figure 4.5  the mapping of a 5tex2html_wrap_inline148455 matrix partitioned into 2tex2html_wrap_inline148452 blocks mapped onto a 2tex2html_wrap_inline148452 process grid (i.e., tex2html_wrap_inline14851, tex2html_wrap_inline14651, and tex2html_wrap_inline14855). The local entries of every matrix column are contiguously stored in the processes' memories.

 figure2490
Figure 4.6: A tex2html_wrap_inline14857 matrix decomposed into tex2html_wrap_inline14859   blocks mapped onto a tex2html_wrap_inline14859 process grid

In figure 4.5, the process of coordinates (0,0) owns four blocks. The matrix entries of the global columns 1, 2 and 5 are contiguously stored in that process's memory. Finally, these columns are themselves continuously stored forming a conventional two-dimensional local array. In that local array A, the entry A(2,3) contains the value of the global matrix entry tex2html_wrap_inline14869. This example would be expressed in HPF  as: 

      REAL :: A( 5, 5 )
!HPF$ PROCESSORS PROC( 2, 2 )
!HPF$ DISTRIBUTE A( CYCLIC( 2 ), CYCLIC( 2 ) ) ONTO PROC

Determining the number of     rows or columns of a global dense matrix that a specific process receives is an essential task for the user. ScaLAPACK provides a tool routine, NUMROC, to perform this function. The notation LOCtex2html_wrap_inline12112() and LOCtex2html_wrap_inline12114() is used to reflect these local quantities throughout the leading comments of the source code and is reflected in the sample argument description in section 4.3.5. The values of LOCtex2html_wrap_inline12112()  and LOCtex2html_wrap_inline12114()  computed by NUMROC are precise calculations.

However, if users want a general idea of the size of a local array, they can perform the following ``back of the envelope'' calculation to receive an upper bound on the quantity.

An upper bound on the value of LOCtex2html_wrap_inline12112() can be calculated as:
displaymath14671
or equivalently as
displaymath14672

Similarly, an upper bound on the value of LOCtex2html_wrap_inline12114() can be calculated as
displaymath14673
or equivalently as
displaymath14674

Note that this calculation can yield a gross overestimate of the amount of space actually required.

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Susan Blackford
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Array Descriptor for In-core Dense Matrices

 

The array descriptor DESC_, whose type is defined as DESC_(DTYPE_)=1 , is an integer array of length 9. It is used for the ScaLAPACK routines solving dense linear systems and eigenvalue problems. All global vector and matrix operands are assumed to be distributed on the process grid according to the one- or two-dimensional block cyclic data distribution scheme. Refer to section 4.3.1 for further details on block cyclic data distribution.

A general M_ by N_ distributed matrix is defined by its dimensions, the size of the elementary MB_ by NB_ block used for its decomposition, the coordinates of the process having in its local memory the first matrix entry (RSRC_,CSRC_), and the BLACS context (CTXT_) in which this matrix is defined. Finally, a local leading dimension LLD_ is associated with the local memory address pointing to the data structure used for the local storage of this distributed matrix.

Let us assume, for example, that we have an array descriptor DESCA for a dense global matrix A. As previously mentioned, the notations x_ used in the entries of the array descriptor denote the attributes of a global array. For readability of the code, we have associated symbolic names for the descriptor entries. For example, M_ denotes the number of rows and M_A specifically denotes the number of rows in global matrix A.

 table2535
Table 4.7: Content of the array descriptor for in-core dense matrices

For a detailed description of LOCtex2html_wrap_inline12112() notation, please refer to section 4.3.2.


Susan Blackford
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Example

 

As mentioned in section 4.3.1, ScaLAPACK assumes a one-dimensional or two-dimensional block-cyclic distribution for the dense matrix computational routines. The block-cyclic distribution is a generalization of the block and cyclic distributions. In one dimension, blocks of rows of size MB or blocks of columns of size NB are cyclically distributed over the processes. In two dimensions, blocks of size tex2html_wrap_inline14817 are distributed cyclically over the processes.   Example programs can be found in section 2.3 and Appendix C.1.

According to the two-dimensional block cyclic data distribution, scheme an M_ by N_  dense matrix is first decomposed into MB_ by NB_  blocks starting at its upper left corner. These blocks are then uniformly distributed in each dimension of the process grid. Thus, every process owns a collection of blocks, which are locally and contiguously stored in a two-dimensional ``column major'' array. The partitioning   of a tex2html_wrap_inline12402 matrix into tex2html_wrap_inline14859 blocks and the mapping of these blocks onto a tex2html_wrap_inline14922 process grid are shown in figure 4.6. The local entries of every matrix column are contiguously stored in the processes' memories.

 figure2562
Figure: 4.6 A tex2html_wrap_inline12402 matrix decomposed into tex2html_wrap_inline14859   blocks mapped onto a tex2html_wrap_inline14922 process grid

The number of rows of a matrix     and the number of columns of a matrix that a specific process owns, denoted LOCtex2html_wrap_inline12112 and LOCtex2html_wrap_inline12114 respectively, may differ from process to process in the process grid. Likewise, there is a local leading dimension LLD_  for each process in the process grid. This value may be different on each process in the process grid. For example, we can see on the right of figure 4.6 that the local array stored in process row 0 must have a local leading dimension LLD_ greater than or equal to 5, and greater than or equal to 4 in the process row 1.

  table2576
Table 4.8: Sizes of the local arrays

Table 4.8 gives the values of the local array sizes associated with figure 4.6.

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Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Submatrix Argument Descriptions

   

As previously mentioned, the ScaLAPACK routines that solve dense linear systems and eigenvalue problems assume that all global arrays are distributed in a one- or two-dimensional block cyclic fashion.           After a global vector or matrix has been block-cyclicly distributed over a process grid, the user may choose to perform an operation on a portion of the global matrix. This subset of the global matrix is referred to as a ``submatrix'' and is referenced through the use of six arguments in the calling sequence: the number of rows of the submatrix M, the number of columns of the submatrix N, the local array A containing the global array, the row index IA, the column index JA and the array descriptor of the global array DESCA. This argument convention allows for a global view of the matrix operands and the global addressing of distributed matrices as illustrated in figure 4.7. This scheme allows the complete specification of the submatrix A(IA:IA+M-1,JA:JA+N-1) on which to be operated.

  figure2606
Figure 4.7: Global view of the matrix operands

The description of a global dense subarray consists of (M, N, A, IA, JA, DESCA)

  • the number of rows and columns M and N of the global subarray,
  • a pointer to the local array containing the entire global array (A, for example),
  • the row and column indices, (IA, JA), in the global array, and
  • the array descriptor, DESCA, for the global array.

The names of the row and column indices for the global array have the form I<array_name>  and J<array_name> , respectively. The array descriptor has a name of the form DESC<array_name> .    The length of the array descriptor is specified by DLEN_ and varies according to the descriptor type DTYPE_.

Included in the leading comments of each subroutine (immediately preceding the Argument section), is a brief note describing the array descriptor    and some commonly used expressions in calculating workspace.

The style of the argument  descriptions for dense matrices is illustrated by the following example. As previously mentioned, the notations x_ used in the entries of the array descriptor denote the attributes of a global array. For readability of the code, we have associated symbolic names for the descriptor entries. For example, M_ denotes the number of rows and M_A specifically denotes the number of rows in global matrix A. Complete details can be found in section 4.3.3.

    M
    (global input) INTEGER
    The number of rows of the matrix A(IA:IA+M-1,JA:JA+N-1) on which to be operated. M tex2html_wrap_inline14966 0 and IA+M-1 tex2html_wrap_inline14970 M_A.
    N
    (global input) INTEGER
    The number of columns of the matrix A(IA:IA+M-1,JA:JA+N-1) on which to be operated. N tex2html_wrap_inline14966 0 and JA+N-1 tex2html_wrap_inline14970 N_A.
    NRHS
    (global input) INTEGER
    The number of right hand side vectors, i.e. the number of columns of the matrix B(IB:IB+N-1,JB:JB+NRHS-1). NRHS tex2html_wrap_inline14966 0.
    A
    (local input/local output) REAL pointer into the local memory to an array of local dimension (LLD_A, LOCtex2html_wrap_inline12114(JA+N-1))
    IA
    (global input) INTEGER
    The row index in the global array A indicating the first row of A(IA:IA+M-1,JA:JA+N-1).
    JA
    (global input) INTEGER
    The column index in the global array A indicating the first column of A(IA:IA+M-1,JA:JA+N-1).
    DESCA
    (global and local input) INTEGER array of dimension DLEN_
    The array descriptor for the global matrix A.
    B
    (local input/local output) REAL pointer into the local memory to an array of local dimension (LLD_B, LOCtex2html_wrap_inline12114(JB+NRHS-1)).
    IB
    (global input) INTEGER
    The row index in the global array B indicating the first row of B(IB:IB+N-1,JB:JB+NRHS-1).
    JB
    (global input) INTEGER
    The column index in the global array B indicating the first column of B(IB:IB+N-1,JB:JB+NRHS-1).
    DESCB
    (global and local input) INTEGER array of dimension DLEN_
    The array descriptor for the global matrix B.

The description of each argument gives

  • A classification of the argument as (local input), (global and local input), (local input/local output), (global input), (local output), (global output), (global input/global output), (local input or local output),gif (local or global input),gif (local workspace), or (local workspace/local output).
  • The type of the argument;
  • For an array, its dimension(s).

    These dimensions are often expressed in terms of LOCtex2html_wrap_inline12112() and LOCtex2html_wrap_inline12114() calculations. For further details, please refer to section 4.3.2.

  • A specification of the value(s) that must be supplied for the argument (if it is an input argument), or of the value(s) returned by the routine (if it is an output argument), or both (if it is an input/output argument). In the last case, the two parts of the description are introduced by the phrases ``On entry'' and ``On exit''.
  • For a scalar input argument, any constraints that the supplied values must satisfy (such as N tex2html_wrap_inline14966 0 in the example above).

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Susan Blackford
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Guide





Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node80.html0100644000056400000620000001243706336113711017446 0ustar pfrauenfstaff Matrix and Vector Storage Conventions next up previous contents index
Next: In-Core Narrow Band and Up: In-core Dense Matrices Previous: Submatrix Argument Descriptions

Matrix and Vector Storage Conventions

Whether a dense coefficient matrix  operand is nonsymmetric, symmetric or Hermitian, the entire two-dimensional global array is distributed onto the process grid.

For symmetric  and Hermitian  matrix operands, only the upper (UPLO='U') triangle or the lower (UPLO='L') triangle of the global array is accessed. For triangular  matrix operands, the argument UPLO defines whether the matrix is upper (UPLO='U') or lower (UPLO='L') triangular. Only the elements of the relevant triangle of the global array are accessed. Some ScaLAPACK routines have an option to handle unit triangular matrix operands (that is, triangular matrices with diagonal elements = 1). This option is specified by an argument DIAG . If DIAG = 'U' (Unit triangular), the local array elements corresponding to the diagonal elements of the matrix are not referenced by the ScaLAPACK routines.

If an input matrix operand is Hermitian , the imaginary parts of the diagonal elements are zero, and thus the imaginary parts of the corresponding local arrays need not be set, but are assumed to be zero. If an output matrix operand is Hermitian, the imaginary parts of the diagonal elements are set to zero (e.g., PCPOTRF and PCHETRD).

Similarly, if the matrix is upper Hessenberg, the local array elements corresponding to global array elements below the first subdiagonal are not referenced.

Vectors can be distributed across       process rows or across process columns. A vector of length N distributed across process rows is distributed the same way that a N-by-1 matrix is. A vector of length N distributed across process columns is distributed the same way that a 1-by-N matrix is.

Within some ScaLAPACK routines, some vectors are replicated in one dimension and distributed in the other dimension. These vectors always aligned with one dimension of another distributed matrix. For example, in PDSYTRD, the vectors D, E, and TAU are replicated across process rows, distributed across process columns, and aligned with the distributed matrix operand A. The data distribution of these replicated vectors is inferred from the distribution of the matrix they are associated with. There is no specific array descriptors for these particular vectors at the present time.      

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Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node81.html0100644000056400000620000001035106336113712017441 0ustar pfrauenfstaff In-Core Narrow Band and Tridiagonal Matrices next up previous contents index
Next: The Block Column and Up: Data Distributions and Software Previous: Matrix and Vector Storage

In-Core Narrow Band and Tridiagonal Matrices

The ScaLAPACK routines solving narrow-band and tridiagonal linear systems assume their operands to be distributed according to the block-column and block-row data distribution schemes. Specifically, the narrow band or tridiagonal coefficient matrix is distributed in a block-column fashion, and the dense matrix of right hand side vectors is distributed in a block-row fashion. This section presents these distributions and demonstrates how the ScaLAPACK software encodes this essential information as well as the related software conventions.

The block data layout has been selected for narrow band matrices. Divide-and-conquer algorithms have been implemented in ScaLAPACK because these algorithms offer a much greater scope for exploiting parallelism than the corresponding adapted dense algorithms. The narrow band or tridiagonal coefficient matrix is partitioned into blocks. The inherent parallelism of these divide-and-conquer methods is limited by the number of these blocks because each block is processed independently; hence, it is necessary to choose the number of blocks at least equal to the desired parallelism. However, because the size of the reduced system is proportional to the number of blocks, and solving this reduced system is the major parallelism bottleneck, it follows that a block layout in which each process has exactly one block allows maximum exploitation of parallelism while minimizing the size of the reduced system.



Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node82.html0100644000056400000620000001327406336113713017452 0ustar pfrauenfstaff The Block Column and Row Distributions next up previous contents index
Next: The Block Mapping Up: In-Core Narrow Band and Previous: In-Core Narrow Band and

The Block Column and Row Distributions

           

ScaLAPACK assumes a one-dimensional block distribution for the band and tridiagonal routines. The block distribution is used when the computational load is distributed homogeneously over the global data. This distribution leads to a highly efficient implementation of the divide-and-conquer algorithms used in ScaLAPACK.

For convenience we will number the processes from 0 to P-1, and the matrix rows from 1 to M and the matrix columns from 1 to N. Figure 4.8 shows the two data layouts used in ScaLAPACK for solving narrow band linear systems. In all cases, each submatrix is labeled with the number of the process that contains it. Process 0 owns the shaded submatrices.

Consider the layout illustrated on the left of figure 4.8, the one-dimensional block column distribution. This distribution   

  figure2693
Figure 4.8: The one-dimensional block-column and block-row distributions

assigns a block of NB contiguous columns of a matrix to successive processes arranged in a tex2html_wrap_inline15088 one-dimensional process grid. Each process receives at most one block of columns of the matrix, i.e., tex2html_wrap_inline15090. Column k is stored on process tex2html_wrap_inline15094. The maximum number of columns stored per process is given by tex2html_wrap_inline15096. In the figure M=N=16 and P=4. This distribution assigns blocks of columns of size NB to successive processes. If the value of P evenly divides the value of N and NB = N / P, then each process owns a block of equal size. However, if this is not the case, then either the last process to receive a portion of the matrix will receive a smaller block than other processes, or some processes may receive an empty portion of the matrix. The transpose of this layout, the one-dimensional block-row distribution,    is shown on the right of figure 4.8.

The block-column distribution scheme is the data layout that is used in the ScaLAPACK library for the coefficient matrix of the narrow band and tridiagonal solvers.

The block-row distribution scheme is the data layout that is used in the ScaLAPACK library for the right-hand-side matrix of the narrow band and tridiagonal solvers.

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Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node83.html0100644000056400000620000003414106336635410017452 0ustar pfrauenfstaff The Block Mapping next up previous contents index
Next: Local Storage Scheme for Up: In-Core Narrow Band and Previous: The Block Column and

The Block Mapping

 

The one-dimensional distribution scheme is a mapping of a set of blocks onto the processes. The previous section informally described this mapping as well as some of its properties. To be complete, we shall describe the precise mapping that associates to a matrix entry identified by its global indexes the coordinates of the process that owns it and its local position within that process's memory.

Suppose we have a two dimensional array A of size tex2html_wrap_inline15127 to be distributed on a tex2html_wrap_inline15088 process grid in a block-column fashion. By convention, the array columns are numbered 1 through N and the processes are numbered 0 through P-1. First, the array is divided into contiguous blocks of NB columns with tex2html_wrap_inline15090. When NB does not divide N evenly, the last block of columns will only contain tex2html_wrap_inline14699 columns instead of NB. By convention, these blocks are numbered starting from zero and dealt out to the processes. In other words, if we assume that the process 0 receives the first block, the tex2html_wrap_inline15153 block is assigned to the process of coordinate (0,p). The mapping of a column of the array globally indexed by J is defined by the following analytical equation:
displaymath15123
where J is a global column index in the array, p is the column coordinate of the process owning that column, and finally x is the column coordinate within that block of columns where the global array column of index J is to be found. It is then fairly easy to establish the analytical relationship between these variables. One obtains:
 equation2708
These equations allow to determine the local information, i.e. the local index x as well as the process column coordinate p corresponding to a global column identified by its global index J and conversely. Table 4.9 illustrates this mapping layout when P=2 and N=16 and NB=8. At most one block is assigned to each process.

  table2713
Table 4.9: One-dimensional block-column mapping example for P=2 and N=16

This example of the one-dimensional block-column distribution mapping    can be expressed in HPF  by using the following statements: 

      REAL :: A( M, N )
!HPF$ PROCESSORS PROC( 1, P )
!HPF$ DISTRIBUTE A( *, BLOCK( NB ) ) ONTO PROC

A similar example of block-row distribution    can easily be constructed. For an tex2html_wrap_inline15189 array B, such an example can be expressed in HPF  by using the following statements: 

      REAL :: B( N, NRHS )
!HPF$ PROCESSORS PROC( P, 1 )
!HPF$ DISTRIBUTE B( BLOCK( NB ), * ) ONTO PROC

There is in fact no real reason to always deal out the blocks starting with the process 0. In fact, it is sometimes useful to start the data distribution with the process of arbitrary coordinate SRC, in which case Equation 4.3 becomes:
 equation2729

Table 4.10 illustrates Equation 4.4 for the block-cyclic layout when tex2html_wrap_inline14791, tex2html_wrap_inline14793, tex2html_wrap_inline14795 and tex2html_wrap_inline14797.

  table2738
Table 4.10: One-dimensional block-column mapping for P=2, SRC=1, N=16 and NB=8

This example of the one-dimensional block-column    distribution mapping can be expressed in HPF  by using the following statements: 

      REAL :: A( M, N )
!HPF$ PROCESSORS PROC( 1, P )
!HPF$ TEMPLATE T( M, N + P*NB )
!HPF$ DISTRIBUTE T( *, BLOCK( NB ) ) ONTO PROC
!HPF$ ALIGN A( I, J ) WITH T( I, SRC*NB + J )

A similar example of block-row distribution     can easily be constructed. For an tex2html_wrap_inline15189 array B, such an example can be expressed in HPF  by using the following statements: 

      REAL :: B( N, NRHS )
!HPF$ PROCESSORS PROC( P, 1 )
!HPF$ TEMPLATE T( N + P*NB, NRHS )
!HPF$ DISTRIBUTE T( BLOCK( NB ), * ) ONTO PROC
!HPF$ ALIGN A( I, J ) WITH T( SRC*NB + I, J )

In ScaLAPACK, the local storage convention of the one-dimensional block distributed matrix in every process's memory is assumed to be Fortran-like, that is, ``column major'' .

Determining the number of rows or columns     of a global band matrix that a specific process receives is an essential task for the user. The notation LOCtex2html_wrap_inline12112() is used for block-row distributions and LOCtex2html_wrap_inline12114() is used for block-column distributions. These local quantities occur throughout the leading comments of the source code, and are reflected in the sample argument description in section 4.4.7.

For block distribution, a matrix can be distributed unevenly. More specifically, one process in the process grid can receive an array that is smaller than other processes. It is also possible that some processes receive no data. For further information on one-dimensional block-column or block-row data distribution, please refer to section 4.4.1.

Block-Column Distribution: LOCtex2html_wrap_inline12114(N_A) denotes the number of columns that a process would receive if N_A columns of a matrix is distributed over tex2html_wrap_inline12162 columns of its process row.

For example, let us assume that the coefficient matrix A is band symmetric of order N and has been block-column distributed on a tex2html_wrap_inline15233 process grid.

In the ideal case where the matrix is evenly distributed to all processes in the process grid, tex2html_wrap_inline15235 and tex2html_wrap_inline15237. Thus, each process receives a block of size tex2html_wrap_inline14839 of the matrix A. Therefore,

LOCtex2html_wrap_inline12114(N_A) = NB_A.

However, if tex2html_wrap_inline15245, at least one of the processes in the process grid will receive a block of size smaller than tex2html_wrap_inline14839. Thus, 

 if ( tex2html_wrap_inline15245 and tex2html_wrap_inline15251 ) then


   		 processes (0,0), ... , (0,K-1) receive


   		           		 LOCtex2html_wrap_inline12114(N_A) = NB_A


   		 and process (0,K) receives


   		           		 LOCtex2html_wrap_inline12114(N_A) = N_A - K tex2html_wrap_inline15263 NB_A.


   		 if tex2html_wrap_inline15265 then processes tex2html_wrap_inline15267 do
      not receive any data.


end if

Block-Row Distribution: LOCtex2html_wrap_inline12112(M_B) denotes the number of rows that a process would receive if M_B rows of a matrix is distributed over tex2html_wrap_inline12172 rows of its process column.

Let us assume that the N-by-NRHS right-hand-side matrix B has been block-row distributed on a tex2html_wrap_inline15275 process grid.

In the ideal case where the matrix is evenly distributed to all processes in the process grid, tex2html_wrap_inline15277 and tex2html_wrap_inline15279. Thus, each process receives a block of size tex2html_wrap_inline15281 of the matrix B. Therefore,

LOCtex2html_wrap_inline12112(M_B) = MB_B.

However, if tex2html_wrap_inline15287, then at least one of the processes in the process grid will receive a block of size smaller than tex2html_wrap_inline15281. Thus, 

 if ( tex2html_wrap_inline15287 and tex2html_wrap_inline15293 ) then


   		 processes t#tex2html_wrap_inline15295# receive


   		           		 LOCtex2html_wrap_inline12112(M_B) = MB_B


   		 and process (K,0) receives


   		           		 LOCtex2html_wrap_inline12112(M_B) = M_B - K tex2html_wrap_inline15263 MB_B.


   		 if tex2html_wrap_inline15307 then processes tex2html_wrap_inline15309 do
      not receive any data.


end if
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Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Local Storage Scheme for Narrow Band Matrices

 

Let us first discuss how to distribute a narrow band matrix A  over a one-dimensional process grid using a block-column distribution. We assume that the coefficient band matrix A   is of size tex2html_wrap_inline15332 (tex2html_wrap_inline15334) with a bandwidth BW=2 if the matrix A is symmetric positive definite, and BWL=2 and BWU=2 if the matrix A is nonsymmetric. The matrix A is represented by the following.


displaymath15326

If we assume that the matrix A is nonsymmetric band, the user may choose to perform partial pivoting or no pivoting during the factorization (PxGBTRF     or PxDBTRF    , respectively). Both strategies assume a block-column distribution of the coefficient matrix, but additional storage is required for fill-in if partial pivoting is selected. First, let us assume that we have selected no pivoting, and we distribute this matrix onto a tex2html_wrap_inline14534 process grid with a block size of tex2html_wrap_inline15352. The processes would contain the local arrays found in figure 4.9. Figure 4.9 also illustrates that the leading dimension of the local arrays containing the coefficient matrix must be at least BWL+1+BWU for the non-pivoting narrow band linear solver.

  figure2837
Figure 4.9: Mapping of local arrays for nonsymmetric band matrix A (no pivoting)

If, however, we select partial pivoting and distribute this same matrix onto a tex2html_wrap_inline14534 process grid with a block size of tex2html_wrap_inline15430, the processes would contain the local arrays found in figure 4.10. The amount of additional storage required for fill-in is represented by F in the figure and is equal to the sum of the lower bandwidth (number of subdiagonals), BWL, and the upper bandwidth (number of superdiagonals), BWU. In this example, BWL=2 and BWU=2. Refer to the leading comments of the routine PxGBTRF for further details. Figure 4.10 also illustrates that the leading dimension of the local arrays containing the coefficient matrix must be at least 2*(BWL+BWU)+1 for the partial pivoting narrow band linear solver.

  figure2886
Figure 4.10: Mapping of local arrays for nonsymmetric band matrix A (partial pivoting)

Let us now assume that the matrix A is symmetric positive definite band with BW=2, and we distribute this matrix assuming lower triangular storage (UPLO='L') onto a tex2html_wrap_inline14534 process grid with a block size tex2html_wrap_inline15430. The processes would contain the local arrays found in figure 4.11. We would then call the routine PxPBTRF     with BW=2 to perform the factorization, for example.

  figure2938
Figure 4.11: Mapping of local arrays for symmetric positive definite band matrix A (UPLO='L')

If we then distributed this same matrix assuming upper triangular storage (UPLO='U') onto a tex2html_wrap_inline14534 process grid with a block size tex2html_wrap_inline15430, the processes would contain the local arrays found in figure 4.12.

  figure2975
Figure 4.12: Mapping of local arrays for symmetric positive definite band matrix A (UPLO='U')

Figures 4.11 and  4.12 also illustrate that the leading dimension of the local arrays containing the coefficient matrix must be at least BW+1 for the symmetric positive definite narrow band linear solver.

The tex2html_wrap_inline15263 notation in figures 4.9,  4.10,  4.11, and 4.12 and the F notation in figure 4.10 signify an entry in which one need not store a value in that position of the local array. These storage positions, however, are required and overwritten during the computation.

The tex2html_wrap_inline15189 matrix of right-hand-side vectors B (for example, used in PxGBTRS    , PxDBTRS    , and PxPBTRS    ) is assumed to be a dense matrix distributed in a block-row manner across the process grid. Thus, consecutive blocks of rows of the matrix B are assigned to successive processes in the process grid, as described in section 4.4.1.

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Susan Blackford
Tue May 13 09:21:01 EDT 1997
scalapack-doc-1.5/html/slug/node85.html0100644000056400000620000001616206336113715017456 0ustar pfrauenfstaff Local Storage Schemes for Tridiagonal Matrices next up previous contents index
Next: Array Descriptor for Narrow Up: In-Core Narrow Band and Previous: Local Storage Scheme for

Local Storage Schemes for Tridiagonal Matrices

 

A global tridiagonal matrix A , represented as three vectors (DL, D, DU), should be distributed over a one-dimensional process grid assuming a block-column data distribution. We assume that the coefficient tridiagonal matrix A   is of size tex2html_wrap_inline15332 (tex2html_wrap_inline15334) and is represented by the following.


displaymath15653

If we first assume that the matrix A is nonsymmetric (diagonally dominant like), and it is known a priori that no pivoting is required for numerical stability, the user may choose to perform no pivoting during the factorization (PxDTTRF    ). If we distribute this matrix (assuming no pivoting) onto a tex2html_wrap_inline14534 process grid with a block size of tex2html_wrap_inline15430, the processes would contain the local arrays found in figure 4.13.

  figure3064
Figure 4.13: Mapping of local arrays for nonsymmetric tridiagonal matrix A

Finally, a global symmetric positive definite tridiagonal matrix A , represented as two vectors (D and E), should be distributed over a one-dimensional    process grid assuming a block-column data distribution.

Let us now assume that this matrix A is symmetric positive definite and that we distribute this matrix assuming lower triangular storage (UPLO='L') onto a tex2html_wrap_inline14534 process grid with a block size tex2html_wrap_inline15430. The processes would contain the local arrays found in figure 4.14. We would then call the routine PxPTTRF     to perform the factorization, for example.

  figure3109
Figure 4.14: Mapping of local arrays for symmetric positive definite tridiagonal matrix A (UPLO='L')

If we then distributed this same matrix assuming upper triangular storage (UPLO='U') onto a tex2html_wrap_inline14534 process grid with a block size tex2html_wrap_inline15430, the processes would contain the local arrays found in figure 4.15.

  figure3141
Figure 4.15: Mapping of local arrays for symmetric positive definite tridiagonal matrix A (UPLO='U')

Note that in the tridiagonal cases, it is not necessary to maintain the empty storage positions as designated by tex2html_wrap_inline15263 in the narrow band routines.

The matrix of right-hand-side vectors B (for example, used in PxDTTRS     and PxPTTRS    ) is assumed to be a dense matrix distributed block-row across the process grid. Thus, consecutive blocks of rows of the matrix B are assigned to successive processes in the process grid, as described in section 4.4.1.

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Susan Blackford
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Array Descriptor for Narrow Band and Tridiagonal Matrices

 

The array descriptor DESC_ whose type is defined as DESC_(DTYPE_)=501 , is an integer array of length 7. This descriptor type is used in the ScaLAPACK narrow band routines and tridiagonal routines to specify a block-column distribution of a global array over a one-dimensional process grid. In the general and symmetric positive definite banded and tridiagonal routines, a one-dimensional block-column distribution is specified for the coefficient matrix . The matrix of right-hand-side vectors must be distributed over a one-dimensional process grid using a block-row data distribution. Refer to section 4.4.1 for further details on block data distribution.

Let us assume, for example, that we have an array descriptor DESCA for a block-column distributed array A. For readability of the code, we have associated symbolic names with the descriptor entries. As previously mentioned, the notations x_ used in the entries of the array descriptor denote the attributes of a global array. For readability of the code, we have associated symbolic names for the descriptor entries. For example, N_ denotes the number of columns and N_A specifically denotes the number of columns in global array A.

 table3188
Table 4.11: Content of the array descriptor for in-core narrow band and tridiagonal coefficient matrices

When A is non-symmetric and factorized without pivoting,      LLD_A must be at least BWL+1+BWU. When A is non-symmetric and factorized with partial pivoting, LLD_A      must be at least 2(BWL+BWU)+1. When A is symmetric positive definite, LLD_A must be at least      BW+1. Finally, when A is tridiagonal, LLD_A is not      referenced.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Array Descriptor for the Matrix of Right-Hand-Side Vectors

 

The array descriptor DESC_ whose type is defined as DESC_(DTYPE_)=502 , is an integer array of length 7. This descriptor type is used in the ScaLAPACK narrow band routines and tridiagonal routines to specify the block-row distribution of a global array containing the right-hand-side vectors over a one-dimensional process grid. In the narrow band and tridiagonal routines, a one-dimensional block-column distribution is specified for the coefficient matrix . The matrix of right-hand-side vectors however must be distributed over a one-dimensional process grid according to a block-row data distribution scheme. Refer to section 4.4.1 for further details on block data distribution.

Let us now assume that we have an array descriptor DESCB for a block-row distributed matrix B. For readability of the code, we have associated symbolic names with the descriptor entries.

 table3220
Table 4.12: Content of the array descriptor for right-hand-side dense matrices for narrow band and tridiagonal solvers

For a detailed description of LOCtex2html_wrap_inline12112() notation, please refer to section 4.4.2.


Susan Blackford
Tue May 13 09:21:01 EDT 1997
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Argument Descriptions for Band and Tridiagonal Routines

           

All ScaLAPACK narrow band and tridiagonal routines assume that the global matrices are distributed in a one-dimensional block data distribution. Thus, each process has at most one block of data. With selective choices for the block size NB_ and the order N_ of the global matrix, it is possible that some processes in the process grid may not receive any data, or the last process receiving data will receive a smaller block of data than the other processes.

For further information on one-dimensional block-column or block-row data distribution, please refer to section 4.4.1.

The description of a block-column distributed band matrix consists of (N, A, JA, DESCA)

  • the order N of the band matrix operand,
  • a pointer to the local array containing the entire global array (A, for example),
  • the column index, JA, of the global array, and
  • the array descriptor, DESCA, of the global array.

The description of a block-row distributed right-hand-side matrix consists of (NRHS, B, IB, DESCB)

  • the number of right-hand-side vectors NRHS in the matrix,
  • a pointer to the local array containing the entire global array (B, for example),
  • the row index, IB, of the global array, and
  • the array descriptor, DESCB, for the global array.

The description of a block-distributed diagonally dominant-like tridiagonal matrix consists of (N, DL, D, DU, JA, DESCA)

  • the order N of the tridiagonal matrix operand,
  • pointer to the local arrays, (DL, D, DU, for example),
  • the column index, JA, of the global array, and
  • the array descriptor, DESCA, for the global array.

The description of a block-distributed symmetric positive definite tridiagonal matrix consists of (N, D, E, JA, DESCA)

  • the order N of the tridiagonal matrix operand,
  • pointer to the local arrays, (D, E, for example),
  • the column index, JA, of the global array,
  • the array descriptor, DESCA, for the global array.

The name of the row or column index for the global array has the form I<array_name>  or J<array_name> , respectively. The array descriptor has a name of the form DESC<array_name> .    The length of the array descriptor is specified by DLEN_ and varies according to the descriptor type DTYPE_.

Included in the leading comments of each subroutine (immediately preceding the Argument section), is a brief note describing the array descriptor    and some commonly used expressions in calculating workspace.

The style of the argument  descriptions for symmetric positive definite narrow band routines (PxPByyy) and diagonally dominant-like narrow band routines (PxDByyy) is illustrated by the following example:

    N
    (global input) INTEGER
    The number of rows and columns of the matrix A(JA:JA+N-1,JA:JA+N-1) on which to be operated. N tex2html_wrap_inline14966 0.
    NRHS
    (global input) INTEGER
    The number of right hand sides, i.e., the number of columns of the matrix B(IB:IB+N-1,*). NRHS tex2html_wrap_inline14966 0.
    A
    (local input/local output) REAL pointer into the local memory to an array of local dimension (LLD_A, LOCtex2html_wrap_inline12114(JA+N-1))
    On entry, the local part of the N-by-N global symmetric band matrix A(JA:JA+N-1,JA:JA+N-1).
    JA
    (global input) INTEGER
    The column index of the global matrix A.
    DESCA
    (global and local input) INTEGER array of dimension DLEN_
    The array descriptor for the global matrix A.
    B
    (local input/local output) REAL array, dimension (LLD_B, NRHS)
    On entry, the local part of the N-by-NRHS right-hand-side matrix.
    IB
    (global input) INTEGER
    The row index of the global matrix B.
    DESCB
    (global and local input) INTEGER array of dimension DLEN_
    The array descriptor for the global matrix B.

The style of the argument  descriptions for diagonally dominant-like tridiagonal routines (PxDTyyy) is illustrated by the following example:

    N
    (global input) INTEGER
    The number of rows and columns of the matrix A(JA:JA+N-1,JA:JA+N-1) on which to be operated. N tex2html_wrap_inline14966 0.
    NRHS
    (global input) INTEGER
    The number of right hand sides, i.e., the number of columns of the matrix B(IB:IB+N-1,*). NRHS tex2html_wrap_inline14966 0.
    DL
    (local input/local output) REAL pointer into the local memory to an array of local dimension (LOCtex2html_wrap_inline12114(JA+N-1))
    On entry, the local part of the subdiagonal entries of the global tridiagonal matrix A(JA:JA+N-1,JA:JA+N-1).
    D
    (local input/local output) REAL pointer into the local memory to an array of local dimension (LOCtex2html_wrap_inline12114(JA+N-1))
    On entry, the local part of the diagonal entries of the global tridiagonal matrix A(JA:JA+N-1,JA:JA+N-1).
    DU
    (local input/local output) REAL pointer into the local memory to an array of local dimension (LOCtex2html_wrap_inline12114(JA+N-1))
    On entry, the local part of the superdiagonal entries of the global tridiagonal matrix A(JA:JA+N-1,JA:JA+N-1).
    JA
    (global input) INTEGER
    The column index of the global matrix A.
    DESCA
    (global and local input) INTEGER array of dimension DLEN_
    The array descriptor for the global matrix A.
    B
    (local input/local output) REAL array, dimension (LLD_B, NRHS)
    On entry, the local part of the N-by-NRHS right-hand-side matrix.
    IB
    (global input) INTEGER
    The row index of the global matrix B.
    DESCB
    (global and local input) INTEGER array of dimension DLEN_
    The array descriptor for the global matrix B.

The style of the argument  descriptions for symmetric positive definite tridiagonal routines (PxPTyyy) is illustrated by the following example:

    N
    (global input) INTEGER
    The number of rows and columns of the matrix A(JA:JA+N-1,JA:JA+N-1) on which to be operated. N tex2html_wrap_inline14966 0.
    NRHS
    (global input) INTEGER
    The number of right hand sides, i.e., the number of columns of the matrix B(IB:IB+N-1,*). NRHS tex2html_wrap_inline14966 0.
    D
    (local input/local output) REAL pointer into the local memory to an array of local dimension (LOCtex2html_wrap_inline12114(JA+N-1))
    On entry, the local part of the diagonal entries of the global tridiagonal matrix A(JA:JA+N-1,JA:JA+N-1).
    E
    (local input/local output) REAL pointer into the local memory to an array of local dimension (LOCtex2html_wrap_inline12114(JA+N-1))
    On entry, the local part of the off-diagonal entries of the global tridiagonal matrix A(JA:JA+N-1,JA:JA+N-1).
    JA
    (global input) INTEGER
    The column index of the global matrix A.
    DESCA
    (global and local input) INTEGER array of dimension DLEN_
    The array descriptor for the global matrix A.
    B
    (local input/local output) REAL array, dimension (LLD_B, NRHS)
    On entry, the local part of the N-by-NRHS right-hand-side matrix.
    IB
    (global input) INTEGER
    The row index of the global matrix B.
    DESCB
    (global and local input) INTEGER array of dimension DLEN_
    The array descriptor for the global matrix B.

The description of each argument contains the following information:

  • A classification of the argument as (local input), (global and local input), (local input/local output), (global input), (local output), (global output), (global input/global output), (local or global input)gif (local workspace), or (local workspace/local output).
  • The type of the argument;
  • For an array, its dimension(s).

    These dimensions are often expressed in terms of LOCtex2html_wrap_inline12112() and LOCtex2html_wrap_inline12114() calculations. For further details, please refer to section 4.4.2.

  • A specification of the value(s) that must be supplied for the argument (if it is an input argument), or of the value(s) returned by the routine (if it is an output argument), or both (if it is an input/output argument). In the last case, the two parts of the description are introduced by the phrases ``On entry'' and ``On exit''.
  • For a scalar input argument, any constraints that the supplied values must satisfy (such as ``N tex2html_wrap_inline14966 0'' in the example above).

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Matrix Storage Conventions for Band and Tridiagonal Matrices

A general  tridiagonal matrix of order n is stored globally in three one-dimensional arrays dl, d, du of length n containing the subdiagonal, diagonal, and superdiagonal elements, respectively. Note the mild change from LAPACK in which dl and du were actually of global length n-1. To make the distribution of the vectors consistent, we have chosen to make them all of length n. Note that dl(1)=du(n)=0.

Similarly, a symmetric  tridiagonal matrix is stored globally in two one-dimensional arrays d, e of length n containing the diagonal and off-diagonal elements, respectively. Again, there is a slight departure from LAPACK in which e was of global length n-1. Here, e(n)=0.

The vectors (DL, D, DU) or (D, E) representing these matrices must be block distributed to a one-dimensional process grid. These vectors can be equivalently distributed block-row or block-column since vectors are one-dimensional data structures. Note that when inputting vectors to these special-purpose low-diagonal routines, LLD_ can be ignored, since it is assumed that the local portions of the vectors are of unit stride.


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Essentials

 



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Out-of-Core Matrices

The ScaLAPACK software library provides routines for solving out-of-core linear systems, in which case the matrices are stored on disk. A particular array descriptor is required to specify such a data storage.



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Array Descriptor for Out-core Dense Matrices

 

The array descriptor DESC_ whose type is defined as DESC_(DTYPE_)=601 , is an integer array of length 11. DESC_(DTYPE_)=601  is used for the ScaLAPACK routines involved in the out-of-core solution of dense linear systems using LU, QR or Cholesky factorizations [55]. The matrix stored on disk is composed of records each record of which corresponds to an MMB tex2html_wrap_inline12420 NNB DESC_(DTYPE_)=1 ScaLAPACK matrix and these records are organized in a column major (Fortran array) manner. The array descriptor for out-of-core matrices has extra fields to store file parameters associated with the matrix, such as the I/O device unit number, MMB and NNB, and the amount of temporary buffer storage available.

Similar to DESC_(DTYPE_)=1 symmetric ScaLAPACK matrices, only the upper (UPLO='U') triangle or the lower (UPLO='L') triangle is accessed. The entire coefficient matrix is stored on disk, regardless of whether the matrix is nonsymmetric or symmetric.

 table3366
Table 4.13: Content of the array descriptor for out-of-core dense matrices


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Design and Documentation of Argument Lists

 

As in LAPACK, the argument lists of all ScaLAPACK routines conform to a single set of conventions for their design and documentation.

Specifications of all ScaLAPACK driver and computational routines are given in Part ii of this users guide. These are derived from the specifications given in the leading comments in the code, but in Part ii the specifications for real and complex versions of each routine have been merged in order to save space.



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Structure of the Documentation

 

The documentation  of each ScaLAPACK routine includes the following:

  • The SUBROUTINE or FUNCTION statement, followed by statements declaring the type and dimensions of the arguments.
  • A summary of the Purpose of the routine.
  • Descriptions of each of the Arguments in the order of the argument list.
  • (optionally) Further Details (only in the code, not in Part ii of this users guide);
  • (optionally) Internal Parameters (only in the code, not in Part ii of this users guide).


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Order of Arguments

 

Arguments  of a ScaLAPACK routine appear in the following order:

  • arguments specifying options,
  • problem dimensions,
  • array or scalar arguments defining the input data; some of them may be overwritten by results,
  • other array or scalar arguments returning results,
  • work arrays (and associated array dimensions), and
  • diagnostic argument INFO.

Note that not every category is present in each of the routines.

When defining each of these categories of arguments, ScaLAPACK distinguishes between local and global data. On entry to a ScaLAPACK routine, local input arguments   may have different values on each process in the process grid. Similarly, local output arguments   may be assigned different values on different processes in the process grid on exit from the ScaLAPACK routine.

Global input arguments   must have the same value on each process in the process grid on entry to a ScaLAPACK routine. If this is not the case, most routines will call PXERBLA and return. Global output arguments   are assigned the same value on all processes in the process grid on exit from a ScaLAPACK routine.


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Option Arguments

 

Arguments  specifying options are usually of type CHARACTERtex2html_wrap_inline152631. The arguments that specify options are character arguments with the names SIDE, TRANS, UPLO, and DIAG. On entry to a ScaLAPACK routine, these arguments are global input and must have the same value on each process in the process grid.

SIDE  is used by the routines as follows:
tabular3422

TRANS  is used by the routines as follows:
tabular3431
In the real case the values `T' and `C' have the same meaning, and in the complex case the value `T' is not allowed.

UPLO  is used by the Hermitian, symmetric, and triangular distributed matrix routines to specify whether the upper or lower triangle is being referenced as follows:
tabular3444

DIAG  is used by the triangular distributed matrix routines to specify whether the distributed matrix is unit triangular, as follows:
tabular3453
When DIAG is supplied as `U', the diagonal elements are not referenced. For example:

    UPLO
    (global input) CHARACTERtex2html_wrap_inline152631
    = 'U': Upper triangle of the matrix A(IA:IA+M-1,JA:JA+N-1);
    = 'L': Lower triangle of the matrix A(IA:IA+M-1,JA:JA+N-1).

The corresponding lower-case characters may be supplied (with the same meaning), but any other value is illegal (see section 4.6.6).

A longer character string can be passed as the actual argument, making the calling program more readable, but only the first character is significant; this is a standard feature of Fortran 77. For example: 

       CALL PSPOTRS('upper', . . . )


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Problem Dimensions

 

The problem  dimensions may be passed as zero, in which case the computation (or part of it) is skipped. Negative dimensions are regarded as erroneous.


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Workspace Issues

 



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WORK Arrays

Many ScaLAPACK routines require one or more work arrays  to be passed as arguments. The name of a work array is usually WORK - sometimes IWORK or RWORK to distinguish work arrays of type integer or real. Immediately following the work array in the argument list is the specified length of the work array, LWORK , LIWORK, or LRWORK, respectively. LWORK is defined as the minimum amount of workspace necessary to perform the operation specified.

The first element of the work array is always used to return the correct value of LWORK for the computation. Whether or not an error is detected, the minimum value of LWORK is placed in WORK(1) on exit from the routine.

If the user passes a value for LWORK that is too small, an input error is detected and INFO is set accordingly (see section 4.6.6), the correct value for LWORK is placed in WORK(1), and the routine PXERBLA is called. The user is thus strongly advised to always check the value of INFO on exit from the called routine.


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LWORK Query

 

If in doubt about the amount of workspace to supply to a ScaLAPACK routine, the user may choose to supply LWORK=-1  and use the returned value in WORK(1) as the correct value for LWORK. Setting LWORK=-1 does not invoke an error message from PXERBLA and is defined as a global query   .


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ScaLAPACK

ScaLAPACK is a library of high-performance linear algebra routines for distributed-memory message-passing MIMD    computers and networks of workstations supporting   PVM [68] and/or MPI [64, 110]. It is a continuation of the LAPACK [3] project, which designed and produced analogous software for workstations, vector supercomputers, and shared-memory parallel computers. Both libraries contain routines for solving systems of linear equations, least squares problems, and eigenvalue problems. The goals of both projects are efficiency (to run as fast as possible), scalability  (as the problem size and number of processors grow), reliability  (including error bounds), portability  (across all important parallel machines), flexibility (so users can construct new routines from well-designed parts), and ease of use (by making the interface to LAPACK and ScaLAPACK look as similar as possible). Many of these goals, particularly portability, are aided by developing and promoting standards , especially for low-level communication and computation routines. We have been successful in attaining these goals, limiting most machine dependencies to two standard libraries called the BLAS, or Basic Linear Algebra Subprograms [57, 59, 74, 93], and BLACS, or Basic Linear Algebra Communication Subprograms [50, 54]. LAPACK will run on any machine where the BLAS are available, and ScaLAPACK will run on any machine where both the BLAS and the BLACS are available.

The library is currently written in Fortran 77 (with the exception of a few symmetric eigenproblem auxiliary routines written in C to exploit IEEE arithmetic) in a Single Program Multiple Data (SPMD)   style using explicit message passing  for interprocessor communication. The name ScaLAPACK is an acronym for Scalable Linear Algebra PACKage, or Scalable LAPACK.


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Next: Contents



ScaLAPACK Users' Guide

  • L. S. Blackford,
  • J. Choi,
  • A. Cleary,
  • E. D'Azevedo,
  • J. Demmel,
  • I. Dhillon,
  • J. Dongarra,
  • S. Hammarling,
  • G. Henry,
  • A. Petitet,
  • K. Stanley,
  • D. Walker,
  • R. C. Whaley

    1 May 1997

    Dedication

    This work is dedicated to the pioneers of high-performance computing who blazed a trail, set standards, and made our job easier.

    Acknowledgment

    We give credit to and thank all of the LAPACK authors for allowing us to reuse large portions of the LAPACK Users' Guide in creating this users guide for ScaLAPACK.

    1997 by the Society for Industrial and Applied Mathematics. Certain derivative work portions have been copyrighted by the Numerical Algorithms Group Ltd.

    The printed version of the ScaLAPACK Users' Guide is available from SIAM.

    • ISBN 0-89871-397-8;
    • The list price for SIAM members is $39.60;
    • the cost for nonmembers is $49.50.
    • Order code SE04.
    Contact SIAM for additional information.

  • click here to send e-mail to service@siam.org
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  • phone: (USA) 800-447-SIAM
  • (outside USA) 215-386-7999
  • mail: SIAM, 3600 University City Science Center, Philadelphia, PA 19104-2688.

    The royalties from the sales of this book are being placed in a fund to help students attend SIAM meetings and other SIAM related activities. This fund is administered by SIAM and qualified individuals are encouraged to write directly to SIAM for guidelines.





    Susan Blackford
    Tue May 13 09:21:01 EDT 1997
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This is the unblocked version of the algorithm, calling Level 2 BLAS. .SH ARGUMENTS .TP 8 M (input) INTEGER The number of rows of the matrix A. M >= 0. .TP 8 N (input) INTEGER The number of columns of the matrix A. N >= 0. .TP 8 KL (input) INTEGER The number of subdiagonals within the band of A. KL >= 0. .TP 8 KU (input) INTEGER The number of superdiagonals within the band of A. KU >= 0. .TP 8 AB (input/output) COMPLEX array, dimension (LDAB,N) On entry, the matrix A in band storage, in rows KL+1 to 2*KL+KU+1; rows 1 to KL of the array need not be set. The j-th column of A is stored in the j-th column of the array AB as follows: AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl) On exit, details of the factorization: U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. See below for further details. .TP 8 LDAB (input) INTEGER The leading dimension of the array AB. LDAB >= 2*KL+KU+1. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 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. .SH FURTHER DETAILS The band storage scheme is illustrated by the following example, when M = N = 6, KL = 2, KU = 1: .br On entry: On exit: .br * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 * a31 a42 a53 a64 * * m31 m42 m53 m64 * * Array elements marked * are not used by the routine; elements marked + need not be set on entry, but are required by the routine to store elements of U, because of fill-in resulting from the row .br interchanges. .br scalapack-doc-1.5/man/manl/cdbtrf.l0100644000056400000620000000435206335610612016666 0ustar pfrauenfstaff.SH NAME CDBTRF - compute an LU factorization of a real m-by-n band matrix A without using partial pivoting or row interchanges .SH SYNOPSIS .TP 19 SUBROUTINE CDBTRF( M, N, KL, KU, AB, LDAB, INFO ) .TP 19 .ti +4 INTEGER INFO, KL, KU, LDAB, M, N .TP 19 .ti +4 COMPLEX AB( LDAB, * ) .SH PURPOSE Cdbtrf computes an LU factorization of a real m-by-n band matrix A without using partial pivoting or row interchanges. This is the blocked version of the algorithm, calling Level 3 BLAS. .SH ARGUMENTS .TP 8 M (input) INTEGER The number of rows of the matrix A. M >= 0. .TP 8 N (input) INTEGER The number of columns of the matrix A. N >= 0. .TP 8 KL (input) INTEGER The number of subdiagonals within the band of A. KL >= 0. .TP 8 KU (input) INTEGER The number of superdiagonals within the band of A. KU >= 0. .TP 8 AB (input/output) REAL array, dimension (LDAB,N) On entry, the matrix A in band storage, in rows KL+1 to 2*KL+KU+1; rows 1 to KL of the array need not be set. The j-th column of A is stored in the j-th column of the array AB as follows: AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl) On exit, details of the factorization: U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. See below for further details. .TP 8 LDAB (input) INTEGER The leading dimension of the array AB. LDAB >= 2*KL+KU+1. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 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. .SH FURTHER DETAILS The band storage scheme is illustrated by the following example, when M = N = 6, KL = 2, KU = 1: .br On entry: On exit: .br * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 * a31 a42 a53 a64 * * m31 m42 m53 m64 * * Array elements marked * are not used by the routine. .br scalapack-doc-1.5/man/manl/cdttrf.l0100644000056400000620000000331706335610612016710 0ustar pfrauenfstaff.TH CDTTRF l "12 May 1997" "modified LAPACK routine" "LAPACK routine (version 2.0)" .SH NAME CDTTRF - compute an LU factorization of a complex tridiagonal matrix A using elimination without partial pivoting .SH SYNOPSIS .TP 19 SUBROUTINE CDTTRF( N, DL, D, DU, INFO ) .TP 19 .ti +4 INTEGER INFO, N .TP 19 .ti +4 COMPLEX D( * ), DL( * ), DU( * ) .SH PURPOSE CDTTRF computes an LU factorization of a complex tridiagonal matrix A using elimination without partial pivoting. The factorization has the form .br A = L * U .br where L is a product of unit lower bidiagonal .br matrices and U is upper triangular with nonzeros in only the main diagonal and first superdiagonal. .br .SH ARGUMENTS .TP 8 N (input) INTEGER The order of the matrix A. N >= 0. .TP 8 DL (input/output) COMPLEX array, dimension (N-1) On entry, DL must contain the (n-1) subdiagonal elements of A. On exit, DL is overwritten by the (n-1) multipliers that define the matrix L from the LU factorization of A. .TP 8 D (input/output) COMPLEX array, dimension (N) On entry, D must contain the diagonal elements of A. On exit, D is overwritten by the n diagonal elements of the upper triangular matrix U from the LU factorization of A. .TP 8 DU (input/output) COMPLEX array, dimension (N-1) On entry, DU must contain the (n-1) superdiagonal elements of A. On exit, DU is overwritten by the (n-1) elements of the first superdiagonal of U. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 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. scalapack-doc-1.5/man/manl/cdttrsv.l0100644000056400000620000000352406335610612017113 0ustar pfrauenfstaff.TH CDTTRSV l "12 May 1997" "modified LAPACK routine" "LAPACK routine (version 2.0)" .SH NAME CDTTRSV - solve one of the systems of equations L * X = B, L**T * X = B, or L**H * X = B, .SH SYNOPSIS .TP 20 SUBROUTINE CDTTRSV( UPLO, TRANS, N, NRHS, DL, D, DU, B, LDB, INFO ) .TP 20 .ti +4 CHARACTER UPLO, TRANS .TP 20 .ti +4 INTEGER INFO, LDB, N, NRHS .TP 20 .ti +4 COMPLEX B( LDB, * ), D( * ), DL( * ), DU( * ) .SH PURPOSE CDTTRSV solves one of the systems of equations L * X = B, L**T * X = B, or L**H * X = B, U * X = B, U**T * X = B, or U**H * X = B, .br with factors of the tridiagonal matrix A from the LU factorization computed by CDTTRF. .br .SH ARGUMENTS .TP 8 UPLO (input) CHARACTER*1 Specifies whether to solve with L or U. .TP 8 TRANS (input) CHARACTER Specifies the form of the system of equations: .br = 'N': A * X = B (No transpose) .br = 'T': A**T * X = B (Transpose) .br = 'C': A**H * X = B (Conjugate transpose) .TP 8 N (input) INTEGER The order of the matrix A. N >= 0. .TP 8 NRHS (input) INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. .TP 8 DL (input) COMPLEX array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A. .TP 8 D (input) COMPLEX array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A. .TP 8 DU (input) COMPLEX array, dimension (N-1) The (n-1) elements of the first superdiagonal of U. .TP 8 B (input/output) COMPLEX array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, B is overwritten by the solution matrix X. .TP 8 LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value scalapack-doc-1.5/man/manl/cpttrsv.l0100644000056400000620000000411406335610612017123 0ustar pfrauenfstaff.TH CPTTRSV l "12 May 1997" "modified LAPACK routine" "LAPACK routine (version 2.0)" .SH NAME CPTTRSV - solve one of the triangular systems L * X = B, or L**H * X = B, .SH SYNOPSIS .TP 20 SUBROUTINE CPTTRSV( UPLO, TRANS, N, NRHS, D, E, B, LDB, INFO ) .TP 20 .ti +4 CHARACTER UPLO, TRANS .TP 20 .ti +4 INTEGER INFO, LDB, N, NRHS .TP 20 .ti +4 REAL D( * ) .TP 20 .ti +4 COMPLEX B( LDB, * ), E( * ) .SH PURPOSE CPTTRSV solves one of the triangular systems L * X = B, or L**H * X = B, U * X = B, or U**H * X = B, .br where L or U is the Cholesky factor of a Hermitian positive definite tridiagonal matrix A such that .br A = U**H*D*U or A = L*D*L**H (computed by CPTTRF). .br .SH ARGUMENTS .TP 8 UPLO (input) CHARACTER*1 Specifies whether the superdiagonal or the subdiagonal of the tridiagonal matrix A is stored and the form of the factorization: .br = 'U': E is the superdiagonal of U, and A = U'*D*U; .br = 'L': E is the subdiagonal of L, and A = L*D*L'. (The two forms are equivalent if A is real.) .TP 8 TRANS (input) CHARACTER Specifies the form of the system of equations: .br = 'N': L * X = B (No transpose) .br = 'N': L * X = B (No transpose) .br = 'C': U**H * X = B (Conjugate transpose) .br = 'C': L**H * X = B (Conjugate transpose) .TP 8 N (input) INTEGER The order of the tridiagonal matrix A. N >= 0. .TP 8 NRHS (input) INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. .TP 8 D (input) REAL array, dimension (N) The n diagonal elements of the diagonal matrix D from the factorization computed by CPTTRF. .TP 8 E (input) COMPLEX array, dimension (N-1) The (n-1) off-diagonal elements of the unit bidiagonal factor U or L from the factorization computed by CPTTRF (see UPLO). .TP 8 B (input/output) COMPLEX array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X. .TP 8 LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value scalapack-doc-1.5/man/manl/ddbtf2.l0100644000056400000620000000464706335610612016576 0ustar pfrauenfstaff.SH NAME DDBTF2 - compute an LU factorization of a real m-by-n band matrix A without using partial pivoting with row interchanges .SH SYNOPSIS .TP 19 SUBROUTINE DDBTF2( M, N, KL, KU, AB, LDAB, INFO ) .TP 19 .ti +4 INTEGER INFO, KL, KU, LDAB, M, N .TP 19 .ti +4 DOUBLE PRECISION AB( LDAB, * ) .SH PURPOSE Ddbtrf computes an LU factorization of a real m-by-n band matrix A without using partial pivoting with row interchanges. This is the unblocked version of the algorithm, calling Level 2 BLAS. .SH ARGUMENTS .TP 8 M (input) INTEGER The number of rows of the matrix A. M >= 0. .TP 8 N (input) INTEGER The number of columns of the matrix A. N >= 0. .TP 8 KL (input) INTEGER The number of subdiagonals within the band of A. KL >= 0. .TP 8 KU (input) INTEGER The number of superdiagonals within the band of A. KU >= 0. .TP 8 AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N) On entry, the matrix A in band storage, in rows KL+1 to 2*KL+KU+1; rows 1 to KL of the array need not be set. The j-th column of A is stored in the j-th column of the array AB as follows: AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl) On exit, details of the factorization: U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. See below for further details. .TP 8 LDAB (input) INTEGER The leading dimension of the array AB. LDAB >= 2*KL+KU+1. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 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. .SH FURTHER DETAILS The band storage scheme is illustrated by the following example, when M = N = 6, KL = 2, KU = 1: .br On entry: On exit: .br * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 * a31 a42 a53 a64 * * m31 m42 m53 m64 * * Array elements marked * are not used by the routine; elements marked + need not be set on entry, but are required by the routine to store elements of U, because of fill-in resulting from the row .br interchanges. .br scalapack-doc-1.5/man/manl/ddbtrf.l0100644000056400000620000000436306335610612016671 0ustar pfrauenfstaff.SH NAME DDBTRF - compute an LU factorization of a real m-by-n band matrix A without using partial pivoting or row interchanges .SH SYNOPSIS .TP 19 SUBROUTINE DDBTRF( M, N, KL, KU, AB, LDAB, INFO ) .TP 19 .ti +4 INTEGER INFO, KL, KU, LDAB, M, N .TP 19 .ti +4 DOUBLE PRECISION AB( LDAB, * ) .SH PURPOSE Ddbtrf computes an LU factorization of a real m-by-n band matrix A without using partial pivoting or row interchanges. This is the blocked version of the algorithm, calling Level 3 BLAS. .SH ARGUMENTS .TP 8 M (input) INTEGER The number of rows of the matrix A. M >= 0. .TP 8 N (input) INTEGER The number of columns of the matrix A. N >= 0. .TP 8 KL (input) INTEGER The number of subdiagonals within the band of A. KL >= 0. .TP 8 KU (input) INTEGER The number of superdiagonals within the band of A. KU >= 0. .TP 8 AB (input/output) REAL array, dimension (LDAB,N) On entry, the matrix A in band storage, in rows KL+1 to 2*KL+KU+1; rows 1 to KL of the array need not be set. The j-th column of A is stored in the j-th column of the array AB as follows: AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl) On exit, details of the factorization: U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. See below for further details. .TP 8 LDAB (input) INTEGER The leading dimension of the array AB. LDAB >= 2*KL+KU+1. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 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. .SH FURTHER DETAILS The band storage scheme is illustrated by the following example, when M = N = 6, KL = 2, KU = 1: .br On entry: On exit: .br * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 * a31 a42 a53 a64 * * m31 m42 m53 m64 * * Array elements marked * are not used by the routine. .br scalapack-doc-1.5/man/manl/ddttrf.l0100644000056400000620000000333006335610612016704 0ustar pfrauenfstaff.TH DDTTRF l "12 May 1997" "modified LAPACK routine" "LAPACK routine (version 2.0)" .SH NAME DDTTRF - compute an LU factorization of a complex tridiagonal matrix A using elimination without partial pivoting .SH SYNOPSIS .TP 19 SUBROUTINE DDTTRF( N, DL, D, DU, INFO ) .TP 19 .ti +4 INTEGER INFO, N .TP 19 .ti +4 DOUBLE PRECISION D( * ), DL( * ), DU( * ) .SH PURPOSE DDTTRF computes an LU factorization of a complex tridiagonal matrix A using elimination without partial pivoting. The factorization has the form .br A = L * U .br where L is a product of unit lower bidiagonal .br matrices and U is upper triangular with nonzeros in only the main diagonal and first superdiagonal. .br .SH ARGUMENTS .TP 8 N (input) INTEGER The order of the matrix A. N >= 0. .TP 8 DL (input/output) COMPLEX array, dimension (N-1) On entry, DL must contain the (n-1) subdiagonal elements of A. On exit, DL is overwritten by the (n-1) multipliers that define the matrix L from the LU factorization of A. .TP 8 D (input/output) COMPLEX array, dimension (N) On entry, D must contain the diagonal elements of A. On exit, D is overwritten by the n diagonal elements of the upper triangular matrix U from the LU factorization of A. .TP 8 DU (input/output) COMPLEX array, dimension (N-1) On entry, DU must contain the (n-1) superdiagonal elements of A. On exit, DU is overwritten by the (n-1) elements of the first superdiagonal of U. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 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. scalapack-doc-1.5/man/manl/ddttrsv.l0100644000056400000620000000353506335610612017116 0ustar pfrauenfstaff.TH DDTTRSV l "12 May 1997" "modified LAPACK routine" "LAPACK routine (version 2.0)" .SH NAME DDTTRSV - solve one of the systems of equations L * X = B, L**T * X = B, or L**H * X = B, .SH SYNOPSIS .TP 20 SUBROUTINE DDTTRSV( UPLO, TRANS, N, NRHS, DL, D, DU, B, LDB, INFO ) .TP 20 .ti +4 CHARACTER UPLO, TRANS .TP 20 .ti +4 INTEGER INFO, LDB, N, NRHS .TP 20 .ti +4 DOUBLE PRECISION B( LDB, * ), D( * ), DL( * ), DU( * ) .SH PURPOSE DDTTRSV solves one of the systems of equations L * X = B, L**T * X = B, or L**H * X = B, U * X = B, U**T * X = B, or U**H * X = B, .br with factors of the tridiagonal matrix A from the LU factorization computed by DDTTRF. .br .SH ARGUMENTS .TP 8 UPLO (input) CHARACTER*1 Specifies whether to solve with L or U. .TP 8 TRANS (input) CHARACTER Specifies the form of the system of equations: .br = 'N': A * X = B (No transpose) .br = 'T': A**T * X = B (Transpose) .br = 'C': A**H * X = B (Conjugate transpose) .TP 8 N (input) INTEGER The order of the matrix A. N >= 0. .TP 8 NRHS (input) INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. .TP 8 DL (input) COMPLEX array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A. .TP 8 D (input) COMPLEX array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A. .TP 8 DU (input) COMPLEX array, dimension (N-1) The (n-1) elements of the first superdiagonal of U. .TP 8 B (input/output) COMPLEX array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, B is overwritten by the solution matrix X. .TP 8 LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value scalapack-doc-1.5/man/manl/dlamsh.l0100644000056400000620000000452406335610612016673 0ustar pfrauenfstaff.TH DLAMSH l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME DLAMSH - send multiple shifts through a small (single node) matrix to see how consecutive small subdiagonal elements are modified by subsequent shifts in an effort to maximize the number of bulges that can be sent through .SH SYNOPSIS .TP 18 SUBROUTINE DLAMSH ( S, LDS, NBULGE, JBLK, H, LDH, N, ULP ) .TP 18 .ti +4 INTEGER LDS, NBULGE, JBLK, LDH, N .TP 18 .ti +4 DOUBLE PRECISION ULP .TP 18 .ti +4 DOUBLE PRECISION S(LDS,*), H(LDH,*) .SH PURPOSE DLAMSH sends multiple shifts through a small (single node) matrix to see how consecutive small subdiagonal elements are modified by subsequent shifts in an effort to maximize the number of bulges that can be sent through. DLAMSH should only be called when there are multiple shifts/bulges (NBULGE > 1) and the first shift is starting in the middle of an unreduced Hessenberg matrix because of two or more consecutive small subdiagonal elements. .br .SH ARGUMENTS .TP 8 S (local input/output) DOUBLE PRECISION array, (LDS,*) On entry, the matrix of shifts. Only the 2x2 diagonal of S is referenced. It is assumed that S has JBLK double shifts (size 2). On exit, the data is rearranged in the best order for applying. .TP 8 LDS (local input) INTEGER On entry, the leading dimension of S. Unchanged on exit. 1 < NBULGE <= JBLK <= LDS/2 .TP 8 NBULGE (local input/output) INTEGER On entry, the number of bulges to send through H ( >1 ). NBULGE should be less than the maximum determined (JBLK). 1 < NBULGE <= JBLK <= LDS/2 On exit, the maximum number of bulges that can be sent through. .TP 8 JBLK (local input) INTEGER On entry, the number of shifts determined for S. Unchanged on exit. .TP 8 H (local input/output) DOUBLE PRECISION array (LDH,N) On entry, the local matrix to apply the shifts on. H should be aligned so that the starting row is 2. On exit, the data is destroyed. .TP 8 LDS (local input) INTEGER On entry, the leading dimension of S. Unchanged on exit. .TP 8 N (local input) INTEGER On entry, the size of H. If all the bulges are expected to go through, N should be at least 4*NBULGE+2. Otherwise, NBULGE may be reduced by this routine. .TP 8 ULP (local input) DOUBLE PRECISION On entry, machine precision Unchanged on exit. Implemented by: G. Henry, May 1, 1997 scalapack-doc-1.5/man/manl/dlaref.l0100644000056400000620000000624606335610612016663 0ustar pfrauenfstaff.TH DLAREF l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME DLAREF - applie one or several Householder reflectors of size 3 to one or two matrices (if column is specified) on either their rows or columns .SH SYNOPSIS .TP 19 SUBROUTINE DLAREF( TYPE, A, LDA, WANTZ, Z, LDZ, BLOCK, IROW1, ICOL1, ISTART, ISTOP, ITMP1, ITMP2, LILOZ, LIHIZ, VECS, V2, V3, T1, T2, T3 ) .TP 19 .ti +4 LOGICAL BLOCK, WANTZ .TP 19 .ti +4 CHARACTER TYPE .TP 19 .ti +4 INTEGER ICOL1, IROW1, ISTART, ISTOP, ITMP1, ITMP2, LDA, LDZ, LIHIZ, LILOZ .TP 19 .ti +4 DOUBLE PRECISION T1, T2, T3, V2, V3 .TP 19 .ti +4 DOUBLE PRECISION A( LDA, * ), VECS( * ), Z( LDZ, * ) .SH PURPOSE DLAREF applies one or several Householder reflectors of size 3 to one or two matrices (if column is specified) on either their rows or columns. .SH ARGUMENTS .TP 8 TYPE (global input) CHARACTER*1 If 'R': Apply reflectors to the rows of the matrix (apply from left) Otherwise: Apply reflectors to the columns of the matrix Unchanged on exit. .TP 8 A (global input/output) DOUBLE PRECISION array, (LDA,*) On entry, the matrix to receive the reflections. The updated matrix on exit. .TP 8 LDA (local input) INTEGER On entry, the leading dimension of A. Unchanged on exit. .TP 8 WANTZ (global input) LOGICAL If .TRUE., then apply any column reflections to Z as well. If .FALSE., then do no additional work on Z. .TP 8 Z (global input/output) DOUBLE PRECISION array, (LDZ,*) On entry, the second matrix to receive column reflections. This is changed only if WANTZ is set. .TP 8 LDZ (local input) INTEGER On entry, the leading dimension of Z. Unchanged on exit. .TP 8 BLOCK (global input) LOGICAL If .TRUE., then apply several reflectors at once and read their data from the VECS array. If .FALSE., apply the single reflector given by V2, V3, T1, T2, and T3. .TP 8 IROW1 (local input/output) INTEGER On entry, the local row element of A. Undefined on output. .TP 8 ICOL1 (local input/output) INTEGER On entry, the local column element of A. Undefined on output. .TP 8 ISTART (global input) INTEGER Specifies the "number" of the first reflector. This is used as an index into VECS if BLOCK is set. ISTART is ignored if BLOCK is .FALSE.. .TP 8 ISTOP (global input) INTEGER Specifies the "number" of the last reflector. This is used as an index into VECS if BLOCK is set. ISTOP is ignored if BLOCK is .FALSE.. .TP 8 ITMP1 (local input) INTEGER Starting range into A. For rows, this is the local first column. For columns, this is the local first row. .TP 8 ITMP2 (local input) INTEGER Ending range into A. For rows, this is the local last column. For columns, this is the local last row. LILOZ LIHIZ (local input) INTEGER These serve the same purpose as ITMP1,ITMP2 but for Z when WANTZ is set. .TP 8 VECS (global input) DOUBLE PRECISION array of size 3*N (matrix size) This holds the size 3 reflectors one after another and this is only accessed when BLOCK is .TRUE. V2 V3 T1 T2 T3 (global input/output) DOUBLE PRECISION This holds information on a single size 3 Householder reflector and is read when BLOCK is .FALSE., and overwritten when BLOCK is .TRUE. Implemented by: G. Henry, May 1, 1997 scalapack-doc-1.5/man/manl/dlasorte.l0100644000056400000620000000266506335610613017245 0ustar pfrauenfstaff.TH DLASORTE l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME DLASORTE - sort eigenpairs so that real eigenpairs are together and complex are together .SH SYNOPSIS .TP 20 SUBROUTINE DLASORTE ( S, LDS, J, OUT, INFO ) .TP 20 .ti +4 INTEGER INFO, J, LDS .TP 20 .ti +4 DOUBLE PRECISION OUT( J, * ), S( LDS, * ) .SH PURPOSE DLASORTE sorts eigenpairs so that real eigenpairs are together and complex are together. This way one can employ 2x2 shifts easily since every 2nd subdiagonal is guaranteed to be zero. .br This routine does no parallel work and makes no calls. .br .SH ARGUMENTS .TP 8 S (local input/output) DOUBLE PRECISION array, dimension LDS On entry, a matrix already in Schur form. On exit, the diagonal blocks of S have been rewritten to pair the eigenvalues. The resulting matrix is no longer similar to the input. .TP 8 LDS (local input) INTEGER On entry, the leading dimension of the local array S. Unchanged on exit. .TP 8 J (local input) INTEGER On entry, the order of the matrix S. Unchanged on exit. .TP 8 OUT (local input/output) DOUBLE PRECISION array, dimension Jx2 This is the work buffer required by this routine. .TP 8 INFO (local input) INTEGER This is set if the input matrix had an odd number of real eigenvalues and things couldn't be paired or if the input matrix S was not originally in Schur form. 0 indicates successful completion. Implemented by: G. Henry, May 1, 1997 scalapack-doc-1.5/man/manl/dlasrt2.l0100644000056400000620000000266206335610613017000 0ustar pfrauenfstaff.TH DLASRT2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME DLASRT2 - the numbers in D in increasing order (if ID = 'I') or in decreasing order (if ID = 'D' ) .SH SYNOPSIS .TP 20 SUBROUTINE DLASRT2( ID, N, D, KEY, INFO ) .TP 20 .ti +4 CHARACTER ID .TP 20 .ti +4 INTEGER INFO, N .TP 20 .ti +4 INTEGER KEY( * ) .TP 20 .ti +4 DOUBLE PRECISION D( * ) .SH PURPOSE Sort the numbers in D in increasing order (if ID = 'I') or in decreasing order (if ID = 'D' ). Use Quick Sort, reverting to Insertion sort on arrays of .br size <= 20. Dimension of STACK limits N to about 2**32. .br .SH ARGUMENTS .TP 8 ID (input) CHARACTER*1 = 'I': sort D in increasing order; .br = 'D': sort D in decreasing order. .TP 8 N (input) INTEGER The length of the array D. .TP 8 D (input/output) DOUBLE PRECISION array, dimension (N) On entry, the array to be sorted. On exit, D has been sorted into increasing order (D(1) <= ... <= D(N) ) or into decreasing order (D(1) >= ... >= D(N) ), depending on ID. .TP 8 KEY (input/output) INTEGER array, dimension (N) On entry, KEY contains a key to each of the entries in D() Typically, KEY(I) = I for all I On exit, KEY is permuted in exactly the same manner as D() was permuted from input to output Therefore, if KEY(I) = I for all I upon input, then D_out(I) = D_in(KEY(I)) .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value scalapack-doc-1.5/man/manl/dpttrsv.l0100644000056400000620000000317406335610613017132 0ustar pfrauenfstaff.TH DPTTRSV l "12 May 1997" "modified LAPACK routine" "LAPACK routine (version 2.0)" .SH NAME DPTTRSV - solve one of the triangular systems L**T* X = B, or L * X = B, .SH SYNOPSIS .TP 20 SUBROUTINE DPTTRSV( TRANS, N, NRHS, D, E, B, LDB, INFO ) .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER INFO, LDB, N, NRHS .TP 20 .ti +4 DOUBLE PRECISION D( * ) .TP 20 .ti +4 DOUBLE PRECISION B( LDB, * ), E( * ) .SH PURPOSE DPTTRSV solves one of the triangular systems L**T* X = B, or L * X = B, where L is the Cholesky factor of a Hermitian positive .br definite tridiagonal matrix A such that .br A = L*D*L**H (computed by DPTTRF). .br .SH ARGUMENTS .TP 8 TRANS (input) CHARACTER Specifies the form of the system of equations: .br = 'N': L * X = B (No transpose) .br = 'T': L**T * X = B (Transpose) .TP 8 N (input) INTEGER The order of the tridiagonal matrix A. N >= 0. .TP 8 NRHS (input) INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. .TP 8 D (input) REAL array, dimension (N) The n diagonal elements of the diagonal matrix D from the factorization computed by DPTTRF. .TP 8 E (input) COMPLEX array, dimension (N-1) The (n-1) off-diagonal elements of the unit bidiagonal factor U or L from the factorization computed by DPTTRF (see UPLO). .TP 8 B (input/output) COMPLEX array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X. .TP 8 LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value scalapack-doc-1.5/man/manl/dstein2.l0100644000056400000620000000702706335610613016775 0ustar pfrauenfstaff.TH DSTEIN2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME DSTEIN2 - compute the eigenvectors of a real symmetric tridiagonal matrix T corresponding to specified eigenvalues, using inverse iteration .SH SYNOPSIS .TP 20 SUBROUTINE DSTEIN2( N, D, E, M, W, IBLOCK, ISPLIT, ORFAC, Z, LDZ, WORK, IWORK, IFAIL, INFO ) .TP 20 .ti +4 INTEGER INFO, LDZ, M, N .TP 20 .ti +4 DOUBLE PRECISION ORFAC .TP 20 .ti +4 INTEGER IBLOCK( * ), IFAIL( * ), ISPLIT( * ), IWORK( * ) .TP 20 .ti +4 DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * ) .SH PURPOSE DSTEIN2 computes the eigenvectors of a real symmetric tridiagonal matrix T corresponding to specified eigenvalues, using inverse iteration. The maximum number of iterations allowed for each eigenvector is specified by an internal parameter MAXITS (currently set to 5). .SH ARGUMENTS .TP 8 N (input) INTEGER The order of the matrix. N >= 0. .TP 8 D (input) DOUBLE PRECISION array, dimension (N) The n diagonal elements of the tridiagonal matrix T. .TP 8 E (input) DOUBLE PRECISION array, dimension (N) The (n-1) subdiagonal elements of the tridiagonal matrix T, in elements 1 to N-1. E(N) need not be set. .TP 8 M (input) INTEGER The number of eigenvectors to be found. 0 <= M <= N. .TP 8 W (input) DOUBLE PRECISION array, dimension (N) The first M elements of W contain the eigenvalues for which eigenvectors are to be computed. The eigenvalues should be grouped by split-off block and ordered from smallest to largest within the block. ( The output array W from DSTEBZ with ORDER = 'B' is expected here. ) .TP 8 IBLOCK (input) INTEGER array, dimension (N) The submatrix indices associated with the corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to the first submatrix from the top, =2 if W(i) belongs to the second submatrix, etc. ( The output array IBLOCK from DSTEBZ is expected here. ) .TP 8 ISPLIT (input) INTEGER array, dimension (N) The splitting points, at which T breaks up into submatrices. The first submatrix consists of rows/columns 1 to ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1 through ISPLIT( 2 ), etc. ( The output array ISPLIT from DSTEBZ is expected here. ) .TP 8 ORFAC (input) DOUBLE PRECISION ORFAC specifies which eigenvectors should be orthogonalized. Eigenvectors that correspond to eigenvalues which are within ORFAC*||T|| of each other are to be orthogonalized. .TP 8 Z (output) DOUBLE PRECISION array, dimension (LDZ, M) The computed eigenvectors. The eigenvector associated with the eigenvalue W(i) is stored in the i-th column of Z. Any vector which fails to converge is set to its current iterate after MAXITS iterations. .TP 8 LDZ (input) INTEGER The leading dimension of the array Z. LDZ >= max(1,N). .TP 8 WORK (workspace) DOUBLE PRECISION array, dimension (5*N) .TP 8 IWORK (workspace) INTEGER array, dimension (N) .TP 8 IFAIL (output) INTEGER array, dimension (M) On normal exit, all elements of IFAIL are zero. If one or more eigenvectors fail to converge after MAXITS iterations, then their indices are stored in array IFAIL. .TP 8 INFO (output) INTEGER = 0: successful exit. .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 0: if INFO = i, then i eigenvectors failed to converge in MAXITS iterations. Their indices are stored in array IFAIL. .SH PARAMETERS .TP 8 MAXITS INTEGER, default = 5 The maximum number of iterations performed. .TP 8 EXTRA INTEGER, default = 2 The number of iterations performed after norm growth criterion is satisfied, should be at least 1. scalapack-doc-1.5/man/manl/dsteqr2.l0100644000056400000620000000523306335610613017006 0ustar pfrauenfstaff.TH DSTEQR2 l "12 May 1997" "modified LAPACK routine" "LAPACK routine (version 2.0)" .SH NAME DSTEQR2 - i a modified version of LAPACK routine DSTEQR .SH SYNOPSIS .TP 20 SUBROUTINE DSTEQR2( COMPZ, N, D, E, Z, LDZ, NR, WORK, INFO ) .TP 20 .ti +4 CHARACTER COMPZ .TP 20 .ti +4 INTEGER INFO, LDZ, N, NR .TP 20 .ti +4 DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * ) .SH PURPOSE DSTEQR2 is a modified version of LAPACK routine DSTEQR. DSTEQR2 computes all eigenvalues and, optionally, eigenvectors of a symmetric tridiagonal matrix using the implicit QL or QR method. DSTEQR2 is modified from DSTEQR to allow each ScaLAPACK process running DSTEQR2 to perform updates on a distributed matrix Q. Proper usage of DSTEQR2 can be gleaned from examination of ScaLAPACK's PDSYEV. .br .SH ARGUMENTS .TP 8 COMPZ (input) CHARACTER*1 = 'N': Compute eigenvalues only. .br = 'I': Compute eigenvalues and eigenvectors of the tridiagonal matrix. Z must be initialized to the identity matrix by PDLASET or DLASET prior to entering this subroutine. .TP 8 N (input) INTEGER The order of the matrix. N >= 0. .TP 8 D (input/output) DOUBLE PRECISION array, dimension (N) On entry, the diagonal elements of the tridiagonal matrix. On exit, if INFO = 0, the eigenvalues in ascending order. .TP 8 E (input/output) DOUBLE PRECISION array, dimension (N-1) On entry, the (n-1) subdiagonal elements of the tridiagonal matrix. On exit, E has been destroyed. .TP 8 Z (local input/local output) DOUBLE PRECISION array, global dimension (N, N), local dimension (LDZ, NR). On entry, if COMPZ = 'V', then Z contains the orthogonal matrix used in the reduction to tridiagonal form. On exit, if INFO = 0, then if COMPZ = 'V', Z contains the orthonormal eigenvectors of the original symmetric matrix, and if COMPZ = 'I', Z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix. If COMPZ = 'N', then Z is not referenced. .TP 8 LDZ (input) INTEGER The leading dimension of the array Z. LDZ >= 1, and if eigenvectors are desired, then LDZ >= max(1,N). .TP 8 NR (input) INTEGER NR = MAX(1, NUMROC( N, NB, MYPROW, 0, NPROCS ) ). If COMPZ = 'N', then NR is not referenced. .TP 8 WORK (workspace) DOUBLE PRECISION array, dimension (max(1,2*N-2)) If COMPZ = 'N', then WORK is not referenced. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 0: the algorithm has failed to find all the eigenvalues in a total of 30*N iterations; if INFO = i, then i elements of E have not converged to zero; on exit, D and E contain the elements of a symmetric tridiagonal matrix which is orthogonally similar to the original matrix. scalapack-doc-1.5/man/manl/pcdbsv.l0100644000056400000620000000141106335610613016675 0ustar pfrauenfstaff.TH PCDBSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCDBSV - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 19 SUBROUTINE PCDBSV( N, BWL, BWU, NRHS, A, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 INTEGER BWL, BWU, IB, INFO, JA, LWORK, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 COMPLEX A( * ), B( * ), WORK( * ) .SH PURPOSE PCDBSV solves a system of linear equations where A(1:N, JA:JA+N-1) is an N-by-N complex .br banded diagonally dominant-like distributed .br matrix with bandwidth BWL, BWU. .br Gaussian elimination without pivoting .br is used to factor a reordering .br of the matrix into L U. .br See PCDBTRF and PCDBTRS for details. .br scalapack-doc-1.5/man/manl/pcdbtrf.l0100644000056400000620000000214306335610613017043 0ustar pfrauenfstaff.TH PCDBTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCDBTRF - compute a LU factorization of an N-by-N complex banded diagonally dominant-like distributed matrix with bandwidth BWL, BWU .SH SYNOPSIS .TP 20 SUBROUTINE PCDBTRF( N, BWL, BWU, A, JA, DESCA, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER BWL, BWU, INFO, JA, LAF, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ), AF( * ), WORK( * ) .SH PURPOSE PCDBTRF computes a LU factorization of an N-by-N complex banded diagonally dominant-like distributed matrix with bandwidth BWL, BWU: A(1:N, JA:JA+N-1). Reordering is used to increase parallelism in the factorization. This reordering results in factors that are DIFFERENT from those produced by equivalent sequential codes. These factors cannot be used directly by users; however, they can be used in .br subsequent calls to PCDBTRS to solve linear systems. .br The factorization has the form .br P A(1:N, JA:JA+N-1) P^T = L U .br where U is a banded upper triangular matrix and L is banded lower triangular, and P is a permutation matrix. .br scalapack-doc-1.5/man/manl/pcdbtrs.l0100644000056400000620000000166306335610613017066 0ustar pfrauenfstaff.TH PCDBTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCDBTRS - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 20 SUBROUTINE PCDBTRS( TRANS, N, BWL, BWU, NRHS, A, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER BWL, BWU, IB, INFO, JA, LAF, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX A( * ), AF( * ), B( * ), WORK( * ) .SH PURPOSE PCDBTRS solves a system of linear equations or .br A(1:N, JA:JA+N-1)' * X = B(IB:IB+N-1, 1:NRHS) .br where A(1:N, JA:JA+N-1) is the matrix used to produce the factors stored in A(1:N,JA:JA+N-1) and AF by PCDBTRF. .br A(1:N, JA:JA+N-1) is an N-by-N complex .br banded diagonally dominant-like distributed .br matrix with bandwidth BWL, BWU. .br Routine PCDBTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pcdbtrsv.l0100644000056400000620000000220706335610613017247 0ustar pfrauenfstaff.TH PCDBTRSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCDBTRSV - solve a banded triangular system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 21 SUBROUTINE PCDBTRSV( UPLO, TRANS, N, BWL, BWU, NRHS, A, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 21 .ti +4 CHARACTER TRANS, UPLO .TP 21 .ti +4 INTEGER BWL, BWU, IB, INFO, JA, LAF, LWORK, N, NRHS .TP 21 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 21 .ti +4 COMPLEX A( * ), AF( * ), B( * ), WORK( * ) .SH PURPOSE PCDBTRSV solves a banded triangular system of linear equations or .br A(1:N, JA:JA+N-1)^H * X = B(IB:IB+N-1, 1:NRHS) where A(1:N, JA:JA+N-1) is a banded .br triangular matrix factor produced by the .br Gaussian elimination code PC@(dom_pre)BTRF .br and is stored in A(1:N,JA:JA+N-1) and AF. .br The matrix stored in A(1:N, JA:JA+N-1) is either .br upper or lower triangular according to UPLO, .br and the choice of solving A(1:N, JA:JA+N-1) or A(1:N, JA:JA+N-1)^H is dictated by the user by the parameter TRANS. .br Routine PCDBTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pcdtsv.l0100644000056400000620000000137406335610613016727 0ustar pfrauenfstaff.TH PCDTSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCDTSV - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 19 SUBROUTINE PCDTSV( N, NRHS, DL, D, DU, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 INTEGER IB, INFO, JA, LWORK, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 COMPLEX B( * ), D( * ), DL( * ), DU( * ), WORK( * ) .SH PURPOSE PCDTSV solves a system of linear equations where A(1:N, JA:JA+N-1) is an N-by-N complex .br tridiagonal diagonally dominant-like distributed .br matrix. .br Gaussian elimination without pivoting .br is used to factor a reordering .br of the matrix into L U. .br See PCDTTRF and PCDTTRS for details. .br scalapack-doc-1.5/man/manl/pcdttrf.l0100644000056400000620000000213606335610613017067 0ustar pfrauenfstaff.TH PCDTTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCDTTRF - compute a LU factorization of an N-by-N complex tridiagonal diagonally dominant-like distributed matrix A(1:N, JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PCDTTRF( N, DL, D, DU, JA, DESCA, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER INFO, JA, LAF, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX AF( * ), D( * ), DL( * ), DU( * ), WORK( * ) .SH PURPOSE PCDTTRF computes a LU factorization of an N-by-N complex tridiagonal diagonally dominant-like distributed matrix A(1:N, JA:JA+N-1). Reordering is used to increase parallelism in the factorization. This reordering results in factors that are DIFFERENT from those produced by equivalent sequential codes. These factors cannot be used directly by users; however, they can be used in .br subsequent calls to PCDTTRS to solve linear systems. .br The factorization has the form .br P A(1:N, JA:JA+N-1) P^T = L U .br where U is a tridiagonal upper triangular matrix and L is tridiagonal lower triangular, and P is a permutation matrix. .br scalapack-doc-1.5/man/manl/pcdttrs.l0100644000056400000620000000164606335610613017111 0ustar pfrauenfstaff.TH PCDTTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCDTTRS - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 20 SUBROUTINE PCDTTRS( TRANS, N, NRHS, DL, D, DU, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER IB, INFO, JA, LAF, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX AF( * ), B( * ), D( * ), DL( * ), DU( * ), WORK( * ) .SH PURPOSE PCDTTRS solves a system of linear equations or .br A(1:N, JA:JA+N-1)' * X = B(IB:IB+N-1, 1:NRHS) .br where A(1:N, JA:JA+N-1) is the matrix used to produce the factors stored in A(1:N,JA:JA+N-1) and AF by PCDTTRF. .br A(1:N, JA:JA+N-1) is an N-by-N complex .br tridiagonal diagonally dominant-like distributed .br matrix. .br Routine PCDTTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pcdttrsv.l0100644000056400000620000000223406335610613017271 0ustar pfrauenfstaff.TH PCDTTRSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCDTTRSV - solve a tridiagonal triangular system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 21 SUBROUTINE PCDTTRSV( UPLO, TRANS, N, NRHS, DL, D, DU, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 21 .ti +4 CHARACTER TRANS, UPLO .TP 21 .ti +4 INTEGER IB, INFO, JA, LAF, LWORK, N, NRHS .TP 21 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 21 .ti +4 COMPLEX AF( * ), B( * ), D( * ), DL( * ), DU( * ), WORK( * ) .SH PURPOSE PCDTTRSV solves a tridiagonal triangular system of linear equations or .br A(1:N, JA:JA+N-1)^H * X = B(IB:IB+N-1, 1:NRHS) where A(1:N, JA:JA+N-1) is a tridiagonal .br triangular matrix factor produced by the .br Gaussian elimination code PC@(dom_pre)TTRF .br and is stored in A(1:N,JA:JA+N-1) and AF. .br The matrix stored in A(1:N, JA:JA+N-1) is either .br upper or lower triangular according to UPLO, .br and the choice of solving A(1:N, JA:JA+N-1) or A(1:N, JA:JA+N-1)^H is dictated by the user by the parameter TRANS. .br Routine PCDTTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pcgbsv.l0100644000056400000620000000140006335610613016676 0ustar pfrauenfstaff.TH PCGBSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGBSV - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 19 SUBROUTINE PCGBSV( N, BWL, BWU, NRHS, A, JA, DESCA, IPIV, B, IB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 INTEGER BWL, BWU, IB, INFO, JA, LWORK, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ), IPIV( * ) .TP 19 .ti +4 COMPLEX A( * ), B( * ), WORK( * ) .SH PURPOSE PCGBSV solves a system of linear equations where A(1:N, JA:JA+N-1) is an N-by-N complex .br banded distributed .br matrix with bandwidth BWL, BWU. .br Gaussian elimination with pivoting .br is used to factor a reordering .br of the matrix into P L U. .br See PCGBTRF and PCGBTRS for details. .br scalapack-doc-1.5/man/manl/pcgbtrf.l0100644000056400000620000000237306335610613017053 0ustar pfrauenfstaff.TH PCGBTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGBTRF - compute a LU factorization of an N-by-N complex banded distributed matrix with bandwidth BWL, BWU .SH SYNOPSIS .TP 20 SUBROUTINE PCGBTRF( N, BWL, BWU, A, JA, DESCA, IPIV, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER BWL, BWU, INFO, JA, LAF, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ) .TP 20 .ti +4 COMPLEX A( * ), AF( * ), WORK( * ) .SH PURPOSE PCGBTRF computes a LU factorization of an N-by-N complex banded distributed matrix with bandwidth BWL, BWU: A(1:N, JA:JA+N-1). Reordering is used to increase parallelism in the factorization. This reordering results in factors that are DIFFERENT from those produced by equivalent sequential codes. These factors cannot be used directly by users; however, they can be used in .br subsequent calls to PCGBTRS to solve linear systems. .br The factorization has the form .br P A(1:N, JA:JA+N-1) Q = L U .br where U is a banded upper triangular matrix and L is banded lower triangular, and P and Q are permutation matrices. .br The matrix Q represents reordering of columns .br for parallelism's sake, while P represents .br reordering of rows for numerical stability using .br classic partial pivoting. .br scalapack-doc-1.5/man/manl/pcgbtrs.l0100644000056400000620000000165106335610614017067 0ustar pfrauenfstaff.TH PCGBTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGBTRS - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 20 SUBROUTINE PCGBTRS( TRANS, N, BWL, BWU, NRHS, A, JA, DESCA, IPIV, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER BWU, BWL, IB, INFO, JA, LAF, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ), IPIV(*) .TP 20 .ti +4 COMPLEX A( * ), AF( * ), B( * ), WORK( * ) .SH PURPOSE PCGBTRS solves a system of linear equations or .br A(1:N, JA:JA+N-1)' * X = B(IB:IB+N-1, 1:NRHS) .br where A(1:N, JA:JA+N-1) is the matrix used to produce the factors stored in A(1:N,JA:JA+N-1) and AF by PCGBTRF. .br A(1:N, JA:JA+N-1) is an N-by-N complex .br banded distributed .br matrix with bandwidth BWL, BWU. .br Routine PCGBTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pcgebd2.l0100644000056400000620000002224206335610614016730 0ustar pfrauenfstaff.TH PCGEBD2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCGEBD2 - reduce a complex general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form B by an unitary transformation .SH SYNOPSIS .TP 20 SUBROUTINE PCGEBD2( M, N, A, IA, JA, DESCA, D, E, TAUQ, TAUP, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL D( * ), E( * ) .TP 20 .ti +4 COMPLEX A( * ), TAUP( * ), TAUQ( * ), WORK( * ) .SH PURPOSE PCGEBD2 reduces a complex general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form B by an unitary transformation: Q' * sub( A ) * P = B. If M >= N, B is upper bidiagonal; if M < N, B is lower bidiagonal. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the general distributed matrix sub( A ). On exit, if M >= N, the diagonal and the first superdiagonal of sub( A ) are overwritten with the upper bidiagonal matrix B; the elements below the diagonal, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. If M < N, the diagonal and the first subdiagonal are overwritten with the lower bidiagonal matrix B; the elements below the first subdiagonal, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) REAL array, dimension LOCc(JA+MIN(M,N)-1) if M >= N; LOCr(IA+MIN(M,N)-1) otherwise. The distributed diagonal elements of the bidiagonal matrix B: D(i) = A(i,i). D is tied to the distributed matrix A. .TP 8 E (local output) REAL array, dimension LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-2) otherwise. The distributed off-diagonal elements of the bidiagonal distributed matrix B: if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. E is tied to the distributed matrix A. .TP 8 TAUQ (local output) COMPLEX array dimension LOCc(JA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the unitary matrix Q. TAUQ is tied to the distributed matrix A. See Further Details. TAUP (local output) COMPLEX array, dimension LOCr(IA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the unitary matrix P. TAUP is tied to the distributed matrix A. See Further Details. WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MAX( MpA0, NqA0 ) where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ) IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, NB, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+IROFFA, NB, MYCOL, IACOL, NPCOL ). INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrices Q and P are represented as products of elementary reflectors: .br If m >= n, .br Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) Each H(i) and G(i) has the form: .br H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are complex scalars, and v and u are complex vectors; .br v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(ia+i:ia+m-1,ja+i-1); .br u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(ia+i-1,ja+i+1:ja+n-1); .br tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br If m < n, .br Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) Each H(i) and G(i) has the form: .br H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are complex scalars, and v and u are complex vectors; .br v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(ia+i+1:ia+m-1,ja+i-1); .br u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(ia+i-1,ja+i:ja+n-1); .br tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br The contents of sub( A ) on exit are illustrated by the following examples: .br m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) ( v1 v2 v3 v4 v5 ) .br where d and e denote diagonal and off-diagonal elements of B, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expressions should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) .br scalapack-doc-1.5/man/manl/pcgebrd.l0100644000056400000620000002226206335610614017032 0ustar pfrauenfstaff.TH PCGEBRD l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGEBRD - reduce a complex general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form B by an unitary transformation .SH SYNOPSIS .TP 20 SUBROUTINE PCGEBRD( M, N, A, IA, JA, DESCA, D, E, TAUQ, TAUP, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL D( * ), E( * ) .TP 20 .ti +4 COMPLEX A( * ), TAUP( * ), TAUQ( * ), WORK( * ) .SH PURPOSE PCGEBRD reduces a complex general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form B by an unitary transformation: Q' * sub( A ) * P = B. If M >= N, B is upper bidiagonal; if M < N, B is lower bidiagonal. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the general distributed matrix sub( A ). On exit, if M >= N, the diagonal and the first superdiagonal of sub( A ) are overwritten with the upper bidiagonal matrix B; the elements below the diagonal, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. If M < N, the diagonal and the first subdiagonal are overwritten with the lower bidiagonal matrix B; the elements below the first subdiagonal, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) REAL array, dimension LOCc(JA+MIN(M,N)-1) if M >= N; LOCr(IA+MIN(M,N)-1) otherwise. The distributed diagonal elements of the bidiagonal matrix B: D(i) = A(i,i). D is tied to the distributed matrix A. .TP 8 E (local output) REAL array, dimension LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-2) otherwise. The distributed off-diagonal elements of the bidiagonal distributed matrix B: if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. E is tied to the distributed matrix A. .TP 8 TAUQ (local output) COMPLEX array dimension LOCc(JA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the unitary matrix Q. TAUQ is tied to the distributed matrix A. See Further Details. TAUP (local output) COMPLEX array, dimension LOCr(IA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the unitary matrix P. TAUP is tied to the distributed matrix A. See Further Details. WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK( 1 ) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB*( MpA0 + NqA0 + 1 ) + NqA0 where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ), ICOFFA = MOD( JA-1, NB ), IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, NB, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB, MYCOL, IACOL, NPCOL ). INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrices Q and P are represented as products of elementary reflectors: .br If m >= n, .br Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) Each H(i) and G(i) has the form: .br H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are complex scalars, and v and u are complex vectors; .br v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(ia+i:ia+m-1,ja+i-1); .br u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(ia+i-1,ja+i+1:ja+n-1); .br tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br If m < n, .br Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) Each H(i) and G(i) has the form: .br H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are complex scalars, and v and u are complex vectors; .br v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(ia+i+1:ia+m-1,ja+i-1); .br u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(ia+i-1,ja+i:ja+n-1); .br tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br The contents of sub( A ) on exit are illustrated by the following examples: .br m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) ( v1 v2 v3 v4 v5 ) .br where d and e denote diagonal and off-diagonal elements of B, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expressions should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) .br scalapack-doc-1.5/man/manl/pcgecon.l0100644000056400000620000001541706335610614017046 0ustar pfrauenfstaff.TH PCGECON l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGECON - estimate the reciprocal of the condition number of a general distributed complex matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm, using the LU factorization computed by PCGETRF .SH SYNOPSIS .TP 20 SUBROUTINE PCGECON( NORM, N, A, IA, JA, DESCA, ANORM, RCOND, WORK, LWORK, RWORK, LRWORK, INFO ) .TP 20 .ti +4 CHARACTER NORM .TP 20 .ti +4 INTEGER IA, INFO, JA, LRWORK, LWORK, N .TP 20 .ti +4 REAL ANORM, RCOND .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL RWORK( * ) .TP 20 .ti +4 COMPLEX A( * ), WORK( * ) .SH PURPOSE PCGECON estimates the reciprocal of the condition number of a general distributed complex matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm, using the LU factorization computed by PCGETRF. An estimate is obtained for norm(inv(A(IA:IA+N-1,JA:JA+N-1))), and the reciprocal of the condition number is computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies whether the 1-norm condition number or the infinity-norm condition number is required: .br = '1' or 'O': 1-norm .br = 'I': Infinity-norm .TP 8 N (global input) INTEGER .br The order of the distributed matrix A(IA:IA+N-1,JA:JA+N-1). N >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension ( LLD_A, LOCc(JA+N-1) ). On entry, this array contains the local pieces of the factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U; the unit diagonal elements of L are not stored. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 ANORM (global input) REAL If NORM = '1' or 'O', the 1-norm of the original distributed matrix A(IA:IA+N-1,JA:JA+N-1). If NORM = 'I', the infinity-norm of the original distributed matrix A(IA:IA+N-1,JA:JA+N-1). .TP 8 RCOND (global output) REAL The reciprocal of the condition number of the distributed matrix A(IA:IA+N-1,JA:JA+N-1), computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * .br norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 2*LOCr(N+MOD(IA-1,MB_A)) + MAX( 2, MAX(NB_A*CEIL(NPROW-1,NPCOL),LOCc(N+MOD(JA-1,NB_A)) + NB_A*CEIL(NPCOL-1,NPROW)) ). LOCr and LOCc values can be computed using the ScaLAPACK tool function NUMROC; NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 RWORK (local workspace/local output) REAL array, dimension (LRWORK) On exit, RWORK(1) returns the minimal and optimal LRWORK. .TP 8 LRWORK (local or global input) INTEGER The dimension of the array RWORK. LRWORK is local input and must be at least LRWORK >= 2*LOCc(N+MOD(JA-1,NB_A)). If LRWORK = -1, then LRWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pcgeequ.l0100644000056400000620000001437206335610614017060 0ustar pfrauenfstaff.TH PCGEEQU l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGEEQU - compute row and column scalings intended to equilibrate an M-by-N distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA:JA+N-1) and reduce its condition number .SH SYNOPSIS .TP 20 SUBROUTINE PCGEEQU( M, N, A, IA, JA, DESCA, R, C, ROWCND, COLCND, AMAX, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, M, N .TP 20 .ti +4 REAL AMAX, COLCND, ROWCND .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL C( * ), R( * ) .TP 20 .ti +4 COMPLEX A( * ) .SH PURPOSE PCGEEQU computes row and column scalings intended to equilibrate an M-by-N distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA:JA+N-1) and reduce its condition number. R returns the row scale factors and C the column scale factors, chosen to try to make the largest entry in each row and column of the distributed matrix B with elements B(i,j) = R(i) * A(i,j) * C(j) have absolute value 1. .br R(i) and C(j) are restricted to be between SMLNUM = smallest safe number and BIGNUM = largest safe number. Use of these scaling factors is not guaranteed to reduce the condition number of sub( A ) but works well in practice. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension ( LLD_A, LOCc(JA+N-1) ), the local pieces of the M-by-N distributed matrix whose equilibration factors are to be computed. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 R (local output) REAL array, dimension LOCr(M_A) If INFO = 0 or INFO > IA+M-1, R(IA:IA+M-1) contains the row scale factors for sub( A ). R is aligned with the distributed matrix A, and replicated across every process column. R is tied to the distributed matrix A. .TP 8 C (local output) REAL array, dimension LOCc(N_A) If INFO = 0, C(JA:JA+N-1) contains the column scale factors for sub( A ). C is aligned with the distributed matrix A, and replicated down every process row. C is tied to the distri- buted matrix A. .TP 8 ROWCND (global output) REAL If INFO = 0 or INFO > IA+M-1, ROWCND contains the ratio of the smallest R(i) to the largest R(i) (IA <= i <= IA+M-1). If ROWCND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by R(IA:IA+M-1). .TP 8 COLCND (global output) REAL If INFO = 0, COLCND contains the ratio of the smallest C(j) to the largest C(j) (JA <= j <= JA+N-1). If COLCND >= 0.1, it is not worth scaling by C(JA:JA+N-1). .TP 8 AMAX (global output) REAL Absolute value of largest distributed matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = i, and i is .br <= M: the i-th row of the distributed matrix sub( A ) is exactly zero, > M: the (i-M)-th column of the distributed matrix sub( A ) is exactly zero. scalapack-doc-1.5/man/manl/pcgehd2.l0100644000056400000620000001651106335610614016740 0ustar pfrauenfstaff.TH PCGEHD2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCGEHD2 - reduce a complex general distributed matrix sub( A ) to upper Hessenberg form H by an unitary similarity transformation .SH SYNOPSIS .TP 20 SUBROUTINE PCGEHD2( N, ILO, IHI, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, IHI, ILO, INFO, JA, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), WORK( * ) .SH PURPOSE PCGEHD2 reduces a complex general distributed matrix sub( A ) to upper Hessenberg form H by an unitary similarity transformation: Q' * sub( A ) * Q = H, where .br sub( A ) = A(IA+N-1:IA+N-1,JA+N-1:JA+N-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 ILO (global input) INTEGER IHI (global input) INTEGER It is assumed that sub( A ) is already upper triangular in rows IA:IA+ILO-2 and IA+IHI:IA+N-1 and columns JA:JA+JLO-2 and JA+JHI:JA+N-1. See Further Details. If N > 0, .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N general distributed matrix sub( A ) to be reduced. On exit, the upper triangle and the first subdiagonal of sub( A ) are overwritten with the upper Hessenberg matrix H, and the ele- ments below the first subdiagonal, with the array TAU, repre- sent the unitary matrix Q as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX array, dimension LOCc(JA+N-2) The scalar factors of the elementary reflectors (see Further Details). Elements JA:JA+ILO-2 and JA+IHI:JA+N-2 of TAU are set to zero. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK( 1 ) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB + MAX( NpA0, NB ) where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ), IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ), NpA0 = NUMROC( IHI+IROFFA, NB, MYROW, IAROW, NPROW ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of (ihi-ilo) elementary reflectors .br Q = H(ilo) H(ilo+1) . . . H(ihi-1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on exit in A(ia+ilo+i:ia+ihi-1,ja+ilo+i-2), and tau in TAU(ja+ilo+i-2). The contents of A(IA:IA+N-1,JA:JA+N-1) are illustrated by the follo- wing example, with n = 7, ilo = 2 and ihi = 6: .br on entry on exit .br ( a a a a a a a ) ( a a h h h h a ) ( a a a a a a ) ( a h h h h a ) ( a a a a a a ) ( h h h h h h ) ( a a a a a a ) ( v2 h h h h h ) ( a a a a a a ) ( v2 v3 h h h h ) ( a a a a a a ) ( v2 v3 v4 h h h ) ( a ) ( a ) where a denotes an element of the original matrix sub( A ), h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(ja+ilo+i-2). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expression should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) .br scalapack-doc-1.5/man/manl/pcgehrd.l0100644000056400000620000001714506335610614017044 0ustar pfrauenfstaff.TH PCGEHRD l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGEHRD - reduce a complex general distributed matrix sub( A ) to upper Hessenberg form H by an unitary similarity transformation .SH SYNOPSIS .TP 20 SUBROUTINE PCGEHRD( N, ILO, IHI, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, IHI, ILO, INFO, JA, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), WORK( * ) .SH PURPOSE PCGEHRD reduces a complex general distributed matrix sub( A ) to upper Hessenberg form H by an unitary similarity transformation: Q' * sub( A ) * Q = H, where .br sub( A ) = A(IA+N-1:IA+N-1,JA+N-1:JA+N-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 ILO (global input) INTEGER IHI (global input) INTEGER It is assumed that sub( A ) is already upper triangular in rows IA:IA+ILO-2 and IA+IHI:IA+N-1 and columns JA:JA+ILO-2 and JA+IHI:JA+N-1. See Further Details. If N > 0, .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N general distributed matrix sub( A ) to be reduced. On exit, the upper triangle and the first subdiagonal of sub( A ) are overwritten with the upper Hessenberg matrix H, and the ele- ments below the first subdiagonal, with the array TAU, repre- sent the unitary matrix Q as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX array, dimension LOCc(JA+N-2) The scalar factors of the elementary reflectors (see Further Details). Elements JA:JA+ILO-2 and JA+IHI:JA+N-2 of TAU are set to zero. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK( 1 ) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB*NB + NB*MAX( IHIP+1, IHLP+INLQ ) where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ), ICOFFA = MOD( JA-1, NB ), IOFF = MOD( IA+ILO-2, NB ), IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ), IHIP = NUMROC( IHI+IROFFA, NB, MYROW, IAROW, NPROW ), ILROW = INDXG2P( IA+ILO-1, NB, MYROW, RSRC_A, NPROW ), IHLP = NUMROC( IHI-ILO+IOFF+1, NB, MYROW, ILROW, NPROW ), ILCOL = INDXG2P( JA+ILO-1, NB, MYCOL, CSRC_A, NPCOL ), INLQ = NUMROC( N-ILO+IOFF+1, NB, MYCOL, ILCOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of (ihi-ilo) elementary reflectors .br Q = H(ilo) H(ilo+1) . . . H(ihi-1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(1:I) = 0, v(I+1) = 1 and v(IHI+1:N) = 0; v(I+2:IHI) is stored on exit in A(IA+ILO+I:IA+IHI-1,JA+ILO+I-2), and tau in TAU(JA+ILO+I-2). The contents of A(IA:IA+N-1,JA:JA+N-1) are illustrated by the follow- ing example, with N = 7, ILO = 2 and IHI = 6: .br on entry on exit .br ( a a a a a a a ) ( a a h h h h a ) ( a a a a a a ) ( a h h h h a ) ( a a a a a a ) ( h h h h h h ) ( a a a a a a ) ( v2 h h h h h ) ( a a a a a a ) ( v2 v3 h h h h ) ( a a a a a a ) ( v2 v3 v4 h h h ) ( a ) ( a ) where a denotes an element of the original matrix sub( A ), H denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(JA+ILO+I-2). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expression should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) .br scalapack-doc-1.5/man/manl/pcgelq2.l0100644000056400000620000001422306335610614016757 0ustar pfrauenfstaff.TH PCGELQ2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGELQ2 - compute a LQ factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = L * Q .SH SYNOPSIS .TP 20 SUBROUTINE PCGELQ2( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), WORK( * ) .SH PURPOSE PCGELQ2 computes a LQ factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = L * Q. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the elements on and below the diagonal of sub( A ) contain the M by min(M,N) lower trapezoidal matrix L (L is lower triangular if M <= N); the elements above the diagonal, with the array TAU, repre- sent the unitary matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX, array, dimension LOCr(IA+MIN(M,N)-1). This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= Nq0 + MAX( 1, Mp0 ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ia+k-1)' H(ia+k-2)' . . . H(ia)', where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; conjg(v(i+1:n)) is stored on exit in A(ia+i-1,ja+i:ja+n-1), and tau in TAU(ia+i-1). .br scalapack-doc-1.5/man/manl/pcgelqf.l0100644000056400000620000001423406335610614017045 0ustar pfrauenfstaff.TH PCGELQF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGELQF - compute a LQ factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = L * Q .SH SYNOPSIS .TP 20 SUBROUTINE PCGELQF( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), WORK( * ) .SH PURPOSE PCGELQF computes a LQ factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = L * Q. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the elements on and below the diagonal of sub( A ) contain the M by min(M,N) lower trapezoidal matrix L (L is lower triangular if M <= N); the elements above the diagonal, with the array TAU, repre- sent the unitary matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX, array, dimension LOCr(IA+MIN(M,N)-1). This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MB_A * ( Mp0 + Nq0 + MB_A ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ia+k-1)' H(ia+k-2)' . . . H(ia)', where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; conjg(v(i+1:n)) is stored on exit in A(ia+i-1,ja+i:ja+n-1), and tau in TAU(ia+i-1). .br scalapack-doc-1.5/man/manl/pcgels.l0100644000056400000620000002221206335610614016674 0ustar pfrauenfstaff.TH PCGELS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGELS - solve overdetermined or underdetermined complex linear systems involving an M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1), .SH SYNOPSIS .TP 19 SUBROUTINE PCGELS( TRANS, M, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 CHARACTER TRANS .TP 19 .ti +4 INTEGER IA, IB, INFO, JA, JB, LWORK, M, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 COMPLEX A( * ), B( * ), WORK( * ) .SH PURPOSE PCGELS solves overdetermined or underdetermined complex linear systems involving an M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1), or its conjugate-transpose, using a QR or LQ factorization of sub( A ). It is assumed that sub( A ) has full rank. .br The following options are provided: .br 1. If TRANS = 'N' and m >= n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem minimize || sub( B ) - sub( A )*X ||. .br 2. If TRANS = 'N' and m < n: find the minimum norm solution of an underdetermined system sub( A ) * X = sub( B ). .br 3. If TRANS = 'C' and m >= n: find the minimum norm solution of an undetermined system sub( A )**H * X = sub( B ). .br 4. If TRANS = 'C' and m < n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem minimize || sub( B ) - sub( A )**H * X ||. where sub( B ) denotes B( IB:IB+M-1, JB:JB+NRHS-1 ) when TRANS = 'N' and B( IB:IB+N-1, JB:JB+NRHS-1 ) otherwise. Several right hand side vectors b and solution vectors x can be handled in a single call; When TRANS = 'N', the solution vectors are stored as the columns of the N-by-NRHS right hand side matrix sub( B ) and the M-by-NRHS right hand side matrix sub( B ) otherwise. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 TRANS (global input) CHARACTER = 'N': the linear system involves sub( A ); .br = 'C': the linear system involves sub( A )**H. .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e. the number of columns of the distributed submatrices sub( B ) and X. NRHS >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of local dimension ( LLD_A, LOCc(JA+N-1) ). On entry, the M-by-N matrix A. if M >= N, sub( A ) is overwritten by details of its QR factorization as returned by PCGEQRF; if M < N, sub( A ) is overwritten by details of its LQ factorization as returned by PCGELQF. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input/local output) COMPLEX pointer into the local memory to an array of local dimension (LLD_B, LOCc(JB+NRHS-1)). On entry, this array contains the local pieces of the distributed matrix B of right hand side vectors, stored columnwise; sub( B ) is M-by-NRHS if TRANS='N', and N-by-NRHS otherwise. On exit, sub( B ) is overwritten by the solution vectors, stored columnwise: if TRANS = 'N' and M >= N, rows 1 to N of sub( B ) contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements N+1 to M in that column; if TRANS = 'N' and M < N, rows 1 to N of sub( B ) contain the minimum norm solution vectors; if TRANS = 'C' and M >= N, rows 1 to M of sub( B ) contain the minimum norm solution vectors; if TRANS = 'C' and M < N, rows 1 to M of sub( B ) contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements M+1 to N in that column. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= LTAU + MAX( LWF, LWS ) where If M >= N, then LTAU = NUMROC( JA+MIN(M,N)-1, NB_A, MYCOL, CSRC_A, NPCOL ), LWF = NB_A * ( MpA0 + NqA0 + NB_A ) LWS = MAX( (NB_A*(NB_A-1))/2, (NRHSqB0 + MpB0)*NB_A ) + NB_A * NB_A Else LTAU = NUMROC( IA+MIN(M,N)-1, MB_A, MYROW, RSRC_A, NPROW ), LWF = MB_A * ( MpA0 + NqA0 + MB_A ) LWS = MAX( (MB_A*(MB_A-1))/2, ( NpB0 + MAX( NqA0 + NUMROC( NUMROC( N+IROFFB, MB_A, 0, 0, NPROW ), MB_A, 0, 0, LCMP ), NRHSqB0 ) )*MB_A ) + MB_A * MB_A End if where LCMP = LCM / NPROW with LCM = ILCM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFB = MOD( IB-1, MB_B ), ICOFFB = MOD( JB-1, NB_B ), IBROW = INDXG2P( IB, MB_B, MYROW, RSRC_B, NPROW ), IBCOL = INDXG2P( JB, NB_B, MYCOL, CSRC_B, NPCOL ), MpB0 = NUMROC( M+IROFFB, MB_B, MYROW, IBROW, NPROW ), NpB0 = NUMROC( N+IROFFB, MB_B, MYROW, IBROW, NPROW ), NRHSqB0 = NUMROC( NRHS+ICOFFB, NB_B, MYCOL, IBCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pcgeql2.l0100644000056400000620000001436506335610614016766 0ustar pfrauenfstaff.TH PCGEQL2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGEQL2 - compute a QL factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * L .SH SYNOPSIS .TP 20 SUBROUTINE PCGEQL2( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), WORK( * ) .SH PURPOSE PCGEQL2 computes a QL factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * L. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, if M >= N, the lower triangle of the distributed submatrix A( IA+M-N:IA+M-1, JA:JA+N-1 ) contains the N-by-N lower triangular matrix L; if M <= N, the elements on and below the (N-M)-th superdiagonal contain the M by N lower trapezoidal matrix L; the remaining elements, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX, array, dimension LOCc(JA+N-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= Mp0 + MAX( 1, Nq0 ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ja+k-1) . . . H(ja+1) H(ja), where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in A(ia:ia+m-k+i-2,ja+n-k+i-1), and tau in TAU(ja+n-k+i-1). .br scalapack-doc-1.5/man/manl/pcgeqlf.l0100644000056400000620000001437606335610614017054 0ustar pfrauenfstaff.TH PCGEQLF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGEQLF - compute a QL factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * L .SH SYNOPSIS .TP 20 SUBROUTINE PCGEQLF( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), WORK( * ) .SH PURPOSE PCGEQLF computes a QL factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * L. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, if M >= N, the lower triangle of the distributed submatrix A( IA+M-N:IA+M-1, JA:JA+N-1 ) contains the N-by-N lower triangular matrix L; if M <= N, the elements on and below the (N-M)-th superdiagonal contain the M by N lower trapezoidal matrix L; the remaining elements, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX, array, dimension LOCc(JA+N-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB_A * ( Mp0 + Nq0 + NB_A ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ja+k-1) . . . H(ja+1) H(ja), where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in A(ia:ia+m-k+i-2,ja+n-k+i-1), and tau in TAU(ja+n-k+i-1). .br scalapack-doc-1.5/man/manl/pcgeqpf.l0100644000056400000620000001624406335610615017055 0ustar pfrauenfstaff.TH PCGEQPF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGEQPF - compute a QR factorization with column pivoting of a M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PCGEQPF( M, N, A, IA, JA, DESCA, IPIV, TAU, WORK, LWORK, RWORK, LRWORK, INFO ) .TP 20 .ti +4 INTEGER IA, JA, INFO, LRWORK, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ) .TP 20 .ti +4 REAL RWORK( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), WORK( * ) .SH PURPOSE PCGEQPF computes a QR factorization with column pivoting of a M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1): sub( A ) * P = Q * R. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the elements on and above the diagonal of sub( A ) contain the min(M,N) by N upper trapezoidal matrix R (R is upper triangular if M >= N); the elements below the diagonal, with the array TAU, repre- sent the unitary matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local output) INTEGER array, dimension LOCc(JA+N-1). On exit, if IPIV(I) = K, the local i-th column of sub( A )*P was the global K-th column of sub( A ). IPIV is tied to the distributed matrix A. .TP 8 TAU (local output) COMPLEX, array, dimension LOCc(JA+MIN(M,N)-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MAX(3,Mp0 + Nq0). If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 RWORK (local workspace/local output) REAL array, dimension (LRWORK) On exit, RWORK(1) returns the minimal and optimal LRWORK. .TP 8 LRWORK (local or global input) INTEGER The dimension of the array RWORK. LRWORK is local input and must be at least LRWORK >= LOCc(JA+N-1)+Nq0. IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), LOCc(JA+N-1) = NUMROC( JA+N-1, NB_A, MYCOL, CSRC_A, NPCOL ) and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LRWORK = -1, then LRWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) . . . H(n) .br Each H(i) has the form .br H = I - tau * v * v' .br 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 .br A(ia+i-1:ia+m-1,ja+i-1). .br The matrix P is represented in jpvt as follows: If .br jpvt(j) = i .br then the jth column of P is the ith canonical unit vector. scalapack-doc-1.5/man/manl/pcgeqr2.l0100644000056400000620000001421506335610615016767 0ustar pfrauenfstaff.TH PCGEQR2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGEQR2 - compute a QR factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * R .SH SYNOPSIS .TP 20 SUBROUTINE PCGEQR2( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), WORK( * ) .SH PURPOSE PCGEQR2 computes a QR factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * R. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the elements on and above the diagonal of sub( A ) contain the min(M,N) by N upper trapezoidal matrix R (R is upper triangular if M >= N); the elements below the diagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX, array, dimension LOCc(JA+MIN(M,N)-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= Mp0 + MAX( 1, Nq0 ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ja) H(ja+1) . . . H(ja+k-1), where k = min(m,n). Each H(i) has the form .br H(j) = I - tau * v * v' .br 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 .br A(ia+i:ia+m-1,ja+i-1), and tau in TAU(ja+i-1). .br scalapack-doc-1.5/man/manl/pcgeqrf.l0100644000056400000620000001422606335610615017055 0ustar pfrauenfstaff.TH PCGEQRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGEQRF - compute a QR factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * R .SH SYNOPSIS .TP 20 SUBROUTINE PCGEQRF( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), WORK( * ) .SH PURPOSE PCGEQRF computes a QR factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * R. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the elements on and above the diagonal of sub( A ) contain the min(M,N) by N upper trapezoidal matrix R (R is upper triangular if M >= N); the elements below the diagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX, array, dimension LOCc(JA+MIN(M,N)-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB_A * ( Mp0 + Nq0 + NB_A ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ja) H(ja+1) . . . H(ja+k-1), where k = min(m,n). Each H(i) has the form .br H(j) = I - tau * v * v' .br 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 .br A(ia+i:ia+m-1,ja+i-1), and tau in TAU(ja+i-1). .br scalapack-doc-1.5/man/manl/pcgerfs.l0100644000056400000620000002330306335610615017053 0ustar pfrauenfstaff.TH PCGERFS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGERFS - improve the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solutions .SH SYNOPSIS .TP 20 SUBROUTINE PCGERFS( TRANS, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, IPIV, B, IB, JB, DESCB, X, IX, JX, DESCX, FERR, BERR, WORK, LWORK, RWORK, LRWORK, INFO ) .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER IA, IAF, IB, IX, INFO, JA, JAF, JB, JX, LRWORK, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * ), IPIV( * ) .TP 20 .ti +4 REAL BERR( * ), FERR( * ), RWORK( * ) .TP 20 .ti +4 COMPLEX A( * ), AF( * ), B( * ), WORK( * ), X( * ) .SH PURPOSE PCGERFS improves the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solutions. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br In the following comments, sub( A ), sub( X ) and sub( B ) denote respectively A(IA:IA+N-1,JA:JA+N-1), X(IX:IX+N-1,JX:JX+NRHS-1) and B(IB:IB+N-1,JB:JB+NRHS-1). .br .SH ARGUMENTS .TP 8 TRANS (global input) CHARACTER*1 Specifies the form of the system of equations. = 'N': sub( A ) * sub( X ) = sub( B ) (No transpose) .br = 'T': sub( A )**T * sub( X ) = sub( B ) (Transpose) .br = 'C': sub( A )**H * sub( X ) = sub( B ) (Conjugate transpose) .TP 8 N (global input) INTEGER The order of the matrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the matrices sub( B ) and sub( X ). NRHS >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1)). This array contains the local pieces of the distributed matrix sub( A ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 AF (local input) COMPLEX pointer into the local memory to an array of local dimension (LLD_AF,LOCc(JA+N-1)). This array contains the local pieces of the distributed factors of the matrix sub( A ) = P * L * U as computed by PCGETRF. .TP 8 IAF (global input) INTEGER The row index in the global array AF indicating the first row of sub( AF ). .TP 8 JAF (global input) INTEGER The column index in the global array AF indicating the first column of sub( AF ). .TP 8 DESCAF (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix AF. .TP 8 IPIV (local input) INTEGER array of dimension LOCr(M_AF)+MB_AF. This array contains the pivoting information as computed by PCGETRF. IPIV(i) -> The global row local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 B (local input) COMPLEX pointer into the local memory to an array of local dimension (LLD_B,LOCc(JB+NRHS-1)). This array contains the local pieces of the distributed matrix of right hand sides sub( B ). .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 X (local input and output) COMPLEX pointer into the local memory to an array of local dimension (LLD_X,LOCc(JX+NRHS-1)). On entry, this array contains the local pieces of the distributed matrix solution sub( X ). On exit, the improved solution vectors. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 FERR (local output) REAL array of local dimension LOCc(JB+NRHS-1). The estimated forward error bound for each solution vector of sub( X ). If XTRUE is the true solution corresponding to sub( X ), FERR is an estimated upper bound for the magnitude of the largest element in (sub( X ) - XTRUE) divided by the magnitude of the largest element in sub( X ). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. This array is tied to the distributed matrix X. .TP 8 BERR (local output) REAL array of local dimension LOCc(JB+NRHS-1). The componentwise relative backward error of each solution vector (i.e., the smallest re- lative change in any entry of sub( A ) or sub( B ) that makes sub( X ) an exact solution). This array is tied to the distributed matrix X. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 2*LOCr( N + MOD(IA-1,MB_A) ) If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 RWORK (local workspace/local output) REAL array, dimension (LRWORK) On exit, RWORK(1) returns the minimal and optimal LRWORK. .TP 8 LRWORK (local or global input) INTEGER The dimension of the array RWORK. LRWORK is local input and must be at least LRWORK >= LOCr( N + MOD(IB-1,MB_B) ). If LRWORK = -1, then LRWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH PARAMETERS ITMAX is the maximum number of steps of iterative refinement. Notes ===== This routine temporarily returns when N <= 1. The distributed submatrices op( A ) and op( AF ) (respectively sub( X ) and sub( B ) ) should be distributed the same way on the same processes. These conditions ensure that sub( A ) and sub( AF ) (resp. sub( X ) and sub( B ) ) are "perfectly" aligned. Moreover, this routine requires the distributed submatrices sub( A ), sub( AF ), sub( X ), and sub( B ) to be aligned on a block boundary, i.e., if f(x,y) = MOD( x-1, y ): f( IA, DESCA( MB_ ) ) = f( JA, DESCA( NB_ ) ) = 0, f( IAF, DESCAF( MB_ ) ) = f( JAF, DESCAF( NB_ ) ) = 0, f( IB, DESCB( MB_ ) ) = f( JB, DESCB( NB_ ) ) = 0, and f( IX, DESCX( MB_ ) ) = f( JX, DESCX( NB_ ) ) = 0. scalapack-doc-1.5/man/manl/pcgerq2.l0100644000056400000620000001433706335610615016774 0ustar pfrauenfstaff.TH PCGERQ2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGERQ2 - compute a RQ factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = R * Q .SH SYNOPSIS .TP 20 SUBROUTINE PCGERQ2( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), WORK( * ) .SH PURPOSE PCGERQ2 computes a RQ factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = R * Q. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, if M <= N, the upper triangle of A( IA:IA+M-1, JA+N-M:JA+N-1 ) 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 unitary matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX, array, dimension LOCr(IA+M-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= Nq0 + MAX( 1, Mp0 ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ia)' H(ia+1)' . . . H(ia+k-1)', where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br 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(ia+m-k+i-1,ja:ja+n-k+i-2), and tau in TAU(ia+m-k+i-1). scalapack-doc-1.5/man/manl/pcgerqf.l0100644000056400000620000001435006335610615017053 0ustar pfrauenfstaff.TH PCGERQF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGERQF - compute a RQ factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = R * Q .SH SYNOPSIS .TP 20 SUBROUTINE PCGERQF( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), WORK( * ) .SH PURPOSE PCGERQF computes a RQ factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = R * Q. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, if M <= N, the upper triangle of A( IA:IA+M-1, JA+N-M:JA+N-1 ) 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 unitary matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX, array, dimension LOCr(IA+M-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MB_A * ( Mp0 + Nq0 + MB_A ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ia)' H(ia+1)' . . . H(ia+k-1)', where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br 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(ia+m-k+i-1,ja:ja+n-k+i-2), and tau in TAU(ia+m-k+i-1). scalapack-doc-1.5/man/manl/pcgesv.l0100644000056400000620000001375706335610615016725 0ustar pfrauenfstaff.TH PCGESV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGESV - compute the solution to a complex system of linear equations sub( A ) * X = sub( B ), .SH SYNOPSIS .TP 19 SUBROUTINE PCGESV( N, NRHS, A, IA, JA, DESCA, IPIV, B, IB, JB, DESCB, INFO ) .TP 19 .ti +4 INTEGER IA, IB, INFO, JA, JB, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ), IPIV( * ) .TP 19 .ti +4 COMPLEX A( * ), B( * ) .SH PURPOSE PCGESV computes the solution to a complex system of linear equations where sub( A ) = A(IA:IA+N-1,JA:JA+N-1) is an N-by-N distributed matrix and X and sub( B ) = B(IB:IB+N-1,JB:JB+NRHS-1) are N-by-NRHS distributed matrices. .br The LU decomposition with partial pivoting and row interchanges is used to factor sub( A ) as sub( A ) = P * L * U, where P is a permu- tation matrix, L is unit lower triangular, and U is upper triangular. L and U are stored in sub( A ). The factored form of sub( A ) is then used to solve the system of equations sub( A ) * X = sub( B ). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block decomposition ( MB_A = NB_A ). .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed submatrix sub( A ). NRHS >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the local pieces of the N-by-N distributed matrix sub( A ) to be factored. On exit, this array contains the local pieces of the factors L and U from the factorization sub( A ) = P*L*U; the unit diagonal elements of L are not stored. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local output) INTEGER array, dimension ( LOCr(M_A)+MB_A ) This array contains the pivoting information. IPIV(i) -> The global row local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 B (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_B,LOCc(JB+NRHS-1)). On entry, the right hand side distributed matrix sub( B ). On exit, if INFO = 0, sub( B ) is overwritten by the solution distributed matrix X. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, U(IA+K-1,JA+K-1) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution could not be computed. scalapack-doc-1.5/man/manl/pcgesvx.l0100644000056400000620000004031706335610615017105 0ustar pfrauenfstaff.TH PCGESVX l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGESVX - use the LU factorization to compute the solution to a complex system of linear equations A(IA:IA+N-1,JA:JA+N-1) * X = B(IB:IB+N-1,JB:JB+NRHS-1), .SH SYNOPSIS .TP 20 SUBROUTINE PCGESVX( FACT, TRANS, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, IPIV, EQUED, R, C, B, IB, JB, DESCB, X, IX, JX, DESCX, RCOND, FERR, BERR, WORK, LWORK, RWORK, LRWORK, INFO ) .TP 20 .ti +4 CHARACTER EQUED, FACT, TRANS .TP 20 .ti +4 INTEGER IA, IAF, IB, INFO, IX, JA, JAF, JB, JX, LRWORK, LWORK, N, NRHS .TP 20 .ti +4 REAL RCOND .TP 20 .ti +4 INTEGER DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * ), IPIV( * ) .TP 20 .ti +4 REAL BERR( * ), C( * ), FERR( * ), R( * ), RWORK( * ) .TP 20 .ti +4 COMPLEX A( * ), AF( * ), B( * ), WORK( * ), X( * ) .SH PURPOSE PCGESVX uses the LU factorization to compute the solution to a complex system of linear equations where A(IA:IA+N-1,JA:JA+N-1) is an N-by-N matrix and X and B(IB:IB+N-1,JB:JB+NRHS-1) are N-by-NRHS matrices. .br Error bounds on the solution and a condition estimate are also provided. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH DESCRIPTION In the following description, A denotes A(IA:IA+N-1,JA:JA+N-1), B denotes B(IB:IB+N-1,JB:JB+NRHS-1) and X denotes .br X(IX:IX+N-1,JX:JX+NRHS-1). .br The following steps are performed: .br 1. If FACT = 'E', real scaling factors are computed to equilibrate the system: .br 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'). .br 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as .br A = P * L * U, .br where P is a permutation matrix, L is a unit lower triangular matrix, and U is upper triangular. .br 3. 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, steps 4-6 are skipped. .br 4. The system of equations is solved for X using the factored form of A. .br 5. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. .br 6. If FACT = 'E' and 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. .br .SH ARGUMENTS .TP 8 FACT (global input) CHARACTER Specifies whether or not the factored form of the matrix A(IA:IA+N-1,JA:JA+N-1) is supplied on entry, and if not, .br whether the matrix A(IA:IA+N-1,JA:JA+N-1) should be equilibrated before it is factored. = 'F': On entry, AF(IAF:IAF+N-1,JAF:JAF+N-1) and IPIV con- .br tain the factored form of A(IA:IA+N-1,JA:JA+N-1). If EQUED is not 'N', the matrix A(IA:IA+N-1,JA:JA+N-1) has been equilibrated with scaling factors given by R and C. A(IA:IA+N-1,JA:JA+N-1), AF(IAF:IAF+N-1,JAF:JAF+N-1), and IPIV are not modified. = 'N': The matrix A(IA:IA+N-1,JA:JA+N-1) will be copied to .br AF(IAF:IAF+N-1,JAF:JAF+N-1) and factored. .br = 'E': The matrix A(IA:IA+N-1,JA:JA+N-1) will be equili- brated if necessary, then copied to AF(IAF:IAF+N-1,JAF:JAF+N-1) and factored. .TP 8 TRANS (global input) CHARACTER .br Specifies the form of the system of equations: .br = 'N': A(IA:IA+N-1,JA:JA+N-1) * X(IX:IX+N-1,JX:JX+NRHS-1) .br = B(IB:IB+N-1,JB:JB+NRHS-1) (No transpose) .br = 'T': A(IA:IA+N-1,JA:JA+N-1)**T * X(IX:IX+N-1,JX:JX+NRHS-1) .br = B(IB:IB+N-1,JB:JB+NRHS-1) (Transpose) .br = 'C': A(IA:IA+N-1,JA:JA+N-1)**H * X(IX:IX+N-1,JX:JX+NRHS-1) .br = B(IB:IB+N-1,JB:JB+NRHS-1) (Conjugate transpose) .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix A(IA:IA+N-1,JA:JA+N-1). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right-hand sides, i.e., the number of columns of the distributed submatrices B(IB:IB+N-1,JB:JB+NRHS-1) and .br X(IX:IX+N-1,JX:JX+NRHS-1). NRHS >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1)). On entry, the N-by-N matrix A(IA:IA+N-1,JA:JA+N-1). If FACT = 'F' and EQUED is not 'N', .br then A(IA:IA+N-1,JA:JA+N-1) must have been equilibrated by .br the scaling factors in R and/or C. A(IA:IA+N-1,JA:JA+N-1) is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. On exit, if EQUED .ne. 'N', A(IA:IA+N-1,JA:JA+N-1) is scaled as follows: .br EQUED = 'R': A(IA:IA+N-1,JA:JA+N-1) := .br diag(R) * A(IA:IA+N-1,JA:JA+N-1) .br EQUED = 'C': A(IA:IA+N-1,JA:JA+N-1) := .br A(IA:IA+N-1,JA:JA+N-1) * diag(C) .br EQUED = 'B': A(IA:IA+N-1,JA:JA+N-1) := .br diag(R) * A(IA:IA+N-1,JA:JA+N-1) * diag(C). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 AF (local input or local output) COMPLEX pointer into the local memory to an array of local dimension (LLD_AF,LOCc(JA+N-1)). If FACT = 'F', then AF(IAF:IAF+N-1,JAF:JAF+N-1) is an input argument and on entry contains the factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U as computed by PCGETRF. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix A(IA:IA+N-1,JA:JA+N-1). If FACT = 'N', then AF(IAF:IAF+N-1,JAF:JAF+N-1) is an output argument and on exit returns the factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the original .br matrix A(IA:IA+N-1,JA:JA+N-1). If FACT = 'E', then AF(IAF:IAF+N-1,JAF:JAF+N-1) is an output argument and on exit returns the factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the equili- .br brated matrix A(IA:IA+N-1,JA:JA+N-1) (see the description of .br A(IA:IA+N-1,JA:JA+N-1) for the form of the equilibrated matrix). .TP 8 IAF (global input) INTEGER The row index in the global array AF indicating the first row of sub( AF ). .TP 8 JAF (global input) INTEGER The column index in the global array AF indicating the first column of sub( AF ). .TP 8 DESCAF (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix AF. .TP 8 IPIV (local input or local output) INTEGER array, dimension LOCr(M_A)+MB_A. If FACT = 'F', then IPIV is an input argu- ment and on entry contains the pivot indices from the fac- torization A(IA:IA+N-1,JA:JA+N-1) = P*L*U as computed by PCGETRF; IPIV(i) -> The global row local row i was swapped with. This array must be aligned with A( IA:IA+N-1, * ). If FACT = 'N', then IPIV is an output argument and on exit contains the pivot indices from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the original matrix .br A(IA:IA+N-1,JA:JA+N-1). If FACT = 'E', then IPIV is an output argument and on exit contains the pivot indices from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the equilibrated matrix .br A(IA:IA+N-1,JA:JA+N-1). .TP 8 EQUED (global input or global output) CHARACTER Specifies the form of equilibration that was done. = 'N': No equilibration (always true if FACT = 'N'). .br = 'R': Row equilibration, i.e., A(IA:IA+N-1,JA:JA+N-1) has been premultiplied by diag(R). = 'C': Column equilibration, i.e., A(IA:IA+N-1,JA:JA+N-1) has been postmultiplied by diag(C). = 'B': Both row and column equilibration, i.e., .br A(IA:IA+N-1,JA:JA+N-1) has been replaced by .br diag(R) * A(IA:IA+N-1,JA:JA+N-1) * diag(C). EQUED is an input variable if FACT = 'F'; otherwise, it is an output variable. .TP 8 R (local input or local output) REAL array, dimension LOCr(M_A). The row scale factors for A(IA:IA+N-1,JA:JA+N-1). .br If EQUED = 'R' or 'B', A(IA:IA+N-1,JA:JA+N-1) is multiplied on the left by diag(R); if EQUED='N' or 'C', R is not acces- sed. R is an input variable if FACT = 'F'; otherwise, R is an output variable. If FACT = 'F' and EQUED = 'R' or 'B', each element of R must be positive. R is replicated in every process column, and is aligned with the distributed matrix A. .TP 8 C (local input or local output) REAL array, dimension LOCc(N_A). The column scale factors for A(IA:IA+N-1,JA:JA+N-1). .br If EQUED = 'C' or 'B', A(IA:IA+N-1,JA:JA+N-1) is multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is not accessed. C is an input variable if FACT = 'F'; otherwise, C is an output variable. If FACT = 'F' and EQUED = 'C' or 'B', each element of C must be positive. C is replicated in every process row, and is aligned with the distributed matrix A. .TP 8 B (local input/local output) COMPLEX pointer into the local memory to an array of local dimension (LLD_B,LOCc(JB+NRHS-1) ). On entry, the N-by-NRHS right-hand side matrix B(IB:IB+N-1,JB:JB+NRHS-1). On exit, if .br EQUED = 'N', B(IB:IB+N-1,JB:JB+NRHS-1) is not modified; if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B(IB:IB+N-1,JB:JB+NRHS-1); if TRANS = 'T' or 'C' .br and EQUED = 'C' or 'B', B(IB:IB+N-1,JB:JB+NRHS-1) is over- .br written by diag(C)*B(IB:IB+N-1,JB:JB+NRHS-1). .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 X (local input/local output) COMPLEX pointer into the local memory to an array of local dimension (LLD_X, LOCc(JX+NRHS-1)). If INFO = 0, the N-by-NRHS solution matrix X(IX:IX+N-1,JX:JX+NRHS-1) to the original .br system of equations. Note that A(IA:IA+N-1,JA:JA+N-1) and .br B(IB:IB+N-1,JB:JB+NRHS-1) are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(C))*X(IX:IX+N-1,JX:JX+NRHS-1) if TRANS = 'N' and EQUED = 'C' or 'B', or inv(diag(R))*X(IX:IX+N-1,JX:JX+NRHS-1) if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 RCOND (global output) REAL The estimate of the reciprocal condition number of the matrix A(IA:IA+N-1,JA:JA+N-1) 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. .TP 8 FERR (local output) REAL array, dimension LOCc(N_B) The estimated forward error bounds for each solution vector X(j) (the j-th column of the solution matrix X(IX:IX+N-1,JX:JX+NRHS-1). If XTRUE is the true solution, FERR(j) bounds the magnitude of the largest entry in (X(j) - XTRUE) divided by the magnitude of the largest entry in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. FERR is replicated in every process row, and is aligned with the matrices B and X. .TP 8 BERR (local output) REAL array, dimension LOCc(N_B). The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any entry of A(IA:IA+N-1,JA:JA+N-1) or .br B(IB:IB+N-1,JB:JB+NRHS-1) that makes X(j) an exact solution). BERR is replicated in every process row, and is aligned with the matrices B and X. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK = MAX( PCGECON( LWORK ), PCGERFS( LWORK ) ) + LOCr( N_A ). If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 RWORK (local workspace/local output) REAL array, dimension (LRWORK) On exit, RWORK(1) returns the minimal and optimal LRWORK. .TP 8 LRWORK (local or global input) INTEGER The dimension of the array RWORK. LRWORK is local input and must be at least LRWORK = 2*LOCc(N_A). If LRWORK = -1, then LRWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 0: if INFO = i, and i is .br <= N: U(IA+I-1,IA+I-1) 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. = N+1: RCOND is less than machine precision. The factorization has been completed, but the matrix is singular to working precision, and the solution and error bounds have not been computed. scalapack-doc-1.5/man/manl/pcgetf2.l0100644000056400000620000001260606335610615016760 0ustar pfrauenfstaff.TH PCGETF2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGETF2 - compute an LU factorization of a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) using partial pivoting with row interchanges .SH SYNOPSIS .TP 20 SUBROUTINE PCGETF2( M, N, A, IA, JA, DESCA, IPIV, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, M, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ) .TP 20 .ti +4 COMPLEX A( * ) .SH PURPOSE PCGETF2 computes an LU factorization of a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) using partial pivoting with row interchanges. The factorization has the form sub( A ) = P * L * U, where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n). .br This is the right-looking Parallel Level 2 BLAS version of the algorithm. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires N <= NB_A-MOD(JA-1, NB_A) and square block decomposition ( MB_A = NB_A ). .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). NB_A-MOD(JA-1, NB_A) >= N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the M-by-N distributed matrix sub( A ). On exit, this array contains the local pieces of the factors L and U from the factoriza- tion sub( A ) = P*L*U; the unit diagonal elements of L are not stored. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local output) INTEGER array, dimension ( LOCr(M_A)+MB_A ) This array contains the pivoting information. IPIV(i) -> The global row local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, U(IA+K-1,JA+K-1) 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. scalapack-doc-1.5/man/manl/pcgetrf.l0100644000056400000620000001257006335610615017060 0ustar pfrauenfstaff.TH PCGETRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGETRF - compute an LU factorization of a general M-by-N distributed matrix sub( A ) = (IA:IA+M-1,JA:JA+N-1) using partial pivoting with row interchanges .SH SYNOPSIS .TP 20 SUBROUTINE PCGETRF( M, N, A, IA, JA, DESCA, IPIV, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, M, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ) .TP 20 .ti +4 COMPLEX A( * ) .SH PURPOSE PCGETRF computes an LU factorization of a general M-by-N distributed matrix sub( A ) = (IA:IA+M-1,JA:JA+N-1) using partial pivoting with row interchanges. The factorization has the form sub( A ) = P * L * U, where P is a permutation matrix, L is lower triangular with unit diagonal ele- ments (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n). L and U are stored in sub( A ). This is the right-looking Parallel Level 3 BLAS version of the algorithm. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block decomposition ( MB_A = NB_A ). .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the M-by-N distributed matrix sub( A ) to be factored. On exit, this array contains the local pieces of the factors L and U from the factorization sub( A ) = P*L*U; the unit diagonal ele- ments of L are not stored. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local output) INTEGER array, dimension ( LOCr(M_A)+MB_A ) This array contains the pivoting information. IPIV(i) -> The global row local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, U(IA+K-1,JA+K-1) 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. scalapack-doc-1.5/man/manl/pcgetri.l0100644000056400000620000001441506335610615017063 0ustar pfrauenfstaff.TH PCGETRI l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGETRI - compute the inverse of a distributed matrix using the LU factorization computed by PCGETRF .SH SYNOPSIS .TP 20 SUBROUTINE PCGETRI( N, A, IA, JA, DESCA, IPIV, WORK, LWORK, IWORK, LIWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LIWORK, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ), IWORK( * ) .TP 20 .ti +4 COMPLEX A( * ), WORK( * ) .SH PURPOSE PCGETRI computes the inverse of a distributed matrix using the LU factorization computed by PCGETRF. This method inverts U and then computes the inverse of sub( A ) = A(IA:IA+N-1,JA:JA+N-1) denoted InvA by solving the system InvA*L = inv(U) for InvA. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the local pieces of the L and U obtained by the factorization sub( A ) = P*L*U computed by PCGETRF. On exit, if INFO = 0, sub( A ) contains the inverse of the original distributed matrix sub( A ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local input) INTEGER array, dimension LOCr(M_A)+MB_A keeps track of the pivoting information. IPIV(i) is the global row index the local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK = LOCr(N+MOD(IA-1,MB_A))*NB_A. WORK is used to keep a copy of at most an entire column block of sub( A ). If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/local output) INTEGER array, dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK. .TP 8 LIWORK (local or global input) INTEGER The dimension of the array IWORK used as workspace for physically transposing the pivots. LIWORK is local input and must be at least if NPROW == NPCOL then LIWORK = LOCc( N_A + MOD(JA-1, NB_A) ) + NB_A, else LIWORK = LOCc( N_A + MOD(JA-1, NB_A) ) + MAX( CEIL(CEIL(LOCr(M_A)/MB_A)/(LCM/NPROW)), NB_A ) where LCM is the least common multiple of process rows and columns (NPROW and NPCOL). end if If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, U(IA+K-1,IA+K-1) is exactly zero; the matrix is singular and its inverse could not be computed. scalapack-doc-1.5/man/manl/pcgetrs.l0100644000056400000620000001330006335610615017065 0ustar pfrauenfstaff.TH PCGETRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGETRS - solve a system of distributed linear equations op( sub( A ) ) * X = sub( B ) with a general N-by-N distributed matrix sub( A ) using the LU factorization computed by PCGETRF .SH SYNOPSIS .TP 20 SUBROUTINE PCGETRS( TRANS, N, NRHS, A, IA, JA, DESCA, IPIV, B, IB, JB, DESCB, INFO ) .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER IA, IB, INFO, JA, JB, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ), IPIV( * ) .TP 20 .ti +4 COMPLEX A( * ), B( * ) .SH PURPOSE PCGETRS solves a system of distributed linear equations sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1), op( A ) = A, A**T or A**H and sub( B ) denotes B(IB:IB+N-1,JB:JB+NRHS-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block data decomposition ( MB_A=NB_A ). .SH ARGUMENTS .TP 8 TRANS (global input) CHARACTER Specifies the form of the system of equations: .br = 'N': sub( A ) * X = sub( B ) (No transpose) .br = 'T': sub( A )**T * X = sub( B ) (Transpose) .br = 'C': sub( A )**H * X = sub( B ) (Conjugate transpose) .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed submatrix sub( B ). NRHS >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the factors L and U from the factorization sub( A ) = P*L*U; the unit diagonal elements of L are not stored. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local input) INTEGER array, dimension ( LOCr(M_A)+MB_A ) This array contains the pivoting information. IPIV(i) -> The global row local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 B (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_B,LOCc(JB+NRHS-1)). On entry, the right hand sides sub( B ). On exit, sub( B ) is overwritten by the solution distributed matrix X. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pcggqrf.l0100644000056400000620000002351406335610616017060 0ustar pfrauenfstaff.TH PCGGQRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGGQRF - compute a generalized QR factorization of an N-by-M matrix sub( A ) = A(IA:IA+N-1,JA:JA+M-1) and an N-by-P matrix sub( B ) = B(IB:IB+N-1,JB:JB+P-1) .SH SYNOPSIS .TP 20 SUBROUTINE PCGGQRF( N, M, P, A, IA, JA, DESCA, TAUA, B, IB, JB, DESCB, TAUB, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, IB, INFO, JA, JB, LWORK, M, N, P .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX A( * ), B( * ), TAUA( * ), TAUB( * ), WORK( * ) .SH PURPOSE PCGGQRF computes a generalized QR factorization of an N-by-M matrix sub( A ) = A(IA:IA+N-1,JA:JA+M-1) and an N-by-P matrix sub( B ) = B(IB:IB+N-1,JB:JB+P-1): sub( A ) = Q*R, sub( B ) = Q*T*Z, .br where Q is an N-by-N unitary matrix, Z is a P-by-P unitary matrix, and R and T assume one of the forms: .br if N >= M, R = ( R11 ) M , or if N < M, R = ( R11 R12 ) N, ( 0 ) N-M N M-N M .br where R11 is upper triangular, and .br if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) N-P, P-N N ( T21 ) P P .br where T12 or T21 is upper triangular. .br In particular, if sub( B ) is square and nonsingular, the GQR factorization of sub( A ) and sub( B ) implicitly gives the QR factorization of inv( sub( B ) )* sub( A ): .br inv( sub( B ) )*sub( A )= Z'*(inv(T)*R) .br where inv( sub( B ) ) denotes the inverse of the matrix sub( B ), and Z' denotes the conjugate transpose of matrix Z. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrices sub( A ) and sub( B ). N >= 0. .TP 8 M (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). M >= 0. .TP 8 P (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( B ). P >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+M-1)). On entry, the local pieces of the N-by-M distributed matrix sub( A ) which is to be factored. On exit, the elements on and above the diagonal of sub( A ) contain the min(N,M) by M upper trapezoidal matrix R (R is upper triangular if N >= M); the elements below the diagonal, with the array TAUA, represent the unitary matrix Q as a product of min(N,M) elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAUA (local output) COMPLEX, array, dimension LOCc(JA+MIN(N,M)-1). This array contains the scalar factors TAUA of the elementary reflectors which represent the unitary matrix Q. TAUA is tied to the distributed matrix A. (see Further Details). B (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+P-1)). On entry, the local pieces of the N-by-P distributed matrix sub( B ) which is to be factored. On exit, if N <= P, the upper triangle of B(IB:IB+N-1,JB+P-N:JB+P-1) contains the N by N upper triangular matrix T; if N > P, the elements on and above the (N-P)-th subdiagonal contain the N by P upper trapezoidal matrix T; the remaining elements, with the array TAUB, represent the unitary matrix Z as a product of elementary reflectors (see Further Details). IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 TAUB (local output) COMPLEX, array, dimension LOCr(IB+N-1) This array contains the scalar factors of the elementary reflectors which represent the unitary matrix Z. TAUB is tied to the distributed matrix B (see Further Details). WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MAX( NB_A * ( NpA0 + MqA0 + NB_A ), MAX( (NB_A*(NB_A-1))/2, (PqB0 + NpB0)*NB_A ) + NB_A * NB_A, MB_B * ( NpB0 + PqB0 + MB_B ) ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), NpA0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), MqA0 = NUMROC( M+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFB = MOD( IB-1, MB_B ), ICOFFB = MOD( JB-1, NB_B ), IBROW = INDXG2P( IB, MB_B, MYROW, RSRC_B, NPROW ), IBCOL = INDXG2P( JB, NB_B, MYCOL, CSRC_B, NPCOL ), NpB0 = NUMROC( N+IROFFB, MB_B, MYROW, IBROW, NPROW ), PqB0 = NUMROC( P+ICOFFB, NB_B, MYCOL, IBCOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ja) H(ja+1) . . . H(ja+k-1), where k = min(n,m). Each H(i) has the form .br H(i) = I - taua * v * v' .br where taua is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in .br A(ia+i:ia+n-1,ja+i-1), and taua in TAUA(ja+i-1). .br To form Q explicitly, use ScaLAPACK subroutine PCUNGQR. .br To use Q to update another matrix, use ScaLAPACK subroutine PCUNMQR. The matrix Z is represented as a product of elementary reflectors Z = H(ib)' H(ib+1)' . . . H(ib+k-1)', where k = min(n,p). Each H(i) has the form .br H(i) = I - taub * v * v' .br where taub is a complex scalar, and v is a complex vector with v(p-k+i+1:p) = 0 and v(p-k+i) = 1; conjg(v(1:p-k+i-1)) is stored on exit in B(ib+n-k+i-1,jb:jb+p-k+i-2), and taub in TAUB(ib+n-k+i-1). To form Z explicitly, use ScaLAPACK subroutine PCUNGRQ. .br To use Z to update another matrix, use ScaLAPACK subroutine PCUNMRQ. Alignment requirements .br ====================== .br The distributed submatrices sub( A ) and sub( B ) must verify some alignment properties, namely the following expression should be true: ( MB_A.EQ.MB_B .AND. IROFFA.EQ.IROFFB .AND. IAROW.EQ.IBROW ) scalapack-doc-1.5/man/manl/pcggrqf.l0100644000056400000620000002341406335610616017057 0ustar pfrauenfstaff.TH PCGGRQF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCGGRQF - compute a generalized RQ factorization of an M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PCGGRQF( M, P, N, A, IA, JA, DESCA, TAUA, B, IB, JB, DESCB, TAUB, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, IB, INFO, JA, JB, LWORK, M, N, P .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX A( * ), B( * ), TAUA( * ), TAUB( * ), WORK( * ) .SH PURPOSE PCGGRQF computes a generalized RQ factorization of an M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) and a P-by-N matrix sub( B ) = B(IB:IB+P-1,JB:JB+N-1): .br sub( A ) = R*Q, sub( B ) = Z*T*Q, .br where Q is an N-by-N unitary matrix, Z is a P-by-P unitary matrix, and R and T assume one of the forms: .br if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N, N-M M ( R21 ) N N .br where R12 or R21 is upper triangular, and .br if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P, ( 0 ) P-N P N-P N .br where T11 is upper triangular. .br In particular, if sub( B ) is square and nonsingular, the GRQ factorization of sub( A ) and sub( B ) implicitly gives the RQ factorization of sub( A )*inv( sub( B ) ): .br sub( A )*inv( sub( B ) ) = (R*inv(T))*Z' .br where inv( sub( B ) ) denotes the inverse of the matrix sub( B ), and Z' denotes the conjugate transpose of matrix Z. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 P (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( B ). P >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrices sub( A ) and sub( B ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, if M <= N, the upper triangle of A( IA:IA+M-1, JA+N-M:JA+N-1 ) 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 TAUA, represent the unitary matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAUA (local output) COMPLEX, array, dimension LOCr(IA+M-1) This array contains the scalar factors of the elementary reflectors which represent the unitary matrix Q. TAUA is tied to the distributed matrix A (see Further Details). B (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+N-1)). On entry, the local pieces of the P-by-N distributed matrix sub( B ) which is to be factored. On exit, the elements on and above the diagonal of sub( B ) contain the min(P,N) by N upper trapezoidal matrix T (T is upper triangular if P >= N); the elements below the diagonal, with the array TAUB, represent the unitary matrix Z as a product of elementary reflectors (see Further Details). IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 TAUB (local output) COMPLEX, array, dimension LOCc(JB+MIN(P,N)-1). This array contains the scalar factors TAUB of the elementary reflectors which represent the unitary matrix Z. TAUB is tied to the distributed matrix B (see Further Details). WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MAX( MB_A * ( MpA0 + NqA0 + MB_A ), MAX( (MB_A*(MB_A-1))/2, (PpB0 + NqB0)*MB_A ) + MB_A * MB_A, NB_B * ( PpB0 + NqB0 + NB_B ) ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFB = MOD( IB-1, MB_B ), ICOFFB = MOD( JB-1, NB_B ), IBROW = INDXG2P( IB, MB_B, MYROW, RSRC_B, NPROW ), IBCOL = INDXG2P( JB, NB_B, MYCOL, CSRC_B, NPCOL ), PpB0 = NUMROC( P+IROFFB, MB_B, MYROW, IBROW, NPROW ), NqB0 = NUMROC( N+ICOFFB, NB_B, MYCOL, IBCOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ia)' H(ia+1)' . . . H(ia+k-1)', where k = min(m,n). Each H(i) has the form .br H(i) = I - taua * v * v' .br where taua 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(ia+m-k+i-1,ja:ja+n-k+i-2), and taua in TAUA(ia+m-k+i-1). To form Q explicitly, use ScaLAPACK subroutine PCUNGRQ. .br To use Q to update another matrix, use ScaLAPACK subroutine PCUNMRQ. The matrix Z is represented as a product of elementary reflectors Z = H(jb) H(jb+1) . . . H(jb+k-1), where k = min(p,n). Each H(i) has the form .br H(i) = I - taub * v * v' .br where taub is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in .br B(ib+i:ib+p-1,jb+i-1), and taub in TAUB(jb+i-1). .br To form Z explicitly, use ScaLAPACK subroutine PCUNGQR. .br To use Z to update another matrix, use ScaLAPACK subroutine PCUNMQR. Alignment requirements .br ====================== .br The distributed submatrices sub( A ) and sub( B ) must verify some alignment properties, namely the following expression should be true: ( NB_A.EQ.NB_B .AND. ICOFFA.EQ.ICOFFB .AND. IACOL.EQ.IBCOL ) scalapack-doc-1.5/man/manl/pcheevx.l0100644000056400000620000003531306335610616017071 0ustar pfrauenfstaff.TH PCHEEVX l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME .SH SYNOPSIS .TP 20 SUBROUTINE PCHEEVX( JOBZ, RANGE, UPLO, N, A, IA, JA, DESCA, VL, VU, IL, IU, ABSTOL, M, NZ, W, ORFAC, Z, IZ, JZ, DESCZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, IFAIL, ICLUSTR, GAP, INFO ) .TP 20 .ti +4 CHARACTER JOBZ, RANGE, UPLO .TP 20 .ti +4 INTEGER IA, IL, INFO, IU, IZ, JA, JZ, LIWORK, LRWORK, LWORK, M, N, NZ .TP 20 .ti +4 REAL ABSTOL, ORFAC, VL, VU .TP 20 .ti +4 INTEGER DESCA( * ), DESCZ( * ), ICLUSTR( * ), IFAIL( * ), IWORK( * ) .TP 20 .ti +4 REAL GAP( * ), RWORK( * ), W( * ) .TP 20 .ti +4 COMPLEX A( * ), WORK( * ), Z( * ) .TP 20 .ti +4 INTEGER BLOCK_CYCLIC_2D, DLEN_, DTYPE_, CTXT_, M_, N_, MB_, NB_, RSRC_, CSRC_, LLD_ .TP 20 .ti +4 PARAMETER ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1, CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6, RSRC_ = 7, CSRC_ = 8, LLD_ = 9 ) .TP 20 .ti +4 REAL ZERO, ONE, TEN, FIVE .TP 20 .ti +4 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TEN = 10.0E+0, FIVE = 5.0E+0 ) .TP 20 .ti +4 INTEGER IERREIN, IERRCLS, IERRSPC, IERREBZ .TP 20 .ti +4 PARAMETER ( IERREIN = 1, IERRCLS = 2, IERRSPC = 4, IERREBZ = 8 ) .TP 20 .ti +4 LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, QUICKRETURN, VALEIG, WANTZ .TP 20 .ti +4 CHARACTER ORDER .TP 20 .ti +4 INTEGER CSRC_A, I, IACOL, IAROW, ICOFFA, IINFO, INDD, INDD2, INDE, INDE2, INDIBL, INDISP, INDRWORK, INDTAU, INDWORK, IROFFA, IROFFZ, ISCALE, ISIZESTEBZ, ISIZESTEIN, IZROW, LALLWORK, LIWMIN, LLRWORK, LLWORK, LRWMIN, LWMIN, MAXEIGS, MB_A, MB_Z, MQ0, MYCOL, MYROW, NB, NB_A, NB_Z, NEIG, NN, NNP, NP0, NPCOL, NPROCS, NPROW, NQ0, NSPLIT, NZZ, OFFSET, RSRC_A, RSRC_Z, SIZEHEEVX, SIZEORMTR, SIZESTEIN .TP 20 .ti +4 REAL ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM, VLL, VUU .TP 20 .ti +4 INTEGER IDUM1( 4 ), IDUM2( 4 ) .TP 20 .ti +4 LOGICAL LSAME .TP 20 .ti +4 INTEGER ICEIL, INDXG2P, NUMROC .TP 20 .ti +4 REAL PCLANHE, PSLAMCH .TP 20 .ti +4 EXTERNAL LSAME, ICEIL, INDXG2P, NUMROC, PCLANHE, PSLAMCH .TP 20 .ti +4 EXTERNAL BLACS_GRIDINFO, CHK1MAT, IGAMN2D, PCELGET, PCHETRD, PCHK2MAT, PCLASCL, PCSTEIN, PCUNMTR, PSLARED1D, PSSTEBZ, PXERBLA, SGEBR2D, SGEBS2D, SLASRT, SSCAL .TP 20 .ti +4 INTRINSIC ABS, CMPLX, ICHAR, MAX, MIN, MOD, REAL, SQRT .TP 20 .ti +4 IF( BLOCK_CYCLIC_2D*CSRC_*CTXT_*DLEN_*DTYPE_*LLD_*MB_*M_*NB_*N_* RSRC_.LT.0 )RETURN .TP 20 .ti +4 QUICKRETURN = ( N.EQ.0 ) .TP 20 .ti +4 CALL BLACS_GRIDINFO( DESCA( CTXT_ ), NPROW, NPCOL, MYROW, MYCOL ) .TP 20 .ti +4 INFO = 0 .TP 20 .ti +4 IF( NPROW.EQ.-1 ) THEN .TP 20 .ti +4 INFO = -( 800+CTXT_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCZ( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+CTXT_ ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 CALL CHK1MAT( N, 4, N, 4, IA, JA, DESCA, 8, INFO ) .TP 20 .ti +4 CALL CHK1MAT( N, 4, N, 4, IZ, JZ, DESCZ, 21, INFO ) .TP 20 .ti +4 IF( INFO.EQ.0 ) THEN .TP 20 .ti +4 SAFMIN = PSLAMCH( DESCA( CTXT_ ), 'Safe minimum' ) .TP 20 .ti +4 EPS = PSLAMCH( DESCA( CTXT_ ), 'Precision' ) .TP 20 .ti +4 SMLNUM = SAFMIN / EPS .TP 20 .ti +4 BIGNUM = ONE / SMLNUM .TP 20 .ti +4 RMIN = SQRT( SMLNUM ) .TP 20 .ti +4 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) .TP 20 .ti +4 NPROCS = NPROW*NPCOL .TP 20 .ti +4 LOWER = LSAME( UPLO, 'L' ) .TP 20 .ti +4 WANTZ = LSAME( JOBZ, 'V' ) .TP 20 .ti +4 ALLEIG = LSAME( RANGE, 'A' ) .TP 20 .ti +4 VALEIG = LSAME( RANGE, 'V' ) .TP 20 .ti +4 INDEIG = LSAME( RANGE, 'I' ) .TP 20 .ti +4 INDTAU = 1 .TP 20 .ti +4 INDWORK = INDTAU + N .TP 20 .ti +4 LLWORK = LWORK - INDWORK + 1 .TP 20 .ti +4 INDE = 1 .TP 20 .ti +4 INDD = INDE + N .TP 20 .ti +4 INDD2 = INDD + N .TP 20 .ti +4 INDE2 = INDD2 + N .TP 20 .ti +4 INDRWORK = INDE2 + N .TP 20 .ti +4 LLRWORK = LRWORK - INDRWORK + 1 .TP 20 .ti +4 ISIZESTEIN = 3*N + NPROCS + 1 .TP 20 .ti +4 ISIZESTEBZ = MAX( 4*N, 14, NPROCS ) .TP 20 .ti +4 INDIBL = ( MAX( ISIZESTEIN, ISIZESTEBZ ) ) + 1 .TP 20 .ti +4 INDISP = INDIBL + N .TP 20 .ti +4 LQUERY = .FALSE. .TP 20 .ti +4 IF( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 .OR. LRWORK.EQ.-1 ) LQUERY = .TRUE. .TP 20 .ti +4 NNP = MAX( N, NPROCS+1, 4 ) .TP 20 .ti +4 LIWMIN = 6*NNP .TP 20 .ti +4 NPROCS = NPROW*NPCOL .TP 20 .ti +4 NB_A = DESCA( NB_ ) .TP 20 .ti +4 MB_A = DESCA( MB_ ) .TP 20 .ti +4 NB_Z = DESCZ( NB_ ) .TP 20 .ti +4 MB_Z = DESCZ( MB_ ) .TP 20 .ti +4 NB = NB_A .TP 20 .ti +4 NN = MAX( N, NB, 2 ) .TP 20 .ti +4 RSRC_A = DESCA( RSRC_ ) .TP 20 .ti +4 CSRC_A = DESCA( CSRC_ ) .TP 20 .ti +4 RSRC_Z = DESCZ( RSRC_ ) .TP 20 .ti +4 IROFFA = MOD( IA-1, MB_A ) .TP 20 .ti +4 ICOFFA = MOD( JA-1, NB_A ) .TP 20 .ti +4 IROFFZ = MOD( IZ-1, MB_A ) .TP 20 .ti +4 IAROW = INDXG2P( 1, NB_A, MYROW, RSRC_A, NPROW ) .TP 20 .ti +4 IACOL = INDXG2P( 1, MB_A, MYCOL, CSRC_A, NPCOL ) .TP 20 .ti +4 IZROW = INDXG2P( 1, NB_A, MYROW, RSRC_Z, NPROW ) .TP 20 .ti +4 NP0 = NUMROC( N+IROFFA, NB_Z, MYROW, IAROW, NPROW ) .TP 20 .ti +4 MQ0 = NUMROC( N+ICOFFA, NB_Z, MYCOL, IACOL, NPCOL ) .TP 20 .ti +4 IF( ( .NOT.WANTZ ) .OR. ( VALEIG .AND. ( .NOT.LQUERY ) ) ) THEN .TP 20 .ti +4 LWMIN = N + MAX( NB*( NP0+1 ), 3 ) .TP 20 .ti +4 LRWMIN = 5*NN + 4*N .TP 20 .ti +4 NEIG = 0 .TP 20 .ti +4 ELSE .TP 20 .ti +4 IF( ALLEIG .OR. VALEIG ) THEN .TP 20 .ti +4 NEIG = N .TP 20 .ti +4 ELSE IF( INDEIG ) THEN .TP 20 .ti +4 NEIG = IU - IL + 1 .TP 20 .ti +4 END IF .TP 20 .ti +4 MQ0 = NUMROC( MAX( NEIG, NB, 2 ), NB, MYCOL, IACOL, NPCOL ) .TP 20 .ti +4 NQ0 = NUMROC( NN, NB, 0, 0, NPCOL ) .TP 20 .ti +4 LWMIN = N + ( NP0+NQ0+NB )*NB .TP 20 .ti +4 LRWMIN = 4*N + MAX( 5*NN, NP0*MQ0 ) + ICEIL( NEIG, NPROW*NPCOL )*NN .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( INFO.EQ.0 ) THEN .TP 20 .ti +4 IF( MYROW.EQ.0 .AND. MYCOL.EQ.0 ) THEN .TP 20 .ti +4 RWORK( 1 ) = ABSTOL .TP 20 .ti +4 IF( VALEIG ) THEN .TP 20 .ti +4 RWORK( 2 ) = VL .TP 20 .ti +4 RWORK( 3 ) = VU .TP 20 .ti +4 ELSE .TP 20 .ti +4 RWORK( 2 ) = ZERO .TP 20 .ti +4 RWORK( 3 ) = ZERO .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL SGEBS2D( DESCA( CTXT_ ), 'ALL', ' ', 3, 1, RWORK, 3 ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 CALL SGEBR2D( DESCA( CTXT_ ), 'ALL', ' ', 3, 1, RWORK, 3, 0, 0 ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN .TP 20 .ti +4 INFO = -1 .TP 20 .ti +4 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN .TP 20 .ti +4 INFO = -2 .TP 20 .ti +4 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN .TP 20 .ti +4 INFO = -3 .TP 20 .ti +4 ELSE IF( VALEIG .AND. N.GT.0 .AND. VU.LE.VL ) THEN .TP 20 .ti +4 INFO = -10 .TP 20 .ti +4 ELSE IF( INDEIG .AND. ( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) ) THEN .TP 20 .ti +4 INFO = -11 .TP 20 .ti +4 ELSE IF( INDEIG .AND. ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) ) THEN .TP 20 .ti +4 INFO = -12 .TP 20 .ti +4 ELSE IF( LWORK.LT.LWMIN .AND. LWORK.NE.-1 ) THEN .TP 20 .ti +4 INFO = -23 .TP 20 .ti +4 ELSE IF( LRWORK.LT.LRWMIN .AND. LRWORK.NE.-1 ) THEN .TP 20 .ti +4 INFO = -25 .TP 20 .ti +4 ELSE IF( LIWORK.LT.LIWMIN .AND. LIWORK.NE.-1 ) THEN .TP 20 .ti +4 INFO = -27 .TP 20 .ti +4 ELSE IF( VALEIG .AND. ( ABS( RWORK( 2 )-VL ).GT.FIVE*EPS* ABS( VL ) ) ) THEN .TP 20 .ti +4 INFO = -9 .TP 20 .ti +4 ELSE IF( VALEIG .AND. ( ABS( RWORK( 3 )-VU ).GT.FIVE*EPS* ABS( VU ) ) ) THEN .TP 20 .ti +4 INFO = -10 .TP 20 .ti +4 ELSE IF( ABS( RWORK( 1 )-ABSTOL ).GT.FIVE*EPS* ABS( ABSTOL ) ) THEN .TP 20 .ti +4 INFO = -13 .TP 20 .ti +4 ELSE IF( IROFFA.NE.IROFFZ ) THEN .TP 20 .ti +4 INFO = -19 .TP 20 .ti +4 ELSE IF( IROFFA.NE.0 ) THEN .TP 20 .ti +4 INFO = -6 .TP 20 .ti +4 ELSE IF( IAROW.NE.IZROW ) THEN .TP 20 .ti +4 INFO = -19 .TP 20 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCA( NB_ ) ) THEN .TP 20 .ti +4 INFO = -( 800+NB_ ) .TP 20 .ti +4 ELSE IF( DESCA( M_ ).NE.DESCZ( M_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+M_ ) .TP 20 .ti +4 ELSE IF( DESCA( N_ ).NE.DESCZ( N_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+N_ ) .TP 20 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCZ( MB_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+MB_ ) .TP 20 .ti +4 ELSE IF( DESCA( NB_ ).NE.DESCZ( NB_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+NB_ ) .TP 20 .ti +4 ELSE IF( DESCA( RSRC_ ).NE.DESCZ( RSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+RSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CSRC_ ).NE.DESCZ( CSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+CSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCZ( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+CTXT_ ) .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 IDUM1( 1 ) = ICHAR( 'V' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 1 ) = ICHAR( 'N' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 1 ) = 1 .TP 20 .ti +4 IF( LOWER ) THEN .TP 20 .ti +4 IDUM1( 2 ) = ICHAR( 'L' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 2 ) = ICHAR( 'U' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 2 ) = 2 .TP 20 .ti +4 IF( ALLEIG ) THEN .TP 20 .ti +4 IDUM1( 3 ) = ICHAR( 'A' ) .TP 20 .ti +4 ELSE IF( INDEIG ) THEN .TP 20 .ti +4 IDUM1( 3 ) = ICHAR( 'I' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 3 ) = ICHAR( 'V' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 3 ) = 3 .TP 20 .ti +4 IF( LQUERY ) THEN .TP 20 .ti +4 IDUM1( 4 ) = -1 .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 4 ) = 1 .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 4 ) = 4 .TP 20 .ti +4 CALL PCHK2MAT( N, 4, N, 4, IA, JA, DESCA, 8, N, 4, N, 4, IZ, JZ, DESCZ, 21, 4, IDUM1, IDUM2, INFO ) .TP 20 .ti +4 WORK( 1 ) = CMPLX( LWMIN ) .TP 20 .ti +4 RWORK( 1 ) = REAL( LRWMIN ) .TP 20 .ti +4 IWORK( 1 ) = LIWMIN .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( INFO.NE.0 ) THEN .TP 20 .ti +4 CALL PXERBLA( DESCA( CTXT_ ), 'PCHEEVX', -INFO ) .TP 20 .ti +4 RETURN .TP 20 .ti +4 ELSE IF( LQUERY ) THEN .TP 20 .ti +4 RETURN .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( QUICKRETURN ) THEN .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 NZ = 0 .TP 20 .ti +4 ICLUSTR( 1 ) = 0 .TP 20 .ti +4 END IF .TP 20 .ti +4 M = 0 .TP 20 .ti +4 WORK( 1 ) = CMPLX( LWMIN ) .TP 20 .ti +4 RWORK( 1 ) = REAL( LRWMIN ) .TP 20 .ti +4 IWORK( 1 ) = LIWMIN .TP 20 .ti +4 RETURN .TP 20 .ti +4 END IF .TP 20 .ti +4 ABSTLL = ABSTOL .TP 20 .ti +4 ISCALE = 0 .TP 20 .ti +4 IF( VALEIG ) THEN .TP 20 .ti +4 VLL = VL .TP 20 .ti +4 VUU = VU .TP 20 .ti +4 ELSE .TP 20 .ti +4 VLL = ZERO .TP 20 .ti +4 VUU = ZERO .TP 20 .ti +4 END IF .TP 20 .ti +4 ANRM = PCLANHE( '1', UPLO, N, A, IA, JA, DESCA, RWORK( INDRWORK ) ) .TP 20 .ti +4 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN .TP 20 .ti +4 ISCALE = 1 .TP 20 .ti +4 SIGMA = RMIN / ANRM .TP 20 .ti +4 ANRM = ANRM*SIGMA .TP 20 .ti +4 ELSE IF( ANRM.GT.RMAX ) THEN .TP 20 .ti +4 ISCALE = 1 .TP 20 .ti +4 SIGMA = RMAX / ANRM .TP 20 .ti +4 ANRM = ANRM*SIGMA .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( ISCALE.EQ.1 ) THEN .TP 20 .ti +4 CALL PCLASCL( UPLO, ONE, SIGMA, N, N, A, IA, JA, DESCA, IINFO ) .TP 20 .ti +4 IF( ABSTOL.GT.0 ) ABSTLL = ABSTOL*SIGMA .TP 20 .ti +4 IF( VALEIG ) THEN .TP 20 .ti +4 VLL = VL*SIGMA .TP 20 .ti +4 VUU = VU*SIGMA .TP 20 .ti +4 IF( VUU.EQ.VLL ) THEN .TP 20 .ti +4 VUU = VUU + 2*MAX( ABS( VUU )*EPS, SAFMIN ) .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 LALLWORK = LLRWORK .TP 20 .ti +4 CALL PCHETRD( UPLO, N, A, IA, JA, DESCA, RWORK( INDD ), RWORK( INDE ), WORK( INDTAU ), WORK( INDWORK ), LLWORK, IINFO ) .TP 20 .ti +4 OFFSET = 0 .TP 20 .ti +4 IF( IA.EQ.1 .AND. JA.EQ.1 .AND. RSRC_A.EQ.0 .AND. CSRC_A.EQ.0 ) THEN .TP 20 .ti +4 CALL PSLARED1D( N, IA, JA, DESCA, RWORK( INDD ), RWORK( INDD2 ), RWORK( INDRWORK ), LLRWORK ) .TP 20 .ti +4 CALL PSLARED1D( N, IA, JA, DESCA, RWORK( INDE ), RWORK( INDE2 ), RWORK( INDRWORK ), LLRWORK ) .TP 20 .ti +4 IF( .NOT.LOWER ) OFFSET = 1 .TP 20 .ti +4 ELSE .TP 20 .ti +4 DO 10 I = 1, N .TP 20 .ti +4 CALL PCELGET( 'A', ' ', WORK( INDD2+I-1 ), A, I+IA-1, I+JA-1, DESCA ) .TP 20 .ti +4 RWORK( INDD2+I-1 ) = REAL( WORK( INDD2+I-1 ) ) .TP 20 .ti +4 10 CONTINUE .TP 20 .ti +4 IF( LSAME( UPLO, 'U' ) ) THEN .TP 20 .ti +4 DO 20 I = 1, N - 1 .TP 20 .ti +4 CALL PCELGET( 'A', ' ', WORK( INDE2+I-1 ), A, I+IA-1, I+JA, DESCA ) .TP 20 .ti +4 RWORK( INDE2+I-1 ) = REAL( WORK( INDE2+I-1 ) ) .TP 20 .ti +4 20 CONTINUE .TP 20 .ti +4 ELSE .TP 20 .ti +4 DO 30 I = 1, N - 1 .TP 20 .ti +4 CALL PCELGET( 'A', ' ', WORK( INDE2+I-1 ), A, I+IA, I+JA-1, DESCA ) .TP 20 .ti +4 RWORK( INDE2+I-1 ) = REAL( WORK( INDE2+I-1 ) ) .TP 20 .ti +4 30 CONTINUE .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 ORDER = 'b' .TP 20 .ti +4 ELSE .TP 20 .ti +4 ORDER = 'e' .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PSSTEBZ( DESCA( CTXT_ ), RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, RWORK( INDD2 ), RWORK( INDE2+OFFSET ), M, NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWORK ), LLRWORK, IWORK( 1 ), ISIZESTEBZ, IINFO ) .TP 20 .ti +4 IF( IINFO.NE.0 ) THEN .TP 20 .ti +4 INFO = INFO + IERREBZ .TP 20 .ti +4 DO 40 I = 1, M .TP 20 .ti +4 IWORK( INDIBL+I-1 ) = ABS( IWORK( INDIBL+I-1 ) ) .TP 20 .ti +4 40 CONTINUE .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 IF( VALEIG ) THEN .TP 20 .ti +4 CALL IGAMN2D( DESCA( CTXT_ ), 'A', ' ', 1, 1, LALLWORK, 1, 1, 1, -1, -1, -1 ) .TP 20 .ti +4 MAXEIGS = DESCZ( N_ ) .TP 20 .ti +4 DO 50 NZ = MIN( MAXEIGS, M ), 0, -1 .TP 20 .ti +4 MQ0 = NUMROC( NZ, NB, 0, 0, NPCOL ) .TP 20 .ti +4 SIZESTEIN = ICEIL( NZ, NPROCS )*N + MAX( 5*N, NP0*MQ0 ) .TP 20 .ti +4 SIZEORMTR = MAX( ( NB*( NB-1 ) ) / 2, ( MQ0+NP0 )*NB ) + NB*NB .TP 20 .ti +4 SIZEHEEVX = MAX( SIZESTEIN, SIZEORMTR ) .TP 20 .ti +4 IF( SIZEHEEVX.LE.LALLWORK ) GO TO 60 .TP 20 .ti +4 50 CONTINUE .TP 20 .ti +4 60 CONTINUE .TP 20 .ti +4 ELSE .TP 20 .ti +4 NZ = M .TP 20 .ti +4 END IF .TP 20 .ti +4 NZ = MAX( NZ, 0 ) .TP 20 .ti +4 IF( NZ.NE.M ) THEN .TP 20 .ti +4 INFO = INFO + IERRSPC .TP 20 .ti +4 DO 70 I = 1, M .TP 20 .ti +4 IFAIL( I ) = 0 .TP 20 .ti +4 70 CONTINUE .TP 20 .ti +4 IF( NSPLIT.GT.1 ) THEN .TP 20 .ti +4 CALL SLASRT( 'I', M, W, IINFO ) .TP 20 .ti +4 IF( NZ.GT.0 ) THEN .TP 20 .ti +4 VUU = W( NZ ) - TEN*( EPS*ANRM+SAFMIN ) .TP 20 .ti +4 IF( VLL.GE.VUU ) THEN .TP 20 .ti +4 NZZ = 0 .TP 20 .ti +4 ELSE .TP 20 .ti +4 CALL PSSTEBZ( DESCA( CTXT_ ), RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, RWORK( INDD2 ), RWORK( INDE2+ OFFSET ), NZZ, NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWORK ), LLRWORK, IWORK( 1 ), ISIZESTEBZ, IINFO ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( MOD( INFO / IERREBZ, 1 ).EQ.0 ) THEN .TP 20 .ti +4 IF( NZZ.GT.NZ .OR. IINFO.NE.0 ) THEN .TP 20 .ti +4 INFO = INFO + IERREBZ .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 NZ = MIN( NZ, NZZ ) .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PCSTEIN( N, RWORK( INDD2 ), RWORK( INDE2+OFFSET ), NZ, W, IWORK( INDIBL ), IWORK( INDISP ), ORFAC, Z, IZ, JZ, DESCZ, RWORK( INDRWORK ), LALLWORK, IWORK( 1 ), ISIZESTEIN, IFAIL, ICLUSTR, GAP, IINFO ) .TP 20 .ti +4 IF( IINFO.GE.NZ+1 ) INFO = INFO + IERRCLS .TP 20 .ti +4 IF( MOD( IINFO, NZ+1 ).NE.0 ) INFO = INFO + IERREIN .TP 20 .ti +4 IF( NZ.GT.0 ) THEN .TP 20 .ti +4 CALL PCUNMTR( 'L', UPLO, 'N', N, NZ, A, IA, JA, DESCA, WORK( INDTAU ), Z, IZ, JZ, DESCZ, WORK( INDWORK ), LLWORK, IINFO ) .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( ISCALE.EQ.1 ) THEN .TP 20 .ti +4 CALL SSCAL( M, ONE / SIGMA, W, 1 ) .TP 20 .ti +4 END IF .TP 20 .ti +4 WORK( 1 ) = CMPLX( LWMIN ) .TP 20 .ti +4 RWORK( 1 ) = REAL( LRWMIN ) .TP 20 .ti +4 IWORK( 1 ) = LIWMIN .TP 20 .ti +4 RETURN .TP 20 .ti +4 END .SH PURPOSE scalapack-doc-1.5/man/manl/pchegs2.l0100644000056400000620000001415706335610616016765 0ustar pfrauenfstaff.TH PCHEGS2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCHEGS2 - reduce a complex Hermitian-definite generalized eigenproblem to standard form .SH SYNOPSIS .TP 20 SUBROUTINE PCHEGS2( IBTYPE, UPLO, N, A, IA, JA, DESCA, B, IB, JB, DESCB, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IB, IBTYPE, INFO, JA, JB, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX A( * ), B( * ) .SH PURPOSE PCHEGS2 reduces a complex Hermitian-definite generalized eigenproblem to standard form. In the following sub( A ) denotes A( IA:IA+N-1, JA:JA+N-1 ) and sub( B ) denotes B( IB:IB+N-1, JB:JB+N-1 ). .br If IBTYPE = 1, the problem is sub( A )*x = lambda*sub( B )*x, and sub( A ) is overwritten by inv(U**H)*sub( A )*inv(U) or inv(L)*sub( A )*inv(L**H) .br If IBTYPE = 2 or 3, the problem is sub( A )*sub( B )*x = lambda*x or sub( B )*sub( A )*x = lambda*x, and sub( A ) is overwritten by U*sub( A )*U**H or L**H*sub( A )*L. .br sub( B ) must have been previously factorized as U**H*U or L*L**H by PCPOTRF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 9 IBTYPE (global input) INTEGER = 1: compute inv(U**H)*sub( A )*inv(U) or inv(L)*sub( A )*inv(L**H); = 2 or 3: compute U*sub( A )*U**H or L**H*sub( A )*L. .TP 8 UPLO (global input) CHARACTER .br = 'U': Upper triangle of sub( A ) is stored and sub( B ) is factored as U**H*U; = 'L': Lower triangle of sub( A ) is stored and sub( B ) is factored as L*L**H. .TP 8 N (global input) INTEGER The order of the matrices sub( A ) and sub( B ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N Hermitian distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced. On exit, if INFO = 0, the transformed matrix, stored in the same format as sub( A ). .TP 8 IA (global input) INTEGER A's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JA (global input) INTEGER A's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+N-1)). On entry, this array contains the local pieces of the triangular factor from the Cholesky factorization of sub( B ), as returned by PCPOTRF. .TP 8 IB (global input) INTEGER B's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JB (global input) INTEGER B's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pchegst.l0100644000056400000620000001457606335610616017074 0ustar pfrauenfstaff.TH PCHEGST l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCHEGST - reduce a complex Hermitian-definite generalized eigenproblem to standard form .SH SYNOPSIS .TP 20 SUBROUTINE PCHEGST( IBTYPE, UPLO, N, A, IA, JA, DESCA, B, IB, JB, DESCB, SCALE, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IB, IBTYPE, INFO, JA, JB, N .TP 20 .ti +4 REAL SCALE .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX A( * ), B( * ) .SH PURPOSE PCHEGST reduces a complex Hermitian-definite generalized eigenproblem to standard form. In the following sub( A ) denotes A( IA:IA+N-1, JA:JA+N-1 ) and sub( B ) denotes B( IB:IB+N-1, JB:JB+N-1 ). .br If IBTYPE = 1, the problem is sub( A )*x = lambda*sub( B )*x, and sub( A ) is overwritten by inv(U**H)*sub( A )*inv(U) or inv(L)*sub( A )*inv(L**H) .br If IBTYPE = 2 or 3, the problem is sub( A )*sub( B )*x = lambda*x or sub( B )*sub( A )*x = lambda*x, and sub( A ) is overwritten by U*sub( A )*U**H or L**H*sub( A )*L. .br sub( B ) must have been previously factorized as U**H*U or L*L**H by PCPOTRF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 9 IBTYPE (global input) INTEGER = 1: compute inv(U**H)*sub( A )*inv(U) or inv(L)*sub( A )*inv(L**H); = 2 or 3: compute U*sub( A )*U**H or L**H*sub( A )*L. .TP 8 UPLO (global input) CHARACTER .br = 'U': Upper triangle of sub( A ) is stored and sub( B ) is factored as U**H*U; = 'L': Lower triangle of sub( A ) is stored and sub( B ) is factored as L*L**H. .TP 8 N (global input) INTEGER The order of the matrices sub( A ) and sub( B ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N Hermitian distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced. On exit, if INFO = 0, the transformed matrix, stored in the same format as sub( A ). .TP 8 IA (global input) INTEGER A's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JA (global input) INTEGER A's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+N-1)). On entry, this array contains the local pieces of the triangular factor from the Cholesky factorization of sub( B ), as returned by PCPOTRF. .TP 8 IB (global input) INTEGER B's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JB (global input) INTEGER B's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 SCALE (global output) REAL Amount by which the eigenvalues should be scaled to compensate for the scaling performed in this routine. At present, SCALE is always returned as 1.0, it is returned here to allow for future enhancement. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pchegvx.l0100644000056400000620000002446406335610616017100 0ustar pfrauenfstaff.TH PCHEGVX l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME .SH SYNOPSIS .TP 20 SUBROUTINE PCHEGVX( IBTYPE, JOBZ, RANGE, UPLO, N, A, IA, JA, DESCA, B, IB, JB, DESCB, VL, VU, IL, IU, ABSTOL, M, NZ, W, ORFAC, Z, IZ, JZ, DESCZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, IFAIL, ICLUSTR, GAP, INFO ) .TP 20 .ti +4 CHARACTER JOBZ, RANGE, UPLO .TP 20 .ti +4 INTEGER IA, IB, IBTYPE, IL, INFO, IU, IZ, JA, JB, JZ, LIWORK, LRWORK, LWORK, M, N, NZ .TP 20 .ti +4 REAL ABSTOL, ORFAC, VL, VU .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ), DESCZ( * ), ICLUSTR( * ), IFAIL( * ), IWORK( * ) .TP 20 .ti +4 REAL GAP( * ), RWORK( * ), W( * ) .TP 20 .ti +4 COMPLEX A( * ), B( * ), WORK( * ), Z( * ) .TP 20 .ti +4 INTEGER BLOCK_CYCLIC_2D, DLEN_, DTYPE_, CTXT_, M_, N_, MB_, NB_, RSRC_, CSRC_, LLD_ .TP 20 .ti +4 PARAMETER ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1, CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6, RSRC_ = 7, CSRC_ = 8, LLD_ = 9 ) .TP 20 .ti +4 COMPLEX ONE .TP 20 .ti +4 PARAMETER ( ONE = 1.0E+0 ) .TP 20 .ti +4 REAL FIVE, ZERO .TP 20 .ti +4 PARAMETER ( FIVE = 5.0E+0, ZERO = 0.0E+0 ) .TP 20 .ti +4 INTEGER IERRNPD .TP 20 .ti +4 PARAMETER ( IERRNPD = 16 ) .TP 20 .ti +4 LOGICAL ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER IACOL, IAROW, IBCOL, IBROW, ICOFFA, ICOFFB, ICTXT, IROFFA, IROFFB, LIWMIN, LRWMIN, LWMIN, MQ0, MYCOL, MYROW, NB, NEIG, NN, NP0, NPCOL, NPROW .TP 20 .ti +4 REAL EPS, SCALE .TP 20 .ti +4 INTEGER IDUM1( 5 ), IDUM2( 5 ) .TP 20 .ti +4 LOGICAL LSAME .TP 20 .ti +4 INTEGER ICEIL, INDXG2P, NUMROC .TP 20 .ti +4 REAL PSLAMCH .TP 20 .ti +4 EXTERNAL LSAME, ICEIL, INDXG2P, NUMROC, PSLAMCH .TP 20 .ti +4 EXTERNAL BLACS_GRIDINFO, CHK1MAT, PCHEEVX, PCHEGST, PCHK1MAT, PCHK2MAT, PCPOTRF, PCTRMM, PCTRSM, PXERBLA, SGEBR2D, SGEBS2D, SSCAL .TP 20 .ti +4 INTRINSIC ABS, CMPLX, ICHAR, MAX, MIN, MOD, REAL .TP 20 .ti +4 IF( BLOCK_CYCLIC_2D*CSRC_*CTXT_*DLEN_*DTYPE_*LLD_*MB_*M_*NB_*N_* RSRC_.LT.0 )RETURN .TP 20 .ti +4 ICTXT = DESCA( CTXT_ ) .TP 20 .ti +4 CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL ) .TP 20 .ti +4 INFO = 0 .TP 20 .ti +4 IF( NPROW.EQ.-1 ) THEN .TP 20 .ti +4 INFO = -( 900+CTXT_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCB( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+CTXT_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCZ( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 2600+CTXT_ ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 EPS = PSLAMCH( DESCA( CTXT_ ), 'Precision' ) .TP 20 .ti +4 WANTZ = LSAME( JOBZ, 'V' ) .TP 20 .ti +4 UPPER = LSAME( UPLO, 'U' ) .TP 20 .ti +4 ALLEIG = LSAME( RANGE, 'A' ) .TP 20 .ti +4 VALEIG = LSAME( RANGE, 'V' ) .TP 20 .ti +4 INDEIG = LSAME( RANGE, 'I' ) .TP 20 .ti +4 CALL CHK1MAT( N, 4, N, 4, IA, JA, DESCA, 9, INFO ) .TP 20 .ti +4 CALL CHK1MAT( N, 4, N, 4, IB, JB, DESCB, 13, INFO ) .TP 20 .ti +4 CALL CHK1MAT( N, 4, N, 4, IZ, JZ, DESCZ, 26, INFO ) .TP 20 .ti +4 IF( INFO.EQ.0 ) THEN .TP 20 .ti +4 IF( MYROW.EQ.0 .AND. MYCOL.EQ.0 ) THEN .TP 20 .ti +4 RWORK( 1 ) = ABSTOL .TP 20 .ti +4 IF( VALEIG ) THEN .TP 20 .ti +4 RWORK( 2 ) = VL .TP 20 .ti +4 RWORK( 3 ) = VU .TP 20 .ti +4 ELSE .TP 20 .ti +4 RWORK( 2 ) = ZERO .TP 20 .ti +4 RWORK( 3 ) = ZERO .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL SGEBS2D( DESCA( CTXT_ ), 'ALL', ' ', 3, 1, RWORK, 3 ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 CALL SGEBR2D( DESCA( CTXT_ ), 'ALL', ' ', 3, 1, RWORK, 3, 0, 0 ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IAROW = INDXG2P( IA, DESCA( MB_ ), MYROW, DESCA( RSRC_ ), NPROW ) .TP 20 .ti +4 IBROW = INDXG2P( IB, DESCB( MB_ ), MYROW, DESCB( RSRC_ ), NPROW ) .TP 20 .ti +4 IACOL = INDXG2P( JA, DESCA( NB_ ), MYCOL, DESCA( CSRC_ ), NPCOL ) .TP 20 .ti +4 IBCOL = INDXG2P( JB, DESCB( NB_ ), MYCOL, DESCB( CSRC_ ), NPCOL ) .TP 20 .ti +4 IROFFA = MOD( IA-1, DESCA( MB_ ) ) .TP 20 .ti +4 ICOFFA = MOD( JA-1, DESCA( NB_ ) ) .TP 20 .ti +4 IROFFB = MOD( IB-1, DESCB( MB_ ) ) .TP 20 .ti +4 ICOFFB = MOD( JB-1, DESCB( NB_ ) ) .TP 20 .ti +4 LQUERY = .FALSE. .TP 20 .ti +4 IF( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 .OR. LRWORK.EQ.-1 ) LQUERY = .TRUE. .TP 20 .ti +4 LIWMIN = 6*MAX( N, ( NPROW*NPCOL )+1, 4 ) .TP 20 .ti +4 NB = DESCA( MB_ ) .TP 20 .ti +4 NN = MAX( N, NB, 2 ) .TP 20 .ti +4 NP0 = NUMROC( NN, NB, 0, 0, NPROW ) .TP 20 .ti +4 IF( ( .NOT.WANTZ ) .OR. ( VALEIG .AND. ( .NOT.LQUERY ) ) ) THEN .TP 20 .ti +4 LWMIN = N + MAX( NB*( NP0+1 ), 3 ) .TP 20 .ti +4 LRWMIN = 5*NN + 4*N .TP 20 .ti +4 NEIG = 0 .TP 20 .ti +4 ELSE .TP 20 .ti +4 IF( ALLEIG .OR. VALEIG ) THEN .TP 20 .ti +4 NEIG = N .TP 20 .ti +4 ELSE IF( INDEIG ) THEN .TP 20 .ti +4 NEIG = IU - IL + 1 .TP 20 .ti +4 END IF .TP 20 .ti +4 MQ0 = NUMROC( MAX( NEIG, NB, 2 ), NB, 0, 0, NPCOL ) .TP 20 .ti +4 LWMIN = N + ( NP0+MQ0+NB )*NB .TP 20 .ti +4 LRWMIN = 4*N + MAX( 5*NN, NP0*MQ0 ) + ICEIL( NEIG, NPROW*NPCOL )*NN .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( IBTYPE.LT.1 .OR. IBTYPE.GT.3 ) THEN .TP 20 .ti +4 INFO = -1 .TP 20 .ti +4 ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN .TP 20 .ti +4 INFO = -2 .TP 20 .ti +4 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN .TP 20 .ti +4 INFO = -3 .TP 20 .ti +4 ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN .TP 20 .ti +4 INFO = -4 .TP 20 .ti +4 ELSE IF( N.LT.0 ) THEN .TP 20 .ti +4 INFO = -5 .TP 20 .ti +4 ELSE IF( IROFFA.NE.0 ) THEN .TP 20 .ti +4 INFO = -7 .TP 20 .ti +4 ELSE IF( ICOFFA.NE.0 ) THEN .TP 20 .ti +4 INFO = -8 .TP 20 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCA( NB_ ) ) THEN .TP 20 .ti +4 INFO = -( 900+NB_ ) .TP 20 .ti +4 ELSE IF( DESCA( M_ ).NE.DESCB( M_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+M_ ) .TP 20 .ti +4 ELSE IF( DESCA( N_ ).NE.DESCB( N_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+N_ ) .TP 20 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCB( MB_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+MB_ ) .TP 20 .ti +4 ELSE IF( DESCA( NB_ ).NE.DESCB( NB_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+NB_ ) .TP 20 .ti +4 ELSE IF( DESCA( RSRC_ ).NE.DESCB( RSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+RSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CSRC_ ).NE.DESCB( CSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+CSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCB( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+CTXT_ ) .TP 20 .ti +4 ELSE IF( DESCA( M_ ).NE.DESCZ( M_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+M_ ) .TP 20 .ti +4 ELSE IF( DESCA( N_ ).NE.DESCZ( N_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+N_ ) .TP 20 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCZ( MB_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+MB_ ) .TP 20 .ti +4 ELSE IF( DESCA( NB_ ).NE.DESCZ( NB_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+NB_ ) .TP 20 .ti +4 ELSE IF( DESCA( RSRC_ ).NE.DESCZ( RSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+RSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CSRC_ ).NE.DESCZ( CSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+CSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCZ( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+CTXT_ ) .TP 20 .ti +4 ELSE IF( IROFFB.NE.0 .OR. IBROW.NE.IAROW ) THEN .TP 20 .ti +4 INFO = -11 .TP 20 .ti +4 ELSE IF( ICOFFB.NE.0 .OR. IBCOL.NE.IACOL ) THEN .TP 20 .ti +4 INFO = -12 .TP 20 .ti +4 ELSE IF( VALEIG .AND. N.GT.0 .AND. VU.LE.VL ) THEN .TP 20 .ti +4 INFO = -15 .TP 20 .ti +4 ELSE IF( INDEIG .AND. ( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) ) THEN .TP 20 .ti +4 INFO = -16 .TP 20 .ti +4 ELSE IF( INDEIG .AND. ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) ) THEN .TP 20 .ti +4 INFO = -17 .TP 20 .ti +4 ELSE IF( VALEIG .AND. ( ABS( RWORK( 2 )-VL ).GT.FIVE*EPS* ABS( VL ) ) ) THEN .TP 20 .ti +4 INFO = -14 .TP 20 .ti +4 ELSE IF( VALEIG .AND. ( ABS( RWORK( 3 )-VU ).GT.FIVE*EPS* ABS( VU ) ) ) THEN .TP 20 .ti +4 INFO = -15 .TP 20 .ti +4 ELSE IF( ABS( RWORK( 1 )-ABSTOL ).GT.FIVE*EPS* ABS( ABSTOL ) ) THEN .TP 20 .ti +4 INFO = -18 .TP 20 .ti +4 ELSE IF( LWORK.LT.LWMIN .AND. LWORK.NE.-1 ) THEN .TP 20 .ti +4 INFO = -28 .TP 20 .ti +4 ELSE IF( LRWORK.LT.LRWMIN .AND. LRWORK.NE.-1 ) THEN .TP 20 .ti +4 INFO = -30 .TP 20 .ti +4 ELSE IF( LIWORK.LT.LIWMIN .AND. LIWORK.NE.-1 ) THEN .TP 20 .ti +4 INFO = -32 .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM1( 1 ) = IBTYPE .TP 20 .ti +4 IDUM2( 1 ) = 1 .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 IDUM1( 2 ) = ICHAR( 'V' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 2 ) = ICHAR( 'N' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 2 ) = 2 .TP 20 .ti +4 IF( UPPER ) THEN .TP 20 .ti +4 IDUM1( 3 ) = ICHAR( 'U' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 3 ) = ICHAR( 'L' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 3 ) = 3 .TP 20 .ti +4 IF( ALLEIG ) THEN .TP 20 .ti +4 IDUM1( 4 ) = ICHAR( 'A' ) .TP 20 .ti +4 ELSE IF( INDEIG ) THEN .TP 20 .ti +4 IDUM1( 4 ) = ICHAR( 'I' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 4 ) = ICHAR( 'V' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 4 ) = 4 .TP 20 .ti +4 IF( LQUERY ) THEN .TP 20 .ti +4 IDUM1( 5 ) = -1 .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 5 ) = 1 .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 5 ) = 5 .TP 20 .ti +4 CALL PCHK2MAT( N, 4, N, 4, IA, JA, DESCA, 9, N, 4, N, 4, IB, JB, DESCB, 13, 5, IDUM1, IDUM2, INFO ) .TP 20 .ti +4 CALL PCHK1MAT( N, 4, N, 4, IZ, JZ, DESCZ, 26, 0, IDUM1, IDUM2, INFO ) .TP 20 .ti +4 END IF .TP 20 .ti +4 WORK( 1 ) = CMPLX( REAL( LWMIN ) ) .TP 20 .ti +4 RWORK( 1 ) = REAL( LRWMIN ) .TP 20 .ti +4 IWORK( 1 ) = LIWMIN .TP 20 .ti +4 IF( INFO.NE.0 ) THEN .TP 20 .ti +4 CALL PXERBLA( ICTXT, 'PCHEGVX ', -INFO ) .TP 20 .ti +4 RETURN .TP 20 .ti +4 ELSE IF( LQUERY ) THEN .TP 20 .ti +4 RETURN .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PCPOTRF( UPLO, N, B, IB, JB, DESCB, INFO ) .TP 20 .ti +4 IF( INFO.NE.0 ) THEN .TP 20 .ti +4 IFAIL( 1 ) = INFO .TP 20 .ti +4 INFO = IERRNPD .TP 20 .ti +4 RETURN .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PCHEGST( IBTYPE, UPLO, N, A, IA, JA, DESCA, B, IB, JB, DESCB, SCALE, INFO ) .TP 20 .ti +4 CALL PCHEEVX( JOBZ, RANGE, UPLO, N, A, IA, JA, DESCA, VL, VU, IL, IU, ABSTOL, M, NZ, W, ORFAC, Z, IZ, JZ, DESCZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, IFAIL, ICLUSTR, GAP, INFO ) .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 NEIG = M .TP 20 .ti +4 IF( IBTYPE.EQ.1 .OR. IBTYPE.EQ.2 ) THEN .TP 20 .ti +4 IF( UPPER ) THEN .TP 20 .ti +4 TRANS = 'N' .TP 20 .ti +4 ELSE .TP 20 .ti +4 TRANS = 'C' .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PCTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, NEIG, ONE, B, IB, JB, DESCB, Z, IZ, JZ, DESCZ ) .TP 20 .ti +4 ELSE IF( IBTYPE.EQ.3 ) THEN .TP 20 .ti +4 IF( UPPER ) THEN .TP 20 .ti +4 TRANS = 'C' .TP 20 .ti +4 ELSE .TP 20 .ti +4 TRANS = 'N' .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PCTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, NEIG, ONE, B, IB, JB, DESCB, Z, IZ, JZ, DESCZ ) .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( SCALE.NE.ONE ) THEN .TP 20 .ti +4 CALL SSCAL( N, SCALE, W, 1 ) .TP 20 .ti +4 END IF .TP 20 .ti +4 RETURN .TP 20 .ti +4 END .SH PURPOSE scalapack-doc-1.5/man/manl/pchetd2.l0100644000056400000620000002023706335610616016757 0ustar pfrauenfstaff.TH PCHETD2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCHETD2 - reduce a complex Hermitian matrix sub( A ) to Hermitian tridiagonal form T by an unitary similarity transformation .SH SYNOPSIS .TP 20 SUBROUTINE PCHETD2( UPLO, N, A, IA, JA, DESCA, D, E, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL D( * ), E( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), WORK( * ) .SH PURPOSE PCHETD2 reduces a complex Hermitian matrix sub( A ) to Hermitian tridiagonal form T by an unitary similarity transformation: Q' * sub( A ) * Q = T, where sub( A ) = A(IA:IA+N-1,JA:JA+N-1). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the upper or lower triangular part of the Hermitian matrix sub( A ) is stored: .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the Hermitian distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced. On exit, if UPLO = 'U', the diagonal and first superdiagonal of sub( A ) are over- written by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of sub( A ) are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) REAL array, dimension LOCc(JA+N-1) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i). D is tied to the distributed matrix A. .TP 8 E (local output) REAL array, dimension LOCc(JA+N-1) if UPLO = 'U', LOCc(JA+N-2) otherwise. The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. E is tied to the distributed matrix A. .TP 8 TAU (local output) COMPLEX, array, dimension LOCc(JA+N-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK( 1 ) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 3*N. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors .br Q = H(n-1) . . . H(2) H(1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in .br A(ia:ia+i-2,ja+i), and tau in TAU(ja+i-1). .br If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors .br Q = H(1) H(2) . . . H(n-1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in .br A(ia+i+1:ia+n-1,ja+i-1), and tau in TAU(ja+i-1). .br The contents of sub( A ) on exit are illustrated by the following examples with n = 5: .br if UPLO = 'U': if UPLO = 'L': .br ( d e v2 v3 v4 ) ( d ) ( d e v3 v4 ) ( e d ) ( d e v4 ) ( v1 e d ) ( d e ) ( v1 v2 e d ) ( d ) ( v1 v2 v3 e d ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expression should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) with .br IROFFA = MOD( IA-1, MB_A ) and ICOFFA = MOD( JA-1, NB_A ). scalapack-doc-1.5/man/manl/pchetrd.l0100644000056400000620000002067606335610616017066 0ustar pfrauenfstaff.TH PCHETRD l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCHETRD - reduce a complex Hermitian matrix sub( A ) to Hermitian tridiagonal form T by an unitary similarity transformation .SH SYNOPSIS .TP 20 SUBROUTINE PCHETRD( UPLO, N, A, IA, JA, DESCA, D, E, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL D( * ), E( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), WORK( * ) .SH PURPOSE PCHETRD reduces a complex Hermitian matrix sub( A ) to Hermitian tridiagonal form T by an unitary similarity transformation: Q' * sub( A ) * Q = T, where sub( A ) = A(IA:IA+N-1,JA:JA+N-1). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the upper or lower triangular part of the Hermitian matrix sub( A ) is stored: .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the Hermitian distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced. On exit, if UPLO = 'U', the diagonal and first superdiagonal of sub( A ) are over- written by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of sub( A ) are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) REAL array, dimension LOCc(JA+N-1) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i). D is tied to the distributed matrix A. .TP 8 E (local output) REAL array, dimension LOCc(JA+N-1) if UPLO = 'U', LOCc(JA+N-2) otherwise. The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. E is tied to the distributed matrix A. .TP 8 TAU (local output) COMPLEX, array, dimension LOCc(JA+N-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK( 1 ) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MAX( NB * ( NP +1 ), 3 * NB ) where NB = MB_A = NB_A, NP = NUMROC( N, NB, MYROW, IAROW, NPROW ), IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ). INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors .br Q = H(n-1) . . . H(2) H(1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in .br A(ia:ia+i-2,ja+i), and tau in TAU(ja+i-1). .br If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors .br Q = H(1) H(2) . . . H(n-1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in .br A(ia+i+1:ia+n-1,ja+i-1), and tau in TAU(ja+i-1). .br The contents of sub( A ) on exit are illustrated by the following examples with n = 5: .br if UPLO = 'U': if UPLO = 'L': .br ( d e v2 v3 v4 ) ( d ) ( d e v3 v4 ) ( e d ) ( d e v4 ) ( v1 e d ) ( d e ) ( v1 v2 e d ) ( d ) ( v1 v2 v3 e d ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expression should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA .AND. IROFFA.EQ.0 ) with IROFFA = MOD( IA-1, MB_A ) and ICOFFA = MOD( JA-1, NB_A ). scalapack-doc-1.5/man/manl/pclabrd.l0100644000056400000620000002351306335610616017035 0ustar pfrauenfstaff.TH PCLABRD l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLABRD - reduce the first NB rows and columns of a complex general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form by an unitary transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transfor- mation to the unreduced part of sub( A ) .SH SYNOPSIS .TP 20 SUBROUTINE PCLABRD( M, N, NB, A, IA, JA, DESCA, D, E, TAUQ, TAUP, X, IX, JX, DESCX, Y, IY, JY, DESCY, WORK ) .TP 20 .ti +4 INTEGER IA, IX, IY, JA, JX, JY, M, N, NB .TP 20 .ti +4 INTEGER DESCA( * ), DESCX( * ), DESCY( * ) .TP 20 .ti +4 REAL D( * ), E( * ) .TP 20 .ti +4 COMPLEX A( * ), TAUP( * ), TAUQ( * ), X( * ), Y( * ), WORK( * ) .SH PURPOSE PCLABRD reduces the first NB rows and columns of a complex general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form by an unitary transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transfor- mation to the unreduced part of sub( A ). If M >= N, sub( A ) is reduced to upper bidiagonal form; if M < N, to lower bidiagonal form. .br This is an auxiliary routine called by PCGEBRD. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 NB (global input) INTEGER The number of leading rows and columns of sub( A ) to be reduced. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the general distributed matrix sub( A ) to be reduced. On exit, the first NB rows and columns of the matrix are overwritten; the rest of the distributed matrix sub( A ) is unchanged. If m >= n, elements on and below the diagonal in the first NB columns, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors. If m < n, elements below the diagonal in the first NB columns, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and elements on and above the diagonal in the first NB rows, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) REAL array, dimension LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-1) otherwise. The distributed diagonal elements of the bidiagonal matrix B: D(i) = A(ia+i-1,ja+i-1). D is tied to the distributed matrix A. .TP 8 E (local output) REAL array, dimension LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-2) otherwise. The distributed off-diagonal elements of the bidiagonal distributed matrix B: if m >= n, E(i) = A(ia+i-1,ja+i) for i = 1,2,...,n-1; if m < n, E(i) = A(ia+i,ja+i-1) for i = 1,2,...,m-1. E is tied to the distributed matrix A. .TP 8 TAUQ (local output) COMPLEX array dimension LOCc(JA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the unitary matrix Q. TAUQ is tied to the distributed matrix A. See Further Details. TAUP (local output) COMPLEX array, dimension LOCr(IA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the unitary matrix P. TAUP is tied to the distributed matrix A. See Further Details. X (local output) COMPLEX pointer into the local memory to an array of dimension (LLD_X,NB). On exit, the local pieces of the distributed M-by-NB matrix X(IX:IX+M-1,JX:JX+NB-1) required to update the unreduced part of sub( A ). .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 Y (local output) COMPLEX pointer into the local memory to an array of dimension (LLD_Y,NB). On exit, the local pieces of the distributed N-by-NB matrix Y(IY:IY+N-1,JY:JY+NB-1) required to update the unreduced part of sub( A ). .TP 8 IY (global input) INTEGER The row index in the global array Y indicating the first row of sub( Y ). .TP 8 JY (global input) INTEGER The column index in the global array Y indicating the first column of sub( Y ). .TP 8 DESCY (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix Y. .TP 8 WORK (local workspace) COMPLEX array, dimension (LWORK) LWORK >= NB_A + NQ, with NQ = NUMROC( N+MOD( IA-1, NB_Y ), NB_Y, MYCOL, IACOL, NPCOL ) IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ) INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. .SH FURTHER DETAILS The matrices Q and P are represented as products of elementary reflectors: .br Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb) Each H(i) and G(i) has the form: .br H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are complex scalars, and v and u are complex vectors. .br If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in A(ia+i-1:ia+m-1,ja+i-1); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in A(ia+i-1,ja+i:ja+n-1); tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in A(ia+i+1:ia+m-1,ja+i-1); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in A(ia+i-1,ja+i:ja+n-1); tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br The elements of the vectors v and u together form the m-by-nb matrix V and the nb-by-n matrix U' which are needed, with X and Y, to apply the transformation to the unreduced part of the matrix, using a block update of the form: sub( A ) := sub( A ) - V*Y' - X*U'. .br The contents of sub( A ) on exit are illustrated by the following examples with nb = 2: .br m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 ) ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 ) ( v1 v2 a a a ) ( v1 1 a a a a ) ( v1 v2 a a a ) ( v1 v2 a a a a ) ( v1 v2 a a a ) ( v1 v2 a a a a ) ( v1 v2 a a a ) .br where a denotes an element of the original matrix which is unchanged, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i). .br scalapack-doc-1.5/man/manl/pclacgv.l0100644000056400000620000001026506335610616017045 0ustar pfrauenfstaff.TH PCLACGV l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLACGV - conjugate a complex vector of length N, sub( X ), where sub( X ) denotes X(IX,JX:JX+N-1) if INCX = DESCX( M_ ) and X(IX:IX+N-1,JX) if INCX = 1, and Notes ===== Each global data object is described by an associated description vector .SH SYNOPSIS .TP 20 SUBROUTINE PCLACGV( N, X, IX, JX, DESCX, INCX ) .TP 20 .ti +4 INTEGER INCX, IX, JX, N .TP 20 .ti +4 INTEGER DESCX( * ) .TP 20 .ti +4 COMPLEX X( * ) .SH PURPOSE PCLACGV conjugates a complex vector of length N, sub( X ), where sub( X ) denotes X(IX,JX:JX+N-1) if INCX = DESCX( M_ ) and X(IX:IX+N-1,JX) if INCX = 1, and the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. .SH ARGUMENTS .TP 8 N (global input) INTEGER The length of the distributed vector sub( X ). .TP 8 X (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_X,*). On entry the vector to be conjugated x( i ) = X(IX+(JX-1)*M_X +(i-1)*INCX ), 1 <= i <= N. On exit the conjugated vector. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 INCX (global input) INTEGER The global increment for the elements of X. Only two values of INCX are supported in this version, namely 1 and M_X. INCX must not be zero. scalapack-doc-1.5/man/manl/pclacon.l0100644000056400000620000001262006335610616017042 0ustar pfrauenfstaff.TH PCLACON l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLACON - estimate the 1-norm of a square, complex distributed matrix A .SH SYNOPSIS .TP 20 SUBROUTINE PCLACON( N, V, IV, JV, DESCV, X, IX, JX, DESCX, EST, KASE ) .TP 20 .ti +4 INTEGER IV, IX, JV, JX, KASE, N .TP 20 .ti +4 REAL EST .TP 20 .ti +4 INTEGER DESCV( * ), DESCX( * ) .TP 20 .ti +4 COMPLEX V( * ), X( * ) .SH PURPOSE PCLACON estimates the 1-norm of a square, complex distributed matrix A. Reverse communication is used for evaluating matrix-vector products. X and V are aligned with the distributed matrix A, this information is implicitly contained within IV, IX, DESCV, and DESCX. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The length of the distributed vectors V and X. N >= 0. .TP 8 V (local workspace) COMPLEX pointer into the local memory to an array of dimension LOCr(N+MOD(IV-1,MB_V)). On the final return, V = A*W, where EST = norm(V)/norm(W) (W is not returned). .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 X (local input/local output) COMPLEX pointer into the local memory to an array of dimension LOCr(N+MOD(IX-1,MB_X)). On an intermediate return, X should be overwritten by A * X, if KASE=1, A' * X, if KASE=2, where A' is the conjugate transpose of A, and PCLACON must be re-called with all the other parameters unchanged. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 EST (global output) REAL An estimate (a lower bound) for norm(A). .TP 8 KASE (local input/local output) INTEGER On the initial call to PCLACON, KASE should be 0. On an intermediate return, KASE will be 1 or 2, indicating whether X should be overwritten by A * X or A' * X. On the final return from PCLACON, KASE will again be 0. .SH FURTHER DETAILS The serial version CLACON has been contributed by Nick Higham, University of Manchester. It was originally named SONEST, dated March 16, 1988. .br Reference: N.J. Higham, "FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation", ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988. scalapack-doc-1.5/man/manl/pclacp2.l0100644000056400000620000001267406335610617016761 0ustar pfrauenfstaff.TH PCLACP2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLACP2 - copie all or part of a distributed matrix A to another distributed matrix B .SH SYNOPSIS .TP 20 SUBROUTINE PCLACP2( UPLO, M, N, A, IA, JA, DESCA, B, IB, JB, DESCB ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IB, JA, JB, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX A( * ), B( * ) .SH PURPOSE PCLACP2 copies all or part of a distributed matrix A to another distributed matrix B. No communication is performed, PCLACP2 performs a local copy sub( A ) := sub( B ), where sub( A ) denotes A(IA:IA+M-1,JA:JA+N-1) and sub( B ) denotes B(IB:IB+M-1,JB:JB+N-1). PCLACP2 requires that only dimension of the matrix operands is distributed. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies the part of the distributed matrix sub( A ) to be copied: .br = 'U': Upper triangular part is copied; the strictly lower triangular part of sub( A ) is not referenced; = 'L': Lower triangular part is copied; the strictly upper triangular part of sub( A ) is not referenced; Otherwise: All of the matrix sub( A ) is copied. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1) ). This array contains the local pieces of the distributed matrix sub( A ) to be copied from. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local output) COMPLEX pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+N-1) ). This array contains on exit the local pieces of the distributed matrix sub( B ) set as follows: if UPLO = 'U', B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), 1<=i<=j, 1<=j<=N; if UPLO = 'L', B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), j<=i<=M, 1<=j<=N; otherwise, B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), 1<=i<=M, 1<=j<=N. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. scalapack-doc-1.5/man/manl/pclacpy.l0100644000056400000620000001255406335610617017065 0ustar pfrauenfstaff.TH PCLACPY l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLACPY - copie all or part of a distributed matrix A to another distributed matrix B .SH SYNOPSIS .TP 20 SUBROUTINE PCLACPY( UPLO, M, N, A, IA, JA, DESCA, B, IB, JB, DESCB ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IB, JA, JB, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX A( * ), B( * ) .SH PURPOSE PCLACPY copies all or part of a distributed matrix A to another distributed matrix B. No communication is performed, PCLACPY performs a local copy sub( A ) := sub( B ), where sub( A ) denotes A(IA:IA+M-1,JA:JA+N-1) and sub( B ) denotes B(IB:IB+M-1,JB:JB+N-1). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies the part of the distributed matrix sub( A ) to be copied: .br = 'U': Upper triangular part is copied; the strictly lower triangular part of sub( A ) is not referenced; = 'L': Lower triangular part is copied; the strictly upper triangular part of sub( A ) is not referenced; Otherwise: All of the matrix sub( A ) is copied. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1) ). This array contains the local pieces of the distributed matrix sub( A ) to be copied from. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local output) COMPLEX pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+N-1) ). This array contains on exit the local pieces of the distributed matrix sub( B ) set as follows: if UPLO = 'U', B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), 1<=i<=j, 1<=j<=N; if UPLO = 'L', B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), j<=i<=M, 1<=j<=N; otherwise, B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), 1<=i<=M, 1<=j<=N. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. scalapack-doc-1.5/man/manl/pclaevswp.l0100644000056400000620000001217206335610617017432 0ustar pfrauenfstaff.TH PCLAEVSWP l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCLAEVSWP - move the eigenvectors (potentially unsorted) from where they are computed, to a ScaLAPACK standard block cyclic array, sorted so that the corresponding eigenvalues are sorted .SH SYNOPSIS .TP 22 SUBROUTINE PCLAEVSWP( N, ZIN, LDZI, Z, IZ, JZ, DESCZ, NVS, KEY, RWORK, LRWORK ) .TP 22 .ti +4 INTEGER IZ, JZ, LDZI, LRWORK, N .TP 22 .ti +4 INTEGER DESCZ( * ), KEY( * ), NVS( * ) .TP 22 .ti +4 REAL RWORK( * ), ZIN( LDZI, * ) .TP 22 .ti +4 COMPLEX Z( * ) .SH PURPOSE PCLAEVSWP moves the eigenvectors (potentially unsorted) from where they are computed, to a ScaLAPACK standard block cyclic array, sorted so that the corresponding eigenvalues are sorted. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS NP = the number of rows local to a given process. NQ = the number of columns local to a given process. .TP 8 N (global input) INTEGER The order of the matrix A. N >= 0. .TP 8 ZIN (local input) REAL array, dimension ( LDZI, NVS(iam) ) The eigenvectors on input. Each eigenvector resides entirely in one process. Each process holds a contiguous set of NVS(iam) eigenvectors. The first eigenvector which the process holds is: sum for i=[0,iam-1) of NVS(i) .TP 8 LDZI (locl input) INTEGER leading dimension of the ZIN array .TP 8 Z (local output) COMPLEX array global dimension (N, N), local dimension (DESCZ(DLEN_), NQ) The eigenvectors on output. The eigenvectors are distributed in a block cyclic manner in both dimensions, with a block size of NB. .TP 8 IZ (global input) INTEGER Z's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JZ (global input) INTEGER Z's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCZ (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix Z. .TP 8 NVS (global input) INTEGER array, dimension( nprocs+1 ) nvs(i) = number of processes number of eigenvectors held by processes [0,i-1) nvs(1) = number of eigen vectors held by [0,1-1) == 0 nvs(nprocs+1) = number of eigen vectors held by [0,nprocs) == total number of eigenvectors .TP 8 KEY (global input) INTEGER array, dimension( N ) Indicates the actual index (after sorting) for each of the eigenvectors. .TP 9 RWORK (local workspace) REAL array, dimension (LRWORK) .TP 9 LRWORK (local input) INTEGER dimension of RWORK scalapack-doc-1.5/man/manl/pclahrd.l0100644000056400000620000001052506335610617017043 0ustar pfrauenfstaff.TH PCLAHRD l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLAHRD - reduce the first NB columns of a complex general N-by-(N-K+1) distributed matrix A(IA:IA+N-1,JA:JA+N-K) so that elements below the k-th subdiagonal are zero .SH SYNOPSIS .TP 20 SUBROUTINE PCLAHRD( N, K, NB, A, IA, JA, DESCA, TAU, T, Y, IY, JY, DESCY, WORK ) .TP 20 .ti +4 INTEGER IA, IY, JA, JY, K, N, NB .TP 20 .ti +4 INTEGER DESCA( * ), DESCY( * ) .TP 20 .ti +4 COMPLEX A( * ), T( * ), TAU( * ), WORK( * ), Y( * ) .SH PURPOSE PCLAHRD reduces the first NB columns of a complex general N-by-(N-K+1) distributed matrix A(IA:IA+N-1,JA:JA+N-K) so that elements below the k-th subdiagonal are zero. The reduction is performed by an unitary similarity transformation Q' * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T. .br This is an auxiliary routine called by PCGEHRD. In the following comments sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1). .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 K (global input) INTEGER The offset for the reduction. Elements below the k-th subdiagonal in the first NB columns are reduced to zero. .TP 8 NB (global input) INTEGER The number of columns to be reduced. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-K)). On entry, this array contains the the local pieces of the N-by-(N-K+1) general distributed matrix A(IA:IA+N-1,JA:JA+N-K). On exit, the elements on and above the k-th subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced distributed matrix; the elements below the k-th subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors. The other columns of A(IA:IA+N-1,JA:JA+N-K) are unchanged. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX array, dimension LOCc(JA+N-2) The scalar factors of the elementary reflectors (see Further Details). TAU is tied to the distributed matrix A. .TP 8 T (local output) COMPLEX array, dimension (NB_A,NB_A) The upper triangular matrix T. .TP 8 Y (local output) COMPLEX pointer into the local memory to an array of dimension (LLD_Y,NB_A). On exit, this array contains the local pieces of the N-by-NB distributed matrix Y. LLD_Y >= LOCr(IA+N-1). .TP 8 IY (global input) INTEGER The row index in the global array Y indicating the first row of sub( Y ). .TP 8 JY (global input) INTEGER The column index in the global array Y indicating the first column of sub( Y ). .TP 8 DESCY (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix Y. .TP 8 WORK (local workspace) COMPLEX array, dimension (NB) .SH FURTHER DETAILS The matrix Q is represented as a product of nb elementary reflectors Q = H(1) H(2) . . . H(nb). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(ia+i+k:ia+n-1,ja+i-1), and tau in TAU(ja+i-1). .br The elements of the vectors v together form the (n-k+1)-by-nb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form: A(ia:ia+n-1,ja:ja+n-k) := (I-V*T*V')*(A(ia:ia+n-1,ja:ja+n-k)-Y*V'). The contents of A(ia:ia+n-1,ja:ja+n-k) on exit are illustrated by the following example with n = 7, k = 3 and nb = 2: .br ( a h a a a ) .br ( a h a a a ) .br ( a h a a a ) .br ( h h a a a ) .br ( v1 h a a a ) .br ( v1 v2 a a a ) .br ( v1 v2 a a a ) .br where a denotes an element of the original matrix .br A(ia:ia+n-1,ja:ja+n-k), h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i). .br scalapack-doc-1.5/man/manl/pclange.l0100644000056400000620000001277706335610617017052 0ustar pfrauenfstaff.TH PCLANGE l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLANGE - return the value of the one norm, or the Frobenius norm, .SH SYNOPSIS .TP 14 REAL FUNCTION PCLANGE( NORM, M, N, A, IA, JA, DESCA, WORK ) .TP 14 .ti +4 CHARACTER NORM .TP 14 .ti +4 INTEGER IA, JA, M, N .TP 14 .ti +4 INTEGER DESCA( * ) .TP 14 .ti +4 REAL WORK( * ) .TP 14 .ti +4 COMPLEX A( * ) .SH PURPOSE PCLANGE returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a distributed matrix sub( A ) = A(IA:IA+M-1, JA:JA+N-1). .br PCLANGE returns the value .br ( max(abs(A(i,j))), NORM = 'M' or 'm' with IA <= i <= IA+M-1, ( and JA <= j <= JA+N-1, ( .br ( norm1( sub( A ) ), NORM = '1', 'O' or 'o' .br ( .br ( normI( sub( A ) ), NORM = 'I' or 'i' .br ( .br ( normF( sub( A ) ), NORM = 'F', 'f', 'E' or 'e' .br where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a matrix norm. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies the value to be returned in PCLANGE as described above. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). When M = 0, PCLANGE is set to zero. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). When N = 0, PCLANGE is set to zero. N >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)) containing the local pieces of the distributed matrix sub( A ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 WORK (local workspace) REAL array dimension (LWORK) LWORK >= 0 if NORM = 'M' or 'm' (not referenced), Nq0 if NORM = '1', 'O' or 'o', Mp0 if NORM = 'I' or 'i', 0 if NORM = 'F', 'f', 'E' or 'e' (not referenced), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. scalapack-doc-1.5/man/manl/pclanhe.l0100644000056400000620000001442506335610617017043 0ustar pfrauenfstaff.TH PCLANHE l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLANHE - return the value of the one norm, or the Frobenius norm, .SH SYNOPSIS .TP 14 REAL FUNCTION PCLANHE( NORM, UPLO, N, A, IA, JA, DESCA, WORK ) .TP 14 .ti +4 CHARACTER NORM, UPLO .TP 14 .ti +4 INTEGER IA, JA, N .TP 14 .ti +4 INTEGER DESCA( * ) .TP 14 .ti +4 REAL WORK( * ) .TP 14 .ti +4 COMPLEX A( * ) .SH PURPOSE PCLANHE returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex hermitian distributed matrix sub(A) = A(IA:IA+N-1,JA:JA+N-1). PCLANHE returns the value .br ( max(abs(A(i,j))), NORM = 'M' or 'm' with IA <= i <= IA+N-1, ( and JA <= j <= JA+N-1, ( .br ( norm1( sub( A ) ), NORM = '1', 'O' or 'o' .br ( .br ( normI( sub( A ) ), NORM = 'I' or 'i' .br ( .br ( normF( sub( A ) ), NORM = 'F', 'f', 'E' or 'e' .br where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a matrix norm. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies the value to be returned in PCLANHE as described above. .TP 8 UPLO (global input) CHARACTER Specifies whether the upper or lower triangular part of the hermitian matrix sub( A ) is to be referenced. = 'U': Upper triangular part of sub( A ) is referenced, .br = 'L': Lower triangular part of sub( A ) is referenced. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on i.e the number of rows and columns of the distributed submatrix sub( A ). When N = 0, PCLANHE is set to zero. N >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)) containing the local pieces of the hermitian distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular matrix which norm is to be computed, and the strictly lower triangular part of this matrix is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular matrix which norm is to be computed, and the strictly upper triangular part of sub( A ) is not referenced. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 WORK (local workspace) REAL array dimension (LWORK) LWORK >= 0 if NORM = 'M' or 'm' (not referenced), 2*Nq0+Np0+LDW if NORM = '1', 'O', 'o', 'I' or 'i', where LDW is given by: IF( NPROW.NE.NPCOL ) THEN LDW = MB_A*CEIL(CEIL(Np0/MB_A)/(LCM/NPROW)) ELSE LDW = 0 END IF 0 if NORM = 'F', 'f', 'E' or 'e' (not referenced), where LCM is the least common multiple of NPROW and NPCOL LCM = ILCM( NPROW, NPCOL ) and CEIL denotes the ceiling operation (ICEIL). IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Np0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), ICEIL, ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. scalapack-doc-1.5/man/manl/pclanhs.l0100644000056400000620000001250706335610617017060 0ustar pfrauenfstaff.TH PCLANHS l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLANHS - return the value of the one norm, or the Frobenius norm, .SH SYNOPSIS .TP 14 REAL FUNCTION PCLANHS( NORM, N, A, IA, JA, DESCA, WORK ) .TP 14 .ti +4 CHARACTER NORM .TP 14 .ti +4 INTEGER IA, JA, N .TP 14 .ti +4 INTEGER DESCA( * ) .TP 14 .ti +4 REAL WORK( * ) .TP 14 .ti +4 COMPLEX A( * ) .SH PURPOSE PCLANHS returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hessenberg distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). PCLANHS returns the value .br ( max(abs(A(i,j))), NORM = 'M' or 'm' with IA <= i <= IA+N-1, ( and JA <= j <= JA+N-1, ( .br ( norm1( sub( A ) ), NORM = '1', 'O' or 'o' .br ( .br ( normI( sub( A ) ), NORM = 'I' or 'i' .br ( .br ( normF( sub( A ) ), NORM = 'F', 'f', 'E' or 'e' .br where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a matrix norm. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies the value to be returned in PCLANHS as described above. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on i.e the number of rows and columns of the distributed submatrix sub( A ). When N = 0, PCLANHS is set to zero. N >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1) ) containing the local pieces of sub( A ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 WORK (local workspace) REAL array dimension (LWORK) LWORK >= 0 if NORM = 'M' or 'm' (not referenced), Nq0 if NORM = '1', 'O' or 'o', Mp0 if NORM = 'I' or 'i', 0 if NORM = 'F', 'f', 'E' or 'e' (not referenced), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Np0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. scalapack-doc-1.5/man/manl/pclansy.l0100644000056400000620000001442506335610617017102 0ustar pfrauenfstaff.TH PCLANSY l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLANSY - return the value of the one norm, or the Frobenius norm, .SH SYNOPSIS .TP 14 REAL FUNCTION PCLANSY( NORM, UPLO, N, A, IA, JA, DESCA, WORK ) .TP 14 .ti +4 CHARACTER NORM, UPLO .TP 14 .ti +4 INTEGER IA, JA, N .TP 14 .ti +4 INTEGER DESCA( * ) .TP 14 .ti +4 REAL WORK( * ) .TP 14 .ti +4 COMPLEX A( * ) .SH PURPOSE PCLANSY returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). PCLANSY returns the value .br ( max(abs(A(i,j))), NORM = 'M' or 'm' with IA <= i <= IA+N-1, ( and JA <= j <= JA+N-1, ( .br ( norm1( sub( A ) ), NORM = '1', 'O' or 'o' .br ( .br ( normI( sub( A ) ), NORM = 'I' or 'i' .br ( .br ( normF( sub( A ) ), NORM = 'F', 'f', 'E' or 'e' .br where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a matrix norm. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies the value to be returned in PCLANSY as described above. .TP 8 UPLO (global input) CHARACTER Specifies whether the upper or lower triangular part of the symmetric matrix sub( A ) is to be referenced. = 'U': Upper triangular part of sub( A ) is referenced, .br = 'L': Lower triangular part of sub( A ) is referenced. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on i.e the number of rows and columns of the distributed submatrix sub( A ). When N = 0, PCLANSY is set to zero. N >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)) containing the local pieces of the symmetric distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular matrix which norm is to be computed, and the strictly lower triangular part of this matrix is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular matrix which norm is to be computed, and the strictly upper triangular part of sub( A ) is not referenced. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 WORK (local workspace) REAL array dimension (LWORK) LWORK >= 0 if NORM = 'M' or 'm' (not referenced), 2*Nq0+Np0+LDW if NORM = '1', 'O', 'o', 'I' or 'i', where LDW is given by: IF( NPROW.NE.NPCOL ) THEN LDW = MB_A*CEIL(CEIL(Np0/MB_A)/(LCM/NPROW)) ELSE LDW = 0 END IF 0 if NORM = 'F', 'f', 'E' or 'e' (not referenced), where LCM is the least common multiple of NPROW and NPCOL LCM = ILCM( NPROW, NPCOL ) and CEIL denotes the ceiling operation (ICEIL). IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Np0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), ICEIL, ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. scalapack-doc-1.5/man/manl/pclantr.l0100644000056400000620000001365706335610617017102 0ustar pfrauenfstaff.TH PCLANTR l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLANTR - return the value of the one norm, or the Frobenius norm, .SH SYNOPSIS .TP 14 REAL FUNCTION PCLANTR( NORM, UPLO, DIAG, M, N, A, IA, JA, DESCA, WORK ) .TP 14 .ti +4 CHARACTER DIAG, NORM, UPLO .TP 14 .ti +4 INTEGER IA, JA, M, N .TP 14 .ti +4 INTEGER DESCA( * ) .TP 14 .ti +4 REAL WORK( * ) .TP 14 .ti +4 COMPLEX A( * ) .SH PURPOSE PCLANTR returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular distributed matrix sub( A ) denoting A(IA:IA+M-1, JA:JA+N-1). .br PCLANTR returns the value .br ( max(abs(A(i,j))), NORM = 'M' or 'm' with ia <= i <= ia+m-1, ( and ja <= j <= ja+n-1, ( .br ( norm1( sub( A ) ), NORM = '1', 'O' or 'o' .br ( .br ( normI( sub( A ) ), NORM = 'I' or 'i' .br ( .br ( normF( sub( A ) ), NORM = 'F', 'f', 'E' or 'e' .br where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a matrix norm. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies the value to be returned in PCLANTR as described above. .TP 8 UPLO (global input) CHARACTER Specifies whether the matrix sub( A ) is upper or lower trapezoidal. = 'U': Upper trapezoidal .br = 'L': Lower trapezoidal Note that sub( A ) is triangular instead of trapezoidal if M = N. .TP 8 DIAG (global input) CHARACTER Specifies whether or not the distributed matrix sub( A ) has unit diagonal. = 'N': Non-unit diagonal .br = 'U': Unit diagonal .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). When M = 0, PCLANTR is set to zero. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). When N = 0, PCLANTR is set to zero. N >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1) ) containing the local pieces of sub( A ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 WORK (local workspace) REAL array dimension (LWORK) LWORK >= 0 if NORM = 'M' or 'm' (not referenced), Nq0 if NORM = '1', 'O' or 'o', Mp0 if NORM = 'I' or 'i', 0 if NORM = 'F', 'f', 'E' or 'e' (not referenced), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. scalapack-doc-1.5/man/manl/pclapiv.l0100644000056400000620000001547606335610617017076 0ustar pfrauenfstaff.TH PCLAPIV l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLAPIV - applie either P (permutation matrix indicated by IPIV) or inv( P ) to a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1), resulting in row or column pivoting .SH SYNOPSIS .TP 20 SUBROUTINE PCLAPIV( DIREC, ROWCOL, PIVROC, M, N, A, IA, JA, DESCA, IPIV, IP, JP, DESCIP, IWORK ) .TP 20 .ti +4 CHARACTER*1 DIREC, PIVROC, ROWCOL .TP 20 .ti +4 INTEGER IA, IP, JA, JP, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCIP( * ), IPIV( * ), IWORK( * ) .TP 20 .ti +4 COMPLEX A( * ) .SH PURPOSE PCLAPIV applies either P (permutation matrix indicated by IPIV) or inv( P ) to a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1), resulting in row or column pivoting. The pivot vector may be distributed across a process row or a column. The pivot vector should be aligned with the distributed matrix A. This routine will transpose the pivot vector if necessary. For example if the row pivots should be applied to the columns of sub( A ), pass ROWCOL='C' and PIVROC='C'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Restrictions .br ============ .br IPIV must always be a distributed vector (not a matrix). Thus: IF( ROWPIV .EQ. 'C' ) THEN .br JP must be 1 .br ELSE .br IP must be 1 .br END IF .br The following restrictions apply when IPIV must be transposed: IF( ROWPIV.EQ.'C' .AND. PIVROC.EQ.'C') THEN .br DESCIP(MB_) must equal DESCA(NB_) .br ELSE IF( ROWPIV.EQ.'R" .AND. PIVROC.EQ.'R') THEN .br DESCIP(NB_) must equal DESCA(MB_) .br END IF .br .SH ARGUMENTS .TP 8 DIREC (global input) CHARACTER*1 Specifies in which order the permutation is applied: = 'F' (Forward) Applies pivots Forward from top of matrix. Computes P*sub( A ). = 'B' (Backward) Applies pivots Backward from bottom of matrix. Computes inv( P )*sub( A ). .TP 8 ROWCOL (global input) CHARACTER*1 Specifies if the rows or columns are to be permuted: = 'R' Rows will be permuted, = 'C' Columns will be permuted. .TP 8 PIVROC (global input) CHARACTER*1 Specifies whether IPIV is distributed over a process row or column: = 'R' IPIV distributed over a process row = 'C' IPIV distributed over a process column .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the distributed submatrix sub( A ) to which the row or column interchanges will be applied. On exit, the local pieces of the permuted distributed submatrix. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local input) INTEGER array, dimension >= LOCr(M_A)+MB_A if ROWCOL='R', otherwise LOCc(N_A)+NB_A. It contains the pivoting information. IPIV(i) is the global row (column), local row (column) i was swapped with. The last piece of the array of size MB_A (resp. NB_A) is used as workspace. This array is tied to the distributed matrix A. .TP 8 IWORK (local workspace) INTEGER array, dimension (LDW) where LDW is equal to the workspace necessary for transposition, and the storage of the tranposed IPIV: Let LCM be the least common multiple of NPROW and NPCOL. IF( ROWCOL.EQ.'R' .AND. PIVROC.EQ.'R' ) THEN IF( NPROW.EQ.NPCOL ) THEN LDW = LOCr( N_P + MOD(JP-1, NB_P) ) + NB_P ELSE LDW = LOCr( N_P + MOD(JP-1, NB_P) ) + NB_P * CEIL( CEIL(LOCc(N_P)/NB_P) / (LCM/NPCOL) ) END IF ELSE IF( ROWCOL.EQ.'C' .AND. PIVROC.EQ.'C' ) THEN IF( NPROW.EQ.NPCOL ) THEN LDW = LOCc( M_P + MOD(IP-1, MB_P) ) + MB_P ELSE LDW = LOCc( M_P + MOD(IP-1, MB_P) ) + MB_P * CEIL( CEIL(LOCr(M_P)/MB_P) / (LCM/NPROW) ) END IF ELSE IWORK is not referenced. END IF scalapack-doc-1.5/man/manl/pclapv2.l0100644000056400000620000001357306335610617017003 0ustar pfrauenfstaff.TH PCLAPV2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLAPV2 - applie either P (permutation matrix indicated by IPIV) or inv( P ) to a M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1), resulting in row or column pivoting .SH SYNOPSIS .TP 20 SUBROUTINE PCLAPV2( DIREC, ROWCOL, M, N, A, IA, JA, DESCA, IPIV, IP, JP, DESCIP ) .TP 20 .ti +4 CHARACTER DIREC, ROWCOL .TP 20 .ti +4 INTEGER IA, IP, JA, JP, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCIP( * ), IPIV( * ) .TP 20 .ti +4 COMPLEX A( * ) .SH PURPOSE PCLAPV2 applies either P (permutation matrix indicated by IPIV) or inv( P ) to a M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1), resulting in row or column pivoting. The pivot vector should be aligned with the distributed matrix A. For pivoting the rows of sub( A ), IPIV should be distributed along a process column and replicated over all process rows. Similarly, IPIV should be distributed along a process row and replicated over all process columns for column pivoting. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 DIREC (global input) CHARACTER Specifies in which order the permutation is applied: = 'F' (Forward) Applies pivots Forward from top of matrix. Computes P * sub( A ); = 'B' (Backward) Applies pivots Backward from bottom of matrix. Computes inv( P ) * sub( A ). .TP 8 ROWCOL (global input) CHARACTER Specifies if the rows or columns are to be permuted: = 'R' Rows will be permuted, = 'C' Columns will be permuted. .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this local array contains the local pieces of the distributed matrix sub( A ) to which the row or columns interchanges will be applied. On exit, this array contains the local pieces of the permuted distributed matrix. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (input) INTEGER array, dimension >= LOCr(M_A)+MB_A if ROWCOL = 'R', LOCc(N_A)+NB_A otherwise. It contains the pivoting information. IPIV(i) is the global row (column), local row (column) i was swapped with. The last piece of the array of size MB_A (resp. NB_A) is used as workspace. IPIV is tied to the distributed matrix A. .TP 8 IP (global input) INTEGER IPIV's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JP (global input) INTEGER IPIV's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCIP (global and local input) INTEGER array of dimension 8 The array descriptor for the distributed matrix IPIV. scalapack-doc-1.5/man/manl/pclaqge.l0100644000056400000620000001366606335610620017045 0ustar pfrauenfstaff.TH PCLAQGE l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLAQGE - equilibrate a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) using the row and scaling factors in the vectors R and C .SH SYNOPSIS .TP 20 SUBROUTINE PCLAQGE( M, N, A, IA, JA, DESCA, R, C, ROWCND, COLCND, AMAX, EQUED ) .TP 20 .ti +4 CHARACTER EQUED .TP 20 .ti +4 INTEGER IA, JA, M, N .TP 20 .ti +4 REAL AMAX, COLCND, ROWCND .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL C( * ), R( * ) .TP 20 .ti +4 COMPLEX A( * ) .SH PURPOSE PCLAQGE equilibrates a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) using the row and scaling factors in the vectors R and C. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)) containing on entry the M-by-N matrix sub( A ). On exit, the equilibrated distributed matrix. See EQUED for the form of the equilibrated distributed submatrix. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 R (local input) REAL array, dimension LOCr(M_A) The row scale factors for sub( A ). R is aligned with the distributed matrix A, and replicated across every process column. R is tied to the distributed matrix A. .TP 8 C (local input) REAL array, dimension LOCc(N_A) The column scale factors of sub( A ). C is aligned with the distributed matrix A, and replicated down every process row. C is tied to the distributed matrix A. .TP 8 ROWCND (global input) REAL The global ratio of the smallest R(i) to the largest R(i), IA <= i <= IA+M-1. .TP 8 COLCND (global input) REAL The global ratio of the smallest C(i) to the largest C(i), JA <= j <= JA+N-1. .TP 8 AMAX (global input) REAL Absolute value of largest distributed submatrix entry. .TP 8 EQUED (global output) CHARACTER Specifies the form of equilibration that was done. = 'N': No equilibration .br = 'R': Row equilibration, i.e., sub( A ) has been pre- .br multiplied by diag(R(IA:IA+M-1)), .br = 'C': Column equilibration, i.e., sub( A ) has been post- .br multiplied by diag(C(JA:JA+N-1)), .br = 'B': Both row and column equilibration, i.e., sub( A ) has been replaced by diag(R(IA:IA+M-1)) * sub( A ) * diag(C(JA:JA+N-1)). .SH PARAMETERS THRESH is a threshold value used to decide if row or column scaling should be done based on the ratio of the row or column scaling factors. If ROWCND < THRESH, row scaling is done, and if COLCND < THRESH, column scaling is done. LARGE and SMALL are threshold values used to decide if row scaling should be done based on the absolute size of the largest matrix element. If AMAX > LARGE or AMAX < SMALL, row scaling is done. scalapack-doc-1.5/man/manl/pclaqsy.l0100644000056400000620000001403506335610620017074 0ustar pfrauenfstaff.TH PCLAQSY l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLAQSY - equilibrate a symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) using the scaling factors in the vectors SR and SC .SH SYNOPSIS .TP 20 SUBROUTINE PCLAQSY( UPLO, N, A, IA, JA, DESCA, SR, SC, SCOND, AMAX, EQUED ) .TP 20 .ti +4 CHARACTER EQUED, UPLO .TP 20 .ti +4 INTEGER IA, JA, N .TP 20 .ti +4 REAL AMAX, SCOND .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL SC( * ), SR( * ) .TP 20 .ti +4 COMPLEX A( * ) .SH PURPOSE PCLAQSY equilibrates a symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) using the scaling factors in the vectors SR and SC. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the upper or lower triangular part of the symmetric distributed matrix sub( A ) is to be referenced: .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (input/output) COMPLEX pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1)). On entry, the local pieces of the distributed symmetric matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and the strictly lower triangular part of sub( A ) is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and the strictly upper trian- gular part of sub( A ) is not referenced. On exit, if EQUED = 'Y', the equilibrated matrix: .br diag(SR(IA:IA+N-1)) * sub( A ) * diag(SC(JA:JA+N-1)). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 SR (local input) REAL array, dimension LOCr(M_A) The scale factors for A(IA:IA+M-1,JA:JA+N-1). SR is aligned with the distributed matrix A, and replicated across every process column. SR is tied to the distributed matrix A. .TP 8 SC (local input) REAL array, dimension LOCc(N_A) The scale factors for sub( A ). SC is aligned with the dis- tributed matrix A, and replicated down every process row. SC is tied to the distributed matrix A. .TP 8 SCOND (global input) REAL Ratio of the smallest SR(i) (respectively SC(j)) to the largest SR(i) (respectively SC(j)), with IA <= i <= IA+N-1 and JA <= j <= JA+N-1. .TP 8 AMAX (global input) REAL Absolute value of the largest distributed submatrix entry. .TP 8 EQUED (output) CHARACTER*1 Specifies whether or not equilibration was done. = 'N': No equilibration. .br = 'Y': Equilibration was done, i.e., sub( A ) has been re- .br placed by: .br diag(SR(IA:IA+N-1)) * sub( A ) * diag(SC(JA:JA+N-1)). .SH PARAMETERS THRESH is a threshold value used to decide if scaling should be done based on the ratio of the scaling factors. If SCOND < THRESH, scaling is done. LARGE and SMALL are threshold values used to decide if scaling should be done based on the absolute size of the largest matrix element. If AMAX > LARGE or AMAX < SMALL, scaling is done. scalapack-doc-1.5/man/manl/pclarf.l0100644000056400000620000001777606335610620016706 0ustar pfrauenfstaff.TH PCLARF l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLARF - applie a complex elementary reflector Q to a complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the left or the right .SH SYNOPSIS .TP 19 SUBROUTINE PCLARF( SIDE, M, N, V, IV, JV, DESCV, INCV, TAU, C, IC, JC, DESCC, WORK ) .TP 19 .ti +4 CHARACTER SIDE .TP 19 .ti +4 INTEGER IC, INCV, IV, JC, JV, M, N .TP 19 .ti +4 INTEGER DESCC( * ), DESCV( * ) .TP 19 .ti +4 COMPLEX C( * ), TAU( * ), V( * ), WORK( * ) .SH PURPOSE PCLARF applies a complex elementary reflector Q to a complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the left or the right. Q is represented in the form Q = I - tau * v * v' .br where tau is a complex scalar and v is a complex vector. .br If tau = 0, then Q is taken to be the unit matrix. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. Restrictions .br ============ .br If SIDE = 'Left' and INCV = 1, then the row process having the first entry V(IV,JV) must also have the first row of sub( C ). Moreover, MOD(IV-1,MB_V) must be equal to MOD(IC-1,MB_C), if INCV=M_V, only the last equality must be satisfied. .br If SIDE = 'Right' and INCV = M_V then the column process having the first entry V(IV,JV) must also have the first column of sub( C ) and MOD(JV-1,NB_V) must be equal to MOD(JC-1,NB_C), if INCV = 1 only the last equality must be satisfied. .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': form Q * sub( C ), .br = 'R': form sub( C ) * Q. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 V (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_V,*) containing the local pieces of the distributed vectors V representing the Householder transformation Q, V(IV:IV+M-1,JV) if SIDE = 'L' and INCV = 1, .br V(IV,JV:JV+M-1) if SIDE = 'L' and INCV = M_V, .br V(IV:IV+N-1,JV) if SIDE = 'R' and INCV = 1, .br V(IV,JV:JV+N-1) if SIDE = 'R' and INCV = M_V, The vector v in the representation of Q. V is not used if TAU = 0. .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 INCV (global input) INTEGER The global increment for the elements of V. Only two values of INCV are supported in this version, namely 1 and M_V. INCV must not be zero. .TP 8 TAU (local input) COMPLEX, array, dimension LOCc(JV) if INCV = 1, and LOCr(IV) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix V. .TP 8 C (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_C, LOCc(JC+N-1) ), containing the local pieces of sub( C ). On exit, sub( C ) is overwritten by the Q * sub( C ) if SIDE = 'L', or sub( C ) * Q if SIDE = 'R'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace) COMPLEX array, dimension (LWORK) If INCV = 1, if SIDE = 'L', if IVCOL = ICCOL, LWORK >= NqC0 else LWORK >= MpC0 + MAX( 1, NqC0 ) end if else if SIDE = 'R', LWORK >= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC( N+ICOFFC,NB_V,0,0,NPCOL ),NB_V,0,0,LCMQ ) ) end if else if INCV = M_V, if SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_V,0,0,NPROW ),MB_V,0,0,LCMP ) ) else if SIDE = 'R', if IVROW = ICROW, LWORK >= MpC0 else LWORK >= NqC0 + MAX( 1, MpC0 ) end if end if end if where LCM is the least common multiple of NPROW and NPCOL and LCM = ILCM( NPROW, NPCOL ), LCMP = LCM / NPROW, LCMQ = LCM / NPCOL, IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. Alignment requirements ====================== The distributed submatrices V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: MB_V = NB_V, If INCV = 1, If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', ( MB_V.EQ.NB_A .AND. MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if INCV = M_V, If SIDE = 'Left', ( MB_V.EQ.NB_V .AND. MB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND. IVCOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/pclarfb.l0100644000056400000620000001765706335610620017046 0ustar pfrauenfstaff.TH PCLARFB l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLARFB - applie a complex block reflector Q or its conjugate transpose Q**H to a complex M-by-N distributed matrix sub( C ) denoting C(IC:IC+M-1,JC:JC+N-1), from the left or the right .SH SYNOPSIS .TP 20 SUBROUTINE PCLARFB( SIDE, TRANS, DIRECT, STOREV, M, N, K, V, IV, JV, DESCV, T, C, IC, JC, DESCC, WORK ) .TP 20 .ti +4 CHARACTER SIDE, TRANS, DIRECT, STOREV .TP 20 .ti +4 INTEGER IC, IV, JC, JV, K, M, N .TP 20 .ti +4 INTEGER DESCC( * ), DESCV( * ) .TP 20 .ti +4 COMPLEX C( * ), T( * ), V( * ), WORK( * ) .SH PURPOSE PCLARFB applies a complex block reflector Q or its conjugate transpose Q**H to a complex M-by-N distributed matrix sub( C ) denoting C(IC:IC+M-1,JC:JC+N-1), from the left or the right. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 DIRECT (global input) CHARACTER Indicates how Q is formed from a product of elementary reflectors = 'F': Q = H(1) H(2) . . . H(k) (Forward) .br = 'B': Q = H(k) . . . H(2) H(1) (Backward) .TP 8 STOREV (global input) CHARACTER Indicates how the vectors which define the elementary reflectors are stored: .br = 'C': Columnwise .br = 'R': Rowwise .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The order of the matrix T (= the number of elementary reflectors whose product defines the block reflector). .TP 8 V (local input) COMPLEX pointer into the local memory to an array of dimension ( LLD_V, LOCc(JV+K-1) ) if STOREV = 'C', ( LLD_V, LOCc(JV+M-1)) if STOREV = 'R' and SIDE = 'L', ( LLD_V, LOCc(JV+N-1) ) if STOREV = 'R' and SIDE = 'R'. It contains the local pieces of the distributed vectors V representing the Householder transformation. See further details. If STOREV = 'C' and SIDE = 'L', LLD_V >= MAX(1,LOCr(IV+M-1)); if STOREV = 'C' and SIDE = 'R', LLD_V >= MAX(1,LOCr(IV+N-1)); if STOREV = 'R', LLD_V >= LOCr(IV+K-1). .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 T (local input) COMPLEX array, dimension MB_V by MB_V if STOREV = 'R' and NB_V by NB_V if STOREV = 'C'. The trian- gular matrix T in the representation of the block reflector. .TP 8 C (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the M-by-N distributed matrix sub( C ). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q or sub( C )*Q'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace) COMPLEX array, dimension (LWORK) If STOREV = 'C', if SIDE = 'L', LWORK >= ( NqC0 + MpC0 ) * K else if SIDE = 'R', LWORK >= ( NqC0 + MAX( NpV0 + NUMROC( NUMROC( N+ICOFFC, NB_V, 0, 0, NPCOL ), NB_V, 0, 0, LCMQ ), MpC0 ) ) * K end if else if STOREV = 'R', if SIDE = 'L', LWORK >= ( MpC0 + MAX( MqV0 + NUMROC( NUMROC( M+IROFFC, MB_V, 0, 0, NPROW ), MB_V, 0, 0, LCMP ), NqC0 ) ) * K else if SIDE = 'R', LWORK >= ( MpC0 + NqC0 ) * K end if end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFV = MOD( IV-1, MB_V ), ICOFFV = MOD( JV-1, NB_V ), IVROW = INDXG2P( IV, MB_V, MYROW, RSRC_V, NPROW ), IVCOL = INDXG2P( JV, NB_V, MYCOL, CSRC_V, NPCOL ), MqV0 = NUMROC( M+ICOFFV, NB_V, MYCOL, IVCOL, NPCOL ), NpV0 = NUMROC( N+IROFFV, MB_V, MYROW, IVROW, NPROW ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NpC0 = NUMROC( N+ICOFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. Alignment requirements ====================== The distributed submatrices V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If STOREV = 'Columnwise' If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', ( MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if STOREV = 'Rowwise' If SIDE = 'Left', ( NB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND. IVCOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/pclarfc.l0100644000056400000620000001776206335610620017044 0ustar pfrauenfstaff.TH PCLARFC l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLARFC - applie a complex elementary reflector Q**H to a complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), .SH SYNOPSIS .TP 20 SUBROUTINE PCLARFC( SIDE, M, N, V, IV, JV, DESCV, INCV, TAU, C, IC, JC, DESCC, WORK ) .TP 20 .ti +4 CHARACTER SIDE .TP 20 .ti +4 INTEGER IC, INCV, IV, JC, JV, M, N .TP 20 .ti +4 INTEGER DESCC( * ), DESCV( * ) .TP 20 .ti +4 COMPLEX C( * ), TAU( * ), V( * ), WORK( * ) .SH PURPOSE PCLARFC applies a complex elementary reflector Q**H to a complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the left or the right. Q is represented in the form Q = I - tau * v * v' .br where tau is a complex scalar and v is a complex vector. .br If tau = 0, then Q is taken to be the unit matrix. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. Restrictions .br ============ .br If SIDE = 'Left' and INCV = 1, then the row process having the first entry V(IV,JV) must also have the first row of sub( C ). Moreover, MOD(IV-1,MB_V) must be equal to MOD(IC-1,MB_C), if INCV=M_V, only the last equality must be satisfied. .br If SIDE = 'Right' and INCV = M_V then the column process having the first entry V(IV,JV) must also have the first column of sub( C ) and MOD(JV-1,NB_V) must be equal to MOD(JC-1,NB_C), if INCV = 1 only the last equality must be satisfied. .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': form Q**H * sub( C ), .br = 'R': form sub( C ) * Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 V (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_V,*) containing the local pieces of the distributed vectors V representing the Householder transformation Q, V(IV:IV+M-1,JV) if SIDE = 'L' and INCV = 1, .br V(IV,JV:JV+M-1) if SIDE = 'L' and INCV = M_V, .br V(IV:IV+N-1,JV) if SIDE = 'R' and INCV = 1, .br V(IV,JV:JV+N-1) if SIDE = 'R' and INCV = M_V, The vector v in the representation of Q. V is not used if TAU = 0. .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 INCV (global input) INTEGER The global increment for the elements of V. Only two values of INCV are supported in this version, namely 1 and M_V. INCV must not be zero. .TP 8 TAU (local input) COMPLEX, array, dimension LOCc(JV) if INCV = 1, and LOCr(IV) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix V. .TP 8 C (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_C, LOCc(JC+N-1) ), containing the local pieces of sub( C ). On exit, sub( C ) is overwritten by the Q**H * sub( C ) if SIDE = 'L', or sub( C ) * Q**H if SIDE = 'R'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace) COMPLEX array, dimension (LWORK) If INCV = 1, if SIDE = 'L', if IVCOL = ICCOL, LWORK >= NqC0 else LWORK >= MpC0 + MAX( 1, NqC0 ) end if else if SIDE = 'R', LWORK >= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC( N+ICOFFC,NB_V,0,0,NPCOL ),NB_V,0,0,LCMQ ) ) end if else if INCV = M_V, if SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_V,0,0,NPROW ),MB_V,0,0,LCMP ) ) else if SIDE = 'R', if IVROW = ICROW, LWORK >= MpC0 else LWORK >= NqC0 + MAX( 1, MpC0 ) end if end if end if where LCM is the least common multiple of NPROW and NPCOL and LCM = ILCM( NPROW, NPCOL ), LCMP = LCM / NPROW, LCMQ = LCM / NPCOL, IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. Alignment requirements ====================== The distributed submatrices V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: MB_V = NB_V, If INCV = 1, If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', ( MB_V.EQ.NB_A .AND. MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if INCV = M_V, If SIDE = 'Left', ( MB_V.EQ.NB_V .AND. MB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND. IVCOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/pclarfg.l0100644000056400000620000001254706335610620017044 0ustar pfrauenfstaff.TH PCLARFG l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLARFG - generate a complex elementary reflector H of order n, such that H * sub( X ) = H * ( x(iax,jax) ) = ( alpha ), H' * H = I .SH SYNOPSIS .TP 20 SUBROUTINE PCLARFG( N, ALPHA, IAX, JAX, X, IX, JX, DESCX, INCX, TAU ) .TP 20 .ti +4 INTEGER IAX, INCX, IX, JAX, JX, N .TP 20 .ti +4 COMPLEX ALPHA .TP 20 .ti +4 INTEGER DESCX( * ) .TP 20 .ti +4 COMPLEX TAU( * ), X( * ) .SH PURPOSE PCLARFG generates a complex elementary reflector H of order n, such that ( x ) ( 0 ) .br where alpha is a real scalar, and sub( X ) is an (N-1)-element complex distributed vector X(IX:IX+N-2,JX) if INCX = 1 and X(IX,JX:JX+N-2) if INCX = DESCX(M_). H is represented in the form H = I - tau * ( 1 ) * ( 1 v' ) , .br ( v ) .br where tau is a complex scalar and v is a complex (N-1)-element vector. Note that H is not Hermitian. .br If the elements of sub( X ) are all zero and X(IAX,JAX) is real, then tau = 0 and H is taken to be the unit matrix. .br Otherwise 1 <= real(tau) <= 2 and abs(tau-1) <= 1. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. .SH ARGUMENTS .TP 8 N (global input) INTEGER The global order of the elementary reflector. N >= 0. .TP 8 ALPHA (local output) COMPLEX On exit, alpha is computed in the process scope having the vector sub( X ). .TP 8 IAX (global input) INTEGER The global row index in X of X(IAX,JAX). .TP 8 JAX (global input) INTEGER The global column index in X of X(IAX,JAX). .TP 8 X (local input/local output) COMPLEX, pointer into the local memory to an array of dimension (LLD_X,*). This array contains the local pieces of the distributed vector sub( X ). Before entry, the incremented array sub( X ) must contain the vector x. On exit, it is overwritten with the vector v. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 INCX (global input) INTEGER The global increment for the elements of X. Only two values of INCX are supported in this version, namely 1 and M_X. INCX must not be zero. .TP 8 TAU (local output) COMPLEX, array, dimension LOCc(JX) if INCX = 1, and LOCr(IX) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix X. scalapack-doc-1.5/man/manl/pclarft.l0100644000056400000620000001506506335610620017057 0ustar pfrauenfstaff.TH PCLARFT l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLARFT - form the triangular factor T of a complex block reflector H of order n, which is defined as a product of k elementary reflectors .SH SYNOPSIS .TP 20 SUBROUTINE PCLARFT( DIRECT, STOREV, N, K, V, IV, JV, DESCV, TAU, T, WORK ) .TP 20 .ti +4 CHARACTER DIRECT, STOREV .TP 20 .ti +4 INTEGER IV, JV, K, N .TP 20 .ti +4 INTEGER DESCV( * ) .TP 20 .ti +4 COMPLEX TAU( * ), T( * ), V( * ), WORK( * ) .SH PURPOSE PCLARFT forms the triangular factor T of a complex block reflector H of order n, which is defined as a product of k elementary reflectors. If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular; If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular. If STOREV = 'C', the vector which defines the elementary reflector H(i) is stored in the i-th column of the distributed matrix V, and H = I - V * T * V' .br If STOREV = 'R', the vector which defines the elementary reflector H(i) is stored in the i-th row of the distributed matrix V, and H = I - V' * T * V .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 DIRECT (global input) CHARACTER*1 Specifies the order in which the elementary reflectors are multiplied to form the block reflector: .br = 'F': H = H(1) H(2) . . . H(k) (Forward) .br = 'B': H = H(k) . . . H(2) H(1) (Backward) .TP 8 STOREV (global input) CHARACTER*1 Specifies how the vectors which define the elementary reflectors are stored (see also Further Details): .br = 'R': rowwise .TP 8 N (global input) INTEGER The order of the block reflector H. N >= 0. .TP 8 K (global input) INTEGER The order of the triangular factor T (= the number of elementary reflectors). 1 <= K <= MB_V (= NB_V). .TP 8 V (input/output) COMPLEX pointer into the local memory to an array of local dimension (LOCr(IV+N-1),LOCc(JV+K-1)) if STOREV = 'C', and (LOCr(IV+K-1),LOCc(JV+N-1)) if STOREV = 'R'. The distributed matrix V contains the Householder vectors. See further details. .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 TAU (local input) COMPLEX, array, dimension LOCr(IV+K-1) if INCV = M_V, and LOCc(JV+K-1) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix V. .TP 8 T (local output) COMPLEX array, dimension (NB_V,NB_V) if STOREV = 'Col', and (MB_V,MB_V) otherwise. It contains the k-by-k triangular factor of the block reflector asso- ciated with V. If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is lower triangular. .TP 8 WORK (local workspace) COMPLEX array, dimension (K*(K-1)/2) .SH FURTHER DETAILS The shape of the matrix V and the storage of the vectors which define the H(i) is best illustrated by the following example with n = 5 and k = 3. The elements equal to 1 are not stored; the corresponding array elements are modified but restored on exit. The rest of the array is not used. .br DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R': V( IV:IV+N-1, ( 1 ) V( IV:IV+K-1, ( 1 v1 v1 v1 v1 ) JV:JV+K-1 ) = ( v1 1 ) JV:JV+N-1 ) = ( 1 v2 v2 v2 ) ( v1 v2 1 ) ( 1 v3 v3 ) ( v1 v2 v3 ) .br ( v1 v2 v3 ) .br DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R': V( IV:IV+N-1, ( v1 v2 v3 ) V( IV:IV+K-1, ( v1 v1 1 ) JV:JV+K-1 ) = ( v1 v2 v3 ) JV:JV+N-1 ) = ( v2 v2 v2 1 ) ( 1 v2 v3 ) ( v3 v3 v3 v3 1 ) ( 1 v3 ) .br ( 1 ) .br scalapack-doc-1.5/man/manl/pclarz.l0100644000056400000620000002041306335610620016710 0ustar pfrauenfstaff.TH PCLARZ l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLARZ - applie a complex elementary reflector Q to a complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the left or the right .SH SYNOPSIS .TP 19 SUBROUTINE PCLARZ( SIDE, M, N, L, V, IV, JV, DESCV, INCV, TAU, C, IC, JC, DESCC, WORK ) .TP 19 .ti +4 CHARACTER SIDE .TP 19 .ti +4 INTEGER IC, INCV, IV, JC, JV, L, M, N .TP 19 .ti +4 INTEGER DESCC( * ), DESCV( * ) .TP 19 .ti +4 COMPLEX C( * ), TAU( * ), V( * ), WORK( * ) .SH PURPOSE PCLARZ applies a complex elementary reflector Q to a complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the left or the right. Q is represented in the form Q = I - tau * v * v' .br where tau is a complex scalar and v is a complex vector. .br If tau = 0, then Q is taken to be the unit matrix. .br Q is a product of k elementary reflectors as returned by PCTZRZF. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. Restrictions .br ============ .br If SIDE = 'Left' and INCV = 1, then the row process having the first entry V(IV,JV) must also own C(IC+M-L,JC:JC+N-1). Moreover, MOD(IV-1,MB_V) must be equal to MOD(IC+N-L-1,MB_C), if INCV=M_V, only the last equality must be satisfied. .br If SIDE = 'Right' and INCV = M_V then the column process having the first entry V(IV,JV) must also own C(IC:IC+M-1,JC+N-L) and MOD(JV-1,NB_V) must be equal to MOD(JC+N-L-1,NB_C), if INCV = 1 only the last equality must be satisfied. .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': form Q * sub( C ), .br = 'R': form sub( C ) * Q. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 L (global input) INTEGER The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. .TP 8 V (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_V,*) containing the local pieces of the distributed vectors V representing the Householder transformation Q, V(IV:IV+L-1,JV) if SIDE = 'L' and INCV = 1, .br V(IV,JV:JV+L-1) if SIDE = 'L' and INCV = M_V, .br V(IV:IV+L-1,JV) if SIDE = 'R' and INCV = 1, .br V(IV,JV:JV+L-1) if SIDE = 'R' and INCV = M_V, The vector v in the representation of Q. V is not used if TAU = 0. .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 INCV (global input) INTEGER The global increment for the elements of V. Only two values of INCV are supported in this version, namely 1 and M_V. INCV must not be zero. .TP 8 TAU (local input) COMPLEX, array, dimension LOCc(JV) if INCV = 1, and LOCr(IV) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix V. .TP 8 C (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_C, LOCc(JC+N-1) ), containing the local pieces of sub( C ). On exit, sub( C ) is overwritten by the Q * sub( C ) if SIDE = 'L', or sub( C ) * Q if SIDE = 'R'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace) COMPLEX array, dimension (LWORK) If INCV = 1, if SIDE = 'L', if IVCOL = ICCOL, LWORK >= NqC0 else LWORK >= MpC0 + MAX( 1, NqC0 ) end if else if SIDE = 'R', LWORK >= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC( N+ICOFFC,NB_V,0,0,NPCOL ),NB_V,0,0,LCMQ ) ) end if else if INCV = M_V, if SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_V,0,0,NPROW ),MB_V,0,0,LCMP ) ) else if SIDE = 'R', if IVROW = ICROW, LWORK >= MpC0 else LWORK >= NqC0 + MAX( 1, MpC0 ) end if end if end if where LCM is the least common multiple of NPROW and NPCOL and LCM = ILCM( NPROW, NPCOL ), LCMP = LCM / NPROW, LCMQ = LCM / NPCOL, IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. Alignment requirements ====================== The distributed submatrices V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: MB_V = NB_V, If INCV = 1, If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', ( MB_V.EQ.NB_A .AND. MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if INCV = M_V, If SIDE = 'Left', ( MB_V.EQ.NB_V .AND. MB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND. IVCOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/pclarzb.l0100644000056400000620000002007206335610620017053 0ustar pfrauenfstaff.TH PCLARZB l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLARZB - applie a complex block reflector Q or its conjugate transpose Q**H to a complex M-by-N distributed matrix sub( C ) denoting C(IC:IC+M-1,JC:JC+N-1), from the left or the right .SH SYNOPSIS .TP 20 SUBROUTINE PCLARZB( SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V, IV, JV, DESCV, T, C, IC, JC, DESCC, WORK ) .TP 20 .ti +4 CHARACTER DIRECT, SIDE, STOREV, TRANS .TP 20 .ti +4 INTEGER IC, IV, JC, JV, K, L, M, N .TP 20 .ti +4 INTEGER DESCC( * ), DESCV( * ) .TP 20 .ti +4 COMPLEX C( * ), T( * ), V( * ), WORK( * ) .SH PURPOSE PCLARZB applies a complex block reflector Q or its conjugate transpose Q**H to a complex M-by-N distributed matrix sub( C ) denoting C(IC:IC+M-1,JC:JC+N-1), from the left or the right. Q is a product of k elementary reflectors as returned by PCTZRZF. Currently, only STOREV = 'R' and DIRECT = 'B' are supported. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 DIRECT (global input) CHARACTER Indicates how H is formed from a product of elementary reflectors = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet) .br = 'B': H = H(k) . . . H(2) H(1) (Backward) .TP 8 STOREV (global input) CHARACTER Indicates how the vectors which define the elementary reflectors are stored: .br = 'C': Columnwise (not supported yet) .br = 'R': Rowwise .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The order of the matrix T (= the number of elementary reflectors whose product defines the block reflector). .TP 8 L (global input) INTEGER The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. .TP 8 V (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_V, LOCc(JV+M-1)) if SIDE = 'L', (LLD_V, LOCc(JV+N-1)) if SIDE = 'R'. It contains the local pieces of the distributed vectors V representing the Householder transformation as returned by PCTZRZF. LLD_V >= LOCr(IV+K-1). .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 T (local input) COMPLEX array, dimension MB_V by MB_V The lower triangular matrix T in the representation of the block reflector. .TP 8 C (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the M-by-N distributed matrix sub( C ). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q or sub( C )*Q'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace) COMPLEX array, dimension (LWORK) If STOREV = 'C', if SIDE = 'L', LWORK >= ( NqC0 + MpC0 ) * K else if SIDE = 'R', LWORK >= ( NqC0 + MAX( NpV0 + NUMROC( NUMROC( N+ICOFFC, NB_V, 0, 0, NPCOL ), NB_V, 0, 0, LCMQ ), MpC0 ) ) * K end if else if STOREV = 'R', if SIDE = 'L', LWORK >= ( MpC0 + MAX( MqV0 + NUMROC( NUMROC( M+IROFFC, MB_V, 0, 0, NPROW ), MB_V, 0, 0, LCMP ), NqC0 ) ) * K else if SIDE = 'R', LWORK >= ( MpC0 + NqC0 ) * K end if end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFV = MOD( IV-1, MB_V ), ICOFFV = MOD( JV-1, NB_V ), IVROW = INDXG2P( IV, MB_V, MYROW, RSRC_V, NPROW ), IVCOL = INDXG2P( JV, NB_V, MYCOL, CSRC_V, NPCOL ), MqV0 = NUMROC( M+ICOFFV, NB_V, MYCOL, IVCOL, NPCOL ), NpV0 = NUMROC( N+IROFFV, MB_V, MYROW, IVROW, NPROW ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NpC0 = NUMROC( N+ICOFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. Alignment requirements ====================== The distributed submatrices V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If STOREV = 'Columnwise' If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', ( MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if STOREV = 'Rowwise' If SIDE = 'Left', ( NB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND. IVCOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/pclarzc.l0100644000056400000620000002037706335610620017064 0ustar pfrauenfstaff.TH PCLARZC l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLARZC - applie a complex elementary reflector Q**H to a complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), .SH SYNOPSIS .TP 20 SUBROUTINE PCLARZC( SIDE, M, N, L, V, IV, JV, DESCV, INCV, TAU, C, IC, JC, DESCC, WORK ) .TP 20 .ti +4 CHARACTER SIDE .TP 20 .ti +4 INTEGER IC, INCV, IV, JC, JV, L, M, N .TP 20 .ti +4 INTEGER DESCC( * ), DESCV( * ) .TP 20 .ti +4 COMPLEX C( * ), TAU( * ), V( * ), WORK( * ) .SH PURPOSE PCLARZC applies a complex elementary reflector Q**H to a complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the left or the right. Q is represented in the form Q = I - tau * v * v' .br where tau is a complex scalar and v is a complex vector. .br If tau = 0, then Q is taken to be the unit matrix. .br Q is a product of k elementary reflectors as returned by PCTZRZF. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. Restrictions .br ============ .br If SIDE = 'Left' and INCV = 1, then the row process having the first entry V(IV,JV) must also own C(IC+M-L,JC:JC+N-1). Moreover, MOD(IV-1,MB_V) must be equal to MOD(IC+N-L-1,MB_C), if INCV=M_V, only the last equality must be satisfied. .br If SIDE = 'Right' and INCV = M_V then the column process having the first entry V(IV,JV) must also own C(IC:IC+M-1,JC+N-L) and MOD(JV-1,NB_V) must be equal to MOD(JC+N-L-1,NB_C), if INCV = 1 only the last equality must be satisfied. .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': form Q**H * sub( C ), .br = 'R': form sub( C ) * Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 L (global input) INTEGER The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. .TP 8 V (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_V,*) containing the local pieces of the distributed vectors V representing the Householder transformation Q, V(IV:IV+L-1,JV) if SIDE = 'L' and INCV = 1, .br V(IV,JV:JV+L-1) if SIDE = 'L' and INCV = M_V, .br V(IV:IV+L-1,JV) if SIDE = 'R' and INCV = 1, .br V(IV,JV:JV+L-1) if SIDE = 'R' and INCV = M_V, The vector v in the representation of Q. V is not used if TAU = 0. .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 INCV (global input) INTEGER The global increment for the elements of V. Only two values of INCV are supported in this version, namely 1 and M_V. INCV must not be zero. .TP 8 TAU (local input) COMPLEX, array, dimension LOCc(JV) if INCV = 1, and LOCr(IV) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix V. .TP 8 C (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_C, LOCc(JC+N-1) ), containing the local pieces of sub( C ). On exit, sub( C ) is overwritten by the Q**H * sub( C ) if SIDE = 'L', or sub( C ) * Q**H if SIDE = 'R'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace) COMPLEX array, dimension (LWORK) If INCV = 1, if SIDE = 'L', if IVCOL = ICCOL, LWORK >= NqC0 else LWORK >= MpC0 + MAX( 1, NqC0 ) end if else if SIDE = 'R', LWORK >= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC( N+ICOFFC,NB_V,0,0,NPCOL ),NB_V,0,0,LCMQ ) ) end if else if INCV = M_V, if SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_V,0,0,NPROW ),MB_V,0,0,LCMP ) ) else if SIDE = 'R', if IVROW = ICROW, LWORK >= MpC0 else LWORK >= NqC0 + MAX( 1, MpC0 ) end if end if end if where LCM is the least common multiple of NPROW and NPCOL and LCM = ILCM( NPROW, NPCOL ), LCMP = LCM / NPROW, LCMQ = LCM / NPCOL, IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. Alignment requirements ====================== The distributed submatrices V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: MB_V = NB_V, If INCV = 1, If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', ( MB_V.EQ.NB_A .AND. MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if INCV = M_V, If SIDE = 'Left', ( MB_V.EQ.NB_V .AND. MB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND. IVCOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/pclarzt.l0100644000056400000620000001565306335610620017106 0ustar pfrauenfstaff.TH PCLARZT l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLARZT - form the triangular factor T of a complex block reflector H of order > n, which is defined as a product of k elementary reflectors as returned by PCTZRZF .SH SYNOPSIS .TP 20 SUBROUTINE PCLARZT( DIRECT, STOREV, N, K, V, IV, JV, DESCV, TAU, T, WORK ) .TP 20 .ti +4 CHARACTER DIRECT, STOREV .TP 20 .ti +4 INTEGER IV, JV, K, N .TP 20 .ti +4 INTEGER DESCV( * ) .TP 20 .ti +4 COMPLEX TAU( * ), T( * ), V( * ), WORK( * ) .SH PURPOSE PCLARZT forms the triangular factor T of a complex block reflector H of order > n, which is defined as a product of k elementary reflectors as returned by PCTZRZF. If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular; If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular. If STOREV = 'C', the vector which defines the elementary reflector H(i) is stored in the i-th column of the array V, and .br H = I - V * T * V' .br If STOREV = 'R', the vector which defines the elementary reflector H(i) is stored in the i-th row of the array V, and .br H = I - V' * T * V .br Currently, only STOREV = 'R' and DIRECT = 'B' are supported. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 DIRECT (global input) CHARACTER Specifies the order in which the elementary reflectors are multiplied to form the block reflector: .br = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet) .br = 'B': H = H(k) . . . H(2) H(1) (Backward) .TP 8 STOREV (global input) CHARACTER Specifies how the vectors which define the elementary reflectors are stored (see also Further Details): .br = 'R': rowwise .TP 8 N (global input) INTEGER The number of meaningful entries of the block reflector H. N >= 0. .TP 8 K (global input) INTEGER The order of the triangular factor T (= the number of elementary reflectors). 1 <= K <= MB_V (= NB_V). .TP 8 V (input/output) COMPLEX pointer into the local memory to an array of local dimension (LOCr(IV+K-1),LOCc(JV+N-1)). The distributed matrix V contains the Householder vectors. See further details. .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 TAU (local input) COMPLEX, array, dimension LOCr(IV+K-1) if INCV = M_V, and LOCc(JV+K-1) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix V. .TP 8 T (local output) COMPLEX array, dimension (MB_V,MB_V) It contains the k-by-k triangular factor of the block reflector associated with V. T is lower triangular. .TP 8 WORK (local workspace) COMPLEX array, dimension (K*(K-1)/2) .SH FURTHER DETAILS The shape of the matrix V and the storage of the vectors which define the H(i) is best illustrated by the following example with n = 5 and k = 3. The elements equal to 1 are not stored; the corresponding array elements are modified but restored on exit. The rest of the array is not used. .br DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R': ______V_____ .br ( v1 v2 v3 ) / \ ( v1 v2 v3 ) ( v1 v1 v1 v1 v1 . . . . 1 ) V = ( v1 v2 v3 ) ( v2 v2 v2 v2 v2 . . . 1 ) ( v1 v2 v3 ) ( v3 v3 v3 v3 v3 . . 1 ) ( v1 v2 v3 ) .br . . . .br . . . .br 1 . . .br 1 . .br 1 .br DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R': ______V_____ 1 / \ . 1 ( 1 . . . . v1 v1 v1 v1 v1 ) . . 1 ( . 1 . . . v2 v2 v2 v2 v2 ) . . . ( . . 1 . . v3 v3 v3 v3 v3 ) . . . .br ( v1 v2 v3 ) .br ( v1 v2 v3 ) .br V = ( v1 v2 v3 ) .br ( v1 v2 v3 ) .br ( v1 v2 v3 ) .br scalapack-doc-1.5/man/manl/pclascl.l0100644000056400000620000001235306335610620017042 0ustar pfrauenfstaff.TH PCLASCL l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLASCL - multiplie the M-by-N complex distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) by the real scalar CTO/CFROM .SH SYNOPSIS .TP 20 SUBROUTINE PCLASCL( TYPE, CFROM, CTO, M, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER TYPE .TP 20 .ti +4 INTEGER IA, INFO, JA, M, N .TP 20 .ti +4 REAL CFROM, CTO .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ) .SH PURPOSE PCLASCL multiplies the M-by-N complex distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) 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 sub( A ) may be full, upper triangular, lower triangular or upper Hessenberg. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 TYPE (global input) CHARACTER TYPE indices the storage type of the input distributed matrix. = 'G': sub( A ) is a full matrix, .br = 'L': sub( A ) is a lower triangular matrix, .br = 'U': sub( A ) is an upper triangular matrix, .br = 'H': sub( A ) is an upper Hessenberg matrix. .TP 8 CFROM (global input) REAL CTO (global input) REAL The distributed matrix sub( 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. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). This array contains the local pieces of the distributed matrix sub( A ). On exit, this array contains the local pieces of the distributed matrix multiplied by CTO/CFROM. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pclase2.l0100644000056400000620000001215706335610621016755 0ustar pfrauenfstaff.TH PCLASE2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLASE2 - initialize an M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) to BETA on the diagonal and ALPHA on the offdiagonals .SH SYNOPSIS .TP 20 SUBROUTINE PCLASE2( UPLO, M, N, ALPHA, BETA, A, IA, JA, DESCA ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, JA, M, N .TP 20 .ti +4 COMPLEX ALPHA, BETA .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ) .SH PURPOSE PCLASE2 initializes an M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) to BETA on the diagonal and ALPHA on the offdiagonals. PCLASE2 requires that only dimension of the matrix operand is distributed. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies the part of the distributed matrix sub( A ) to be set: .br = 'U': Upper triangular part is set; the strictly lower triangular part of sub( A ) is not changed; = 'L': Lower triangular part is set; the strictly upper triangular part of sub( A ) is not changed; Otherwise: All of the matrix sub( A ) is set. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 ALPHA (global input) COMPLEX The constant to which the offdiagonal elements are to be set. .TP 8 BETA (global input) COMPLEX The constant to which the diagonal elements are to be set. .TP 8 A (local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). This array contains the local pieces of the distributed matrix sub( A ) to be set. On exit, the leading M-by-N submatrix sub( A ) is set as follows: if UPLO = 'U', A(IA+i-1,JA+j-1) = ALPHA, 1<=i<=j-1, 1<=j<=N, if UPLO = 'L', A(IA+i-1,JA+j-1) = ALPHA, j+1<=i<=M, 1<=j<=N, otherwise, A(IA+i-1,JA+j-1) = ALPHA, 1<=i<=M, 1<=j<=N, IA+i.NE.JA+j, and, for all UPLO, A(IA+i-1,JA+i-1) = BETA, 1<=i<=min(M,N). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. scalapack-doc-1.5/man/manl/pclaset.l0100644000056400000620000001203706335610621017054 0ustar pfrauenfstaff.TH PCLASET l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLASET - initialize an M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) to BETA on the diagonal and ALPHA on the offdiagonals .SH SYNOPSIS .TP 20 SUBROUTINE PCLASET( UPLO, M, N, ALPHA, BETA, A, IA, JA, DESCA ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, JA, M, N .TP 20 .ti +4 COMPLEX ALPHA, BETA .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ) .SH PURPOSE PCLASET initializes an M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) to BETA on the diagonal and ALPHA on the offdiagonals. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies the part of the distributed matrix sub( A ) to be set: .br = 'U': Upper triangular part is set; the strictly lower triangular part of sub( A ) is not changed; = 'L': Lower triangular part is set; the strictly upper triangular part of sub( A ) is not changed; Otherwise: All of the matrix sub( A ) is set. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 ALPHA (global input) COMPLEX The constant to which the offdiagonal elements are to be set. .TP 8 BETA (global input) COMPLEX The constant to which the diagonal elements are to be set. .TP 8 A (local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). This array contains the local pieces of the distributed matrix sub( A ) to be set. On exit, the leading M-by-N submatrix sub( A ) is set as follows: if UPLO = 'U', A(IA+i-1,JA+j-1) = ALPHA, 1<=i<=j-1, 1<=j<=N, if UPLO = 'L', A(IA+i-1,JA+j-1) = ALPHA, j+1<=i<=M, 1<=j<=N, otherwise, A(IA+i-1,JA+j-1) = ALPHA, 1<=i<=M, 1<=j<=N, IA+i.NE.JA+j, and, for all UPLO, A(IA+i-1,JA+i-1) = BETA, 1<=i<=min(M,N). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. scalapack-doc-1.5/man/manl/pclassq.l0100644000056400000620000001244206335610621017067 0ustar pfrauenfstaff.TH PCLASSQ l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLASSQ - return the values scl and smsq such that ( scl**2 )*smsq = x( 1 )**2 +...+ x( n )**2 + ( scale**2 )*sumsq, .SH SYNOPSIS .TP 20 SUBROUTINE PCLASSQ( N, X, IX, JX, DESCX, INCX, SCALE, SUMSQ ) .TP 20 .ti +4 INTEGER IX, INCX, JX, N .TP 20 .ti +4 REAL SCALE, SUMSQ .TP 20 .ti +4 INTEGER DESCX( * ) .TP 20 .ti +4 COMPLEX X( * ) .SH PURPOSE PCLASSQ returns the values scl and smsq such that where x( i ) = sub( X ) = abs( X( IX+(JX-1)*DESCX(M_)+(i-1)*INCX ) ). The value of sumsq is assumed to be at least unity and the value of ssq will then satisfy .br 1.0 .le. ssq .le. ( sumsq + 2*n ). .br scale is assumed to be non-negative and scl returns the value scl = max( scale, abs( real( x( i ) ) ), abs( aimag( x( i ) ) ) ), i .br scale and sumsq must be supplied in SCALE and SUMSQ respectively. SCALE and SUMSQ are overwritten by scl and ssq respectively. The routine makes only one pass through the vector sub( X ). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. The result are only available in the scope of sub( X ), i.e if sub( X ) is distributed along a process row, the correct results are only available in this process row of the grid. Similarly if sub( X ) is distributed along a process column, the correct results are only available in this process column of the grid. .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The length of the distributed vector sub( X ). .TP 8 X (input) COMPLEX The vector for which a scaled sum of squares is computed. x( i ) = X(IX+(JX-1)*M_X +(i-1)*INCX ), 1 <= i <= n. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 INCX (global input) INTEGER The global increment for the elements of X. Only two values of INCX are supported in this version, namely 1 and M_X. INCX must not be zero. .TP 8 SCALE (local input/local output) REAL On entry, the value scale in the equation above. On exit, SCALE is overwritten with scl , the scaling factor for the sum of squares. .TP 8 SUMSQ (local input/local output) REAL On entry, the value sumsq in the equation above. On exit, SUMSQ is overwritten with smsq , the basic sum of squares from which scl has been factored out. scalapack-doc-1.5/man/manl/pclaswp.l0100644000056400000620000001224706335610621017075 0ustar pfrauenfstaff.TH PCLASWP l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLASWP - perform a series of row or column interchanges on the distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PCLASWP( DIREC, ROWCOL, N, A, IA, JA, DESCA, K1, K2, IPIV ) .TP 20 .ti +4 CHARACTER DIREC, ROWCOL .TP 20 .ti +4 INTEGER IA, JA, K1, K2, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ) .TP 20 .ti +4 COMPLEX A( * ) .SH PURPOSE PCLASWP performs a series of row or column interchanges on the distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1). One interchange is initiated for each of rows or columns K1 trough K2 of sub( A ). This routine assumes that the pivoting information has already been broadcast along the process row or column. .br Also note that this routine will only work for K1-K2 being in the same MB (or NB) block. If you want to pivot a full matrix, use PCLAPIV. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 DIREC (global input) CHARACTER Specifies in which order the permutation is applied: = 'F' (Forward) = 'B' (Backward) .TP 8 ROWCOL (global input) CHARACTER Specifies if the rows or columns are permuted: = 'R' (Rows) = 'C' (Columns) .TP 8 N (global input) INTEGER If ROWCOL = 'R', the length of the rows of the distributed matrix A(*,JA:JA+N-1) to be permuted; If ROWCOL = 'C', the length of the columns of the distributed matrix A(IA:IA+N-1,*) to be permuted. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, * ). On entry, this array contains the local pieces of the distri- buted matrix to which the row/columns interchanges will be applied. On exit the permuted distributed matrix. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 K1 (global input) INTEGER The first element of IPIV for which a row or column inter- change will be done. .TP 8 K2 (global input) INTEGER The last element of IPIV for which a row or column inter- change will be done. .TP 8 IPIV (local input) INTEGER array, dimension LOCr(M_A)+MB_A for row pivoting and LOCc(N_A)+NB_A for column pivoting. This array is tied to the matrix A, IPIV(K) = L implies rows (or columns) K and L are to be interchanged. scalapack-doc-1.5/man/manl/pclatra.l0100644000056400000620000000766506335610621017062 0ustar pfrauenfstaff.TH PCLATRA l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLATRA - compute the trace of an N-by-N distributed matrix sub( A ) denoting A( IA:IA+N-1, JA:JA+N-1 ) .SH SYNOPSIS .TP 17 COMPLEX FUNCTION PCLATRA( N, A, IA, JA, DESCA ) .TP 17 .ti +4 INTEGER IA, JA, N .TP 17 .ti +4 INTEGER DESCA( * ) .TP 17 .ti +4 COMPLEX A( * ) .SH PURPOSE PCLATRA computes the trace of an N-by-N distributed matrix sub( A ) denoting A( IA:IA+N-1, JA:JA+N-1 ). The result is left on every process of the grid. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on i.e the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension ( LLD_A, LOCc(JA+N-1) ). This array contains the local pieces of the distributed matrix the trace is to be computed. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. scalapack-doc-1.5/man/manl/pclatrd.l0100644000056400000620000002131606335610621017052 0ustar pfrauenfstaff.TH PCLATRD l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLATRD - reduce NB rows and columns of a complex Hermitian distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) to complex tridiagonal form by an unitary similarity transformation Q' * sub( A ) * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of sub( A ) .SH SYNOPSIS .TP 20 SUBROUTINE PCLATRD( UPLO, N, NB, A, IA, JA, DESCA, D, E, TAU, W, IW, JW, DESCW, WORK ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IW, JA, JW, N, NB .TP 20 .ti +4 INTEGER DESCA( * ), DESCW( * ) .TP 20 .ti +4 REAL D( * ), E( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), W( * ), WORK( * ) .SH PURPOSE PCLATRD reduces NB rows and columns of a complex Hermitian distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) to complex tridiagonal form by an unitary similarity transformation Q' * sub( A ) * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of sub( A ). If UPLO = 'U', PCLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; .br if UPLO = 'L', PCLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied. .br This is an auxiliary routine called by PCHETRD. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the upper or lower triangular part of the Hermitian matrix sub( A ) is stored: .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NB (global input) INTEGER The number of rows and columns to be reduced. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the Hermitian distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced. On exit, if UPLO = 'U', the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of sub( A ); the elements above the diagonal with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. If UPLO = 'L', the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of sub( A ); the elements below the diagonal with the array TAU, represent the unitary matrix Q as a product of elementary reflectors; See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) REAL array, dimension LOCc(JA+N-1) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i). D is tied to the distributed matrix A. .TP 8 E (local output) REAL array, dimension LOCc(JA+N-1) if UPLO = 'U', LOCc(JA+N-2) otherwise. The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. E is tied to the distributed matrix A. .TP 8 TAU (local output) COMPLEX, array, dimension LOCc(JA+N-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 W (local output) COMPLEX pointer into the local memory to an array of dimension (LLD_W,NB_W), This array contains the local pieces of the N-by-NB_W matrix W required to update the unreduced part of sub( A ). .TP 8 IW (global input) INTEGER The row index in the global array W indicating the first row of sub( W ). .TP 8 JW (global input) INTEGER The column index in the global array W indicating the first column of sub( W ). .TP 8 DESCW (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix W. .TP 8 WORK (local workspace) COMPLEX array, dimension (NB_A) .SH FURTHER DETAILS If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors .br Q = H(n) H(n-1) . . . H(n-nb+1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in .br A(ia:ia+i-2,ja+i), and tau in TAU(ja+i-1). .br If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors .br Q = H(1) H(2) . . . H(nb). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in .br A(ia+i+1:ia+n-1,ja+i-1), and tau in TAU(ja+i-1). .br The elements of the vectors v together form the N-by-NB matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a Hermitian rank-2k update of the form: sub( A ) := sub( A ) - V*W' - W*V'. .br The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2: .br if UPLO = 'U': if UPLO = 'L': .br ( a a a v4 v5 ) ( d ) ( a a v4 v5 ) ( 1 d ) ( a 1 v5 ) ( v1 1 a ) ( d 1 ) ( v1 v2 a a ) ( d ) ( v1 v2 a a a ) where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i). .br scalapack-doc-1.5/man/manl/pclatrs.l0100644000056400000620000000117006335610621017065 0ustar pfrauenfstaff.TH PCLATRS l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLATRS - solve a triangular system .SH SYNOPSIS .TP 20 SUBROUTINE PCLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, IA, JA, DESCA, X, IX, JX, DESCX, SCALE, CNORM, WORK ) .TP 20 .ti +4 CHARACTER DIAG, NORMIN, TRANS, UPLO .TP 20 .ti +4 INTEGER IA, IX, JA, JX, N .TP 20 .ti +4 REAL SCALE .TP 20 .ti +4 INTEGER DESCA( * ), DESCX( * ) .TP 20 .ti +4 REAL CNORM( * ) .TP 20 .ti +4 COMPLEX A( * ), X( * ), WORK( * ) .SH PURPOSE PCLATRS solves a triangular system. This routine in unfinished at this time, but will be part of the next release. .br scalapack-doc-1.5/man/manl/pclatrz.l0100644000056400000620000001453606335610621017106 0ustar pfrauenfstaff.TH PCLATRZ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCLATRZ - reduce the M-by-N ( M<=N ) complex upper trapezoidal matrix sub( A ) = [A(IA:IA+M-1,JA:JA+M-1) A(IA:IA+M-1,JA+N-L:JA+N-1)] .SH SYNOPSIS .TP 20 SUBROUTINE PCLATRZ( M, N, L, A, IA, JA, DESCA, TAU, WORK ) .TP 20 .ti +4 INTEGER IA, JA, L, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), WORK( * ) .SH PURPOSE PCLATRZ reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix sub( A ) = [A(IA:IA+M-1,JA:JA+M-1) A(IA:IA+M-1,JA+N-L:JA+N-1)] to upper triangular form by means of unitary transformations. The upper trapezoidal matrix sub( A ) is factored as .br sub( A ) = ( R 0 ) * Z, .br where Z is an N-by-N unitary matrix and R is an M-by-M upper triangular matrix. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 L (global input) INTEGER The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. L > 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the leading M-by-M upper triangular part of sub( A ) contains the upper trian- gular matrix R, and elements N-L+1 to N of the first M rows of sub( A ), with the array TAU, represent the unitary matrix Z as a product of M elementary reflectors. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX, array, dimension LOCr(IA+M-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace) COMPLEX array, dimension (LWORK) LWORK >= Nq0 + MAX( 1, Mp0 ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. .SH FURTHER DETAILS The factorization is obtained by Householder's method. The kth transformation matrix, Z( k ), whose conjugate transpose is used to introduce zeros into the (m - k + 1)th row of sub( A ), is given in the form .br Z( k ) = ( I 0 ), .br ( 0 T( k ) ) .br where .br T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ), ( 0 ) ( z( k ) ) tau is a scalar and z( k ) is an ( n - m ) element vector. tau and z( k ) are chosen to annihilate the elements of the kth row of sub( A ). .br The scalar tau is returned in the kth element of TAU and the vector u( k ) in the kth row of sub( A ), such that the elements of z( k ) are in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in the upper triangular part of sub( A ). .br Z is given by .br Z = Z( 1 ) * Z( 2 ) * ... * Z( m ). .br scalapack-doc-1.5/man/manl/pclauu2.l0100644000056400000620000001160206335610621016771 0ustar pfrauenfstaff.TH PCLAUU2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLAUU2 - compute the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PCLAUU2( UPLO, N, A, IA, JA, DESCA ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ) .SH PURPOSE PCLAUU2 computes the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). If UPLO = 'U' or 'u' then the upper triangle of the result is stored, overwriting the factor U in sub( A ). .br If UPLO = 'L' or 'l' then the lower triangle of the result is stored, overwriting the factor L in sub( A ). .br This is the unblocked form of the algorithm, calling Level 2 BLAS. No communication is performed by this routine, the matrix to operate on should be strictly local to one process. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 Specifies whether the triangular factor stored in the matrix sub( A ) is upper or lower triangular: .br = 'U': Upper triangular, .br = 'L': Lower triangular. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the order of the triangular factor U or L. N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the triangular factor L or U. On exit, if UPLO = 'U', the upper triangle of the distributed matrix sub( A ) is overwritten with the upper triangle of the product U * U'; if UPLO = 'L', the lower triangle of sub( A ) is overwritten with the lower triangle of the product L' * L. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. scalapack-doc-1.5/man/manl/pclauum.l0100644000056400000620000001144006335610621017064 0ustar pfrauenfstaff.TH PCLAUUM l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCLAUUM - compute the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PCLAUUM( UPLO, N, A, IA, JA, DESCA ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ) .SH PURPOSE PCLAUUM computes the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). If UPLO = 'U' or 'u' then the upper triangle of the result is stored, overwriting the factor U in sub( A ). .br If UPLO = 'L' or 'l' then the lower triangle of the result is stored, overwriting the factor L in sub( A ). .br This is the blocked form of the algorithm, calling Level 3 PBLAS. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 Specifies whether the triangular factor stored in the distributed matrix sub( A ) is upper or lower triangular: .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the triangular factor U or L. N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the triangular factor L or U. On exit, if UPLO = 'U', the upper triangle of the distributed matrix sub( A ) is overwritten with the upper triangle of the product U * U'; if UPLO = 'L', the lower triangle of sub( A ) is overwritten with the lower triangle of the product L' * L. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. scalapack-doc-1.5/man/manl/pcmax1.l0100644000056400000620000001336406335610621016616 0ustar pfrauenfstaff.TH PCMAX1 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCMAX1 - compute the global index of the maximum element in absolute value of a distributed vector sub( X ) .SH SYNOPSIS .TP 19 SUBROUTINE PCMAX1( N, AMAX, INDX, X, IX, JX, DESCX, INCX ) .TP 19 .ti +4 INTEGER INDX, INCX, IX, JX, N .TP 19 .ti +4 COMPLEX AMAX .TP 19 .ti +4 INTEGER DESCX( * ) .TP 19 .ti +4 COMPLEX X( * ) .SH PURPOSE PCMAX1 computes the global index of the maximum element in absolute value of a distributed vector sub( X ). The global index is returned in INDX and the value is returned in AMAX, .br where sub( X ) denotes X(IX:IX+N-1,JX) if INCX = 1, .br X(IX,JX:JX+N-1) if INCX = M_X. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. When the result of a vector-oriented PBLAS call is a scalar, it will be made available only within the scope which owns the vector(s) being operated on. Let X be a generic term for the input vector(s). Then, the processes which receive the answer will be (note that if an operation involves more than one vector, the processes which re- ceive the result will be the union of the following calculation for each vector): .br If N = 1, M_X = 1 and INCX = 1, then one can't determine if a process row or process column owns the vector operand, therefore only the process of coordinate {RSRC_X, CSRC_X} receives the result; If INCX = M_X, then sub( X ) is a vector distributed over a process row. Each process part of this row receives the result; .br If INCX = 1, then sub( X ) is a vector distributed over a process column. Each process part of this column receives the result; Based on PCAMAX from Level 1 PBLAS. The change is to use the 'genuine' absolute value. .br The serial version was contributed to LAPACK by Nick Higham for use with CLACON. .br .SH ARGUMENTS .TP 8 N (global input) pointer to INTEGER The number of components of the distributed vector sub( X ). N >= 0. .TP 8 AMAX (global output) pointer to REAL The absolute value of the largest entry of the distributed vector sub( X ) only in the scope of sub( X ). .TP 8 INDX (global output) pointer to INTEGER The global index of the element of the distributed vector sub( X ) whose real part has maximum absolute value. .TP 8 X (local input) COMPLEX array containing the local pieces of a distributed matrix of dimension of at least ( (JX-1)*M_X + IX + ( N - 1 )*abs( INCX ) ) This array contains the entries of the distributed vector sub( X ). .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 INCX (global input) INTEGER The global increment for the elements of X. Only two values of INCX are supported in this version, namely 1 and M_X. INCX must not be zero. scalapack-doc-1.5/man/manl/pcpbsv.l0100644000056400000620000000141306335610621016712 0ustar pfrauenfstaff.TH PCPBSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCPBSV - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 19 SUBROUTINE PCPBSV( UPLO, N, BW, NRHS, A, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 CHARACTER UPLO .TP 19 .ti +4 INTEGER BW, IB, INFO, JA, LWORK, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 COMPLEX A( * ), B( * ), WORK( * ) .SH PURPOSE PCPBSV solves a system of linear equations where A(1:N, JA:JA+N-1) is an N-by-N complex .br banded symmetric positive definite distributed .br matrix with bandwidth BW. .br Cholesky factorization is used to factor a reordering of .br the matrix into L L'. .br See PCPBTRF and PCPBTRS for details. .br scalapack-doc-1.5/man/manl/pcpbtrf.l0100644000056400000620000000231506335610621017057 0ustar pfrauenfstaff.TH PCPBTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCPBTRF - compute a Cholesky factorization of an N-by-N complex banded symmetric positive definite distributed matrix with bandwidth BW .SH SYNOPSIS .TP 20 SUBROUTINE PCPBTRF( UPLO, N, BW, A, JA, DESCA, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER BW, INFO, JA, LAF, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ), AF( * ), WORK( * ) .SH PURPOSE PCPBTRF computes a Cholesky factorization of an N-by-N complex banded symmetric positive definite distributed matrix with bandwidth BW: A(1:N, JA:JA+N-1). Reordering is used to increase parallelism in the factorization. This reordering results in factors that are DIFFERENT from those produced by equivalent sequential codes. These factors cannot be used directly by users; however, they can be used in .br subsequent calls to PCPBTRS to solve linear systems. .br The factorization has the form .br P A(1:N, JA:JA+N-1) P^T = U' U , if UPLO = 'U', or P A(1:N, JA:JA+N-1) P^T = L L', if UPLO = 'L' .br where U is a banded upper triangular matrix and L is banded lower triangular, and P is a permutation matrix. .br scalapack-doc-1.5/man/manl/pcpbtrs.l0100644000056400000620000000167506335610621017104 0ustar pfrauenfstaff.TH PCPBTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCPBTRS - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 20 SUBROUTINE PCPBTRS( UPLO, N, BW, NRHS, A, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER BW, IB, INFO, JA, LAF, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX A( * ), AF( * ), B( * ), WORK( * ) .SH PURPOSE PCPBTRS solves a system of linear equations where A(1:N, JA:JA+N-1) is the matrix used to produce the factors stored in A(1:N,JA:JA+N-1) and AF by PCPBTRF. .br A(1:N, JA:JA+N-1) is an N-by-N complex .br banded symmetric positive definite distributed .br matrix with bandwidth BW. .br Depending on the value of UPLO, A stores either U or L in the equn A(1:N, JA:JA+N-1) = U'*U or L*L' as computed by PCPBTRF. .br Routine PCPBTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pcpbtrsv.l0100644000056400000620000000216406335610622017265 0ustar pfrauenfstaff.TH PCPBTRSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCPBTRSV - solve a banded triangular system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 21 SUBROUTINE PCPBTRSV( UPLO, TRANS, N, BW, NRHS, A, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 21 .ti +4 CHARACTER TRANS, UPLO .TP 21 .ti +4 INTEGER BW, IB, INFO, JA, LAF, LWORK, N, NRHS .TP 21 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 21 .ti +4 COMPLEX A( * ), AF( * ), B( * ), WORK( * ) .SH PURPOSE PCPBTRSV solves a banded triangular system of linear equations or .br A(1:N, JA:JA+N-1)^H * X = B(IB:IB+N-1, 1:NRHS) where A(1:N, JA:JA+N-1) is a banded .br triangular matrix factor produced by the .br Cholesky factorization code PCPBTRF .br and is stored in A(1:N,JA:JA+N-1) and AF. .br The matrix stored in A(1:N, JA:JA+N-1) is either .br upper or lower triangular according to UPLO, .br and the choice of solving A(1:N, JA:JA+N-1) or A(1:N, JA:JA+N-1)^H is dictated by the user by the parameter TRANS. .br Routine PCPBTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pcpocon.l0100644000056400000620000001500406335610622017060 0ustar pfrauenfstaff.TH PCPOCON l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCPOCON - estimate the reciprocal of the condition number (in the 1-norm) of a complex Hermitian positive definite distributed matrix using the Cholesky factorization A = U**H*U or A = L*L**H computed by PCPOTRF .SH SYNOPSIS .TP 20 SUBROUTINE PCPOCON( UPLO, N, A, IA, JA, DESCA, ANORM, RCOND, WORK, LWORK, RWORK, LRWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, LRWORK, LWORK, N .TP 20 .ti +4 REAL ANORM, RCOND .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL RWORK( * ) .TP 20 .ti +4 COMPLEX A( * ), WORK( * ) .SH PURPOSE PCPOCON estimates the reciprocal of the condition number (in the 1-norm) of a complex Hermitian positive definite distributed matrix using the Cholesky factorization A = U**H*U or A = L*L**H computed by PCPOTRF. An estimate is obtained for norm(inv(A(IA:IA+N-1,JA:JA+N-1))), and the reciprocal of the condition number is computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the factor stored in A(IA:IA+N-1,JA:JA+N-1) is upper or lower triangular. .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER .br The order of the distributed matrix A(IA:IA+N-1,JA:JA+N-1). N >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension ( LLD_A, LOCc(JA+N-1) ). On entry, this array contains the local pieces of the factors L or U from the Cholesky factorization A(IA:IA+N-1,JA:JA+N-1) = U'*U or L*L', as computed by PCPOTRF. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 ANORM (global input) REAL The 1-norm (or infinity-norm) of the hermitian distributed matrix A(IA:IA+N-1,JA:JA+N-1). .TP 8 RCOND (global output) REAL The reciprocal of the condition number of the distributed matrix A(IA:IA+N-1,JA:JA+N-1), computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * .br norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 2*LOCr(N+MOD(IA-1,MB_A)) + MAX( 2, MAX(NB_A*MAX(1,CEIL(P-1,Q)),LOCc(N+MOD(JA-1,NB_A)) + NB_A*MAX(1,CEIL(Q-1,P))) ). If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 RWORK (local workspace/local output) REAL array, dimension (LRWORK) On exit, RWORK(1) returns the minimal and optimal LRWORK. .TP 8 LRWORK (local or global input) INTEGER The dimension of the array RWORK. LRWORK is local input and must be at least LRWORK >= 2*LOCc(N+MOD(JA-1,NB_A)). If LRWORK = -1, then LRWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pcpoequ.l0100644000056400000620000001402306335610622017073 0ustar pfrauenfstaff.TH PCPOEQU l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCPOEQU - compute row and column scalings intended to equilibrate a distributed Hermitian positive definite matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) and reduce its condition number (with respect to the two-norm) .SH SYNOPSIS .TP 20 SUBROUTINE PCPOEQU( N, A, IA, JA, DESCA, SR, SC, SCOND, AMAX, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 REAL AMAX, SCOND .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL SC( * ), SR( * ) .TP 20 .ti +4 COMPLEX A( * ) .SH PURPOSE PCPOEQU computes row and column scalings intended to equilibrate a distributed Hermitian positive definite matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) and reduce its condition number (with respect to the two-norm). SR and SC contain the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled distri- buted matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of SR and SC puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings. .br The scaling factor are stored along process rows in SR and along process columns in SC. The duplication of information simplifies greatly the application of the factors. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on i.e the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of local dimension ( LLD_A, LOCc(JA+N-1) ), the N-by-N Hermitian positive definite distributed matrix sub( A ) whose scaling factors are to be computed. Only the diagonal elements of sub( A ) are referenced. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 SR (local output) REAL array, dimension LOCr(M_A) If INFO = 0, SR(IA:IA+N-1) contains the row scale factors for sub( A ). SR is aligned with the distributed matrix A, and replicated across every process column. SR is tied to the distributed matrix A. .TP 8 SC (local output) REAL array, dimension LOCc(N_A) If INFO = 0, SC(JA:JA+N-1) contains the column scale factors .br for A(IA:IA+M-1,JA:JA+N-1). SC is aligned with the distribu- ted matrix A, and replicated down every process row. SC is tied to the distributed matrix A. .TP 8 SCOND (global output) REAL If INFO = 0, SCOND contains the ratio of the smallest SR(i) (or SC(j)) to the largest SR(i) (or SC(j)), with IA <= i <= IA+N-1 and JA <= j <= JA+N-1. If SCOND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by SR (or SC). .TP 8 AMAX (global output) REAL Absolute value of largest matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, the K-th diagonal entry of sub( A ) is nonpositive. scalapack-doc-1.5/man/manl/pcporfs.l0100644000056400000620000002355406335610622017104 0ustar pfrauenfstaff.TH PCPORFS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCPORFS - improve the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite and provides error bounds and backward error estimates for the solutions .SH SYNOPSIS .TP 20 SUBROUTINE PCPORFS( UPLO, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, B, IB, JB, DESCB, X, IX, JX, DESCX, FERR, BERR, WORK, LWORK, RWORK, LRWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IAF, IB, INFO, IX, JA, JAF, JB, JX, LRWORK, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * ) .TP 20 .ti +4 COMPLEX A( * ), AF( * ), B( * ), BERR( * ), FERR( * ), WORK( * ), X( * ) .TP 20 .ti +4 REAL RWORK( * ) .SH PURPOSE PCPORFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite and provides error bounds and backward error estimates for the solutions. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br In the following comments, sub( A ), sub( X ) and sub( B ) denote respectively A(IA:IA+N-1,JA:JA+N-1), X(IX:IX+N-1,JX:JX+NRHS-1) and B(IB:IB+N-1,JB:JB+NRHS-1). .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix sub( A ) is stored. = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The order of the matrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the matrices sub( B ) and sub( X ). NRHS >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1) ). This array contains the local pieces of the N-by-N Hermitian distributed matrix sub( A ) to be factored. If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the distribu- ted matrix, and its strictly upper triangular part is not referenced. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 AF (local input) COMPLEX pointer into the local memory to an array of local dimension (LLD_AF,LOCc(JA+N-1)). On entry, this array contains the factors L or U from the Cholesky factorization sub( A ) = L*L**H or U**H*U, as computed by PCPOTRF. .TP 8 IAF (global input) INTEGER The row index in the global array AF indicating the first row of sub( AF ). .TP 8 JAF (global input) INTEGER The column index in the global array AF indicating the first column of sub( AF ). .TP 8 DESCAF (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix AF. .TP 8 B (local input) COMPLEX pointer into the local memory to an array of local dimension (LLD_B, LOCc(JB+NRHS-1) ). On entry, this array contains the the local pieces of the right hand sides sub( B ). .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 X (local input) COMPLEX pointer into the local memory to an array of local dimension (LLD_X, LOCc(JX+NRHS-1) ). On entry, this array contains the the local pieces of the solution vectors sub( X ). On exit, it contains the improved solution vectors. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 FERR (local output) REAL array of local dimension LOCc(JB+NRHS-1). The estimated forward error bound for each solution vector of sub( X ). If XTRUE is the true solution corresponding to sub( X ), FERR is an estimated upper bound for the magnitude of the largest element in (sub( X ) - XTRUE) divided by the magnitude of the largest element in sub( X ). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. This array is tied to the distributed matrix X. .TP 8 BERR (local output) REAL array of local dimension LOCc(JB+NRHS-1). The componentwise relative backward error of each solution vector (i.e., the smallest re- lative change in any entry of sub( A ) or sub( B ) that makes sub( X ) an exact solution). This array is tied to the distributed matrix X. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 2*LOCr( N + MOD( IA-1, MB_A ) ) If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 RWORK (local workspace/local output) REAL array, dimension (LRWORK) On exit, RWORK(1) returns the minimal and optimal LRWORK. .TP 8 LRWORK (local or global input) INTEGER The dimension of the array RWORK. LRWORK is local input and must be at least LRWORK >= LOCr( N + MOD( IB-1, MB_B ) ). If LRWORK = -1, then LRWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH PARAMETERS ITMAX is the maximum number of steps of iterative refinement. Notes ===== This routine temporarily returns when N <= 1. The distributed submatrices op( A ) and op( AF ) (respectively sub( X ) and sub( B ) ) should be distributed the same way on the same processes. These conditions ensure that sub( A ) and sub( AF ) (resp. sub( X ) and sub( B ) ) are "perfectly" aligned. Moreover, this routine requires the distributed submatrices sub( A ), sub( AF ), sub( X ), and sub( B ) to be aligned on a block boundary, i.e., if f(x,y) = MOD( x-1, y ): f( IA, DESCA( MB_ ) ) = f( JA, DESCA( NB_ ) ) = 0, f( IAF, DESCAF( MB_ ) ) = f( JAF, DESCAF( NB_ ) ) = 0, f( IB, DESCB( MB_ ) ) = f( JB, DESCB( NB_ ) ) = 0, and f( IX, DESCX( MB_ ) ) = f( JX, DESCX( NB_ ) ) = 0. scalapack-doc-1.5/man/manl/pcposv.l0100644000056400000620000001460106335610622016733 0ustar pfrauenfstaff.TH PCPOSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCPOSV - compute the solution to a complex system of linear equations sub( A ) * X = sub( B ), .SH SYNOPSIS .TP 19 SUBROUTINE PCPOSV( UPLO, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, INFO ) .TP 19 .ti +4 CHARACTER UPLO .TP 19 .ti +4 INTEGER IA, IB, INFO, JA, JB, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 COMPLEX A( * ), B( * ) .SH PURPOSE PCPOSV computes the solution to a complex system of linear equations where sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1) and is an N-by-N hermitian distributed positive definite matrix and X and sub( B ) denoting B(IB:IB+N-1,JB:JB+NRHS-1) are N-by-NRHS distributed matrices. .br The Cholesky decomposition is used to factor sub( A ) as .br sub( A ) = U**H * U, if UPLO = 'U', or sub( A ) = L * L**H, if UPLO = 'L', .br where U is an upper triangular matrix and L is a lower triangular matrix. The factored form of sub( A ) is then used to solve the system of equations. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block decomposition ( MB_A = NB_A ). .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER = 'U': Upper triangle of sub( A ) is stored; .br = 'L': Lower triangle of sub( A ) is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed submatrix sub( B ). NRHS >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N symmetric distributed matrix sub( A ) to be factored. If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the distribu- ted matrix, and its strictly upper triangular part is not referenced. On exit, if INFO = 0, this array contains the local pieces of the factor U or L from the Cholesky factori- zation sub( A ) = U**H*U or L*L**H. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_B,LOC(JB+NRHS-1)). On entry, the local pieces of the right hand sides distribu- ted matrix sub( B ). On exit, if INFO = 0, sub( B ) is over- written with the solution distributed matrix X. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, the leading minor of order K, .br A(IA:IA+K-1,JA:JA+K-1) is not positive definite, and the factorization could not be completed, and the solution has not been computed. scalapack-doc-1.5/man/manl/pcposvx.l0100644000056400000620000003312106335610622017121 0ustar pfrauenfstaff.TH PCPOSVX l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCPOSVX - use the Cholesky factorization A = U**H*U or A = L*L**H to compute the solution to a complex system of linear equations A(IA:IA+N-1,JA:JA+N-1) * X = B(IB:IB+N-1,JB:JB+NRHS-1), .SH SYNOPSIS .TP 20 SUBROUTINE PCPOSVX( FACT, UPLO, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, EQUED, SR, SC, B, IB, JB, DESCB, X, IX, JX, DESCX, RCOND, FERR, BERR, WORK, LWORK, RWORK, LRWORK, INFO ) .TP 20 .ti +4 CHARACTER EQUED, FACT, UPLO .TP 20 .ti +4 INTEGER IA, IAF, IB, INFO, IX, JA, JAF, JB, JX, LRWORK, LWORK, N, NRHS .TP 20 .ti +4 REAL RCOND .TP 20 .ti +4 INTEGER DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * ) .TP 20 .ti +4 REAL BERR( * ), FERR( * ), SC( * ), SR( * ), RWORK( * ) .TP 20 .ti +4 COMPLEX A( * ), AF( * ), B( * ), WORK( * ), X( * ) .SH PURPOSE PCPOSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to compute the solution to a complex system of linear equations where A(IA:IA+N-1,JA:JA+N-1) is an N-by-N matrix and X and B(IB:IB+N-1,JB:JB+NRHS-1) are N-by-NRHS matrices. .br Error bounds on the solution and a condition estimate are also provided. In the following comments Y denotes Y(IY:IY+M-1,JY:JY+K-1) a M-by-K matrix where Y can be A, AF, B and X. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH DESCRIPTION The following steps are performed: .br 1. If FACT = 'E', real scaling factors are computed to equilibrate the system: .br diag(SR) * A * diag(SC) * inv(diag(SC)) * X = diag(SR) * 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(SR)*A*diag(SC) and B by diag(SR)*B. 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as A = U**T* U, if UPLO = 'U', or .br A = L * L**T, if UPLO = 'L', .br where U is an upper triangular matrix and L is a lower triangular matrix. .br 3. 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, steps 4-6 are skipped. .br 4. The system of equations is solved for X using the factored form of A. .br 5. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. .br 6. If equilibration was used, the matrix X is premultiplied by diag(SR) so that it solves the original system before .br equilibration. .br .SH ARGUMENTS .TP 8 FACT (global input) CHARACTER 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 contains the factored form of A. If EQUED = 'Y', the matrix A has been equilibrated with scaling factors given by S. A and AF will not be modified. = 'N': The matrix A will be copied to AF and factored. .br = 'E': The matrix A will be equilibrated if necessary, then copied to AF and factored. .TP 8 UPLO (global input) CHARACTER = 'U': Upper triangle of A is stored; .br = 'L': Lower triangle of A is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix A(IA:IA+N-1,JA:JA+N-1). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed submatrices B and X. NRHS >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of local dimension ( LLD_A, LOCc(JA+N-1) ). On entry, the Hermitian matrix A, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(SR)*A*diag(SC). 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. A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(SR)*A*diag(SC). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 AF (local input or local output) COMPLEX pointer into the local memory to an array of local dimension ( LLD_AF, LOCc(JA+N-1)). If FACT = 'F', then AF is an input argument and on entry contains the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, in the same storage format as A. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix diag(SR)*A*diag(SC). If FACT = 'N', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the original matrix A. If FACT = 'E', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix). .TP 8 IAF (global input) INTEGER The row index in the global array AF indicating the first row of sub( AF ). .TP 8 JAF (global input) INTEGER The column index in the global array AF indicating the first column of sub( AF ). .TP 8 DESCAF (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix AF. .TP 8 EQUED (global input/global output) CHARACTER Specifies the form of equilibration that was done. = 'N': No equilibration (always true if FACT = 'N'). .br = 'Y': Equilibration was done, i.e., A has been replaced by diag(SR) * A * diag(SC). EQUED is an input variable if FACT = 'F'; otherwise, it is an output variable. .TP 8 SR (local input/local output) COMPLEX array, dimension (LLD_A) The scale factors for A distributed across process rows; not accessed if EQUED = 'N'. SR is an input variable if FACT = 'F'; otherwise, SR is an output variable. If FACT = 'F' and EQUED = 'Y', each element of SR must be positive. .TP 8 SC (local input/local output) COMPLEX array, dimension (LOC(N_A)) The scale factors for A distributed across process columns; not accessed if EQUED = 'N'. SC is an input variable if FACT = 'F'; otherwise, SC is an output variable. If FACT = 'F' and EQUED = 'Y', each element of SC must be positive. .TP 8 B (local input/local output) COMPLEX pointer into the local memory to an array of local dimension ( LLD_B, LOCc(JB+NRHS-1) ). On entry, the N-by-NRHS right-hand side matrix B. On exit, if EQUED = 'N', B is not modified; if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B; if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is overwritten by diag(C)*B. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 X (local input/local output) COMPLEX pointer into the local memory to an array of local dimension ( LLD_X, LOCc(JX+NRHS-1) ). If INFO = 0, the N-by-NRHS 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(SC))*X if TRANS = 'N' and EQUED = 'C' or 'B', or inv(diag(SR))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 RCOND (global output) REAL 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, and the solution and error bounds are not computed. .TP 8 FERR (local output) REAL array, dimension (LOC(N_B)) The estimated forward error bounds for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution, FERR(j) bounds the magnitude of the largest entry in (X(j) - XTRUE) divided by the magnitude of the largest entry in X(j). The quality of the error bound depends on the quality of the estimate of norm(inv(A)) computed in the code; if the estimate of norm(inv(A)) is accurate, the error bound is guaranteed. .TP 8 BERR (local output) REAL array, dimension (LOC(N_B)) The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any entry of A or B that makes X(j) an exact solution). .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK = MAX( PCPOCON( LWORK ), PCPORFS( LWORK ) ) + LOCr( N_A ). LWORK = 3*DESCA( LLD_ ) If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 RWORK (local workspace/local output) REAL array, dimension (LRWORK) On exit, RWORK(1) returns the minimal and optimal LRWORK. .TP 8 LRWORK (local or global input) INTEGER The dimension of the array RWORK. LRWORK is local input and must be at least LRWORK = 2*LOCc(N_A). If LRWORK = -1, then LRWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 0: if INFO = i, and i is .br <= N: if INFO = i, the leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution and error bounds could not be computed. = N+1: RCOND is less than machine precision. The factorization has been completed, but the matrix is singular to working precision, and the solution and error bounds have not been computed. scalapack-doc-1.5/man/manl/pcpotf2.l0100644000056400000620000001260206335610622016775 0ustar pfrauenfstaff.TH PCPOTF2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCPOTF2 - compute the Cholesky factorization of a complex hermitian positive definite distributed matrix sub( A )=A(IA:IA+N-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PCPOTF2( UPLO, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ) .SH PURPOSE PCPOTF2 computes the Cholesky factorization of a complex hermitian positive definite distributed matrix sub( A )=A(IA:IA+N-1,JA:JA+N-1). The factorization has the form .br sub( A ) = U' * U , if UPLO = 'U', or .br sub( A ) = L * L', if UPLO = 'L', .br where U is an upper triangular matrix and L is lower triangular. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires N <= NB_A-MOD(JA-1, NB_A) and square block decomposition ( MB_A = NB_A ). .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER = 'U': Upper triangle of sub( A ) is stored; .br = 'L': Lower triangle of sub( A ) is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N symmetric distributed matrix sub( A ) to be factored. If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the distribu- ted matrix, and its strictly upper triangular part is not referenced. On exit, if UPLO = 'U', the upper triangular part of the distributed matrix contains the Cholesky factor U, if UPLO = 'L', the lower triangular part of the distribu- ted matrix contains the Cholesky factor L. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, the leading minor of order K, .br A(IA:IA+K-1,JA:JA+K-1) is not positive definite, and the factorization could not be completed. scalapack-doc-1.5/man/manl/pcpotrf.l0100644000056400000620000001260406335610622017077 0ustar pfrauenfstaff.TH PCPOTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCPOTRF - compute the Cholesky factorization of an N-by-N complex hermitian positive definite distributed matrix sub( A ) denoting A(IA:IA+N-1, JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PCPOTRF( UPLO, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ) .SH PURPOSE PCPOTRF computes the Cholesky factorization of an N-by-N complex hermitian positive definite distributed matrix sub( A ) denoting A(IA:IA+N-1, JA:JA+N-1). The factorization has the form .br sub( A ) = U' * U , if UPLO = 'U', or .br sub( A ) = L * L', if UPLO = 'L', .br where U is an upper triangular matrix and L is lower triangular. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block decomposition ( MB_A = NB_A ). .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER = 'U': Upper triangle of sub( A ) is stored; .br = 'L': Lower triangle of sub( A ) is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N Hermitian distributed matrix sub( A ) to be factored. If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the distribu- ted matrix, and its strictly upper triangular part is not referenced. On exit, if UPLO = 'U', the upper triangular part of the distributed matrix contains the Cholesky factor U, if UPLO = 'L', the lower triangular part of the distribu- ted matrix contains the Cholesky factor L. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, the leading minor of order K, .br A(IA:IA+K-1,JA:JA+K-1) is not positive definite, and the factorization could not be completed. scalapack-doc-1.5/man/manl/pcpotri.l0100644000056400000620000001145506335610622017105 0ustar pfrauenfstaff.TH PCPOTRI l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCPOTRI - compute the inverse of a complex Hermitian positive definite distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) using the Cholesky factorization sub( A ) = U**H*U or L*L**H computed by PCPOTRF .SH SYNOPSIS .TP 20 SUBROUTINE PCPOTRI( UPLO, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ) .SH PURPOSE PCPOTRI computes the inverse of a complex Hermitian positive definite distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) using the Cholesky factorization sub( A ) = U**H*U or L*L**H computed by PCPOTRF. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 = 'U': Upper triangle of sub( A ) is stored; .br = 'L': Lower triangle of sub( A ) is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the triangular factor U or L from the Cholesky factorization of the distributed matrix sub( A ) = U**H*U or L*L**H, as computed by PCPOTRF. On exit, the local pieces of the upper or lower triangle of the (Hermitian) inverse of sub( A ), overwriting the input factor U or L. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = i, the (i,i) element of the factor U or L is zero, and the inverse could not be computed. scalapack-doc-1.5/man/manl/pcpotrs.l0100644000056400000620000001267306335610622017122 0ustar pfrauenfstaff.TH PCPOTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCPOTRS - solve a system of linear equations sub( A ) * X = sub( B ) A(IA:IA+N-1,JA:JA+N-1)*X = B(IB:IB+N-1,JB:JB+NRHS-1) .SH SYNOPSIS .TP 20 SUBROUTINE PCPOTRS( UPLO, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IB, INFO, JA, JB, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX A( * ), B( * ) .SH PURPOSE PCPOTRS solves a system of linear equations where sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1) and is a N-by-N hermitian positive definite distributed matrix using the Cholesky factorization sub( A ) = U**H*U or L*L**H computed by PCPOTRF. sub( B ) denotes the distributed matrix B(IB:IB+N-1,JB:JB+NRHS-1). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block decomposition ( MB_A = NB_A ). .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER = 'U': Upper triangle of sub( A ) is stored; .br = 'L': Lower triangle of sub( A ) is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed submatrix sub( B ). NRHS >= 0. .TP 8 A (local input) COMPLEX pointer into local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the factors L or U from the Cholesky facto- rization sub( A ) = L*L**H or U**H*U, as computed by PCPOTRF. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input/local output) COMPLEX pointer into the local memory to an array of local dimension (LLD_B,LOCc(JB+NRHS-1)). On entry, this array contains the the local pieces of the right hand sides sub( B ). On exit, this array contains the local pieces of the solution distributed matrix X. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pcptsv.l0100644000056400000620000000142306335610622016736 0ustar pfrauenfstaff.TH PCPTSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCPTSV - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 19 SUBROUTINE PCPTSV( UPLO, N, NRHS, D, E, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 CHARACTER UPLO .TP 19 .ti +4 INTEGER IB, INFO, JA, LWORK, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 COMPLEX B( * ), E( * ), WORK( * ) .TP 19 .ti +4 REAL D( * ) .SH PURPOSE PCPTSV solves a system of linear equations where A(1:N, JA:JA+N-1) is an N-by-N complex .br tridiagonal symmetric positive definite distributed .br matrix. .br Cholesky factorization is used to factor a reordering of .br the matrix into L L'. .br See PCPTTRF and PCPTTRS for details. .br scalapack-doc-1.5/man/manl/pcpttrf.l0100644000056400000620000000225106335610622017101 0ustar pfrauenfstaff.TH PCPTTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCPTTRF - compute a Cholesky factorization of an N-by-N complex tridiagonal symmetric positive definite distributed matrix A(1:N, JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PCPTTRF( N, D, E, JA, DESCA, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER INFO, JA, LAF, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX AF( * ), E( * ), WORK( * ) .TP 20 .ti +4 REAL D( * ) .SH PURPOSE PCPTTRF computes a Cholesky factorization of an N-by-N complex tridiagonal symmetric positive definite distributed matrix A(1:N, JA:JA+N-1). Reordering is used to increase parallelism in the factorization. This reordering results in factors that are DIFFERENT from those produced by equivalent sequential codes. These factors cannot be used directly by users; however, they can be used in .br subsequent calls to PCPTTRS to solve linear systems. .br The factorization has the form .br P A(1:N, JA:JA+N-1) P^T = U' D U or .br P A(1:N, JA:JA+N-1) P^T = L D L', .br where U is a tridiagonal upper triangular matrix and L is tridiagonal lower triangular, and P is a permutation matrix. .br scalapack-doc-1.5/man/manl/pcpttrs.l0100644000056400000620000000170506335610622017121 0ustar pfrauenfstaff.TH PCPTTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCPTTRS - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 20 SUBROUTINE PCPTTRS( UPLO, N, NRHS, D, E, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IB, INFO, JA, LAF, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX AF( * ), B( * ), E( * ), WORK( * ) .TP 20 .ti +4 REAL D( * ) .SH PURPOSE PCPTTRS solves a system of linear equations where A(1:N, JA:JA+N-1) is the matrix used to produce the factors stored in A(1:N,JA:JA+N-1) and AF by PCPTTRF. .br A(1:N, JA:JA+N-1) is an N-by-N complex .br tridiagonal symmetric positive definite distributed .br matrix. .br Depending on the value of UPLO, A stores either U or L in the equn A(1:N, JA:JA+N-1) = U'D *U or L*D L' as computed by PCPTTRF. Routine PCPTTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pcpttrsv.l0100644000056400000620000000223006335610623017302 0ustar pfrauenfstaff.TH PCPTTRSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCPTTRSV - solve a tridiagonal triangular system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 21 SUBROUTINE PCPTTRSV( UPLO, TRANS, N, NRHS, D, E, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 21 .ti +4 CHARACTER TRANS, UPLO .TP 21 .ti +4 INTEGER IB, INFO, JA, LAF, LWORK, N, NRHS .TP 21 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 21 .ti +4 COMPLEX AF( * ), B( * ), E( * ), WORK( * ) .TP 21 .ti +4 REAL D( * ) .SH PURPOSE PCPTTRSV solves a tridiagonal triangular system of linear equations or .br A(1:N, JA:JA+N-1)^H * X = B(IB:IB+N-1, 1:NRHS) where A(1:N, JA:JA+N-1) is a tridiagonal .br triangular matrix factor produced by the .br Cholesky factorization code PCPTTRF .br and is stored in A(1:N,JA:JA+N-1) and AF. .br The matrix stored in A(1:N, JA:JA+N-1) is either .br upper or lower triangular according to UPLO, .br and the choice of solving A(1:N, JA:JA+N-1) or A(1:N, JA:JA+N-1)^H is dictated by the user by the parameter TRANS. .br Routine PCPTTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pcsrscl.l0100644000056400000620000001110706335610623017071 0ustar pfrauenfstaff.TH PCSRSCL l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCSRSCL - multiplie an N-element complex distributed vector sub( X ) by the real scalar 1/a .SH SYNOPSIS .TP 20 SUBROUTINE PCSRSCL( N, SA, SX, IX, JX, DESCX, INCX ) .TP 20 .ti +4 INTEGER IX, INCX, JX, N .TP 20 .ti +4 REAL SA .TP 20 .ti +4 INTEGER DESCX( * ) .TP 20 .ti +4 COMPLEX SX( * ) .SH PURPOSE PCSRSCL multiplies an N-element complex distributed vector sub( X ) by the real scalar 1/a. This is done without overflow or underflow as long as the final sub( X )/a does not overflow or underflow. .br where sub( X ) denotes X(IX:IX+N-1,JX:JX), if INCX = 1, .br X(IX:IX,JX:JX+N-1), if INCX = M_X. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector descA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DT_A (global) descA[ DT_ ] The descriptor type. In this case, DT_A = 1. .br CTXT_A (global) descA[ CTXT_ ] The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) descA[ M_ ] The number of rows in the global array A. .br N_A (global) descA[ N_ ] The number of columns in the global array A. .br MB_A (global) descA[ MB_ ] The blocking factor used to distribu- te the rows of the array. .br NB_A (global) descA[ NB_ ] The blocking factor used to distribu- te the columns of the array. RSRC_A (global) descA[ RSRC_ ] The process row over which the first row of the array A is distributed. CSRC_A (global) descA[ CSRC_ ] The process column over which the first column of the array A is distributed. .br LLD_A (local) descA[ LLD_ ] The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be seen as particular matrices, a distributed vector is considered to be a distributed matrix. .br .SH ARGUMENTS .TP 8 N (global input) pointer to INTEGER The number of components of the distributed vector sub( X ). N >= 0. .TP 8 SA (global input) REAL The scalar a which is used to divide each component of sub( X ). SA must be >= 0, or the subroutine will divide by zero. .TP 8 SX (local input/local output) COMPLEX array containing the local pieces of a distributed matrix of dimension of at least ( (JX-1)*M_X + IX + ( N - 1 )*abs( INCX ) ) This array contains the entries of the distributed vector sub( X ). .TP 8 IX (global input) pointer to INTEGER The global row index of the submatrix of the distributed matrix X to operate on. .TP 8 JX (global input) pointer to INTEGER The global column index of the submatrix of the distributed matrix X to operate on. .TP 8 DESCX (global and local input) INTEGER array of dimension 8. The array descriptor of the distributed matrix X. .TP 8 INCX (global input) pointer to INTEGER The global increment for the elements of X. Only two values of INCX are supported in this version, namely 1 and M_X. scalapack-doc-1.5/man/manl/pcstein.l0100644000056400000620000002563606335610623017101 0ustar pfrauenfstaff.TH PCSTEIN l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCSTEIN - compute the eigenvectors of a symmetric tridiagonal matrix in parallel, using inverse iteration .SH SYNOPSIS .TP 20 SUBROUTINE PCSTEIN( N, D, E, M, W, IBLOCK, ISPLIT, ORFAC, Z, IZ, JZ, DESCZ, WORK, LWORK, IWORK, LIWORK, IFAIL, ICLUSTR, GAP, INFO ) .TP 20 .ti +4 INTEGER INFO, IZ, JZ, LIWORK, LWORK, M, N .TP 20 .ti +4 REAL ORFAC .TP 20 .ti +4 INTEGER DESCZ( * ), IBLOCK( * ), ICLUSTR( * ), IFAIL( * ), ISPLIT( * ), IWORK( * ) .TP 20 .ti +4 REAL D( * ), E( * ), GAP( * ), W( * ), WORK( * ) .TP 20 .ti +4 COMPLEX Z( * ) .SH PURPOSE PCSTEIN computes the eigenvectors of a symmetric tridiagonal matrix in parallel, using inverse iteration. The eigenvectors found correspond to user specified eigenvalues. PCSTEIN does not orthogonalize vectors that are on different processes. The extent of orthogonalization is controlled by the input parameter LWORK. Eigenvectors that are to be orthogonalized are computed by the same process. PCSTEIN decides on the allocation of work among the processes and then calls SSTEIN2 (modified LAPACK routine) on each individual process. If insufficient workspace is allocated, the expected orthogonalization may not be done. .br Note : If the eigenvectors obtained are not orthogonal, increase LWORK and run the code again. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS P = NPROW * NPCOL is the total number of processes .TP 8 N (global input) INTEGER The order of the tridiagonal matrix T. N >= 0. .TP 8 D (global input) REAL array, dimension (N) The n diagonal elements of the tridiagonal matrix T. .TP 8 E (global input) REAL array, dimension (N-1) The (n-1) off-diagonal elements of the tridiagonal matrix T. .TP 8 M (global input) INTEGER The total number of eigenvectors to be found. 0 <= M <= N. .TP 8 W (global input/global output) REAL array, dim (M) On input, the first M elements of W contain all the eigenvalues for which eigenvectors are to be computed. The eigenvalues should be grouped by split-off block and ordered from smallest to largest within the block (The output array W from PSSTEBZ with ORDER='b' is expected here). This array should be replicated on all processes. On output, the first M elements contain the input eigenvalues in ascending order. Note : To obtain orthogonal vectors, it is best if eigenvalues are computed to highest accuracy ( this can be done by setting ABSTOL to the underflow threshold = SLAMCH('U') --- ABSTOL is an input parameter to PSSTEBZ ) .TP 8 IBLOCK (global input) INTEGER array, dimension (N) The submatrix indices associated with the corresponding eigenvalues in W -- 1 for eigenvalues belonging to the first submatrix from the top, 2 for those belonging to the second submatrix, etc. (The output array IBLOCK from PSSTEBZ is expected here). .TP 8 ISPLIT (global input) INTEGER array, dimension (N) The splitting points, at which T breaks up into submatrices. The first submatrix consists of rows/columns 1 to ISPLIT(1), the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N (The output array ISPLIT from PSSTEBZ is expected here.) .TP 8 ORFAC (global input) REAL ORFAC specifies which eigenvectors should be orthogonalized. Eigenvectors that correspond to eigenvalues which are within ORFAC*||T|| of each other are to be orthogonalized. However, if the workspace is insufficient (see LWORK), this tolerance may be decreased until all eigenvectors to be orthogonalized can be stored in one process. No orthogonalization will be done if ORFAC equals zero. A default value of 10^-3 is used if ORFAC is negative. ORFAC should be identical on all processes. .TP 8 Z (local output) COMPLEX array, dimension (DESCZ(DLEN_), N/npcol + NB) Z contains the computed eigenvectors associated with the specified eigenvalues. Any vector which fails to converge is set to its current iterate after MAXITS iterations ( See SSTEIN2 ). On output, Z is distributed across the P processes in block cyclic format. .TP 8 IZ (global input) INTEGER Z's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JZ (global input) INTEGER Z's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCZ (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix Z. .TP 8 WORK (local workspace/global output) REAL array, dimension ( LWORK ) On output, WORK(1) gives a lower bound on the workspace ( LWORK ) that guarantees the user desired orthogonalization (see ORFAC). Note that this may overestimate the minimum workspace needed. .TP 8 LWORK (local input) integer LWORK controls the extent of orthogonalization which can be done. The number of eigenvectors for which storage is allocated on each process is NVEC = floor(( LWORK- max(5*N,NP00*MQ00) )/N). Eigenvectors corresponding to eigenvalue clusters of size NVEC - ceil(M/P) + 1 are guaranteed to be orthogonal ( the orthogonality is similar to that obtained from CSTEIN2). Note : LWORK must be no smaller than: max(5*N,NP00*MQ00) + ceil(M/P)*N, and should have the same input value on all processes. It is the minimum value of LWORK input on different processes that is significant. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/global output) INTEGER array, dimension ( 3*N+P+1 ) On return, IWORK(1) contains the amount of integer workspace required. On return, the IWORK(2) through IWORK(P+2) indicate the eigenvectors computed by each process. Process I computes eigenvectors indexed IWORK(I+2)+1 thru' IWORK(I+3). .TP 8 LIWORK (local input) INTEGER Size of array IWORK. Must be >= 3*N + P + 1 If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IFAIL (global output) integer array, dimension (M) On normal exit, all elements of IFAIL are zero. If one or more eigenvectors fail to converge after MAXITS iterations (as in CSTEIN), then INFO > 0 is returned. If mod(INFO,M+1)>0, then for I=1 to mod(INFO,M+1), the eigenvector corresponding to the eigenvalue W(IFAIL(I)) failed to converge ( W refers to the array of eigenvalues on output ). ICLUSTR (global output) integer array, dimension (2*P) This output array contains indices of eigenvectors corresponding to a cluster of eigenvalues that could not be orthogonalized due to insufficient workspace (see LWORK, ORFAC and INFO). Eigenvectors corresponding to clusters of eigenvalues indexed ICLUSTR(2*I-1) to ICLUSTR(2*I), I = 1 to INFO/(M+1), could not be orthogonalized due to lack of workspace. Hence the eigenvectors corresponding to these clusters may not be orthogonal. ICLUSTR is a zero terminated array --- ( ICLUSTR(2*K).NE.0 .AND. ICLUSTR(2*K+1).EQ.0 ) if and only if K is the number of clusters. .TP 8 GAP (global output) REAL array, dimension (P) This output array contains the gap between eigenvalues whose eigenvectors could not be orthogonalized. The INFO/M output values in this array correspond to the INFO/(M+1) clusters indicated by the array ICLUSTR. As a result, the dot product between eigenvectors corresponding to the I^th cluster may be as high as ( O(n)*macheps ) / GAP(I). .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. < 0 : if INFO = -I, the I-th argument had an illegal value .br > 0 : if mod(INFO,M+1) = I, then I eigenvectors failed to converge in MAXITS iterations. Their indices are stored in the array IFAIL. if INFO/(M+1) = I, then eigenvectors corresponding to I clusters of eigenvalues could not be orthogonalized due to insufficient workspace. The indices of the clusters are stored in the array ICLUSTR. scalapack-doc-1.5/man/manl/pctrcon.l0100644000056400000620000001603206335610623017072 0ustar pfrauenfstaff.TH PCTRCON l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCTRCON - estimate the reciprocal of the condition number of a triangular distributed matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm .SH SYNOPSIS .TP 20 SUBROUTINE PCTRCON( NORM, UPLO, DIAG, N, A, IA, JA, DESCA, RCOND, WORK, LWORK, RWORK, LRWORK, INFO ) .TP 20 .ti +4 CHARACTER DIAG, NORM, UPLO .TP 20 .ti +4 INTEGER IA, JA, INFO, LRWORK, LWORK, N .TP 20 .ti +4 REAL RCOND .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL RWORK( * ) .TP 20 .ti +4 COMPLEX A( * ), WORK( * ) .SH PURPOSE PCTRCON estimates the reciprocal of the condition number of a triangular distributed matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm. The norm of A(IA:IA+N-1,JA:JA+N-1) is computed and an estimate is obtained for norm(inv(A(IA:IA+N-1,JA:JA+N-1))), then the reciprocal of the condition number is computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies whether the 1-norm condition number or the infinity-norm condition number is required: .br = '1' or 'O': 1-norm; .br = 'I': Infinity-norm. .TP 8 UPLO (global input) CHARACTER .br = 'U': A(IA:IA+N-1,JA:JA+N-1) is upper triangular; .br = 'L': A(IA:IA+N-1,JA:JA+N-1) is lower triangular. .TP 8 DIAG (global input) CHARACTER .br = 'N': A(IA:IA+N-1,JA:JA+N-1) is non-unit triangular; .br = 'U': A(IA:IA+N-1,JA:JA+N-1) is unit triangular. .TP 8 N (global input) INTEGER .br The order of the distributed matrix A(IA:IA+N-1,JA:JA+N-1). N >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension ( LLD_A, LOCc(JA+N-1) ). This array contains the local pieces of the triangular distributed matrix A(IA:IA+N-1,JA:JA+N-1). If UPLO = 'U', the leading N-by-N upper triangular part of this distributed matrix con- tains the upper triangular matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of this ditributed matrix contains the lower triangular matrix, and the strictly upper triangular part is not referenced. If DIAG = 'U', the diagonal elements of A(IA:IA+N-1,JA:JA+N-1) are also not referenced and are assumed to be 1. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 RCOND (global output) REAL The reciprocal of the condition number of the distributed matrix A(IA:IA+N-1,JA:JA+N-1), computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * .br norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 2*LOCr(N+MOD(IA-1,MB_A)) + MAX( 2, MAX(NB_A*CEIL(P-1,Q),LOCc(N+MOD(JA-1,NB_A)) + NB_A*CEIL(Q-1,P)) ). If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 RWORK (local workspace/local output) REAL array, dimension (LRWORK) On exit, RWORK(1) returns the minimal and optimal LRWORK. .TP 8 LRWORK (local or global input) INTEGER The dimension of the array RWORK. LRWORK is local input and must be at least LRWORK >= LOCc(N+MOD(JA-1,NB_A)). If LRWORK = -1, then LRWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pctrrfs.l0100644000056400000620000002264506335610623017114 0ustar pfrauenfstaff.TH PCTRRFS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCTRRFS - provide error bounds and backward error estimates for the solution to a system of linear equations with a triangular coefficient matrix .SH SYNOPSIS .TP 20 SUBROUTINE PCTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, X, IX, JX, DESCX, FERR, BERR, WORK, LWORK, RWORK, LRWORK, INFO ) .TP 20 .ti +4 CHARACTER DIAG, TRANS, UPLO .TP 20 .ti +4 INTEGER INFO, IA, IB, IX, JA, JB, JX, LRWORK, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ), DESCX( * ) .TP 20 .ti +4 REAL BERR( * ), FERR( * ), RWORK( * ) .TP 20 .ti +4 COMPLEX A( * ), B( * ), WORK( * ), X( * ) .SH PURPOSE PCTRRFS provides error bounds and backward error estimates for the solution to a system of linear equations with a triangular coefficient matrix. The solution matrix X must be computed by PCTRTRS or some other means before entering this routine. PCTRRFS does not do iterative refinement because doing so cannot improve the backward error. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br In the following comments, sub( A ), sub( X ) and sub( B ) denote respectively A(IA:IA+N-1,JA:JA+N-1), X(IX:IX+N-1,JX:JX+NRHS-1) and B(IB:IB+N-1,JB:JB+NRHS-1). .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 = 'U': sub( A ) is upper triangular; .br = 'L': sub( A ) is lower triangular. .TP 8 TRANS (global input) CHARACTER*1 Specifies the form of the system of equations. = 'N': sub( A ) * sub( X ) = sub( B ) (No transpose) .br = 'T': sub( A )**T * sub( X ) = sub( B ) (Transpose) .br = 'C': sub( A )**H * sub( X ) = sub( B ) (Conjugate transpose) .TP 8 DIAG (global input) CHARACTER*1 = 'N': sub( A ) is non-unit triangular; .br = 'U': sub( A ) is unit triangular. .TP 8 N (global input) INTEGER The order of the matrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the matrices sub( B ) and sub( X ). NRHS >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1) ). This array contains the local pieces of the original triangular distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the distribu- ted matrix, and its strictly upper triangular part is not referenced. If DIAG = 'U', the diagonal elements of sub( A ) are also not referenced and are assumed to be 1. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input) COMPLEX pointer into the local memory to an array of local dimension (LLD_B, LOCc(JB+NRHS-1) ). On entry, this array contains the the local pieces of the right hand sides sub( B ). .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 X (local input) COMPLEX pointer into the local memory to an array of local dimension (LLD_X, LOCc(JX+NRHS-1) ). On entry, this array contains the the local pieces of the solution vectors sub( X ). .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 FERR (local output) REAL array of local dimension LOCc(JB+NRHS-1). The estimated forward error bounds for each solution vector of sub( X ). If XTRUE is the true solution, FERR bounds the magnitude of the largest entry in (sub( X ) - XTRUE) divided by the magnitude of the largest entry in sub( X ). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. This array is tied to the distributed matrix X. .TP 8 BERR (local output) REAL array of local dimension LOCc(JB+NRHS-1). The componentwise relative backward error of each solution vector (i.e., the smallest re- lative change in any entry of sub( A ) or sub( B ) that makes sub( X ) an exact solution). This array is tied to the distributed matrix X. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 2*LOCr( N + MOD( IA-1, MB_A ) ). If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 RWORK (local workspace/local output) REAL array, dimension (LRWORK) On exit, RWORK(1) returns the minimal and optimal LRWORK. .TP 8 LRWORK (local or global input) INTEGER The dimension of the array RWORK. LRWORK is local input and must be at least LRWORK >= LOCr( N + MOD( IB-1, MB_B ) ). If LRWORK = -1, then LRWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Notes ===== This routine temporarily returns when N <= 1. The distributed submatrices sub( X ) and sub( B ) should be distributed the same way on the same processes. These conditions ensure that sub( X ) and sub( B ) are "perfectly" aligned. Moreover, this routine requires the distributed submatrices sub( A ), sub( X ), and sub( B ) to be aligned on a block boundary, i.e., if f(x,y) = MOD( x-1, y ): f( IA, DESCA( MB_ ) ) = f( JA, DESCA( NB_ ) ) = 0, f( IB, DESCB( MB_ ) ) = f( JB, DESCB( NB_ ) ) = 0, and f( IX, DESCX( MB_ ) ) = f( JX, DESCX( NB_ ) ) = 0. scalapack-doc-1.5/man/manl/pctrti2.l0100644000056400000620000001205306335610623017010 0ustar pfrauenfstaff.TH PCTRTI2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCTRTI2 - compute the inverse of a complex upper or lower triangular block matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PCTRTI2( UPLO, DIAG, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER DIAG, UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ) .SH PURPOSE PCTRTI2 computes the inverse of a complex upper or lower triangular block matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). This matrix should be contained in one and only one process memory space (local operation). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 = 'U': sub( A ) is upper triangular; .br = 'L': sub( A ) is lower triangular. .TP 8 DIAG (global input) CHARACTER*1 .br = 'N': sub( A ) is non-unit triangular .br = 'U': sub( A ) is unit triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)), this array contains the local pieces of the triangular matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of the matrix sub( A ) contains the upper triangular matrix, and the strictly lower triangular part of sub( A ) is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of the matrix sub( A ) contains the lower triangular matrix, and the strictly upper triangular part of sub( A ) is not referenced. If DIAG = 'U', the diagonal elements of sub( A ) are also not referenced and are assumed to be 1. On exit, the (triangular) inverse of the original matrix, in the same storage format. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pctrtri.l0100644000056400000620000001213306335610623017107 0ustar pfrauenfstaff.TH PCTRTRI l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCTRTRI - compute the inverse of a upper or lower triangular distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PCTRTRI( UPLO, DIAG, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER DIAG, UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ) .SH PURPOSE PCTRTRI computes the inverse of a upper or lower triangular distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the distributed matrix sub( A ) is upper or lower triangular: .br = 'U': Upper triangular, .br = 'L': Lower triangular. .TP 8 DIAG (global input) CHARACTER Specifies whether or not the distributed matrix sub( A ) is unit triangular: .br = 'N': Non-unit triangular, .br = 'U': Unit triangular. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the triangular matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of the matrix sub( A ) contains the upper triangular matrix to be inverted, and the strictly lower triangular part of sub( A ) is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of the matrix sub( A ) contains the lower triangular matrix, and the strictly upper triangular part of sub( A ) is not referenced. On exit, the (triangular) inverse of the original matrix. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, A(IA+K-1,JA+K-1) is exactly zero. The triangular matrix sub( A ) is singular and its inverse can not be computed. scalapack-doc-1.5/man/manl/pctrtrs.l0100644000056400000620000001446406335610623017132 0ustar pfrauenfstaff.TH PCTRTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PCTRTRS - solve a triangular system of the form sub( A ) * X = sub( B ) or sub( A )**T * X = sub( B ) or sub( A )**H * X = sub( B ), .SH SYNOPSIS .TP 20 SUBROUTINE PCTRTRS( UPLO, TRANS, DIAG, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, INFO ) .TP 20 .ti +4 CHARACTER DIAG, TRANS, UPLO .TP 20 .ti +4 INTEGER IA, IB, INFO, JA, JB, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX A( * ), B( * ) .SH PURPOSE PCTRTRS solves a triangular system of the form where sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1) and is a triangular distributed matrix of order N, and B(IB:IB+N-1,JB:JB+NRHS-1) is an N-by-NRHS distributed matrix denoted by sub( B ). A check is made to verify that sub( A ) is nonsingular. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER = 'U': sub( A ) is upper triangular; .br = 'L': sub( A ) is lower triangular. .TP 8 TRANS (global input) CHARACTER .br Specifies the form of the system of equations: .br = 'N': Solve sub( A ) * X = sub( B ) (No transpose) .br = 'T': Solve sub( A )**T * X = sub( B ) (Transpose) .br = 'C': Solve sub( A )**H * X = sub( B ) (Conjugate transpose) .TP 8 DIAG (global input) CHARACTER .br = 'N': sub( A ) is non-unit triangular; .br = 'U': sub( A ) is unit triangular. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on i.e the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed matrix sub( B ). NRHS >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1) ). This array contains the local pieces of the distributed triangular matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular matrix, and the strictly lower triangular part of sub( A ) is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular matrix, and the strictly upper triangular part of sub( A ) is not referenced. If DIAG = 'U', the diagonal elements of sub( A ) are also not referenced and are assumed to be 1. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_B,LOCc(JB+NRHS-1)). On entry, this array contains the local pieces of the right hand side distributed matrix sub( B ). On exit, if INFO = 0, sub( B ) is overwritten by the solution matrix X. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = i, the i-th diagonal element of sub( A ) is zero, indicating that the submatrix is singular and the solutions X have not been computed. scalapack-doc-1.5/man/manl/pctzrzf.l0100644000056400000620000001565006335610623017131 0ustar pfrauenfstaff.TH PCTZRZF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCTZRZF - reduce the M-by-N ( M<=N ) complex upper trapezoidal matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper triangular form by means of unitary transformations .SH SYNOPSIS .TP 20 SUBROUTINE PCTZRZF( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), WORK( * ) .SH PURPOSE PCTZRZF reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper triangular form by means of unitary transformations. The upper trapezoidal matrix sub( A ) is factored as .br sub( A ) = ( R 0 ) * Z, .br where Z is an N-by-N unitary matrix and R is an M-by-M upper triangular matrix. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the leading M-by-M upper triangular part of sub( A ) contains the upper trian- gular matrix R, and elements M+1 to N of the first M rows of sub( A ), with the array TAU, represent the unitary matrix Z as a product of M elementary reflectors. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX, array, dimension LOCr(IA+M-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MB_A * ( Mp0 + Nq0 + MB_A ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The factorization is obtained by Householder's method. The kth transformation matrix, Z( k ), whose conjugate transpose is used to introduce zeros into the (m - k + 1)th row of sub( A ), is given in the form .br Z( k ) = ( I 0 ), .br ( 0 T( k ) ) .br where .br T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ), ( 0 ) ( z( k ) ) tau is a scalar and z( k ) is an ( n - m ) element vector. tau and z( k ) are chosen to annihilate the elements of the kth row of sub( A ). .br The scalar tau is returned in the kth element of TAU and the vector u( k ) in the kth row of sub( A ), such that the elements of z( k ) are in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in the upper triangular part of sub( A ). .br Z is given by .br Z = Z( 1 ) * Z( 2 ) * ... * Z( m ). .br scalapack-doc-1.5/man/manl/pcung2l.l0100644000056400000620000001403706335610623016777 0ustar pfrauenfstaff.TH PCUNG2L l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCUNG2L - generate an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the last N columns of a product of K elementary reflectors of order M Q = H(k) .SH SYNOPSIS .TP 20 SUBROUTINE PCUNG2L( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), WORK( * ) .SH PURPOSE PCUNG2L generates an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the last N columns of a product of K elementary reflectors of order M as returned by PCGEQLF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. M >= N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the j-th column must contain the vector which defines the elementary reflector H(j), JA+N-K <= j <= JA+N-1, as returned by PCGEQLF in the K columns of its distributed matrix argument A(IA:*,JA+N-K:JA+N-1). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX, array, dimension LOCc(JA+N-1) This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PCGEQLF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MpA0 + MAX( 1, NqA0 ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pcung2r.l0100644000056400000620000001402206335610623016777 0ustar pfrauenfstaff.TH PCUNG2R l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCUNG2R - generate an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M Q = H(1) H(2) .SH SYNOPSIS .TP 20 SUBROUTINE PCUNG2R( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), WORK( * ) .SH PURPOSE PCUNG2R generates an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M as returned by PCGEQRF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. M >= N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the j-th column must contain the vector which defines the elementary reflector H(j), JA <= j <= JA+K-1, as returned by PCGEQRF in the K columns of its array argument A(IA:*,JA:JA+K-1). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX, array, dimension LOCc(JA+K-1). This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PCGEQRF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MpA0 + MAX( 1, NqA0 ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pcungl2.l0100644000056400000620000001401106335610623016767 0ustar pfrauenfstaff.TH PCUNGL2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCUNGL2 - generate an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N Q = H(k)' .SH SYNOPSIS .TP 20 SUBROUTINE PCUNGL2( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), WORK( * ) .SH PURPOSE PCUNGL2 generates an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N as returned by PCGELQF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. N >= M >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PCGELQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX, array, dimension LOCr(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PCGELQF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NqA0 + MAX( 1, MpA0 ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pcunglq.l0100644000056400000620000001402206335610623017070 0ustar pfrauenfstaff.TH PCUNGLQ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCUNGLQ - generate an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N Q = H(k)' .SH SYNOPSIS .TP 20 SUBROUTINE PCUNGLQ( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), WORK( * ) .SH PURPOSE PCUNGLQ generates an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N as returned by PCGELQF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. N >= M >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PCGELQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX, array, dimension LOCr(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PCGELQF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MB_A * ( MpA0 + NqA0 + MB_A ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pcungql.l0100644000056400000620000001405006335610624017072 0ustar pfrauenfstaff.TH PCUNGQL l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCUNGQL - generate an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the last N columns of a product of K elementary reflectors of order M Q = H(k) .SH SYNOPSIS .TP 20 SUBROUTINE PCUNGQL( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), WORK( * ) .SH PURPOSE PCUNGQL generates an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the last N columns of a product of K elementary reflectors of order M as returned by PCGEQLF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. M >= N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the j-th column must contain the vector which defines the elementary reflector H(j), JA+N-K <= j <= JA+N-1, as returned by PCGEQLF in the K columns of its distributed matrix argument A(IA:*,JA+N-K:JA+N-1). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX, array, dimension LOCc(JA+N-1) This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PCGEQLF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB_A * ( NqA0 + MpA0 + NB_A ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pcungqr.l0100644000056400000620000001404706335610624017106 0ustar pfrauenfstaff.TH PCUNGQR l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCUNGQR - generate an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M Q = H(1) H(2) .SH SYNOPSIS .TP 20 SUBROUTINE PCUNGQR( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), WORK( * ) .SH PURPOSE PCUNGQR generates an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M as returned by PCGEQRF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. M >= N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the j-th column must contain the vector which defines the elementary reflector H(j), JA <= j <= JA+K-1, as returned by PCGEQRF in the K columns of its distributed matrix argument A(IA:*,JA:JA+K-1). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX, array, dimension LOCc(JA+K-1) This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PCGEQRF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB_A * ( NqA0 + MpA0 + NB_A ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pcungr2.l0100644000056400000620000001402406335610624017002 0ustar pfrauenfstaff.TH PCUNGR2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCUNGR2 - generate an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N Q = H(1)' H(2)' .SH SYNOPSIS .TP 20 SUBROUTINE PCUNGR2( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), WORK( * ) .SH PURPOSE PCUNGR2 generates an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N as returned by PCGERQF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. N >= M >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA+M-K <= i <= IA+M-1, as returned by PCGERQF in the K rows of its distributed matrix argument A(IA+M-K:IA+M-1,JA:*). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX, array, dimension LOCr(IA+M-1) This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PCGERQF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NqA0 + MAX( 1, MpA0 ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pcungrq.l0100644000056400000620000001403506335610624017103 0ustar pfrauenfstaff.TH PCUNGRQ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCUNGRQ - generate an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N Q = H(1)' H(2)' .SH SYNOPSIS .TP 20 SUBROUTINE PCUNGRQ( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX A( * ), TAU( * ), WORK( * ) .SH PURPOSE PCUNGRQ generates an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N as returned by PCGERQF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. N >= M >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0. .TP 8 A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA+M-K <= i <= IA+M-1, as returned by PCGERQF in the K rows of its distributed matrix argument A(IA+M-K:IA+M-1,JA:*). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX, array, dimension LOCr(IA+M-1) This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PCGERQF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MB_A * ( MpA0 + NqA0 + MB_A ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pcunm2l.l0100644000056400000620000001726306335610624017012 0ustar pfrauenfstaff.TH PCUNM2L l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCUNM2L - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PCUNM2L( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PCUNM2L overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C ) sub( C ) * Q**H .br where Q is a complex unitary distributed matrix defined as the product of K elementary reflectors .br Q = H(k) . . . H(2) H(1) .br as returned by PCGEQLF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+K-1)). On entry, the j-th column must contain the vector which defines the elemen- tary reflector H(j), JA <= j <= JA+K-1, as returned by PCGEQLF in the K columns of its distributed matrix argument A(IA:*,JA:JA+K-1). A(IA:*,JA:JA+K-1) is modified by the routine but restored on exit. If SIDE = 'L', LLD_A >= MAX( 1, LOCr(IA+M-1) ), if SIDE = 'R', LLD_A >= MAX( 1, LOCr(IA+N-1) ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX, array, dimension LOCc(JA+N-1) This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PCGEQLF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MpC0 + MAX( 1, NqC0 ); if SIDE = 'R', LWORK >= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC( N+ICOFFC,NB_A,0,0,NPCOL ),NB_A,0,0,LCMQ ) ); where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/pcunm2r.l0100644000056400000620000001726506335610624017022 0ustar pfrauenfstaff.TH PCUNM2R l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCUNM2R - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PCUNM2R( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PCUNM2R overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C ) sub( C ) * Q**H .br where Q is a complex unitary distributed matrix defined as the product of k elementary reflectors .br Q = H(1) H(2) . . . H(k) .br as returned by PCGEQRF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+K-1)). On entry, the j-th column must contain the vector which defines the elemen- tary reflector H(j), JA <= j <= JA+K-1, as returned by PCGEQRF in the K columns of its distributed matrix argument A(IA:*,JA:JA+K-1). A(IA:*,JA:JA+K-1) is modified by the routine but restored on exit. If SIDE = 'L', LLD_A >= MAX( 1, LOCr(IA+M-1) ); if SIDE = 'R', LLD_A >= MAX( 1, LOCr(IA+N-1) ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX, array, dimension LOCc(JA+K-1). This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PCGEQRF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MpC0 + MAX( 1, NqC0 ); if SIDE = 'R', LWORK >= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC( N+ICOFFC,NB_A,0,0,NPCOL ),NB_A,0,0,LCMQ ) ); where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/pcunmbr.l0100644000056400000620000002402206335610624017067 0ustar pfrauenfstaff.TH PCUNMBR l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCUNMBR - VECT = 'Q', PCUNMBR overwrites the general complex distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PCUNMBR( VECT, SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS, VECT .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE If VECT = 'Q', PCUNMBR overwrites the general complex distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C ) sub( C ) * Q**H .br If VECT = 'P', PCUNMBR overwrites sub( C ) with .br SIDE = 'L' SIDE = 'R' .br TRANS = 'N': P * sub( C ) sub( C ) * P .br TRANS = 'C': P**H * sub( C ) sub( C ) * P**H .br Here Q and P**H are the unitary distributed matrices determined by PCGEBRD when reducing a complex distributed matrix A(IA:*,JA:*) to bidiagonal form: A(IA:*,JA:*) = Q * B * P**H. Q and P**H are defined as products of elementary reflectors H(i) and G(i) respectively. Let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the order of the unitary matrix Q or P**H that is applied. .br If VECT = 'Q', A(IA:*,JA:*) is assumed to have been an NQ-by-K matrix: .br if nq >= k, Q = H(1) H(2) . . . H(k); .br if nq < k, Q = H(1) H(2) . . . H(nq-1). .br If VECT = 'P', A(IA:*,JA:*) is assumed to have been a K-by-NQ matrix: .br if k < nq, P = G(1) G(2) . . . G(k); .br if k >= nq, P = G(1) G(2) . . . G(nq-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 VECT (global input) CHARACTER = 'Q': apply Q or Q**H; .br = 'P': apply P or P**H. .TP 8 SIDE (global input) CHARACTER .br = 'L': apply Q, Q**H, P or P**H from the Left; .br = 'R': apply Q, Q**H, P or P**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q or P; .br = 'C': Conjugate transpose, apply Q**H or P**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER If VECT = 'Q', the number of columns in the original distributed matrix reduced by PCGEBRD. If VECT = 'P', the number of rows in the original distributed matrix reduced by PCGEBRD. K >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+MIN(NQ,K)-1)) if VECT='Q', and (LLD_A,LOCc(JA+NQ-1)) if VECT = 'P'. NQ = M if SIDE = 'L', and NQ = N otherwise. The vectors which define the elementary reflectors H(i) and G(i), whose products determine the matrices Q and P, as returned by PCGEBRD. If VECT = 'Q', LLD_A >= max(1,LOCr(IA+NQ-1)); if VECT = 'P', LLD_A >= max(1,LOCr(IA+MIN(NQ,K)-1)). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX array, dimension LOCc(JA+MIN(NQ,K)-1) if VECT = 'Q', LOCr(IA+MIN(NQ,K)-1) if VECT = 'P', TAU(i) must contain the scalar factor of the elementary reflector H(i) or G(i), which determines Q or P, as returned by PDGEBRD in its array argument TAUQ or TAUP. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, if VECT='Q', sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q; if VECT='P, sub( C ) is overwritten by P*sub( C ) or P'*sub( C ) or sub( C )*P or sub( C )*P'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', NQ = M; if( (VECT = 'Q' and NQ >= K) or (VECT <> 'Q' and NQ > K) ), IAA=IA; JAA=JA; MI=M; NI=N; ICC=IC; JCC=JC; else IAA=IA+1; JAA=JA; MI=M-1; NI=N; ICC=IC+1; JCC=JC; end if else if SIDE = 'R', NQ = N; if( (VECT = 'Q' and NQ >= K) or (VECT <> 'Q' and NQ > K) ), IAA=IA; JAA=JA; MI=M; NI=N; ICC=IC; JCC=JC; else IAA=IA; JAA=JA+1; MI=M; NI=N-1; ICC=IC; JCC=JC+1; end if end if If VECT = 'Q', If SIDE = 'L', LWORK >= MAX( (NB_A*(NB_A-1))/2, (NqC0 + MpC0)*NB_A ) + NB_A * NB_A else if SIDE = 'R', LWORK >= MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX( NpA0 + NUMROC( NUMROC( NI+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0 ) )*NB_A ) + NB_A * NB_A end if else if VECT <> 'Q', if SIDE = 'L', LWORK >= MAX( (MB_A*(MB_A-1))/2, ( MpC0 + MAX( MqA0 + NUMROC( NUMROC( MI+IROFFC, MB_A, 0, 0, NPROW ), MB_A, 0, 0, LCMP ), NqC0 ) )*MB_A ) + MB_A * MB_A else if SIDE = 'R', LWORK >= MAX( (MB_A*(MB_A-1))/2, (MpC0 + NqC0)*MB_A ) + MB_A * MB_A end if end if where LCMP = LCM / NPROW, LCMQ = LCM / NPCOL, with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IAA-1, MB_A ), ICOFFA = MOD( JAA-1, NB_A ), IAROW = INDXG2P( IAA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JAA, NB_A, MYCOL, CSRC_A, NPCOL ), MqA0 = NUMROC( MI+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), NpA0 = NUMROC( NI+IROFFA, MB_A, MYROW, IAROW, NPROW ), IROFFC = MOD( ICC-1, MB_C ), ICOFFC = MOD( JCC-1, NB_C ), ICROW = INDXG2P( ICC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JCC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( MI+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( NI+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If VECT = 'Q', If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) else If SIDE = 'L', ( MB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/pcunmhr.l0100644000056400000620000002014506335610624017077 0ustar pfrauenfstaff.TH PCUNMHR l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCUNMHR - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PCUNMHR( SIDE, TRANS, M, N, ILO, IHI, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, IHI, ILO, INFO, JA, JC, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PCUNMHR overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C ) sub( C ) * Q**H .br where Q is a complex unitary distributed matrix of order nq, with nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of IHI-ILO elementary reflectors, as returned by PCGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 ILO (global input) INTEGER IHI (global input) INTEGER ILO and IHI must have the same values as in the previous call of PCGEHRD. Q is equal to the unit matrix except in the distributed submatrix Q(ia+ilo:ia+ihi-1,ia+ilo:ja+ihi-1). If SIDE = 'L', 1 <= ILO <= IHI <= max(1,M); if SIDE = 'R', 1 <= ILO <= IHI <= max(1,N); ILO and IHI are relative indexes. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE = 'R'. The vectors which define the elementary reflectors, as returned by PCGEHRD. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX, array, dimension LOCc(JA+M-2) if SIDE = 'L', and LOCc(JA+N-2) if SIDE = 'R'. This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PCGEHRD. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least IAA = IA + ILO; JAA = JA+ILO-1; If SIDE = 'L', MI = IHI-ILO; NI = N; ICC = IC + ILO; JCC = JC; LWORK >= MAX( (NB_A*(NB_A-1))/2, (NqC0 + MpC0)*NB_A ) + NB_A * NB_A else if SIDE = 'R', MI = M; NI = IHI-ILO; ICC = IC; JCC = JC + ILO; LWORK >= MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX( NpA0 + NUMROC( NUMROC( NI+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0 ) )*NB_A ) + NB_A * NB_A end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IAA-1, MB_A ), ICOFFA = MOD( JAA-1, NB_A ), IAROW = INDXG2P( IAA, MB_A, MYROW, RSRC_A, NPROW ), NpA0 = NUMROC( NI+IROFFA, MB_A, MYROW, IAROW, NPROW ), IROFFC = MOD( ICC-1, MB_C ), ICOFFC = MOD( JCC-1, NB_C ), ICROW = INDXG2P( ICC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JCC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( MI+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( NI+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/pcunml2.l0100644000056400000620000001725106335610624017007 0ustar pfrauenfstaff.TH PCUNML2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCUNML2 - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PCUNML2( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PCUNML2 overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C sub( C ) * Q**H .br where Q is a complex unitary distributed matrix defined as the product of K elementary reflectors .br Q = H(k)' . . . H(2)' H(1)' .br as returned by PCGELQF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= max(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PCGELQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PCGELQF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_A,0,0,NPROW ),MB_A,0,0,LCMP ) ); if SIDE = 'R', LWORK >= NqC0 + MAX( 1, MpC0 ); where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/pcunmlq.l0100644000056400000620000001766106335610624017113 0ustar pfrauenfstaff.TH PCUNMLQ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCUNMLQ - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PCUNMLQ( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PCUNMLQ overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C sub( C ) * Q**H .br where Q is a complex unitary distributed matrix defined as the product of K elementary reflectors .br Q = H(k)' . . . H(2)' H(1)' .br as returned by PCGELQF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= max(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PCGELQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PCGELQF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least if SIDE = 'L', LWORK >= MAX( (MB_A*(MB_A-1))/2, ( MpC0 + MAX( MqA0 + NUMROC( NUMROC( M+IROFFC, MB_A, 0, 0, NPROW ), MB_A, 0, 0, LCMP ), NqC0 ) )*MB_A ) + MB_A * MB_A else if SIDE = 'R', LWORK >= MAX( (MB_A*(MB_A-1))/2, (MpC0 + NqC0)*MB_A ) + MB_A * MB_A end if where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MqA0 = NUMROC( M+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/pcunmql.l0100644000056400000620000001767306335610624017116 0ustar pfrauenfstaff.TH PCUNMQL l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCUNMQL - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PCUNMQL( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PCUNMQL overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C ) sub( C ) * Q**H .br where Q is a complex unitary distributed matrix defined as the product of K elementary reflectors .br Q = H(k) . . . H(2) H(1) .br as returned by PCGEQLF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+K-1)). On entry, the j-th column must contain the vector which defines the elemen- tary reflector H(j), JA <= j <= JA+K-1, as returned by PCGEQLF in the K columns of its distributed matrix argument A(IA:*,JA:JA+K-1). A(IA:*,JA:JA+K-1) is modified by the routine but restored on exit. If SIDE = 'L', LLD_A >= MAX( 1, LOCr(IA+M-1) ), if SIDE = 'R', LLD_A >= MAX( 1, LOCr(IA+N-1) ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX, array, dimension LOCc(JA+N-1) This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PCGEQLF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MAX( (NB_A*(NB_A-1))/2, (NqC0 + MpC0)*NB_A ) + NB_A * NB_A else if SIDE = 'R', LWORK >= MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX( NpA0 + NUMROC( NUMROC( N+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0 ) )*NB_A ) + NB_A * NB_A end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), NpA0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/pcunmqr.l0100644000056400000620000001767506335610624017126 0ustar pfrauenfstaff.TH PCUNMQR l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCUNMQR - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PCUNMQR( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PCUNMQR overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C ) sub( C ) * Q**H .br where Q is a complex unitary distributed matrix defined as the product of k elementary reflectors .br Q = H(1) H(2) . . . H(k) .br as returned by PCGEQRF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+K-1)). On entry, the j-th column must contain the vector which defines the elemen- tary reflector H(j), JA <= j <= JA+K-1, as returned by PCGEQRF in the K columns of its distributed matrix argument A(IA:*,JA:JA+K-1). A(IA:*,JA:JA+K-1) is modified by the routine but restored on exit. If SIDE = 'L', LLD_A >= MAX( 1, LOCr(IA+M-1) ); if SIDE = 'R', LLD_A >= MAX( 1, LOCr(IA+N-1) ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX, array, dimension LOCc(JA+K-1). This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PCGEQRF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MAX( (NB_A*(NB_A-1))/2, (NqC0 + MpC0)*NB_A ) + NB_A * NB_A else if SIDE = 'R', LWORK >= MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX( NpA0 + NUMROC( NUMROC( N+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0 ) )*NB_A ) + NB_A * NB_A end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), NpA0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/pcunmr2.l0100644000056400000620000001725106335610624017015 0ustar pfrauenfstaff.TH PCUNMR2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCUNMR2 - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PCUNMR2( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PCUNMR2 overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C ) sub( C ) * Q**H .br where Q is a complex unitary distributed matrix defined as the product of K elementary reflectors .br Q = H(1)' H(2)' . . . H(k)' .br as returned by PCGERQF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= MAX(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PCGERQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PCGERQF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_A,0,0,NPROW ),MB_A,0,0,LCMP ) ); if SIDE = 'R', LWORK >= NqC0 + MAX( 1, MpC0 ); where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/pcunmr3.l0100644000056400000620000001757406335610624017026 0ustar pfrauenfstaff.TH PCUNMR3 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCUNMR3 - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PCUNMR3( SIDE, TRANS, M, N, K, L, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, L, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PCUNMR3 overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C ) sub( C ) * Q**H .br where Q is a complex unitary distributed matrix defined as the product of K elementary reflectors .br Q = H(1)' H(2)' . . . H(k)' .br as returned by PCTZRZF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 L (global input) INTEGER The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= MAX(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PCTZRZF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PCTZRZF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_A,0,0,NPROW ),MB_A,0,0,LCMP ) ); if SIDE = 'R', LWORK >= NqC0 + MAX( 1, MpC0 ); where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/pcunmrq.l0100644000056400000620000001766206335610625017123 0ustar pfrauenfstaff.TH PCUNMRQ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCUNMRQ - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PCUNMRQ( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PCUNMRQ overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C ) sub( C ) * Q**H .br where Q is a complex unitary distributed matrix defined as the product of K elementary reflectors .br Q = H(1)' H(2)' . . . H(k)' .br as returned by PCGERQF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= MAX(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PCGERQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PCGERQF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least if SIDE = 'L', LWORK >= MAX( (MB_A*(MB_A-1))/2, ( MpC0 + MAX( MqA0 + NUMROC( NUMROC( M+IROFFC, MB_A, 0, 0, NPROW ), MB_A, 0, 0, LCMP ), NqC0 ) )*MB_A ) + MB_A * MB_A else if SIDE = 'R', LWORK >= MAX( (MB_A*(MB_A-1))/2, (MpC0 + NqC0)*MB_A ) + MB_A * MB_A end if where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MqA0 = NUMROC( M+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/pcunmrz.l0100644000056400000620000002020506335610625017117 0ustar pfrauenfstaff.TH PCUNMRZ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCUNMRZ - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PCUNMRZ( SIDE, TRANS, M, N, K, L, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, L, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PCUNMRZ overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C ) sub( C ) * Q**H .br where Q is a complex unitary distributed matrix defined as the product of K elementary reflectors .br Q = H(1)' H(2)' . . . H(k)' .br as returned by PCTZRZF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 L (global input) INTEGER The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= MAX(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PCTZRZF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PCTZRZF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least if SIDE = 'L', LWORK >= MAX( (MB_A*(MB_A-1))/2, ( MpC0 + MAX( MqA0 + NUMROC( NUMROC( M+IROFFC, MB_A, 0, 0, NPROW ), MB_A, 0, 0, LCMP ), NqC0 ) )*MB_A ) + MB_A * MB_A else if SIDE = 'R', LWORK >= MAX( (MB_A*(MB_A-1))/2, (MpC0 + NqC0)*MB_A ) + MB_A * MB_A end if where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MqA0 = NUMROC( M+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/pcunmtr.l0100644000056400000620000002045406335610625017117 0ustar pfrauenfstaff.TH PCUNMTR l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PCUNMTR - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PCUNMTR( SIDE, UPLO, TRANS, M, N, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS, UPLO .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PCUNMTR overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C ) sub( C ) * Q**H .br where Q is a complex unitary distributed matrix of order nq, with nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of nq-1 elementary reflectors, as returned by PCHETRD: if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1); .br if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 UPLO (global input) CHARACTER .br = 'U': Upper triangle of A(IA:*,JA:*) contains elementary reflectors from PCHETRD; = 'L': Lower triangle of A(IA:*,JA:*) contains elementary reflectors from PCHETRD. .TP 8 TRANS (global input) CHARACTER = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 A (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', or (LLD_A,LOCc(JA+N-1)) if SIDE = 'R'. The vectors which define the elementary reflectors, as returned by PCHETRD. If SIDE = 'L', LLD_A >= max(1,LOCr(IA+M-1)); if SIDE = 'R', LLD_A >= max(1,LOCr(IA+N-1)). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX array, dimension LTAU, where if SIDE = 'L' and UPLO = 'U', LTAU = LOCc(M_A), if SIDE = 'L' and UPLO = 'L', LTAU = LOCc(JA+M-2), if SIDE = 'R' and UPLO = 'U', LTAU = LOCc(N_A), if SIDE = 'R' and UPLO = 'L', LTAU = LOCc(JA+N-2). TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by PCHETRD. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If UPLO = 'U', IAA = IA, JAA = JA+1, ICC = IC, JCC = JC; else UPLO = 'L', IAA = IA+1, JAA = JA; if SIDE = 'L', ICC = IC+1; JCC = JC; else ICC = IC; JCC = JC+1; end if end if If SIDE = 'L', MI = M-1; NI = N; LWORK >= MAX( (NB_A*(NB_A-1))/2, (NqC0 + MpC0)*NB_A ) + NB_A * NB_A else if SIDE = 'R', MI = M; MI = N-1; LWORK >= MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX( NpA0 + NUMROC( NUMROC( NI+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0 ) )*NB_A ) + NB_A * NB_A end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IAA-1, MB_A ), ICOFFA = MOD( JAA-1, NB_A ), IAROW = INDXG2P( IAA, MB_A, MYROW, RSRC_A, NPROW ), NpA0 = NUMROC( NI+IROFFA, MB_A, MYROW, IAROW, NPROW ), IROFFC = MOD( ICC-1, MB_C ), ICOFFC = MOD( JCC-1, NB_C ), ICROW = INDXG2P( ICC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JCC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( MI+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( NI+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/pddbsv.l0100644000056400000620000000141706335610625016707 0ustar pfrauenfstaff.TH PDDBSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDDBSV - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 19 SUBROUTINE PDDBSV( N, BWL, BWU, NRHS, A, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 INTEGER BWL, BWU, IB, INFO, JA, LWORK, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 DOUBLE PRECISION A( * ), B( * ), WORK( * ) .SH PURPOSE PDDBSV solves a system of linear equations where A(1:N, JA:JA+N-1) is an N-by-N real .br banded diagonally dominant-like distributed .br matrix with bandwidth BWL, BWU. .br Gaussian elimination without pivoting .br is used to factor a reordering .br of the matrix into L U. .br See PDDBTRF and PDDBTRS for details. .br scalapack-doc-1.5/man/manl/pddbtrf.l0100644000056400000620000000214606335610625017052 0ustar pfrauenfstaff.TH PDDBTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDDBTRF - compute a LU factorization of an N-by-N real banded diagonally dominant-like distributed matrix with bandwidth BWL, BWU .SH SYNOPSIS .TP 20 SUBROUTINE PDDBTRF( N, BWL, BWU, A, JA, DESCA, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER BWL, BWU, INFO, JA, LAF, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), AF( * ), WORK( * ) .SH PURPOSE PDDBTRF computes a LU factorization of an N-by-N real banded diagonally dominant-like distributed matrix with bandwidth BWL, BWU: A(1:N, JA:JA+N-1). Reordering is used to increase parallelism in the factorization. This reordering results in factors that are DIFFERENT from those produced by equivalent sequential codes. These factors cannot be used directly by users; however, they can be used in .br subsequent calls to PDDBTRS to solve linear systems. .br The factorization has the form .br P A(1:N, JA:JA+N-1) P^T = L U .br where U is a banded upper triangular matrix and L is banded lower triangular, and P is a permutation matrix. .br scalapack-doc-1.5/man/manl/pddbtrs.l0100644000056400000620000000167106335610625017071 0ustar pfrauenfstaff.TH PDDBTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDDBTRS - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 20 SUBROUTINE PDDBTRS( TRANS, N, BWL, BWU, NRHS, A, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER BWL, BWU, IB, INFO, JA, LAF, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), AF( * ), B( * ), WORK( * ) .SH PURPOSE PDDBTRS solves a system of linear equations or .br A(1:N, JA:JA+N-1)' * X = B(IB:IB+N-1, 1:NRHS) .br where A(1:N, JA:JA+N-1) is the matrix used to produce the factors stored in A(1:N,JA:JA+N-1) and AF by PDDBTRF. .br A(1:N, JA:JA+N-1) is an N-by-N real .br banded diagonally dominant-like distributed .br matrix with bandwidth BWL, BWU. .br Routine PDDBTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pddbtrsv.l0100644000056400000620000000222006335610625017246 0ustar pfrauenfstaff.TH PDDBTRSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDDBTRSV - solve a banded triangular system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 21 SUBROUTINE PDDBTRSV( UPLO, TRANS, N, BWL, BWU, NRHS, A, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 21 .ti +4 CHARACTER TRANS, UPLO .TP 21 .ti +4 INTEGER BWL, BWU, IB, INFO, JA, LAF, LWORK, N, NRHS .TP 21 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 21 .ti +4 DOUBLE PRECISION A( * ), AF( * ), B( * ), WORK( * ) .SH PURPOSE PDDBTRSV solves a banded triangular system of linear equations or .br A(1:N, JA:JA+N-1)^T * X = B(IB:IB+N-1, 1:NRHS) where A(1:N, JA:JA+N-1) is a banded .br triangular matrix factor produced by the .br Gaussian elimination code PD@(dom_pre)BTRF .br and is stored in A(1:N,JA:JA+N-1) and AF. .br The matrix stored in A(1:N, JA:JA+N-1) is either .br upper or lower triangular according to UPLO, .br and the choice of solving A(1:N, JA:JA+N-1) or A(1:N, JA:JA+N-1)^T is dictated by the user by the parameter TRANS. .br Routine PDDBTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pddtsv.l0100644000056400000620000000140206335610625016723 0ustar pfrauenfstaff.TH PDDTSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDDTSV - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 19 SUBROUTINE PDDTSV( N, NRHS, DL, D, DU, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 INTEGER IB, INFO, JA, LWORK, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 DOUBLE PRECISION B( * ), D( * ), DL( * ), DU( * ), WORK( * ) .SH PURPOSE PDDTSV solves a system of linear equations where A(1:N, JA:JA+N-1) is an N-by-N real .br tridiagonal diagonally dominant-like distributed .br matrix. .br Gaussian elimination without pivoting .br is used to factor a reordering .br of the matrix into L U. .br See PDDTTRF and PDDTTRS for details. .br scalapack-doc-1.5/man/manl/pddttrf.l0100644000056400000620000000214106335610625017067 0ustar pfrauenfstaff.TH PDDTTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDDTTRF - compute a LU factorization of an N-by-N real tridiagonal diagonally dominant-like distributed matrix A(1:N, JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PDDTTRF( N, DL, D, DU, JA, DESCA, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER INFO, JA, LAF, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION AF( * ), D( * ), DL( * ), DU( * ), WORK( * ) .SH PURPOSE PDDTTRF computes a LU factorization of an N-by-N real tridiagonal diagonally dominant-like distributed matrix A(1:N, JA:JA+N-1). Reordering is used to increase parallelism in the factorization. This reordering results in factors that are DIFFERENT from those produced by equivalent sequential codes. These factors cannot be used directly by users; however, they can be used in .br subsequent calls to PDDTTRS to solve linear systems. .br The factorization has the form .br P A(1:N, JA:JA+N-1) P^T = L U .br where U is a tridiagonal upper triangular matrix and L is tridiagonal lower triangular, and P is a permutation matrix. .br scalapack-doc-1.5/man/manl/pddttrs.l0100644000056400000620000000165406335610625017114 0ustar pfrauenfstaff.TH PDDTTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDDTTRS - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 20 SUBROUTINE PDDTTRS( TRANS, N, NRHS, DL, D, DU, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER IB, INFO, JA, LAF, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 DOUBLE PRECISION AF( * ), B( * ), D( * ), DL( * ), DU( * ), WORK( * ) .SH PURPOSE PDDTTRS solves a system of linear equations or .br A(1:N, JA:JA+N-1)' * X = B(IB:IB+N-1, 1:NRHS) .br where A(1:N, JA:JA+N-1) is the matrix used to produce the factors stored in A(1:N,JA:JA+N-1) and AF by PDDTTRF. .br A(1:N, JA:JA+N-1) is an N-by-N real .br tridiagonal diagonally dominant-like distributed .br matrix. .br Routine PDDTTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pddttrsv.l0100644000056400000620000000224506335610625017277 0ustar pfrauenfstaff.TH PDDTTRSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDDTTRSV - solve a tridiagonal triangular system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 21 SUBROUTINE PDDTTRSV( UPLO, TRANS, N, NRHS, DL, D, DU, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 21 .ti +4 CHARACTER TRANS, UPLO .TP 21 .ti +4 INTEGER IB, INFO, JA, LAF, LWORK, N, NRHS .TP 21 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 21 .ti +4 DOUBLE PRECISION AF( * ), B( * ), D( * ), DL( * ), DU( * ), WORK( * ) .SH PURPOSE PDDTTRSV solves a tridiagonal triangular system of linear equations or .br A(1:N, JA:JA+N-1)^T * X = B(IB:IB+N-1, 1:NRHS) where A(1:N, JA:JA+N-1) is a tridiagonal .br triangular matrix factor produced by the .br Gaussian elimination code PD@(dom_pre)TTRF .br and is stored in A(1:N,JA:JA+N-1) and AF. .br The matrix stored in A(1:N, JA:JA+N-1) is either .br upper or lower triangular according to UPLO, .br and the choice of solving A(1:N, JA:JA+N-1) or A(1:N, JA:JA+N-1)^T is dictated by the user by the parameter TRANS. .br Routine PDDTTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pdgbsv.l0100644000056400000620000000140606335610625016710 0ustar pfrauenfstaff.TH PDGBSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGBSV - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 19 SUBROUTINE PDGBSV( N, BWL, BWU, NRHS, A, JA, DESCA, IPIV, B, IB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 INTEGER BWL, BWU, IB, INFO, JA, LWORK, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ), IPIV( * ) .TP 19 .ti +4 DOUBLE PRECISION A( * ), B( * ), WORK( * ) .SH PURPOSE PDGBSV solves a system of linear equations where A(1:N, JA:JA+N-1) is an N-by-N real .br banded distributed .br matrix with bandwidth BWL, BWU. .br Gaussian elimination with pivoting .br is used to factor a reordering .br of the matrix into P L U. .br See PDGBTRF and PDGBTRS for details. .br scalapack-doc-1.5/man/manl/pdgbtrf.l0100644000056400000620000000237606335610625017062 0ustar pfrauenfstaff.TH PDGBTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGBTRF - compute a LU factorization of an N-by-N real banded distributed matrix with bandwidth BWL, BWU .SH SYNOPSIS .TP 20 SUBROUTINE PDGBTRF( N, BWL, BWU, A, JA, DESCA, IPIV, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER BWL, BWU, INFO, JA, LAF, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), AF( * ), WORK( * ) .SH PURPOSE PDGBTRF computes a LU factorization of an N-by-N real banded distributed matrix with bandwidth BWL, BWU: A(1:N, JA:JA+N-1). Reordering is used to increase parallelism in the factorization. This reordering results in factors that are DIFFERENT from those produced by equivalent sequential codes. These factors cannot be used directly by users; however, they can be used in .br subsequent calls to PDGBTRS to solve linear systems. .br The factorization has the form .br P A(1:N, JA:JA+N-1) Q = L U .br where U is a banded upper triangular matrix and L is banded lower triangular, and P and Q are permutation matrices. .br The matrix Q represents reordering of columns .br for parallelism's sake, while P represents .br reordering of rows for numerical stability using .br classic partial pivoting. .br scalapack-doc-1.5/man/manl/pdgbtrs.l0100644000056400000620000000165706335610625017100 0ustar pfrauenfstaff.TH PDGBTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGBTRS - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 20 SUBROUTINE PDGBTRS( TRANS, N, BWL, BWU, NRHS, A, JA, DESCA, IPIV, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER BWU, BWL, IB, INFO, JA, LAF, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ), IPIV(*) .TP 20 .ti +4 DOUBLE PRECISION A( * ), AF( * ), B( * ), WORK( * ) .SH PURPOSE PDGBTRS solves a system of linear equations or .br A(1:N, JA:JA+N-1)' * X = B(IB:IB+N-1, 1:NRHS) .br where A(1:N, JA:JA+N-1) is the matrix used to produce the factors stored in A(1:N,JA:JA+N-1) and AF by PDGBTRF. .br A(1:N, JA:JA+N-1) is an N-by-N real .br banded distributed .br matrix with bandwidth BWL, BWU. .br Routine PDGBTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pdgebd2.l0100644000056400000620000002231506335610626016735 0ustar pfrauenfstaff.TH PDGEBD2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDGEBD2 - reduce a real general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form B by an orthogonal transformation .SH SYNOPSIS .TP 20 SUBROUTINE PDGEBD2( M, N, A, IA, JA, DESCA, D, E, TAUQ, TAUP, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), D( * ), E( * ), TAUP( * ), TAUQ( * ), WORK( * ) .SH PURPOSE PDGEBD2 reduces a real general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form B by an orthogonal transformation: Q' * sub( A ) * P = B. If M >= N, B is upper bidiagonal; if M < N, B is lower bidiagonal. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the general distributed matrix sub( A ). On exit, if M >= N, the diagonal and the first superdiagonal of sub( A ) are overwritten with the upper bidiagonal matrix B; the elements below the diagonal, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. If M < N, the diagonal and the first subdiagonal are overwritten with the lower bidiagonal matrix B; the elements below the first subdiagonal, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) DOUBLE PRECISION array, dimension LOCc(JA+MIN(M,N)-1) if M >= N; LOCr(IA+MIN(M,N)-1) otherwise. The distributed diagonal elements of the bidiagonal matrix B: D(i) = A(i,i). D is tied to the distributed matrix A. .TP 8 E (local output) DOUBLE PRECISION array, dimension LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-2) otherwise. The distributed off-diagonal elements of the bidiagonal distributed matrix B: if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. E is tied to the distributed matrix A. .TP 8 TAUQ (local output) DOUBLE PRECISION array dimension LOCc(JA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. TAUQ is tied to the distributed matrix A. See Further Details. TAUP (local output) DOUBLE PRECISION array, dimension LOCr(IA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the orthogonal matrix P. TAUP is tied to the distributed matrix A. See Further Details. WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MAX( MpA0, NqA0 ) where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ) IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, NB, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+IROFFA, NB, MYCOL, IACOL, NPCOL ). INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrices Q and P are represented as products of elementary reflectors: .br If m >= n, .br Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) Each H(i) and G(i) has the form: .br H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are real scalars, and v and u are real vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(ia+i:ia+m-1,ja+i-1); .br u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(ia+i-1,ja+i+1:ja+n-1); .br tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br If m < n, .br Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) Each H(i) and G(i) has the form: .br H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are real scalars, and v and u are real vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(ia+i+1:ia+m-1,ja+i-1); .br u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(ia+i-1,ja+i:ja+n-1); .br tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br The contents of sub( A ) on exit are illustrated by the following examples: .br m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) ( v1 v2 v3 v4 v5 ) .br where d and e denote diagonal and off-diagonal elements of B, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expressions should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) .br scalapack-doc-1.5/man/manl/pdgebrd.l0100644000056400000620000002233506335610626017037 0ustar pfrauenfstaff.TH PDGEBRD l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGEBRD - reduce a real general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form B by an orthogonal transformation .SH SYNOPSIS .TP 20 SUBROUTINE PDGEBRD( M, N, A, IA, JA, DESCA, D, E, TAUQ, TAUP, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), D( * ), E( * ), TAUP( * ), TAUQ( * ), WORK( * ) .SH PURPOSE PDGEBRD reduces a real general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form B by an orthogonal transformation: Q' * sub( A ) * P = B. If M >= N, B is upper bidiagonal; if M < N, B is lower bidiagonal. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the general distributed matrix sub( A ). On exit, if M >= N, the diagonal and the first superdiagonal of sub( A ) are overwritten with the upper bidiagonal matrix B; the elements below the diagonal, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. If M < N, the diagonal and the first subdiagonal are overwritten with the lower bidiagonal matrix B; the elements below the first subdiagonal, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) DOUBLE PRECISION array, dimension LOCc(JA+MIN(M,N)-1) if M >= N; LOCr(IA+MIN(M,N)-1) otherwise. The distributed diagonal elements of the bidiagonal matrix B: D(i) = A(i,i). D is tied to the distributed matrix A. .TP 8 E (local output) DOUBLE PRECISION array, dimension LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-2) otherwise. The distributed off-diagonal elements of the bidiagonal distributed matrix B: if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. E is tied to the distributed matrix A. .TP 8 TAUQ (local output) DOUBLE PRECISION array dimension LOCc(JA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. TAUQ is tied to the distributed matrix A. See Further Details. TAUP (local output) DOUBLE PRECISION array, dimension LOCr(IA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the orthogonal matrix P. TAUP is tied to the distributed matrix A. See Further Details. WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK( 1 ) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB*( MpA0 + NqA0 + 1 ) + NqA0 where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ), ICOFFA = MOD( JA-1, NB ), IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, NB, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB, MYCOL, IACOL, NPCOL ). INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrices Q and P are represented as products of elementary reflectors: .br If m >= n, .br Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) Each H(i) and G(i) has the form: .br H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are real scalars, and v and u are real vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(ia+i:ia+m-1,ja+i-1); .br u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(ia+i-1,ja+i+1:ja+n-1); .br tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br If m < n, .br Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) Each H(i) and G(i) has the form: .br H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are real scalars, and v and u are real vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(ia+i+1:ia+m-1,ja+i-1); .br u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(ia+i-1,ja+i:ja+n-1); .br tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br The contents of sub( A ) on exit are illustrated by the following examples: .br m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) ( v1 v2 v3 v4 v5 ) .br where d and e denote diagonal and off-diagonal elements of B, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expressions should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) .br scalapack-doc-1.5/man/manl/pdgecon.l0100644000056400000620000001554706335610626017056 0ustar pfrauenfstaff.TH PDGECON l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGECON - estimate the reciprocal of the condition number of a general distributed real matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm, using the LU factorization computed by PDGETRF .SH SYNOPSIS .TP 20 SUBROUTINE PDGECON( NORM, N, A, IA, JA, DESCA, ANORM, RCOND, WORK, LWORK, IWORK, LIWORK, INFO ) .TP 20 .ti +4 CHARACTER NORM .TP 20 .ti +4 INTEGER IA, INFO, JA, LIWORK, LWORK, N .TP 20 .ti +4 DOUBLE PRECISION ANORM, RCOND .TP 20 .ti +4 INTEGER DESCA( * ), IWORK( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), WORK( * ) .SH PURPOSE PDGECON estimates the reciprocal of the condition number of a general distributed real matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm, using the LU factorization computed by PDGETRF. An estimate is obtained for norm(inv(A(IA:IA+N-1,JA:JA+N-1))), and the reciprocal of the condition number is computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies whether the 1-norm condition number or the infinity-norm condition number is required: .br = '1' or 'O': 1-norm .br = 'I': Infinity-norm .TP 8 N (global input) INTEGER .br The order of the distributed matrix A(IA:IA+N-1,JA:JA+N-1). N >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension ( LLD_A, LOCc(JA+N-1) ). On entry, this array contains the local pieces of the factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U; the unit diagonal elements of L are not stored. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 ANORM (global input) DOUBLE PRECISION If NORM = '1' or 'O', the 1-norm of the original distributed matrix A(IA:IA+N-1,JA:JA+N-1). If NORM = 'I', the infinity-norm of the original distributed matrix A(IA:IA+N-1,JA:JA+N-1). .TP 8 RCOND (global output) DOUBLE PRECISION The reciprocal of the condition number of the distributed matrix A(IA:IA+N-1,JA:JA+N-1), computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * .br norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 2*LOCr(N+MOD(IA-1,MB_A)) + 2*LOCc(N+MOD(JA-1,NB_A)) + MAX( 2, MAX( NB_A*MAX( 1, CEIL(NPROW-1,NPCOL) ), LOCc(N+MOD(JA-1,NB_A)) + NB_A*MAX( 1, CEIL(NPCOL-1,NPROW) ) ). LOCr and LOCc values can be computed using the ScaLAPACK tool function NUMROC; NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/local output) INTEGER array, dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK. .TP 8 LIWORK (local or global input) INTEGER The dimension of the array IWORK. LIWORK is local input and must be at least LIWORK >= LOCr(N+MOD(IA-1,MB_A)). If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pdgeequ.l0100644000056400000620000001450206335610626017057 0ustar pfrauenfstaff.TH PDGEEQU l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGEEQU - compute row and column scalings intended to equilibrate an M-by-N distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA:JA+N-1) and reduce its condition number .SH SYNOPSIS .TP 20 SUBROUTINE PDGEEQU( M, N, A, IA, JA, DESCA, R, C, ROWCND, COLCND, AMAX, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, M, N .TP 20 .ti +4 DOUBLE PRECISION AMAX, COLCND, ROWCND .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), C( * ), R( * ) .SH PURPOSE PDGEEQU computes row and column scalings intended to equilibrate an M-by-N distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA:JA+N-1) and reduce its condition number. R returns the row scale factors and C the column scale factors, chosen to try to make the largest entry in each row and column of the distributed matrix B with elements B(i,j) = R(i) * A(i,j) * C(j) have absolute value 1. .br R(i) and C(j) are restricted to be between SMLNUM = smallest safe number and BIGNUM = largest safe number. Use of these scaling factors is not guaranteed to reduce the condition number of sub( A ) but works well in practice. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension ( LLD_A, LOCc(JA+N-1) ), the local pieces of the M-by-N distributed matrix whose equilibration factors are to be computed. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 R (local output) DOUBLE PRECISION array, dimension LOCr(M_A) If INFO = 0 or INFO > IA+M-1, R(IA:IA+M-1) contains the row scale factors for sub( A ). R is aligned with the distributed matrix A, and replicated across every process column. R is tied to the distributed matrix A. .TP 8 C (local output) DOUBLE PRECISION array, dimension LOCc(N_A) If INFO = 0, C(JA:JA+N-1) contains the column scale factors for sub( A ). C is aligned with the distributed matrix A, and replicated down every process row. C is tied to the distri- buted matrix A. .TP 8 ROWCND (global output) DOUBLE PRECISION If INFO = 0 or INFO > IA+M-1, ROWCND contains the ratio of the smallest R(i) to the largest R(i) (IA <= i <= IA+M-1). If ROWCND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by R(IA:IA+M-1). .TP 8 COLCND (global output) DOUBLE PRECISION If INFO = 0, COLCND contains the ratio of the smallest C(j) to the largest C(j) (JA <= j <= JA+N-1). If COLCND >= 0.1, it is not worth scaling by C(JA:JA+N-1). .TP 8 AMAX (global output) DOUBLE PRECISION Absolute value of largest distributed matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = i, and i is .br <= M: the i-th row of the distributed matrix sub( A ) is exactly zero, > M: the (i-M)-th column of the distributed matrix sub( A ) is exactly zero. scalapack-doc-1.5/man/manl/pdgehd2.l0100644000056400000620000001655706335610626016756 0ustar pfrauenfstaff.TH PDGEHD2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDGEHD2 - reduce a real general distributed matrix sub( A ) to upper Hessenberg form H by an orthogonal similarity transforma- tion .SH SYNOPSIS .TP 20 SUBROUTINE PDGEHD2( N, ILO, IHI, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, IHI, ILO, INFO, JA, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), TAU( * ), WORK( * ) .SH PURPOSE PDGEHD2 reduces a real general distributed matrix sub( A ) to upper Hessenberg form H by an orthogonal similarity transforma- tion: Q' * sub( A ) * Q = H, where sub( A ) = A(IA+N-1:IA+N-1,JA+N-1:JA+N-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 ILO (global input) INTEGER IHI (global input) INTEGER It is assumed that sub( A ) is already upper triangular in rows IA:IA+ILO-2 and IA+IHI:IA+N-1 and columns JA:JA+JLO-2 and JA+JHI:JA+N-1. See Further Details. If N > 0, .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N general distributed matrix sub( A ) to be reduced. On exit, the upper triangle and the first subdiagonal of sub( A ) are overwritten with the upper Hessenberg matrix H, and the ele- ments below the first subdiagonal, with the array TAU, repre- sent the orthogonal matrix Q as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-2) The scalar factors of the elementary reflectors (see Further Details). Elements JA:JA+ILO-2 and JA+IHI:JA+N-2 of TAU are set to zero. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK( 1 ) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB + MAX( NpA0, NB ) where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ), IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ), NpA0 = NUMROC( IHI+IROFFA, NB, MYROW, IAROW, NPROW ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of (ihi-ilo) elementary reflectors .br Q = H(ilo) H(ilo+1) . . . H(ihi-1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on exit in A(ia+ilo+i:ia+ihi-1,ja+ilo+i-2), and tau in TAU(ja+ilo+i-2). The contents of A(IA:IA+N-1,JA:JA+N-1) are illustrated by the follo- wing example, with n = 7, ilo = 2 and ihi = 6: .br on entry on exit .br ( a a a a a a a ) ( a a h h h h a ) ( a a a a a a ) ( a h h h h a ) ( a a a a a a ) ( h h h h h h ) ( a a a a a a ) ( v2 h h h h h ) ( a a a a a a ) ( v2 v3 h h h h ) ( a a a a a a ) ( v2 v3 v4 h h h ) ( a ) ( a ) where a denotes an element of the original matrix sub( A ), h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(ja+ilo+i-2). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expression should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) .br scalapack-doc-1.5/man/manl/pdgehrd.l0100644000056400000620000001721306335610626017044 0ustar pfrauenfstaff.TH PDGEHRD l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGEHRD - reduce a real general distributed matrix sub( A ) to upper Hessenberg form H by an orthogonal similarity transforma- tion .SH SYNOPSIS .TP 20 SUBROUTINE PDGEHRD( N, ILO, IHI, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, IHI, ILO, INFO, JA, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), TAU( * ), WORK( * ) .SH PURPOSE PDGEHRD reduces a real general distributed matrix sub( A ) to upper Hessenberg form H by an orthogonal similarity transforma- tion: Q' * sub( A ) * Q = H, where sub( A ) = A(IA+N-1:IA+N-1,JA+N-1:JA+N-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 ILO (global input) INTEGER IHI (global input) INTEGER It is assumed that sub( A ) is already upper triangular in rows IA:IA+ILO-2 and IA+IHI:IA+N-1 and columns JA:JA+ILO-2 and JA+IHI:JA+N-1. See Further Details. If N > 0, .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N general distributed matrix sub( A ) to be reduced. On exit, the upper triangle and the first subdiagonal of sub( A ) are overwritten with the upper Hessenberg matrix H, and the ele- ments below the first subdiagonal, with the array TAU, repre- sent the orthogonal matrix Q as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-2) The scalar factors of the elementary reflectors (see Further Details). Elements JA:JA+ILO-2 and JA+IHI:JA+N-2 of TAU are set to zero. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK( 1 ) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB*NB + NB*MAX( IHIP+1, IHLP+INLQ ) where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ), ICOFFA = MOD( JA-1, NB ), IOFF = MOD( IA+ILO-2, NB ), IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ), IHIP = NUMROC( IHI+IROFFA, NB, MYROW, IAROW, NPROW ), ILROW = INDXG2P( IA+ILO-1, NB, MYROW, RSRC_A, NPROW ), IHLP = NUMROC( IHI-ILO+IOFF+1, NB, MYROW, ILROW, NPROW ), ILCOL = INDXG2P( JA+ILO-1, NB, MYCOL, CSRC_A, NPCOL ), INLQ = NUMROC( N-ILO+IOFF+1, NB, MYCOL, ILCOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of (ihi-ilo) elementary reflectors .br Q = H(ilo) H(ilo+1) . . . H(ihi-1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br v(1:I) = 0, v(I+1) = 1 and v(IHI+1:N) = 0; v(I+2:IHI) is stored on exit in A(IA+ILO+I:IA+IHI-1,JA+ILO+I-2), and tau in TAU(JA+ILO+I-2). The contents of A(IA:IA+N-1,JA:JA+N-1) are illustrated by the follow- ing example, with N = 7, ILO = 2 and IHI = 6: .br on entry on exit .br ( a a a a a a a ) ( a a h h h h a ) ( a a a a a a ) ( a h h h h a ) ( a a a a a a ) ( h h h h h h ) ( a a a a a a ) ( v2 h h h h h ) ( a a a a a a ) ( v2 v3 h h h h ) ( a a a a a a ) ( v2 v3 v4 h h h ) ( a ) ( a ) where a denotes an element of the original matrix sub( A ), H denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(JA+ILO+I-2). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expression should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) .br scalapack-doc-1.5/man/manl/pdgelq2.l0100644000056400000620000001424206335610626016764 0ustar pfrauenfstaff.TH PDGELQ2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGELQ2 - compute a LQ factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = L * Q .SH SYNOPSIS .TP 20 SUBROUTINE PDGELQ2( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), TAU( * ), WORK( * ) .SH PURPOSE PDGELQ2 computes a LQ factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = L * Q. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the elements on and below the diagonal of sub( A ) contain the M by min(M,N) lower trapezoidal matrix L (L is lower triangular if M <= N); the elements above the diagonal, with the array TAU, repre- sent the orthogonal matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) DOUBLE PRECISION, array, dimension LOCr(IA+MIN(M,N)-1). This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= Nq0 + MAX( 1, Mp0 ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ia+k-1) H(ia+k-2) . . . H(ia), where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br 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 A(ia+i-1,ja+i:ja+n-1), and tau in TAU(ia+i-1). .br scalapack-doc-1.5/man/manl/pdgelqf.l0100644000056400000620000001425306335610626017052 0ustar pfrauenfstaff.TH PDGELQF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGELQF - compute a LQ factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = L * Q .SH SYNOPSIS .TP 20 SUBROUTINE PDGELQF( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), TAU( * ), WORK( * ) .SH PURPOSE PDGELQF computes a LQ factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = L * Q. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the elements on and below the diagonal of sub( A ) contain the M by min(M,N) lower trapezoidal matrix L (L is lower triangular if M <= N); the elements above the diagonal, with the array TAU, repre- sent the orthogonal matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) DOUBLE PRECISION, array, dimension LOCr(IA+MIN(M,N)-1). This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MB_A * ( Mp0 + Nq0 + MB_A ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ia+k-1) H(ia+k-2) . . . H(ia), where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br 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 A(ia+i-1,ja+i:ja+n-1), and tau in TAU(ia+i-1). .br scalapack-doc-1.5/man/manl/pdgels.l0100644000056400000620000002223606335610626016706 0ustar pfrauenfstaff.TH PDGELS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGELS - solve overdetermined or underdetermined real linear systems involving an M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1), .SH SYNOPSIS .TP 19 SUBROUTINE PDGELS( TRANS, M, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 CHARACTER TRANS .TP 19 .ti +4 INTEGER IA, IB, INFO, JA, JB, LWORK, M, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 DOUBLE PRECISION A( * ), B( * ), WORK( * ) .SH PURPOSE PDGELS solves overdetermined or underdetermined real linear systems involving an M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1), or its transpose, using a QR or LQ factorization of sub( A ). It is assumed that sub( A ) has full rank. .br The following options are provided: .br 1. If TRANS = 'N' and m >= n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem minimize || sub( B ) - sub( A )*X ||. .br 2. If TRANS = 'N' and m < n: find the minimum norm solution of an underdetermined system sub( A ) * X = sub( B ). .br 3. If TRANS = 'T' and m >= n: find the minimum norm solution of an undetermined system sub( A )**T * X = sub( B ). .br 4. If TRANS = 'T' and m < n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem minimize || sub( B ) - sub( A )**T * X ||. where sub( B ) denotes B( IB:IB+M-1, JB:JB+NRHS-1 ) when TRANS = 'N' and B( IB:IB+N-1, JB:JB+NRHS-1 ) otherwise. Several right hand side vectors b and solution vectors x can be handled in a single call; When TRANS = 'N', the solution vectors are stored as the columns of the N-by-NRHS right hand side matrix sub( B ) and the M-by-NRHS right hand side matrix sub( B ) otherwise. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 TRANS (global input) CHARACTER = 'N': the linear system involves sub( A ); .br = 'T': the linear system involves sub( A )**T. .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e. the number of columns of the distributed submatrices sub( B ) and X. NRHS >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of local dimension ( LLD_A, LOCc(JA+N-1) ). On entry, the M-by-N matrix A. if M >= N, sub( A ) is overwritten by details of its QR factorization as returned by PDGEQRF; if M < N, sub( A ) is overwritten by details of its LQ factorization as returned by PDGELQF. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of local dimension (LLD_B, LOCc(JB+NRHS-1)). On entry, this array contains the local pieces of the distributed matrix B of right hand side vectors, stored columnwise; sub( B ) is M-by-NRHS if TRANS='N', and N-by-NRHS otherwise. On exit, sub( B ) is overwritten by the solution vectors, stored columnwise: if TRANS = 'N' and M >= N, rows 1 to N of sub( B ) contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements N+1 to M in that column; if TRANS = 'N' and M < N, rows 1 to N of sub( B ) contain the minimum norm solution vectors; if TRANS = 'T' and M >= N, rows 1 to M of sub( B ) contain the minimum norm solution vectors; if TRANS = 'T' and M < N, rows 1 to M of sub( B ) contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements M+1 to N in that column. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= LTAU + MAX( LWF, LWS ) where If M >= N, then LTAU = NUMROC( JA+MIN(M,N)-1, NB_A, MYCOL, CSRC_A, NPCOL ), LWF = NB_A * ( MpA0 + NqA0 + NB_A ) LWS = MAX( (NB_A*(NB_A-1))/2, (NRHSqB0 + MpB0)*NB_A ) + NB_A * NB_A Else LTAU = NUMROC( IA+MIN(M,N)-1, MB_A, MYROW, RSRC_A, NPROW ), LWF = MB_A * ( MpA0 + NqA0 + MB_A ) LWS = MAX( (MB_A*(MB_A-1))/2, ( NpB0 + MAX( NqA0 + NUMROC( NUMROC( N+IROFFB, MB_A, 0, 0, NPROW ), MB_A, 0, 0, LCMP ), NRHSqB0 ) )*MB_A ) + MB_A * MB_A End if where LCMP = LCM / NPROW with LCM = ILCM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFB = MOD( IB-1, MB_B ), ICOFFB = MOD( JB-1, NB_B ), IBROW = INDXG2P( IB, MB_B, MYROW, RSRC_B, NPROW ), IBCOL = INDXG2P( JB, NB_B, MYCOL, CSRC_B, NPCOL ), MpB0 = NUMROC( M+IROFFB, MB_B, MYROW, IBROW, NPROW ), NpB0 = NUMROC( N+IROFFB, MB_B, MYROW, IBROW, NPROW ), NRHSqB0 = NUMROC( NRHS+ICOFFB, NB_B, MYCOL, IBCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pdgeql2.l0100644000056400000620000001442406335610626016766 0ustar pfrauenfstaff.TH PDGEQL2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGEQL2 - compute a QL factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * L .SH SYNOPSIS .TP 20 SUBROUTINE PDGEQL2( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), TAU( * ), WORK( * ) .SH PURPOSE PDGEQL2 computes a QL factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * L. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, if M >= N, the lower triangle of the distributed submatrix A( IA+M-N:IA+M-1, JA:JA+N-1 ) contains the N-by-N lower triangular matrix L; if M <= N, the elements on and below the (N-M)-th superdiagonal contain the M by N lower trapezoidal matrix L; the remaining elements, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) DOUBLE PRECISION, array, dimension LOCc(JA+N-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= Mp0 + MAX( 1, Nq0 ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ja+k-1) . . . H(ja+1) H(ja), where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in A(ia:ia+m-k+i-2,ja+n-k+i-1), and tau in TAU(ja+n-k+i-1). .br scalapack-doc-1.5/man/manl/pdgeqlf.l0100644000056400000620000001443506335610626017054 0ustar pfrauenfstaff.TH PDGEQLF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGEQLF - compute a QL factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * L .SH SYNOPSIS .TP 20 SUBROUTINE PDGEQLF( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), TAU( * ), WORK( * ) .SH PURPOSE PDGEQLF computes a QL factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * L. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, if M >= N, the lower triangle of the distributed submatrix A( IA+M-N:IA+M-1, JA:JA+N-1 ) contains the N-by-N lower triangular matrix L; if M <= N, the elements on and below the (N-M)-th superdiagonal contain the M by N lower trapezoidal matrix L; the remaining elements, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) DOUBLE PRECISION, array, dimension LOCc(JA+N-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB_A * ( Mp0 + Nq0 + NB_A ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ja+k-1) . . . H(ja+1) H(ja), where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in A(ia:ia+m-k+i-2,ja+n-k+i-1), and tau in TAU(ja+n-k+i-1). .br scalapack-doc-1.5/man/manl/pdgeqpf.l0100644000056400000620000001514106335610626017053 0ustar pfrauenfstaff.TH PDGEQPF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGEQPF - compute a QR factorization with column pivoting of a M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PDGEQPF( M, N, A, IA, JA, DESCA, IPIV, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, JA, INFO, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), TAU( * ), WORK( * ) .SH PURPOSE PDGEQPF computes a QR factorization with column pivoting of a M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1): sub( A ) * P = Q * R. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the elements on and above the diagonal of sub( A ) contain the min(M,N) by N upper trapezoidal matrix R (R is upper triangular if M >= N); the elements below the diagonal, with the array TAU, repre- sent the orthogonal matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local output) INTEGER array, dimension LOCc(JA+N-1). On exit, if IPIV(I) = K, the local i-th column of sub( A )*P was the global K-th column of sub( A ). IPIV is tied to the distributed matrix A. .TP 8 TAU (local output) DOUBLE PRECISION, array, dimension LOCc(JA+MIN(M,N)-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MAX(3,Mp0 + Nq0) + LOCc(JA+N-1)+Nq0. IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), LOCc(JA+N-1) = NUMROC( JA+N-1, NB_A, MYCOL, CSRC_A, NPCOL ) and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) . . . H(n) .br Each H(i) has the form .br H = I - tau * v * v' .br 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(ia+i-1:ia+m-1,ja+i-1). The matrix P is represented in jpvt as follows: If .br jpvt(j) = i .br then the jth column of P is the ith canonical unit vector. scalapack-doc-1.5/man/manl/pdgeqr2.l0100644000056400000620000001424406335610627016775 0ustar pfrauenfstaff.TH PDGEQR2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGEQR2 - compute a QR factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * R .SH SYNOPSIS .TP 20 SUBROUTINE PDGEQR2( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), TAU( * ), WORK( * ) .SH PURPOSE PDGEQR2 computes a QR factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * R. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the elements on and above the diagonal of sub( A ) contain the min(M,N) by N upper trapezoidal matrix R (R is upper triangular if M >= N); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) DOUBLE PRECISION, array, dimension LOCc(JA+MIN(M,N)-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= Mp0 + MAX( 1, Nq0 ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ja) H(ja+1) . . . H(ja+k-1), where k = min(m,n). Each H(i) has the form .br H(j) = I - tau * v * v' .br 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(ia+i:ia+m-1,ja+i-1), and tau in TAU(ja+i-1). .br scalapack-doc-1.5/man/manl/pdgeqrf.l0100644000056400000620000001425506335610627017063 0ustar pfrauenfstaff.TH PDGEQRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGEQRF - compute a QR factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * R .SH SYNOPSIS .TP 20 SUBROUTINE PDGEQRF( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), TAU( * ), WORK( * ) .SH PURPOSE PDGEQRF computes a QR factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * R. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the elements on and above the diagonal of sub( A ) contain the min(M,N) by N upper trapezoidal matrix R (R is upper triangular if M >= N); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) DOUBLE PRECISION, array, dimension LOCc(JA+MIN(M,N)-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB_A * ( Mp0 + Nq0 + NB_A ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ja) H(ja+1) . . . H(ja+k-1), where k = min(m,n). Each H(i) has the form .br H(j) = I - tau * v * v' .br 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(ia+i:ia+m-1,ja+i-1), and tau in TAU(ja+i-1). .br scalapack-doc-1.5/man/manl/pdgerfs.l0100644000056400000620000002341506335610627017063 0ustar pfrauenfstaff.TH PDGERFS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGERFS - improve the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solutions .SH SYNOPSIS .TP 20 SUBROUTINE PDGERFS( TRANS, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, IPIV, B, IB, JB, DESCB, X, IX, JX, DESCX, FERR, BERR, WORK, LWORK, IWORK, LIWORK, INFO ) .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER IA, IAF, IB, IX, INFO, JA, JAF, JB, JX, LIWORK, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * ),IPIV( * ), IWORK( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), AF( * ), B( * ), BERR( * ), FERR( * ), WORK( * ), X( * ) .SH PURPOSE PDGERFS improves the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solutions. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br In the following comments, sub( A ), sub( X ) and sub( B ) denote respectively A(IA:IA+N-1,JA:JA+N-1), X(IX:IX+N-1,JX:JX+NRHS-1) and B(IB:IB+N-1,JB:JB+NRHS-1). .br .SH ARGUMENTS .TP 8 TRANS (global input) CHARACTER*1 Specifies the form of the system of equations. = 'N': sub( A ) * sub( X ) = sub( B ) (No transpose) .br = 'T': sub( A )**T * sub( X ) = sub( B ) (Transpose) .br = 'C': sub( A )**T * sub( X ) = sub( B ) (Conjugate transpose = Transpose) .TP 8 N (global input) INTEGER The order of the matrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the matrices sub( B ) and sub( X ). NRHS >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1)). This array contains the local pieces of the distributed matrix sub( A ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 AF (local input) DOUBLE PRECISION pointer into the local memory to an array of local dimension (LLD_AF,LOCc(JA+N-1)). This array contains the local pieces of the distributed factors of the matrix sub( A ) = P * L * U as computed by PDGETRF. .TP 8 IAF (global input) INTEGER The row index in the global array AF indicating the first row of sub( AF ). .TP 8 JAF (global input) INTEGER The column index in the global array AF indicating the first column of sub( AF ). .TP 8 DESCAF (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix AF. .TP 8 IPIV (local input) INTEGER array of dimension LOCr(M_AF)+MB_AF. This array contains the pivoting information as computed by PDGETRF. IPIV(i) -> The global row local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 B (local input) DOUBLE PRECISION pointer into the local memory to an array of local dimension (LLD_B,LOCc(JB+NRHS-1)). This array contains the local pieces of the distributed matrix of right hand sides sub( B ). .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 X (local input and output) DOUBLE PRECISION pointer into the local memory to an array of local dimension (LLD_X,LOCc(JX+NRHS-1)). On entry, this array contains the local pieces of the distributed matrix solution sub( X ). On exit, the improved solution vectors. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 FERR (local output) DOUBLE PRECISION array of local dimension LOCc(JB+NRHS-1). The estimated forward error bound for each solution vector of sub( X ). If XTRUE is the true solution corresponding to sub( X ), FERR is an estimated upper bound for the magnitude of the largest element in (sub( X ) - XTRUE) divided by the magnitude of the largest element in sub( X ). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. This array is tied to the distributed matrix X. .TP 8 BERR (local output) DOUBLE PRECISION array of local dimension LOCc(JB+NRHS-1). The componentwise relative backward error of each solution vector (i.e., the smallest re- lative change in any entry of sub( A ) or sub( B ) that makes sub( X ) an exact solution). This array is tied to the distributed matrix X. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 3*LOCr( N + MOD(IA-1,MB_A) ) If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/local output) INTEGER array, dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK. .TP 8 LIWORK (local or global input) INTEGER The dimension of the array IWORK. LIWORK is local input and must be at least LIWORK >= LOCr( N + MOD(IB-1,MB_B) ). If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH PARAMETERS ITMAX is the maximum number of steps of iterative refinement. Notes ===== This routine temporarily returns when N <= 1. The distributed submatrices op( A ) and op( AF ) (respectively sub( X ) and sub( B ) ) should be distributed the same way on the same processes. These conditions ensure that sub( A ) and sub( AF ) (resp. sub( X ) and sub( B ) ) are "perfectly" aligned. Moreover, this routine requires the distributed submatrices sub( A ), sub( AF ), sub( X ), and sub( B ) to be aligned on a block boundary, i.e., if f(x,y) = MOD( x-1, y ): f( IA, DESCA( MB_ ) ) = f( JA, DESCA( NB_ ) ) = 0, f( IAF, DESCAF( MB_ ) ) = f( JAF, DESCAF( NB_ ) ) = 0, f( IB, DESCB( MB_ ) ) = f( JB, DESCB( NB_ ) ) = 0, and f( IX, DESCX( MB_ ) ) = f( JX, DESCX( NB_ ) ) = 0. scalapack-doc-1.5/man/manl/pdgerq2.l0100644000056400000620000001437006335610627016775 0ustar pfrauenfstaff.TH PDGERQ2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGERQ2 - compute a RQ factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = R * Q .SH SYNOPSIS .TP 20 SUBROUTINE PDGERQ2( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), TAU( * ), WORK( * ) .SH PURPOSE PDGERQ2 computes a RQ factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = R * Q. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, if M <= N, the upper triangle of A( IA:IA+M-1, JA+N-M:JA+N-1 ) 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 elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) DOUBLE PRECISION, array, dimension LOCr(IA+M-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= Nq0 + MAX( 1, Mp0 ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ia) H(ia+1) . . . H(ia+k-1), where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br 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(ia+m-k+i-1,ja:ja+n-k+i-2), and tau in TAU(ia+m-k+i-1). .br scalapack-doc-1.5/man/manl/pdgerqf.l0100644000056400000620000001440106335610627017054 0ustar pfrauenfstaff.TH PDGERQF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGERQF - compute a RQ factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = R * Q .SH SYNOPSIS .TP 20 SUBROUTINE PDGERQF( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), TAU( * ), WORK( * ) .SH PURPOSE PDGERQF computes a RQ factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = R * Q. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, if M <= N, the upper triangle of A( IA:IA+M-1, JA+N-M:JA+N-1 ) 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 elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) DOUBLE PRECISION, array, dimension LOCr(IA+M-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MB_A * ( Mp0 + Nq0 + MB_A ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ia) H(ia+1) . . . H(ia+k-1), where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br 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(ia+m-k+i-1,ja:ja+n-k+i-2), and tau in TAU(ia+m-k+i-1). .br scalapack-doc-1.5/man/manl/pdgesv.l0100644000056400000620000001400406335610627016713 0ustar pfrauenfstaff.TH PDGESV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGESV - compute the solution to a real system of linear equations sub( A ) * X = sub( B ), .SH SYNOPSIS .TP 19 SUBROUTINE PDGESV( N, NRHS, A, IA, JA, DESCA, IPIV, B, IB, JB, DESCB, INFO ) .TP 19 .ti +4 INTEGER IA, IB, INFO, JA, JB, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ), IPIV( * ) .TP 19 .ti +4 DOUBLE PRECISION A( * ), B( * ) .SH PURPOSE PDGESV computes the solution to a real system of linear equations where sub( A ) = A(IA:IA+N-1,JA:JA+N-1) is an N-by-N distributed matrix and X and sub( B ) = B(IB:IB+N-1,JB:JB+NRHS-1) are N-by-NRHS distributed matrices. .br The LU decomposition with partial pivoting and row interchanges is used to factor sub( A ) as sub( A ) = P * L * U, where P is a permu- tation matrix, L is unit lower triangular, and U is upper triangular. L and U are stored in sub( A ). The factored form of sub( A ) is then used to solve the system of equations sub( A ) * X = sub( B ). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block decomposition ( MB_A = NB_A ). .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed submatrix sub( A ). NRHS >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the local pieces of the N-by-N distributed matrix sub( A ) to be factored. On exit, this array contains the local pieces of the factors L and U from the factorization sub( A ) = P*L*U; the unit diagonal elements of L are not stored. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local output) INTEGER array, dimension ( LOCr(M_A)+MB_A ) This array contains the pivoting information. IPIV(i) -> The global row local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 B (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_B,LOCc(JB+NRHS-1)). On entry, the right hand side distributed matrix sub( B ). On exit, if INFO = 0, sub( B ) is overwritten by the solution distributed matrix X. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, U(IA+K-1,JA+K-1) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution could not be computed. scalapack-doc-1.5/man/manl/pdgesvd.l0100644000056400000620000002256206335610627017067 0ustar pfrauenfstaff.TH PDGESVD l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGESVD - compute the singular value decomposition (SVD) of an M-by-N matrix A, optionally computing the left and/or right singular vectors .SH SYNOPSIS .TP 20 SUBROUTINE PDGESVD( JOBU, JOBVT, M, N, A, IA, JA, DESCA, S, U, IU, JU, DESCU, VT, IVT, JVT, DESCVT, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER JOBU, JOBVT .TP 20 .ti +4 INTEGER IA, INFO, IU, IVT, JA, JU, JVT, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCU( * ), DESCVT( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), S( * ), U( * ), VT( * ), WORK( * ) .SH PURPOSE PDGESVD computes the singular value decomposition (SVD) of an M-by-N matrix A, optionally computing the left and/or right singular vectors. The SVD is written as A = U * SIGMA * transpose(V) .br where SIGMA is an M-by-N matrix which is zero except for its min(M,N) diagonal elements, U is an M-by-M orthogonal matrix, and V is an N-by-N orthogonal matrix. The diagonal elements of SIGMA are the singular values of A and the columns of U and V are the corresponding right and left singular vectors, respectively. The singular values are returned in array S in decreasing order and only the first min(M,N) columns of U and rows of VT = V**T are computed. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS MP = number of local rows in A and U NQ = number of local columns in A and VT SIZE = min( M, N ) SIZEQ = number of local columns in U SIZEP = number of local rows in VT .TP 8 JOBU (global input) CHARACTER*1 Specifies options for computing all or part of the matrix U: .br = 'V': the first SIZE columns of U (the left singular vectors) are returned in the array U; = 'N': no columns of U (no left singular vectors) are computed. .TP 8 JOBVT (global input) CHARACTER*1 Specifies options for computing all or part of the matrix V**T: .br = 'V': the first SIZE rows of V**T (the right singular vectors) are returned in the array VT; = 'N': no rows of V**T (no right singular vectors) are computed. .TP 8 M (global input) INTEGER The number of rows of the input matrix A. M >= 0. .TP 8 N (global input) INTEGER The number of columns of the input matrix A. N >= 0. .TP 8 A (local input/workspace) block cyclic DOUBLE PRECISION array, global dimension (M, N), local dimension (MP, NQ) On exit, the contents of A are destroyed. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global input) INTEGER array of dimension DLEN_ The array descriptor for the distributed matrix A. .TP 8 S (global output) DOUBLE PRECISION array, dimension SIZE The singular values of A, sorted so that S(i) >= S(i+1). .TP 8 U (local output) DOUBLE PRECISION array, local dimension (MP, SIZEQ), global dimension (M, SIZE) if JOBU = 'V', U contains the first min(m,n) columns of U if JOBU = 'N', U is not referenced. .TP 8 IU (global input) INTEGER The row index in the global array U indicating the first row of sub( U ). .TP 8 JU (global input) INTEGER The column index in the global array U indicating the first column of sub( U ). .TP 8 DESCU (global input) INTEGER array of dimension DLEN_ The array descriptor for the distributed matrix U. .TP 8 VT (local output) DOUBLE PRECISION array, local dimension (SIZEP, NQ), global dimension (SIZE, N). If JOBVT = 'V', VT contains the first SIZE rows of V**T. If JOBVT = 'N', VT is not referenced. .TP 8 IVT (global input) INTEGER The row index in the global array VT indicating the first row of sub( VT ). .TP 8 JVT (global input) INTEGER The column index in the global array VT indicating the first column of sub( VT ). .TP 9 DESCVT (global input) INTEGER array of dimension DLEN_ The array descriptor for the distributed matrix VT. .TP 8 WORK (local workspace/output) DOUBLE PRECISION array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK; .TP 8 LWORK (local input) INTEGER The dimension of the array WORK. LWORK > 2 + 6*SIZEB + MAX(WATOBD, WBDTOSVD), where SIZEB = MAX(M,N), and WATOBD and WBDTOSVD refer, respectively, to the workspace required to bidiagonalize the matrix A and to go from the bidiagonal matrix to the singular value decomposition U*S*VT. For WATOBD, the following holds: WATOBD = MAX(MAX(WPDLANGE,WPDGEBRD), MAX(WPDLARED2D,WPDLARED1D)), where WPDLANGE, WPDLARED1D, WPDLARED2D, WPDGEBRD are the workspaces required respectively for the subprograms PDLANGE, PDLARED1D, PDLARED2D, PDGEBRD. Using the standard notation MP = NUMROC( M, MB, MYROW, DESCA( CTXT_ ), NPROW), NQ = NUMROC( N, NB, MYCOL, DESCA( LLD_ ), NPCOL), the workspaces required for the above subprograms are WPDLANGE = MP, WPDLARED1D = NQ0, WPDLARED2D = MP0, WPDGEBRD = NB*(MP + NQ + 1) + NQ, where NQ0 and MP0 refer, respectively, to the values obtained at MYCOL = 0 and MYROW = 0. In general, the upper limit for the workspace is given by a workspace required on processor (0,0): WATOBD <= NB*(MP0 + NQ0 + 1) + NQ0. In case of a homogeneous process grid this upper limit can be used as an estimate of the minimum workspace for every processor. For WBDTOSVD, the following holds: WBDTOSVD = SIZE*(WANTU*NRU + WANTVT*NCVT) + MAX(WDBDSQR, MAX(WANTU*WPDORMBRQLN, WANTVT*WPDORMBRPRT)), .TP -1 where 1, if left(right) singular vectors are wanted WANTU(WANTVT) = 0, otherwise and WDBDSQR, WPDORMBRQLN and WPDORMBRPRT refer respectively to the workspace required for the subprograms DBDSQR, PDORMBR(QLN), and PDORMBR(PRT), where QLN and PRT are the values of the arguments VECT, SIDE, and TRANS in the call to PDORMBR. NRU is equal to the local number of rows of the matrix U when distributed 1-dimensional "column" of processes. Analogously, NCVT is equal to the local number of columns of the matrix VT when distributed across 1-dimensional "row" of processes. Calling the LAPACK procedure DBDSQR requires WDBDSQR = MAX(1, 2*SIZE + (2*SIZE - 4)*MAX(WANTU, WANTVT)) on every processor. Finally, WPDORMBRQLN = MAX( (NB*(NB-1))/2, (SIZEQ+MP)*NB)+NB*NB, WPDORMBRPRT = MAX( (MB*(MB-1))/2, (SIZEP+NQ)*MB )+MB*MB, If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum size for the work array. The required workspace is returned as the first element of WORK and no error message is issued by PXERBLA. .TP 8 INFO (output) INTEGER = 0: successful exit. .br < 0: if INFO = -i, the i-th argument had an illegal value. .br > 0: if SBDSQR did not converge If INFO = MIN(M,N) + 1, then PDGESVD has detected heterogeneity by finding that eigenvalues were not identical across the process grid. In this case, the accuracy of the results from PDGESVD cannot be guaranteed. scalapack-doc-1.5/man/manl/pdgesvx.l0100644000056400000620000004045406335610627017113 0ustar pfrauenfstaff.TH PDGESVX l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGESVX - use the LU factorization to compute the solution to a real system of linear equations A(IA:IA+N-1,JA:JA+N-1) * X = B(IB:IB+N-1,JB:JB+NRHS-1), .SH SYNOPSIS .TP 20 SUBROUTINE PDGESVX( FACT, TRANS, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, IPIV, EQUED, R, C, B, IB, JB, DESCB, X, IX, JX, DESCX, RCOND, FERR, BERR, WORK, LWORK, IWORK, LIWORK, INFO ) .TP 20 .ti +4 CHARACTER EQUED, FACT, TRANS .TP 20 .ti +4 INTEGER IA, IAF, IB, INFO, IX, JA, JAF, JB, JX, LIWORK, LWORK, N, NRHS .TP 20 .ti +4 DOUBLE PRECISION RCOND .TP 20 .ti +4 INTEGER DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * ), IPIV( * ), IWORK( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), AF( * ), B( * ), BERR( * ), C( * ), FERR( * ), R( * ), WORK( * ), X( * ) .SH PURPOSE PDGESVX uses the LU factorization to compute the solution to a real system of linear equations where A(IA:IA+N-1,JA:JA+N-1) is an N-by-N matrix and X and B(IB:IB+N-1,JB:JB+NRHS-1) are N-by-NRHS matrices. .br Error bounds on the solution and a condition estimate are also provided. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH DESCRIPTION In the following description, A denotes A(IA:IA+N-1,JA:JA+N-1), B denotes B(IB:IB+N-1,JB:JB+NRHS-1) and X denotes .br X(IX:IX+N-1,JX:JX+NRHS-1). .br The following steps are performed: .br 1. If FACT = 'E', real scaling factors are computed to equilibrate the system: .br 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'). .br 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as .br A = P * L * U, .br where P is a permutation matrix, L is a unit lower triangular matrix, and U is upper triangular. .br 3. 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, steps 4-6 are skipped. .br 4. The system of equations is solved for X using the factored form of A. .br 5. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. .br 6. If FACT = 'E' and 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. .br .SH ARGUMENTS .TP 8 FACT (global input) CHARACTER Specifies whether or not the factored form of the matrix A(IA:IA+N-1,JA:JA+N-1) is supplied on entry, and if not, .br whether the matrix A(IA:IA+N-1,JA:JA+N-1) should be equilibrated before it is factored. = 'F': On entry, AF(IAF:IAF+N-1,JAF:JAF+N-1) and IPIV con- .br tain the factored form of A(IA:IA+N-1,JA:JA+N-1). If EQUED is not 'N', the matrix A(IA:IA+N-1,JA:JA+N-1) has been equilibrated with scaling factors given by R and C. A(IA:IA+N-1,JA:JA+N-1), AF(IAF:IAF+N-1,JAF:JAF+N-1), and IPIV are not modified. = 'N': The matrix A(IA:IA+N-1,JA:JA+N-1) will be copied to .br AF(IAF:IAF+N-1,JAF:JAF+N-1) and factored. .br = 'E': The matrix A(IA:IA+N-1,JA:JA+N-1) will be equili- brated if necessary, then copied to AF(IAF:IAF+N-1,JAF:JAF+N-1) and factored. .TP 8 TRANS (global input) CHARACTER .br Specifies the form of the system of equations: .br = 'N': A(IA:IA+N-1,JA:JA+N-1) * X(IX:IX+N-1,JX:JX+NRHS-1) .br = B(IB:IB+N-1,JB:JB+NRHS-1) (No transpose) .br = 'T': A(IA:IA+N-1,JA:JA+N-1)**T * X(IX:IX+N-1,JX:JX+NRHS-1) .br = B(IB:IB+N-1,JB:JB+NRHS-1) (Transpose) .br = 'C': A(IA:IA+N-1,JA:JA+N-1)**H * X(IX:IX+N-1,JX:JX+NRHS-1) .br = B(IB:IB+N-1,JB:JB+NRHS-1) (Transpose) .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix A(IA:IA+N-1,JA:JA+N-1). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right-hand sides, i.e., the number of columns of the distributed submatrices B(IB:IB+N-1,JB:JB+NRHS-1) and .br X(IX:IX+N-1,JX:JX+NRHS-1). NRHS >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1)). On entry, the N-by-N matrix A(IA:IA+N-1,JA:JA+N-1). If FACT = 'F' and EQUED is not 'N', .br then A(IA:IA+N-1,JA:JA+N-1) must have been equilibrated by .br the scaling factors in R and/or C. A(IA:IA+N-1,JA:JA+N-1) is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. On exit, if EQUED .ne. 'N', A(IA:IA+N-1,JA:JA+N-1) is scaled as follows: .br EQUED = 'R': A(IA:IA+N-1,JA:JA+N-1) := .br diag(R) * A(IA:IA+N-1,JA:JA+N-1) .br EQUED = 'C': A(IA:IA+N-1,JA:JA+N-1) := .br A(IA:IA+N-1,JA:JA+N-1) * diag(C) .br EQUED = 'B': A(IA:IA+N-1,JA:JA+N-1) := .br diag(R) * A(IA:IA+N-1,JA:JA+N-1) * diag(C). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 AF (local input or local output) DOUBLE PRECISION pointer into the local memory to an array of local dimension (LLD_AF,LOCc(JA+N-1)). If FACT = 'F', then AF(IAF:IAF+N-1,JAF:JAF+N-1) is an input argument and on entry contains the factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U as computed by PDGETRF. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix A(IA:IA+N-1,JA:JA+N-1). If FACT = 'N', then AF(IAF:IAF+N-1,JAF:JAF+N-1) is an output argument and on exit returns the factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the original .br matrix A(IA:IA+N-1,JA:JA+N-1). If FACT = 'E', then AF(IAF:IAF+N-1,JAF:JAF+N-1) is an output argument and on exit returns the factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the equili- .br brated matrix A(IA:IA+N-1,JA:JA+N-1) (see the description of .br A(IA:IA+N-1,JA:JA+N-1) for the form of the equilibrated matrix). .TP 8 IAF (global input) INTEGER The row index in the global array AF indicating the first row of sub( AF ). .TP 8 JAF (global input) INTEGER The column index in the global array AF indicating the first column of sub( AF ). .TP 8 DESCAF (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix AF. .TP 8 IPIV (local input or local output) INTEGER array, dimension LOCr(M_A)+MB_A. If FACT = 'F', then IPIV is an input argu- ment and on entry contains the pivot indices from the fac- torization A(IA:IA+N-1,JA:JA+N-1) = P*L*U as computed by PDGETRF; IPIV(i) -> The global row local row i was swapped with. This array must be aligned with A( IA:IA+N-1, * ). If FACT = 'N', then IPIV is an output argument and on exit contains the pivot indices from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the original matrix .br A(IA:IA+N-1,JA:JA+N-1). If FACT = 'E', then IPIV is an output argument and on exit contains the pivot indices from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the equilibrated matrix .br A(IA:IA+N-1,JA:JA+N-1). .TP 8 EQUED (global input or global output) CHARACTER Specifies the form of equilibration that was done. = 'N': No equilibration (always true if FACT = 'N'). .br = 'R': Row equilibration, i.e., A(IA:IA+N-1,JA:JA+N-1) has been premultiplied by diag(R). = 'C': Column equilibration, i.e., A(IA:IA+N-1,JA:JA+N-1) has been postmultiplied by diag(C). = 'B': Both row and column equilibration, i.e., .br A(IA:IA+N-1,JA:JA+N-1) has been replaced by .br diag(R) * A(IA:IA+N-1,JA:JA+N-1) * diag(C). EQUED is an input variable if FACT = 'F'; otherwise, it is an output variable. .TP 8 R (local input or local output) DOUBLE PRECISION array, dimension LOCr(M_A). The row scale factors for A(IA:IA+N-1,JA:JA+N-1). .br If EQUED = 'R' or 'B', A(IA:IA+N-1,JA:JA+N-1) is multiplied on the left by diag(R); if EQUED='N' or 'C', R is not acces- sed. R is an input variable if FACT = 'F'; otherwise, R is an output variable. If FACT = 'F' and EQUED = 'R' or 'B', each element of R must be positive. R is replicated in every process column, and is aligned with the distributed matrix A. .TP 8 C (local input or local output) DOUBLE PRECISION array, dimension LOCc(N_A). The column scale factors for A(IA:IA+N-1,JA:JA+N-1). .br If EQUED = 'C' or 'B', A(IA:IA+N-1,JA:JA+N-1) is multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is not accessed. C is an input variable if FACT = 'F'; otherwise, C is an output variable. If FACT = 'F' and EQUED = 'C' or 'B', each element of C must be positive. C is replicated in every process row, and is aligned with the distributed matrix A. .TP 8 B (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of local dimension (LLD_B,LOCc(JB+NRHS-1) ). On entry, the N-by-NRHS right-hand side matrix B(IB:IB+N-1,JB:JB+NRHS-1). On exit, if .br EQUED = 'N', B(IB:IB+N-1,JB:JB+NRHS-1) is not modified; if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B(IB:IB+N-1,JB:JB+NRHS-1); if TRANS = 'T' or 'C' .br and EQUED = 'C' or 'B', B(IB:IB+N-1,JB:JB+NRHS-1) is over- .br written by diag(C)*B(IB:IB+N-1,JB:JB+NRHS-1). .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 X (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of local dimension (LLD_X, LOCc(JX+NRHS-1)). If INFO = 0, the N-by-NRHS solution matrix X(IX:IX+N-1,JX:JX+NRHS-1) to the original .br system of equations. Note that A(IA:IA+N-1,JA:JA+N-1) and .br B(IB:IB+N-1,JB:JB+NRHS-1) are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(C))*X(IX:IX+N-1,JX:JX+NRHS-1) if TRANS = 'N' and EQUED = 'C' or 'B', or inv(diag(R))*X(IX:IX+N-1,JX:JX+NRHS-1) if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 RCOND (global output) DOUBLE PRECISION The estimate of the reciprocal condition number of the matrix A(IA:IA+N-1,JA:JA+N-1) 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. .TP 8 FERR (local output) DOUBLE PRECISION array, dimension LOCc(N_B) The estimated forward error bounds for each solution vector X(j) (the j-th column of the solution matrix X(IX:IX+N-1,JX:JX+NRHS-1). If XTRUE is the true solution, FERR(j) bounds the magnitude of the largest entry in (X(j) - XTRUE) divided by the magnitude of the largest entry in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. FERR is replicated in every process row, and is aligned with the matrices B and X. .TP 8 BERR (local output) DOUBLE PRECISION array, dimension LOCc(N_B). The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any entry of A(IA:IA+N-1,JA:JA+N-1) or .br B(IB:IB+N-1,JB:JB+NRHS-1) that makes X(j) an exact solution). BERR is replicated in every process row, and is aligned with the matrices B and X. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK = MAX( PDGECON( LWORK ), PDGERFS( LWORK ) ) + LOCr( N_A ). If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/local output) INTEGER array, dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK. .TP 8 LIWORK (local or global input) INTEGER The dimension of the array IWORK. LIWORK is local input and must be at least LIWORK = LOCr(N_A). If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 0: if INFO = i, and i is .br <= N: U(IA+I-1,IA+I-1) 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. = N+1: RCOND is less than machine precision. The factorization has been completed, but the matrix is singular to working precision, and the solution and error bounds have not been computed. scalapack-doc-1.5/man/manl/pdgetf2.l0100644000056400000620000001263006335610627016761 0ustar pfrauenfstaff.TH PDGETF2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGETF2 - compute an LU factorization of a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) using partial pivoting with row interchanges .SH SYNOPSIS .TP 20 SUBROUTINE PDGETF2( M, N, A, IA, JA, DESCA, IPIV, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, M, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ) .SH PURPOSE PDGETF2 computes an LU factorization of a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) using partial pivoting with row interchanges. The factorization has the form sub( A ) = P * L * U, where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n). .br This is the right-looking Parallel Level 2 BLAS version of the algorithm. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires N <= NB_A-MOD(JA-1, NB_A) and square block decomposition ( MB_A = NB_A ). .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). NB_A-MOD(JA-1, NB_A) >= N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the M-by-N distributed matrix sub( A ). On exit, this array contains the local pieces of the factors L and U from the factoriza- tion sub( A ) = P*L*U; the unit diagonal elements of L are not stored. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local output) INTEGER array, dimension ( LOCr(M_A)+MB_A ) This array contains the pivoting information. IPIV(i) -> The global row local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, U(IA+K-1,JA+K-1) 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. scalapack-doc-1.5/man/manl/pdgetrf.l0100644000056400000620000001261206335610627017061 0ustar pfrauenfstaff.TH PDGETRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGETRF - compute an LU factorization of a general M-by-N distributed matrix sub( A ) = (IA:IA+M-1,JA:JA+N-1) using partial pivoting with row interchanges .SH SYNOPSIS .TP 20 SUBROUTINE PDGETRF( M, N, A, IA, JA, DESCA, IPIV, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, M, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ) .SH PURPOSE PDGETRF computes an LU factorization of a general M-by-N distributed matrix sub( A ) = (IA:IA+M-1,JA:JA+N-1) using partial pivoting with row interchanges. The factorization has the form sub( A ) = P * L * U, where P is a permutation matrix, L is lower triangular with unit diagonal ele- ments (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n). L and U are stored in sub( A ). This is the right-looking Parallel Level 3 BLAS version of the algorithm. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block decomposition ( MB_A = NB_A ). .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the M-by-N distributed matrix sub( A ) to be factored. On exit, this array contains the local pieces of the factors L and U from the factorization sub( A ) = P*L*U; the unit diagonal ele- ments of L are not stored. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local output) INTEGER array, dimension ( LOCr(M_A)+MB_A ) This array contains the pivoting information. IPIV(i) -> The global row local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, U(IA+K-1,JA+K-1) 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. scalapack-doc-1.5/man/manl/pdgetri.l0100644000056400000620000001445006335610627017066 0ustar pfrauenfstaff.TH PDGETRI l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGETRI - compute the inverse of a distributed matrix using the LU factorization computed by PDGETRF .SH SYNOPSIS .TP 20 SUBROUTINE PDGETRI( N, A, IA, JA, DESCA, IPIV, WORK, LWORK, IWORK, LIWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LIWORK, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ), IWORK( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), WORK( * ) .SH PURPOSE PDGETRI computes the inverse of a distributed matrix using the LU factorization computed by PDGETRF. This method inverts U and then computes the inverse of sub( A ) = A(IA:IA+N-1,JA:JA+N-1) denoted InvA by solving the system InvA*L = inv(U) for InvA. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the local pieces of the L and U obtained by the factorization sub( A ) = P*L*U computed by PDGETRF. On exit, if INFO = 0, sub( A ) contains the inverse of the original distributed matrix sub( A ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local input) INTEGER array, dimension LOCr(M_A)+MB_A keeps track of the pivoting information. IPIV(i) is the global row index the local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK = LOCr(N+MOD(IA-1,MB_A))*NB_A. WORK is used to keep a copy of at most an entire column block of sub( A ). If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/local output) INTEGER array, dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK. .TP 8 LIWORK (local or global input) INTEGER The dimension of the array IWORK used as workspace for physically transposing the pivots. LIWORK is local input and must be at least if NPROW == NPCOL then LIWORK = LOCc( N_A + MOD(JA-1, NB_A) ) + NB_A, else LIWORK = LOCc( N_A + MOD(JA-1, NB_A) ) + MAX( CEIL(CEIL(LOCr(M_A)/MB_A)/(LCM/NPROW)), NB_A ) where LCM is the least common multiple of process rows and columns (NPROW and NPCOL). end if If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, U(IA+K-1,IA+K-1) is exactly zero; the matrix is singular and its inverse could not be computed. scalapack-doc-1.5/man/manl/pdgetrs.l0100644000056400000620000001331306335610627017075 0ustar pfrauenfstaff.TH PDGETRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGETRS - solve a system of distributed linear equations op( sub( A ) ) * X = sub( B ) with a general N-by-N distributed matrix sub( A ) using the LU factorization computed by PDGETRF .SH SYNOPSIS .TP 20 SUBROUTINE PDGETRS( TRANS, N, NRHS, A, IA, JA, DESCA, IPIV, B, IB, JB, DESCB, INFO ) .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER IA, IB, INFO, JA, JB, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ), IPIV( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), B( * ) .SH PURPOSE PDGETRS solves a system of distributed linear equations sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1), op( A ) = A or A**T and sub( B ) denotes B(IB:IB+N-1,JB:JB+NRHS-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block data decomposition ( MB_A=NB_A ). .SH ARGUMENTS .TP 8 TRANS (global input) CHARACTER Specifies the form of the system of equations: .br = 'N': sub( A ) * X = sub( B ) (No transpose) .br = 'T': sub( A )**T * X = sub( B ) (Transpose) .br = 'C': sub( A )**T * X = sub( B ) (Transpose) .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed submatrix sub( B ). NRHS >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the factors L and U from the factorization sub( A ) = P*L*U; the unit diagonal elements of L are not stored. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local input) INTEGER array, dimension ( LOCr(M_A)+MB_A ) This array contains the pivoting information. IPIV(i) -> The global row local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 B (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_B,LOCc(JB+NRHS-1)). On entry, the right hand sides sub( B ). On exit, sub( B ) is overwritten by the solution distributed matrix X. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pdggqrf.l0100644000056400000620000002361206335610630017054 0ustar pfrauenfstaff.TH PDGGQRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGGQRF - compute a generalized QR factorization of an N-by-M matrix sub( A ) = A(IA:IA+N-1,JA:JA+M-1) and an N-by-P matrix sub( B ) = B(IB:IB+N-1,JB:JB+P-1) .SH SYNOPSIS .TP 20 SUBROUTINE PDGGQRF( N, M, P, A, IA, JA, DESCA, TAUA, B, IB, JB, DESCB, TAUB, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, IB, INFO, JA, JB, LWORK, M, N, P .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), B( * ), TAUA( * ), TAUB( * ), WORK( * ) .SH PURPOSE PDGGQRF computes a generalized QR factorization of an N-by-M matrix sub( A ) = A(IA:IA+N-1,JA:JA+M-1) and an N-by-P matrix sub( B ) = B(IB:IB+N-1,JB:JB+P-1): sub( A ) = Q*R, sub( B ) = Q*T*Z, .br where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal matrix, and R and T assume one of the forms: .br if N >= M, R = ( R11 ) M , or if N < M, R = ( R11 R12 ) N, ( 0 ) N-M N M-N M .br where R11 is upper triangular, and .br if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) N-P, P-N N ( T21 ) P P .br where T12 or T21 is upper triangular. .br In particular, if sub( B ) is square and nonsingular, the GQR factorization of sub( A ) and sub( B ) implicitly gives the QR factorization of inv( sub( B ) )* sub( A ): .br inv( sub( B ) )*sub( A )= Z'*(inv(T)*R) .br where inv( sub( B ) ) denotes the inverse of the matrix sub( B ), and Z' denotes the transpose of matrix Z. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrices sub( A ) and sub( B ). N >= 0. .TP 8 M (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). M >= 0. .TP 8 P (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( B ). P >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+M-1)). On entry, the local pieces of the N-by-M distributed matrix sub( A ) which is to be factored. On exit, the elements on and above the diagonal of sub( A ) contain the min(N,M) by M upper trapezoidal matrix R (R is upper triangular if N >= M); the elements below the diagonal, with the array TAUA, represent the orthogonal matrix Q as a product of min(N,M) elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAUA (local output) DOUBLE PRECISION, array, dimension LOCc(JA+MIN(N,M)-1). This array contains the scalar factors TAUA of the elementary reflectors which represent the orthogonal matrix Q. TAUA is tied to the distributed matrix A. (see Further Details). B (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+P-1)). On entry, the local pieces of the N-by-P distributed matrix sub( B ) which is to be factored. On exit, if N <= P, the upper triangle of B(IB:IB+N-1,JB+P-N:JB+P-1) contains the N by N upper triangular matrix T; if N > P, the elements on and above the (N-P)-th subdiagonal contain the N by P upper trapezoidal matrix T; the remaining elements, with the array TAUB, represent the orthogonal matrix Z as a product of elementary reflectors (see Further Details). IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 TAUB (local output) DOUBLE PRECISION, array, dimension LOCr(IB+N-1) This array contains the scalar factors of the elementary reflectors which represent the orthogonal unitary matrix Z. TAUB is tied to the distributed matrix B (see Further Details). .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MAX( NB_A * ( NpA0 + MqA0 + NB_A ), MAX( (NB_A*(NB_A-1))/2, (PqB0 + NpB0)*NB_A ) + NB_A * NB_A, MB_B * ( NpB0 + PqB0 + MB_B ) ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), NpA0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), MqA0 = NUMROC( M+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFB = MOD( IB-1, MB_B ), ICOFFB = MOD( JB-1, NB_B ), IBROW = INDXG2P( IB, MB_B, MYROW, RSRC_B, NPROW ), IBCOL = INDXG2P( JB, NB_B, MYCOL, CSRC_B, NPCOL ), NpB0 = NUMROC( N+IROFFB, MB_B, MYROW, IBROW, NPROW ), PqB0 = NUMROC( P+ICOFFB, NB_B, MYCOL, IBCOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ja) H(ja+1) . . . H(ja+k-1), where k = min(n,m). Each H(i) has the form .br H(i) = I - taua * v * v' .br where taua is a real scalar, and v is a real vector with .br v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in .br A(ia+i:ia+n-1,ja+i-1), and taua in TAUA(ja+i-1). .br To form Q explicitly, use ScaLAPACK subroutine PDORGQR. .br To use Q to update another matrix, use ScaLAPACK subroutine PDORMQR. The matrix Z is represented as a product of elementary reflectors Z = H(ib) H(ib+1) . . . H(ib+k-1), where k = min(n,p). Each H(i) has the form .br H(i) = I - taub * v * v' .br where taub is a real scalar, and v is a real vector with .br v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in B(ib+n-k+i-1,jb:jb+p-k+i-2), and taub in TAUB(ib+n-k+i-1). To form Z explicitly, use ScaLAPACK subroutine PDORGRQ. .br To use Z to update another matrix, use ScaLAPACK subroutine PDORMRQ. Alignment requirements .br ====================== .br The distributed submatrices sub( A ) and sub( B ) must verify some alignment properties, namely the following expression should be true: ( MB_A.EQ.MB_B .AND. IROFFA.EQ.IROFFB .AND. IAROW.EQ.IBROW ) scalapack-doc-1.5/man/manl/pdggrqf.l0100644000056400000620000002351206335610630017053 0ustar pfrauenfstaff.TH PDGGRQF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDGGRQF - compute a generalized RQ factorization of an M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PDGGRQF( M, P, N, A, IA, JA, DESCA, TAUA, B, IB, JB, DESCB, TAUB, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, IB, INFO, JA, JB, LWORK, M, N, P .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), B( * ), TAUA( * ), TAUB( * ), WORK( * ) .SH PURPOSE PDGGRQF computes a generalized RQ factorization of an M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) and a P-by-N matrix sub( B ) = B(IB:IB+P-1,JB:JB+N-1): .br sub( A ) = R*Q, sub( B ) = Z*T*Q, .br where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal matrix, and R and T assume one of the forms: .br if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N, N-M M ( R21 ) N N .br where R12 or R21 is upper triangular, and .br if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P, ( 0 ) P-N P N-P N .br where T11 is upper triangular. .br In particular, if sub( B ) is square and nonsingular, the GRQ factorization of sub( A ) and sub( B ) implicitly gives the RQ factorization of sub( A )*inv( sub( B ) ): .br sub( A )*inv( sub( B ) ) = (R*inv(T))*Z' .br where inv( sub( B ) ) denotes the inverse of the matrix sub( B ), and Z' denotes the transpose of matrix Z. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 P (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( B ). P >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrices sub( A ) and sub( B ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, if M <= N, the upper triangle of A( IA:IA+M-1, JA+N-M:JA+N-1 ) 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 TAUA, represent the orthogonal matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAUA (local output) DOUBLE PRECISION, array, dimension LOCr(IA+M-1) This array contains the scalar factors of the elementary reflectors which represent the orthogonal unitary matrix Q. TAUA is tied to the distributed matrix A (see Further Details). .TP 8 B (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+N-1)). On entry, the local pieces of the P-by-N distributed matrix sub( B ) which is to be factored. On exit, the elements on and above the diagonal of sub( B ) contain the min(P,N) by N upper trapezoidal matrix T (T is upper triangular if P >= N); the elements below the diagonal, with the array TAUB, represent the orthogonal matrix Z as a product of elementary reflectors (see Further Details). IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 TAUB (local output) DOUBLE PRECISION, array, dimension LOCc(JB+MIN(P,N)-1). This array contains the scalar factors TAUB of the elementary reflectors which represent the orthogonal matrix Z. TAUB is tied to the distributed matrix B (see Further Details). WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MAX( MB_A * ( MpA0 + NqA0 + MB_A ), MAX( (MB_A*(MB_A-1))/2, (PpB0 + NqB0)*MB_A ) + MB_A * MB_A, NB_B * ( PpB0 + NqB0 + NB_B ) ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFB = MOD( IB-1, MB_B ), ICOFFB = MOD( JB-1, NB_B ), IBROW = INDXG2P( IB, MB_B, MYROW, RSRC_B, NPROW ), IBCOL = INDXG2P( JB, NB_B, MYCOL, CSRC_B, NPCOL ), PpB0 = NUMROC( P+IROFFB, MB_B, MYROW, IBROW, NPROW ), NqB0 = NUMROC( N+ICOFFB, NB_B, MYCOL, IBCOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ia) H(ia+1) . . . H(ia+k-1), where k = min(m,n). Each H(i) has the form .br H(i) = I - taua * v * v' .br where taua is a real scalar, and v is a real vector with .br 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(ia+m-k+i-1,ja:ja+n-k+i-2), and taua in TAUA(ia+m-k+i-1). To form Q explicitly, use ScaLAPACK subroutine PDORGRQ. .br To use Q to update another matrix, use ScaLAPACK subroutine PDORMRQ. The matrix Z is represented as a product of elementary reflectors Z = H(jb) H(jb+1) . . . H(jb+k-1), where k = min(p,n). Each H(i) has the form .br H(i) = I - taub * v * v' .br where taub is a real scalar, and v is a real vector with .br v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in .br B(ib+i:ib+p-1,jb+i-1), and taub in TAUB(jb+i-1). .br To form Z explicitly, use ScaLAPACK subroutine PDORGQR. .br To use Z to update another matrix, use ScaLAPACK subroutine PDORMQR. Alignment requirements .br ====================== .br The distributed submatrices sub( A ) and sub( B ) must verify some alignment properties, namely the following expression should be true: ( NB_A.EQ.NB_B .AND. ICOFFA.EQ.ICOFFB .AND. IACOL.EQ.IBCOL ) scalapack-doc-1.5/man/manl/pdlabad.l0100644000056400000620000000313406335610630017006 0ustar pfrauenfstaff.TH PDLABAD l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLABAD - take as input the values computed by PDLAMCH for underflow and overflow, and returns the square root of each of these values if the log of LARGE is sufficiently large .SH SYNOPSIS .TP 20 SUBROUTINE PDLABAD( ICTXT, SMALL, LARGE ) .TP 20 .ti +4 INTEGER ICTXT .TP 20 .ti +4 DOUBLE PRECISION LARGE, SMALL .SH PURPOSE PDLABAD takes as input the values computed by PDLAMCH for underflow and overflow, and returns the square root of each of these values if the log of LARGE is sufficiently large. This subroutine is intended to identify machines with a large exponent range, such as the Crays, and redefine the underflow and overflow limits to be the square roots of the values computed by PDLAMCH. This subroutine is needed because PDLAMCH does not compensate for poor arithmetic in the upper half of the exponent range, as is found on a Cray. .br In addition, this routine performs a global minimization and maximi- zation on these values, to support heterogeneous computing networks. .SH ARGUMENTS .TP 8 ICTXT (global input) INTEGER The BLACS context handle in which the computation takes place. .TP 8 SMALL (local input/local output) DOUBLE PRECISION On entry, the underflow threshold as computed by PDLAMCH. On exit, if LOG10(LARGE) is sufficiently large, the square root of SMALL, otherwise unchanged. .TP 8 LARGE (local input/local output) DOUBLE PRECISION On entry, the overflow threshold as computed by PDLAMCH. On exit, if LOG10(LARGE) is sufficiently large, the square root of LARGE, otherwise unchanged. scalapack-doc-1.5/man/manl/pdlabrd.l0100644000056400000620000002345006335610630017032 0ustar pfrauenfstaff.TH PDLABRD l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLABRD - reduce the first NB rows and columns of a real general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form by an orthogonal transformation Q' * A * P, .SH SYNOPSIS .TP 20 SUBROUTINE PDLABRD( M, N, NB, A, IA, JA, DESCA, D, E, TAUQ, TAUP, X, IX, JX, DESCX, Y, IY, JY, DESCY, WORK ) .TP 20 .ti +4 INTEGER IA, IX, IY, JA, JX, JY, M, N, NB .TP 20 .ti +4 INTEGER DESCA( * ), DESCX( * ), DESCY( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), D( * ), E( * ), TAUP( * ), TAUQ( * ), X( * ), Y( * ), WORK( * ) .SH PURPOSE PDLABRD reduces the first NB rows and columns of a real general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form by an orthogonal transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of sub( A ). .br If M >= N, sub( A ) is reduced to upper bidiagonal form; if M < N, to lower bidiagonal form. .br This is an auxiliary routine called by PDGEBRD. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 NB (global input) INTEGER The number of leading rows and columns of sub( A ) to be reduced. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the general distributed matrix sub( A ) to be reduced. On exit, the first NB rows and columns of the matrix are overwritten; the rest of the distributed matrix sub( A ) is unchanged. If m >= n, elements on and below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. If m < n, elements below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and elements on and above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) DOUBLE PRECISION array, dimension LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-1) otherwise. The distributed diagonal elements of the bidiagonal matrix B: D(i) = A(ia+i-1,ja+i-1). D is tied to the distributed matrix A. .TP 8 E (local output) DOUBLE PRECISION array, dimension LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-2) otherwise. The distributed off-diagonal elements of the bidiagonal distributed matrix B: if m >= n, E(i) = A(ia+i-1,ja+i) for i = 1,2,...,n-1; if m < n, E(i) = A(ia+i,ja+i-1) for i = 1,2,...,m-1. E is tied to the distributed matrix A. .TP 8 TAUQ (local output) DOUBLE PRECISION array dimension LOCc(JA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. TAUQ is tied to the distributed matrix A. See Further Details. TAUP (local output) DOUBLE PRECISION array, dimension LOCr(IA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the orthogonal matrix P. TAUP is tied to the distributed matrix A. See Further Details. X (local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_X,NB). On exit, the local pieces of the distributed M-by-NB matrix X(IX:IX+M-1,JX:JX+NB-1) required to update the unreduced part of sub( A ). .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 Y (local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_Y,NB). On exit, the local pieces of the distributed N-by-NB matrix Y(IY:IY+N-1,JY:JY+NB-1) required to update the unreduced part of sub( A ). .TP 8 IY (global input) INTEGER The row index in the global array Y indicating the first row of sub( Y ). .TP 8 JY (global input) INTEGER The column index in the global array Y indicating the first column of sub( Y ). .TP 8 DESCY (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix Y. .TP 8 WORK (local workspace) DOUBLE PRECISION array, dimension (LWORK) LWORK >= NB_A + NQ, with NQ = NUMROC( N+MOD( IA-1, NB_Y ), NB_Y, MYCOL, IACOL, NPCOL ) IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ) INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. .SH FURTHER DETAILS The matrices Q and P are represented as products of elementary reflectors: .br Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb) Each H(i) and G(i) has the form: .br H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are real scalars, and v and u are real vectors. If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in A(ia+i-1:ia+m-1,ja+i-1); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in A(ia+i-1,ja+i:ja+n-1); tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in A(ia+i+1:ia+m-1,ja+i-1); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in A(ia+i-1,ja+i:ja+n-1); tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br The elements of the vectors v and u together form the m-by-nb matrix V and the nb-by-n matrix U' which are needed, with X and Y, to apply the transformation to the unreduced part of the matrix, using a block update of the form: sub( A ) := sub( A ) - V*Y' - X*U'. .br The contents of sub( A ) on exit are illustrated by the following examples with nb = 2: .br m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 ) ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 ) ( v1 v2 a a a ) ( v1 1 a a a a ) ( v1 v2 a a a ) ( v1 v2 a a a a ) ( v1 v2 a a a ) ( v1 v2 a a a a ) ( v1 v2 a a a ) .br where a denotes an element of the original matrix which is unchanged, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i). .br scalapack-doc-1.5/man/manl/pdlacon.l0100644000056400000620000001302306335610630017035 0ustar pfrauenfstaff.TH PDLACON l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLACON - estimate the 1-norm of a square, real distributed matrix A .SH SYNOPSIS .TP 20 SUBROUTINE PDLACON( N, V, IV, JV, DESCV, X, IX, JX, DESCX, ISGN, EST, KASE ) .TP 20 .ti +4 INTEGER IV, IX, JV, JX, KASE, N .TP 20 .ti +4 DOUBLE PRECISION EST .TP 20 .ti +4 INTEGER DESCV( * ), DESCX( * ), ISGN( * ) .TP 20 .ti +4 DOUBLE PRECISION V( * ), X( * ) .SH PURPOSE PDLACON estimates the 1-norm of a square, real distributed matrix A. Reverse communication is used for evaluating matrix-vector products. X and V are aligned with the distributed matrix A, this information is implicitly contained within IV, IX, DESCV, and DESCX. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The length of the distributed vectors V and X. N >= 0. .TP 8 V (local workspace) DOUBLE PRECISION pointer into the local memory to an array of dimension LOCr(N+MOD(IV-1,MB_V)). On the final return, V = A*W, where EST = norm(V)/norm(W) (W is not returned). .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 X (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension LOCr(N+MOD(IX-1,MB_X)). On an intermediate return, X should be overwritten by A * X, if KASE=1, A' * X, if KASE=2, PDLACON must be re-called with all the other parameters unchanged. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 ISGN (local workspace) INTEGER array, dimension LOCr(N+MOD(IX-1,MB_X)). ISGN is aligned with X and V. .TP 8 EST (global output) DOUBLE PRECISION An estimate (a lower bound) for norm(A). .TP 8 KASE (local input/local output) INTEGER On the initial call to PDLACON, KASE should be 0. On an intermediate return, KASE will be 1 or 2, indicating whether X should be overwritten by A * X or A' * X. On the final return from PDLACON, KASE will again be 0. .SH FURTHER DETAILS The serial version DLACON has been contributed by Nick Higham, University of Manchester. It was originally named SONEST, dated March 16, 1988. .br Reference: N.J. Higham, "FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation", ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988. scalapack-doc-1.5/man/manl/pdlaconsb.l0100644000056400000620000001362306335610630017370 0ustar pfrauenfstaff.TH PDLACONSB l "12 May 1997" "LAPACK version 1.5 " "LAPACK routine (version 1.5 )" .SH NAME PDLACONSB - look for two consecutive small subdiagonal elements by seeing the effect of starting a double shift QR iteration given by H44, H33, & H43H34 and see if this would make a subdiagonal negligible .SH SYNOPSIS .TP 22 SUBROUTINE PDLACONSB( A, DESCA, I, L, M, H44, H33, H43H34, BUF, LWORK ) .TP 22 .ti +4 INTEGER I, L, LWORK, M .TP 22 .ti +4 DOUBLE PRECISION H33, H43H34, H44 .TP 22 .ti +4 INTEGER DESCA( * ) .TP 22 .ti +4 DOUBLE PRECISION A( * ), BUF( * ) .SH PURPOSE PDLACONSB looks for two consecutive small subdiagonal elements by seeing the effect of starting a double shift QR iteration given by H44, H33, & H43H34 and see if this would make a subdiagonal negligible. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 A (global input) DOUBLE PRECISION array, dimension (DESCA(LLD_),*) On entry, the Hessenberg matrix whose tridiagonal part is being scanned. Unchanged on exit. .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 I (global input) INTEGER The global location of the bottom of the unreduced submatrix of A. Unchanged on exit. .TP 8 L (global input) INTEGER The global location of the top of the unreduced submatrix of A. Unchanged on exit. .TP 8 M (global output) INTEGER On exit, this yields the starting location of the QR double shift. This will satisfy: L <= M <= I-2. H44 H33 H43H34 (global input) DOUBLE PRECISION These three values are for the double shift QR iteration. .TP 8 BUF (local output) DOUBLE PRECISION array of size LWORK. .TP 8 LWORK (global input) INTEGER On exit, LWORK is the size of the work buffer. This must be at least 7*Ceil( Ceil( (I-L)/HBL ) / LCM(NPROW,NPCOL) ) Here LCM is least common multiple, and NPROWxNPCOL is the logical grid size. Logic: ====== Two consecutive small subdiagonal elements will stall convergence of a double shift if their product is small relatively even if each is not very small. Thus it is necessary to scan the "tridiagonal portion of the matrix." In the LAPACK algorithm DLAHQR, a loop of M goes from I-2 down to L and examines H(m,m),H(m+1,m+1),H(m+1,m),H(m,m+1),H(m-1,m-1),H(m,m-1), and H(m+2,m-1). Since these elements may be on separate processors, the first major loop (10) goes over the tridiagonal and has each node store whatever values of the 7 it has that the node owning H(m,m) does not. This will occur on a border and can happen in no more than 3 locations per block assuming square blocks. There are 5 buffers that each node stores these values: a buffer to send diagonally down and right, a buffer to send up, a buffer to send left, a buffer to send diagonally up and left and a buffer to send right. Each of these buffers is actually stored in one buffer BUF where BUF(ISTR1+1) starts the first buffer, BUF(ISTR2+1) starts the second, etc.. After the values are stored, if there are any values that a node needs, they will be sent and received. Then the next major loop passes over the data and searches for two consecutive small subdiagonals. Notes: This routine does a global maximum and must be called by all processes. Implemented by: G. Henry, November 17, 1996 scalapack-doc-1.5/man/manl/pdlacp2.l0100644000056400000620000001272706335610630016754 0ustar pfrauenfstaff.TH PDLACP2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLACP2 - copie all or part of a distributed matrix A to another distributed matrix B .SH SYNOPSIS .TP 20 SUBROUTINE PDLACP2( UPLO, M, N, A, IA, JA, DESCA, B, IB, JB, DESCB ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IB, JA, JB, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), B( * ) .SH PURPOSE PDLACP2 copies all or part of a distributed matrix A to another distributed matrix B. No communication is performed, PDLACP2 performs a local copy sub( A ) := sub( B ), where sub( A ) denotes A(IA:IA+M-1,JA:JA+N-1) and sub( B ) denotes B(IB:IB+M-1,JB:JB+N-1). PDLACP2 requires that only dimension of the matrix operands is distributed. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies the part of the distributed matrix sub( A ) to be copied: .br = 'U': Upper triangular part is copied; the strictly lower triangular part of sub( A ) is not referenced; = 'L': Lower triangular part is copied; the strictly upper triangular part of sub( A ) is not referenced; Otherwise: All of the matrix sub( A ) is copied. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1) ). This array contains the local pieces of the distributed matrix sub( A ) to be copied from. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+N-1) ). This array contains on exit the local pieces of the distributed matrix sub( B ) set as follows: if UPLO = 'U', B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), 1<=i<=j, 1<=j<=N; if UPLO = 'L', B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), j<=i<=M, 1<=j<=N; otherwise, B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), 1<=i<=M, 1<=j<=N. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. scalapack-doc-1.5/man/manl/pdlacp3.l0100644000056400000620000001210706335610630016745 0ustar pfrauenfstaff.TH PDLACP3 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDLACP3 - i an auxiliary routine that copies from a global parallel array into a local replicated array or vise versa .SH SYNOPSIS .TP 20 SUBROUTINE PDLACP3( M, I, A, DESCA, B, LDB, II, JJ, REV ) .TP 20 .ti +4 INTEGER I, II, JJ, LDB, M, REV .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), B( LDB, * ) .SH PURPOSE PDLACP3 is an auxiliary routine that copies from a global parallel array into a local replicated array or vise versa. Notice that the entire submatrix that is copied gets placed on one node or more. The receiving node can be specified precisely, or all nodes can receive, or just one row or column of nodes. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER M is the order of the square submatrix that is copied. M >= 0. Unchanged on exit .TP 8 I (global input) INTEGER A(I,I) is the global location that the copying starts from. Unchanged on exit. .TP 8 A (global input/output) DOUBLE PRECISION array, dimension (DESCA(LLD_),*) On entry, the parallel matrix to be copied into or from. On exit, if REV=1, the copied data. Unchanged on exit if REV=0. .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input/output) DOUBLE PRECISION array of size (LDB,M) If REV=0, this is the global portion of the array A(I:I+M-1,I:I+M-1). If REV=1, this is the unchanged on exit. .TP 8 LDB (local input) INTEGER The leading dimension of B. .TP 8 II (global input) INTEGER By using REV 0 & 1, data can be sent out and returned again. If REV=0, then II is destination row index for the node(s) receiving the replicated B. If II>=0,JJ>=0, then node (II,JJ) receives the data If II=-1,JJ>=0, then all rows in column JJ receive the data If II>=0,JJ=-1, then all cols in row II receive the data If II=-1,JJ=-1, then all nodes receive the data If REV<>0, then II is the source row index for the node(s) sending the replicated B. .TP 8 JJ (global input) INTEGER Similar description as II above .TP 8 REV (global input) INTEGER Use REV = 0 to send global A into locally replicated B (on node (II,JJ)). Use REV <> 0 to send locally replicated B from node (II,JJ) to its owner (which changes depending on its location in A) into the global A. Implemented by: G. Henry, May 1, 1997 scalapack-doc-1.5/man/manl/pdlacpy.l0100644000056400000620000001260706335610630017060 0ustar pfrauenfstaff.TH PDLACPY l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLACPY - copie all or part of a distributed matrix A to another distributed matrix B .SH SYNOPSIS .TP 20 SUBROUTINE PDLACPY( UPLO, M, N, A, IA, JA, DESCA, B, IB, JB, DESCB ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IB, JA, JB, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), B( * ) .SH PURPOSE PDLACPY copies all or part of a distributed matrix A to another distributed matrix B. No communication is performed, PDLACPY performs a local copy sub( A ) := sub( B ), where sub( A ) denotes A(IA:IA+M-1,JA:JA+N-1) and sub( B ) denotes B(IB:IB+M-1,JB:JB+N-1). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies the part of the distributed matrix sub( A ) to be copied: .br = 'U': Upper triangular part is copied; the strictly lower triangular part of sub( A ) is not referenced; = 'L': Lower triangular part is copied; the strictly upper triangular part of sub( A ) is not referenced; Otherwise: All of the matrix sub( A ) is copied. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1) ). This array contains the local pieces of the distributed matrix sub( A ) to be copied from. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+N-1) ). This array contains on exit the local pieces of the distributed matrix sub( B ) set as follows: if UPLO = 'U', B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), 1<=i<=j, 1<=j<=N; if UPLO = 'L', B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), j<=i<=M, 1<=j<=N; otherwise, B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), 1<=i<=M, 1<=j<=N. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. scalapack-doc-1.5/man/manl/pdlaevswp.l0100644000056400000620000001221206335610630017421 0ustar pfrauenfstaff.TH PDLAEVSWP l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDLAEVSWP - move the eigenvectors (potentially unsorted) from where they are computed, to a ScaLAPACK standard block cyclic array, sorted so that the corresponding eigenvalues are sorted .SH SYNOPSIS .TP 22 SUBROUTINE PDLAEVSWP( N, ZIN, LDZI, Z, IZ, JZ, DESCZ, NVS, KEY, WORK, LWORK ) .TP 22 .ti +4 INTEGER IZ, JZ, LDZI, LWORK, N .TP 22 .ti +4 INTEGER DESCZ( * ), KEY( * ), NVS( * ) .TP 22 .ti +4 DOUBLE PRECISION WORK( * ), Z( * ), ZIN( LDZI, * ) .SH PURPOSE PDLAEVSWP moves the eigenvectors (potentially unsorted) from where they are computed, to a ScaLAPACK standard block cyclic array, sorted so that the corresponding eigenvalues are sorted. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS NP = the number of rows local to a given process. NQ = the number of columns local to a given process. .TP 8 N (global input) INTEGER The order of the matrix A. N >= 0. .TP 8 ZIN (local input) DOUBLE PRECISION array, dimension ( LDZI, NVS(iam) ) The eigenvectors on input. Each eigenvector resides entirely in one process. Each process holds a contiguous set of NVS(iam) eigenvectors. The first eigenvector which the process holds is: sum for i=[0,iam-1) of NVS(i) .TP 8 LDZI (locl input) INTEGER leading dimension of the ZIN array .TP 8 Z (local output) DOUBLE PRECISION array global dimension (N, N), local dimension (DESCZ(DLEN_), NQ) The eigenvectors on output. The eigenvectors are distributed in a block cyclic manner in both dimensions, with a block size of NB. .TP 8 IZ (global input) INTEGER Z's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JZ (global input) INTEGER Z's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCZ (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix Z. .TP 8 NVS (global input) INTEGER array, dimension( nprocs+1 ) nvs(i) = number of processes number of eigenvectors held by processes [0,i-1) nvs(1) = number of eigen vectors held by [0,1-1) == 0 nvs(nprocs+1) = number of eigen vectors held by [0,nprocs) == total number of eigenvectors .TP 8 KEY (global input) INTEGER array, dimension( N ) Indicates the actual index (after sorting) for each of the eigenvectors. .TP 8 WORK (local workspace) DOUBLE PRECISION array, dimension (LWORK) .TP 8 LWORK (local input) INTEGER dimension of WORK scalapack-doc-1.5/man/manl/pdlahqr.l0100644000056400000620000002470506335610630017061 0ustar pfrauenfstaff.TH PDLAHQR l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLAHQR - i an auxiliary routine used to find the Schur decomposition and or eigenvalues of a matrix already in Hessenberg form from cols ILO to IHI .SH SYNOPSIS .TP 20 SUBROUTINE PDLAHQR( WANTT, WANTZ, N, ILO, IHI, A, DESCA, WR, WI, ILOZ, IHIZ, Z, DESCZ, WORK, LWORK, IWORK, ILWORK, INFO ) .TP 20 .ti +4 LOGICAL WANTT, WANTZ .TP 20 .ti +4 INTEGER IHI, IHIZ, ILO, ILOZ, ILWORK, INFO, LWORK, N, ROTN .TP 20 .ti +4 INTEGER DESCA( * ), DESCZ( * ), IWORK( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), WI( * ), WORK( * ), WR( * ), Z( * ) .SH PURPOSE PDLAHQR is an auxiliary routine used to find the Schur decomposition and or eigenvalues of a matrix already in Hessenberg form from cols ILO to IHI. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 WANTT (global input) LOGICAL = .TRUE. : the full Schur form T is required; .br = .FALSE.: only eigenvalues are required. .TP 8 WANTZ (global input) LOGICAL .br = .TRUE. : the matrix of Schur vectors Z is required; .br = .FALSE.: Schur vectors are not required. .TP 8 N (global input) INTEGER The order of the Hessenberg matrix A (and Z if WANTZ). N >= 0. .TP 8 ILO (global input) INTEGER IHI (global input) INTEGER It is assumed that A is already upper quasi-triangular in rows and columns IHI+1:N, and that A(ILO,ILO-1) = 0 (unless ILO = 1). PDLAHQR works primarily with the Hessenberg submatrix in rows and columns ILO to IHI, but applies transformations to all of H if WANTT is .TRUE.. 1 <= ILO <= max(1,IHI); IHI <= N. .TP 8 A (global input/output) DOUBLE PRECISION array, dimension (DESCA(LLD_),*) On entry, the upper Hessenberg matrix A. On exit, if WANTT is .TRUE., A is upper quasi-triangular in rows and columns ILO:IHI, with any 2-by-2 or larger diagonal blocks not yet in standard form. If WANTT is .FALSE., the contents of A are unspecified on exit. .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 WR (global replicated output) DOUBLE PRECISION array, dimension (N) WI (global replicated 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 WANTT is .TRUE., the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in A. A may be returned with larger diagonal blocks until the next release. .TP 8 ILOZ (global input) INTEGER IHIZ (global input) INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= ILO; IHI <= IHIZ <= N. .TP 8 Z (global input/output) DOUBLE PRECISION array. If WANTZ is .TRUE., on entry Z must contain the current matrix Z of transformations accumulated by PDHSEQR, and on exit Z has been updated; transformations are applied only to the submatrix Z(ILOZ:IHIZ,ILO:IHI). If WANTZ is .FALSE., Z is not referenced. .TP 8 DESCZ (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix Z. .TP 8 WORK (local output) DOUBLE PRECISION array of size LWORK (Unless LWORK=-1, in which case WORK must be at least size 1) .TP 8 LWORK (local input) INTEGER WORK(LWORK) is a local array and LWORK is assumed big enough so that LWORK >= 3*N + MAX( 2*MAX(DESCZ(LLD_),DESCA(LLD_)) + 2*LOCc(N), 7*Ceil(N/HBL)/LCM(NPROW,NPCOL)) + MAX( 2*N, (8*LCM(NPROW,NPCOL)+2)**2 ) If LWORK=-1, then WORK(1) gets set to the above number and the code returns immediately. .TP 8 IWORK (global and local input) INTEGER array of size ILWORK This will hold some of the IBLK integer arrays. This is held as a place holder for a future release. Currently unreferenced. .TP 8 ILWORK (local input) INTEGER This will hold the size of the IWORK array. This is held as a place holder for a future release. Currently unreferenced. .TP 8 INFO (global output) INTEGER < 0: parameter number -INFO incorrect or inconsistent .br = 0: successful exit .br > 0: PDLAHQR failed to compute all the eigenvalues ILO to IHI in a total of 30*(IHI-ILO+1) iterations; if INFO = i, elements i+1:ihi of WR and WI contain those eigenvalues which have been successfully computed. Logic: This algorithm is very similar to _LAHQR. Unlike _LAHQR, instead of sending one double shift through the largest unreduced submatrix, this algorithm sends multiple double shifts and spaces them apart so that there can be parallelism across several processor row/columns. Another critical difference is that this algorithm aggregrates multiple transforms together in order to apply them in a block fashion. Important Local Variables: IBLK = The maximum number of bulges that can be computed. Currently fixed. Future releases this won't be fixed. HBL = The square block size (HBL=DESCA(MB_)=DESCA(NB_)) ROTN = The number of transforms to block together NBULGE = The number of bulges that will be attempted on the current submatrix. IBULGE = The current number of bulges started. K1(*),K2(*) = The current bulge loops from K1(*) to K2(*). Subroutines: From LAPACK, this routine calls: DLAHQR -> Serial QR used to determine shifts and eigenvalues DLARFG -> Determine the Householder transforms This ScaLAPACK, this routine calls: PDLACONSB -> To determine where to start each iteration DLAMSH -> Sends multiple shifts through a small submatrix to see how the consecutive subdiagonals change (if PDLACONSB indicates we can start a run in the middle) PDLAWIL -> Given the shift, get the transformation DLASORTE -> Pair up eigenvalues so that reals are paired. PDLACP3 -> Parallel array to local replicated array copy & back. DLAREF -> Row/column reflector applier. Core routine here. PDLASMSUB -> Finds negligible subdiagonal elements. Current Notes and/or Restrictions: 1.) This code requires the distributed block size to be square and at least six (6); unlike simpler codes like LU, this algorithm is extremely sensitive to block size. Unwise choices of too small a block size can lead to bad performance. 2.) This code requires A and Z to be distributed identically and have identical contxts. A future version may allow Z to have a different contxt to 1D row map it to all nodes (so no communication on Z is necessary.) 3.) This release currently does not have a routine for resolving the Schur blocks into regular 2x2 form after this code is completed. Because of this, a significant performance impact is required while the deflation is done by sometimes a single column of processors. 4.) This code does not currently block the initial transforms so that none of the rows or columns for any bulge are completed until all are started. To offset pipeline start-up it is recommended that at least 2*LCM(NPROW,NPCOL) bulges are used (if possible) 5.) The maximum number of bulges currently supported is fixed at 32. In future versions this will be limited only by the incoming WORK and IWORK array. 6.) The matrix A must be in upper Hessenberg form. If elements below the subdiagonal are nonzero, the resulting transforms may be nonsimilar. This is also true with the LAPACK routine DLAHQR. 7.) For this release, this code has only been tested for RSRC_=CSRC_=0, but it has been written for the general case. 8.) Currently, all the eigenvalues are distributed to all the nodes. Future releases will probably distribute the eigenvalues by the column partitioning. 9.) The internals of this routine are subject to change. 10.) To optimize this for your architecture, try tuning DLAREF. 11.) This code has only been tested for WANTZ = .TRUE. and may behave unpredictably for WANTZ set to .FALSE. Implemented by: G. Henry, May 1, 1997 scalapack-doc-1.5/man/manl/pdlahrd.l0100644000056400000620000001061006335610631017033 0ustar pfrauenfstaff.TH PDLAHRD l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLAHRD - reduce the first NB columns of a real general N-by-(N-K+1) distributed matrix A(IA:IA+N-1,JA:JA+N-K) so that elements below the k-th subdiagonal are zero .SH SYNOPSIS .TP 20 SUBROUTINE PDLAHRD( N, K, NB, A, IA, JA, DESCA, TAU, T, Y, IY, JY, DESCY, WORK ) .TP 20 .ti +4 INTEGER IA, IY, JA, JY, K, N, NB .TP 20 .ti +4 INTEGER DESCA( * ), DESCY( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), T( * ), TAU( * ), WORK( * ), Y( * ) .SH PURPOSE PDLAHRD reduces the first NB columns of a real general N-by-(N-K+1) distributed matrix A(IA:IA+N-1,JA:JA+N-K) so that elements below the k-th subdiagonal are zero. The reduction is performed by an orthogo- nal similarity transformation Q' * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T. .br This is an auxiliary routine called by PDGEHRD. In the following comments sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1). .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 K (global input) INTEGER The offset for the reduction. Elements below the k-th subdiagonal in the first NB columns are reduced to zero. .TP 8 NB (global input) INTEGER The number of columns to be reduced. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-K)). On entry, this array contains the the local pieces of the N-by-(N-K+1) general distributed matrix A(IA:IA+N-1,JA:JA+N-K). On exit, the elements on and above the k-th subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced distributed matrix; the elements below the k-th subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors. The other columns of A(IA:IA+N-1,JA:JA+N-K) are unchanged. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-2) The scalar factors of the elementary reflectors (see Further Details). TAU is tied to the distributed matrix A. .TP 8 T (local output) DOUBLE PRECISION array, dimension (NB_A,NB_A) The upper triangular matrix T. .TP 8 Y (local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_Y,NB_A). On exit, this array contains the local pieces of the N-by-NB distributed matrix Y. LLD_Y >= LOCr(IA+N-1). .TP 8 IY (global input) INTEGER The row index in the global array Y indicating the first row of sub( Y ). .TP 8 JY (global input) INTEGER The column index in the global array Y indicating the first column of sub( Y ). .TP 8 DESCY (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix Y. .TP 8 WORK (local workspace) DOUBLE PRECISION array, dimension (NB) .SH FURTHER DETAILS The matrix Q is represented as a product of nb elementary reflectors Q = H(1) H(2) . . . H(nb). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(ia+i+k:ia+n-1,ja+i-1), and tau in TAU(ja+i-1). .br The elements of the vectors v together form the (n-k+1)-by-nb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form: A(ia:ia+n-1,ja:ja+n-k) := (I-V*T*V')*(A(ia:ia+n-1,ja:ja+n-k)-Y*V'). The contents of A(ia:ia+n-1,ja:ja+n-k) on exit are illustrated by the following example with n = 7, k = 3 and nb = 2: .br ( a h a a a ) .br ( a h a a a ) .br ( a h a a a ) .br ( h h a a a ) .br ( v1 h a a a ) .br ( v1 v2 a a a ) .br ( v1 v2 a a a ) .br where a denotes an element of the original matrix .br A(ia:ia+n-1,ja:ja+n-k), h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i). .br scalapack-doc-1.5/man/manl/pdlamch.l0100644000056400000620000000252206335610631017030 0ustar pfrauenfstaff.TH PDLAMCH l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLAMCH - determine double precision machine parameters .SH SYNOPSIS .TP 17 DOUBLE PRECISION FUNCTION PDLAMCH( ICTXT, CMACH ) .TP 17 .ti +4 CHARACTER CMACH .TP 17 .ti +4 INTEGER ICTXT .SH PURPOSE PDLAMCH determines double precision machine parameters. .SH ARGUMENTS .TP 8 ICTXT (global input) INTEGER The BLACS context handle in which the computation takes place. .TP 8 CMACH (global input) CHARACTER*1 Specifies the value to be returned by PDLAMCH: .br = 'E' or 'e', PDLAMCH := eps .br = 'S' or 's , PDLAMCH := sfmin .br = 'B' or 'b', PDLAMCH := base .br = 'P' or 'p', PDLAMCH := eps*base .br = 'N' or 'n', PDLAMCH := t .br = 'R' or 'r', PDLAMCH := rnd .br = 'M' or 'm', PDLAMCH := emin .br = 'U' or 'u', PDLAMCH := rmin .br = 'L' or 'l', PDLAMCH := emax .br = 'O' or 'o', PDLAMCH := rmax where .TP 6 eps = relative machine precision sfmin = safe minimum, such that 1/sfmin does not overflow base = base of the machine prec = eps*base t = number of (base) digits in the mantissa rnd = 1.0 when rounding occurs in addition, 0.0 otherwise emin = minimum exponent before (gradual) underflow rmin = underflow threshold - base**(emin-1) emax = largest exponent before overflow rmax = overflow threshold - (base**emax)*(1-eps) scalapack-doc-1.5/man/manl/pdlange.l0100644000056400000620000001302706335610631017034 0ustar pfrauenfstaff.TH PDLANGE l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLANGE - return the value of the one norm, or the Frobenius norm, .SH SYNOPSIS .TP 17 DOUBLE PRECISION FUNCTION PDLANGE( NORM, M, N, A, IA, JA, DESCA, WORK ) .TP 17 .ti +4 CHARACTER NORM .TP 17 .ti +4 INTEGER IA, JA, M, N .TP 17 .ti +4 INTEGER DESCA( * ) .TP 17 .ti +4 DOUBLE PRECISION A( * ), WORK( * ) .SH PURPOSE PDLANGE returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a distributed matrix sub( A ) = A(IA:IA+M-1, JA:JA+N-1). .br PDLANGE returns the value .br ( max(abs(A(i,j))), NORM = 'M' or 'm' with IA <= i <= IA+M-1, ( and JA <= j <= JA+N-1, ( .br ( norm1( sub( A ) ), NORM = '1', 'O' or 'o' .br ( .br ( normI( sub( A ) ), NORM = 'I' or 'i' .br ( .br ( normF( sub( A ) ), NORM = 'F', 'f', 'E' or 'e' .br where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a matrix norm. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies the value to be returned in PDLANGE as described above. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). When M = 0, PDLANGE is set to zero. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). When N = 0, PDLANGE is set to zero. N >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)) containing the local pieces of the distributed matrix sub( A ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 WORK (local workspace) DOUBLE PRECISION array dimension (LWORK) LWORK >= 0 if NORM = 'M' or 'm' (not referenced), Nq0 if NORM = '1', 'O' or 'o', Mp0 if NORM = 'I' or 'i', 0 if NORM = 'F', 'f', 'E' or 'e' (not referenced), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. scalapack-doc-1.5/man/manl/pdlanhs.l0100644000056400000620000001253706335610631017060 0ustar pfrauenfstaff.TH PDLANHS l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLANHS - return the value of the one norm, or the Frobenius norm, .SH SYNOPSIS .TP 17 DOUBLE PRECISION FUNCTION PDLANHS( NORM, N, A, IA, JA, DESCA, WORK ) .TP 17 .ti +4 CHARACTER NORM .TP 17 .ti +4 INTEGER IA, JA, N .TP 17 .ti +4 INTEGER DESCA( * ) .TP 17 .ti +4 DOUBLE PRECISION A( * ), WORK( * ) .SH PURPOSE PDLANHS returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hessenberg distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). PDLANHS returns the value .br ( max(abs(A(i,j))), NORM = 'M' or 'm' with IA <= i <= IA+N-1, ( and JA <= j <= JA+N-1, ( .br ( norm1( sub( A ) ), NORM = '1', 'O' or 'o' .br ( .br ( normI( sub( A ) ), NORM = 'I' or 'i' .br ( .br ( normF( sub( A ) ), NORM = 'F', 'f', 'E' or 'e' .br where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a matrix norm. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies the value to be returned in PDLANHS as described above. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on i.e the number of rows and columns of the distributed submatrix sub( A ). When N = 0, PDLANHS is set to zero. N >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1) ) containing the local pieces of sub( A ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 WORK (local workspace) DOUBLE PRECISION array dimension (LWORK) LWORK >= 0 if NORM = 'M' or 'm' (not referenced), Nq0 if NORM = '1', 'O' or 'o', Mp0 if NORM = 'I' or 'i', 0 if NORM = 'F', 'f', 'E' or 'e' (not referenced), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Np0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. scalapack-doc-1.5/man/manl/pdlansy.l0100644000056400000620000001445506335610631017102 0ustar pfrauenfstaff.TH PDLANSY l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLANSY - return the value of the one norm, or the Frobenius norm, .SH SYNOPSIS .TP 17 DOUBLE PRECISION FUNCTION PDLANSY( NORM, UPLO, N, A, IA, JA, DESCA, WORK ) .TP 17 .ti +4 CHARACTER NORM, UPLO .TP 17 .ti +4 INTEGER IA, JA, N .TP 17 .ti +4 INTEGER DESCA( * ) .TP 17 .ti +4 DOUBLE PRECISION A( * ), WORK( * ) .SH PURPOSE PDLANSY returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). PDLANSY returns the value .br ( max(abs(A(i,j))), NORM = 'M' or 'm' with IA <= i <= IA+N-1, ( and JA <= j <= JA+N-1, ( .br ( norm1( sub( A ) ), NORM = '1', 'O' or 'o' .br ( .br ( normI( sub( A ) ), NORM = 'I' or 'i' .br ( .br ( normF( sub( A ) ), NORM = 'F', 'f', 'E' or 'e' .br where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a matrix norm. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies the value to be returned in PDLANSY as described above. .TP 8 UPLO (global input) CHARACTER Specifies whether the upper or lower triangular part of the symmetric matrix sub( A ) is to be referenced. = 'U': Upper triangular part of sub( A ) is referenced, .br = 'L': Lower triangular part of sub( A ) is referenced. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on i.e the number of rows and columns of the distributed submatrix sub( A ). When N = 0, PDLANSY is set to zero. N >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)) containing the local pieces of the symmetric distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular matrix which norm is to be computed, and the strictly lower triangular part of this matrix is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular matrix which norm is to be computed, and the strictly upper triangular part of sub( A ) is not referenced. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 WORK (local workspace) DOUBLE PRECISION array dimension (LWORK) LWORK >= 0 if NORM = 'M' or 'm' (not referenced), 2*Nq0+Np0+LDW if NORM = '1', 'O', 'o', 'I' or 'i', where LDW is given by: IF( NPROW.NE.NPCOL ) THEN LDW = MB_A*CEIL(CEIL(Np0/MB_A)/(LCM/NPROW)) ELSE LDW = 0 END IF 0 if NORM = 'F', 'f', 'E' or 'e' (not referenced), where LCM is the least common multiple of NPROW and NPCOL LCM = ILCM( NPROW, NPCOL ) and CEIL denotes the ceiling operation (ICEIL). IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Np0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), ICEIL, ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. scalapack-doc-1.5/man/manl/pdlantr.l0100644000056400000620000001370706335610631017073 0ustar pfrauenfstaff.TH PDLANTR l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLANTR - return the value of the one norm, or the Frobenius norm, .SH SYNOPSIS .TP 17 DOUBLE PRECISION FUNCTION PDLANTR( NORM, UPLO, DIAG, M, N, A, IA, JA, DESCA, WORK ) .TP 17 .ti +4 CHARACTER DIAG, NORM, UPLO .TP 17 .ti +4 INTEGER IA, JA, M, N .TP 17 .ti +4 INTEGER DESCA( * ) .TP 17 .ti +4 DOUBLE PRECISION A( * ), WORK( * ) .SH PURPOSE PDLANTR returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular distributed matrix sub( A ) denoting A(IA:IA+M-1, JA:JA+N-1). .br PDLANTR returns the value .br ( max(abs(A(i,j))), NORM = 'M' or 'm' with ia <= i <= ia+m-1, ( and ja <= j <= ja+n-1, ( .br ( norm1( sub( A ) ), NORM = '1', 'O' or 'o' .br ( .br ( normI( sub( A ) ), NORM = 'I' or 'i' .br ( .br ( normF( sub( A ) ), NORM = 'F', 'f', 'E' or 'e' .br where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a matrix norm. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies the value to be returned in PDLANTR as described above. .TP 8 UPLO (global input) CHARACTER Specifies whether the matrix sub( A ) is upper or lower trapezoidal. = 'U': Upper trapezoidal .br = 'L': Lower trapezoidal Note that sub( A ) is triangular instead of trapezoidal if M = N. .TP 8 DIAG (global input) CHARACTER Specifies whether or not the distributed matrix sub( A ) has unit diagonal. = 'N': Non-unit diagonal .br = 'U': Unit diagonal .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). When M = 0, PDLANTR is set to zero. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). When N = 0, PDLANTR is set to zero. N >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1) ) containing the local pieces of sub( A ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 WORK (local workspace) DOUBLE PRECISION array dimension (LWORK) LWORK >= 0 if NORM = 'M' or 'm' (not referenced), Nq0 if NORM = '1', 'O' or 'o', Mp0 if NORM = 'I' or 'i', 0 if NORM = 'F', 'f', 'E' or 'e' (not referenced), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. scalapack-doc-1.5/man/manl/pdlapiv.l0100644000056400000620000001552006335610631017061 0ustar pfrauenfstaff.TH PDLAPIV l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLAPIV - applie either P (permutation matrix indicated by IPIV) or inv( P ) to a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1), resulting in row or column pivoting .SH SYNOPSIS .TP 20 SUBROUTINE PDLAPIV( DIREC, ROWCOL, PIVROC, M, N, A, IA, JA, DESCA, IPIV, IP, JP, DESCIP, IWORK ) .TP 20 .ti +4 CHARACTER*1 DIREC, PIVROC, ROWCOL .TP 20 .ti +4 INTEGER IA, IP, JA, JP, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCIP( * ), IPIV( * ), IWORK( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ) .SH PURPOSE PDLAPIV applies either P (permutation matrix indicated by IPIV) or inv( P ) to a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1), resulting in row or column pivoting. The pivot vector may be distributed across a process row or a column. The pivot vector should be aligned with the distributed matrix A. This routine will transpose the pivot vector if necessary. For example if the row pivots should be applied to the columns of sub( A ), pass ROWCOL='C' and PIVROC='C'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Restrictions .br ============ .br IPIV must always be a distributed vector (not a matrix). Thus: IF( ROWPIV .EQ. 'C' ) THEN .br JP must be 1 .br ELSE .br IP must be 1 .br END IF .br The following restrictions apply when IPIV must be transposed: IF( ROWPIV.EQ.'C' .AND. PIVROC.EQ.'C') THEN .br DESCIP(MB_) must equal DESCA(NB_) .br ELSE IF( ROWPIV.EQ.'R" .AND. PIVROC.EQ.'R') THEN .br DESCIP(NB_) must equal DESCA(MB_) .br END IF .br .SH ARGUMENTS .TP 8 DIREC (global input) CHARACTER*1 Specifies in which order the permutation is applied: = 'F' (Forward) Applies pivots Forward from top of matrix. Computes P*sub( A ). = 'B' (Backward) Applies pivots Backward from bottom of matrix. Computes inv( P )*sub( A ). .TP 8 ROWCOL (global input) CHARACTER*1 Specifies if the rows or columns are to be permuted: = 'R' Rows will be permuted, = 'C' Columns will be permuted. .TP 8 PIVROC (global input) CHARACTER*1 Specifies whether IPIV is distributed over a process row or column: = 'R' IPIV distributed over a process row = 'C' IPIV distributed over a process column .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the distributed submatrix sub( A ) to which the row or column interchanges will be applied. On exit, the local pieces of the permuted distributed submatrix. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local input) INTEGER array, dimension >= LOCr(M_A)+MB_A if ROWCOL='R', otherwise LOCc(N_A)+NB_A. It contains the pivoting information. IPIV(i) is the global row (column), local row (column) i was swapped with. The last piece of the array of size MB_A (resp. NB_A) is used as workspace. This array is tied to the distributed matrix A. .TP 8 IWORK (local workspace) INTEGER array, dimension (LDW) where LDW is equal to the workspace necessary for transposition, and the storage of the tranposed IPIV: Let LCM be the least common multiple of NPROW and NPCOL. IF( ROWCOL.EQ.'R' .AND. PIVROC.EQ.'R' ) THEN IF( NPROW.EQ.NPCOL ) THEN LDW = LOCr( N_P + MOD(JP-1, NB_P) ) + NB_P ELSE LDW = LOCr( N_P + MOD(JP-1, NB_P) ) + NB_P * CEIL( CEIL(LOCc(N_P)/NB_P) / (LCM/NPCOL) ) END IF ELSE IF( ROWCOL.EQ.'C' .AND. PIVROC.EQ.'C' ) THEN IF( NPROW.EQ.NPCOL ) THEN LDW = LOCc( M_P + MOD(IP-1, MB_P) ) + MB_P ELSE LDW = LOCc( M_P + MOD(IP-1, MB_P) ) + MB_P * CEIL( CEIL(LOCr(M_P)/MB_P) / (LCM/NPROW) ) END IF ELSE IWORK is not referenced. END IF scalapack-doc-1.5/man/manl/pdlapv2.l0100644000056400000620000001361506335610631016775 0ustar pfrauenfstaff.TH PDLAPV2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLAPV2 - applie either P (permutation matrix indicated by IPIV) or inv( P ) to a M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1), resulting in row or column pivoting .SH SYNOPSIS .TP 20 SUBROUTINE PDLAPV2( DIREC, ROWCOL, M, N, A, IA, JA, DESCA, IPIV, IP, JP, DESCIP ) .TP 20 .ti +4 CHARACTER DIREC, ROWCOL .TP 20 .ti +4 INTEGER IA, IP, JA, JP, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCIP( * ), IPIV( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ) .SH PURPOSE PDLAPV2 applies either P (permutation matrix indicated by IPIV) or inv( P ) to a M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1), resulting in row or column pivoting. The pivot vector should be aligned with the distributed matrix A. For pivoting the rows of sub( A ), IPIV should be distributed along a process column and replicated over all process rows. Similarly, IPIV should be distributed along a process row and replicated over all process columns for column pivoting. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 DIREC (global input) CHARACTER Specifies in which order the permutation is applied: = 'F' (Forward) Applies pivots Forward from top of matrix. Computes P * sub( A ); = 'B' (Backward) Applies pivots Backward from bottom of matrix. Computes inv( P ) * sub( A ). .TP 8 ROWCOL (global input) CHARACTER Specifies if the rows or columns are to be permuted: = 'R' Rows will be permuted, = 'C' Columns will be permuted. .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this local array contains the local pieces of the distributed matrix sub( A ) to which the row or columns interchanges will be applied. On exit, this array contains the local pieces of the permuted distributed matrix. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (input) INTEGER array, dimension >= LOCr(M_A)+MB_A if ROWCOL = 'R', LOCc(N_A)+NB_A otherwise. It contains the pivoting information. IPIV(i) is the global row (column), local row (column) i was swapped with. The last piece of the array of size MB_A (resp. NB_A) is used as workspace. IPIV is tied to the distributed matrix A. .TP 8 IP (global input) INTEGER IPIV's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JP (global input) INTEGER IPIV's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCIP (global and local input) INTEGER array of dimension 8 The array descriptor for the distributed matrix IPIV. scalapack-doc-1.5/man/manl/pdlaqge.l0100644000056400000620000001377606335610631017052 0ustar pfrauenfstaff.TH PDLAQGE l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLAQGE - equilibrate a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) using the row and scaling factors in the vectors R and C .SH SYNOPSIS .TP 20 SUBROUTINE PDLAQGE( M, N, A, IA, JA, DESCA, R, C, ROWCND, COLCND, AMAX, EQUED ) .TP 20 .ti +4 CHARACTER EQUED .TP 20 .ti +4 INTEGER IA, JA, M, N .TP 20 .ti +4 DOUBLE PRECISION AMAX, COLCND, ROWCND .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), C( * ), R( * ) .SH PURPOSE PDLAQGE equilibrates a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) using the row and scaling factors in the vectors R and C. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)) containing on entry the M-by-N matrix sub( A ). On exit, the equilibrated distributed matrix. See EQUED for the form of the equilibrated distributed submatrix. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 R (local input) DOUBLE PRECISION array, dimension LOCr(M_A) The row scale factors for sub( A ). R is aligned with the distributed matrix A, and replicated across every process column. R is tied to the distributed matrix A. .TP 8 C (local input) DOUBLE PRECISION array, dimension LOCc(N_A) The column scale factors of sub( A ). C is aligned with the distributed matrix A, and replicated down every process row. C is tied to the distributed matrix A. .TP 8 ROWCND (global input) DOUBLE PRECISION The global ratio of the smallest R(i) to the largest R(i), IA <= i <= IA+M-1. .TP 8 COLCND (global input) DOUBLE PRECISION The global ratio of the smallest C(i) to the largest C(i), JA <= j <= JA+N-1. .TP 8 AMAX (global input) DOUBLE PRECISION Absolute value of largest distributed submatrix entry. .TP 8 EQUED (global output) CHARACTER Specifies the form of equilibration that was done. = 'N': No equilibration .br = 'R': Row equilibration, i.e., sub( A ) has been pre- .br multiplied by diag(R(IA:IA+M-1)), .br = 'C': Column equilibration, i.e., sub( A ) has been post- .br multiplied by diag(C(JA:JA+N-1)), .br = 'B': Both row and column equilibration, i.e., sub( A ) has been replaced by diag(R(IA:IA+M-1)) * sub( A ) * diag(C(JA:JA+N-1)). .SH PARAMETERS THRESH is a threshold value used to decide if row or column scaling should be done based on the ratio of the row or column scaling factors. If ROWCND < THRESH, row scaling is done, and if COLCND < THRESH, column scaling is done. LARGE and SMALL are threshold values used to decide if row scaling should be done based on the absolute size of the largest matrix element. If AMAX > LARGE or AMAX < SMALL, row scaling is done. scalapack-doc-1.5/man/manl/pdlaqsy.l0100644000056400000620000001413106335610631017074 0ustar pfrauenfstaff.TH PDLAQSY l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLAQSY - equilibrate a symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) using the scaling factors in the vectors SR and SC .SH SYNOPSIS .TP 20 SUBROUTINE PDLAQSY( UPLO, N, A, IA, JA, DESCA, SR, SC, SCOND, AMAX, EQUED ) .TP 20 .ti +4 CHARACTER EQUED, UPLO .TP 20 .ti +4 INTEGER IA, JA, N .TP 20 .ti +4 DOUBLE PRECISION AMAX, SCOND .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), SC( * ), SR( * ) .SH PURPOSE PDLAQSY equilibrates a symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) using the scaling factors in the vectors SR and SC. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the upper or lower triangular part of the symmetric distributed matrix sub( A ) is to be referenced: .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (input/output) DOUBLE PRECISION pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1)). On entry, the local pieces of the distributed symmetric matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and the strictly lower triangular part of sub( A ) is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and the strictly upper trian- gular part of sub( A ) is not referenced. On exit, if EQUED = 'Y', the equilibrated matrix: .br diag(SR(IA:IA+N-1)) * sub( A ) * diag(SC(JA:JA+N-1)). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 SR (local input) DOUBLE PRECISION array, dimension LOCr(M_A) The scale factors for A(IA:IA+M-1,JA:JA+N-1). SR is aligned with the distributed matrix A, and replicated across every process column. SR is tied to the distributed matrix A. .TP 8 SC (local input) DOUBLE PRECISION array, dimension LOCc(N_A) The scale factors for sub( A ). SC is aligned with the dis- tributed matrix A, and replicated down every process row. SC is tied to the distributed matrix A. .TP 8 SCOND (global input) DOUBLE PRECISION Ratio of the smallest SR(i) (respectively SC(j)) to the largest SR(i) (respectively SC(j)), with IA <= i <= IA+N-1 and JA <= j <= JA+N-1. .TP 8 AMAX (global input) DOUBLE PRECISION Absolute value of the largest distributed submatrix entry. .TP 8 EQUED (output) CHARACTER*1 Specifies whether or not equilibration was done. = 'N': No equilibration. .br = 'Y': Equilibration was done, i.e., sub( A ) has been re- .br placed by: .br diag(SR(IA:IA+N-1)) * sub( A ) * diag(SC(JA:JA+N-1)). .SH PARAMETERS THRESH is a threshold value used to decide if scaling should be done based on the ratio of the scaling factors. If SCOND < THRESH, scaling is done. LARGE and SMALL are threshold values used to decide if scaling should be done based on the absolute size of the largest matrix element. If AMAX > LARGE or AMAX < SMALL, scaling is done. scalapack-doc-1.5/man/manl/pdlared1d.l0100644000056400000620000001060506335610631017261 0ustar pfrauenfstaff.TH PDLARED1D l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDLARED1D - redistribute a 1D array It assumes that the input array, BYCOL, is distributed across rows and that all process column contain the same copy of BYCOL .SH SYNOPSIS .TP 22 SUBROUTINE PDLARED1D( N, IA, JA, DESC, BYCOL, BYALL, WORK, LWORK ) .TP 22 .ti +4 INTEGER IA, JA, LWORK, N .TP 22 .ti +4 INTEGER DESC( * ) .TP 22 .ti +4 DOUBLE PRECISION BYALL( * ), BYCOL( * ), WORK( LWORK ) .SH PURPOSE PDLARED1D redistributes a 1D array and will contain the entire array. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS NP = Number of local rows in BYCOL() .TP 8 N (global input) INTEGER The number of elements to be redistributed. N >= 0. .TP 8 IA (global input) INTEGER IA must be equal to 1 .TP 8 JA (global input) INTEGER JA must be equal to 1 .TP 8 DESC (global/local input) INTEGER Array of dimension 8 A 2D array descirptor, which describes BYCOL .TP 8 BYCOL (local input) distributed block cyclic DOUBLE PRECISION array global dimension (N), local dimension NP BYCOL is distributed across the process rows All process columns are assumed to contain the same value .TP 8 BYALL (global output) DOUBLE PRECISION global dimension( N ) local dimension (N) BYALL is exactly duplicated on all processes It contains the same values as BYCOL, but it is replicated across all processes rather than being distributed BYALL(i) = BYCOL( NUMROC(i,NB,MYROW,0,NPROW ) on the procs whose MYROW == mod((i-1)/NB,NPROW) .TP 8 WORK (local workspace) DOUBLE PRECISION dimension (LWORK) Used to hold the buffers sent from one process to another .TP 8 LWORK (local input) INTEGER size of WORK array LWORK >= NUMROC(N, DESC( NB_ ), 0, 0, NPCOL) scalapack-doc-1.5/man/manl/pdlared2d.l0100644000056400000620000001062306335610631017262 0ustar pfrauenfstaff.TH PDLARED2D l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDLARED2D - redistribute a 1D array It assumes that the input array, BYROW, is distributed across columns and that all process rows contain the same copy of BYROW .SH SYNOPSIS .TP 22 SUBROUTINE PDLARED2D( N, IA, JA, DESC, BYROW, BYALL, WORK, LWORK ) .TP 22 .ti +4 INTEGER IA, JA, LWORK, N .TP 22 .ti +4 INTEGER DESC( * ) .TP 22 .ti +4 DOUBLE PRECISION BYALL( * ), BYROW( * ), WORK( LWORK ) .SH PURPOSE PDLARED2D redistributes a 1D array and will contain the entire array. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS NP = Number of local rows in BYROW() .TP 8 N (global input) INTEGER The number of elements to be redistributed. N >= 0. .TP 8 IA (global input) INTEGER IA must be equal to 1 .TP 8 JA (global input) INTEGER JA must be equal to 1 .TP 8 DESC (global/local input) INTEGER Array of dimension DLEN_ A 2D array descriptor, which describes BYROW .TP 8 BYROW (local input) distributed block cyclic DOUBLE PRECISION array global dimension (N), local dimension NP BYCOL is distributed across the process columns All process rows are assumed to contain the same value .TP 8 BYALL (global output) DOUBLE PRECISION global dimension( N ) local dimension (N) BYALL is exactly duplicated on all processes It contains the same values as BYCOL, but it is replicated across all processes rather than being distributed BYALL(i) = BYCOL( NUMROC(i,NB,MYROW,0,NPROW ) on the procs whose MYROW == mod((i-1)/NB,NPROW) .TP 8 WORK (local workspace) DOUBLE PRECISION dimension (LWORK) Used to hold the buffers sent from one process to another .TP 8 LWORK (local input) INTEGER size of WORK array LWORK >= LWORK >= NUMROC(N, DESC( NB_ ), 0, 0, NPCOL) scalapack-doc-1.5/man/manl/pdlarf.l0100644000056400000620000002006706335610631016674 0ustar pfrauenfstaff.TH PDLARF l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLARF - applie a real elementary reflector Q (or Q**T) to a real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the left or the right .SH SYNOPSIS .TP 19 SUBROUTINE PDLARF( SIDE, M, N, V, IV, JV, DESCV, INCV, TAU, C, IC, JC, DESCC, WORK ) .TP 19 .ti +4 CHARACTER SIDE .TP 19 .ti +4 INTEGER IC, INCV, IV, JC, JV, M, N .TP 19 .ti +4 INTEGER DESCC( * ), DESCV( * ) .TP 19 .ti +4 DOUBLE PRECISION C( * ), TAU( * ), V( * ), WORK( * ) .SH PURPOSE PDLARF applies a real elementary reflector Q (or Q**T) to a real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the left or the right. Q is represented in the form Q = I - tau * v * v' .br where tau is a real scalar and v is a real vector. .br If tau = 0, then Q is taken to be the unit matrix. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. Restrictions .br ============ .br If SIDE = 'Left' and INCV = 1, then the row process having the first entry V(IV,JV) must also have the first row of sub( C ). Moreover, MOD(IV-1,MB_V) must be equal to MOD(IC-1,MB_C), if INCV=M_V, only the last equality must be satisfied. .br If SIDE = 'Right' and INCV = M_V then the column process having the first entry V(IV,JV) must also have the first column of sub( C ) and MOD(JV-1,NB_V) must be equal to MOD(JC-1,NB_C), if INCV = 1 only the last equality must be satisfied. .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': form Q * sub( C ), .br = 'R': form sub( C ) * Q, Q = Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 V (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_V,*) containing the local pieces of the distributed vectors V representing the Householder transformation Q, V(IV:IV+M-1,JV) if SIDE = 'L' and INCV = 1, .br V(IV,JV:JV+M-1) if SIDE = 'L' and INCV = M_V, .br V(IV:IV+N-1,JV) if SIDE = 'R' and INCV = 1, .br V(IV,JV:JV+N-1) if SIDE = 'R' and INCV = M_V, The vector v in the representation of Q. V is not used if TAU = 0. .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 INCV (global input) INTEGER The global increment for the elements of V. Only two values of INCV are supported in this version, namely 1 and M_V. INCV must not be zero. .TP 8 TAU (local input) DOUBLE PRECISION, array, dimension LOCc(JV) if INCV = 1, and LOCr(IV) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix V. .TP 8 C (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_C, LOCc(JC+N-1) ), containing the local pieces of sub( C ). On exit, sub( C ) is overwritten by the Q * sub( C ) if SIDE = 'L', or sub( C ) * Q if SIDE = 'R'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace) DOUBLE PRECISION array, dimension (LWORK) If INCV = 1, if SIDE = 'L', if IVCOL = ICCOL, LWORK >= NqC0 else LWORK >= MpC0 + MAX( 1, NqC0 ) end if else if SIDE = 'R', LWORK >= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC( N+ICOFFC,NB_V,0,0,NPCOL ),NB_V,0,0,LCMQ ) ) end if else if INCV = M_V, if SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_V,0,0,NPROW ),MB_V,0,0,LCMP ) ) else if SIDE = 'R', if IVROW = ICROW, LWORK >= MpC0 else LWORK >= NqC0 + MAX( 1, MpC0 ) end if end if end if where LCM is the least common multiple of NPROW and NPCOL and LCM = ILCM( NPROW, NPCOL ), LCMP = LCM / NPROW, LCMQ = LCM / NPCOL, IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. Alignment requirements ====================== The distributed submatrices V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: MB_V = NB_V, If INCV = 1, If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', ( MB_V.EQ.NB_A .AND. MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if INCV = M_V, If SIDE = 'Left', ( MB_V.EQ.NB_V .AND. MB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND. IVCOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/pdlarfb.l0100644000056400000620000001761306335610632017042 0ustar pfrauenfstaff.TH PDLARFB l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLARFB - applie a real block reflector Q or its transpose Q**T to a real distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PDLARFB( SIDE, TRANS, DIRECT, STOREV, M, N, K, V, IV, JV, DESCV, T, C, IC, JC, DESCC, WORK ) .TP 20 .ti +4 CHARACTER SIDE, TRANS, DIRECT, STOREV .TP 20 .ti +4 INTEGER IC, IV, JC, JV, K, M, N .TP 20 .ti +4 INTEGER DESCC( * ), DESCV( * ) .TP 20 .ti +4 DOUBLE PRECISION C( * ), T( * ), V( * ), WORK( * ) .SH PURPOSE PDLARFB applies a real block reflector Q or its transpose Q**T to a real distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) from the left or the right. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 DIRECT (global input) CHARACTER Indicates how Q is formed from a product of elementary reflectors = 'F': Q = H(1) H(2) . . . H(k) (Forward) .br = 'B': Q = H(k) . . . H(2) H(1) (Backward) .TP 8 STOREV (global input) CHARACTER Indicates how the vectors which define the elementary reflectors are stored: .br = 'C': Columnwise .br = 'R': Rowwise .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The order of the matrix T (= the number of elementary reflectors whose product defines the block reflector). .TP 8 V (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension ( LLD_V, LOCc(JV+K-1) ) if STOREV = 'C', ( LLD_V, LOCc(JV+M-1)) if STOREV = 'R' and SIDE = 'L', ( LLD_V, LOCc(JV+N-1) ) if STOREV = 'R' and SIDE = 'R'. It contains the local pieces of the distributed vectors V representing the Householder transformation. See further details. If STOREV = 'C' and SIDE = 'L', LLD_V >= MAX(1,LOCr(IV+M-1)); if STOREV = 'C' and SIDE = 'R', LLD_V >= MAX(1,LOCr(IV+N-1)); if STOREV = 'R', LLD_V >= LOCr(IV+K-1). .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 T (local input) DOUBLE PRECISION array, dimension MB_V by MB_V if STOREV = 'R' and NB_V by NB_V if STOREV = 'C'. The trian- gular matrix T in the representation of the block reflector. .TP 8 C (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the M-by-N distributed matrix sub( C ). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q or sub( C )*Q'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace) DOUBLE PRECISION array, dimension (LWORK) If STOREV = 'C', if SIDE = 'L', LWORK >= ( NqC0 + MpC0 ) * K else if SIDE = 'R', LWORK >= ( NqC0 + MAX( NpV0 + NUMROC( NUMROC( N+ICOFFC, NB_V, 0, 0, NPCOL ), NB_V, 0, 0, LCMQ ), MpC0 ) ) * K end if else if STOREV = 'R', if SIDE = 'L', LWORK >= ( MpC0 + MAX( MqV0 + NUMROC( NUMROC( M+IROFFC, MB_V, 0, 0, NPROW ), MB_V, 0, 0, LCMP ), NqC0 ) ) * K else if SIDE = 'R', LWORK >= ( MpC0 + NqC0 ) * K end if end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFV = MOD( IV-1, MB_V ), ICOFFV = MOD( JV-1, NB_V ), IVROW = INDXG2P( IV, MB_V, MYROW, RSRC_V, NPROW ), IVCOL = INDXG2P( JV, NB_V, MYCOL, CSRC_V, NPCOL ), MqV0 = NUMROC( M+ICOFFV, NB_V, MYCOL, IVCOL, NPCOL ), NpV0 = NUMROC( N+IROFFV, MB_V, MYROW, IVROW, NPROW ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NpC0 = NUMROC( N+ICOFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. Alignment requirements ====================== The distributed submatrices V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If STOREV = 'Columnwise' If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', ( MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if STOREV = 'Rowwise' If SIDE = 'Left', ( NB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND. IVCOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/pdlarfg.l0100644000056400000620000001247106335610632017044 0ustar pfrauenfstaff.TH PDLARFG l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLARFG - generate a real elementary reflector H of order n, such that H * sub( X ) = H * ( x(iax,jax) ) = ( alpha ), H' * H = I .SH SYNOPSIS .TP 20 SUBROUTINE PDLARFG( N, ALPHA, IAX, JAX, X, IX, JX, DESCX, INCX, TAU ) .TP 20 .ti +4 INTEGER IAX, INCX, IX, JAX, JX, N .TP 20 .ti +4 DOUBLE PRECISION ALPHA .TP 20 .ti +4 INTEGER DESCX( * ) .TP 20 .ti +4 DOUBLE PRECISION TAU( * ), X( * ) .SH PURPOSE PDLARFG generates a real elementary reflector H of order n, such that ( x ) ( 0 ) .br where alpha is a scalar, and sub( X ) is an (N-1)-element real distributed vector X(IX:IX+N-2,JX) if INCX = 1 and X(IX,JX:JX+N-2) if INCX = DESCX(M_). H is represented in the form .br H = I - tau * ( 1 ) * ( 1 v' ) , .br ( v ) .br where tau is a real scalar and v is a real (N-1)-element .br vector. .br If the elements of sub( X ) are all zero, then tau = 0 and H is taken to be the unit matrix. .br Otherwise 1 <= tau <= 2. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. .SH ARGUMENTS .TP 8 N (global input) INTEGER The global order of the elementary reflector. N >= 0. .TP 8 ALPHA (local output) DOUBLE PRECISION On exit, alpha is computed in the process scope having the vector sub( X ). .TP 8 IAX (global input) INTEGER The global row index in X of X(IAX,JAX). .TP 8 JAX (global input) INTEGER The global column index in X of X(IAX,JAX). .TP 8 X (local input/local output) DOUBLE PRECISION, pointer into the local memory to an array of dimension (LLD_X,*). This array contains the local pieces of the distributed vector sub( X ). Before entry, the incremented array sub( X ) must contain the vector x. On exit, it is overwritten with the vector v. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 INCX (global input) INTEGER The global increment for the elements of X. Only two values of INCX are supported in this version, namely 1 and M_X. INCX must not be zero. .TP 8 TAU (local output) DOUBLE PRECISION, array, dimension LOCc(JX) if INCX = 1, and LOCr(IX) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix X. scalapack-doc-1.5/man/manl/pdlarft.l0100644000056400000620000001513406335610632017060 0ustar pfrauenfstaff.TH PDLARFT l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLARFT - form the triangular factor T of a real block reflector H of order n, which is defined as a product of k elementary reflectors .SH SYNOPSIS .TP 20 SUBROUTINE PDLARFT( DIRECT, STOREV, N, K, V, IV, JV, DESCV, TAU, T, WORK ) .TP 20 .ti +4 CHARACTER DIRECT, STOREV .TP 20 .ti +4 INTEGER IV, JV, K, N .TP 20 .ti +4 INTEGER DESCV( * ) .TP 20 .ti +4 DOUBLE PRECISION TAU( * ), T( * ), V( * ), WORK( * ) .SH PURPOSE PDLARFT forms the triangular factor T of a real block reflector H of order n, which is defined as a product of k elementary reflectors. If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular; If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular. If STOREV = 'C', the vector which defines the elementary reflector H(i) is stored in the i-th column of the distributed matrix V, and H = I - V * T * V' .br If STOREV = 'R', the vector which defines the elementary reflector H(i) is stored in the i-th row of the distributed matrix V, and H = I - V' * T * V .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 DIRECT (global input) CHARACTER*1 Specifies the order in which the elementary reflectors are multiplied to form the block reflector: .br = 'F': H = H(1) H(2) . . . H(k) (Forward) .br = 'B': H = H(k) . . . H(2) H(1) (Backward) .TP 8 STOREV (global input) CHARACTER*1 Specifies how the vectors which define the elementary reflectors are stored (see also Further Details): .br = 'R': rowwise .TP 8 N (global input) INTEGER The order of the block reflector H. N >= 0. .TP 8 K (global input) INTEGER The order of the triangular factor T (= the number of elementary reflectors). 1 <= K <= MB_V (= NB_V). .TP 8 V (input/output) DOUBLE PRECISION pointer into the local memory to an array of local dimension (LOCr(IV+N-1),LOCc(JV+K-1)) if STOREV = 'C', and (LOCr(IV+K-1),LOCc(JV+N-1)) if STOREV = 'R'. The distributed matrix V contains the Householder vectors. See further details. .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 TAU (local input) DOUBLE PRECISION, array, dimension LOCr(IV+K-1) if INCV = M_V, and LOCc(JV+K-1) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix V. .TP 8 T (local output) DOUBLE PRECISION array, dimension (NB_V,NB_V) if STOREV = 'Col', and (MB_V,MB_V) otherwise. It contains the k-by-k triangular factor of the block reflector asso- ciated with V. If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is lower triangular. .TP 8 WORK (local workspace) DOUBLE PRECISION array, dimension (K*(K-1)/2) .SH FURTHER DETAILS The shape of the matrix V and the storage of the vectors which define the H(i) is best illustrated by the following example with n = 5 and k = 3. The elements equal to 1 are not stored; the corresponding array elements are modified but restored on exit. The rest of the array is not used. .br DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R': V( IV:IV+N-1, ( 1 ) V( IV:IV+K-1, ( 1 v1 v1 v1 v1 ) JV:JV+K-1 ) = ( v1 1 ) JV:JV+N-1 ) = ( 1 v2 v2 v2 ) ( v1 v2 1 ) ( 1 v3 v3 ) ( v1 v2 v3 ) .br ( v1 v2 v3 ) .br DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R': V( IV:IV+N-1, ( v1 v2 v3 ) V( IV:IV+K-1, ( v1 v1 1 ) JV:JV+K-1 ) = ( v1 v2 v3 ) JV:JV+N-1 ) = ( v2 v2 v2 1 ) ( 1 v2 v3 ) ( v3 v3 v3 v3 1 ) ( 1 v3 ) .br ( 1 ) .br scalapack-doc-1.5/man/manl/pdlarz.l0100644000056400000620000002050406335610632016715 0ustar pfrauenfstaff.TH PDLARZ l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLARZ - applie a real elementary reflector Q (or Q**T) to a real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the left or the right .SH SYNOPSIS .TP 19 SUBROUTINE PDLARZ( SIDE, M, N, L, V, IV, JV, DESCV, INCV, TAU, C, IC, JC, DESCC, WORK ) .TP 19 .ti +4 CHARACTER SIDE .TP 19 .ti +4 INTEGER IC, INCV, IV, JC, JV, L, M, N .TP 19 .ti +4 INTEGER DESCC( * ), DESCV( * ) .TP 19 .ti +4 DOUBLE PRECISION C( * ), TAU( * ), V( * ), WORK( * ) .SH PURPOSE PDLARZ applies a real elementary reflector Q (or Q**T) to a real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the left or the right. Q is represented in the form Q = I - tau * v * v' .br where tau is a real scalar and v is a real vector. .br If tau = 0, then Q is taken to be the unit matrix. .br Q is a product of k elementary reflectors as returned by PDTZRZF. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. Restrictions .br ============ .br If SIDE = 'Left' and INCV = 1, then the row process having the first entry V(IV,JV) must also own C(IC+M-L,JC:JC+N-1). Moreover, MOD(IV-1,MB_V) must be equal to MOD(IC+N-L-1,MB_C), if INCV=M_V, only the last equality must be satisfied. .br If SIDE = 'Right' and INCV = M_V then the column process having the first entry V(IV,JV) must also own C(IC:IC+M-1,JC+N-L) and MOD(JV-1,NB_V) must be equal to MOD(JC+N-L-1,NB_C), if INCV = 1 only the last equality must be satisfied. .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': form Q * sub( C ), .br = 'R': form sub( C ) * Q, Q = Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 L (global input) INTEGER The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. .TP 8 V (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_V,*) containing the local pieces of the distributed vectors V representing the Householder transformation Q, V(IV:IV+L-1,JV) if SIDE = 'L' and INCV = 1, .br V(IV,JV:JV+L-1) if SIDE = 'L' and INCV = M_V, .br V(IV:IV+L-1,JV) if SIDE = 'R' and INCV = 1, .br V(IV,JV:JV+L-1) if SIDE = 'R' and INCV = M_V, The vector v in the representation of Q. V is not used if TAU = 0. .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 INCV (global input) INTEGER The global increment for the elements of V. Only two values of INCV are supported in this version, namely 1 and M_V. INCV must not be zero. .TP 8 TAU (local input) DOUBLE PRECISION, array, dimension LOCc(JV) if INCV = 1, and LOCr(IV) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix V. .TP 8 C (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_C, LOCc(JC+N-1) ), containing the local pieces of sub( C ). On exit, sub( C ) is overwritten by the Q * sub( C ) if SIDE = 'L', or sub( C ) * Q if SIDE = 'R'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace) DOUBLE PRECISION array, dimension (LWORK) If INCV = 1, if SIDE = 'L', if IVCOL = ICCOL, LWORK >= NqC0 else LWORK >= MpC0 + MAX( 1, NqC0 ) end if else if SIDE = 'R', LWORK >= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC( N+ICOFFC,NB_V,0,0,NPCOL ),NB_V,0,0,LCMQ ) ) end if else if INCV = M_V, if SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_V,0,0,NPROW ),MB_V,0,0,LCMP ) ) else if SIDE = 'R', if IVROW = ICROW, LWORK >= MpC0 else LWORK >= NqC0 + MAX( 1, MpC0 ) end if end if end if where LCM is the least common multiple of NPROW and NPCOL and LCM = ILCM( NPROW, NPCOL ), LCMP = LCM / NPROW, LCMQ = LCM / NPCOL, IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. Alignment requirements ====================== The distributed submatrices V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: MB_V = NB_V, If INCV = 1, If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', ( MB_V.EQ.NB_A .AND. MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if INCV = M_V, If SIDE = 'Left', ( MB_V.EQ.NB_V .AND. MB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND. IVCOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/pdlarzb.l0100644000056400000620000002002606335610632017056 0ustar pfrauenfstaff.TH PDLARZB l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLARZB - applie a real block reflector Q or its transpose Q**T to a real distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PDLARZB( SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V, IV, JV, DESCV, T, C, IC, JC, DESCC, WORK ) .TP 20 .ti +4 CHARACTER DIRECT, SIDE, STOREV, TRANS .TP 20 .ti +4 INTEGER IC, IV, JC, JV, K, L, M, N .TP 20 .ti +4 INTEGER DESCC( * ), DESCV( * ) .TP 20 .ti +4 DOUBLE PRECISION C( * ), T( * ), V( * ), WORK( * ) .SH PURPOSE PDLARZB applies a real block reflector Q or its transpose Q**T to a real distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) from the left or the right. .br Q is a product of k elementary reflectors as returned by PDTZRZF. Currently, only STOREV = 'R' and DIRECT = 'B' are supported. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 DIRECT (global input) CHARACTER Indicates how H is formed from a product of elementary reflectors = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet) .br = 'B': H = H(k) . . . H(2) H(1) (Backward) .TP 8 STOREV (global input) CHARACTER Indicates how the vectors which define the elementary reflectors are stored: .br = 'C': Columnwise (not supported yet) .br = 'R': Rowwise .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The order of the matrix T (= the number of elementary reflectors whose product defines the block reflector). .TP 8 L (global input) INTEGER The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. .TP 8 V (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_V, LOCc(JV+M-1)) if SIDE = 'L', (LLD_V, LOCc(JV+N-1)) if SIDE = 'R'. It contains the local pieces of the distributed vectors V representing the Householder transformation as returned by PDTZRZF. LLD_V >= LOCr(IV+K-1). .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 T (local input) DOUBLE PRECISION array, dimension MB_V by MB_V The lower triangular matrix T in the representation of the block reflector. .TP 8 C (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the M-by-N distributed matrix sub( C ). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q or sub( C )*Q'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace) DOUBLE PRECISION array, dimension (LWORK) If STOREV = 'C', if SIDE = 'L', LWORK >= ( NqC0 + MpC0 ) * K else if SIDE = 'R', LWORK >= ( NqC0 + MAX( NpV0 + NUMROC( NUMROC( N+ICOFFC, NB_V, 0, 0, NPCOL ), NB_V, 0, 0, LCMQ ), MpC0 ) ) * K end if else if STOREV = 'R', if SIDE = 'L', LWORK >= ( MpC0 + MAX( MqV0 + NUMROC( NUMROC( M+IROFFC, MB_V, 0, 0, NPROW ), MB_V, 0, 0, LCMP ), NqC0 ) ) * K else if SIDE = 'R', LWORK >= ( MpC0 + NqC0 ) * K end if end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFV = MOD( IV-1, MB_V ), ICOFFV = MOD( JV-1, NB_V ), IVROW = INDXG2P( IV, MB_V, MYROW, RSRC_V, NPROW ), IVCOL = INDXG2P( JV, NB_V, MYCOL, CSRC_V, NPCOL ), MqV0 = NUMROC( M+ICOFFV, NB_V, MYCOL, IVCOL, NPCOL ), NpV0 = NUMROC( N+IROFFV, MB_V, MYROW, IVROW, NPROW ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NpC0 = NUMROC( N+ICOFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. Alignment requirements ====================== The distributed submatrices V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If STOREV = 'Columnwise' If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', ( MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if STOREV = 'Rowwise' If SIDE = 'Left', ( NB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND. IVCOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/pdlarzt.l0100644000056400000620000001572206335610632017107 0ustar pfrauenfstaff.TH PDLARZT l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLARZT - form the triangular factor T of a real block reflector H of order > n, which is defined as a product of k elementary reflectors as returned by PDTZRZF .SH SYNOPSIS .TP 20 SUBROUTINE PDLARZT( DIRECT, STOREV, N, K, V, IV, JV, DESCV, TAU, T, WORK ) .TP 20 .ti +4 CHARACTER DIRECT, STOREV .TP 20 .ti +4 INTEGER IV, JV, K, N .TP 20 .ti +4 INTEGER DESCV( * ) .TP 20 .ti +4 DOUBLE PRECISION TAU( * ), T( * ), V( * ), WORK( * ) .SH PURPOSE PDLARZT forms the triangular factor T of a real block reflector H of order > n, which is defined as a product of k elementary reflectors as returned by PDTZRZF. If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular; If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular. If STOREV = 'C', the vector which defines the elementary reflector H(i) is stored in the i-th column of the array V, and .br H = I - V * T * V' .br If STOREV = 'R', the vector which defines the elementary reflector H(i) is stored in the i-th row of the array V, and .br H = I - V' * T * V .br Currently, only STOREV = 'R' and DIRECT = 'B' are supported. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 DIRECT (global input) CHARACTER Specifies the order in which the elementary reflectors are multiplied to form the block reflector: .br = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet) .br = 'B': H = H(k) . . . H(2) H(1) (Backward) .TP 8 STOREV (global input) CHARACTER Specifies how the vectors which define the elementary reflectors are stored (see also Further Details): .br = 'R': rowwise .TP 8 N (global input) INTEGER The number of meaningful entries of the block reflector H. N >= 0. .TP 8 K (global input) INTEGER The order of the triangular factor T (= the number of elementary reflectors). 1 <= K <= MB_V (= NB_V). .TP 8 V (input/output) DOUBLE PRECISION pointer into the local memory to an array of local dimension (LOCr(IV+K-1),LOCc(JV+N-1)). The distributed matrix V contains the Householder vectors. See further details. .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 TAU (local input) DOUBLE PRECISION, array, dimension LOCr(IV+K-1) if INCV = M_V, and LOCc(JV+K-1) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix V. .TP 8 T (local output) DOUBLE PRECISION array, dimension (MB_V,MB_V) It contains the k-by-k triangular factor of the block reflector associated with V. T is lower triangular. .TP 8 WORK (local workspace) DOUBLE PRECISION array, dimension (K*(K-1)/2) .SH FURTHER DETAILS The shape of the matrix V and the storage of the vectors which define the H(i) is best illustrated by the following example with n = 5 and k = 3. The elements equal to 1 are not stored; the corresponding array elements are modified but restored on exit. The rest of the array is not used. .br DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R': ______V_____ .br ( v1 v2 v3 ) / \ ( v1 v2 v3 ) ( v1 v1 v1 v1 v1 . . . . 1 ) V = ( v1 v2 v3 ) ( v2 v2 v2 v2 v2 . . . 1 ) ( v1 v2 v3 ) ( v3 v3 v3 v3 v3 . . 1 ) ( v1 v2 v3 ) .br . . . .br . . . .br 1 . . .br 1 . .br 1 .br DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R': ______V_____ 1 / \ . 1 ( 1 . . . . v1 v1 v1 v1 v1 ) . . 1 ( . 1 . . . v2 v2 v2 v2 v2 ) . . . ( . . 1 . . v3 v3 v3 v3 v3 ) . . . .br ( v1 v2 v3 ) .br ( v1 v2 v3 ) .br V = ( v1 v2 v3 ) .br ( v1 v2 v3 ) .br ( v1 v2 v3 ) .br scalapack-doc-1.5/man/manl/pdlascl.l0100644000056400000620000001243306335610632017045 0ustar pfrauenfstaff.TH PDLASCL l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLASCL - multiplie the M-by-N real distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) by the real scalar CTO/CFROM .SH SYNOPSIS .TP 20 SUBROUTINE PDLASCL( TYPE, CFROM, CTO, M, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER TYPE .TP 20 .ti +4 INTEGER IA, INFO, JA, M, N .TP 20 .ti +4 DOUBLE PRECISION CFROM, CTO .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ) .SH PURPOSE PDLASCL multiplies the M-by-N real distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) 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 sub( A ) may be full, upper triangular, lower triangular or upper Hessenberg. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 TYPE (global input) CHARACTER TYPE indices the storage type of the input distributed matrix. = 'G': sub( A ) is a full matrix, .br = 'L': sub( A ) is a lower triangular matrix, .br = 'U': sub( A ) is an upper triangular matrix, .br = 'H': sub( A ) is an upper Hessenberg matrix. .TP 8 CFROM (global input) DOUBLE PRECISION CTO (global input) DOUBLE PRECISION The distributed matrix sub( 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. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). This array contains the local pieces of the distributed matrix sub( A ). On exit, this array contains the local pieces of the distributed matrix multiplied by CTO/CFROM. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pdlase2.l0100644000056400000620000001223406335610632016754 0ustar pfrauenfstaff.TH PDLASE2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLASE2 - initialize an M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) to BETA on the diagonal and ALPHA on the offdiagonals .SH SYNOPSIS .TP 20 SUBROUTINE PDLASE2( UPLO, M, N, ALPHA, BETA, A, IA, JA, DESCA ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, JA, M, N .TP 20 .ti +4 DOUBLE PRECISION ALPHA, BETA .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ) .SH PURPOSE PDLASE2 initializes an M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) to BETA on the diagonal and ALPHA on the offdiagonals. PDLASE2 requires that only dimension of the matrix operand is distributed. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies the part of the distributed matrix sub( A ) to be set: .br = 'U': Upper triangular part is set; the strictly lower triangular part of sub( A ) is not changed; = 'L': Lower triangular part is set; the strictly upper triangular part of sub( A ) is not changed; Otherwise: All of the matrix sub( A ) is set. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 ALPHA (global input) DOUBLE PRECISION The constant to which the offdiagonal elements are to be set. .TP 8 BETA (global input) DOUBLE PRECISION The constant to which the diagonal elements are to be set. .TP 8 A (local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). This array contains the local pieces of the distributed matrix sub( A ) to be set. On exit, the leading M-by-N submatrix sub( A ) is set as follows: if UPLO = 'U', A(IA+i-1,JA+j-1) = ALPHA, 1<=i<=j-1, 1<=j<=N, if UPLO = 'L', A(IA+i-1,JA+j-1) = ALPHA, j+1<=i<=M, 1<=j<=N, otherwise, A(IA+i-1,JA+j-1) = ALPHA, 1<=i<=M, 1<=j<=N, IA+i.NE.JA+j, and, for all UPLO, A(IA+i-1,JA+i-1) = BETA, 1<=i<=min(M,N). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. scalapack-doc-1.5/man/manl/pdpbsv.l0100644000056400000620000000142106335610635016717 0ustar pfrauenfstaff.TH PDPBSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDPBSV - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 19 SUBROUTINE PDPBSV( UPLO, N, BW, NRHS, A, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 CHARACTER UPLO .TP 19 .ti +4 INTEGER BW, IB, INFO, JA, LWORK, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 DOUBLE PRECISION A( * ), B( * ), WORK( * ) .SH PURPOSE PDPBSV solves a system of linear equations where A(1:N, JA:JA+N-1) is an N-by-N real .br banded symmetric positive definite distributed .br matrix with bandwidth BW. .br Cholesky factorization is used to factor a reordering of .br the matrix into L L'. .br See PDPBTRF and PDPBTRS for details. .br scalapack-doc-1.5/man/manl/pdlaset.l0100644000056400000620000001211406335610632017053 0ustar pfrauenfstaff.TH PDLASET l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLASET - initialize an M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) to BETA on the diagonal and ALPHA on the offdiagonals .SH SYNOPSIS .TP 20 SUBROUTINE PDLASET( UPLO, M, N, ALPHA, BETA, A, IA, JA, DESCA ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, JA, M, N .TP 20 .ti +4 DOUBLE PRECISION ALPHA, BETA .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ) .SH PURPOSE PDLASET initializes an M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) to BETA on the diagonal and ALPHA on the offdiagonals. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies the part of the distributed matrix sub( A ) to be set: .br = 'U': Upper triangular part is set; the strictly lower triangular part of sub( A ) is not changed; = 'L': Lower triangular part is set; the strictly upper triangular part of sub( A ) is not changed; Otherwise: All of the matrix sub( A ) is set. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 ALPHA (global input) DOUBLE PRECISION The constant to which the offdiagonal elements are to be set. .TP 8 BETA (global input) DOUBLE PRECISION The constant to which the diagonal elements are to be set. .TP 8 A (local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). This array contains the local pieces of the distributed matrix sub( A ) to be set. On exit, the leading M-by-N submatrix sub( A ) is set as follows: if UPLO = 'U', A(IA+i-1,JA+j-1) = ALPHA, 1<=i<=j-1, 1<=j<=N, if UPLO = 'L', A(IA+i-1,JA+j-1) = ALPHA, j+1<=i<=M, 1<=j<=N, otherwise, A(IA+i-1,JA+j-1) = ALPHA, 1<=i<=M, 1<=j<=N, IA+i.NE.JA+j, and, for all UPLO, A(IA+i-1,JA+i-1) = BETA, 1<=i<=min(M,N). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. scalapack-doc-1.5/man/manl/pdlasmsub.l0100644000056400000620000001153106335610632017413 0ustar pfrauenfstaff.TH PDLASMSUB l "12 May 1997" "LAPACK version 1.5 " "LAPACK routine (version 1.5 )" .SH NAME PDLASMSUB - look for a small subdiagonal element from the bottom of the matrix that it can safely set to zero .SH SYNOPSIS .TP 22 SUBROUTINE PDLASMSUB( A, DESCA, I, L, K, SMLNUM, BUF, LWORK ) .TP 22 .ti +4 INTEGER I, K, L, LWORK .TP 22 .ti +4 DOUBLE PRECISION SMLNUM .TP 22 .ti +4 INTEGER DESCA( * ) .TP 22 .ti +4 DOUBLE PRECISION A( * ), BUF( * ) .SH PURPOSE PDLASMSUB looks for a small subdiagonal element from the bottom of the matrix that it can safely set to zero. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 A (global input) DOUBLE PRECISION array, dimension (DESCA(LLD_),*) On entry, the Hessenberg matrix whose tridiagonal part is being scanned. Unchanged on exit. .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 I (global input) INTEGER The global location of the bottom of the unreduced submatrix of A. Unchanged on exit. .TP 8 L (global input) INTEGER The global location of the top of the unreduced submatrix of A. Unchanged on exit. .TP 8 K (global output) INTEGER On exit, this yields the bottom portion of the unreduced submatrix. This will satisfy: L <= M <= I-1. .TP 8 SMLNUM (global input) DOUBLE PRECISION On entry, a "small number" for the given matrix. Unchanged on exit. .TP 8 BUF (local output) DOUBLE PRECISION array of size LWORK. .TP 8 LWORK (global input) INTEGER On exit, LWORK is the size of the work buffer. This must be at least 2*Ceil( Ceil( (I-L)/HBL ) / LCM(NPROW,NPCOL) ) Here LCM is least common multiple, and NPROWxNPCOL is the logical grid size. Notes: This routine does a global maximum and must be called by all processes. This code is basically a parallelization of the following snip of LAPACK code from DLAHQR: Look for a single small subdiagonal element. DO 20 K = I, L + 1, -1 TST1 = ABS( H( K-1, K-1 ) ) + ABS( H( K, K ) ) IF( TST1.EQ.ZERO ) $ TST1 = DLANHS( '1', I-L+1, H( L, L ), LDH, WORK ) IF( ABS( H( K, K-1 ) ).LE.MAX( ULP*TST1, SMLNUM ) ) $ GO TO 30 20 CONTINUE 30 CONTINUE Implemented by: G. Henry, November 17, 1996 scalapack-doc-1.5/man/manl/pdlassq.l0100644000056400000620000001225506335610632017074 0ustar pfrauenfstaff.TH PDLASSQ l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLASSQ - return the values scl and smsq such that ( scl**2 )*smsq = x( 1 )**2 +...+ x( n )**2 + ( scale**2 )*sumsq, .SH SYNOPSIS .TP 20 SUBROUTINE PDLASSQ( N, X, IX, JX, DESCX, INCX, SCALE, SUMSQ ) .TP 20 .ti +4 INTEGER IX, INCX, JX, N .TP 20 .ti +4 DOUBLE PRECISION SCALE, SUMSQ .TP 20 .ti +4 INTEGER DESCX( * ) .TP 20 .ti +4 DOUBLE PRECISION X( * ) .SH PURPOSE PDLASSQ returns the values scl and smsq such that where x( i ) = sub( X ) = X( IX+(JX-1)*DESCX(M_)+(i-1)*INCX ). The value of sumsq is assumed to be non-negative and scl returns the value .br scl = max( scale, abs( x( i ) ) ). .br scale and sumsq must be supplied in SCALE and SUMSQ respectively. SCALE and SUMSQ are overwritten by scl and ssq respectively. The routine makes only one pass through the vector sub( X ). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. The result are only available in the scope of sub( X ), i.e if sub( X ) is distributed along a process row, the correct results are only available in this process row of the grid. Similarly if sub( X ) is distributed along a process column, the correct results are only available in this process column of the grid. .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The length of the distributed vector sub( X ). .TP 8 X (input) DOUBLE PRECISION The vector for which a scaled sum of squares is computed. x( i ) = X(IX+(JX-1)*M_X +(i-1)*INCX ), 1 <= i <= n. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 INCX (global input) INTEGER The global increment for the elements of X. Only two values of INCX are supported in this version, namely 1 and M_X. INCX must not be zero. .TP 8 SCALE (local input/local output) DOUBLE PRECISION On entry, the value scale in the equation above. On exit, SCALE is overwritten with scl , the scaling factor for the sum of squares. .TP 8 SUMSQ (local input/local output) DOUBLE PRECISION On entry, the value sumsq in the equation above. On exit, SUMSQ is overwritten with smsq , the basic sum of squares from which scl has been factored out. scalapack-doc-1.5/man/manl/pdlaswp.l0100644000056400000620000001227106335610633017076 0ustar pfrauenfstaff.TH PDLASWP l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLASWP - perform a series of row or column interchanges on the distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PDLASWP( DIREC, ROWCOL, N, A, IA, JA, DESCA, K1, K2, IPIV ) .TP 20 .ti +4 CHARACTER DIREC, ROWCOL .TP 20 .ti +4 INTEGER IA, JA, K1, K2, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ) .SH PURPOSE PDLASWP performs a series of row or column interchanges on the distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1). One interchange is initiated for each of rows or columns K1 trough K2 of sub( A ). This routine assumes that the pivoting information has already been broadcast along the process row or column. .br Also note that this routine will only work for K1-K2 being in the same MB (or NB) block. If you want to pivot a full matrix, use PDLAPIV. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 DIREC (global input) CHARACTER Specifies in which order the permutation is applied: = 'F' (Forward) = 'B' (Backward) .TP 8 ROWCOL (global input) CHARACTER Specifies if the rows or columns are permuted: = 'R' (Rows) = 'C' (Columns) .TP 8 N (global input) INTEGER If ROWCOL = 'R', the length of the rows of the distributed matrix A(*,JA:JA+N-1) to be permuted; If ROWCOL = 'C', the length of the columns of the distributed matrix A(IA:IA+N-1,*) to be permuted. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, * ). On entry, this array contains the local pieces of the distri- buted matrix to which the row/columns interchanges will be applied. On exit the permuted distributed matrix. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 K1 (global input) INTEGER The first element of IPIV for which a row or column inter- change will be done. .TP 8 K2 (global input) INTEGER The last element of IPIV for which a row or column inter- change will be done. .TP 8 IPIV (local input) INTEGER array, dimension LOCr(M_A)+MB_A for row pivoting and LOCc(N_A)+NB_A for column pivoting. This array is tied to the matrix A, IPIV(K) = L implies rows (or columns) K and L are to be interchanged. scalapack-doc-1.5/man/manl/pdlatra.l0100644000056400000620000000772006335610633017056 0ustar pfrauenfstaff.TH PDLATRA l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLATRA - compute the trace of an N-by-N distributed matrix sub( A ) denoting A( IA:IA+N-1, JA:JA+N-1 ) .SH SYNOPSIS .TP 17 DOUBLE PRECISION FUNCTION PDLATRA( N, A, IA, JA, DESCA ) .TP 17 .ti +4 INTEGER IA, JA, N .TP 17 .ti +4 INTEGER DESCA( * ) .TP 17 .ti +4 DOUBLE PRECISION A( * ) .SH PURPOSE PDLATRA computes the trace of an N-by-N distributed matrix sub( A ) denoting A( IA:IA+N-1, JA:JA+N-1 ). The result is left on every process of the grid. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on i.e the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension ( LLD_A, LOCc(JA+N-1) ). This array contains the local pieces of the distributed matrix the trace is to be computed. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. scalapack-doc-1.5/man/manl/pdlatrd.l0100644000056400000620000002123306335610633017054 0ustar pfrauenfstaff.TH PDLATRD l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLATRD - reduce NB rows and columns of a real symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) to symmetric tridiagonal form by an orthogonal similarity transformation Q' * sub( A ) * Q, .SH SYNOPSIS .TP 20 SUBROUTINE PDLATRD( UPLO, N, NB, A, IA, JA, DESCA, D, E, TAU, W, IW, JW, DESCW, WORK ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IW, JA, JW, N, NB .TP 20 .ti +4 INTEGER DESCA( * ), DESCW( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), D( * ), E( * ), TAU( * ), W( * ), WORK( * ) .SH PURPOSE PDLATRD reduces NB rows and columns of a real symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) to symmetric tridiagonal form by an orthogonal similarity transformation Q' * sub( A ) * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of sub( A ). .br If UPLO = 'U', PDLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; .br if UPLO = 'L', PDLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied. .br This is an auxiliary routine called by PDSYTRD. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the upper or lower triangular part of the symmetric matrix sub( A ) is stored: .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NB (global input) INTEGER The number of rows and columns to be reduced. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the symmetric distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced. On exit, if UPLO = 'U', the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of sub( A ); the elements above the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. If UPLO = 'L', the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of sub( A ); the elements below the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-1) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i). D is tied to the distributed matrix A. .TP 8 E (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-1) if UPLO = 'U', LOCc(JA+N-2) otherwise. The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. E is tied to the distributed matrix A. .TP 8 TAU (local output) DOUBLE PRECISION, array, dimension LOCc(JA+N-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 W (local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_W,NB_W), This array contains the local pieces of the N-by-NB_W matrix W required to update the unreduced part of sub( A ). .TP 8 IW (global input) INTEGER The row index in the global array W indicating the first row of sub( W ). .TP 8 JW (global input) INTEGER The column index in the global array W indicating the first column of sub( W ). .TP 8 DESCW (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix W. .TP 8 WORK (local workspace) DOUBLE PRECISION array, dimension (NB_A) .SH FURTHER DETAILS If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors .br Q = H(n) H(n-1) . . . H(n-nb+1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in .br A(ia:ia+i-2,ja+i), and tau in TAU(ja+i-1). .br If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors .br Q = H(1) H(2) . . . H(nb). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in .br A(ia+i+1:ia+n-1,ja+i-1), and tau in TAU(ja+i-1). .br The elements of the vectors v together form the N-by-NB matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a symmetric rank-2k update of the form: sub( A ) := sub( A ) - V*W' - W*V'. .br The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2: .br if UPLO = 'U': if UPLO = 'L': .br ( a a a v4 v5 ) ( d ) ( a a v4 v5 ) ( 1 d ) ( a 1 v5 ) ( v1 1 a ) ( d 1 ) ( v1 v2 a a ) ( d ) ( v1 v2 a a a ) where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i). .br scalapack-doc-1.5/man/manl/pdlatrs.l0100644000056400000620000000117306335610633017074 0ustar pfrauenfstaff.TH PDLATRS l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLATRS - solve a triangular system .SH SYNOPSIS .TP 20 SUBROUTINE PDLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, IA, JA, DESCA, X, IX, JX, DESCX, SCALE, CNORM, WORK ) .TP 20 .ti +4 CHARACTER DIAG, NORMIN, TRANS, UPLO .TP 20 .ti +4 INTEGER IA, IX, JA, JX, N .TP 20 .ti +4 DOUBLE PRECISION SCALE .TP 20 .ti +4 INTEGER DESCA( * ), DESCX( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), CNORM( * ), X( * ), WORK( * ) .SH PURPOSE PDLATRS solves a triangular system. This routine in unfinished at this time, but will be part of the next release. .br scalapack-doc-1.5/man/manl/pdlatrz.l0100644000056400000620000001466506335610633017115 0ustar pfrauenfstaff.TH PDLATRZ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDLATRZ - reduce the M-by-N ( M<=N ) real upper trapezoidal matrix sub( A ) = [ A(IA:IA+M-1,JA:JA+M-1) A(IA:IA+M-1,JA+N-L:JA+N-1) ] to upper triangular form by means of orthogonal transformations .SH SYNOPSIS .TP 20 SUBROUTINE PDLATRZ( M, N, L, A, IA, JA, DESCA, TAU, WORK ) .TP 20 .ti +4 INTEGER IA, JA, L, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), TAU( * ), WORK( * ) .SH PURPOSE PDLATRZ reduces the M-by-N ( M<=N ) real upper trapezoidal matrix sub( A ) = [ A(IA:IA+M-1,JA:JA+M-1) A(IA:IA+M-1,JA+N-L:JA+N-1) ] to upper triangular form by means of orthogonal transformations. The upper trapezoidal matrix sub( A ) is factored as .br sub( A ) = ( R 0 ) * Z, .br where Z is an N-by-N orthogonal matrix and R is an M-by-M upper triangular matrix. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 L (global input) INTEGER The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. L > 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the leading M-by-M upper triangular part of sub( A ) contains the upper trian- gular matrix R, and elements N-L+1 to N of the first M rows of sub( A ), with the array TAU, represent the orthogonal matrix Z as a product of M elementary reflectors. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) DOUBLE PRECISION, array, dimension LOCr(IA+M-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace) DOUBLE PRECISION array, dimension (LWORK) LWORK >= Nq0 + MAX( 1, Mp0 ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. .SH FURTHER DETAILS The factorization is obtained by Householder's method. The kth transformation matrix, Z( k ), which is used to introduce zeros into the (m - k + 1)th row of sub( A ), is given in the form .br Z( k ) = ( I 0 ), .br ( 0 T( k ) ) .br where .br T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ), ( 0 ) ( z( k ) ) tau is a scalar and z( k ) is an ( n - m ) element vector. tau and z( k ) are chosen to annihilate the elements of the kth row of sub( A ). .br The scalar tau is returned in the kth element of TAU and the vector u( k ) in the kth row of sub( A ), such that the elements of z( k ) are in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in the upper triangular part of sub( A ). .br Z is given by .br Z = Z( 1 ) * Z( 2 ) * ... * Z( m ). .br scalapack-doc-1.5/man/manl/pdlauu2.l0100644000056400000620000001162406335610633017001 0ustar pfrauenfstaff.TH PDLAUU2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLAUU2 - compute the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PDLAUU2( UPLO, N, A, IA, JA, DESCA ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ) .SH PURPOSE PDLAUU2 computes the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). If UPLO = 'U' or 'u' then the upper triangle of the result is stored, overwriting the factor U in sub( A ). .br If UPLO = 'L' or 'l' then the lower triangle of the result is stored, overwriting the factor L in sub( A ). .br This is the unblocked form of the algorithm, calling Level 2 BLAS. No communication is performed by this routine, the matrix to operate on should be strictly local to one process. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 Specifies whether the triangular factor stored in the matrix sub( A ) is upper or lower triangular: .br = 'U': Upper triangular, .br = 'L': Lower triangular. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the order of the triangular factor U or L. N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the triangular factor L or U. On exit, if UPLO = 'U', the upper triangle of the distributed matrix sub( A ) is overwritten with the upper triangle of the product U * U'; if UPLO = 'L', the lower triangle of sub( A ) is overwritten with the lower triangle of the product L' * L. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. scalapack-doc-1.5/man/manl/pdlauum.l0100644000056400000620000001146206335610633017074 0ustar pfrauenfstaff.TH PDLAUUM l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDLAUUM - compute the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PDLAUUM( UPLO, N, A, IA, JA, DESCA ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ) .SH PURPOSE PDLAUUM computes the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). If UPLO = 'U' or 'u' then the upper triangle of the result is stored, overwriting the factor U in sub( A ). .br If UPLO = 'L' or 'l' then the lower triangle of the result is stored, overwriting the factor L in sub( A ). .br This is the blocked form of the algorithm, calling Level 3 PBLAS. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 Specifies whether the triangular factor stored in the distributed matrix sub( A ) is upper or lower triangular: .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the triangular factor U or L. N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the triangular factor L or U. On exit, if UPLO = 'U', the upper triangle of the distributed matrix sub( A ) is overwritten with the upper triangle of the product U * U'; if UPLO = 'L', the lower triangle of sub( A ) is overwritten with the lower triangle of the product L' * L. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. scalapack-doc-1.5/man/manl/pdlawil.l0100644000056400000620000000774206335610633017067 0ustar pfrauenfstaff.TH PDLAWIL l "12 May 1997" "LAPACK version 1.5 " "LAPACK routine (version 1.5 )" .SH NAME PDLAWIL - get the transform given by H44,H33, & H43H34 into V starting at row M .SH SYNOPSIS .TP 20 SUBROUTINE PDLAWIL( II, JJ, M, A, DESCA, H44, H33, H43H34, V ) .TP 20 .ti +4 INTEGER II, JJ, M .TP 20 .ti +4 DOUBLE PRECISION H33, H43H34, H44 .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), V( * ) .SH PURPOSE PDLAWIL gets the transform given by H44,H33, & H43H34 into V starting at row M. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 II (global input) INTEGER Row owner of H(M+2,M+2) .TP 8 JJ (global input) INTEGER Column owner of H(M+2,M+2) .TP 8 M (global input) INTEGER On entry, this is where the transform starts (row M.) Unchanged on exit. .TP 8 A (global input) DOUBLE PRECISION array, dimension (DESCA(LLD_),*) On entry, the Hessenberg matrix. Unchanged on exit. .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. Unchanged on exit. H44 H33 H43H34 (global input) DOUBLE PRECISION These three values are for the double shift QR iteration. Unchanged on exit. .TP 8 V (global output) DOUBLE PRECISION array of size 3. Contains the transform on ouput. Implemented by: G. Henry, November 17, 1996 scalapack-doc-1.5/man/manl/pdorg2l.l0100644000056400000620000001407506335610633017001 0ustar pfrauenfstaff.TH PDORG2L l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDORG2L - generate an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the last N columns of a product of K elementary reflectors of order M Q = H(k) .SH SYNOPSIS .TP 20 SUBROUTINE PDORG2L( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), TAU( * ), WORK( * ) .SH PURPOSE PDORG2L generates an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the last N columns of a product of K elementary reflectors of order M as returned by PDGEQLF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. M >= N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the j-th column must contain the vector which defines the elementary reflector H(j), JA+N-K <= j <= JA+N-1, as returned by PDGEQLF in the K columns of its distributed matrix argument A(IA:*,JA+N-K:JA+N-1). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) DOUBLE PRECISION, array, dimension LOCc(JA+N-1) This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PDGEQLF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MpA0 + MAX( 1, NqA0 ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pdorg2r.l0100644000056400000620000001406006335610633017001 0ustar pfrauenfstaff.TH PDORG2R l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDORG2R - generate an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M Q = H(1) H(2) .SH SYNOPSIS .TP 20 SUBROUTINE PDORG2R( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), TAU( * ), WORK( * ) .SH PURPOSE PDORG2R generates an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M as returned by PDGEQRF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. M >= N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the j-th column must contain the vector which defines the elementary reflector H(j), JA <= j <= JA+K-1, as returned by PDGEQRF in the K columns of its array argument A(IA:*,JA:JA+K-1). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) DOUBLE PRECISION, array, dimension LOCc(JA+K-1). This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PDGEQRF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MpA0 + MAX( 1, NqA0 ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pdorgl2.l0100644000056400000620000001404606335610633016777 0ustar pfrauenfstaff.TH PDORGL2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDORGL2 - generate an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N Q = H(k) .SH SYNOPSIS .TP 20 SUBROUTINE PDORGL2( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), TAU( * ), WORK( * ) .SH PURPOSE PDORGL2 generates an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N as returned by PDGELQF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. N >= M >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PDGELQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) DOUBLE PRECISION, array, dimension LOCr(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PDGELQF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NqA0 + MAX( 1, MpA0 ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pdorglq.l0100644000056400000620000001405706335610633017100 0ustar pfrauenfstaff.TH PDORGLQ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDORGLQ - generate an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N Q = H(k) .SH SYNOPSIS .TP 20 SUBROUTINE PDORGLQ( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), TAU( * ), WORK( * ) .SH PURPOSE PDORGLQ generates an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N as returned by PDGELQF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. N >= M >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PDGELQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) DOUBLE PRECISION, array, dimension LOCr(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PDGELQF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MB_A * ( MpA0 + NqA0 + MB_A ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pdorgql.l0100644000056400000620000001410606335610634017074 0ustar pfrauenfstaff.TH PDORGQL l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDORGQL - generate an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the last N columns of a product of K elementary reflectors of order M Q = H(k) .SH SYNOPSIS .TP 20 SUBROUTINE PDORGQL( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), TAU( * ), WORK( * ) .SH PURPOSE PDORGQL generates an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the last N columns of a product of K elementary reflectors of order M as returned by PDGEQLF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. M >= N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the j-th column must contain the vector which defines the elementary reflector H(j), JA+N-K <= j <= JA+N-1, as returned by PDGEQLF in the K columns of its distributed matrix argument A(IA:*,JA+N-K:JA+N-1). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) DOUBLE PRECISION, array, dimension LOCc(JA+N-1) This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PDGEQLF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB_A * ( NqA0 + MpA0 + NB_A ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pdorgqr.l0100644000056400000620000001410506335610634017101 0ustar pfrauenfstaff.TH PDORGQR l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDORGQR - generate an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M Q = H(1) H(2) .SH SYNOPSIS .TP 20 SUBROUTINE PDORGQR( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), TAU( * ), WORK( * ) .SH PURPOSE PDORGQR generates an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M as returned by PDGEQRF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. M >= N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the j-th column must contain the vector which defines the elementary reflector H(j), JA <= j <= JA+K-1, as returned by PDGEQRF in the K columns of its distributed matrix argument A(IA:*,JA:JA+K-1). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) DOUBLE PRECISION, array, dimension LOCc(JA+K-1) This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PDGEQRF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB_A * ( NqA0 + MpA0 + NB_A ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pdorgr2.l0100644000056400000620000001406006335610634017002 0ustar pfrauenfstaff.TH PDORGR2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDORGR2 - generate an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N Q = H(1) H(2) .SH SYNOPSIS .TP 20 SUBROUTINE PDORGR2( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), TAU( * ), WORK( * ) .SH PURPOSE PDORGR2 generates an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N as returned by PDGERQF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. N >= M >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA+M-K <= i <= IA+M-1, as returned by PDGERQF in the K rows of its distributed matrix argument A(IA+M-K:IA+M-1,JA:*). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) DOUBLE PRECISION, array, dimension LOCr(IA+M-1) This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PDGERQF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NqA0 + MAX( 1, MpA0 ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pdorgrq.l0100644000056400000620000001407106335610634017103 0ustar pfrauenfstaff.TH PDORGRQ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDORGRQ - generate an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N Q = H(1) H(2) .SH SYNOPSIS .TP 20 SUBROUTINE PDORGRQ( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), TAU( * ), WORK( * ) .SH PURPOSE PDORGRQ generates an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N as returned by PDGERQF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. N >= M >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA+M-K <= i <= IA+M-1, as returned by PDGERQF in the K rows of its distributed matrix argument A(IA+M-K:IA+M-1,JA:*). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) DOUBLE PRECISION, array, dimension LOCr(IA+M-1) This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PDGERQF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MB_A * ( MpA0 + NqA0 + MB_A ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pdorm2l.l0100644000056400000620000001732106335610634017005 0ustar pfrauenfstaff.TH PDORM2L l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDORM2L - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PDORM2L( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PDORM2L overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix defined as the product of K elementary reflectors .br Q = H(k) . . . H(2) H(1) .br as returned by PDGEQLF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+K-1)). On entry, the j-th column must contain the vector which defines the elemen- tary reflector H(j), JA <= j <= JA+K-1, as returned by PDGEQLF in the K columns of its distributed matrix argument A(IA:*,JA:JA+K-1). A(IA:*,JA:JA+K-1) is modified by the routine but restored on exit. If SIDE = 'L', LLD_A >= MAX( 1, LOCr(IA+M-1) ), if SIDE = 'R', LLD_A >= MAX( 1, LOCr(IA+N-1) ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) DOUBLE PRECISION, array, dimension LOCc(JA+N-1) This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PDGEQLF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MpC0 + MAX( 1, NqC0 ); if SIDE = 'R', LWORK >= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC( N+ICOFFC,NB_A,0,0,NPCOL ),NB_A,0,0,LCMQ ) ); where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/pdorm2r.l0100644000056400000620000001732206335610634017014 0ustar pfrauenfstaff.TH PDORM2R l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDORM2R - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PDORM2R( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PDORM2R overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix defined as the product of k elementary reflectors .br Q = H(1) H(2) . . . H(k) .br as returned by PDGEQRF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+K-1)). On entry, the j-th column must contain the vector which defines the elemen- tary reflector H(j), JA <= j <= JA+K-1, as returned by PDGEQRF in the K columns of its distributed matrix argument A(IA:*,JA:JA+K-1). A(IA:*,JA:JA+K-1) is modified by the routine but restored on exit. If SIDE = 'L', LLD_A >= MAX( 1, LOCr(IA+M-1) ); if SIDE = 'R', LLD_A >= MAX( 1, LOCr(IA+N-1) ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) DOUBLE PRECISION, array, dimension LOCc(JA+K-1). This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PDGEQRF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MpC0 + MAX( 1, NqC0 ); if SIDE = 'R', LWORK >= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC( N+ICOFFC,NB_A,0,0,NPCOL ),NB_A,0,0,LCMQ ) ); where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/pdormbr.l0100644000056400000620000002405606335610634017076 0ustar pfrauenfstaff.TH PDORMBR l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDORMBR - VECT = 'Q', PDORMBR overwrites the general real distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PDORMBR( VECT, SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS, VECT .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE If VECT = 'Q', PDORMBR overwrites the general real distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br If VECT = 'P', PDORMBR overwrites sub( C ) with .br SIDE = 'L' SIDE = 'R' .br TRANS = 'N': P * sub( C ) sub( C ) * P .br TRANS = 'T': P**T * sub( C ) sub( C ) * P**T .br Here Q and P**T are the orthogonal distributed matrices determined by PDGEBRD when reducing a real distributed matrix A(IA:*,JA:*) to bidiagonal form: A(IA:*,JA:*) = Q * B * P**T. Q and P**T are defined as products of elementary reflectors H(i) and G(i) respectively. Let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the order of the orthogonal matrix Q or P**T that is applied. If VECT = 'Q', A(IA:*,JA:*) is assumed to have been an NQ-by-K matrix: .br if nq >= k, Q = H(1) H(2) . . . H(k); .br if nq < k, Q = H(1) H(2) . . . H(nq-1). .br If VECT = 'P', A(IA:*,JA:*) is assumed to have been a K-by-NQ matrix: .br if k < nq, P = G(1) G(2) . . . G(k); .br if k >= nq, P = G(1) G(2) . . . G(nq-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 VECT (global input) CHARACTER = 'Q': apply Q or Q**T; .br = 'P': apply P or P**T. .TP 8 SIDE (global input) CHARACTER .br = 'L': apply Q, Q**T, P or P**T from the Left; .br = 'R': apply Q, Q**T, P or P**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q or P; .br = 'T': Transpose, apply Q**T or P**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER If VECT = 'Q', the number of columns in the original distributed matrix reduced by PDGEBRD. If VECT = 'P', the number of rows in the original distributed matrix reduced by PDGEBRD. K >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+MIN(NQ,K)-1)) if VECT='Q', and (LLD_A,LOCc(JA+NQ-1)) if VECT = 'P'. NQ = M if SIDE = 'L', and NQ = N otherwise. The vectors which define the elementary reflectors H(i) and G(i), whose products determine the matrices Q and P, as returned by PDGEBRD. If VECT = 'Q', LLD_A >= max(1,LOCr(IA+NQ-1)); if VECT = 'P', LLD_A >= max(1,LOCr(IA+MIN(NQ,K)-1)). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) DOUBLE PRECISION array, dimension LOCc(JA+MIN(NQ,K)-1) if VECT = 'Q', LOCr(IA+MIN(NQ,K)-1) if VECT = 'P', TAU(i) must contain the scalar factor of the elementary reflector H(i) or G(i), which determines Q or P, as returned by PDGEBRD in its array argument TAUQ or TAUP. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, if VECT='Q', sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q; if VECT='P, sub( C ) is overwritten by P*sub( C ) or P'*sub( C ) or sub( C )*P or sub( C )*P'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', NQ = M; if( (VECT = 'Q' and NQ >= K) or (VECT <> 'Q' and NQ > K) ), IAA=IA; JAA=JA; MI=M; NI=N; ICC=IC; JCC=JC; else IAA=IA+1; JAA=JA; MI=M-1; NI=N; ICC=IC+1; JCC=JC; end if else if SIDE = 'R', NQ = N; if( (VECT = 'Q' and NQ >= K) or (VECT <> 'Q' and NQ > K) ), IAA=IA; JAA=JA; MI=M; NI=N; ICC=IC; JCC=JC; else IAA=IA; JAA=JA+1; MI=M; NI=N-1; ICC=IC; JCC=JC+1; end if end if If VECT = 'Q', If SIDE = 'L', LWORK >= MAX( (NB_A*(NB_A-1))/2, (NqC0 + MpC0)*NB_A ) + NB_A * NB_A else if SIDE = 'R', LWORK >= MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX( NpA0 + NUMROC( NUMROC( NI+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0 ) )*NB_A ) + NB_A * NB_A end if else if VECT <> 'Q', if SIDE = 'L', LWORK >= MAX( (MB_A*(MB_A-1))/2, ( MpC0 + MAX( MqA0 + NUMROC( NUMROC( MI+IROFFC, MB_A, 0, 0, NPROW ), MB_A, 0, 0, LCMP ), NqC0 ) )*MB_A ) + MB_A * MB_A else if SIDE = 'R', LWORK >= MAX( (MB_A*(MB_A-1))/2, (MpC0 + NqC0)*MB_A ) + MB_A * MB_A end if end if where LCMP = LCM / NPROW, LCMQ = LCM / NPCOL, with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IAA-1, MB_A ), ICOFFA = MOD( JAA-1, NB_A ), IAROW = INDXG2P( IAA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JAA, NB_A, MYCOL, CSRC_A, NPCOL ), MqA0 = NUMROC( MI+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), NpA0 = NUMROC( NI+IROFFA, MB_A, MYROW, IAROW, NPROW ), IROFFC = MOD( ICC-1, MB_C ), ICOFFC = MOD( JCC-1, NB_C ), ICROW = INDXG2P( ICC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JCC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( MI+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( NI+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If VECT = 'Q', If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) else If SIDE = 'L', ( MB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/pdormhr.l0100644000056400000620000002020206335610634017071 0ustar pfrauenfstaff.TH PDORMHR l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDORMHR - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PDORMHR( SIDE, TRANS, M, N, ILO, IHI, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, IHI, ILO, INFO, JA, JC, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PDORMHR overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix of order nq, with nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of IHI-ILO elementary reflectors, as returned by PDGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 ILO (global input) INTEGER IHI (global input) INTEGER ILO and IHI must have the same values as in the previous call of PDGEHRD. Q is equal to the unit matrix except in the distributed submatrix Q(ia+ilo:ia+ihi-1,ia+ilo:ja+ihi-1). If SIDE = 'L', 1 <= ILO <= IHI <= max(1,M); if SIDE = 'R', 1 <= ILO <= IHI <= max(1,N); ILO and IHI are relative indexes. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE = 'R'. The vectors which define the elementary reflectors, as returned by PDGEHRD. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) DOUBLE PRECISION, array, dimension LOCc(JA+M-2) if SIDE = 'L', and LOCc(JA+N-2) if SIDE = 'R'. This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PDGEHRD. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least IAA = IA + ILO; JAA = JA+ILO-1; If SIDE = 'L', MI = IHI-ILO; NI = N; ICC = IC + ILO; JCC = JC; LWORK >= MAX( (NB_A*(NB_A-1))/2, (NqC0 + MpC0)*NB_A ) + NB_A * NB_A else if SIDE = 'R', MI = M; NI = IHI-ILO; ICC = IC; JCC = JC + ILO; LWORK >= MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX( NpA0 + NUMROC( NUMROC( NI+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0 ) )*NB_A ) + NB_A * NB_A end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IAA-1, MB_A ), ICOFFA = MOD( JAA-1, NB_A ), IAROW = INDXG2P( IAA, MB_A, MYROW, RSRC_A, NPROW ), NpA0 = NUMROC( NI+IROFFA, MB_A, MYROW, IAROW, NPROW ), IROFFC = MOD( ICC-1, MB_C ), ICOFFC = MOD( JCC-1, NB_C ), ICROW = INDXG2P( ICC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JCC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( MI+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( NI+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/pdorml2.l0100644000056400000620000001730306335610634017005 0ustar pfrauenfstaff.TH PDORML2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDORML2 - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PDORML2( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PDORML2 overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix defined as the product of K elementary reflectors .br Q = H(k) . . . H(2) H(1) .br as returned by PDGELQF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= max(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PDGELQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) DOUBLE PRECISION, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PDGELQF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_A,0,0,NPROW ),MB_A,0,0,LCMP ) ); if SIDE = 'R', LWORK >= NqC0 + MAX( 1, MpC0 ); where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/pdormlq.l0100644000056400000620000001771306335610634017111 0ustar pfrauenfstaff.TH PDORMLQ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDORMLQ - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PDORMLQ( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PDORMLQ overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix defined as the product of K elementary reflectors .br Q = H(k) . . . H(2) H(1) .br as returned by PDGELQF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= max(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PDGELQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) DOUBLE PRECISION, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PDGELQF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least if SIDE = 'L', LWORK >= MAX( (MB_A*(MB_A-1))/2, ( MpC0 + MAX( MqA0 + NUMROC( NUMROC( M+IROFFC, MB_A, 0, 0, NPROW ), MB_A, 0, 0, LCMP ), NqC0 ) )*MB_A ) + MB_A * MB_A else if SIDE = 'R', LWORK >= MAX( (MB_A*(MB_A-1))/2, (MpC0 + NqC0)*MB_A ) + MB_A * MB_A end if where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MqA0 = NUMROC( M+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/pdormql.l0100644000056400000620000001773106335610634017111 0ustar pfrauenfstaff.TH PDORMQL l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDORMQL - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PDORMQL( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PDORMQL overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix defined as the product of K elementary reflectors .br Q = H(k) . . . H(2) H(1) .br as returned by PDGEQLF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+K-1)). On entry, the j-th column must contain the vector which defines the elemen- tary reflector H(j), JA <= j <= JA+K-1, as returned by PDGEQLF in the K columns of its distributed matrix argument A(IA:*,JA:JA+K-1). A(IA:*,JA:JA+K-1) is modified by the routine but restored on exit. If SIDE = 'L', LLD_A >= MAX( 1, LOCr(IA+M-1) ), if SIDE = 'R', LLD_A >= MAX( 1, LOCr(IA+N-1) ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) DOUBLE PRECISION, array, dimension LOCc(JA+N-1) This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PDGEQLF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MAX( (NB_A*(NB_A-1))/2, (NqC0 + MpC0)*NB_A ) + NB_A * NB_A else if SIDE = 'R', LWORK >= MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX( NpA0 + NUMROC( NUMROC( N+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0 ) )*NB_A ) + NB_A * NB_A end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), NpA0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/pdormqr.l0100644000056400000620000001773206335610634017120 0ustar pfrauenfstaff.TH PDORMQR l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDORMQR - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PDORMQR( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PDORMQR overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix defined as the product of k elementary reflectors .br Q = H(1) H(2) . . . H(k) .br as returned by PDGEQRF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+K-1)). On entry, the j-th column must contain the vector which defines the elemen- tary reflector H(j), JA <= j <= JA+K-1, as returned by PDGEQRF in the K columns of its distributed matrix argument A(IA:*,JA:JA+K-1). A(IA:*,JA:JA+K-1) is modified by the routine but restored on exit. If SIDE = 'L', LLD_A >= MAX( 1, LOCr(IA+M-1) ); if SIDE = 'R', LLD_A >= MAX( 1, LOCr(IA+N-1) ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) DOUBLE PRECISION, array, dimension LOCc(JA+K-1). This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PDGEQRF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MAX( (NB_A*(NB_A-1))/2, (NqC0 + MpC0)*NB_A ) + NB_A * NB_A else if SIDE = 'R', LWORK >= MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX( NpA0 + NUMROC( NUMROC( N+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0 ) )*NB_A ) + NB_A * NB_A end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), NpA0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/pdormr2.l0100644000056400000620000001730306335610635017014 0ustar pfrauenfstaff.TH PDORMR2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDORMR2 - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PDORMR2( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PDORMR2 overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix defined as the product of K elementary reflectors .br Q = H(1) H(2) . . . H(k) .br as returned by PDGERQF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= MAX(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PDGERQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) DOUBLE PRECISION, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PDGERQF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_A,0,0,NPROW ),MB_A,0,0,LCMP ) ); if SIDE = 'R', LWORK >= NqC0 + MAX( 1, MpC0 ); where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/pdormr3.l0100644000056400000620000001762606335610635017025 0ustar pfrauenfstaff.TH PDORMR3 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDORMR3 - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PDORMR3( SIDE, TRANS, M, N, K, L, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, L, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PDORMR3 overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix defined as the product of K elementary reflectors .br Q = H(1) H(2) . . . H(k) .br as returned by PDTZRZF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 L (global input) INTEGER The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= MAX(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PDTZRZF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) DOUBLE PRECISION, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PDTZRZF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_A,0,0,NPROW ),MB_A,0,0,LCMP ) ); if SIDE = 'R', LWORK >= NqC0 + MAX( 1, MpC0 ); where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/pdormrq.l0100644000056400000620000001771406335610635017121 0ustar pfrauenfstaff.TH PDORMRQ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDORMRQ - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PDORMRQ( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PDORMRQ overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix defined as the product of K elementary reflectors .br Q = H(1) H(2) . . . H(k) .br as returned by PDGERQF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= MAX(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PDGERQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) DOUBLE PRECISION, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PDGERQF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least if SIDE = 'L', LWORK >= MAX( (MB_A*(MB_A-1))/2, ( MpC0 + MAX( MqA0 + NUMROC( NUMROC( M+IROFFC, MB_A, 0, 0, NPROW ), MB_A, 0, 0, LCMP ), NqC0 ) )*MB_A ) + MB_A * MB_A else if SIDE = 'R', LWORK >= MAX( (MB_A*(MB_A-1))/2, (MpC0 + NqC0)*MB_A ) + MB_A * MB_A end if where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MqA0 = NUMROC( M+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/pdormrz.l0100644000056400000620000002023706335610635017124 0ustar pfrauenfstaff.TH PDORMRZ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDORMRZ - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PDORMRZ( SIDE, TRANS, M, N, K, L, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, L, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PDORMRZ overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix defined as the product of K elementary reflectors .br Q = H(1) H(2) . . . H(k) .br as returned by PDTZRZF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 L (global input) INTEGER The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= MAX(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PDTZRZF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) DOUBLE PRECISION, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PDTZRZF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least if SIDE = 'L', LWORK >= MAX( (MB_A*(MB_A-1))/2, ( MpC0 + MAX( MqA0 + NUMROC( NUMROC( M+IROFFC, MB_A, 0, 0, NPROW ), MB_A, 0, 0, LCMP ), NqC0 ) )*MB_A ) + MB_A * MB_A else if SIDE = 'R', LWORK >= MAX( (MB_A*(MB_A-1))/2, (MpC0 + NqC0)*MB_A ) + MB_A * MB_A end if where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MqA0 = NUMROC( M+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/pdormtr.l0100644000056400000620000002051106335610635017111 0ustar pfrauenfstaff.TH PDORMTR l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDORMTR - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PDORMTR( SIDE, UPLO, TRANS, M, N, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS, UPLO .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PDORMTR overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix of order nq, with nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of nq-1 elementary reflectors, as returned by PDSYTRD: if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1); .br if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 UPLO (global input) CHARACTER .br = 'U': Upper triangle of A(IA:*,JA:*) contains elementary reflectors from PDSYTRD; = 'L': Lower triangle of A(IA:*,JA:*) contains elementary reflectors from PDSYTRD. .TP 8 TRANS (global input) CHARACTER = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', or (LLD_A,LOCc(JA+N-1)) if SIDE = 'R'. The vectors which define the elementary reflectors, as returned by PDSYTRD. If SIDE = 'L', LLD_A >= max(1,LOCr(IA+M-1)); if SIDE = 'R', LLD_A >= max(1,LOCr(IA+N-1)). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) DOUBLE PRECISION array, dimension LTAU, where if SIDE = 'L' and UPLO = 'U', LTAU = LOCc(M_A), if SIDE = 'L' and UPLO = 'L', LTAU = LOCc(JA+M-2), if SIDE = 'R' and UPLO = 'U', LTAU = LOCc(N_A), if SIDE = 'R' and UPLO = 'L', LTAU = LOCc(JA+N-2). TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by PDSYTRD. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If UPLO = 'U', IAA = IA, JAA = JA+1, ICC = IC, JCC = JC; else UPLO = 'L', IAA = IA+1, JAA = JA; if SIDE = 'L', ICC = IC+1; JCC = JC; else ICC = IC; JCC = JC+1; end if end if If SIDE = 'L', MI = M-1; NI = N; LWORK >= MAX( (NB_A*(NB_A-1))/2, (NqC0 + MpC0)*NB_A ) + NB_A * NB_A else if SIDE = 'R', MI = M; MI = N-1; LWORK >= MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX( NpA0 + NUMROC( NUMROC( NI+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0 ) )*NB_A ) + NB_A * NB_A end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IAA-1, MB_A ), ICOFFA = MOD( JAA-1, NB_A ), IAROW = INDXG2P( IAA, MB_A, MYROW, RSRC_A, NPROW ), NpA0 = NUMROC( NI+IROFFA, MB_A, MYROW, IAROW, NPROW ), IROFFC = MOD( ICC-1, MB_C ), ICOFFC = MOD( JCC-1, NB_C ), ICROW = INDXG2P( ICC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JCC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( MI+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( NI+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/pdpbtrf.l0100644000056400000620000000232006335610635017061 0ustar pfrauenfstaff.TH PDPBTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDPBTRF - compute a Cholesky factorization of an N-by-N real banded symmetric positive definite distributed matrix with bandwidth BW .SH SYNOPSIS .TP 20 SUBROUTINE PDPBTRF( UPLO, N, BW, A, JA, DESCA, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER BW, INFO, JA, LAF, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), AF( * ), WORK( * ) .SH PURPOSE PDPBTRF computes a Cholesky factorization of an N-by-N real banded symmetric positive definite distributed matrix with bandwidth BW: A(1:N, JA:JA+N-1). Reordering is used to increase parallelism in the factorization. This reordering results in factors that are DIFFERENT from those produced by equivalent sequential codes. These factors cannot be used directly by users; however, they can be used in .br subsequent calls to PDPBTRS to solve linear systems. .br The factorization has the form .br P A(1:N, JA:JA+N-1) P^T = U' U , if UPLO = 'U', or P A(1:N, JA:JA+N-1) P^T = L L', if UPLO = 'L' .br where U is a banded upper triangular matrix and L is banded lower triangular, and P is a permutation matrix. .br scalapack-doc-1.5/man/manl/pdpbtrs.l0100644000056400000620000000170306335610635017102 0ustar pfrauenfstaff.TH PDPBTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDPBTRS - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 20 SUBROUTINE PDPBTRS( UPLO, N, BW, NRHS, A, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER BW, IB, INFO, JA, LAF, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), AF( * ), B( * ), WORK( * ) .SH PURPOSE PDPBTRS solves a system of linear equations where A(1:N, JA:JA+N-1) is the matrix used to produce the factors stored in A(1:N,JA:JA+N-1) and AF by PDPBTRF. .br A(1:N, JA:JA+N-1) is an N-by-N real .br banded symmetric positive definite distributed .br matrix with bandwidth BW. .br Depending on the value of UPLO, A stores either U or L in the equn A(1:N, JA:JA+N-1) = U'*U or L*L' as computed by PDPBTRF. .br Routine PDPBTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pdpbtrsv.l0100644000056400000620000000217506335610635017274 0ustar pfrauenfstaff.TH PDPBTRSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDPBTRSV - solve a banded triangular system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 21 SUBROUTINE PDPBTRSV( UPLO, TRANS, N, BW, NRHS, A, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 21 .ti +4 CHARACTER TRANS, UPLO .TP 21 .ti +4 INTEGER BW, IB, INFO, JA, LAF, LWORK, N, NRHS .TP 21 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 21 .ti +4 DOUBLE PRECISION A( * ), AF( * ), B( * ), WORK( * ) .SH PURPOSE PDPBTRSV solves a banded triangular system of linear equations or .br A(1:N, JA:JA+N-1)^T * X = B(IB:IB+N-1, 1:NRHS) where A(1:N, JA:JA+N-1) is a banded .br triangular matrix factor produced by the .br Cholesky factorization code PDPBTRF .br and is stored in A(1:N,JA:JA+N-1) and AF. .br The matrix stored in A(1:N, JA:JA+N-1) is either .br upper or lower triangular according to UPLO, .br and the choice of solving A(1:N, JA:JA+N-1) or A(1:N, JA:JA+N-1)^T is dictated by the user by the parameter TRANS. .br Routine PDPBTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pdpocon.l0100644000056400000620000001511106335610635017064 0ustar pfrauenfstaff.TH PDPOCON l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDPOCON - estimate the reciprocal of the condition number (in the 1-norm) of a real symmetric positive definite distributed matrix using the Cholesky factorization A = U**T*U or A = L*L**T computed by PDPOTRF .SH SYNOPSIS .TP 20 SUBROUTINE PDPOCON( UPLO, N, A, IA, JA, DESCA, ANORM, RCOND, WORK, LWORK, IWORK, LIWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, LIWORK, LWORK, N .TP 20 .ti +4 DOUBLE PRECISION ANORM, RCOND .TP 20 .ti +4 INTEGER DESCA( * ), IWORK( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), WORK( * ) .SH PURPOSE PDPOCON estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric positive definite distributed matrix using the Cholesky factorization A = U**T*U or A = L*L**T computed by PDPOTRF. An estimate is obtained for norm(inv(A(IA:IA+N-1,JA:JA+N-1))), and the reciprocal of the condition number is computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the factor stored in A(IA:IA+N-1,JA:JA+N-1) is upper or lower triangular. .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER .br The order of the distributed matrix A(IA:IA+N-1,JA:JA+N-1). N >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension ( LLD_A, LOCc(JA+N-1) ). On entry, this array contains the local pieces of the factors L or U from the Cholesky factorization A(IA:IA+N-1,JA:JA+N-1) = U'*U or L*L', as computed by PDPOTRF. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 ANORM (global input) DOUBLE PRECISION The 1-norm (or infinity-norm) of the symmetric distributed matrix A(IA:IA+N-1,JA:JA+N-1). .TP 8 RCOND (global output) DOUBLE PRECISION The reciprocal of the condition number of the distributed matrix A(IA:IA+N-1,JA:JA+N-1), computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * .br norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 2*LOCr(N+MOD(IA-1,MB_A)) + 2*LOCc(N+MOD(JA-1,NB_A)) + MAX( 2, MAX(NB_A*CEIL(NPROW-1,NPCOL),LOCc(N+MOD(JA-1,NB_A)) + NB_A*CEIL(NPCOL-1,NPROW)) ). If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/local output) INTEGER array, dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK. .TP 8 LIWORK (local or global input) INTEGER The dimension of the array IWORK. LIWORK is local input and must be at least LIWORK >= LOCr(N+MOD(IA-1,MB_A)). If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pdpoequ.l0100644000056400000620000001411706335610635017104 0ustar pfrauenfstaff.TH PDPOEQU l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDPOEQU - compute row and column scalings intended to equilibrate a distributed symmetric positive definite matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) and reduce its condition number (with respect to the two-norm) .SH SYNOPSIS .TP 20 SUBROUTINE PDPOEQU( N, A, IA, JA, DESCA, SR, SC, SCOND, AMAX, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 DOUBLE PRECISION AMAX, SCOND .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), SC( * ), SR( * ) .SH PURPOSE PDPOEQU computes row and column scalings intended to equilibrate a distributed symmetric positive definite matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) and reduce its condition number (with respect to the two-norm). SR and SC contain the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled distri- buted matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of SR and SC puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings. .br The scaling factor are stored along process rows in SR and along process columns in SC. The duplication of information simplifies greatly the application of the factors. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on i.e the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of local dimension ( LLD_A, LOCc(JA+N-1) ), the N-by-N symmetric positive definite distributed matrix sub( A ) whose scaling factors are to be computed. Only the diagonal elements of sub( A ) are referenced. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 SR (local output) DOUBLE PRECISION array, dimension LOCr(M_A) If INFO = 0, SR(IA:IA+N-1) contains the row scale factors for sub( A ). SR is aligned with the distributed matrix A, and replicated across every process column. SR is tied to the distributed matrix A. .TP 8 SC (local output) DOUBLE PRECISION array, dimension LOCc(N_A) If INFO = 0, SC(JA:JA+N-1) contains the column scale factors .br for A(IA:IA+M-1,JA:JA+N-1). SC is aligned with the distribu- ted matrix A, and replicated down every process row. SC is tied to the distributed matrix A. .TP 8 SCOND (global output) DOUBLE PRECISION If INFO = 0, SCOND contains the ratio of the smallest SR(i) (or SC(j)) to the largest SR(i) (or SC(j)), with IA <= i <= IA+N-1 and JA <= j <= JA+N-1. If SCOND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by SR (or SC). .TP 8 AMAX (global output) DOUBLE PRECISION Absolute value of largest matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, the K-th diagonal entry of sub( A ) is nonpositive. scalapack-doc-1.5/man/manl/pdporfs.l0100644000056400000620000002365306335610635017111 0ustar pfrauenfstaff.TH PDPORFS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDPORFS - improve the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite and provides error bounds and backward error estimates for the solutions .SH SYNOPSIS .TP 20 SUBROUTINE PDPORFS( UPLO, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, B, IB, JB, DESCB, X, IX, JX, DESCX, FERR, BERR, WORK, LWORK, IWORK, LIWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IAF, IB, INFO, IX, JA, JAF, JB, JX, LIWORK, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * ), IWORK( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), AF( * ), B( * ), BERR( * ), FERR( * ), WORK( * ), X( * ) .SH PURPOSE PDPORFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite and provides error bounds and backward error estimates for the solutions. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br In the following comments, sub( A ), sub( X ) and sub( B ) denote respectively A(IA:IA+N-1,JA:JA+N-1), X(IX:IX+N-1,JX:JX+NRHS-1) and B(IB:IB+N-1,JB:JB+NRHS-1). .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix sub( A ) is stored. = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The order of the matrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the matrices sub( B ) and sub( X ). NRHS >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1) ). This array contains the local pieces of the N-by-N symmetric distributed matrix sub( A ) to be factored. If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the distribu- ted matrix, and its strictly upper triangular part is not referenced. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 AF (local input) DOUBLE PRECISION pointer into the local memory to an array of local dimension (LLD_AF,LOCc(JA+N-1)). On entry, this array contains the factors L or U from the Cholesky factorization sub( A ) = L*L**T or U**T*U, as computed by PDPOTRF. .TP 8 IAF (global input) INTEGER The row index in the global array AF indicating the first row of sub( AF ). .TP 8 JAF (global input) INTEGER The column index in the global array AF indicating the first column of sub( AF ). .TP 8 DESCAF (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix AF. .TP 8 B (local input) DOUBLE PRECISION pointer into the local memory to an array of local dimension (LLD_B, LOCc(JB+NRHS-1) ). On entry, this array contains the the local pieces of the right hand sides sub( B ). .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 X (local input) DOUBLE PRECISION pointer into the local memory to an array of local dimension (LLD_X, LOCc(JX+NRHS-1) ). On entry, this array contains the the local pieces of the solution vectors sub( X ). On exit, it contains the improved solution vectors. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 FERR (local output) DOUBLE PRECISION array of local dimension LOCc(JB+NRHS-1). The estimated forward error bound for each solution vector of sub( X ). If XTRUE is the true solution corresponding to sub( X ), FERR is an estimated upper bound for the magnitude of the largest element in (sub( X ) - XTRUE) divided by the magnitude of the largest element in sub( X ). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. This array is tied to the distributed matrix X. .TP 8 BERR (local output) DOUBLE PRECISION array of local dimension LOCc(JB+NRHS-1). The componentwise relative backward error of each solution vector (i.e., the smallest re- lative change in any entry of sub( A ) or sub( B ) that makes sub( X ) an exact solution). This array is tied to the distributed matrix X. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 3*LOCr( N + MOD( IA-1, MB_A ) ) If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/local output) INTEGER array, dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK. .TP 8 LIWORK (local or global input) INTEGER The dimension of the array IWORK. LIWORK is local input and must be at least LIWORK >= LOCr( N + MOD( IB-1, MB_B ) ). If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH PARAMETERS ITMAX is the maximum number of steps of iterative refinement. Notes ===== This routine temporarily returns when N <= 1. The distributed submatrices op( A ) and op( AF ) (respectively sub( X ) and sub( B ) ) should be distributed the same way on the same processes. These conditions ensure that sub( A ) and sub( AF ) (resp. sub( X ) and sub( B ) ) are "perfectly" aligned. Moreover, this routine requires the distributed submatrices sub( A ), sub( AF ), sub( X ), and sub( B ) to be aligned on a block boundary, i.e., if f(x,y) = MOD( x-1, y ): f( IA, DESCA( MB_ ) ) = f( JA, DESCA( NB_ ) ) = 0, f( IAF, DESCAF( MB_ ) ) = f( JAF, DESCAF( NB_ ) ) = 0, f( IB, DESCB( MB_ ) ) = f( JB, DESCB( NB_ ) ) = 0, and f( IX, DESCX( MB_ ) ) = f( JX, DESCX( NB_ ) ) = 0. scalapack-doc-1.5/man/manl/pdposv.l0100644000056400000620000001462606335610636016750 0ustar pfrauenfstaff.TH PDPOSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDPOSV - compute the solution to a real system of linear equations sub( A ) * X = sub( B ), .SH SYNOPSIS .TP 19 SUBROUTINE PDPOSV( UPLO, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, INFO ) .TP 19 .ti +4 CHARACTER UPLO .TP 19 .ti +4 INTEGER IA, IB, INFO, JA, JB, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 DOUBLE PRECISION A( * ), B( * ) .SH PURPOSE PDPOSV computes the solution to a real system of linear equations where sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1) and is an N-by-N symmetric distributed positive definite matrix and X and sub( B ) denoting B(IB:IB+N-1,JB:JB+NRHS-1) are N-by-NRHS distributed matrices. .br The Cholesky decomposition is used to factor sub( A ) as .br sub( A ) = U**T * U, if UPLO = 'U', or sub( A ) = L * L**T, if UPLO = 'L', .br where U is an upper triangular matrix and L is a lower triangular matrix. The factored form of sub( A ) is then used to solve the system of equations. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block decomposition ( MB_A = NB_A ). .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER = 'U': Upper triangle of sub( A ) is stored; .br = 'L': Lower triangle of sub( A ) is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed submatrix sub( B ). NRHS >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N symmetric distributed matrix sub( A ) to be factored. If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the distribu- ted matrix, and its strictly upper triangular part is not referenced. On exit, if INFO = 0, this array contains the local pieces of the factor U or L from the Cholesky factori- zation sub( A ) = U**T*U or L*L**T. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_B,LOC(JB+NRHS-1)). On entry, the local pieces of the right hand sides distribu- ted matrix sub( B ). On exit, if INFO = 0, sub( B ) is over- written with the solution distributed matrix X. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, the leading minor of order K, .br A(IA:IA+K-1,JA:JA+K-1) is not positive definite, and the factorization could not be completed, and the solution has not been computed. scalapack-doc-1.5/man/manl/pdposvx.l0100644000056400000620000003331106335610636017130 0ustar pfrauenfstaff.TH PDPOSVX l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDPOSVX - use the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system of linear equations A(IA:IA+N-1,JA:JA+N-1) * X = B(IB:IB+N-1,JB:JB+NRHS-1), .SH SYNOPSIS .TP 20 SUBROUTINE PDPOSVX( FACT, UPLO, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, EQUED, SR, SC, B, IB, JB, DESCB, X, IX, JX, DESCX, RCOND, FERR, BERR, WORK, LWORK, IWORK, LIWORK, INFO ) .TP 20 .ti +4 CHARACTER EQUED, FACT, UPLO .TP 20 .ti +4 INTEGER IA, IAF, IB, INFO, IX, JA, JAF, JB, JX, LIWORK, LWORK, N, NRHS .TP 20 .ti +4 DOUBLE PRECISION RCOND .TP 20 .ti +4 INTEGER DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * ), IWORK( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), AF( * ), B( * ), BERR( * ), FERR( * ), SC( * ), SR( * ), WORK( * ), X( * ) .SH PURPOSE PDPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system of linear equations where A(IA:IA+N-1,JA:JA+N-1) is an N-by-N matrix and X and B(IB:IB+N-1,JB:JB+NRHS-1) are N-by-NRHS matrices. .br Error bounds on the solution and a condition estimate are also provided. In the following comments Y denotes Y(IY:IY+M-1,JY:JY+K-1) a M-by-K matrix where Y can be A, AF, B and X. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH DESCRIPTION The following steps are performed: .br 1. If FACT = 'E', real scaling factors are computed to equilibrate the system: .br diag(SR) * A * diag(SC) * inv(diag(SC)) * X = diag(SR) * 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(SR)*A*diag(SC) and B by diag(SR)*B. 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as A = U**T* U, if UPLO = 'U', or .br A = L * L**T, if UPLO = 'L', .br where U is an upper triangular matrix and L is a lower triangular matrix. .br 3. 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, steps 4-6 are skipped. .br 4. The system of equations is solved for X using the factored form of A. .br 5. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. .br 6. If equilibration was used, the matrix X is premultiplied by diag(SR) so that it solves the original system before .br equilibration. .br .SH ARGUMENTS .TP 8 FACT (global input) CHARACTER 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 contains the factored form of A. If EQUED = 'Y', the matrix A has been equilibrated with scaling factors given by S. A and AF will not be modified. = 'N': The matrix A will be copied to AF and factored. .br = 'E': The matrix A will be equilibrated if necessary, then copied to AF and factored. .TP 8 UPLO (global input) CHARACTER = 'U': Upper triangle of A is stored; .br = 'L': Lower triangle of A is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix A(IA:IA+N-1,JA:JA+N-1). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed submatrices B and X. NRHS >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of local dimension ( LLD_A, LOCc(JA+N-1) ). On entry, the symmetric matrix A, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(SR)*A*diag(SC). 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. A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(SR)*A*diag(SC). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 AF (local input or local output) DOUBLE PRECISION pointer into the local memory to an array of local dimension ( LLD_AF, LOCc(JA+N-1)). If FACT = 'F', then AF is an input argument and on entry contains the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, in the same storage format as A. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix diag(SR)*A*diag(SC). If FACT = 'N', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the original matrix A. If FACT = 'E', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix). .TP 8 IAF (global input) INTEGER The row index in the global array AF indicating the first row of sub( AF ). .TP 8 JAF (global input) INTEGER The column index in the global array AF indicating the first column of sub( AF ). .TP 8 DESCAF (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix AF. .TP 8 EQUED (global input/global output) CHARACTER Specifies the form of equilibration that was done. = 'N': No equilibration (always true if FACT = 'N'). .br = 'Y': Equilibration was done, i.e., A has been replaced by diag(SR) * A * diag(SC). EQUED is an input variable if FACT = 'F'; otherwise, it is an output variable. .TP 8 SR (local input/local output) DOUBLE PRECISION array, dimension (LLD_A) The scale factors for A distributed across process rows; not accessed if EQUED = 'N'. SR is an input variable if FACT = 'F'; otherwise, SR is an output variable. If FACT = 'F' and EQUED = 'Y', each element of SR must be positive. .TP 8 SC (local input/local output) DOUBLE PRECISION array, dimension (LOC(N_A)) The scale factors for A distributed across process columns; not accessed if EQUED = 'N'. SC is an input variable if FACT = 'F'; otherwise, SC is an output variable. If FACT = 'F' and EQUED = 'Y', each element of SC must be positive. .TP 8 B (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of local dimension ( LLD_B, LOCc(JB+NRHS-1) ). On entry, the N-by-NRHS right-hand side matrix B. On exit, if EQUED = 'N', B is not modified; if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B; if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is overwritten by diag(C)*B. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 X (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of local dimension ( LLD_X, LOCc(JX+NRHS-1) ). If INFO = 0, the N-by-NRHS 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(SC))*X if TRANS = 'N' and EQUED = 'C' or 'B', or inv(diag(SR))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 RCOND (global 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, and the solution and error bounds are not computed. .TP 8 FERR (local output) DOUBLE PRECISION array, dimension (LOC(N_B)) The estimated forward error bounds for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution, FERR(j) bounds the magnitude of the largest entry in (X(j) - XTRUE) divided by the magnitude of the largest entry in X(j). The quality of the error bound depends on the quality of the estimate of norm(inv(A)) computed in the code; if the estimate of norm(inv(A)) is accurate, the error bound is guaranteed. .TP 8 BERR (local output) DOUBLE PRECISION array, dimension (LOC(N_B)) The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any entry of A or B that makes X(j) an exact solution). .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK = MAX( PDPOCON( LWORK ), PDPORFS( LWORK ) ) + LOCr( N_A ). LWORK = 3*DESCA( LLD_ ) If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/local output) INTEGER array, dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK. .TP 8 LIWORK (local or global input) INTEGER The dimension of the array IWORK. LIWORK is local input and must be at least LIWORK = DESCA( LLD_ ) LIWORK = LOCr(N_A). If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 0: if INFO = i, and i is .br <= N: if INFO = i, the leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution and error bounds could not be computed. = N+1: RCOND is less than machine precision. The factorization has been completed, but the matrix is singular to working precision, and the solution and error bounds have not been computed. scalapack-doc-1.5/man/manl/pdpotf2.l0100644000056400000620000001261606335610636017010 0ustar pfrauenfstaff.TH PDPOTF2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDPOTF2 - compute the Cholesky factorization of a real symmetric positive definite distributed matrix sub( A )=A(IA:IA+N-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PDPOTF2( UPLO, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ) .SH PURPOSE PDPOTF2 computes the Cholesky factorization of a real symmetric positive definite distributed matrix sub( A )=A(IA:IA+N-1,JA:JA+N-1). The factorization has the form .br sub( A ) = U' * U , if UPLO = 'U', or .br sub( A ) = L * L', if UPLO = 'L', .br where U is an upper triangular matrix and L is lower triangular. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires N <= NB_A-MOD(JA-1, NB_A) and square block decomposition ( MB_A = NB_A ). .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER = 'U': Upper triangle of sub( A ) is stored; .br = 'L': Lower triangle of sub( A ) is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N symmetric distributed matrix sub( A ) to be factored. If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the distribu- ted matrix, and its strictly upper triangular part is not referenced. On exit, if UPLO = 'U', the upper triangular part of the distributed matrix contains the Cholesky factor U, if UPLO = 'L', the lower triangular part of the distribu- ted matrix contains the Cholesky factor L. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, the leading minor of order K, .br A(IA:IA+K-1,JA:JA+K-1) is not positive definite, and the factorization could not be completed. scalapack-doc-1.5/man/manl/pdpotrf.l0100644000056400000620000001262006335610636017103 0ustar pfrauenfstaff.TH PDPOTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDPOTRF - compute the Cholesky factorization of an N-by-N real symmetric positive definite distributed matrix sub( A ) denoting A(IA:IA+N-1, JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PDPOTRF( UPLO, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ) .SH PURPOSE PDPOTRF computes the Cholesky factorization of an N-by-N real symmetric positive definite distributed matrix sub( A ) denoting A(IA:IA+N-1, JA:JA+N-1). The factorization has the form .br sub( A ) = U' * U , if UPLO = 'U', or .br sub( A ) = L * L', if UPLO = 'L', .br where U is an upper triangular matrix and L is lower triangular. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block decomposition ( MB_A = NB_A ). .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER = 'U': Upper triangle of sub( A ) is stored; .br = 'L': Lower triangle of sub( A ) is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N symmetric distributed matrix sub( A ) to be factored. If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the distribu- ted matrix, and its strictly upper triangular part is not referenced. On exit, if UPLO = 'U', the upper triangular part of the distributed matrix contains the Cholesky factor U, if UPLO = 'L', the lower triangular part of the distribu- ted matrix contains the Cholesky factor L. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, the leading minor of order K, .br A(IA:IA+K-1,JA:JA+K-1) is not positive definite, and the factorization could not be completed. scalapack-doc-1.5/man/manl/pdpotri.l0100644000056400000620000001147106335610636017111 0ustar pfrauenfstaff.TH PDPOTRI l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDPOTRI - compute the inverse of a real symmetric positive definite distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) using the Cholesky factorization sub( A ) = U**T*U or L*L**T computed by PDPOTRF .SH SYNOPSIS .TP 20 SUBROUTINE PDPOTRI( UPLO, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ) .SH PURPOSE PDPOTRI computes the inverse of a real symmetric positive definite distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) using the Cholesky factorization sub( A ) = U**T*U or L*L**T computed by PDPOTRF. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 = 'U': Upper triangle of sub( A ) is stored; .br = 'L': Lower triangle of sub( A ) is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the triangular factor U or L from the Cholesky factorization of the distributed matrix sub( A ) = U**T*U or L*L**T, as computed by PDPOTRF. On exit, the local pieces of the upper or lower triangle of the (symmetric) inverse of sub( A ), overwriting the input factor U or L. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = i, the (i,i) element of the factor U or L is zero, and the inverse could not be computed. scalapack-doc-1.5/man/manl/pdpotrs.l0100644000056400000620000001272606335610636017127 0ustar pfrauenfstaff.TH PDPOTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDPOTRS - solve a system of linear equations sub( A ) * X = sub( B ) A(IA:IA+N-1,JA:JA+N-1)*X = B(IB:IB+N-1,JB:JB+NRHS-1) .SH SYNOPSIS .TP 20 SUBROUTINE PDPOTRS( UPLO, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IB, INFO, JA, JB, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), B( * ) .SH PURPOSE PDPOTRS solves a system of linear equations where sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1) and is a N-by-N symmetric positive definite distributed matrix using the Cholesky factorization sub( A ) = U**T*U or L*L**T computed by PDPOTRF. sub( B ) denotes the distributed matrix B(IB:IB+N-1,JB:JB+NRHS-1). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block decomposition ( MB_A = NB_A ). .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER = 'U': Upper triangle of sub( A ) is stored; .br = 'L': Lower triangle of sub( A ) is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed submatrix sub( B ). NRHS >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the factors L or U from the Cholesky facto- rization sub( A ) = L*L**T or U**T*U, as computed by PDPOTRF. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of local dimension (LLD_B,LOCc(JB+NRHS-1)). On entry, this array contains the the local pieces of the right hand sides sub( B ). On exit, this array contains the local pieces of the solution distributed matrix X. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pdptsv.l0100644000056400000620000000134406335610636016746 0ustar pfrauenfstaff.TH PDPTSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDPTSV - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 19 SUBROUTINE PDPTSV( N, NRHS, D, E, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 INTEGER IB, INFO, JA, LWORK, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 DOUBLE PRECISION B( * ), D( * ), E( * ), WORK( * ) .SH PURPOSE PDPTSV solves a system of linear equations where A(1:N, JA:JA+N-1) is an N-by-N real .br tridiagonal symmetric positive definite distributed .br matrix. .br Cholesky factorization is used to factor a reordering of .br the matrix into L L'. .br See PDPTTRF and PDPTTRS for details. .br scalapack-doc-1.5/man/manl/pdpttrf.l0100644000056400000620000000223206335610636017106 0ustar pfrauenfstaff.TH PDPTTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDPTTRF - compute a Cholesky factorization of an N-by-N real tridiagonal symmetric positive definite distributed matrix A(1:N, JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PDPTTRF( N, D, E, JA, DESCA, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER INFO, JA, LAF, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION AF( * ), D( * ), E( * ), WORK( * ) .SH PURPOSE PDPTTRF computes a Cholesky factorization of an N-by-N real tridiagonal symmetric positive definite distributed matrix A(1:N, JA:JA+N-1). Reordering is used to increase parallelism in the factorization. This reordering results in factors that are DIFFERENT from those produced by equivalent sequential codes. These factors cannot be used directly by users; however, they can be used in .br subsequent calls to PDPTTRS to solve linear systems. .br The factorization has the form .br P A(1:N, JA:JA+N-1) P^T = U' D U or .br P A(1:N, JA:JA+N-1) P^T = L D L', .br where U is a tridiagonal upper triangular matrix and L is tridiagonal lower triangular, and P is a permutation matrix. .br scalapack-doc-1.5/man/manl/pdpttrs.l0100644000056400000620000000142606335610636017127 0ustar pfrauenfstaff.TH PDPTTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDPTTRS - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 20 SUBROUTINE PDPTTRS( N, NRHS, D, E, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IB, INFO, JA, LAF, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 DOUBLE PRECISION AF( * ), B( * ), D( * ), E( * ), WORK( * ) .SH PURPOSE PDPTTRS solves a system of linear equations where A(1:N, JA:JA+N-1) is the matrix used to produce the factors stored in A(1:N,JA:JA+N-1) and AF by PDPTTRF. .br A(1:N, JA:JA+N-1) is an N-by-N real .br tridiagonal symmetric positive definite distributed .br matrix. .br Routine PDPTTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pdpttrsv.l0100644000056400000620000000220106335610636017305 0ustar pfrauenfstaff.TH PDPTTRSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDPTTRSV - solve a tridiagonal triangular system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 21 SUBROUTINE PDPTTRSV( UPLO, N, NRHS, D, E, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 21 .ti +4 CHARACTER UPLO .TP 21 .ti +4 INTEGER IB, INFO, JA, LAF, LWORK, N, NRHS .TP 21 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 21 .ti +4 DOUBLE PRECISION AF( * ), B( * ), D( * ), E( * ), WORK( * ) .SH PURPOSE PDPTTRSV solves a tridiagonal triangular system of linear equations or .br A(1:N, JA:JA+N-1)^T * X = B(IB:IB+N-1, 1:NRHS) where A(1:N, JA:JA+N-1) is a tridiagonal .br triangular matrix factor produced by the .br Cholesky factorization code PDPTTRF .br and is stored in A(1:N,JA:JA+N-1) and AF. .br The matrix stored in A(1:N, JA:JA+N-1) is either .br upper or lower triangular according to UPLO, .br and the choice of solving A(1:N, JA:JA+N-1) or A(1:N, JA:JA+N-1)^T is dictated by the user by the parameter TRANS. .br Routine PDPTTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pdrscl.l0100644000056400000620000001115206335610636016713 0ustar pfrauenfstaff.TH PDRSCL l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDRSCL - multiplie an N-element real distributed vector sub( X ) by the real scalar 1/a .SH SYNOPSIS .TP 19 SUBROUTINE PDRSCL( N, SA, SX, IX, JX, DESCX, INCX ) .TP 19 .ti +4 INTEGER IX, INCX, JX, N .TP 19 .ti +4 DOUBLE PRECISION SA .TP 19 .ti +4 INTEGER DESCX( * ) .TP 19 .ti +4 DOUBLE PRECISION SX( * ) .SH PURPOSE PDRSCL multiplies an N-element real distributed vector sub( X ) by the real scalar 1/a. This is done without overflow or underflow as long as the final result sub( X )/a does not overflow or underflow. where sub( X ) denotes X(IX:IX+N-1,JX:JX), if INCX = 1, .br X(IX:IX,JX:JX+N-1), if INCX = M_X. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector descA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DT_A (global) descA[ DT_ ] The descriptor type. In this case, DT_A = 1. .br CTXT_A (global) descA[ CTXT_ ] The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) descA[ M_ ] The number of rows in the global array A. .br N_A (global) descA[ N_ ] The number of columns in the global array A. .br MB_A (global) descA[ MB_ ] The blocking factor used to distribu- te the rows of the array. .br NB_A (global) descA[ NB_ ] The blocking factor used to distribu- te the columns of the array. RSRC_A (global) descA[ RSRC_ ] The process row over which the first row of the array A is distributed. CSRC_A (global) descA[ CSRC_ ] The process column over which the first column of the array A is distributed. .br LLD_A (local) descA[ LLD_ ] The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be seen as particular matrices, a distributed vector is considered to be a distributed matrix. .br .SH ARGUMENTS .TP 8 N (global input) pointer to INTEGER The number of components of the distributed vector sub( X ). N >= 0. .TP 8 SA (global input) DOUBLE PRECISION The scalar a which is used to divide each component of sub( X ). SA must be >= 0, or the subroutine will divide by zero. .TP 8 SX (local input/local output) DOUBLE PRECISION array containing the local pieces of a distributed matrix of dimension of at least ( (JX-1)*M_X + IX + ( N - 1 )*abs( INCX ) ) This array contains the entries of the distributed vector sub( X ). .TP 8 IX (global input) pointer to INTEGER The global row index of the submatrix of the distributed matrix X to operate on. .TP 8 JX (global input) pointer to INTEGER The global column index of the submatrix of the distributed matrix X to operate on. .TP 8 DESCX (global and local input) INTEGER array of dimension 8. The array descriptor of the distributed matrix X. .TP 8 INCX (global input) pointer to INTEGER The global increment for the elements of X. Only two values of INCX are supported in this version, namely 1 and M_X. scalapack-doc-1.5/man/manl/pdstebz.l0100644000056400000620000002003306335610636017075 0ustar pfrauenfstaff.TH PDSTEBZ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDSTEBZ - compute the eigenvalues of a symmetric tridiagonal matrix in parallel .SH SYNOPSIS .TP 20 SUBROUTINE PDSTEBZ( ICTXT, RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E, M, NSPLIT, W, IBLOCK, ISPLIT, WORK, LWORK, IWORK, LIWORK, INFO ) .TP 20 .ti +4 CHARACTER ORDER, RANGE .TP 20 .ti +4 INTEGER ICTXT, IL, INFO, IU, LIWORK, LWORK, M, N, NSPLIT .TP 20 .ti +4 DOUBLE PRECISION ABSTOL, VL, VU .TP 20 .ti +4 INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * ) .TP 20 .ti +4 DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ) .SH PURPOSE PDSTEBZ computes the eigenvalues of a symmetric tridiagonal matrix in parallel. The user may ask for all eigenvalues, all eigenvalues in the interval [VL, VU], or the eigenvalues indexed IL through IU. A static partitioning of work is done at the beginning of PDSTEBZ which results in all processes finding an (almost) equal number of eigenvalues. .br NOTE : It is assumed that the user is on an IEEE machine. If the user is not on an IEEE mchine, set the compile time flag NO_IEEE to 1 (in SLmake.inc). The features of IEEE arithmetic that are needed for the "fast" Sturm Count are : (a) infinity arithmetic (b) the sign bit of a single precision floating point number is assumed be in the 32nd bit position (c) the sign of negative zero. .br See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix", Report CS41, Computer Science Dept., Stanford .br University, July 21, 1966. .br .SH ARGUMENTS .TP 8 ICTXT (global input) INTEGER The BLACS context handle. .TP 8 RANGE (global input) CHARACTER Specifies which eigenvalues are to be found. = 'A': ("All") all eigenvalues will be found. .br = 'V': ("Value") all eigenvalues in the interval [VL, VU] will be found. = 'I': ("Index") the IL-th through IU-th eigenvalues (of the entire matrix) will be found. .TP 8 ORDER (global input) CHARACTER Specifies the order in which the eigenvalues and their block numbers are stored in W and IBLOCK. = 'B': ("By Block") the eigenvalues will be grouped by split-off block (see IBLOCK, ISPLIT) and ordered from smallest to largest within the block. = 'E': ("Entire matrix") the eigenvalues for the entire matrix will be ordered from smallest to largest. .TP 8 N (global input) INTEGER The order of the tridiagonal matrix T. N >= 0. .TP 8 VL (global input) DOUBLE PRECISION If RANGE='V', the lower bound of the interval to be searched for eigenvalues. Eigenvalues less than VL will not be returned. Not referenced if RANGE='A' or 'I'. .TP 8 VU (global input) DOUBLE PRECISION If RANGE='V', the upper bound of the interval to be searched for eigenvalues. Eigenvalues greater than VU will not be returned. VU must be greater than VL. Not referenced if RANGE='A' or 'I'. .TP 8 IL (global input) INTEGER If RANGE='I', the index (from smallest to largest) of the smallest eigenvalue to be returned. IL must be at least 1. Not referenced if RANGE='A' or 'V'. .TP 8 IU (global input) INTEGER If RANGE='I', the index (from smallest to largest) of the largest eigenvalue to be returned. IU must be at least IL and no greater than N. Not referenced if RANGE='A' or 'V'. .TP 8 ABSTOL (global input) DOUBLE PRECISION The absolute tolerance for the eigenvalues. An eigenvalue (or cluster) is considered to be located if it has been determined to lie in an interval whose width is ABSTOL or less. If ABSTOL is less than or equal to zero, then ULP*|T| will be used, where |T| means the 1-norm of T. Eigenvalues will be computed most accurately when ABSTOL is set to the underflow threshold DLAMCH('U'), not zero. Note : If eigenvectors are desired later by inverse iteration ( PDSTEIN ), ABSTOL should be set to 2*PDLAMCH('S'). .TP 8 D (global input) DOUBLE PRECISION array, dimension (N) The n diagonal elements of the tridiagonal matrix T. To avoid overflow, the matrix must be scaled so that its largest entry is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. .TP 8 E (global input) DOUBLE PRECISION array, dimension (N-1) The (n-1) off-diagonal elements of the tridiagonal matrix T. To avoid overflow, the matrix must be scaled so that its largest entry is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. .TP 8 M (global output) INTEGER The actual number of eigenvalues found. 0 <= M <= N. (See also the description of INFO=2) .TP 8 NSPLIT (global output) INTEGER The number of diagonal blocks in the matrix T. 1 <= NSPLIT <= N. .TP 8 W (global output) DOUBLE PRECISION array, dimension (N) On exit, the first M elements of W contain the eigenvalues on all processes. .TP 8 IBLOCK (global output) INTEGER array, dimension (N) At each row/column j where E(j) is zero or small, the matrix T is considered to split into a block diagonal matrix. On exit IBLOCK(i) specifies which block (from 1 to the number of blocks) the eigenvalue W(i) belongs to. NOTE: in the (theoretically impossible) event that bisection does not converge for some or all eigenvalues, INFO is set to 1 and the ones for which it did not are identified by a negative block number. .TP 8 ISPLIT (global output) INTEGER array, dimension (N) The splitting points, at which T breaks up into submatrices. The first submatrix consists of rows/columns 1 to ISPLIT(1), the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N. (Only the first NSPLIT elements will actually be used, but since the user cannot know a priori what value NSPLIT will have, N words must be reserved for ISPLIT.) .TP 8 WORK (local workspace) DOUBLE PRECISION array, dimension ( MAX( 5*N, 7 ) ) .TP 8 LWORK (local input) INTEGER size of array WORK must be >= MAX( 5*N, 7 ) If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace) INTEGER array, dimension ( MAX( 4*N, 14 ) ) .TP 8 LIWORK (local input) INTEGER size of array IWORK must be >= MAX( 4*N, 14, NPROCS ) If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0 : successful exit .br < 0 : if INFO = -i, the i-th argument had an illegal value .br > 0 : some or all of the eigenvalues failed to converge or .br were not computed: .br = 1 : Bisection failed to converge for some eigenvalues; these eigenvalues are flagged by a negative block number. The effect is that the eigenvalues may not be as accurate as the absolute and relative tolerances. This is generally caused by arithmetic which is less accurate than PDLAMCH says. = 2 : There is a mismatch between the number of eigenvalues output and the number desired. = 3 : RANGE='i', and the Gershgorin interval initially used was incorrect. No eigenvalues were computed. Probable cause: your machine has sloppy floating point arithmetic. Cure: Increase the PARAMETER "FUDGE", recompile, and try again. .SH PARAMETERS .TP 8 RELFAC DOUBLE PRECISION, default = 2.0 The relative tolerance. An interval [a,b] lies within "relative tolerance" if b-a < RELFAC*ulp*max(|a|,|b|), where "ulp" is the machine precision (distance from 1 to the next larger floating point number.) .TP 8 FUDGE DOUBLE PRECISION, default = 2.0 A "fudge factor" to widen the Gershgorin intervals. Ideally, a value of 1 should work, but on machines with sloppy arithmetic, this needs to be larger. The default for publicly released versions should be large enough to handle the worst machine around. Note that this has no effect on the accuracy of the solution. scalapack-doc-1.5/man/manl/pdstein.l0100644000056400000620000002576206335610637017107 0ustar pfrauenfstaff.TH PDSTEIN l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDSTEIN - compute the eigenvectors of a symmetric tridiagonal matrix in parallel, using inverse iteration .SH SYNOPSIS .TP 20 SUBROUTINE PDSTEIN( N, D, E, M, W, IBLOCK, ISPLIT, ORFAC, Z, IZ, JZ, DESCZ, WORK, LWORK, IWORK, LIWORK, IFAIL, ICLUSTR, GAP, INFO ) .TP 20 .ti +4 INTEGER INFO, IZ, JZ, LIWORK, LWORK, M, N .TP 20 .ti +4 DOUBLE PRECISION ORFAC .TP 20 .ti +4 INTEGER DESCZ( * ), IBLOCK( * ), ICLUSTR( * ), IFAIL( * ), ISPLIT( * ), IWORK( * ) .TP 20 .ti +4 DOUBLE PRECISION D( * ), E( * ), GAP( * ), W( * ), WORK( * ), Z( * ) .SH PURPOSE PDSTEIN computes the eigenvectors of a symmetric tridiagonal matrix in parallel, using inverse iteration. The eigenvectors found correspond to user specified eigenvalues. PDSTEIN does not orthogonalize vectors that are on different processes. The extent of orthogonalization is controlled by the input parameter LWORK. Eigenvectors that are to be orthogonalized are computed by the same process. PDSTEIN decides on the allocation of work among the processes and then calls DSTEIN2 (modified LAPACK routine) on each individual process. If insufficient workspace is allocated, the expected orthogonalization may not be done. .br Note : If the eigenvectors obtained are not orthogonal, increase LWORK and run the code again. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS P = NPROW * NPCOL is the total number of processes .TP 8 N (global input) INTEGER The order of the tridiagonal matrix T. N >= 0. .TP 8 D (global input) DOUBLE PRECISION array, dimension (N) The n diagonal elements of the tridiagonal matrix T. .TP 8 E (global input) DOUBLE PRECISION array, dimension (N-1) The (n-1) off-diagonal elements of the tridiagonal matrix T. .TP 8 M (global input) INTEGER The total number of eigenvectors to be found. 0 <= M <= N. .TP 8 W (global input/global output) DOUBLE PRECISION array, dim (M) On input, the first M elements of W contain all the eigenvalues for which eigenvectors are to be computed. The eigenvalues should be grouped by split-off block and ordered from smallest to largest within the block (The output array W from PDSTEBZ with ORDER='b' is expected here). This array should be replicated on all processes. On output, the first M elements contain the input eigenvalues in ascending order. Note : To obtain orthogonal vectors, it is best if eigenvalues are computed to highest accuracy ( this can be done by setting ABSTOL to the underflow threshold = DLAMCH('U') --- ABSTOL is an input parameter to PDSTEBZ ) .TP 8 IBLOCK (global input) INTEGER array, dimension (N) The submatrix indices associated with the corresponding eigenvalues in W -- 1 for eigenvalues belonging to the first submatrix from the top, 2 for those belonging to the second submatrix, etc. (The output array IBLOCK from PDSTEBZ is expected here). .TP 8 ISPLIT (global input) INTEGER array, dimension (N) The splitting points, at which T breaks up into submatrices. The first submatrix consists of rows/columns 1 to ISPLIT(1), the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N (The output array ISPLIT from PDSTEBZ is expected here.) .TP 8 ORFAC (global input) DOUBLE PRECISION ORFAC specifies which eigenvectors should be orthogonalized. Eigenvectors that correspond to eigenvalues which are within ORFAC*||T|| of each other are to be orthogonalized. However, if the workspace is insufficient (see LWORK), this tolerance may be decreased until all eigenvectors to be orthogonalized can be stored in one process. No orthogonalization will be done if ORFAC equals zero. A default value of 10^-3 is used if ORFAC is negative. ORFAC should be identical on all processes. .TP 8 Z (local output) DOUBLE PRECISION array, dimension (DESCZ(DLEN_), N/npcol + NB) Z contains the computed eigenvectors associated with the specified eigenvalues. Any vector which fails to converge is set to its current iterate after MAXITS iterations ( See DSTEIN2 ). On output, Z is distributed across the P processes in block cyclic format. .TP 8 IZ (global input) INTEGER Z's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JZ (global input) INTEGER Z's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCZ (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix Z. .TP 8 WORK (local workspace/global output) DOUBLE PRECISION array, dimension ( LWORK ) On output, WORK(1) gives a lower bound on the workspace ( LWORK ) that guarantees the user desired orthogonalization (see ORFAC). Note that this may overestimate the minimum workspace needed. .TP 8 LWORK (local input) integer LWORK controls the extent of orthogonalization which can be done. The number of eigenvectors for which storage is allocated on each process is NVEC = floor(( LWORK- max(5*N,NP00*MQ00) )/N). Eigenvectors corresponding to eigenvalue clusters of size NVEC - ceil(M/P) + 1 are guaranteed to be orthogonal ( the orthogonality is similar to that obtained from DSTEIN2). Note : LWORK must be no smaller than: max(5*N,NP00*MQ00) + ceil(M/P)*N, and should have the same input value on all processes. It is the minimum value of LWORK input on different processes that is significant. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/global output) INTEGER array, dimension ( 3*N+P+1 ) On return, IWORK(1) contains the amount of integer workspace required. On return, the IWORK(2) through IWORK(P+2) indicate the eigenvectors computed by each process. Process I computes eigenvectors indexed IWORK(I+2)+1 thru' IWORK(I+3). .TP 8 LIWORK (local input) INTEGER Size of array IWORK. Must be >= 3*N + P + 1 If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IFAIL (global output) integer array, dimension (M) On normal exit, all elements of IFAIL are zero. If one or more eigenvectors fail to converge after MAXITS iterations (as in DSTEIN), then INFO > 0 is returned. If mod(INFO,M+1)>0, then for I=1 to mod(INFO,M+1), the eigenvector corresponding to the eigenvalue W(IFAIL(I)) failed to converge ( W refers to the array of eigenvalues on output ). ICLUSTR (global output) integer array, dimension (2*P) This output array contains indices of eigenvectors corresponding to a cluster of eigenvalues that could not be orthogonalized due to insufficient workspace (see LWORK, ORFAC and INFO). Eigenvectors corresponding to clusters of eigenvalues indexed ICLUSTR(2*I-1) to ICLUSTR(2*I), I = 1 to INFO/(M+1), could not be orthogonalized due to lack of workspace. Hence the eigenvectors corresponding to these clusters may not be orthogonal. ICLUSTR is a zero terminated array --- ( ICLUSTR(2*K).NE.0 .AND. ICLUSTR(2*K+1).EQ.0 ) if and only if K is the number of clusters. .TP 8 GAP (global output) DOUBLE PRECISION array, dimension (P) This output array contains the gap between eigenvalues whose eigenvectors could not be orthogonalized. The INFO/M output values in this array correspond to the INFO/(M+1) clusters indicated by the array ICLUSTR. As a result, the dot product between eigenvectors corresponding to the I^th cluster may be as high as ( O(n)*macheps ) / GAP(I). .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. < 0 : if INFO = -I, the I-th argument had an illegal value .br > 0 : if mod(INFO,M+1) = I, then I eigenvectors failed to converge in MAXITS iterations. Their indices are stored in the array IFAIL. if INFO/(M+1) = I, then eigenvectors corresponding to I clusters of eigenvalues could not be orthogonalized due to insufficient workspace. The indices of the clusters are stored in the array ICLUSTR. scalapack-doc-1.5/man/manl/pdsyev.l0100644000056400000620000002321406335610637016741 0ustar pfrauenfstaff.TH PDSYEV l "12 May 1997" "LAPACK version 1.3" "LAPACK routine (version 1.3)" .SH NAME .SH SYNOPSIS .TP 19 SUBROUTINE PDSYEV( JOBZ, UPLO, N, A, IA, JA, DESCA, W, Z, IZ, JZ, DESCZ, WORK, LWORK, INFO ) .TP 19 .ti +4 CHARACTER JOBZ, UPLO .TP 19 .ti +4 INTEGER IA, INFO, IZ, JA, JZ, LWORK, N .TP 19 .ti +4 INTEGER DESCA( * ), DESCZ( * ) .TP 19 .ti +4 DOUBLE PRECISION A( * ), W( * ), WORK( * ), Z( * ) .TP 19 .ti +4 INTEGER BLOCK_CYCLIC_2D, DLEN_, DTYPE_, CTXT_, M_, N_, MB_, NB_, RSRC_, CSRC_, LLD_ .TP 19 .ti +4 PARAMETER ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1, CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6, RSRC_ = 7, CSRC_ = 8, LLD_ = 9 ) .TP 19 .ti +4 DOUBLE PRECISION ZERO, ONE .TP 19 .ti +4 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) .TP 19 .ti +4 INTEGER ITHVAL .TP 19 .ti +4 PARAMETER ( ITHVAL = 10 ) .TP 19 .ti +4 LOGICAL LOWER, WANTZ .TP 19 .ti +4 INTEGER CONTEXTC, CSRC_A, I, IACOL, IAROW, ICOFFA, IINFO, INDD, INDD2, INDE, INDE2, INDTAU, INDWORK, INDWORK2, IROFFA, IROFFZ, ISCALE, IZROW, J, K, LCM, LCMQ, LDC, LLWORK, LWMIN, MB_A, MB_Z, MYCOL, MYPCOLC, MYPROWC, MYROW, NB, NB_A, NB_Z, NN, NP, NPCOL, NPCOLC, NPROCS, NPROW, NPROWC, NQ, NRC, QRMEM, RSRC_A, RSRC_Z, SIZEMQRLEFT, SIZEMQRRIGHT .TP 19 .ti +4 DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM .TP 19 .ti +4 INTEGER DESCQR( 10 ), IDUM1( 3 ), IDUM2( 3 ) .TP 19 .ti +4 LOGICAL LSAME .TP 19 .ti +4 INTEGER ILCM, INDXG2P, NUMROC, SL_GRIDRESHAPE .TP 19 .ti +4 DOUBLE PRECISION PDLAMCH, PDLANSY .TP 19 .ti +4 EXTERNAL LSAME, ILCM, INDXG2P, NUMROC, SL_GRIDRESHAPE, PDLAMCH, PDLANSY .TP 19 .ti +4 EXTERNAL BLACS_GRIDEXIT, BLACS_GRIDINFO, CHK1MAT, DCOPY, DESCINIT, DGAMN2D, DGAMX2D, DSCAL, DSTEQR2, PCHK2MAT, PDELGET, PDGEMR2D, PDLASCL, PDLASET, PDORMTR, PDSYTRD, PXERBLA .TP 19 .ti +4 INTRINSIC DBLE, ICHAR, MAX, MIN, MOD, SQRT .TP 19 .ti +4 IF( BLOCK_CYCLIC_2D*CSRC_*CTXT_*DLEN_*DTYPE_*LLD_*MB_*M_*NB_*N_* RSRC_.LT.0 )RETURN .TP 19 .ti +4 IF( N.EQ.0 ) RETURN .TP 19 .ti +4 CALL BLACS_GRIDINFO( DESCA( CTXT_ ), NPROW, NPCOL, MYROW, MYCOL ) .TP 19 .ti +4 INFO = 0 .TP 19 .ti +4 IF( NPROW.EQ.-1 ) THEN .TP 19 .ti +4 INFO = -( 700+CTXT_ ) .TP 19 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCZ( CTXT_ ) ) THEN .TP 19 .ti +4 INFO = -( 1200+CTXT_ ) .TP 19 .ti +4 ELSE .TP 19 .ti +4 CALL CHK1MAT( N, 3, N, 3, IA, JA, DESCA, 7, INFO ) .TP 19 .ti +4 CALL CHK1MAT( N, 3, N, 3, IZ, JZ, DESCZ, 12, INFO ) .TP 19 .ti +4 IF( INFO.EQ.0 ) THEN .TP 19 .ti +4 SAFMIN = PDLAMCH( DESCA( CTXT_ ), 'Safe minimum' ) .TP 19 .ti +4 EPS = PDLAMCH( DESCA( CTXT_ ), 'Precision' ) .TP 19 .ti +4 SMLNUM = SAFMIN / EPS .TP 19 .ti +4 BIGNUM = ONE / SMLNUM .TP 19 .ti +4 RMIN = SQRT( SMLNUM ) .TP 19 .ti +4 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) .TP 19 .ti +4 NPROCS = NPROW*NPCOL .TP 19 .ti +4 NB_A = DESCA( NB_ ) .TP 19 .ti +4 MB_A = DESCA( MB_ ) .TP 19 .ti +4 NB_Z = DESCZ( NB_ ) .TP 19 .ti +4 MB_Z = DESCZ( MB_ ) .TP 19 .ti +4 NB = NB_A .TP 19 .ti +4 LOWER = LSAME( UPLO, 'L' ) .TP 19 .ti +4 WANTZ = LSAME( JOBZ, 'V' ) .TP 19 .ti +4 RSRC_A = DESCA( RSRC_ ) .TP 19 .ti +4 CSRC_A = DESCA( CSRC_ ) .TP 19 .ti +4 RSRC_Z = DESCZ( RSRC_ ) .TP 19 .ti +4 LCM = ILCM( NPROW, NPCOL ) .TP 19 .ti +4 LCMQ = LCM / NPCOL .TP 19 .ti +4 IROFFA = MOD( IA-1, MB_A ) .TP 19 .ti +4 ICOFFA = MOD( JA-1, NB_A ) .TP 19 .ti +4 IROFFZ = MOD( IZ-1, MB_A ) .TP 19 .ti +4 IAROW = INDXG2P( 1, NB_A, MYROW, RSRC_A, NPROW ) .TP 19 .ti +4 IACOL = INDXG2P( 1, MB_A, MYCOL, CSRC_A, NPCOL ) .TP 19 .ti +4 IZROW = INDXG2P( 1, NB_A, MYROW, RSRC_Z, NPROW ) .TP 19 .ti +4 NP = NUMROC( N+IROFFA, NB_Z, MYROW, IAROW, NPROW ) .TP 19 .ti +4 NQ = NUMROC( N+ICOFFA, NB_Z, MYCOL, IACOL, NPCOL ) .TP 19 .ti +4 SIZEMQRLEFT = MAX( ( NB_A*( NB_A-1 ) ) / 2, ( NP+NQ )*NB_A ) + NB_A*NB_A .TP 19 .ti +4 SIZEMQRRIGHT = MAX( ( NB_A*( NB_A-1 ) ) / 2, ( NQ+MAX( NP+NUMROC( NUMROC( N+ICOFFA, NB_A, 0, 0, NPCOL ), NB, 0, 0, LCMQ ), NP ) )* NB_A ) + NB_A*NB_A .TP 19 .ti +4 LDC = 0 .TP 19 .ti +4 IF( WANTZ ) THEN .TP 19 .ti +4 CONTEXTC = SL_GRIDRESHAPE( DESCA( CTXT_ ), 0, 1, 1, NPROCS, 1 ) .TP 19 .ti +4 CALL BLACS_GRIDINFO( CONTEXTC, NPROWC, NPCOLC, MYPROWC, MYPCOLC ) .TP 19 .ti +4 NRC = NUMROC( N, NB_A, MYPROWC, 0, NPROCS ) .TP 19 .ti +4 LDC = MAX( 1, NRC ) .TP 19 .ti +4 CALL DESCINIT( DESCQR, N, N, NB, NB, 0, 0, CONTEXTC, LDC, INFO ) .TP 19 .ti +4 END IF .TP 19 .ti +4 INDTAU = 1 .TP 19 .ti +4 INDE = INDTAU + N .TP 19 .ti +4 INDD = INDE + N .TP 19 .ti +4 INDD2 = INDD + N .TP 19 .ti +4 INDE2 = INDD2 + N .TP 19 .ti +4 INDWORK = INDE2 + N .TP 19 .ti +4 INDWORK2 = INDWORK + N*LDC .TP 19 .ti +4 LLWORK = LWORK - INDWORK + 1 .TP 19 .ti +4 NN = MAX( N, NB, 2 ) .TP 19 .ti +4 IF( WANTZ ) THEN .TP 19 .ti +4 QRMEM = 5*N + MAX( 2*NP+NQ+NB*NN, 2*NN-2 ) + N*LDC .TP 19 .ti +4 LWMIN = MAX( SIZEMQRLEFT, SIZEMQRRIGHT, QRMEM ) .TP 19 .ti +4 ELSE .TP 19 .ti +4 LWMIN = 5*N + 2*NP + NQ + NB*NN .TP 19 .ti +4 END IF .TP 19 .ti +4 END IF .TP 19 .ti +4 IF( INFO.EQ.0 ) THEN .TP 19 .ti +4 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN .TP 19 .ti +4 INFO = -1 .TP 19 .ti +4 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN .TP 19 .ti +4 INFO = -2 .TP 19 .ti +4 ELSE IF( LWORK.LT.LWMIN .AND. LWORK.NE.-1 ) THEN .TP 19 .ti +4 INFO = -14 .TP 19 .ti +4 ELSE IF( IROFFA.NE.IROFFZ ) THEN .TP 19 .ti +4 INFO = -10 .TP 19 .ti +4 ELSE IF( IROFFA.NE.0 ) THEN .TP 19 .ti +4 INFO = -5 .TP 19 .ti +4 ELSE IF( IAROW.NE.IZROW ) THEN .TP 19 .ti +4 INFO = -10 .TP 19 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCA( NB_ ) ) THEN .TP 19 .ti +4 INFO = -( 700+NB_ ) .TP 19 .ti +4 ELSE IF( DESCA( M_ ).NE.DESCZ( M_ ) ) THEN .TP 19 .ti +4 INFO = -( 1200+M_ ) .TP 19 .ti +4 ELSE IF( DESCA( N_ ).NE.DESCZ( N_ ) ) THEN .TP 19 .ti +4 INFO = -( 1200+N_ ) .TP 19 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCZ( MB_ ) ) THEN .TP 19 .ti +4 INFO = -( 1200+MB_ ) .TP 19 .ti +4 ELSE IF( DESCA( NB_ ).NE.DESCZ( NB_ ) ) THEN .TP 19 .ti +4 INFO = -( 1200+NB_ ) .TP 19 .ti +4 ELSE IF( DESCA( RSRC_ ).NE.DESCZ( RSRC_ ) ) THEN .TP 19 .ti +4 INFO = -( 1200+RSRC_ ) .TP 19 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCZ( CTXT_ ) ) THEN .TP 19 .ti +4 INFO = -( 1200+CTXT_ ) .TP 19 .ti +4 END IF .TP 19 .ti +4 END IF .TP 19 .ti +4 IF( WANTZ ) THEN .TP 19 .ti +4 IDUM1( 1 ) = ICHAR( 'V' ) .TP 19 .ti +4 ELSE .TP 19 .ti +4 IDUM1( 1 ) = ICHAR( 'N' ) .TP 19 .ti +4 END IF .TP 19 .ti +4 IDUM2( 1 ) = 1 .TP 19 .ti +4 IF( LOWER ) THEN .TP 19 .ti +4 IDUM1( 2 ) = ICHAR( 'L' ) .TP 19 .ti +4 ELSE .TP 19 .ti +4 IDUM1( 2 ) = ICHAR( 'U' ) .TP 19 .ti +4 END IF .TP 19 .ti +4 IDUM2( 2 ) = 2 .TP 19 .ti +4 IF( LWORK.EQ.-1 ) THEN .TP 19 .ti +4 IDUM1( 3 ) = -1 .TP 19 .ti +4 ELSE .TP 19 .ti +4 IDUM1( 3 ) = 1 .TP 19 .ti +4 END IF .TP 19 .ti +4 IDUM2( 3 ) = 3 .TP 19 .ti +4 CALL PCHK2MAT( N, 3, N, 3, IA, JA, DESCA, 7, N, 3, N, 3, IZ, JZ, DESCZ, 12, 3, IDUM1, IDUM2, INFO ) .TP 19 .ti +4 WORK( 1 ) = DBLE( LWMIN ) .TP 19 .ti +4 END IF .TP 19 .ti +4 IF( INFO.NE.0 ) THEN .TP 19 .ti +4 CALL PXERBLA( DESCA( CTXT_ ), 'PDSYEV', -INFO ) .TP 19 .ti +4 IF( WANTZ ) CALL BLACS_GRIDEXIT( CONTEXTC ) .TP 19 .ti +4 RETURN .TP 19 .ti +4 ELSE IF( LWORK.EQ.-1 ) THEN .TP 19 .ti +4 IF( WANTZ ) CALL BLACS_GRIDEXIT( CONTEXTC ) .TP 19 .ti +4 RETURN .TP 19 .ti +4 END IF .TP 19 .ti +4 ISCALE = 0 .TP 19 .ti +4 ANRM = PDLANSY( '1', UPLO, N, A, IA, JA, DESCA, WORK( INDWORK ) ) .TP 19 .ti +4 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN .TP 19 .ti +4 ISCALE = 1 .TP 19 .ti +4 SIGMA = RMIN / ANRM .TP 19 .ti +4 ELSE IF( ANRM.GT.RMAX ) THEN .TP 19 .ti +4 ISCALE = 1 .TP 19 .ti +4 SIGMA = RMAX / ANRM .TP 19 .ti +4 END IF .TP 19 .ti +4 IF( ISCALE.EQ.1 ) THEN .TP 19 .ti +4 CALL PDLASCL( UPLO, ONE, SIGMA, N, N, A, IA, JA, DESCA, IINFO ) .TP 19 .ti +4 END IF .TP 19 .ti +4 CALL PDSYTRD( UPLO, N, A, IA, JA, DESCA, WORK( INDD ), WORK( INDE ), WORK( INDTAU ), WORK( INDWORK ), LLWORK, IINFO ) .TP 19 .ti +4 DO 10 I = 1, N .TP 19 .ti +4 CALL PDELGET( 'A', ' ', WORK( INDD2+I-1 ), A, I+IA-1, I+JA-1, DESCA ) .TP 19 .ti +4 10 CONTINUE .TP 19 .ti +4 IF( LSAME( UPLO, 'U' ) ) THEN .TP 19 .ti +4 DO 20 I = 1, N - 1 .TP 19 .ti +4 CALL PDELGET( 'A', ' ', WORK( INDE2+I-1 ), A, I+IA-1, I+JA, DESCA ) .TP 19 .ti +4 20 CONTINUE .TP 19 .ti +4 ELSE .TP 19 .ti +4 DO 30 I = 1, N - 1 .TP 19 .ti +4 CALL PDELGET( 'A', ' ', WORK( INDE2+I-1 ), A, I+IA, I+JA-1, DESCA ) .TP 19 .ti +4 30 CONTINUE .TP 19 .ti +4 END IF .TP 19 .ti +4 IF( WANTZ ) THEN .TP 19 .ti +4 CALL PDLASET( 'Full', N, N, ZERO, ONE, WORK( INDWORK ), 1, 1, DESCQR ) .TP 19 .ti +4 CALL DSTEQR2( 'I', N, WORK( INDD2 ), WORK( INDE2 ), WORK( INDWORK ), LDC, NRC, WORK( INDWORK2 ), INFO ) .TP 19 .ti +4 CALL PDGEMR2D( N, N, WORK( INDWORK ), 1, 1, DESCQR, Z, 1, 1, DESCZ, CONTEXTC ) .TP 19 .ti +4 CALL PDORMTR( 'L', UPLO, 'N', N, N, A, IA, JA, DESCA, WORK( INDTAU ), Z, IZ, JZ, DESCZ, WORK( INDWORK ), LLWORK, IINFO ) .TP 19 .ti +4 ELSE .TP 19 .ti +4 CALL DSTEQR2( 'N', N, WORK( INDD2 ), WORK( INDE2 ), WORK( INDWORK ), 1, 1, WORK( INDWORK2 ), INFO ) .TP 19 .ti +4 END IF .TP 19 .ti +4 CALL DCOPY( N, WORK( INDD2 ), 1, W, 1 ) .TP 19 .ti +4 IF( ISCALE.EQ.1 ) THEN .TP 19 .ti +4 CALL DSCAL( N, ONE / SIGMA, W, 1 ) .TP 19 .ti +4 END IF .TP 19 .ti +4 WORK( 1 ) = DBLE( LWMIN ) .TP 19 .ti +4 IF( WANTZ ) THEN .TP 19 .ti +4 CALL BLACS_GRIDEXIT( CONTEXTC ) .TP 19 .ti +4 END IF .TP 19 .ti +4 IF( N.LE.ITHVAL ) THEN .TP 19 .ti +4 J = N .TP 19 .ti +4 K = 1 .TP 19 .ti +4 ELSE .TP 19 .ti +4 J = N / ITHVAL .TP 19 .ti +4 K = ITHVAL .TP 19 .ti +4 END IF .TP 19 .ti +4 DO 40 I = 1, J .TP 19 .ti +4 WORK( I+INDTAU ) = W( ( I-1 )*K+1 ) .TP 19 .ti +4 WORK( I+INDE ) = W( ( I-1 )*K+1 ) .TP 19 .ti +4 40 CONTINUE .TP 19 .ti +4 CALL DGAMN2D( DESCA( CTXT_ ), 'a', ' ', J, 1, WORK( 1+INDTAU ), J, 1, 1, -1, -1, 0 ) .TP 19 .ti +4 CALL DGAMX2D( DESCA( CTXT_ ), 'a', ' ', J, 1, WORK( 1+INDE ), J, 1, 1, -1, -1, 0 ) .TP 19 .ti +4 DO 50 I = 1, J .TP 19 .ti +4 IF( INFO.EQ.0 .AND. ( WORK( I+INDTAU )-WORK( I+INDE ).NE. ZERO ) ) THEN .TP 19 .ti +4 INFO = N + 1 .TP 19 .ti +4 END IF .TP 19 .ti +4 50 CONTINUE .TP 19 .ti +4 RETURN .TP 19 .ti +4 END .SH PURPOSE scalapack-doc-1.5/man/manl/pdsyevx.l0100644000056400000620000003403506335610637017134 0ustar pfrauenfstaff.TH PDSYEVX l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME .SH SYNOPSIS .TP 20 SUBROUTINE PDSYEVX( JOBZ, RANGE, UPLO, N, A, IA, JA, DESCA, VL, VU, IL, IU, ABSTOL, M, NZ, W, ORFAC, Z, IZ, JZ, DESCZ, WORK, LWORK, IWORK, LIWORK, IFAIL, ICLUSTR, GAP, INFO ) .TP 20 .ti +4 CHARACTER JOBZ, RANGE, UPLO .TP 20 .ti +4 INTEGER IA, IL, INFO, IU, IZ, JA, JZ, LIWORK, LWORK, M, N, NZ .TP 20 .ti +4 DOUBLE PRECISION ABSTOL, ORFAC, VL, VU .TP 20 .ti +4 INTEGER DESCA( * ), DESCZ( * ), ICLUSTR( * ), IFAIL( * ), IWORK( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), GAP( * ), W( * ), WORK( * ), Z( * ) .TP 20 .ti +4 INTEGER BLOCK_CYCLIC_2D, DLEN_, DTYPE_, CTXT_, M_, N_, MB_, NB_, RSRC_, CSRC_, LLD_ .TP 20 .ti +4 PARAMETER ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1, CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6, RSRC_ = 7, CSRC_ = 8, LLD_ = 9 ) .TP 20 .ti +4 DOUBLE PRECISION ZERO, ONE, TEN, FIVE .TP 20 .ti +4 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TEN = 10.0D+0, FIVE = 5.0D+0 ) .TP 20 .ti +4 INTEGER IERREIN, IERRCLS, IERRSPC, IERREBZ .TP 20 .ti +4 PARAMETER ( IERREIN = 1, IERRCLS = 2, IERRSPC = 4, IERREBZ = 8 ) .TP 20 .ti +4 LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, QUICKRETURN, VALEIG, WANTZ .TP 20 .ti +4 CHARACTER ORDER .TP 20 .ti +4 INTEGER CSRC_A, I, IACOL, IAROW, ICOFFA, IINFO, INDD, INDD2, INDE, INDE2, INDIBL, INDISP, INDTAU, INDWORK, IROFFA, IROFFZ, ISCALE, ISIZESTEBZ, ISIZESTEIN, IZROW, LALLWORK, LIWMIN, LLWORK, LWMIN, MAXEIGS, MB_A, MB_Z, MQ0, MYCOL, MYROW, NB, NB_A, NB_Z, NEIG, NN, NNP, NP0, NPCOL, NPROCS, NPROW, NSPLIT, NZZ, OFFSET, RSRC_A, RSRC_Z, SIZEORMTR, SIZESTEIN, SIZESYEVX .TP 20 .ti +4 DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM, VLL, VUU .TP 20 .ti +4 INTEGER IDUM1( 4 ), IDUM2( 4 ) .TP 20 .ti +4 LOGICAL LSAME .TP 20 .ti +4 INTEGER ICEIL, INDXG2P, NUMROC .TP 20 .ti +4 DOUBLE PRECISION PDLAMCH, PDLANSY .TP 20 .ti +4 EXTERNAL LSAME, ICEIL, INDXG2P, NUMROC, PDLAMCH, PDLANSY .TP 20 .ti +4 EXTERNAL BLACS_GRIDINFO, CHK1MAT, DGEBR2D, DGEBS2D, DLASRT, DSCAL, IGAMN2D, PCHK2MAT, PDELGET, PDLARED1D, PDLASCL, PDORMTR, PDSTEBZ, PDSTEIN, PDSYTRD, PXERBLA .TP 20 .ti +4 INTRINSIC ABS, DBLE, ICHAR, MAX, MIN, MOD, SQRT .TP 20 .ti +4 IF( BLOCK_CYCLIC_2D*CSRC_*CTXT_*DLEN_*DTYPE_*LLD_*MB_*M_*NB_*N_* RSRC_.LT.0 )RETURN .TP 20 .ti +4 QUICKRETURN = ( N.EQ.0 ) .TP 20 .ti +4 CALL BLACS_GRIDINFO( DESCA( CTXT_ ), NPROW, NPCOL, MYROW, MYCOL ) .TP 20 .ti +4 INFO = 0 .TP 20 .ti +4 IF( NPROW.EQ.-1 ) THEN .TP 20 .ti +4 INFO = -( 800+CTXT_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCZ( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+CTXT_ ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 CALL CHK1MAT( N, 4, N, 4, IA, JA, DESCA, 8, INFO ) .TP 20 .ti +4 CALL CHK1MAT( N, 4, N, 4, IZ, JZ, DESCZ, 21, INFO ) .TP 20 .ti +4 IF( INFO.EQ.0 ) THEN .TP 20 .ti +4 SAFMIN = PDLAMCH( DESCA( CTXT_ ), 'Safe minimum' ) .TP 20 .ti +4 EPS = PDLAMCH( DESCA( CTXT_ ), 'Precision' ) .TP 20 .ti +4 SMLNUM = SAFMIN / EPS .TP 20 .ti +4 BIGNUM = ONE / SMLNUM .TP 20 .ti +4 RMIN = SQRT( SMLNUM ) .TP 20 .ti +4 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) .TP 20 .ti +4 NPROCS = NPROW*NPCOL .TP 20 .ti +4 LOWER = LSAME( UPLO, 'L' ) .TP 20 .ti +4 WANTZ = LSAME( JOBZ, 'V' ) .TP 20 .ti +4 ALLEIG = LSAME( RANGE, 'A' ) .TP 20 .ti +4 VALEIG = LSAME( RANGE, 'V' ) .TP 20 .ti +4 INDEIG = LSAME( RANGE, 'I' ) .TP 20 .ti +4 INDTAU = 1 .TP 20 .ti +4 INDE = INDTAU + N .TP 20 .ti +4 INDD = INDE + N .TP 20 .ti +4 INDD2 = INDD + N .TP 20 .ti +4 INDE2 = INDD2 + N .TP 20 .ti +4 INDWORK = INDE2 + N .TP 20 .ti +4 LLWORK = LWORK - INDWORK + 1 .TP 20 .ti +4 ISIZESTEIN = 3*N + NPROCS + 1 .TP 20 .ti +4 ISIZESTEBZ = MAX( 4*N, 14, NPROCS ) .TP 20 .ti +4 INDIBL = ( MAX( ISIZESTEIN, ISIZESTEBZ ) ) + 1 .TP 20 .ti +4 INDISP = INDIBL + N .TP 20 .ti +4 LQUERY = .FALSE. .TP 20 .ti +4 IF( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) LQUERY = .TRUE. .TP 20 .ti +4 NNP = MAX( N, NPROCS+1, 4 ) .TP 20 .ti +4 LIWMIN = 6*NNP .TP 20 .ti +4 NPROCS = NPROW*NPCOL .TP 20 .ti +4 NB_A = DESCA( NB_ ) .TP 20 .ti +4 MB_A = DESCA( MB_ ) .TP 20 .ti +4 NB_Z = DESCZ( NB_ ) .TP 20 .ti +4 MB_Z = DESCZ( MB_ ) .TP 20 .ti +4 NB = NB_A .TP 20 .ti +4 NN = MAX( N, NB, 2 ) .TP 20 .ti +4 RSRC_A = DESCA( RSRC_ ) .TP 20 .ti +4 CSRC_A = DESCA( CSRC_ ) .TP 20 .ti +4 RSRC_Z = DESCZ( RSRC_ ) .TP 20 .ti +4 IROFFA = MOD( IA-1, MB_A ) .TP 20 .ti +4 ICOFFA = MOD( JA-1, NB_A ) .TP 20 .ti +4 IROFFZ = MOD( IZ-1, MB_A ) .TP 20 .ti +4 IAROW = INDXG2P( 1, NB_A, MYROW, RSRC_A, NPROW ) .TP 20 .ti +4 IACOL = INDXG2P( 1, MB_A, MYCOL, CSRC_A, NPCOL ) .TP 20 .ti +4 IZROW = INDXG2P( 1, NB_A, MYROW, RSRC_Z, NPROW ) .TP 20 .ti +4 NP0 = NUMROC( N+IROFFA, NB_Z, MYROW, IAROW, NPROW ) .TP 20 .ti +4 MQ0 = NUMROC( N+ICOFFA, NB_Z, MYCOL, IACOL, NPCOL ) .TP 20 .ti +4 IF( ( .NOT.WANTZ ) .OR. ( VALEIG .AND. ( .NOT.LQUERY ) ) ) THEN .TP 20 .ti +4 LWMIN = 5*N + MAX( 5*NN, NB*( NP0+1 ) ) .TP 20 .ti +4 NEIG = 0 .TP 20 .ti +4 ELSE .TP 20 .ti +4 IF( ALLEIG .OR. VALEIG ) THEN .TP 20 .ti +4 NEIG = N .TP 20 .ti +4 ELSE IF( INDEIG ) THEN .TP 20 .ti +4 NEIG = IU - IL + 1 .TP 20 .ti +4 END IF .TP 20 .ti +4 MQ0 = NUMROC( MAX( NEIG, NB, 2 ), NB, MYCOL, IACOL, NPCOL ) .TP 20 .ti +4 LWMIN = 5*N + MAX( 5*NN, NP0*MQ0+2*NB*NB ) + ICEIL( NEIG, NPROW*NPCOL )*NN .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( INFO.EQ.0 ) THEN .TP 20 .ti +4 IF( MYROW.EQ.0 .AND. MYCOL.EQ.0 ) THEN .TP 20 .ti +4 WORK( 1 ) = ABSTOL .TP 20 .ti +4 IF( VALEIG ) THEN .TP 20 .ti +4 WORK( 2 ) = VL .TP 20 .ti +4 WORK( 3 ) = VU .TP 20 .ti +4 ELSE .TP 20 .ti +4 WORK( 2 ) = ZERO .TP 20 .ti +4 WORK( 3 ) = ZERO .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL DGEBS2D( DESCA( CTXT_ ), 'ALL', ' ', 3, 1, WORK, 3 ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 CALL DGEBR2D( DESCA( CTXT_ ), 'ALL', ' ', 3, 1, WORK, 3, 0, 0 ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN .TP 20 .ti +4 INFO = -1 .TP 20 .ti +4 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN .TP 20 .ti +4 INFO = -2 .TP 20 .ti +4 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN .TP 20 .ti +4 INFO = -3 .TP 20 .ti +4 ELSE IF( VALEIG .AND. N.GT.0 .AND. VU.LE.VL ) THEN .TP 20 .ti +4 INFO = -10 .TP 20 .ti +4 ELSE IF( INDEIG .AND. ( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) ) THEN .TP 20 .ti +4 INFO = -11 .TP 20 .ti +4 ELSE IF( INDEIG .AND. ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) ) THEN .TP 20 .ti +4 INFO = -12 .TP 20 .ti +4 ELSE IF( LWORK.LT.LWMIN .AND. LWORK.NE.-1 ) THEN .TP 20 .ti +4 INFO = -23 .TP 20 .ti +4 ELSE IF( LIWORK.LT.LIWMIN .AND. LIWORK.NE.-1 ) THEN .TP 20 .ti +4 INFO = -25 .TP 20 .ti +4 ELSE IF( VALEIG .AND. ( ABS( WORK( 2 )-VL ).GT.FIVE*EPS* ABS( VL ) ) ) THEN .TP 20 .ti +4 INFO = -9 .TP 20 .ti +4 ELSE IF( VALEIG .AND. ( ABS( WORK( 3 )-VU ).GT.FIVE*EPS* ABS( VU ) ) ) THEN .TP 20 .ti +4 INFO = -10 .TP 20 .ti +4 ELSE IF( ABS( WORK( 1 )-ABSTOL ).GT.FIVE*EPS*ABS( ABSTOL ) ) THEN .TP 20 .ti +4 INFO = -13 .TP 20 .ti +4 ELSE IF( IROFFA.NE.IROFFZ ) THEN .TP 20 .ti +4 INFO = -19 .TP 20 .ti +4 ELSE IF( IROFFA.NE.0 ) THEN .TP 20 .ti +4 INFO = -6 .TP 20 .ti +4 ELSE IF( IAROW.NE.IZROW ) THEN .TP 20 .ti +4 INFO = -19 .TP 20 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCA( NB_ ) ) THEN .TP 20 .ti +4 INFO = -( 800+NB_ ) .TP 20 .ti +4 ELSE IF( DESCA( M_ ).NE.DESCZ( M_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+M_ ) .TP 20 .ti +4 ELSE IF( DESCA( N_ ).NE.DESCZ( N_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+N_ ) .TP 20 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCZ( MB_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+MB_ ) .TP 20 .ti +4 ELSE IF( DESCA( NB_ ).NE.DESCZ( NB_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+NB_ ) .TP 20 .ti +4 ELSE IF( DESCA( RSRC_ ).NE.DESCZ( RSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+RSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CSRC_ ).NE.DESCZ( CSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+CSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCZ( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+CTXT_ ) .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 IDUM1( 1 ) = ICHAR( 'V' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 1 ) = ICHAR( 'N' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 1 ) = 1 .TP 20 .ti +4 IF( LOWER ) THEN .TP 20 .ti +4 IDUM1( 2 ) = ICHAR( 'L' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 2 ) = ICHAR( 'U' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 2 ) = 2 .TP 20 .ti +4 IF( ALLEIG ) THEN .TP 20 .ti +4 IDUM1( 3 ) = ICHAR( 'A' ) .TP 20 .ti +4 ELSE IF( INDEIG ) THEN .TP 20 .ti +4 IDUM1( 3 ) = ICHAR( 'I' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 3 ) = ICHAR( 'V' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 3 ) = 3 .TP 20 .ti +4 IF( LQUERY ) THEN .TP 20 .ti +4 IDUM1( 4 ) = -1 .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 4 ) = 1 .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 4 ) = 4 .TP 20 .ti +4 CALL PCHK2MAT( N, 4, N, 4, IA, JA, DESCA, 8, N, 4, N, 4, IZ, JZ, DESCZ, 21, 4, IDUM1, IDUM2, INFO ) .TP 20 .ti +4 WORK( 1 ) = DBLE( LWMIN ) .TP 20 .ti +4 IWORK( 1 ) = LIWMIN .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( INFO.NE.0 ) THEN .TP 20 .ti +4 CALL PXERBLA( DESCA( CTXT_ ), 'PDSYEVX', -INFO ) .TP 20 .ti +4 RETURN .TP 20 .ti +4 ELSE IF( LQUERY ) THEN .TP 20 .ti +4 RETURN .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( QUICKRETURN ) THEN .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 NZ = 0 .TP 20 .ti +4 ICLUSTR( 1 ) = 0 .TP 20 .ti +4 END IF .TP 20 .ti +4 M = 0 .TP 20 .ti +4 WORK( 1 ) = DBLE( LWMIN ) .TP 20 .ti +4 IWORK( 1 ) = LIWMIN .TP 20 .ti +4 RETURN .TP 20 .ti +4 END IF .TP 20 .ti +4 ABSTLL = ABSTOL .TP 20 .ti +4 ISCALE = 0 .TP 20 .ti +4 IF( VALEIG ) THEN .TP 20 .ti +4 VLL = VL .TP 20 .ti +4 VUU = VU .TP 20 .ti +4 ELSE .TP 20 .ti +4 VLL = ZERO .TP 20 .ti +4 VUU = ZERO .TP 20 .ti +4 END IF .TP 20 .ti +4 ANRM = PDLANSY( '1', UPLO, N, A, IA, JA, DESCA, WORK( INDWORK ) ) .TP 20 .ti +4 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN .TP 20 .ti +4 ISCALE = 1 .TP 20 .ti +4 SIGMA = RMIN / ANRM .TP 20 .ti +4 ANRM = ANRM*SIGMA .TP 20 .ti +4 ELSE IF( ANRM.GT.RMAX ) THEN .TP 20 .ti +4 ISCALE = 1 .TP 20 .ti +4 SIGMA = RMAX / ANRM .TP 20 .ti +4 ANRM = ANRM*SIGMA .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( ISCALE.EQ.1 ) THEN .TP 20 .ti +4 CALL PDLASCL( UPLO, ONE, SIGMA, N, N, A, IA, JA, DESCA, IINFO ) .TP 20 .ti +4 IF( ABSTOL.GT.0 ) ABSTLL = ABSTOL*SIGMA .TP 20 .ti +4 IF( VALEIG ) THEN .TP 20 .ti +4 VLL = VL*SIGMA .TP 20 .ti +4 VUU = VU*SIGMA .TP 20 .ti +4 IF( VUU.EQ.VLL ) THEN .TP 20 .ti +4 VUU = VUU + 2*MAX( ABS( VUU )*EPS, SAFMIN ) .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 LALLWORK = LLWORK .TP 20 .ti +4 CALL PDSYTRD( UPLO, N, A, IA, JA, DESCA, WORK( INDD ), WORK( INDE ), WORK( INDTAU ), WORK( INDWORK ), LLWORK, IINFO ) .TP 20 .ti +4 OFFSET = 0 .TP 20 .ti +4 IF( IA.EQ.1 .AND. JA.EQ.1 .AND. RSRC_A.EQ.0 .AND. CSRC_A.EQ.0 ) THEN .TP 20 .ti +4 CALL PDLARED1D( N, IA, JA, DESCA, WORK( INDD ), WORK( INDD2 ), WORK( INDWORK ), LLWORK ) .TP 20 .ti +4 CALL PDLARED1D( N, IA, JA, DESCA, WORK( INDE ), WORK( INDE2 ), WORK( INDWORK ), LLWORK ) .TP 20 .ti +4 IF( .NOT.LOWER ) OFFSET = 1 .TP 20 .ti +4 ELSE .TP 20 .ti +4 DO 10 I = 1, N .TP 20 .ti +4 CALL PDELGET( 'A', ' ', WORK( INDD2+I-1 ), A, I+IA-1, I+JA-1, DESCA ) .TP 20 .ti +4 10 CONTINUE .TP 20 .ti +4 IF( LSAME( UPLO, 'U' ) ) THEN .TP 20 .ti +4 DO 20 I = 1, N - 1 .TP 20 .ti +4 CALL PDELGET( 'A', ' ', WORK( INDE2+I-1 ), A, I+IA-1, I+JA, DESCA ) .TP 20 .ti +4 20 CONTINUE .TP 20 .ti +4 ELSE .TP 20 .ti +4 DO 30 I = 1, N - 1 .TP 20 .ti +4 CALL PDELGET( 'A', ' ', WORK( INDE2+I-1 ), A, I+IA, I+JA-1, DESCA ) .TP 20 .ti +4 30 CONTINUE .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 ORDER = 'b' .TP 20 .ti +4 ELSE .TP 20 .ti +4 ORDER = 'e' .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PDSTEBZ( DESCA( CTXT_ ), RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, WORK( INDD2 ), WORK( INDE2+OFFSET ), M, NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWORK ), LLWORK, IWORK( 1 ), ISIZESTEBZ, IINFO ) .TP 20 .ti +4 IF( IINFO.NE.0 ) THEN .TP 20 .ti +4 INFO = INFO + IERREBZ .TP 20 .ti +4 DO 40 I = 1, M .TP 20 .ti +4 IWORK( INDIBL+I-1 ) = ABS( IWORK( INDIBL+I-1 ) ) .TP 20 .ti +4 40 CONTINUE .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 IF( VALEIG ) THEN .TP 20 .ti +4 CALL IGAMN2D( DESCA( CTXT_ ), 'A', ' ', 1, 1, LALLWORK, 1, 1, 1, -1, -1, -1 ) .TP 20 .ti +4 MAXEIGS = DESCZ( N_ ) .TP 20 .ti +4 DO 50 NZ = MIN( MAXEIGS, M ), 0, -1 .TP 20 .ti +4 MQ0 = NUMROC( NZ, NB, 0, 0, NPCOL ) .TP 20 .ti +4 SIZESTEIN = ICEIL( NZ, NPROCS )*N + MAX( 5*N, NP0*MQ0 ) .TP 20 .ti +4 SIZEORMTR = MAX( ( NB*( NB-1 ) ) / 2, ( MQ0+NP0 )*NB ) + NB*NB .TP 20 .ti +4 SIZESYEVX = MAX( SIZESTEIN, SIZEORMTR ) .TP 20 .ti +4 IF( SIZESYEVX.LE.LALLWORK ) GO TO 60 .TP 20 .ti +4 50 CONTINUE .TP 20 .ti +4 60 CONTINUE .TP 20 .ti +4 ELSE .TP 20 .ti +4 NZ = M .TP 20 .ti +4 END IF .TP 20 .ti +4 NZ = MAX( NZ, 0 ) .TP 20 .ti +4 IF( NZ.NE.M ) THEN .TP 20 .ti +4 INFO = INFO + IERRSPC .TP 20 .ti +4 DO 70 I = 1, M .TP 20 .ti +4 IFAIL( I ) = 0 .TP 20 .ti +4 70 CONTINUE .TP 20 .ti +4 IF( NSPLIT.GT.1 ) THEN .TP 20 .ti +4 CALL DLASRT( 'I', M, W, IINFO ) .TP 20 .ti +4 IF( NZ.GT.0 ) THEN .TP 20 .ti +4 VUU = W( NZ ) - TEN*( EPS*ANRM+SAFMIN ) .TP 20 .ti +4 IF( VLL.GE.VUU ) THEN .TP 20 .ti +4 NZZ = 0 .TP 20 .ti +4 ELSE .TP 20 .ti +4 CALL PDSTEBZ( DESCA( CTXT_ ), RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, WORK( INDD2 ), WORK( INDE2+OFFSET ), NZZ, NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWORK ), LLWORK, IWORK( 1 ), ISIZESTEBZ, IINFO ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( MOD( INFO / IERREBZ, 1 ).EQ.0 ) THEN .TP 20 .ti +4 IF( NZZ.GT.NZ .OR. IINFO.NE.0 ) THEN .TP 20 .ti +4 INFO = INFO + IERREBZ .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 NZ = MIN( NZ, NZZ ) .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PDSTEIN( N, WORK( INDD2 ), WORK( INDE2+OFFSET ), NZ, W, IWORK( INDIBL ), IWORK( INDISP ), ORFAC, Z, IZ, JZ, DESCZ, WORK( INDWORK ), LALLWORK, IWORK( 1 ), ISIZESTEIN, IFAIL, ICLUSTR, GAP, IINFO ) .TP 20 .ti +4 IF( IINFO.GE.NZ+1 ) INFO = INFO + IERRCLS .TP 20 .ti +4 IF( MOD( IINFO, NZ+1 ).NE.0 ) INFO = INFO + IERREIN .TP 20 .ti +4 IF( NZ.GT.0 ) THEN .TP 20 .ti +4 CALL PDORMTR( 'L', UPLO, 'N', N, NZ, A, IA, JA, DESCA, WORK( INDTAU ), Z, IZ, JZ, DESCZ, WORK( INDWORK ), LLWORK, IINFO ) .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( ISCALE.EQ.1 ) THEN .TP 20 .ti +4 CALL DSCAL( M, ONE / SIGMA, W, 1 ) .TP 20 .ti +4 END IF .TP 20 .ti +4 WORK( 1 ) = DBLE( LWMIN ) .TP 20 .ti +4 IWORK( 1 ) = LIWMIN .TP 20 .ti +4 RETURN .TP 20 .ti +4 END .SH PURPOSE scalapack-doc-1.5/man/manl/pdsygs2.l0100644000056400000620000001420406335610637017021 0ustar pfrauenfstaff.TH PDSYGS2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDSYGS2 - reduce a real symmetric-definite generalized eigenproblem to standard form .SH SYNOPSIS .TP 20 SUBROUTINE PDSYGS2( IBTYPE, UPLO, N, A, IA, JA, DESCA, B, IB, JB, DESCB, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IB, IBTYPE, INFO, JA, JB, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), B( * ) .SH PURPOSE PDSYGS2 reduces a real symmetric-definite generalized eigenproblem to standard form. In the following sub( A ) denotes A( IA:IA+N-1, JA:JA+N-1 ) and sub( B ) denotes B( IB:IB+N-1, JB:JB+N-1 ). .br If IBTYPE = 1, the problem is sub( A )*x = lambda*sub( B )*x, and sub( A ) is overwritten by inv(U**T)*sub( A )*inv(U) or inv(L)*sub( A )*inv(L**T) .br If IBTYPE = 2 or 3, the problem is sub( A )*sub( B )*x = lambda*x or sub( B )*sub( A )*x = lambda*x, and sub( A ) is overwritten by U*sub( A )*U**T or L**T*sub( A )*L. .br sub( B ) must have been previously factorized as U**T*U or L*L**T by PDPOTRF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 9 IBTYPE (global input) INTEGER = 1: compute inv(U**T)*sub( A )*inv(U) or inv(L)*sub( A )*inv(L**T); = 2 or 3: compute U*sub( A )*U**T or L**T*sub( A )*L. .TP 8 UPLO (global input) CHARACTER .br = 'U': Upper triangle of sub( A ) is stored and sub( B ) is factored as U**T*U; = 'L': Lower triangle of sub( A ) is stored and sub( B ) is factored as L*L**T. .TP 8 N (global input) INTEGER The order of the matrices sub( A ) and sub( B ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N symmetric distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced. On exit, if INFO = 0, the transformed matrix, stored in the same format as sub( A ). .TP 8 IA (global input) INTEGER A's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JA (global input) INTEGER A's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+N-1)). On entry, this array contains the local pieces of the triangular factor from the Cholesky factorization of sub( B ), as returned by PDPOTRF. .TP 8 IB (global input) INTEGER B's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JB (global input) INTEGER B's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pdsygst.l0100644000056400000620000001465306335610637017133 0ustar pfrauenfstaff.TH PDSYGST l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDSYGST - reduce a real symmetric-definite generalized eigenproblem to standard form .SH SYNOPSIS .TP 20 SUBROUTINE PDSYGST( IBTYPE, UPLO, N, A, IA, JA, DESCA, B, IB, JB, DESCB, SCALE, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IB, IBTYPE, INFO, JA, JB, N .TP 20 .ti +4 DOUBLE PRECISION SCALE .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), B( * ) .SH PURPOSE PDSYGST reduces a real symmetric-definite generalized eigenproblem to standard form. In the following sub( A ) denotes A( IA:IA+N-1, JA:JA+N-1 ) and sub( B ) denotes B( IB:IB+N-1, JB:JB+N-1 ). .br If IBTYPE = 1, the problem is sub( A )*x = lambda*sub( B )*x, and sub( A ) is overwritten by inv(U**T)*sub( A )*inv(U) or inv(L)*sub( A )*inv(L**T) .br If IBTYPE = 2 or 3, the problem is sub( A )*sub( B )*x = lambda*x or sub( B )*sub( A )*x = lambda*x, and sub( A ) is overwritten by U*sub( A )*U**T or L**T*sub( A )*L. .br sub( B ) must have been previously factorized as U**T*U or L*L**T by PDPOTRF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 9 IBTYPE (global input) INTEGER = 1: compute inv(U**T)*sub( A )*inv(U) or inv(L)*sub( A )*inv(L**T); = 2 or 3: compute U*sub( A )*U**T or L**T*sub( A )*L. .TP 8 UPLO (global input) CHARACTER .br = 'U': Upper triangle of sub( A ) is stored and sub( B ) is factored as U**T*U; = 'L': Lower triangle of sub( A ) is stored and sub( B ) is factored as L*L**T. .TP 8 N (global input) INTEGER The order of the matrices sub( A ) and sub( B ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N symmetric distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced. On exit, if INFO = 0, the transformed matrix, stored in the same format as sub( A ). .TP 8 IA (global input) INTEGER A's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JA (global input) INTEGER A's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+N-1)). On entry, this array contains the local pieces of the triangular factor from the Cholesky factorization of sub( B ), as returned by PDPOTRF. .TP 8 IB (global input) INTEGER B's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JB (global input) INTEGER B's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 SCALE (global output) DOUBLE PRECISION Amount by which the eigenvalues should be scaled to compensate for the scaling performed in this routine. At present, SCALE is always returned as 1.0, it is returned here to allow for future enhancement. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pdsygvx.l0100644000056400000620000002406606335610637017141 0ustar pfrauenfstaff.TH PDSYGVX l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME .SH SYNOPSIS .TP 20 SUBROUTINE PDSYGVX( IBTYPE, JOBZ, RANGE, UPLO, N, A, IA, JA, DESCA, B, IB, JB, DESCB, VL, VU, IL, IU, ABSTOL, M, NZ, W, ORFAC, Z, IZ, JZ, DESCZ, WORK, LWORK, IWORK, LIWORK, IFAIL, ICLUSTR, GAP, INFO ) .TP 20 .ti +4 CHARACTER JOBZ, RANGE, UPLO .TP 20 .ti +4 INTEGER IA, IB, IBTYPE, IL, INFO, IU, IZ, JA, JB, JZ, LIWORK, LWORK, M, N, NZ .TP 20 .ti +4 DOUBLE PRECISION ABSTOL, ORFAC, VL, VU .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ), DESCZ( * ), ICLUSTR( * ), IFAIL( * ), IWORK( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), B( * ), GAP( * ), W( * ), WORK( * ), Z( * ) .TP 20 .ti +4 INTEGER BLOCK_CYCLIC_2D, DLEN_, DTYPE_, CTXT_, M_, N_, MB_, NB_, RSRC_, CSRC_, LLD_ .TP 20 .ti +4 PARAMETER ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1, CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6, RSRC_ = 7, CSRC_ = 8, LLD_ = 9 ) .TP 20 .ti +4 DOUBLE PRECISION ONE .TP 20 .ti +4 PARAMETER ( ONE = 1.0D+0 ) .TP 20 .ti +4 DOUBLE PRECISION FIVE, ZERO .TP 20 .ti +4 PARAMETER ( FIVE = 5.0D+0, ZERO = 0.0D+0 ) .TP 20 .ti +4 INTEGER IERRNPD .TP 20 .ti +4 PARAMETER ( IERRNPD = 16 ) .TP 20 .ti +4 LOGICAL ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER IACOL, IAROW, IBCOL, IBROW, ICOFFA, ICOFFB, ICTXT, IROFFA, IROFFB, LIWMIN, LWMIN, MQ0, MYCOL, MYROW, NB, NEIG, NN, NP0, NPCOL, NPROW .TP 20 .ti +4 DOUBLE PRECISION EPS, SCALE .TP 20 .ti +4 INTEGER IDUM1( 5 ), IDUM2( 5 ) .TP 20 .ti +4 LOGICAL LSAME .TP 20 .ti +4 INTEGER ICEIL, INDXG2P, NUMROC .TP 20 .ti +4 DOUBLE PRECISION PDLAMCH .TP 20 .ti +4 EXTERNAL LSAME, ICEIL, INDXG2P, NUMROC, PDLAMCH .TP 20 .ti +4 EXTERNAL BLACS_GRIDINFO, CHK1MAT, DGEBR2D, DGEBS2D, DSCAL, PCHK1MAT, PCHK2MAT, PDPOTRF, PDSYEVX, PDSYGST, PDTRMM, PDTRSM, PXERBLA .TP 20 .ti +4 INTRINSIC ABS, DBLE, ICHAR, MAX, MIN, MOD .TP 20 .ti +4 IF( BLOCK_CYCLIC_2D*CSRC_*CTXT_*DLEN_*DTYPE_*LLD_*MB_*M_*NB_*N_* RSRC_.LT.0 )RETURN .TP 20 .ti +4 ICTXT = DESCA( CTXT_ ) .TP 20 .ti +4 CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL ) .TP 20 .ti +4 INFO = 0 .TP 20 .ti +4 IF( NPROW.EQ.-1 ) THEN .TP 20 .ti +4 INFO = -( 900+CTXT_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCB( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+CTXT_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCZ( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 2600+CTXT_ ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 EPS = PDLAMCH( DESCA( CTXT_ ), 'Precision' ) .TP 20 .ti +4 WANTZ = LSAME( JOBZ, 'V' ) .TP 20 .ti +4 UPPER = LSAME( UPLO, 'U' ) .TP 20 .ti +4 ALLEIG = LSAME( RANGE, 'A' ) .TP 20 .ti +4 VALEIG = LSAME( RANGE, 'V' ) .TP 20 .ti +4 INDEIG = LSAME( RANGE, 'I' ) .TP 20 .ti +4 CALL CHK1MAT( N, 4, N, 4, IA, JA, DESCA, 9, INFO ) .TP 20 .ti +4 CALL CHK1MAT( N, 4, N, 4, IB, JB, DESCB, 13, INFO ) .TP 20 .ti +4 CALL CHK1MAT( N, 4, N, 4, IZ, JZ, DESCZ, 26, INFO ) .TP 20 .ti +4 IF( INFO.EQ.0 ) THEN .TP 20 .ti +4 IF( MYROW.EQ.0 .AND. MYCOL.EQ.0 ) THEN .TP 20 .ti +4 WORK( 1 ) = ABSTOL .TP 20 .ti +4 IF( VALEIG ) THEN .TP 20 .ti +4 WORK( 2 ) = VL .TP 20 .ti +4 WORK( 3 ) = VU .TP 20 .ti +4 ELSE .TP 20 .ti +4 WORK( 2 ) = ZERO .TP 20 .ti +4 WORK( 3 ) = ZERO .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL DGEBS2D( DESCA( CTXT_ ), 'ALL', ' ', 3, 1, WORK, 3 ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 CALL DGEBR2D( DESCA( CTXT_ ), 'ALL', ' ', 3, 1, WORK, 3, 0, 0 ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IAROW = INDXG2P( IA, DESCA( MB_ ), MYROW, DESCA( RSRC_ ), NPROW ) .TP 20 .ti +4 IBROW = INDXG2P( IB, DESCB( MB_ ), MYROW, DESCB( RSRC_ ), NPROW ) .TP 20 .ti +4 IACOL = INDXG2P( JA, DESCA( NB_ ), MYCOL, DESCA( CSRC_ ), NPCOL ) .TP 20 .ti +4 IBCOL = INDXG2P( JB, DESCB( NB_ ), MYCOL, DESCB( CSRC_ ), NPCOL ) .TP 20 .ti +4 IROFFA = MOD( IA-1, DESCA( MB_ ) ) .TP 20 .ti +4 ICOFFA = MOD( JA-1, DESCA( NB_ ) ) .TP 20 .ti +4 IROFFB = MOD( IB-1, DESCB( MB_ ) ) .TP 20 .ti +4 ICOFFB = MOD( JB-1, DESCB( NB_ ) ) .TP 20 .ti +4 LQUERY = .FALSE. .TP 20 .ti +4 IF( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) LQUERY = .TRUE. .TP 20 .ti +4 LIWMIN = 6*MAX( N, ( NPROW*NPCOL )+1, 4 ) .TP 20 .ti +4 NB = DESCA( MB_ ) .TP 20 .ti +4 NN = MAX( N, NB, 2 ) .TP 20 .ti +4 NP0 = NUMROC( NN, NB, 0, 0, NPROW ) .TP 20 .ti +4 IF( ( .NOT.WANTZ ) .OR. ( VALEIG .AND. ( .NOT.LQUERY ) ) ) THEN .TP 20 .ti +4 LWMIN = 5*N + MAX( 5*NN, NB*( NP0+1 ) ) .TP 20 .ti +4 NEIG = 0 .TP 20 .ti +4 ELSE .TP 20 .ti +4 IF( ALLEIG .OR. VALEIG ) THEN .TP 20 .ti +4 NEIG = N .TP 20 .ti +4 ELSE IF( INDEIG ) THEN .TP 20 .ti +4 NEIG = IU - IL + 1 .TP 20 .ti +4 END IF .TP 20 .ti +4 MQ0 = NUMROC( MAX( NEIG, NB, 2 ), NB, 0, 0, NPCOL ) .TP 20 .ti +4 LWMIN = 5*N + MAX( 5*NN, NP0*MQ0+2*NB*NB ) + ICEIL( NEIG, NPROW*NPCOL )*NN .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( IBTYPE.LT.1 .OR. IBTYPE.GT.3 ) THEN .TP 20 .ti +4 INFO = -1 .TP 20 .ti +4 ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN .TP 20 .ti +4 INFO = -2 .TP 20 .ti +4 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN .TP 20 .ti +4 INFO = -3 .TP 20 .ti +4 ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN .TP 20 .ti +4 INFO = -4 .TP 20 .ti +4 ELSE IF( N.LT.0 ) THEN .TP 20 .ti +4 INFO = -5 .TP 20 .ti +4 ELSE IF( IROFFA.NE.0 ) THEN .TP 20 .ti +4 INFO = -7 .TP 20 .ti +4 ELSE IF( ICOFFA.NE.0 ) THEN .TP 20 .ti +4 INFO = -8 .TP 20 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCA( NB_ ) ) THEN .TP 20 .ti +4 INFO = -( 900+NB_ ) .TP 20 .ti +4 ELSE IF( DESCA( M_ ).NE.DESCB( M_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+M_ ) .TP 20 .ti +4 ELSE IF( DESCA( N_ ).NE.DESCB( N_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+N_ ) .TP 20 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCB( MB_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+MB_ ) .TP 20 .ti +4 ELSE IF( DESCA( NB_ ).NE.DESCB( NB_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+NB_ ) .TP 20 .ti +4 ELSE IF( DESCA( RSRC_ ).NE.DESCB( RSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+RSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CSRC_ ).NE.DESCB( CSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+CSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCB( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+CTXT_ ) .TP 20 .ti +4 ELSE IF( DESCA( M_ ).NE.DESCZ( M_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+M_ ) .TP 20 .ti +4 ELSE IF( DESCA( N_ ).NE.DESCZ( N_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+N_ ) .TP 20 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCZ( MB_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+MB_ ) .TP 20 .ti +4 ELSE IF( DESCA( NB_ ).NE.DESCZ( NB_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+NB_ ) .TP 20 .ti +4 ELSE IF( DESCA( RSRC_ ).NE.DESCZ( RSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+RSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CSRC_ ).NE.DESCZ( CSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+CSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCZ( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+CTXT_ ) .TP 20 .ti +4 ELSE IF( IROFFB.NE.0 .OR. IBROW.NE.IAROW ) THEN .TP 20 .ti +4 INFO = -11 .TP 20 .ti +4 ELSE IF( ICOFFB.NE.0 .OR. IBCOL.NE.IACOL ) THEN .TP 20 .ti +4 INFO = -12 .TP 20 .ti +4 ELSE IF( VALEIG .AND. N.GT.0 .AND. VU.LE.VL ) THEN .TP 20 .ti +4 INFO = -15 .TP 20 .ti +4 ELSE IF( INDEIG .AND. ( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) ) THEN .TP 20 .ti +4 INFO = -16 .TP 20 .ti +4 ELSE IF( INDEIG .AND. ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) ) THEN .TP 20 .ti +4 INFO = -17 .TP 20 .ti +4 ELSE IF( VALEIG .AND. ( ABS( WORK( 2 )-VL ).GT.FIVE*EPS* ABS( VL ) ) ) THEN .TP 20 .ti +4 INFO = -14 .TP 20 .ti +4 ELSE IF( VALEIG .AND. ( ABS( WORK( 3 )-VU ).GT.FIVE*EPS* ABS( VU ) ) ) THEN .TP 20 .ti +4 INFO = -15 .TP 20 .ti +4 ELSE IF( ABS( WORK( 1 )-ABSTOL ).GT.FIVE*EPS*ABS( ABSTOL ) ) THEN .TP 20 .ti +4 INFO = -18 .TP 20 .ti +4 ELSE IF( LWORK.LT.LWMIN .AND. LWORK.NE.-1 ) THEN .TP 20 .ti +4 INFO = -28 .TP 20 .ti +4 ELSE IF( LIWORK.LT.LIWMIN .AND. LIWORK.NE.-1 ) THEN .TP 20 .ti +4 INFO = -30 .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM1( 1 ) = IBTYPE .TP 20 .ti +4 IDUM2( 1 ) = 1 .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 IDUM1( 2 ) = ICHAR( 'V' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 2 ) = ICHAR( 'N' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 2 ) = 2 .TP 20 .ti +4 IF( UPPER ) THEN .TP 20 .ti +4 IDUM1( 3 ) = ICHAR( 'U' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 3 ) = ICHAR( 'L' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 3 ) = 3 .TP 20 .ti +4 IF( ALLEIG ) THEN .TP 20 .ti +4 IDUM1( 4 ) = ICHAR( 'A' ) .TP 20 .ti +4 ELSE IF( INDEIG ) THEN .TP 20 .ti +4 IDUM1( 4 ) = ICHAR( 'I' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 4 ) = ICHAR( 'V' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 4 ) = 4 .TP 20 .ti +4 IF( LQUERY ) THEN .TP 20 .ti +4 IDUM1( 5 ) = -1 .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 5 ) = 1 .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 5 ) = 5 .TP 20 .ti +4 CALL PCHK2MAT( N, 4, N, 4, IA, JA, DESCA, 9, N, 4, N, 4, IB, JB, DESCB, 13, 5, IDUM1, IDUM2, INFO ) .TP 20 .ti +4 CALL PCHK1MAT( N, 4, N, 4, IZ, JZ, DESCZ, 26, 0, IDUM1, IDUM2, INFO ) .TP 20 .ti +4 END IF .TP 20 .ti +4 WORK( 1 ) = DBLE( LWMIN ) .TP 20 .ti +4 IWORK( 1 ) = LIWMIN .TP 20 .ti +4 IF( INFO.NE.0 ) THEN .TP 20 .ti +4 CALL PXERBLA( ICTXT, 'PDSYGVX ', -INFO ) .TP 20 .ti +4 RETURN .TP 20 .ti +4 ELSE IF( LQUERY ) THEN .TP 20 .ti +4 RETURN .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PDPOTRF( UPLO, N, B, IB, JB, DESCB, INFO ) .TP 20 .ti +4 IF( INFO.NE.0 ) THEN .TP 20 .ti +4 IFAIL( 1 ) = INFO .TP 20 .ti +4 INFO = IERRNPD .TP 20 .ti +4 RETURN .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PDSYGST( IBTYPE, UPLO, N, A, IA, JA, DESCA, B, IB, JB, DESCB, SCALE, INFO ) .TP 20 .ti +4 CALL PDSYEVX( JOBZ, RANGE, UPLO, N, A, IA, JA, DESCA, VL, VU, IL, IU, ABSTOL, M, NZ, W, ORFAC, Z, IZ, JZ, DESCZ, WORK, LWORK, IWORK, LIWORK, IFAIL, ICLUSTR, GAP, INFO ) .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 NEIG = M .TP 20 .ti +4 IF( IBTYPE.EQ.1 .OR. IBTYPE.EQ.2 ) THEN .TP 20 .ti +4 IF( UPPER ) THEN .TP 20 .ti +4 TRANS = 'N' .TP 20 .ti +4 ELSE .TP 20 .ti +4 TRANS = 'T' .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PDTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, NEIG, ONE, B, IB, JB, DESCB, Z, IZ, JZ, DESCZ ) .TP 20 .ti +4 ELSE IF( IBTYPE.EQ.3 ) THEN .TP 20 .ti +4 IF( UPPER ) THEN .TP 20 .ti +4 TRANS = 'T' .TP 20 .ti +4 ELSE .TP 20 .ti +4 TRANS = 'N' .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PDTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, NEIG, ONE, B, IB, JB, DESCB, Z, IZ, JZ, DESCZ ) .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( SCALE.NE.ONE ) THEN .TP 20 .ti +4 CALL DSCAL( N, SCALE, W, 1 ) .TP 20 .ti +4 END IF .TP 20 .ti +4 RETURN .TP 20 .ti +4 END .SH PURPOSE scalapack-doc-1.5/man/manl/pdsytd2.l0100644000056400000620000002031306335610637017015 0ustar pfrauenfstaff.TH PDSYTD2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDSYTD2 - reduce a real symmetric matrix sub( A ) to symmetric tridiagonal form T by an orthogonal similarity transformation .SH SYNOPSIS .TP 20 SUBROUTINE PDSYTD2( UPLO, N, A, IA, JA, DESCA, D, E, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), D( * ), E( * ), TAU( * ), WORK( * ) .SH PURPOSE PDSYTD2 reduces a real symmetric matrix sub( A ) to symmetric tridiagonal form T by an orthogonal similarity transformation: Q' * sub( A ) * Q = T, where sub( A ) = A(IA:IA+N-1,JA:JA+N-1). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the upper or lower triangular part of the symmetric matrix sub( A ) is stored: .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the symmetric distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced. On exit, if UPLO = 'U', the diagonal and first superdiagonal of sub( A ) are over- written by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of sub( A ) are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-1) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i). D is tied to the distributed matrix A. .TP 8 E (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-1) if UPLO = 'U', LOCc(JA+N-2) otherwise. The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. E is tied to the distributed matrix A. .TP 8 TAU (local output) DOUBLE PRECISION, array, dimension LOCc(JA+N-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK( 1 ) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 3*N. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors .br Q = H(n-1) . . . H(2) H(1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in .br A(ia:ia+i-2,ja+i), and tau in TAU(ja+i-1). .br If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors .br Q = H(1) H(2) . . . H(n-1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in .br A(ia+i+1:ia+n-1,ja+i-1), and tau in TAU(ja+i-1). .br The contents of sub( A ) on exit are illustrated by the following examples with n = 5: .br if UPLO = 'U': if UPLO = 'L': .br ( d e v2 v3 v4 ) ( d ) ( d e v3 v4 ) ( e d ) ( d e v4 ) ( v1 e d ) ( d e ) ( v1 v2 e d ) ( d ) ( v1 v2 v3 e d ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expression should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) with .br IROFFA = MOD( IA-1, MB_A ) and ICOFFA = MOD( JA-1, NB_A ). scalapack-doc-1.5/man/manl/pdsytrd.l0100644000056400000620000002075206335610637017124 0ustar pfrauenfstaff.TH PDSYTRD l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDSYTRD - reduce a real symmetric matrix sub( A ) to symmetric tridiagonal form T by an orthogonal similarity transformation .SH SYNOPSIS .TP 20 SUBROUTINE PDSYTRD( UPLO, N, A, IA, JA, DESCA, D, E, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), D( * ), E( * ), TAU( * ), WORK( * ) .SH PURPOSE PDSYTRD reduces a real symmetric matrix sub( A ) to symmetric tridiagonal form T by an orthogonal similarity transformation: Q' * sub( A ) * Q = T, where sub( A ) = A(IA:IA+N-1,JA:JA+N-1). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the upper or lower triangular part of the symmetric matrix sub( A ) is stored: .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the symmetric distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced. On exit, if UPLO = 'U', the diagonal and first superdiagonal of sub( A ) are over- written by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of sub( A ) are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-1) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i). D is tied to the distributed matrix A. .TP 8 E (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-1) if UPLO = 'U', LOCc(JA+N-2) otherwise. The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. E is tied to the distributed matrix A. .TP 8 TAU (local output) DOUBLE PRECISION, array, dimension LOCc(JA+N-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK( 1 ) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MAX( NB * ( NP +1 ), 3 * NB ) where NB = MB_A = NB_A, NP = NUMROC( N, NB, MYROW, IAROW, NPROW ), IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ). INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors .br Q = H(n-1) . . . H(2) H(1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in .br A(ia:ia+i-2,ja+i), and tau in TAU(ja+i-1). .br If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors .br Q = H(1) H(2) . . . H(n-1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in .br A(ia+i+1:ia+n-1,ja+i-1), and tau in TAU(ja+i-1). .br The contents of sub( A ) on exit are illustrated by the following examples with n = 5: .br if UPLO = 'U': if UPLO = 'L': .br ( d e v2 v3 v4 ) ( d ) ( d e v3 v4 ) ( e d ) ( d e v4 ) ( v1 e d ) ( d e ) ( v1 v2 e d ) ( d ) ( v1 v2 v3 e d ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expression should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA .AND. IROFFA.EQ.0 ) with IROFFA = MOD( IA-1, MB_A ) and ICOFFA = MOD( JA-1, NB_A ). scalapack-doc-1.5/man/manl/pdtrcon.l0100644000056400000620000001617406335610637017107 0ustar pfrauenfstaff.TH PDTRCON l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDTRCON - estimate the reciprocal of the condition number of a triangular distributed matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm .SH SYNOPSIS .TP 20 SUBROUTINE PDTRCON( NORM, UPLO, DIAG, N, A, IA, JA, DESCA, RCOND, WORK, LWORK, IWORK, LIWORK, INFO ) .TP 20 .ti +4 CHARACTER DIAG, NORM, UPLO .TP 20 .ti +4 INTEGER IA, JA, INFO, LIWORK, LWORK, N .TP 20 .ti +4 DOUBLE PRECISION RCOND .TP 20 .ti +4 INTEGER DESCA( * ), IWORK( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), WORK( * ) .SH PURPOSE PDTRCON estimates the reciprocal of the condition number of a triangular distributed matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm. The norm of A(IA:IA+N-1,JA:JA+N-1) is computed and an estimate is obtained for norm(inv(A(IA:IA+N-1,JA:JA+N-1))), then the reciprocal of the condition number is computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies whether the 1-norm condition number or the infinity-norm condition number is required: .br = '1' or 'O': 1-norm; .br = 'I': Infinity-norm. .TP 8 UPLO (global input) CHARACTER .br = 'U': A(IA:IA+N-1,JA:JA+N-1) is upper triangular; .br = 'L': A(IA:IA+N-1,JA:JA+N-1) is lower triangular. .TP 8 DIAG (global input) CHARACTER .br = 'N': A(IA:IA+N-1,JA:JA+N-1) is non-unit triangular; .br = 'U': A(IA:IA+N-1,JA:JA+N-1) is unit triangular. .TP 8 N (global input) INTEGER .br The order of the distributed matrix A(IA:IA+N-1,JA:JA+N-1). N >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension ( LLD_A, LOCc(JA+N-1) ). This array contains the local pieces of the triangular distributed matrix A(IA:IA+N-1,JA:JA+N-1). If UPLO = 'U', the leading N-by-N upper triangular part of this distributed matrix con- tains the upper triangular matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of this ditributed matrix contains the lower triangular matrix, and the strictly upper triangular part is not referenced. If DIAG = 'U', the diagonal elements of A(IA:IA+N-1,JA:JA+N-1) are also not referenced and are assumed to be 1. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 RCOND (global output) DOUBLE PRECISION The reciprocal of the condition number of the distributed matrix A(IA:IA+N-1,JA:JA+N-1), computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * .br norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 2*LOCr(N+MOD(IA-1,MB_A)) + LOCc(N+MOD(JA-1,NB_A)) + MAX( 2, MAX( NB_A*MAX( 1, CEIL(NPROW-1,NPCOL) ), LOCc(N+MOD(JA-1,NB_A)) + NB_A*MAX( 1, CEIL(NPCOL-1,NPROW) ) ). If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/local output) INTEGER array, dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK. .TP 8 LIWORK (local or global input) INTEGER The dimension of the array IWORK. LIWORK is local input and must be at least LIWORK >= LOCr(N+MOD(IA-1,MB_A)). If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pdtrrfs.l0100644000056400000620000002274706335610637017125 0ustar pfrauenfstaff.TH PDTRRFS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDTRRFS - provide error bounds and backward error estimates for the solution to a system of linear equations with a triangular coefficient matrix .SH SYNOPSIS .TP 20 SUBROUTINE PDTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, X, IX, JX, DESCX, FERR, BERR, WORK, LWORK, IWORK, LIWORK, INFO ) .TP 20 .ti +4 CHARACTER DIAG, TRANS, UPLO .TP 20 .ti +4 INTEGER INFO, IA, IB, IX, JA, JB, JX, LIWORK, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ), DESCX( * ), IWORK( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), B( * ), BERR( * ), FERR( * ), WORK( * ), X( * ) .SH PURPOSE PDTRRFS provides error bounds and backward error estimates for the solution to a system of linear equations with a triangular coefficient matrix. The solution matrix X must be computed by PDTRTRS or some other means before entering this routine. PDTRRFS does not do iterative refinement because doing so cannot improve the backward error. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br In the following comments, sub( A ), sub( X ) and sub( B ) denote respectively A(IA:IA+N-1,JA:JA+N-1), X(IX:IX+N-1,JX:JX+NRHS-1) and B(IB:IB+N-1,JB:JB+NRHS-1). .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 = 'U': sub( A ) is upper triangular; .br = 'L': sub( A ) is lower triangular. .TP 8 TRANS (global input) CHARACTER*1 Specifies the form of the system of equations. = 'N': sub( A ) * sub( X ) = sub( B ) (No transpose) .br = 'T': sub( A )**T * sub( X ) = sub( B ) (Transpose) .br = 'C': sub( A )**T * sub( X ) = sub( B ) (Conjugate transpose = Transpose) .TP 8 DIAG (global input) CHARACTER*1 = 'N': sub( A ) is non-unit triangular; .br = 'U': sub( A ) is unit triangular. .TP 8 N (global input) INTEGER The order of the matrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the matrices sub( B ) and sub( X ). NRHS >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1) ). This array contains the local pieces of the original triangular distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the distribu- ted matrix, and its strictly upper triangular part is not referenced. If DIAG = 'U', the diagonal elements of sub( A ) are also not referenced and are assumed to be 1. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input) DOUBLE PRECISION pointer into the local memory to an array of local dimension (LLD_B, LOCc(JB+NRHS-1) ). On entry, this array contains the the local pieces of the right hand sides sub( B ). .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 X (local input) DOUBLE PRECISION pointer into the local memory to an array of local dimension (LLD_X, LOCc(JX+NRHS-1) ). On entry, this array contains the the local pieces of the solution vectors sub( X ). .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 FERR (local output) DOUBLE PRECISION array of local dimension LOCc(JB+NRHS-1). The estimated forward error bounds for each solution vector of sub( X ). If XTRUE is the true solution, FERR bounds the magnitude of the largest entry in (sub( X ) - XTRUE) divided by the magnitude of the largest entry in sub( X ). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. This array is tied to the distributed matrix X. .TP 8 BERR (local output) DOUBLE PRECISION array of local dimension LOCc(JB+NRHS-1). The componentwise relative backward error of each solution vector (i.e., the smallest re- lative change in any entry of sub( A ) or sub( B ) that makes sub( X ) an exact solution). This array is tied to the distributed matrix X. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 3*LOCr( N + MOD( IA-1, MB_A ) ). If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/local output) INTEGER array, dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK. .TP 8 LIWORK (local or global input) INTEGER The dimension of the array IWORK. LIWORK is local input and must be at least LIWORK >= LOCr( N + MOD( IB-1, MB_B ) ). If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Notes ===== This routine temporarily returns when N <= 1. The distributed submatrices sub( X ) and sub( B ) should be distributed the same way on the same processes. These conditions ensure that sub( X ) and sub( B ) are "perfectly" aligned. Moreover, this routine requires the distributed submatrices sub( A ), sub( X ), and sub( B ) to be aligned on a block boundary, i.e., if f(x,y) = MOD( x-1, y ): f( IA, DESCA( MB_ ) ) = f( JA, DESCA( NB_ ) ) = 0, f( IB, DESCB( MB_ ) ) = f( JB, DESCB( NB_ ) ) = 0, and f( IX, DESCX( MB_ ) ) = f( JX, DESCX( NB_ ) ) = 0. scalapack-doc-1.5/man/manl/pdtrti2.l0100644000056400000620000001206706335610640017015 0ustar pfrauenfstaff.TH PDTRTI2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDTRTI2 - compute the inverse of a real upper or lower triangular block matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PDTRTI2( UPLO, DIAG, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER DIAG, UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ) .SH PURPOSE PDTRTI2 computes the inverse of a real upper or lower triangular block matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). This matrix should be contained in one and only one process memory space (local operation). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 = 'U': sub( A ) is upper triangular; .br = 'L': sub( A ) is lower triangular. .TP 8 DIAG (global input) CHARACTER*1 .br = 'N': sub( A ) is non-unit triangular .br = 'U': sub( A ) is unit triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)), this array contains the local pieces of the triangular matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of the matrix sub( A ) contains the upper triangular matrix, and the strictly lower triangular part of sub( A ) is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of the matrix sub( A ) contains the lower triangular matrix, and the strictly upper triangular part of sub( A ) is not referenced. If DIAG = 'U', the diagonal elements of sub( A ) are also not referenced and are assumed to be 1. On exit, the (triangular) inverse of the original matrix, in the same storage format. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pdtrtri.l0100644000056400000620000001215506335610640017113 0ustar pfrauenfstaff.TH PDTRTRI l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDTRTRI - compute the inverse of a upper or lower triangular distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PDTRTRI( UPLO, DIAG, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER DIAG, UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ) .SH PURPOSE PDTRTRI computes the inverse of a upper or lower triangular distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the distributed matrix sub( A ) is upper or lower triangular: .br = 'U': Upper triangular, .br = 'L': Lower triangular. .TP 8 DIAG (global input) CHARACTER Specifies whether or not the distributed matrix sub( A ) is unit triangular: .br = 'N': Non-unit triangular, .br = 'U': Unit triangular. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the triangular matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of the matrix sub( A ) contains the upper triangular matrix to be inverted, and the strictly lower triangular part of sub( A ) is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of the matrix sub( A ) contains the lower triangular matrix, and the strictly upper triangular part of sub( A ) is not referenced. On exit, the (triangular) inverse of the original matrix. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, A(IA+K-1,JA+K-1) is exactly zero. The triangular matrix sub( A ) is singular and its inverse can not be computed. scalapack-doc-1.5/man/manl/pdtrtrs.l0100644000056400000620000001444506335610640017131 0ustar pfrauenfstaff.TH PDTRTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDTRTRS - solve a triangular system of the form sub( A ) * X = sub( B ) or sub( A )**T * X = sub( B ), .SH SYNOPSIS .TP 20 SUBROUTINE PDTRTRS( UPLO, TRANS, DIAG, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, INFO ) .TP 20 .ti +4 CHARACTER DIAG, TRANS, UPLO .TP 20 .ti +4 INTEGER IA, IB, INFO, JA, JB, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), B( * ) .SH PURPOSE PDTRTRS solves a triangular system of the form where sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1) and is a triangular distributed matrix of order N, and B(IB:IB+N-1,JB:JB+NRHS-1) is an N-by-NRHS distributed matrix denoted by sub( B ). A check is made to verify that sub( A ) is nonsingular. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER = 'U': sub( A ) is upper triangular; .br = 'L': sub( A ) is lower triangular. .TP 8 TRANS (global input) CHARACTER .br Specifies the form of the system of equations: .br = 'N': Solve sub( A ) * X = sub( B ) (No transpose) .br = 'T': Solve sub( A )**T * X = sub( B ) (Transpose) .br = 'C': Solve sub( A )**T * X = sub( B ) (Transpose) .TP 8 DIAG (global input) CHARACTER .br = 'N': sub( A ) is non-unit triangular; .br = 'U': sub( A ) is unit triangular. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on i.e the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed matrix sub( B ). NRHS >= 0. .TP 8 A (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1) ). This array contains the local pieces of the distributed triangular matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular matrix, and the strictly lower triangular part of sub( A ) is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular matrix, and the strictly upper triangular part of sub( A ) is not referenced. If DIAG = 'U', the diagonal elements of sub( A ) are also not referenced and are assumed to be 1. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_B,LOCc(JB+NRHS-1)). On entry, this array contains the local pieces of the right hand side distributed matrix sub( B ). On exit, if INFO = 0, sub( B ) is overwritten by the solution matrix X. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = i, the i-th diagonal element of sub( A ) is zero, indicating that the submatrix is singular and the solutions X have not been computed. scalapack-doc-1.5/man/manl/pdtzrzf.l0100644000056400000620000001567606335610640017141 0ustar pfrauenfstaff.TH PDTZRZF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PDTZRZF - reduce the M-by-N ( M<=N ) real upper trapezoidal matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper triangular form by means of orthogonal transformations .SH SYNOPSIS .TP 20 SUBROUTINE PDTZRZF( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION A( * ), TAU( * ), WORK( * ) .SH PURPOSE PDTZRZF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper triangular form by means of orthogonal transformations. The upper trapezoidal matrix sub( A ) is factored as .br sub( A ) = ( R 0 ) * Z, .br where Z is an N-by-N orthogonal matrix and R is an M-by-M upper triangular matrix. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the leading M-by-M upper triangular part of sub( A ) contains the upper trian- gular matrix R, and elements M+1 to N of the first M rows of sub( A ), with the array TAU, represent the orthogonal matrix Z as a product of M elementary reflectors. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) DOUBLE PRECISION, array, dimension LOCr(IA+M-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MB_A * ( Mp0 + Nq0 + MB_A ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The factorization is obtained by Householder's method. The kth transformation matrix, Z( k ), which is used to introduce zeros into the (m - k + 1)th row of sub( A ), is given in the form .br Z( k ) = ( I 0 ), .br ( 0 T( k ) ) .br where .br T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ), ( 0 ) ( z( k ) ) tau is a scalar and z( k ) is an ( n - m ) element vector. tau and z( k ) are chosen to annihilate the elements of the kth row of sub( A ). .br The scalar tau is returned in the kth element of TAU and the vector u( k ) in the kth row of sub( A ), such that the elements of z( k ) are in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in the upper triangular part of sub( A ). .br Z is given by .br Z = Z( 1 ) * Z( 2 ) * ... * Z( m ). .br scalapack-doc-1.5/man/manl/pdzsum1.l0100644000056400000620000001302106335610640017017 0ustar pfrauenfstaff.TH PDZSUM1 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PDZSUM1 - return the sum of absolute values of a complex distributed vector sub( X ) in ASUM, .SH SYNOPSIS .TP 20 SUBROUTINE PDZSUM1( N, ASUM, X, IX, JX, DESCX, INCX ) .TP 20 .ti +4 INTEGER IX, INCX, JX, N .TP 20 .ti +4 DOUBLE PRECISION ASUM .TP 20 .ti +4 INTEGER DESCX( * ) .TP 20 .ti +4 COMPLEX*16 X( * ) .SH PURPOSE PDZSUM1 returns the sum of absolute values of a complex distributed vector sub( X ) in ASUM, where sub( X ) denotes X(IX:IX+N-1,JX:JX), if INCX = 1, .br X(IX:IX,JX:JX+N-1), if INCX = M_X. Based on PDZASUM from the Level 1 PBLAS. The change is .br to use the 'genuine' absolute value. .br The serial version of this routine was originally contributed by Nick Higham for use with ZLACON. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. When the result of a vector-oriented PBLAS call is a scalar, it will be made available only within the scope which owns the vector(s) being operated on. Let X be a generic term for the input vector(s). Then, the processes which receive the answer will be (note that if an operation involves more than one vector, the processes which re- ceive the result will be the union of the following calculation for each vector): .br If N = 1, M_X = 1 and INCX = 1, then one can't determine if a process row or process column owns the vector operand, therefore only the process of coordinate {RSRC_X, CSRC_X} receives the result; If INCX = M_X, then sub( X ) is a vector distributed over a process row. Each process part of this row receives the result; .br If INCX = 1, then sub( X ) is a vector distributed over a process column. Each process part of this column receives the result; .SH PARAMETERS .TP 8 N (global input) pointer to INTEGER The number of components of the distributed vector sub( X ). N >= 0. .TP 8 ASUM (local output) pointer to DOUBLE PRECISION The sum of absolute values of the distributed vector sub( X ) only in its scope. .TP 8 X (local input) COMPLEX*16 array containing the local pieces of a distributed matrix of dimension of at least ( (JX-1)*M_X + IX + ( N - 1 )*abs( INCX ) ) This array contains the entries of the distributed vector sub( X ). .TP 8 IX (global input) pointer to INTEGER The global row index of the submatrix of the distributed matrix X to operate on. .TP 8 JX (global input) pointer to INTEGER The global column index of the submatrix of the distributed matrix X to operate on. .TP 8 DESCX (global and local input) INTEGER array of dimension 8. The array descriptor of the distributed matrix X. .TP 8 INCX (global input) pointer to INTEGER The global increment for the elements of X. Only two values of INCX are supported in this version, namely 1 and M_X. scalapack-doc-1.5/man/manl/pscsum1.l0100644000056400000620000001276306335610640017023 0ustar pfrauenfstaff.TH PSCSUM1 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSCSUM1 - return the sum of absolute values of a complex distributed vector sub( X ) in ASUM, .SH SYNOPSIS .TP 20 SUBROUTINE PSCSUM1( N, ASUM, X, IX, JX, DESCX, INCX ) .TP 20 .ti +4 INTEGER IX, INCX, JX, N .TP 20 .ti +4 REAL ASUM .TP 20 .ti +4 INTEGER DESCX( * ) .TP 20 .ti +4 COMPLEX X( * ) .SH PURPOSE PSCSUM1 returns the sum of absolute values of a complex distributed vector sub( X ) in ASUM, where sub( X ) denotes X(IX:IX+N-1,JX:JX), if INCX = 1, .br X(IX:IX,JX:JX+N-1), if INCX = M_X. Based on PSCASUM from the Level 1 PBLAS. The change is .br to use the 'genuine' absolute value. .br The serial version of this routine was originally contributed by Nick Higham for use with CLACON. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. When the result of a vector-oriented PBLAS call is a scalar, it will be made available only within the scope which owns the vector(s) being operated on. Let X be a generic term for the input vector(s). Then, the processes which receive the answer will be (note that if an operation involves more than one vector, the processes which re- ceive the result will be the union of the following calculation for each vector): .br If N = 1, M_X = 1 and INCX = 1, then one can't determine if a process row or process column owns the vector operand, therefore only the process of coordinate {RSRC_X, CSRC_X} receives the result; If INCX = M_X, then sub( X ) is a vector distributed over a process row. Each process part of this row receives the result; .br If INCX = 1, then sub( X ) is a vector distributed over a process column. Each process part of this column receives the result; .SH PARAMETERS .TP 8 N (global input) pointer to INTEGER The number of components of the distributed vector sub( X ). N >= 0. .TP 8 ASUM (local output) pointer to REAL The sum of absolute values of the distributed vector sub( X ) only in its scope. .TP 8 X (local input) COMPLEX array containing the local pieces of a distributed matrix of dimension of at least ( (JX-1)*M_X + IX + ( N - 1 )*abs( INCX ) ) This array contains the entries of the distributed vector sub( X ). .TP 8 IX (global input) pointer to INTEGER The global row index of the submatrix of the distributed matrix X to operate on. .TP 8 JX (global input) pointer to INTEGER The global column index of the submatrix of the distributed matrix X to operate on. .TP 8 DESCX (global and local input) INTEGER array of dimension 8. The array descriptor of the distributed matrix X. .TP 8 INCX (global input) pointer to INTEGER The global increment for the elements of X. Only two values of INCX are supported in this version, namely 1 and M_X. scalapack-doc-1.5/man/manl/psdbsv.l0100644000056400000620000000140306335610640016716 0ustar pfrauenfstaff.TH PSDBSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSDBSV - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 19 SUBROUTINE PSDBSV( N, BWL, BWU, NRHS, A, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 INTEGER BWL, BWU, IB, INFO, JA, LWORK, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 REAL A( * ), B( * ), WORK( * ) .SH PURPOSE PSDBSV solves a system of linear equations where A(1:N, JA:JA+N-1) is an N-by-N real .br banded diagonally dominant-like distributed .br matrix with bandwidth BWL, BWU. .br Gaussian elimination without pivoting .br is used to factor a reordering .br of the matrix into L U. .br See PSDBTRF and PSDBTRS for details. .br scalapack-doc-1.5/man/manl/psdbtrf.l0100644000056400000620000000213206335610640017061 0ustar pfrauenfstaff.TH PSDBTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSDBTRF - compute a LU factorization of an N-by-N real banded diagonally dominant-like distributed matrix with bandwidth BWL, BWU .SH SYNOPSIS .TP 20 SUBROUTINE PSDBTRF( N, BWL, BWU, A, JA, DESCA, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER BWL, BWU, INFO, JA, LAF, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), AF( * ), WORK( * ) .SH PURPOSE PSDBTRF computes a LU factorization of an N-by-N real banded diagonally dominant-like distributed matrix with bandwidth BWL, BWU: A(1:N, JA:JA+N-1). Reordering is used to increase parallelism in the factorization. This reordering results in factors that are DIFFERENT from those produced by equivalent sequential codes. These factors cannot be used directly by users; however, they can be used in .br subsequent calls to PSDBTRS to solve linear systems. .br The factorization has the form .br P A(1:N, JA:JA+N-1) P^T = L U .br where U is a banded upper triangular matrix and L is banded lower triangular, and P is a permutation matrix. .br scalapack-doc-1.5/man/manl/psdbtrs.l0100644000056400000620000000165506335610640017107 0ustar pfrauenfstaff.TH PSDBTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSDBTRS - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 20 SUBROUTINE PSDBTRS( TRANS, N, BWL, BWU, NRHS, A, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER BWL, BWU, IB, INFO, JA, LAF, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 REAL A( * ), AF( * ), B( * ), WORK( * ) .SH PURPOSE PSDBTRS solves a system of linear equations or .br A(1:N, JA:JA+N-1)' * X = B(IB:IB+N-1, 1:NRHS) .br where A(1:N, JA:JA+N-1) is the matrix used to produce the factors stored in A(1:N,JA:JA+N-1) and AF by PSDBTRF. .br A(1:N, JA:JA+N-1) is an N-by-N real .br banded diagonally dominant-like distributed .br matrix with bandwidth BWL, BWU. .br Routine PSDBTRF MUST be called first. .br scalapack-doc-1.5/man/manl/psdbtrsv.l0100644000056400000620000000220406335610640017264 0ustar pfrauenfstaff.TH PSDBTRSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSDBTRSV - solve a banded triangular system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 21 SUBROUTINE PSDBTRSV( UPLO, TRANS, N, BWL, BWU, NRHS, A, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 21 .ti +4 CHARACTER TRANS, UPLO .TP 21 .ti +4 INTEGER BWL, BWU, IB, INFO, JA, LAF, LWORK, N, NRHS .TP 21 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 21 .ti +4 REAL A( * ), AF( * ), B( * ), WORK( * ) .SH PURPOSE PSDBTRSV solves a banded triangular system of linear equations or .br A(1:N, JA:JA+N-1)^T * X = B(IB:IB+N-1, 1:NRHS) where A(1:N, JA:JA+N-1) is a banded .br triangular matrix factor produced by the .br Gaussian elimination code PS@(dom_pre)BTRF .br and is stored in A(1:N,JA:JA+N-1) and AF. .br The matrix stored in A(1:N, JA:JA+N-1) is either .br upper or lower triangular according to UPLO, .br and the choice of solving A(1:N, JA:JA+N-1) or A(1:N, JA:JA+N-1)^T is dictated by the user by the parameter TRANS. .br Routine PSDBTRF MUST be called first. .br scalapack-doc-1.5/man/manl/psdtsv.l0100644000056400000620000000136606335610640016750 0ustar pfrauenfstaff.TH PSDTSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSDTSV - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 19 SUBROUTINE PSDTSV( N, NRHS, DL, D, DU, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 INTEGER IB, INFO, JA, LWORK, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 REAL B( * ), D( * ), DL( * ), DU( * ), WORK( * ) .SH PURPOSE PSDTSV solves a system of linear equations where A(1:N, JA:JA+N-1) is an N-by-N real .br tridiagonal diagonally dominant-like distributed .br matrix. .br Gaussian elimination without pivoting .br is used to factor a reordering .br of the matrix into L U. .br See PSDTTRF and PSDTTRS for details. .br scalapack-doc-1.5/man/manl/psdttrf.l0100644000056400000620000000212506335610640017105 0ustar pfrauenfstaff.TH PSDTTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSDTTRF - compute a LU factorization of an N-by-N real tridiagonal diagonally dominant-like distributed matrix A(1:N, JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PSDTTRF( N, DL, D, DU, JA, DESCA, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER INFO, JA, LAF, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL AF( * ), D( * ), DL( * ), DU( * ), WORK( * ) .SH PURPOSE PSDTTRF computes a LU factorization of an N-by-N real tridiagonal diagonally dominant-like distributed matrix A(1:N, JA:JA+N-1). Reordering is used to increase parallelism in the factorization. This reordering results in factors that are DIFFERENT from those produced by equivalent sequential codes. These factors cannot be used directly by users; however, they can be used in .br subsequent calls to PSDTTRS to solve linear systems. .br The factorization has the form .br P A(1:N, JA:JA+N-1) P^T = L U .br where U is a tridiagonal upper triangular matrix and L is tridiagonal lower triangular, and P is a permutation matrix. .br scalapack-doc-1.5/man/manl/psdttrs.l0100644000056400000620000000164006335610640017123 0ustar pfrauenfstaff.TH PSDTTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSDTTRS - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 20 SUBROUTINE PSDTTRS( TRANS, N, NRHS, DL, D, DU, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER IB, INFO, JA, LAF, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 REAL AF( * ), B( * ), D( * ), DL( * ), DU( * ), WORK( * ) .SH PURPOSE PSDTTRS solves a system of linear equations or .br A(1:N, JA:JA+N-1)' * X = B(IB:IB+N-1, 1:NRHS) .br where A(1:N, JA:JA+N-1) is the matrix used to produce the factors stored in A(1:N,JA:JA+N-1) and AF by PSDTTRF. .br A(1:N, JA:JA+N-1) is an N-by-N real .br tridiagonal diagonally dominant-like distributed .br matrix. .br Routine PSDTTRF MUST be called first. .br scalapack-doc-1.5/man/manl/psdttrsv.l0100644000056400000620000000223106335610641017307 0ustar pfrauenfstaff.TH PSDTTRSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSDTTRSV - solve a tridiagonal triangular system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 21 SUBROUTINE PSDTTRSV( UPLO, TRANS, N, NRHS, DL, D, DU, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 21 .ti +4 CHARACTER TRANS, UPLO .TP 21 .ti +4 INTEGER IB, INFO, JA, LAF, LWORK, N, NRHS .TP 21 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 21 .ti +4 REAL AF( * ), B( * ), D( * ), DL( * ), DU( * ), WORK( * ) .SH PURPOSE PSDTTRSV solves a tridiagonal triangular system of linear equations or .br A(1:N, JA:JA+N-1)^T * X = B(IB:IB+N-1, 1:NRHS) where A(1:N, JA:JA+N-1) is a tridiagonal .br triangular matrix factor produced by the .br Gaussian elimination code PS@(dom_pre)TTRF .br and is stored in A(1:N,JA:JA+N-1) and AF. .br The matrix stored in A(1:N, JA:JA+N-1) is either .br upper or lower triangular according to UPLO, .br and the choice of solving A(1:N, JA:JA+N-1) or A(1:N, JA:JA+N-1)^T is dictated by the user by the parameter TRANS. .br Routine PSDTTRF MUST be called first. .br scalapack-doc-1.5/man/manl/psgbsv.l0100644000056400000620000000137206335610641016727 0ustar pfrauenfstaff.TH PSGBSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGBSV - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 19 SUBROUTINE PSGBSV( N, BWL, BWU, NRHS, A, JA, DESCA, IPIV, B, IB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 INTEGER BWL, BWU, IB, INFO, JA, LWORK, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ), IPIV( * ) .TP 19 .ti +4 REAL A( * ), B( * ), WORK( * ) .SH PURPOSE PSGBSV solves a system of linear equations where A(1:N, JA:JA+N-1) is an N-by-N real .br banded distributed .br matrix with bandwidth BWL, BWU. .br Gaussian elimination with pivoting .br is used to factor a reordering .br of the matrix into P L U. .br See PSGBTRF and PSGBTRS for details. .br scalapack-doc-1.5/man/manl/psgbtrf.l0100644000056400000620000000236206335610641017072 0ustar pfrauenfstaff.TH PSGBTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGBTRF - compute a LU factorization of an N-by-N real banded distributed matrix with bandwidth BWL, BWU .SH SYNOPSIS .TP 20 SUBROUTINE PSGBTRF( N, BWL, BWU, A, JA, DESCA, IPIV, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER BWL, BWU, INFO, JA, LAF, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ) .TP 20 .ti +4 REAL A( * ), AF( * ), WORK( * ) .SH PURPOSE PSGBTRF computes a LU factorization of an N-by-N real banded distributed matrix with bandwidth BWL, BWU: A(1:N, JA:JA+N-1). Reordering is used to increase parallelism in the factorization. This reordering results in factors that are DIFFERENT from those produced by equivalent sequential codes. These factors cannot be used directly by users; however, they can be used in .br subsequent calls to PSGBTRS to solve linear systems. .br The factorization has the form .br P A(1:N, JA:JA+N-1) Q = L U .br where U is a banded upper triangular matrix and L is banded lower triangular, and P and Q are permutation matrices. .br The matrix Q represents reordering of columns .br for parallelism's sake, while P represents .br reordering of rows for numerical stability using .br classic partial pivoting. .br scalapack-doc-1.5/man/manl/psgbtrs.l0100644000056400000620000000164306335610641017110 0ustar pfrauenfstaff.TH PSGBTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGBTRS - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 20 SUBROUTINE PSGBTRS( TRANS, N, BWL, BWU, NRHS, A, JA, DESCA, IPIV, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER BWU, BWL, IB, INFO, JA, LAF, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ), IPIV(*) .TP 20 .ti +4 REAL A( * ), AF( * ), B( * ), WORK( * ) .SH PURPOSE PSGBTRS solves a system of linear equations or .br A(1:N, JA:JA+N-1)' * X = B(IB:IB+N-1, 1:NRHS) .br where A(1:N, JA:JA+N-1) is the matrix used to produce the factors stored in A(1:N,JA:JA+N-1) and AF by PSGBTRF. .br A(1:N, JA:JA+N-1) is an N-by-N real .br banded distributed .br matrix with bandwidth BWL, BWU. .br Routine PSGBTRF MUST be called first. .br scalapack-doc-1.5/man/manl/psgebd2.l0100644000056400000620000002217106335610641016751 0ustar pfrauenfstaff.TH PSGEBD2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSGEBD2 - reduce a real general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form B by an orthogonal transformation .SH SYNOPSIS .TP 20 SUBROUTINE PSGEBD2( M, N, A, IA, JA, DESCA, D, E, TAUQ, TAUP, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), D( * ), E( * ), TAUP( * ), TAUQ( * ), WORK( * ) .SH PURPOSE PSGEBD2 reduces a real general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form B by an orthogonal transformation: Q' * sub( A ) * P = B. If M >= N, B is upper bidiagonal; if M < N, B is lower bidiagonal. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the general distributed matrix sub( A ). On exit, if M >= N, the diagonal and the first superdiagonal of sub( A ) are overwritten with the upper bidiagonal matrix B; the elements below the diagonal, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. If M < N, the diagonal and the first subdiagonal are overwritten with the lower bidiagonal matrix B; the elements below the first subdiagonal, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) REAL array, dimension LOCc(JA+MIN(M,N)-1) if M >= N; LOCr(IA+MIN(M,N)-1) otherwise. The distributed diagonal elements of the bidiagonal matrix B: D(i) = A(i,i). D is tied to the distributed matrix A. .TP 8 E (local output) REAL array, dimension LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-2) otherwise. The distributed off-diagonal elements of the bidiagonal distributed matrix B: if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. E is tied to the distributed matrix A. .TP 8 TAUQ (local output) REAL array dimension LOCc(JA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. TAUQ is tied to the distributed matrix A. See Further Details. TAUP (local output) REAL array, dimension LOCr(IA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the orthogonal matrix P. TAUP is tied to the distributed matrix A. See Further Details. WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MAX( MpA0, NqA0 ) where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ) IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, NB, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+IROFFA, NB, MYCOL, IACOL, NPCOL ). INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrices Q and P are represented as products of elementary reflectors: .br If m >= n, .br Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) Each H(i) and G(i) has the form: .br H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are real scalars, and v and u are real vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(ia+i:ia+m-1,ja+i-1); .br u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(ia+i-1,ja+i+1:ja+n-1); .br tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br If m < n, .br Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) Each H(i) and G(i) has the form: .br H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are real scalars, and v and u are real vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(ia+i+1:ia+m-1,ja+i-1); .br u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(ia+i-1,ja+i:ja+n-1); .br tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br The contents of sub( A ) on exit are illustrated by the following examples: .br m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) ( v1 v2 v3 v4 v5 ) .br where d and e denote diagonal and off-diagonal elements of B, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expressions should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) .br scalapack-doc-1.5/man/manl/psgebrd.l0100644000056400000620000002221106335610641017044 0ustar pfrauenfstaff.TH PSGEBRD l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGEBRD - reduce a real general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form B by an orthogonal transformation .SH SYNOPSIS .TP 20 SUBROUTINE PSGEBRD( M, N, A, IA, JA, DESCA, D, E, TAUQ, TAUP, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), D( * ), E( * ), TAUP( * ), TAUQ( * ), WORK( * ) .SH PURPOSE PSGEBRD reduces a real general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form B by an orthogonal transformation: Q' * sub( A ) * P = B. If M >= N, B is upper bidiagonal; if M < N, B is lower bidiagonal. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the general distributed matrix sub( A ). On exit, if M >= N, the diagonal and the first superdiagonal of sub( A ) are overwritten with the upper bidiagonal matrix B; the elements below the diagonal, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. If M < N, the diagonal and the first subdiagonal are overwritten with the lower bidiagonal matrix B; the elements below the first subdiagonal, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) REAL array, dimension LOCc(JA+MIN(M,N)-1) if M >= N; LOCr(IA+MIN(M,N)-1) otherwise. The distributed diagonal elements of the bidiagonal matrix B: D(i) = A(i,i). D is tied to the distributed matrix A. .TP 8 E (local output) REAL array, dimension LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-2) otherwise. The distributed off-diagonal elements of the bidiagonal distributed matrix B: if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. E is tied to the distributed matrix A. .TP 8 TAUQ (local output) REAL array dimension LOCc(JA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. TAUQ is tied to the distributed matrix A. See Further Details. TAUP (local output) REAL array, dimension LOCr(IA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the orthogonal matrix P. TAUP is tied to the distributed matrix A. See Further Details. WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK( 1 ) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB*( MpA0 + NqA0 + 1 ) + NqA0 where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ), ICOFFA = MOD( JA-1, NB ), IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, NB, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB, MYCOL, IACOL, NPCOL ). INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrices Q and P are represented as products of elementary reflectors: .br If m >= n, .br Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) Each H(i) and G(i) has the form: .br H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are real scalars, and v and u are real vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(ia+i:ia+m-1,ja+i-1); .br u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(ia+i-1,ja+i+1:ja+n-1); .br tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br If m < n, .br Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) Each H(i) and G(i) has the form: .br H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are real scalars, and v and u are real vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(ia+i+1:ia+m-1,ja+i-1); .br u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(ia+i-1,ja+i:ja+n-1); .br tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br The contents of sub( A ) on exit are illustrated by the following examples: .br m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) ( v1 v2 v3 v4 v5 ) .br where d and e denote diagonal and off-diagonal elements of B, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expressions should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) .br scalapack-doc-1.5/man/manl/psgecon.l0100644000056400000620000001543706335610641017070 0ustar pfrauenfstaff.TH PSGECON l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGECON - estimate the reciprocal of the condition number of a general distributed real matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm, using the LU factorization computed by PSGETRF .SH SYNOPSIS .TP 20 SUBROUTINE PSGECON( NORM, N, A, IA, JA, DESCA, ANORM, RCOND, WORK, LWORK, IWORK, LIWORK, INFO ) .TP 20 .ti +4 CHARACTER NORM .TP 20 .ti +4 INTEGER IA, INFO, JA, LIWORK, LWORK, N .TP 20 .ti +4 REAL ANORM, RCOND .TP 20 .ti +4 INTEGER DESCA( * ), IWORK( * ) .TP 20 .ti +4 REAL A( * ), WORK( * ) .SH PURPOSE PSGECON estimates the reciprocal of the condition number of a general distributed real matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm, using the LU factorization computed by PSGETRF. An estimate is obtained for norm(inv(A(IA:IA+N-1,JA:JA+N-1))), and the reciprocal of the condition number is computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies whether the 1-norm condition number or the infinity-norm condition number is required: .br = '1' or 'O': 1-norm .br = 'I': Infinity-norm .TP 8 N (global input) INTEGER .br The order of the distributed matrix A(IA:IA+N-1,JA:JA+N-1). N >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension ( LLD_A, LOCc(JA+N-1) ). On entry, this array contains the local pieces of the factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U; the unit diagonal elements of L are not stored. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 ANORM (global input) REAL If NORM = '1' or 'O', the 1-norm of the original distributed matrix A(IA:IA+N-1,JA:JA+N-1). If NORM = 'I', the infinity-norm of the original distributed matrix A(IA:IA+N-1,JA:JA+N-1). .TP 8 RCOND (global output) REAL The reciprocal of the condition number of the distributed matrix A(IA:IA+N-1,JA:JA+N-1), computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * .br norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 2*LOCr(N+MOD(IA-1,MB_A)) + 2*LOCc(N+MOD(JA-1,NB_A)) + MAX( 2, MAX( NB_A*MAX( 1, CEIL(NPROW-1,NPCOL) ), LOCc(N+MOD(JA-1,NB_A)) + NB_A*MAX( 1, CEIL(NPCOL-1,NPROW) ) ). LOCr and LOCc values can be computed using the ScaLAPACK tool function NUMROC; NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/local output) INTEGER array, dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK. .TP 8 LIWORK (local or global input) INTEGER The dimension of the array IWORK. LIWORK is local input and must be at least LIWORK >= LOCr(N+MOD(IA-1,MB_A)). If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/psgeequ.l0100644000056400000620000001434206335610641017075 0ustar pfrauenfstaff.TH PSGEEQU l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGEEQU - compute row and column scalings intended to equilibrate an M-by-N distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA:JA+N-1) and reduce its condition number .SH SYNOPSIS .TP 20 SUBROUTINE PSGEEQU( M, N, A, IA, JA, DESCA, R, C, ROWCND, COLCND, AMAX, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, M, N .TP 20 .ti +4 REAL AMAX, COLCND, ROWCND .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), C( * ), R( * ) .SH PURPOSE PSGEEQU computes row and column scalings intended to equilibrate an M-by-N distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA:JA+N-1) and reduce its condition number. R returns the row scale factors and C the column scale factors, chosen to try to make the largest entry in each row and column of the distributed matrix B with elements B(i,j) = R(i) * A(i,j) * C(j) have absolute value 1. .br R(i) and C(j) are restricted to be between SMLNUM = smallest safe number and BIGNUM = largest safe number. Use of these scaling factors is not guaranteed to reduce the condition number of sub( A ) but works well in practice. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension ( LLD_A, LOCc(JA+N-1) ), the local pieces of the M-by-N distributed matrix whose equilibration factors are to be computed. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 R (local output) REAL array, dimension LOCr(M_A) If INFO = 0 or INFO > IA+M-1, R(IA:IA+M-1) contains the row scale factors for sub( A ). R is aligned with the distributed matrix A, and replicated across every process column. R is tied to the distributed matrix A. .TP 8 C (local output) REAL array, dimension LOCc(N_A) If INFO = 0, C(JA:JA+N-1) contains the column scale factors for sub( A ). C is aligned with the distributed matrix A, and replicated down every process row. C is tied to the distri- buted matrix A. .TP 8 ROWCND (global output) REAL If INFO = 0 or INFO > IA+M-1, ROWCND contains the ratio of the smallest R(i) to the largest R(i) (IA <= i <= IA+M-1). If ROWCND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by R(IA:IA+M-1). .TP 8 COLCND (global output) REAL If INFO = 0, COLCND contains the ratio of the smallest C(j) to the largest C(j) (JA <= j <= JA+N-1). If COLCND >= 0.1, it is not worth scaling by C(JA:JA+N-1). .TP 8 AMAX (global output) REAL Absolute value of largest distributed matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = i, and i is .br <= M: the i-th row of the distributed matrix sub( A ) is exactly zero, > M: the (i-M)-th column of the distributed matrix sub( A ) is exactly zero. scalapack-doc-1.5/man/manl/psgehd2.l0100644000056400000620000001647706335610641016773 0ustar pfrauenfstaff.TH PSGEHD2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSGEHD2 - reduce a real general distributed matrix sub( A ) to upper Hessenberg form H by an orthogonal similarity transforma- tion .SH SYNOPSIS .TP 20 SUBROUTINE PSGEHD2( N, ILO, IHI, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, IHI, ILO, INFO, JA, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), TAU( * ), WORK( * ) .SH PURPOSE PSGEHD2 reduces a real general distributed matrix sub( A ) to upper Hessenberg form H by an orthogonal similarity transforma- tion: Q' * sub( A ) * Q = H, where sub( A ) = A(IA+N-1:IA+N-1,JA+N-1:JA+N-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 ILO (global input) INTEGER IHI (global input) INTEGER It is assumed that sub( A ) is already upper triangular in rows IA:IA+ILO-2 and IA+IHI:IA+N-1 and columns JA:JA+JLO-2 and JA+JHI:JA+N-1. See Further Details. If N > 0, .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N general distributed matrix sub( A ) to be reduced. On exit, the upper triangle and the first subdiagonal of sub( A ) are overwritten with the upper Hessenberg matrix H, and the ele- ments below the first subdiagonal, with the array TAU, repre- sent the orthogonal matrix Q as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) REAL array, dimension LOCc(JA+N-2) The scalar factors of the elementary reflectors (see Further Details). Elements JA:JA+ILO-2 and JA+IHI:JA+N-2 of TAU are set to zero. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK( 1 ) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB + MAX( NpA0, NB ) where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ), IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ), NpA0 = NUMROC( IHI+IROFFA, NB, MYROW, IAROW, NPROW ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of (ihi-ilo) elementary reflectors .br Q = H(ilo) H(ilo+1) . . . H(ihi-1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on exit in A(ia+ilo+i:ia+ihi-1,ja+ilo+i-2), and tau in TAU(ja+ilo+i-2). The contents of A(IA:IA+N-1,JA:JA+N-1) are illustrated by the follo- wing example, with n = 7, ilo = 2 and ihi = 6: .br on entry on exit .br ( a a a a a a a ) ( a a h h h h a ) ( a a a a a a ) ( a h h h h a ) ( a a a a a a ) ( h h h h h h ) ( a a a a a a ) ( v2 h h h h h ) ( a a a a a a ) ( v2 v3 h h h h ) ( a a a a a a ) ( v2 v3 v4 h h h ) ( a ) ( a ) where a denotes an element of the original matrix sub( A ), h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(ja+ilo+i-2). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expression should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) .br scalapack-doc-1.5/man/manl/psgehrd.l0100644000056400000620000001713306335610641017061 0ustar pfrauenfstaff.TH PSGEHRD l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGEHRD - reduce a real general distributed matrix sub( A ) to upper Hessenberg form H by an orthogonal similarity transforma- tion .SH SYNOPSIS .TP 20 SUBROUTINE PSGEHRD( N, ILO, IHI, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, IHI, ILO, INFO, JA, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), TAU( * ), WORK( * ) .SH PURPOSE PSGEHRD reduces a real general distributed matrix sub( A ) to upper Hessenberg form H by an orthogonal similarity transforma- tion: Q' * sub( A ) * Q = H, where sub( A ) = A(IA+N-1:IA+N-1,JA+N-1:JA+N-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 ILO (global input) INTEGER IHI (global input) INTEGER It is assumed that sub( A ) is already upper triangular in rows IA:IA+ILO-2 and IA+IHI:IA+N-1 and columns JA:JA+ILO-2 and JA+IHI:JA+N-1. See Further Details. If N > 0, .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N general distributed matrix sub( A ) to be reduced. On exit, the upper triangle and the first subdiagonal of sub( A ) are overwritten with the upper Hessenberg matrix H, and the ele- ments below the first subdiagonal, with the array TAU, repre- sent the orthogonal matrix Q as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) REAL array, dimension LOCc(JA+N-2) The scalar factors of the elementary reflectors (see Further Details). Elements JA:JA+ILO-2 and JA+IHI:JA+N-2 of TAU are set to zero. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK( 1 ) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB*NB + NB*MAX( IHIP+1, IHLP+INLQ ) where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ), ICOFFA = MOD( JA-1, NB ), IOFF = MOD( IA+ILO-2, NB ), IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ), IHIP = NUMROC( IHI+IROFFA, NB, MYROW, IAROW, NPROW ), ILROW = INDXG2P( IA+ILO-1, NB, MYROW, RSRC_A, NPROW ), IHLP = NUMROC( IHI-ILO+IOFF+1, NB, MYROW, ILROW, NPROW ), ILCOL = INDXG2P( JA+ILO-1, NB, MYCOL, CSRC_A, NPCOL ), INLQ = NUMROC( N-ILO+IOFF+1, NB, MYCOL, ILCOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of (ihi-ilo) elementary reflectors .br Q = H(ilo) H(ilo+1) . . . H(ihi-1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br v(1:I) = 0, v(I+1) = 1 and v(IHI+1:N) = 0; v(I+2:IHI) is stored on exit in A(IA+ILO+I:IA+IHI-1,JA+ILO+I-2), and tau in TAU(JA+ILO+I-2). The contents of A(IA:IA+N-1,JA:JA+N-1) are illustrated by the follow- ing example, with N = 7, ILO = 2 and IHI = 6: .br on entry on exit .br ( a a a a a a a ) ( a a h h h h a ) ( a a a a a a ) ( a h h h h a ) ( a a a a a a ) ( h h h h h h ) ( a a a a a a ) ( v2 h h h h h ) ( a a a a a a ) ( v2 v3 h h h h ) ( a a a a a a ) ( v2 v3 v4 h h h ) ( a ) ( a ) where a denotes an element of the original matrix sub( A ), H denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(JA+ILO+I-2). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expression should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) .br scalapack-doc-1.5/man/manl/psgelq2.l0100644000056400000620000001416206335610641017001 0ustar pfrauenfstaff.TH PSGELQ2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGELQ2 - compute a LQ factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = L * Q .SH SYNOPSIS .TP 20 SUBROUTINE PSGELQ2( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), TAU( * ), WORK( * ) .SH PURPOSE PSGELQ2 computes a LQ factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = L * Q. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the elements on and below the diagonal of sub( A ) contain the M by min(M,N) lower trapezoidal matrix L (L is lower triangular if M <= N); the elements above the diagonal, with the array TAU, repre- sent the orthogonal matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) REAL, array, dimension LOCr(IA+MIN(M,N)-1). This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= Nq0 + MAX( 1, Mp0 ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ia+k-1) H(ia+k-2) . . . H(ia), where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br 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 A(ia+i-1,ja+i:ja+n-1), and tau in TAU(ia+i-1). .br scalapack-doc-1.5/man/manl/psgelqf.l0100644000056400000620000001417306335610641017067 0ustar pfrauenfstaff.TH PSGELQF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGELQF - compute a LQ factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = L * Q .SH SYNOPSIS .TP 20 SUBROUTINE PSGELQF( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), TAU( * ), WORK( * ) .SH PURPOSE PSGELQF computes a LQ factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = L * Q. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the elements on and below the diagonal of sub( A ) contain the M by min(M,N) lower trapezoidal matrix L (L is lower triangular if M <= N); the elements above the diagonal, with the array TAU, repre- sent the orthogonal matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) REAL, array, dimension LOCr(IA+MIN(M,N)-1). This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MB_A * ( Mp0 + Nq0 + MB_A ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ia+k-1) H(ia+k-2) . . . H(ia), where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br 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 A(ia+i-1,ja+i:ja+n-1), and tau in TAU(ia+i-1). .br scalapack-doc-1.5/man/manl/psgels.l0100644000056400000620000002215606335610642016724 0ustar pfrauenfstaff.TH PSGELS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGELS - solve overdetermined or underdetermined real linear systems involving an M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1), .SH SYNOPSIS .TP 19 SUBROUTINE PSGELS( TRANS, M, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 CHARACTER TRANS .TP 19 .ti +4 INTEGER IA, IB, INFO, JA, JB, LWORK, M, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 REAL A( * ), B( * ), WORK( * ) .SH PURPOSE PSGELS solves overdetermined or underdetermined real linear systems involving an M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1), or its transpose, using a QR or LQ factorization of sub( A ). It is assumed that sub( A ) has full rank. .br The following options are provided: .br 1. If TRANS = 'N' and m >= n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem minimize || sub( B ) - sub( A )*X ||. .br 2. If TRANS = 'N' and m < n: find the minimum norm solution of an underdetermined system sub( A ) * X = sub( B ). .br 3. If TRANS = 'T' and m >= n: find the minimum norm solution of an undetermined system sub( A )**T * X = sub( B ). .br 4. If TRANS = 'T' and m < n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem minimize || sub( B ) - sub( A )**T * X ||. where sub( B ) denotes B( IB:IB+M-1, JB:JB+NRHS-1 ) when TRANS = 'N' and B( IB:IB+N-1, JB:JB+NRHS-1 ) otherwise. Several right hand side vectors b and solution vectors x can be handled in a single call; When TRANS = 'N', the solution vectors are stored as the columns of the N-by-NRHS right hand side matrix sub( B ) and the M-by-NRHS right hand side matrix sub( B ) otherwise. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 TRANS (global input) CHARACTER = 'N': the linear system involves sub( A ); .br = 'T': the linear system involves sub( A )**T. .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e. the number of columns of the distributed submatrices sub( B ) and X. NRHS >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of local dimension ( LLD_A, LOCc(JA+N-1) ). On entry, the M-by-N matrix A. if M >= N, sub( A ) is overwritten by details of its QR factorization as returned by PSGEQRF; if M < N, sub( A ) is overwritten by details of its LQ factorization as returned by PSGELQF. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input/local output) REAL pointer into the local memory to an array of local dimension (LLD_B, LOCc(JB+NRHS-1)). On entry, this array contains the local pieces of the distributed matrix B of right hand side vectors, stored columnwise; sub( B ) is M-by-NRHS if TRANS='N', and N-by-NRHS otherwise. On exit, sub( B ) is overwritten by the solution vectors, stored columnwise: if TRANS = 'N' and M >= N, rows 1 to N of sub( B ) contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements N+1 to M in that column; if TRANS = 'N' and M < N, rows 1 to N of sub( B ) contain the minimum norm solution vectors; if TRANS = 'T' and M >= N, rows 1 to M of sub( B ) contain the minimum norm solution vectors; if TRANS = 'T' and M < N, rows 1 to M of sub( B ) contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements M+1 to N in that column. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= LTAU + MAX( LWF, LWS ) where If M >= N, then LTAU = NUMROC( JA+MIN(M,N)-1, NB_A, MYCOL, CSRC_A, NPCOL ), LWF = NB_A * ( MpA0 + NqA0 + NB_A ) LWS = MAX( (NB_A*(NB_A-1))/2, (NRHSqB0 + MpB0)*NB_A ) + NB_A * NB_A Else LTAU = NUMROC( IA+MIN(M,N)-1, MB_A, MYROW, RSRC_A, NPROW ), LWF = MB_A * ( MpA0 + NqA0 + MB_A ) LWS = MAX( (MB_A*(MB_A-1))/2, ( NpB0 + MAX( NqA0 + NUMROC( NUMROC( N+IROFFB, MB_A, 0, 0, NPROW ), MB_A, 0, 0, LCMP ), NRHSqB0 ) )*MB_A ) + MB_A * MB_A End if where LCMP = LCM / NPROW with LCM = ILCM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFB = MOD( IB-1, MB_B ), ICOFFB = MOD( JB-1, NB_B ), IBROW = INDXG2P( IB, MB_B, MYROW, RSRC_B, NPROW ), IBCOL = INDXG2P( JB, NB_B, MYCOL, CSRC_B, NPCOL ), MpB0 = NUMROC( M+IROFFB, MB_B, MYROW, IBROW, NPROW ), NpB0 = NUMROC( N+IROFFB, MB_B, MYROW, IBROW, NPROW ), NRHSqB0 = NUMROC( NRHS+ICOFFB, NB_B, MYCOL, IBCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/psgeql2.l0100644000056400000620000001434406335610642017004 0ustar pfrauenfstaff.TH PSGEQL2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGEQL2 - compute a QL factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * L .SH SYNOPSIS .TP 20 SUBROUTINE PSGEQL2( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), TAU( * ), WORK( * ) .SH PURPOSE PSGEQL2 computes a QL factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * L. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, if M >= N, the lower triangle of the distributed submatrix A( IA+M-N:IA+M-1, JA:JA+N-1 ) contains the N-by-N lower triangular matrix L; if M <= N, the elements on and below the (N-M)-th superdiagonal contain the M by N lower trapezoidal matrix L; the remaining elements, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) REAL, array, dimension LOCc(JA+N-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= Mp0 + MAX( 1, Nq0 ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ja+k-1) . . . H(ja+1) H(ja), where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in A(ia:ia+m-k+i-2,ja+n-k+i-1), and tau in TAU(ja+n-k+i-1). .br scalapack-doc-1.5/man/manl/psgeqlf.l0100644000056400000620000001435506335610642017072 0ustar pfrauenfstaff.TH PSGEQLF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGEQLF - compute a QL factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * L .SH SYNOPSIS .TP 20 SUBROUTINE PSGEQLF( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), TAU( * ), WORK( * ) .SH PURPOSE PSGEQLF computes a QL factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * L. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, if M >= N, the lower triangle of the distributed submatrix A( IA+M-N:IA+M-1, JA:JA+N-1 ) contains the N-by-N lower triangular matrix L; if M <= N, the elements on and below the (N-M)-th superdiagonal contain the M by N lower trapezoidal matrix L; the remaining elements, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) REAL, array, dimension LOCc(JA+N-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB_A * ( Mp0 + Nq0 + NB_A ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ja+k-1) . . . H(ja+1) H(ja), where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in A(ia:ia+m-k+i-2,ja+n-k+i-1), and tau in TAU(ja+n-k+i-1). .br scalapack-doc-1.5/man/manl/psgeqpf.l0100644000056400000620000001506106335610642017071 0ustar pfrauenfstaff.TH PSGEQPF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGEQPF - compute a QR factorization with column pivoting of a M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PSGEQPF( M, N, A, IA, JA, DESCA, IPIV, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, JA, INFO, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ) .TP 20 .ti +4 REAL A( * ), TAU( * ), WORK( * ) .SH PURPOSE PSGEQPF computes a QR factorization with column pivoting of a M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1): sub( A ) * P = Q * R. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the elements on and above the diagonal of sub( A ) contain the min(M,N) by N upper trapezoidal matrix R (R is upper triangular if M >= N); the elements below the diagonal, with the array TAU, repre- sent the orthogonal matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local output) INTEGER array, dimension LOCc(JA+N-1). On exit, if IPIV(I) = K, the local i-th column of sub( A )*P was the global K-th column of sub( A ). IPIV is tied to the distributed matrix A. .TP 8 TAU (local output) REAL, array, dimension LOCc(JA+MIN(M,N)-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MAX(3,Mp0 + Nq0) + LOCc(JA+N-1)+Nq0. IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), LOCc(JA+N-1) = NUMROC( JA+N-1, NB_A, MYCOL, CSRC_A, NPCOL ) and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) . . . H(n) .br Each H(i) has the form .br H = I - tau * v * v' .br 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(ia+i-1:ia+m-1,ja+i-1). The matrix P is represented in jpvt as follows: If .br jpvt(j) = i .br then the jth column of P is the ith canonical unit vector. scalapack-doc-1.5/man/manl/psgeqr2.l0100644000056400000620000001416406335610642017012 0ustar pfrauenfstaff.TH PSGEQR2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGEQR2 - compute a QR factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * R .SH SYNOPSIS .TP 20 SUBROUTINE PSGEQR2( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), TAU( * ), WORK( * ) .SH PURPOSE PSGEQR2 computes a QR factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * R. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the elements on and above the diagonal of sub( A ) contain the min(M,N) by N upper trapezoidal matrix R (R is upper triangular if M >= N); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) REAL, array, dimension LOCc(JA+MIN(M,N)-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= Mp0 + MAX( 1, Nq0 ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ja) H(ja+1) . . . H(ja+k-1), where k = min(m,n). Each H(i) has the form .br H(j) = I - tau * v * v' .br 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(ia+i:ia+m-1,ja+i-1), and tau in TAU(ja+i-1). .br scalapack-doc-1.5/man/manl/psgeqrf.l0100644000056400000620000001417506335610642017100 0ustar pfrauenfstaff.TH PSGEQRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGEQRF - compute a QR factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * R .SH SYNOPSIS .TP 20 SUBROUTINE PSGEQRF( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), TAU( * ), WORK( * ) .SH PURPOSE PSGEQRF computes a QR factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * R. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the elements on and above the diagonal of sub( A ) contain the min(M,N) by N upper trapezoidal matrix R (R is upper triangular if M >= N); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) REAL, array, dimension LOCc(JA+MIN(M,N)-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB_A * ( Mp0 + Nq0 + NB_A ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ja) H(ja+1) . . . H(ja+k-1), where k = min(m,n). Each H(i) has the form .br H(j) = I - tau * v * v' .br 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(ia+i:ia+m-1,ja+i-1), and tau in TAU(ja+i-1). .br scalapack-doc-1.5/man/manl/psgerfs.l0100644000056400000620000002325506335610642017101 0ustar pfrauenfstaff.TH PSGERFS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGERFS - improve the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solutions .SH SYNOPSIS .TP 20 SUBROUTINE PSGERFS( TRANS, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, IPIV, B, IB, JB, DESCB, X, IX, JX, DESCX, FERR, BERR, WORK, LWORK, IWORK, LIWORK, INFO ) .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER IA, IAF, IB, IX, INFO, JA, JAF, JB, JX, LIWORK, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * ),IPIV( * ), IWORK( * ) .TP 20 .ti +4 REAL A( * ), AF( * ), B( * ), BERR( * ), FERR( * ), WORK( * ), X( * ) .SH PURPOSE PSGERFS improves the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solutions. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br In the following comments, sub( A ), sub( X ) and sub( B ) denote respectively A(IA:IA+N-1,JA:JA+N-1), X(IX:IX+N-1,JX:JX+NRHS-1) and B(IB:IB+N-1,JB:JB+NRHS-1). .br .SH ARGUMENTS .TP 8 TRANS (global input) CHARACTER*1 Specifies the form of the system of equations. = 'N': sub( A ) * sub( X ) = sub( B ) (No transpose) .br = 'T': sub( A )**T * sub( X ) = sub( B ) (Transpose) .br = 'C': sub( A )**T * sub( X ) = sub( B ) (Conjugate transpose = Transpose) .TP 8 N (global input) INTEGER The order of the matrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the matrices sub( B ) and sub( X ). NRHS >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1)). This array contains the local pieces of the distributed matrix sub( A ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 AF (local input) REAL pointer into the local memory to an array of local dimension (LLD_AF,LOCc(JA+N-1)). This array contains the local pieces of the distributed factors of the matrix sub( A ) = P * L * U as computed by PSGETRF. .TP 8 IAF (global input) INTEGER The row index in the global array AF indicating the first row of sub( AF ). .TP 8 JAF (global input) INTEGER The column index in the global array AF indicating the first column of sub( AF ). .TP 8 DESCAF (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix AF. .TP 8 IPIV (local input) INTEGER array of dimension LOCr(M_AF)+MB_AF. This array contains the pivoting information as computed by PSGETRF. IPIV(i) -> The global row local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 B (local input) REAL pointer into the local memory to an array of local dimension (LLD_B,LOCc(JB+NRHS-1)). This array contains the local pieces of the distributed matrix of right hand sides sub( B ). .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 X (local input and output) REAL pointer into the local memory to an array of local dimension (LLD_X,LOCc(JX+NRHS-1)). On entry, this array contains the local pieces of the distributed matrix solution sub( X ). On exit, the improved solution vectors. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 FERR (local output) REAL array of local dimension LOCc(JB+NRHS-1). The estimated forward error bound for each solution vector of sub( X ). If XTRUE is the true solution corresponding to sub( X ), FERR is an estimated upper bound for the magnitude of the largest element in (sub( X ) - XTRUE) divided by the magnitude of the largest element in sub( X ). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. This array is tied to the distributed matrix X. .TP 8 BERR (local output) REAL array of local dimension LOCc(JB+NRHS-1). The componentwise relative backward error of each solution vector (i.e., the smallest re- lative change in any entry of sub( A ) or sub( B ) that makes sub( X ) an exact solution). This array is tied to the distributed matrix X. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 3*LOCr( N + MOD(IA-1,MB_A) ) If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/local output) INTEGER array, dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK. .TP 8 LIWORK (local or global input) INTEGER The dimension of the array IWORK. LIWORK is local input and must be at least LIWORK >= LOCr( N + MOD(IB-1,MB_B) ). If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH PARAMETERS ITMAX is the maximum number of steps of iterative refinement. Notes ===== This routine temporarily returns when N <= 1. The distributed submatrices op( A ) and op( AF ) (respectively sub( X ) and sub( B ) ) should be distributed the same way on the same processes. These conditions ensure that sub( A ) and sub( AF ) (resp. sub( X ) and sub( B ) ) are "perfectly" aligned. Moreover, this routine requires the distributed submatrices sub( A ), sub( AF ), sub( X ), and sub( B ) to be aligned on a block boundary, i.e., if f(x,y) = MOD( x-1, y ): f( IA, DESCA( MB_ ) ) = f( JA, DESCA( NB_ ) ) = 0, f( IAF, DESCAF( MB_ ) ) = f( JAF, DESCAF( NB_ ) ) = 0, f( IB, DESCB( MB_ ) ) = f( JB, DESCB( NB_ ) ) = 0, and f( IX, DESCX( MB_ ) ) = f( JX, DESCX( NB_ ) ) = 0. scalapack-doc-1.5/man/manl/psgerq2.l0100644000056400000620000001431006335610642017003 0ustar pfrauenfstaff.TH PSGERQ2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGERQ2 - compute a RQ factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = R * Q .SH SYNOPSIS .TP 20 SUBROUTINE PSGERQ2( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), TAU( * ), WORK( * ) .SH PURPOSE PSGERQ2 computes a RQ factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = R * Q. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, if M <= N, the upper triangle of A( IA:IA+M-1, JA+N-M:JA+N-1 ) 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 elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) REAL, array, dimension LOCr(IA+M-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= Nq0 + MAX( 1, Mp0 ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ia) H(ia+1) . . . H(ia+k-1), where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br 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(ia+m-k+i-1,ja:ja+n-k+i-2), and tau in TAU(ia+m-k+i-1). .br scalapack-doc-1.5/man/manl/psgerqf.l0100644000056400000620000001432106335610642017071 0ustar pfrauenfstaff.TH PSGERQF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGERQF - compute a RQ factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = R * Q .SH SYNOPSIS .TP 20 SUBROUTINE PSGERQF( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), TAU( * ), WORK( * ) .SH PURPOSE PSGERQF computes a RQ factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = R * Q. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, if M <= N, the upper triangle of A( IA:IA+M-1, JA+N-M:JA+N-1 ) 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 elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) REAL, array, dimension LOCr(IA+M-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MB_A * ( Mp0 + Nq0 + MB_A ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ia) H(ia+1) . . . H(ia+k-1), where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br 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(ia+m-k+i-1,ja:ja+n-k+i-2), and tau in TAU(ia+m-k+i-1). .br scalapack-doc-1.5/man/manl/psgesv.l0100644000056400000620000001374006335610642016735 0ustar pfrauenfstaff.TH PSGESV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGESV - compute the solution to a real system of linear equations sub( A ) * X = sub( B ), .SH SYNOPSIS .TP 19 SUBROUTINE PSGESV( N, NRHS, A, IA, JA, DESCA, IPIV, B, IB, JB, DESCB, INFO ) .TP 19 .ti +4 INTEGER IA, IB, INFO, JA, JB, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ), IPIV( * ) .TP 19 .ti +4 REAL A( * ), B( * ) .SH PURPOSE PSGESV computes the solution to a real system of linear equations where sub( A ) = A(IA:IA+N-1,JA:JA+N-1) is an N-by-N distributed matrix and X and sub( B ) = B(IB:IB+N-1,JB:JB+NRHS-1) are N-by-NRHS distributed matrices. .br The LU decomposition with partial pivoting and row interchanges is used to factor sub( A ) as sub( A ) = P * L * U, where P is a permu- tation matrix, L is unit lower triangular, and U is upper triangular. L and U are stored in sub( A ). The factored form of sub( A ) is then used to solve the system of equations sub( A ) * X = sub( B ). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block decomposition ( MB_A = NB_A ). .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed submatrix sub( A ). NRHS >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the local pieces of the N-by-N distributed matrix sub( A ) to be factored. On exit, this array contains the local pieces of the factors L and U from the factorization sub( A ) = P*L*U; the unit diagonal elements of L are not stored. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local output) INTEGER array, dimension ( LOCr(M_A)+MB_A ) This array contains the pivoting information. IPIV(i) -> The global row local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 B (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_B,LOCc(JB+NRHS-1)). On entry, the right hand side distributed matrix sub( B ). On exit, if INFO = 0, sub( B ) is overwritten by the solution distributed matrix X. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, U(IA+K-1,JA+K-1) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution could not be computed. scalapack-doc-1.5/man/manl/psgesvd.l0100644000056400000620000002253306335610642017101 0ustar pfrauenfstaff.TH PSGESVD l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGESVD - compute the singular value decomposition (SVD) of an M-by-N matrix A, optionally computing the left and/or right singular vectors .SH SYNOPSIS .TP 20 SUBROUTINE PSGESVD( JOBU, JOBVT, M, N, A, IA, JA, DESCA, S, U, IU, JU, DESCU, VT, IVT, JVT, DESCVT, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER JOBU, JOBVT .TP 20 .ti +4 INTEGER IA, INFO, IU, IVT, JA, JU, JVT, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCU( * ), DESCVT( * ) .TP 20 .ti +4 REAL A( * ), S( * ), U( * ), VT( * ), WORK( * ) .SH PURPOSE PSGESVD computes the singular value decomposition (SVD) of an M-by-N matrix A, optionally computing the left and/or right singular vectors. The SVD is written as A = U * SIGMA * transpose(V) .br where SIGMA is an M-by-N matrix which is zero except for its min(M,N) diagonal elements, U is an M-by-M orthogonal matrix, and V is an N-by-N orthogonal matrix. The diagonal elements of SIGMA are the singular values of A and the columns of U and V are the corresponding right and left singular vectors, respectively. The singular values are returned in array S in decreasing order and only the first min(M,N) columns of U and rows of VT = V**T are computed. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS MP = number of local rows in A and U NQ = number of local columns in A and VT SIZE = min( M, N ) SIZEQ = number of local columns in U SIZEP = number of local rows in VT .TP 8 JOBU (global input) CHARACTER*1 Specifies options for computing all or part of the matrix U: .br = 'V': the first SIZE columns of U (the left singular vectors) are returned in the array U; = 'N': no columns of U (no left singular vectors) are computed. .TP 8 JOBVT (global input) CHARACTER*1 Specifies options for computing all or part of the matrix V**T: .br = 'V': the first SIZE rows of V**T (the right singular vectors) are returned in the array VT; = 'N': no rows of V**T (no right singular vectors) are computed. .TP 8 M (global input) INTEGER The number of rows of the input matrix A. M >= 0. .TP 8 N (global input) INTEGER The number of columns of the input matrix A. N >= 0. .TP 8 A (local input/workspace) block cyclic REAL array, global dimension (M, N), local dimension (MP, NQ) On exit, the contents of A are destroyed. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global input) INTEGER array of dimension DLEN_ The array descriptor for the distributed matrix A. .TP 8 S (global output) REAL array, dimension SIZE The singular values of A, sorted so that S(i) >= S(i+1). .TP 8 U (local output) REAL array, local dimension (MP, SIZEQ), global dimension (M, SIZE) if JOBU = 'V', U contains the first min(m,n) columns of U if JOBU = 'N', U is not referenced. .TP 8 IU (global input) INTEGER The row index in the global array U indicating the first row of sub( U ). .TP 8 JU (global input) INTEGER The column index in the global array U indicating the first column of sub( U ). .TP 8 DESCU (global input) INTEGER array of dimension DLEN_ The array descriptor for the distributed matrix U. .TP 8 VT (local output) REAL array, local dimension (SIZEP, NQ), global dimension (SIZE, N). If JOBVT = 'V', VT contains the first SIZE rows of V**T. If JOBVT = 'N', VT is not referenced. .TP 8 IVT (global input) INTEGER The row index in the global array VT indicating the first row of sub( VT ). .TP 8 JVT (global input) INTEGER The column index in the global array VT indicating the first column of sub( VT ). .TP 9 DESCVT (global input) INTEGER array of dimension DLEN_ The array descriptor for the distributed matrix VT. .TP 8 WORK (local workspace/output) REAL array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK; .TP 8 LWORK (local input) INTEGER The dimension of the array WORK. LWORK > 2 + 6*SIZEB + MAX(WATOBD, WBDTOSVD), where SIZEB = MAX(M,N), and WATOBD and WBDTOSVD refer, respectively, to the workspace required to bidiagonalize the matrix A and to go from the bidiagonal matrix to the singular value decomposition U*S*VT. For WATOBD, the following holds: WATOBD = MAX(MAX(WPSLANGE,WPSGEBRD), MAX(WPSLARED2D,WPSLARED1D)), where WPSLANGE, WPSLARED1D, WPSLARED2D, WPSGEBRD are the workspaces required respectively for the subprograms PSLANGE, PSLARED1D, PSLARED2D, PSGEBRD. Using the standard notation MP = NUMROC( M, MB, MYROW, DESCA( CTXT_ ), NPROW), NQ = NUMROC( N, NB, MYCOL, DESCA( LLD_ ), NPCOL), the workspaces required for the above subprograms are WPSLANGE = MP, WPSLARED1D = NQ0, WPSLARED2D = MP0, WPSGEBRD = NB*(MP + NQ + 1) + NQ, where NQ0 and MP0 refer, respectively, to the values obtained at MYCOL = 0 and MYROW = 0. In general, the upper limit for the workspace is given by a workspace required on processor (0,0): WATOBD <= NB*(MP0 + NQ0 + 1) + NQ0. In case of a homogeneous process grid this upper limit can be used as an estimate of the minimum workspace for every processor. For WBDTOSVD, the following holds: WBDTOSVD = SIZE*(WANTU*NRU + WANTVT*NCVT) + MAX(WDBDSQR, MAX(WANTU*WPSORMBRQLN, WANTVT*WPSORMBRPRT)), .TP -1 where 1, if left(right) singular vectors are wanted WANTU(WANTVT) = 0, otherwise and WDBDSQR, WPSORMBRQLN and WPSORMBRPRT refer respectively to the workspace required for the subprograms DBDSQR, PSORMBR(QLN), and PSORMBR(PRT), where QLN and PRT are the values of the arguments VECT, SIDE, and TRANS in the call to PSORMBR. NRU is equal to the local number of rows of the matrix U when distributed 1-dimensional "column" of processes. Analogously, NCVT is equal to the local number of columns of the matrix VT when distributed across 1-dimensional "row" of processes. Calling the LAPACK procedure DBDSQR requires WDBDSQR = MAX(1, 2*SIZE + (2*SIZE - 4)*MAX(WANTU, WANTVT)) on every processor. Finally, WPSORMBRQLN = MAX( (NB*(NB-1))/2, (SIZEQ+MP)*NB)+NB*NB, WPSORMBRPRT = MAX( (MB*(MB-1))/2, (SIZEP+NQ)*MB )+MB*MB, If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum size for the work array. The required workspace is returned as the first element of WORK and no error message is issued by PXERBLA. .TP 8 INFO (output) INTEGER = 0: successful exit. .br < 0: if INFO = -i, the i-th argument had an illegal value. .br > 0: if SBDSQR did not converge If INFO = MIN(M,N) + 1, then PSSYEV has detected heterogeneity by finding that eigenvalues were not identical across the process grid. In this case, the accuracy of the results from PSSYEV cannot be guaranteed. scalapack-doc-1.5/man/manl/psgesvx.l0100644000056400000620000004023406335610642017123 0ustar pfrauenfstaff.TH PSGESVX l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGESVX - use the LU factorization to compute the solution to a real system of linear equations A(IA:IA+N-1,JA:JA+N-1) * X = B(IB:IB+N-1,JB:JB+NRHS-1), .SH SYNOPSIS .TP 20 SUBROUTINE PSGESVX( FACT, TRANS, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, IPIV, EQUED, R, C, B, IB, JB, DESCB, X, IX, JX, DESCX, RCOND, FERR, BERR, WORK, LWORK, IWORK, LIWORK, INFO ) .TP 20 .ti +4 CHARACTER EQUED, FACT, TRANS .TP 20 .ti +4 INTEGER IA, IAF, IB, INFO, IX, JA, JAF, JB, JX, LIWORK, LWORK, N, NRHS .TP 20 .ti +4 REAL RCOND .TP 20 .ti +4 INTEGER DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * ), IPIV( * ), IWORK( * ) .TP 20 .ti +4 REAL A( * ), AF( * ), B( * ), BERR( * ), C( * ), FERR( * ), R( * ), WORK( * ), X( * ) .SH PURPOSE PSGESVX uses the LU factorization to compute the solution to a real system of linear equations where A(IA:IA+N-1,JA:JA+N-1) is an N-by-N matrix and X and B(IB:IB+N-1,JB:JB+NRHS-1) are N-by-NRHS matrices. .br Error bounds on the solution and a condition estimate are also provided. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH DESCRIPTION In the following description, A denotes A(IA:IA+N-1,JA:JA+N-1), B denotes B(IB:IB+N-1,JB:JB+NRHS-1) and X denotes .br X(IX:IX+N-1,JX:JX+NRHS-1). .br The following steps are performed: .br 1. If FACT = 'E', real scaling factors are computed to equilibrate the system: .br 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'). .br 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as .br A = P * L * U, .br where P is a permutation matrix, L is a unit lower triangular matrix, and U is upper triangular. .br 3. 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, steps 4-6 are skipped. .br 4. The system of equations is solved for X using the factored form of A. .br 5. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. .br 6. If FACT = 'E' and 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. .br .SH ARGUMENTS .TP 8 FACT (global input) CHARACTER Specifies whether or not the factored form of the matrix A(IA:IA+N-1,JA:JA+N-1) is supplied on entry, and if not, .br whether the matrix A(IA:IA+N-1,JA:JA+N-1) should be equilibrated before it is factored. = 'F': On entry, AF(IAF:IAF+N-1,JAF:JAF+N-1) and IPIV con- .br tain the factored form of A(IA:IA+N-1,JA:JA+N-1). If EQUED is not 'N', the matrix A(IA:IA+N-1,JA:JA+N-1) has been equilibrated with scaling factors given by R and C. A(IA:IA+N-1,JA:JA+N-1), AF(IAF:IAF+N-1,JAF:JAF+N-1), and IPIV are not modified. = 'N': The matrix A(IA:IA+N-1,JA:JA+N-1) will be copied to .br AF(IAF:IAF+N-1,JAF:JAF+N-1) and factored. .br = 'E': The matrix A(IA:IA+N-1,JA:JA+N-1) will be equili- brated if necessary, then copied to AF(IAF:IAF+N-1,JAF:JAF+N-1) and factored. .TP 8 TRANS (global input) CHARACTER .br Specifies the form of the system of equations: .br = 'N': A(IA:IA+N-1,JA:JA+N-1) * X(IX:IX+N-1,JX:JX+NRHS-1) .br = B(IB:IB+N-1,JB:JB+NRHS-1) (No transpose) .br = 'T': A(IA:IA+N-1,JA:JA+N-1)**T * X(IX:IX+N-1,JX:JX+NRHS-1) .br = B(IB:IB+N-1,JB:JB+NRHS-1) (Transpose) .br = 'C': A(IA:IA+N-1,JA:JA+N-1)**H * X(IX:IX+N-1,JX:JX+NRHS-1) .br = B(IB:IB+N-1,JB:JB+NRHS-1) (Transpose) .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix A(IA:IA+N-1,JA:JA+N-1). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right-hand sides, i.e., the number of columns of the distributed submatrices B(IB:IB+N-1,JB:JB+NRHS-1) and .br X(IX:IX+N-1,JX:JX+NRHS-1). NRHS >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1)). On entry, the N-by-N matrix A(IA:IA+N-1,JA:JA+N-1). If FACT = 'F' and EQUED is not 'N', .br then A(IA:IA+N-1,JA:JA+N-1) must have been equilibrated by .br the scaling factors in R and/or C. A(IA:IA+N-1,JA:JA+N-1) is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. On exit, if EQUED .ne. 'N', A(IA:IA+N-1,JA:JA+N-1) is scaled as follows: .br EQUED = 'R': A(IA:IA+N-1,JA:JA+N-1) := .br diag(R) * A(IA:IA+N-1,JA:JA+N-1) .br EQUED = 'C': A(IA:IA+N-1,JA:JA+N-1) := .br A(IA:IA+N-1,JA:JA+N-1) * diag(C) .br EQUED = 'B': A(IA:IA+N-1,JA:JA+N-1) := .br diag(R) * A(IA:IA+N-1,JA:JA+N-1) * diag(C). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 AF (local input or local output) REAL pointer into the local memory to an array of local dimension (LLD_AF,LOCc(JA+N-1)). If FACT = 'F', then AF(IAF:IAF+N-1,JAF:JAF+N-1) is an input argument and on entry contains the factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U as computed by PSGETRF. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix A(IA:IA+N-1,JA:JA+N-1). If FACT = 'N', then AF(IAF:IAF+N-1,JAF:JAF+N-1) is an output argument and on exit returns the factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the original .br matrix A(IA:IA+N-1,JA:JA+N-1). If FACT = 'E', then AF(IAF:IAF+N-1,JAF:JAF+N-1) is an output argument and on exit returns the factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the equili- .br brated matrix A(IA:IA+N-1,JA:JA+N-1) (see the description of .br A(IA:IA+N-1,JA:JA+N-1) for the form of the equilibrated matrix). .TP 8 IAF (global input) INTEGER The row index in the global array AF indicating the first row of sub( AF ). .TP 8 JAF (global input) INTEGER The column index in the global array AF indicating the first column of sub( AF ). .TP 8 DESCAF (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix AF. .TP 8 IPIV (local input or local output) INTEGER array, dimension LOCr(M_A)+MB_A. If FACT = 'F', then IPIV is an input argu- ment and on entry contains the pivot indices from the fac- torization A(IA:IA+N-1,JA:JA+N-1) = P*L*U as computed by PSGETRF; IPIV(i) -> The global row local row i was swapped with. This array must be aligned with A( IA:IA+N-1, * ). If FACT = 'N', then IPIV is an output argument and on exit contains the pivot indices from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the original matrix .br A(IA:IA+N-1,JA:JA+N-1). If FACT = 'E', then IPIV is an output argument and on exit contains the pivot indices from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the equilibrated matrix .br A(IA:IA+N-1,JA:JA+N-1). .TP 8 EQUED (global input or global output) CHARACTER Specifies the form of equilibration that was done. = 'N': No equilibration (always true if FACT = 'N'). .br = 'R': Row equilibration, i.e., A(IA:IA+N-1,JA:JA+N-1) has been premultiplied by diag(R). = 'C': Column equilibration, i.e., A(IA:IA+N-1,JA:JA+N-1) has been postmultiplied by diag(C). = 'B': Both row and column equilibration, i.e., .br A(IA:IA+N-1,JA:JA+N-1) has been replaced by .br diag(R) * A(IA:IA+N-1,JA:JA+N-1) * diag(C). EQUED is an input variable if FACT = 'F'; otherwise, it is an output variable. .TP 8 R (local input or local output) REAL array, dimension LOCr(M_A). The row scale factors for A(IA:IA+N-1,JA:JA+N-1). .br If EQUED = 'R' or 'B', A(IA:IA+N-1,JA:JA+N-1) is multiplied on the left by diag(R); if EQUED='N' or 'C', R is not acces- sed. R is an input variable if FACT = 'F'; otherwise, R is an output variable. If FACT = 'F' and EQUED = 'R' or 'B', each element of R must be positive. R is replicated in every process column, and is aligned with the distributed matrix A. .TP 8 C (local input or local output) REAL array, dimension LOCc(N_A). The column scale factors for A(IA:IA+N-1,JA:JA+N-1). .br If EQUED = 'C' or 'B', A(IA:IA+N-1,JA:JA+N-1) is multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is not accessed. C is an input variable if FACT = 'F'; otherwise, C is an output variable. If FACT = 'F' and EQUED = 'C' or 'B', each element of C must be positive. C is replicated in every process row, and is aligned with the distributed matrix A. .TP 8 B (local input/local output) REAL pointer into the local memory to an array of local dimension (LLD_B,LOCc(JB+NRHS-1) ). On entry, the N-by-NRHS right-hand side matrix B(IB:IB+N-1,JB:JB+NRHS-1). On exit, if .br EQUED = 'N', B(IB:IB+N-1,JB:JB+NRHS-1) is not modified; if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B(IB:IB+N-1,JB:JB+NRHS-1); if TRANS = 'T' or 'C' .br and EQUED = 'C' or 'B', B(IB:IB+N-1,JB:JB+NRHS-1) is over- .br written by diag(C)*B(IB:IB+N-1,JB:JB+NRHS-1). .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 X (local input/local output) REAL pointer into the local memory to an array of local dimension (LLD_X, LOCc(JX+NRHS-1)). If INFO = 0, the N-by-NRHS solution matrix X(IX:IX+N-1,JX:JX+NRHS-1) to the original .br system of equations. Note that A(IA:IA+N-1,JA:JA+N-1) and .br B(IB:IB+N-1,JB:JB+NRHS-1) are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(C))*X(IX:IX+N-1,JX:JX+NRHS-1) if TRANS = 'N' and EQUED = 'C' or 'B', or inv(diag(R))*X(IX:IX+N-1,JX:JX+NRHS-1) if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 RCOND (global output) REAL The estimate of the reciprocal condition number of the matrix A(IA:IA+N-1,JA:JA+N-1) 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. .TP 8 FERR (local output) REAL array, dimension LOCc(N_B) The estimated forward error bounds for each solution vector X(j) (the j-th column of the solution matrix X(IX:IX+N-1,JX:JX+NRHS-1). If XTRUE is the true solution, FERR(j) bounds the magnitude of the largest entry in (X(j) - XTRUE) divided by the magnitude of the largest entry in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. FERR is replicated in every process row, and is aligned with the matrices B and X. .TP 8 BERR (local output) REAL array, dimension LOCc(N_B). The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any entry of A(IA:IA+N-1,JA:JA+N-1) or .br B(IB:IB+N-1,JB:JB+NRHS-1) that makes X(j) an exact solution). BERR is replicated in every process row, and is aligned with the matrices B and X. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK = MAX( PSGECON( LWORK ), PSGERFS( LWORK ) ) + LOCr( N_A ). If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/local output) INTEGER array, dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK. .TP 8 LIWORK (local or global input) INTEGER The dimension of the array IWORK. LIWORK is local input and must be at least LIWORK = LOCr(N_A). If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 0: if INFO = i, and i is .br <= N: U(IA+I-1,IA+I-1) 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. = N+1: RCOND is less than machine precision. The factorization has been completed, but the matrix is singular to working precision, and the solution and error bounds have not been computed. scalapack-doc-1.5/man/manl/psgetf2.l0100644000056400000620000001260006335610643016773 0ustar pfrauenfstaff.TH PSGETF2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGETF2 - compute an LU factorization of a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) using partial pivoting with row interchanges .SH SYNOPSIS .TP 20 SUBROUTINE PSGETF2( M, N, A, IA, JA, DESCA, IPIV, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, M, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ) .TP 20 .ti +4 REAL A( * ) .SH PURPOSE PSGETF2 computes an LU factorization of a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) using partial pivoting with row interchanges. The factorization has the form sub( A ) = P * L * U, where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n). .br This is the right-looking Parallel Level 2 BLAS version of the algorithm. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires N <= NB_A-MOD(JA-1, NB_A) and square block decomposition ( MB_A = NB_A ). .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). NB_A-MOD(JA-1, NB_A) >= N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the M-by-N distributed matrix sub( A ). On exit, this array contains the local pieces of the factors L and U from the factoriza- tion sub( A ) = P*L*U; the unit diagonal elements of L are not stored. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local output) INTEGER array, dimension ( LOCr(M_A)+MB_A ) This array contains the pivoting information. IPIV(i) -> The global row local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, U(IA+K-1,JA+K-1) 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. scalapack-doc-1.5/man/manl/psgetrf.l0100644000056400000620000001256206335610643017102 0ustar pfrauenfstaff.TH PSGETRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGETRF - compute an LU factorization of a general M-by-N distributed matrix sub( A ) = (IA:IA+M-1,JA:JA+N-1) using partial pivoting with row interchanges .SH SYNOPSIS .TP 20 SUBROUTINE PSGETRF( M, N, A, IA, JA, DESCA, IPIV, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, M, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ) .TP 20 .ti +4 REAL A( * ) .SH PURPOSE PSGETRF computes an LU factorization of a general M-by-N distributed matrix sub( A ) = (IA:IA+M-1,JA:JA+N-1) using partial pivoting with row interchanges. The factorization has the form sub( A ) = P * L * U, where P is a permutation matrix, L is lower triangular with unit diagonal ele- ments (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n). L and U are stored in sub( A ). This is the right-looking Parallel Level 3 BLAS version of the algorithm. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block decomposition ( MB_A = NB_A ). .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the M-by-N distributed matrix sub( A ) to be factored. On exit, this array contains the local pieces of the factors L and U from the factorization sub( A ) = P*L*U; the unit diagonal ele- ments of L are not stored. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local output) INTEGER array, dimension ( LOCr(M_A)+MB_A ) This array contains the pivoting information. IPIV(i) -> The global row local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, U(IA+K-1,JA+K-1) 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. scalapack-doc-1.5/man/manl/psgetri.l0100644000056400000620000001440406335610643017102 0ustar pfrauenfstaff.TH PSGETRI l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGETRI - compute the inverse of a distributed matrix using the LU factorization computed by PSGETRF .SH SYNOPSIS .TP 20 SUBROUTINE PSGETRI( N, A, IA, JA, DESCA, IPIV, WORK, LWORK, IWORK, LIWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LIWORK, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ), IWORK( * ) .TP 20 .ti +4 REAL A( * ), WORK( * ) .SH PURPOSE PSGETRI computes the inverse of a distributed matrix using the LU factorization computed by PSGETRF. This method inverts U and then computes the inverse of sub( A ) = A(IA:IA+N-1,JA:JA+N-1) denoted InvA by solving the system InvA*L = inv(U) for InvA. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the local pieces of the L and U obtained by the factorization sub( A ) = P*L*U computed by PSGETRF. On exit, if INFO = 0, sub( A ) contains the inverse of the original distributed matrix sub( A ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local input) INTEGER array, dimension LOCr(M_A)+MB_A keeps track of the pivoting information. IPIV(i) is the global row index the local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK = LOCr(N+MOD(IA-1,MB_A))*NB_A. WORK is used to keep a copy of at most an entire column block of sub( A ). If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/local output) INTEGER array, dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK. .TP 8 LIWORK (local or global input) INTEGER The dimension of the array IWORK used as workspace for physically transposing the pivots. LIWORK is local input and must be at least if NPROW == NPCOL then LIWORK = LOCc( N_A + MOD(JA-1, NB_A) ) + NB_A, else LIWORK = LOCc( N_A + MOD(JA-1, NB_A) ) + MAX( CEIL(CEIL(LOCr(M_A)/MB_A)/(LCM/NPROW)), NB_A ) where LCM is the least common multiple of process rows and columns (NPROW and NPCOL). end if If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, U(IA+K-1,IA+K-1) is exactly zero; the matrix is singular and its inverse could not be computed. scalapack-doc-1.5/man/manl/psgetrs.l0100644000056400000620000001324706335610643017120 0ustar pfrauenfstaff.TH PSGETRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGETRS - solve a system of distributed linear equations op( sub( A ) ) * X = sub( B ) with a general N-by-N distributed matrix sub( A ) using the LU factorization computed by PSGETRF .SH SYNOPSIS .TP 20 SUBROUTINE PSGETRS( TRANS, N, NRHS, A, IA, JA, DESCA, IPIV, B, IB, JB, DESCB, INFO ) .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER IA, IB, INFO, JA, JB, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ), IPIV( * ) .TP 20 .ti +4 REAL A( * ), B( * ) .SH PURPOSE PSGETRS solves a system of distributed linear equations sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1), op( A ) = A or A**T and sub( B ) denotes B(IB:IB+N-1,JB:JB+NRHS-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block data decomposition ( MB_A=NB_A ). .SH ARGUMENTS .TP 8 TRANS (global input) CHARACTER Specifies the form of the system of equations: .br = 'N': sub( A ) * X = sub( B ) (No transpose) .br = 'T': sub( A )**T * X = sub( B ) (Transpose) .br = 'C': sub( A )**T * X = sub( B ) (Transpose) .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed submatrix sub( B ). NRHS >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the factors L and U from the factorization sub( A ) = P*L*U; the unit diagonal elements of L are not stored. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local input) INTEGER array, dimension ( LOCr(M_A)+MB_A ) This array contains the pivoting information. IPIV(i) -> The global row local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 B (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_B,LOCc(JB+NRHS-1)). On entry, the right hand sides sub( B ). On exit, sub( B ) is overwritten by the solution distributed matrix X. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/psggqrf.l0100644000056400000620000002350206335610643017075 0ustar pfrauenfstaff.TH PSGGQRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGGQRF - compute a generalized QR factorization of an N-by-M matrix sub( A ) = A(IA:IA+N-1,JA:JA+M-1) and an N-by-P matrix sub( B ) = B(IB:IB+N-1,JB:JB+P-1) .SH SYNOPSIS .TP 20 SUBROUTINE PSGGQRF( N, M, P, A, IA, JA, DESCA, TAUA, B, IB, JB, DESCB, TAUB, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, IB, INFO, JA, JB, LWORK, M, N, P .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 REAL A( * ), B( * ), TAUA( * ), TAUB( * ), WORK( * ) .SH PURPOSE PSGGQRF computes a generalized QR factorization of an N-by-M matrix sub( A ) = A(IA:IA+N-1,JA:JA+M-1) and an N-by-P matrix sub( B ) = B(IB:IB+N-1,JB:JB+P-1): sub( A ) = Q*R, sub( B ) = Q*T*Z, .br where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal matrix, and R and T assume one of the forms: .br if N >= M, R = ( R11 ) M , or if N < M, R = ( R11 R12 ) N, ( 0 ) N-M N M-N M .br where R11 is upper triangular, and .br if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) N-P, P-N N ( T21 ) P P .br where T12 or T21 is upper triangular. .br In particular, if sub( B ) is square and nonsingular, the GQR factorization of sub( A ) and sub( B ) implicitly gives the QR factorization of inv( sub( B ) )* sub( A ): .br inv( sub( B ) )*sub( A )= Z'*(inv(T)*R) .br where inv( sub( B ) ) denotes the inverse of the matrix sub( B ), and Z' denotes the transpose of matrix Z. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrices sub( A ) and sub( B ). N >= 0. .TP 8 M (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). M >= 0. .TP 8 P (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( B ). P >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+M-1)). On entry, the local pieces of the N-by-M distributed matrix sub( A ) which is to be factored. On exit, the elements on and above the diagonal of sub( A ) contain the min(N,M) by M upper trapezoidal matrix R (R is upper triangular if N >= M); the elements below the diagonal, with the array TAUA, represent the orthogonal matrix Q as a product of min(N,M) elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAUA (local output) REAL, array, dimension LOCc(JA+MIN(N,M)-1). This array contains the scalar factors TAUA of the elementary reflectors which represent the orthogonal matrix Q. TAUA is tied to the distributed matrix A. (see Further Details). B (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+P-1)). On entry, the local pieces of the N-by-P distributed matrix sub( B ) which is to be factored. On exit, if N <= P, the upper triangle of B(IB:IB+N-1,JB+P-N:JB+P-1) contains the N by N upper triangular matrix T; if N > P, the elements on and above the (N-P)-th subdiagonal contain the N by P upper trapezoidal matrix T; the remaining elements, with the array TAUB, represent the orthogonal matrix Z as a product of elementary reflectors (see Further Details). IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 TAUB (local output) REAL, array, dimension LOCr(IB+N-1) This array contains the scalar factors of the elementary reflectors which represent the orthogonal unitary matrix Z. TAUB is tied to the distributed matrix B (see Further Details). .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MAX( NB_A * ( NpA0 + MqA0 + NB_A ), MAX( (NB_A*(NB_A-1))/2, (PqB0 + NpB0)*NB_A ) + NB_A * NB_A, MB_B * ( NpB0 + PqB0 + MB_B ) ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), NpA0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), MqA0 = NUMROC( M+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFB = MOD( IB-1, MB_B ), ICOFFB = MOD( JB-1, NB_B ), IBROW = INDXG2P( IB, MB_B, MYROW, RSRC_B, NPROW ), IBCOL = INDXG2P( JB, NB_B, MYCOL, CSRC_B, NPCOL ), NpB0 = NUMROC( N+IROFFB, MB_B, MYROW, IBROW, NPROW ), PqB0 = NUMROC( P+ICOFFB, NB_B, MYCOL, IBCOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ja) H(ja+1) . . . H(ja+k-1), where k = min(n,m). Each H(i) has the form .br H(i) = I - taua * v * v' .br where taua is a real scalar, and v is a real vector with .br v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in .br A(ia+i:ia+n-1,ja+i-1), and taua in TAUA(ja+i-1). .br To form Q explicitly, use ScaLAPACK subroutine PSORGQR. .br To use Q to update another matrix, use ScaLAPACK subroutine PSORMQR. The matrix Z is represented as a product of elementary reflectors Z = H(ib) H(ib+1) . . . H(ib+k-1), where k = min(n,p). Each H(i) has the form .br H(i) = I - taub * v * v' .br where taub is a real scalar, and v is a real vector with .br v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in B(ib+n-k+i-1,jb:jb+p-k+i-2), and taub in TAUB(ib+n-k+i-1). To form Z explicitly, use ScaLAPACK subroutine PSORGRQ. .br To use Z to update another matrix, use ScaLAPACK subroutine PSORMRQ. Alignment requirements .br ====================== .br The distributed submatrices sub( A ) and sub( B ) must verify some alignment properties, namely the following expression should be true: ( MB_A.EQ.MB_B .AND. IROFFA.EQ.IROFFB .AND. IAROW.EQ.IBROW ) scalapack-doc-1.5/man/manl/psggrqf.l0100644000056400000620000002340206335610643017074 0ustar pfrauenfstaff.TH PSGGRQF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSGGRQF - compute a generalized RQ factorization of an M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PSGGRQF( M, P, N, A, IA, JA, DESCA, TAUA, B, IB, JB, DESCB, TAUB, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, IB, INFO, JA, JB, LWORK, M, N, P .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 REAL A( * ), B( * ), TAUA( * ), TAUB( * ), WORK( * ) .SH PURPOSE PSGGRQF computes a generalized RQ factorization of an M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) and a P-by-N matrix sub( B ) = B(IB:IB+P-1,JB:JB+N-1): .br sub( A ) = R*Q, sub( B ) = Z*T*Q, .br where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal matrix, and R and T assume one of the forms: .br if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N, N-M M ( R21 ) N N .br where R12 or R21 is upper triangular, and .br if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P, ( 0 ) P-N P N-P N .br where T11 is upper triangular. .br In particular, if sub( B ) is square and nonsingular, the GRQ factorization of sub( A ) and sub( B ) implicitly gives the RQ factorization of sub( A )*inv( sub( B ) ): .br sub( A )*inv( sub( B ) ) = (R*inv(T))*Z' .br where inv( sub( B ) ) denotes the inverse of the matrix sub( B ), and Z' denotes the transpose of matrix Z. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 P (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( B ). P >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrices sub( A ) and sub( B ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, if M <= N, the upper triangle of A( IA:IA+M-1, JA+N-M:JA+N-1 ) 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 TAUA, represent the orthogonal matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAUA (local output) REAL, array, dimension LOCr(IA+M-1) This array contains the scalar factors of the elementary reflectors which represent the orthogonal unitary matrix Q. TAUA is tied to the distributed matrix A (see Further Details). .TP 8 B (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+N-1)). On entry, the local pieces of the P-by-N distributed matrix sub( B ) which is to be factored. On exit, the elements on and above the diagonal of sub( B ) contain the min(P,N) by N upper trapezoidal matrix T (T is upper triangular if P >= N); the elements below the diagonal, with the array TAUB, represent the orthogonal matrix Z as a product of elementary reflectors (see Further Details). IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 TAUB (local output) REAL, array, dimension LOCc(JB+MIN(P,N)-1). This array contains the scalar factors TAUB of the elementary reflectors which represent the orthogonal matrix Z. TAUB is tied to the distributed matrix B (see Further Details). WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MAX( MB_A * ( MpA0 + NqA0 + MB_A ), MAX( (MB_A*(MB_A-1))/2, (PpB0 + NqB0)*MB_A ) + MB_A * MB_A, NB_B * ( PpB0 + NqB0 + NB_B ) ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFB = MOD( IB-1, MB_B ), ICOFFB = MOD( JB-1, NB_B ), IBROW = INDXG2P( IB, MB_B, MYROW, RSRC_B, NPROW ), IBCOL = INDXG2P( JB, NB_B, MYCOL, CSRC_B, NPCOL ), PpB0 = NUMROC( P+IROFFB, MB_B, MYROW, IBROW, NPROW ), NqB0 = NUMROC( N+ICOFFB, NB_B, MYCOL, IBCOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ia) H(ia+1) . . . H(ia+k-1), where k = min(m,n). Each H(i) has the form .br H(i) = I - taua * v * v' .br where taua is a real scalar, and v is a real vector with .br 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(ia+m-k+i-1,ja:ja+n-k+i-2), and taua in TAUA(ia+m-k+i-1). To form Q explicitly, use ScaLAPACK subroutine PSORGRQ. .br To use Q to update another matrix, use ScaLAPACK subroutine PSORMRQ. The matrix Z is represented as a product of elementary reflectors Z = H(jb) H(jb+1) . . . H(jb+k-1), where k = min(p,n). Each H(i) has the form .br H(i) = I - taub * v * v' .br where taub is a real scalar, and v is a real vector with .br v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in .br B(ib+i:ib+p-1,jb+i-1), and taub in TAUB(jb+i-1). .br To form Z explicitly, use ScaLAPACK subroutine PSORGQR. .br To use Z to update another matrix, use ScaLAPACK subroutine PSORMQR. Alignment requirements .br ====================== .br The distributed submatrices sub( A ) and sub( B ) must verify some alignment properties, namely the following expression should be true: ( NB_A.EQ.NB_B .AND. ICOFFA.EQ.ICOFFB .AND. IACOL.EQ.IBCOL ) scalapack-doc-1.5/man/manl/pslabad.l0100644000056400000620000000307006335610643017030 0ustar pfrauenfstaff.TH PSLABAD l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLABAD - take as input the values computed by PSLAMCH for underflow and overflow, and returns the square root of each of these values if the log of LARGE is sufficiently large .SH SYNOPSIS .TP 20 SUBROUTINE PSLABAD( ICTXT, SMALL, LARGE ) .TP 20 .ti +4 INTEGER ICTXT .TP 20 .ti +4 REAL LARGE, SMALL .SH PURPOSE PSLABAD takes as input the values computed by PSLAMCH for underflow and overflow, and returns the square root of each of these values if the log of LARGE is sufficiently large. This subroutine is intended to identify machines with a large exponent range, such as the Crays, and redefine the underflow and overflow limits to be the square roots of the values computed by PSLAMCH. This subroutine is needed because PSLAMCH does not compensate for poor arithmetic in the upper half of the exponent range, as is found on a Cray. .br In addition, this routine performs a global minimization and maximi- zation on these values, to support heterogeneous computing networks. .SH ARGUMENTS .TP 8 ICTXT (global input) INTEGER The BLACS context handle in which the computation takes place. .TP 8 SMALL (local input/local output) REAL On entry, the underflow threshold as computed by PSLAMCH. On exit, if LOG10(LARGE) is sufficiently large, the square root of SMALL, otherwise unchanged. .TP 8 LARGE (local input/local output) REAL On entry, the overflow threshold as computed by PSLAMCH. On exit, if LOG10(LARGE) is sufficiently large, the square root of LARGE, otherwise unchanged. scalapack-doc-1.5/man/manl/pslabrd.l0100644000056400000620000002327406335610643017061 0ustar pfrauenfstaff.TH PSLABRD l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLABRD - reduce the first NB rows and columns of a real general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form by an orthogonal transformation Q' * A * P, .SH SYNOPSIS .TP 20 SUBROUTINE PSLABRD( M, N, NB, A, IA, JA, DESCA, D, E, TAUQ, TAUP, X, IX, JX, DESCX, Y, IY, JY, DESCY, WORK ) .TP 20 .ti +4 INTEGER IA, IX, IY, JA, JX, JY, M, N, NB .TP 20 .ti +4 INTEGER DESCA( * ), DESCX( * ), DESCY( * ) .TP 20 .ti +4 REAL A( * ), D( * ), E( * ), TAUP( * ), TAUQ( * ), X( * ), Y( * ), WORK( * ) .SH PURPOSE PSLABRD reduces the first NB rows and columns of a real general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form by an orthogonal transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of sub( A ). .br If M >= N, sub( A ) is reduced to upper bidiagonal form; if M < N, to lower bidiagonal form. .br This is an auxiliary routine called by PSGEBRD. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 NB (global input) INTEGER The number of leading rows and columns of sub( A ) to be reduced. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the general distributed matrix sub( A ) to be reduced. On exit, the first NB rows and columns of the matrix are overwritten; the rest of the distributed matrix sub( A ) is unchanged. If m >= n, elements on and below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. If m < n, elements below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and elements on and above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) REAL array, dimension LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-1) otherwise. The distributed diagonal elements of the bidiagonal matrix B: D(i) = A(ia+i-1,ja+i-1). D is tied to the distributed matrix A. .TP 8 E (local output) REAL array, dimension LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-2) otherwise. The distributed off-diagonal elements of the bidiagonal distributed matrix B: if m >= n, E(i) = A(ia+i-1,ja+i) for i = 1,2,...,n-1; if m < n, E(i) = A(ia+i,ja+i-1) for i = 1,2,...,m-1. E is tied to the distributed matrix A. .TP 8 TAUQ (local output) REAL array dimension LOCc(JA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. TAUQ is tied to the distributed matrix A. See Further Details. TAUP (local output) REAL array, dimension LOCr(IA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the orthogonal matrix P. TAUP is tied to the distributed matrix A. See Further Details. X (local output) REAL pointer into the local memory to an array of dimension (LLD_X,NB). On exit, the local pieces of the distributed M-by-NB matrix X(IX:IX+M-1,JX:JX+NB-1) required to update the unreduced part of sub( A ). .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 Y (local output) REAL pointer into the local memory to an array of dimension (LLD_Y,NB). On exit, the local pieces of the distributed N-by-NB matrix Y(IY:IY+N-1,JY:JY+NB-1) required to update the unreduced part of sub( A ). .TP 8 IY (global input) INTEGER The row index in the global array Y indicating the first row of sub( Y ). .TP 8 JY (global input) INTEGER The column index in the global array Y indicating the first column of sub( Y ). .TP 8 DESCY (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix Y. .TP 8 WORK (local workspace) REAL array, dimension (LWORK) LWORK >= NB_A + NQ, with NQ = NUMROC( N+MOD( IA-1, NB_Y ), NB_Y, MYCOL, IACOL, NPCOL ) IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ) INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. .SH FURTHER DETAILS The matrices Q and P are represented as products of elementary reflectors: .br Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb) Each H(i) and G(i) has the form: .br H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are real scalars, and v and u are real vectors. If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in A(ia+i-1:ia+m-1,ja+i-1); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in A(ia+i-1,ja+i:ja+n-1); tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in A(ia+i+1:ia+m-1,ja+i-1); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in A(ia+i-1,ja+i:ja+n-1); tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br The elements of the vectors v and u together form the m-by-nb matrix V and the nb-by-n matrix U' which are needed, with X and Y, to apply the transformation to the unreduced part of the matrix, using a block update of the form: sub( A ) := sub( A ) - V*Y' - X*U'. .br The contents of sub( A ) on exit are illustrated by the following examples with nb = 2: .br m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 ) ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 ) ( v1 v2 a a a ) ( v1 1 a a a a ) ( v1 v2 a a a ) ( v1 v2 a a a a ) ( v1 v2 a a a ) ( v1 v2 a a a a ) ( v1 v2 a a a ) .br where a denotes an element of the original matrix which is unchanged, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i). .br scalapack-doc-1.5/man/manl/pslacon.l0100644000056400000620000001272706335610643017072 0ustar pfrauenfstaff.TH PSLACON l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLACON - estimate the 1-norm of a square, real distributed matrix A .SH SYNOPSIS .TP 20 SUBROUTINE PSLACON( N, V, IV, JV, DESCV, X, IX, JX, DESCX, ISGN, EST, KASE ) .TP 20 .ti +4 INTEGER IV, IX, JV, JX, KASE, N .TP 20 .ti +4 REAL EST .TP 20 .ti +4 INTEGER DESCV( * ), DESCX( * ), ISGN( * ) .TP 20 .ti +4 REAL V( * ), X( * ) .SH PURPOSE PSLACON estimates the 1-norm of a square, real distributed matrix A. Reverse communication is used for evaluating matrix-vector products. X and V are aligned with the distributed matrix A, this information is implicitly contained within IV, IX, DESCV, and DESCX. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The length of the distributed vectors V and X. N >= 0. .TP 8 V (local workspace) REAL pointer into the local memory to an array of dimension LOCr(N+MOD(IV-1,MB_V)). On the final return, V = A*W, where EST = norm(V)/norm(W) (W is not returned). .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 X (local input/local output) REAL pointer into the local memory to an array of dimension LOCr(N+MOD(IX-1,MB_X)). On an intermediate return, X should be overwritten by A * X, if KASE=1, A' * X, if KASE=2, PSLACON must be re-called with all the other parameters unchanged. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 ISGN (local workspace) INTEGER array, dimension LOCr(N+MOD(IX-1,MB_X)). ISGN is aligned with X and V. .TP 8 EST (global output) REAL An estimate (a lower bound) for norm(A). .TP 8 KASE (local input/local output) INTEGER On the initial call to PSLACON, KASE should be 0. On an intermediate return, KASE will be 1 or 2, indicating whether X should be overwritten by A * X or A' * X. On the final return from PSLACON, KASE will again be 0. .SH FURTHER DETAILS The serial version SLACON has been contributed by Nick Higham, University of Manchester. It was originally named SONEST, dated March 16, 1988. .br Reference: N.J. Higham, "FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation", ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988. scalapack-doc-1.5/man/manl/pslaconsb.l0100644000056400000620000001352706335610643017416 0ustar pfrauenfstaff.TH PSLACONSB l "12 May 1997" "LAPACK version 1.5 " "LAPACK routine (version 1.5 )" .SH NAME PSLACONSB - look for two consecutive small subdiagonal elements by seeing the effect of starting a double shift QR iteration given by H44, H33, & H43H34 and see if this would make a subdiagonal negligible .SH SYNOPSIS .TP 22 SUBROUTINE PSLACONSB( A, DESCA, I, L, M, H44, H33, H43H34, BUF, LWORK ) .TP 22 .ti +4 INTEGER I, L, LWORK, M .TP 22 .ti +4 REAL H33, H43H34, H44 .TP 22 .ti +4 INTEGER DESCA( * ) .TP 22 .ti +4 REAL A( * ), BUF( * ) .SH PURPOSE PSLACONSB looks for two consecutive small subdiagonal elements by seeing the effect of starting a double shift QR iteration given by H44, H33, & H43H34 and see if this would make a subdiagonal negligible. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 A (global input) REAL array, dimension (DESCA(LLD_),*) On entry, the Hessenberg matrix whose tridiagonal part is being scanned. Unchanged on exit. .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 I (global input) INTEGER The global location of the bottom of the unreduced submatrix of A. Unchanged on exit. .TP 8 L (global input) INTEGER The global location of the top of the unreduced submatrix of A. Unchanged on exit. .TP 8 M (global output) INTEGER On exit, this yields the starting location of the QR double shift. This will satisfy: L <= M <= I-2. H44 H33 H43H34 (global input) REAL These three values are for the double shift QR iteration. .TP 8 BUF (local output) REAL array of size LWORK. .TP 8 LWORK (global input) INTEGER On exit, LWORK is the size of the work buffer. This must be at least 7*Ceil( Ceil( (I-L)/HBL ) / LCM(NPROW,NPCOL) ) Here LCM is least common multiple, and NPROWxNPCOL is the logical grid size. Logic: ====== Two consecutive small subdiagonal elements will stall convergence of a double shift if their product is small relatively even if each is not very small. Thus it is necessary to scan the "tridiagonal portion of the matrix." In the LAPACK algorithm DLAHQR, a loop of M goes from I-2 down to L and examines H(m,m),H(m+1,m+1),H(m+1,m),H(m,m+1),H(m-1,m-1),H(m,m-1), and H(m+2,m-1). Since these elements may be on separate processors, the first major loop (10) goes over the tridiagonal and has each node store whatever values of the 7 it has that the node owning H(m,m) does not. This will occur on a border and can happen in no more than 3 locations per block assuming square blocks. There are 5 buffers that each node stores these values: a buffer to send diagonally down and right, a buffer to send up, a buffer to send left, a buffer to send diagonally up and left and a buffer to send right. Each of these buffers is actually stored in one buffer BUF where BUF(ISTR1+1) starts the first buffer, BUF(ISTR2+1) starts the second, etc.. After the values are stored, if there are any values that a node needs, they will be sent and received. Then the next major loop passes over the data and searches for two consecutive small subdiagonals. Notes: This routine does a global maximum and must be called by all processes. Implemented by: G. Henry, November 17, 1996 scalapack-doc-1.5/man/manl/pslacp2.l0100644000056400000620000001266306335610643016776 0ustar pfrauenfstaff.TH PSLACP2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLACP2 - copie all or part of a distributed matrix A to another distributed matrix B .SH SYNOPSIS .TP 20 SUBROUTINE PSLACP2( UPLO, M, N, A, IA, JA, DESCA, B, IB, JB, DESCB ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IB, JA, JB, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 REAL A( * ), B( * ) .SH PURPOSE PSLACP2 copies all or part of a distributed matrix A to another distributed matrix B. No communication is performed, PSLACP2 performs a local copy sub( A ) := sub( B ), where sub( A ) denotes A(IA:IA+M-1,JA:JA+N-1) and sub( B ) denotes B(IB:IB+M-1,JB:JB+N-1). PSLACP2 requires that only dimension of the matrix operands is distributed. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies the part of the distributed matrix sub( A ) to be copied: .br = 'U': Upper triangular part is copied; the strictly lower triangular part of sub( A ) is not referenced; = 'L': Lower triangular part is copied; the strictly upper triangular part of sub( A ) is not referenced; Otherwise: All of the matrix sub( A ) is copied. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1) ). This array contains the local pieces of the distributed matrix sub( A ) to be copied from. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local output) REAL pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+N-1) ). This array contains on exit the local pieces of the distributed matrix sub( B ) set as follows: if UPLO = 'U', B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), 1<=i<=j, 1<=j<=N; if UPLO = 'L', B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), j<=i<=M, 1<=j<=N; otherwise, B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), 1<=i<=M, 1<=j<=N. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. scalapack-doc-1.5/man/manl/pslacp3.l0100644000056400000620000001204306335610643016767 0ustar pfrauenfstaff.TH PSLACP3 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSLACP3 - i an auxiliary routine that copies from a global parallel array into a local replicated array or vise versa .SH SYNOPSIS .TP 20 SUBROUTINE PSLACP3( M, I, A, DESCA, B, LDB, II, JJ, REV ) .TP 20 .ti +4 INTEGER I, II, JJ, LDB, M, REV .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), B( LDB, * ) .SH PURPOSE PSLACP3 is an auxiliary routine that copies from a global parallel array into a local replicated array or vise versa. Notice that the entire submatrix that is copied gets placed on one node or more. The receiving node can be specified precisely, or all nodes can receive, or just one row or column of nodes. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER M is the order of the square submatrix that is copied. M >= 0. Unchanged on exit .TP 8 I (global input) INTEGER A(I,I) is the global location that the copying starts from. Unchanged on exit. .TP 8 A (global input/output) REAL array, dimension (DESCA(LLD_),*) On entry, the parallel matrix to be copied into or from. On exit, if REV=1, the copied data. Unchanged on exit if REV=0. .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input/output) REAL array of size (LDB,M) If REV=0, this is the global portion of the array A(I:I+M-1,I:I+M-1). If REV=1, this is the unchanged on exit. .TP 8 LDB (local input) INTEGER The leading dimension of B. .TP 8 II (global input) INTEGER By using REV 0 & 1, data can be sent out and returned again. If REV=0, then II is destination row index for the node(s) receiving the replicated B. If II>=0,JJ>=0, then node (II,JJ) receives the data If II=-1,JJ>=0, then all rows in column JJ receive the data If II>=0,JJ=-1, then all cols in row II receive the data If II=-1,JJ=-1, then all nodes receive the data If REV<>0, then II is the source row index for the node(s) sending the replicated B. .TP 8 JJ (global input) INTEGER Similar description as II above .TP 8 REV (global input) INTEGER Use REV = 0 to send global A into locally replicated B (on node (II,JJ)). Use REV <> 0 to send locally replicated B from node (II,JJ) to its owner (which changes depending on its location in A) into the global A. Implemented by: G. Henry, May 1, 1997 scalapack-doc-1.5/man/manl/pslacpy.l0100644000056400000620000001254306335610644017103 0ustar pfrauenfstaff.TH PSLACPY l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLACPY - copie all or part of a distributed matrix A to another distributed matrix B .SH SYNOPSIS .TP 20 SUBROUTINE PSLACPY( UPLO, M, N, A, IA, JA, DESCA, B, IB, JB, DESCB ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IB, JA, JB, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 REAL A( * ), B( * ) .SH PURPOSE PSLACPY copies all or part of a distributed matrix A to another distributed matrix B. No communication is performed, PSLACPY performs a local copy sub( A ) := sub( B ), where sub( A ) denotes A(IA:IA+M-1,JA:JA+N-1) and sub( B ) denotes B(IB:IB+M-1,JB:JB+N-1). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies the part of the distributed matrix sub( A ) to be copied: .br = 'U': Upper triangular part is copied; the strictly lower triangular part of sub( A ) is not referenced; = 'L': Lower triangular part is copied; the strictly upper triangular part of sub( A ) is not referenced; Otherwise: All of the matrix sub( A ) is copied. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1) ). This array contains the local pieces of the distributed matrix sub( A ) to be copied from. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local output) REAL pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+N-1) ). This array contains on exit the local pieces of the distributed matrix sub( B ) set as follows: if UPLO = 'U', B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), 1<=i<=j, 1<=j<=N; if UPLO = 'L', B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), j<=i<=M, 1<=j<=N; otherwise, B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), 1<=i<=M, 1<=j<=N. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. scalapack-doc-1.5/man/manl/pslaevswp.l0100644000056400000620000001213206335610644017446 0ustar pfrauenfstaff.TH PSLAEVSWP l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSLAEVSWP - move the eigenvectors (potentially unsorted) from where they are computed, to a ScaLAPACK standard block cyclic array, sorted so that the corresponding eigenvalues are sorted .SH SYNOPSIS .TP 22 SUBROUTINE PSLAEVSWP( N, ZIN, LDZI, Z, IZ, JZ, DESCZ, NVS, KEY, WORK, LWORK ) .TP 22 .ti +4 INTEGER IZ, JZ, LDZI, LWORK, N .TP 22 .ti +4 INTEGER DESCZ( * ), KEY( * ), NVS( * ) .TP 22 .ti +4 REAL WORK( * ), Z( * ), ZIN( LDZI, * ) .SH PURPOSE PSLAEVSWP moves the eigenvectors (potentially unsorted) from where they are computed, to a ScaLAPACK standard block cyclic array, sorted so that the corresponding eigenvalues are sorted. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS NP = the number of rows local to a given process. NQ = the number of columns local to a given process. .TP 8 N (global input) INTEGER The order of the matrix A. N >= 0. .TP 8 ZIN (local input) REAL array, dimension ( LDZI, NVS(iam) ) The eigenvectors on input. Each eigenvector resides entirely in one process. Each process holds a contiguous set of NVS(iam) eigenvectors. The first eigenvector which the process holds is: sum for i=[0,iam-1) of NVS(i) .TP 8 LDZI (locl input) INTEGER leading dimension of the ZIN array .TP 8 Z (local output) REAL array global dimension (N, N), local dimension (DESCZ(DLEN_), NQ) The eigenvectors on output. The eigenvectors are distributed in a block cyclic manner in both dimensions, with a block size of NB. .TP 8 IZ (global input) INTEGER Z's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JZ (global input) INTEGER Z's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCZ (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix Z. .TP 8 NVS (global input) INTEGER array, dimension( nprocs+1 ) nvs(i) = number of processes number of eigenvectors held by processes [0,i-1) nvs(1) = number of eigen vectors held by [0,1-1) == 0 nvs(nprocs+1) = number of eigen vectors held by [0,nprocs) == total number of eigenvectors .TP 8 KEY (global input) INTEGER array, dimension( N ) Indicates the actual index (after sorting) for each of the eigenvectors. .TP 8 WORK (local workspace) REAL array, dimension (LWORK) .TP 8 LWORK (local input) INTEGER dimension of WORK scalapack-doc-1.5/man/manl/pslahqr.l0100644000056400000620000002457506335610644017112 0ustar pfrauenfstaff.TH PSLAHQR l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLAHQR - i an auxiliary routine used to find the Schur decomposition and or eigenvalues of a matrix already in Hessenberg form from cols ILO to IHI .SH SYNOPSIS .TP 20 SUBROUTINE PSLAHQR( WANTT, WANTZ, N, ILO, IHI, A, DESCA, WR, WI, ILOZ, IHIZ, Z, DESCZ, WORK, LWORK, IWORK, ILWORK, INFO ) .TP 20 .ti +4 LOGICAL WANTT, WANTZ .TP 20 .ti +4 INTEGER IHI, IHIZ, ILO, ILOZ, ILWORK, INFO, LWORK, N, ROTN .TP 20 .ti +4 INTEGER DESCA( * ), DESCZ( * ), IWORK( * ) .TP 20 .ti +4 REAL A( * ), WI( * ), WORK( * ), WR( * ), Z( * ) .SH PURPOSE PSLAHQR is an auxiliary routine used to find the Schur decomposition and or eigenvalues of a matrix already in Hessenberg form from cols ILO to IHI. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 WANTT (global input) LOGICAL = .TRUE. : the full Schur form T is required; .br = .FALSE.: only eigenvalues are required. .TP 8 WANTZ (global input) LOGICAL .br = .TRUE. : the matrix of Schur vectors Z is required; .br = .FALSE.: Schur vectors are not required. .TP 8 N (global input) INTEGER The order of the Hessenberg matrix A (and Z if WANTZ). N >= 0. .TP 8 ILO (global input) INTEGER IHI (global input) INTEGER It is assumed that A is already upper quasi-triangular in rows and columns IHI+1:N, and that A(ILO,ILO-1) = 0 (unless ILO = 1). PSLAHQR works primarily with the Hessenberg submatrix in rows and columns ILO to IHI, but applies transformations to all of H if WANTT is .TRUE.. 1 <= ILO <= max(1,IHI); IHI <= N. .TP 8 A (global input/output) REAL array, dimension (DESCA(LLD_),*) On entry, the upper Hessenberg matrix A. On exit, if WANTT is .TRUE., A is upper quasi-triangular in rows and columns ILO:IHI, with any 2-by-2 or larger diagonal blocks not yet in standard form. If WANTT is .FALSE., the contents of A are unspecified on exit. .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 WR (global replicated output) REAL array, dimension (N) WI (global replicated output) REAL 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 WANTT is .TRUE., the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in A. A may be returned with larger diagonal blocks until the next release. .TP 8 ILOZ (global input) INTEGER IHIZ (global input) INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= ILO; IHI <= IHIZ <= N. .TP 8 Z (global input/output) REAL array. If WANTZ is .TRUE., on entry Z must contain the current matrix Z of transformations accumulated by PDHSEQR, and on exit Z has been updated; transformations are applied only to the submatrix Z(ILOZ:IHIZ,ILO:IHI). If WANTZ is .FALSE., Z is not referenced. .TP 8 DESCZ (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix Z. .TP 8 WORK (local output) REAL array of size LWORK (Unless LWORK=-1, in which case WORK must be at least size 1) .TP 8 LWORK (local input) INTEGER WORK(LWORK) is a local array and LWORK is assumed big enough so that LWORK >= 3*N + MAX( 2*MAX(DESCZ(LLD_),DESCA(LLD_)) + 2*LOCc(N), 7*Ceil(N/HBL)/LCM(NPROW,NPCOL)) + MAX( 2*N, (8*LCM(NPROW,NPCOL)+2)**2 ) If LWORK=-1, then WORK(1) gets set to the above number and the code returns immediately. .TP 8 IWORK (global and local input) INTEGER array of size ILWORK This will hold some of the IBLK integer arrays. This is held as a place holder for a future release. Currently unreferenced. .TP 8 ILWORK (local input) INTEGER This will hold the size of the IWORK array. This is held as a place holder for a future release. Currently unreferenced. .TP 8 INFO (global output) INTEGER < 0: parameter number -INFO incorrect or inconsistent .br = 0: successful exit .br > 0: PSLAHQR failed to compute all the eigenvalues ILO to IHI in a total of 30*(IHI-ILO+1) iterations; if INFO = i, elements i+1:ihi of WR and WI contain those eigenvalues which have been successfully computed. Logic: This algorithm is very similar to _LAHQR. Unlike _LAHQR, instead of sending one double shift through the largest unreduced submatrix, this algorithm sends multiple double shifts and spaces them apart so that there can be parallelism across several processor row/columns. Another critical difference is that this algorithm aggregrates multiple transforms together in order to apply them in a block fashion. Important Local Variables: IBLK = The maximum number of bulges that can be computed. Currently fixed. Future releases this won't be fixed. HBL = The square block size (HBL=DESCA(MB_)=DESCA(NB_)) ROTN = The number of transforms to block together NBULGE = The number of bulges that will be attempted on the current submatrix. IBULGE = The current number of bulges started. K1(*),K2(*) = The current bulge loops from K1(*) to K2(*). Subroutines: From LAPACK, this routine calls: SLAHQR -> Serial QR used to determine shifts and eigenvalues SLARFG -> Determine the Householder transforms This ScaLAPACK, this routine calls: PSLACONSB -> To determine where to start each iteration SLAMSH -> Sends multiple shifts through a small submatrix to see how the consecutive subdiagonals change (if PSLACONSB indicates we can start a run in the middle) PSLAWIL -> Given the shift, get the transformation SLASORTE -> Pair up eigenvalues so that reals are paired. PSLACP3 -> Parallel array to local replicated array copy & back. SLAREF -> Row/column reflector applier. Core routine here. PSLASMSUB -> Finds negligible subdiagonal elements. Current Notes and/or Restrictions: 1.) This code requires the distributed block size to be square and at least six (6); unlike simpler codes like LU, this algorithm is extremely sensitive to block size. Unwise choices of too small a block size can lead to bad performance. 2.) This code requires A and Z to be distributed identically and have identical contxts. A future version may allow Z to have a different contxt to 1D row map it to all nodes (so no communication on Z is necessary.) 3.) This release currently does not have a routine for resolving the Schur blocks into regular 2x2 form after this code is completed. Because of this, a significant performance impact is required while the deflation is done by sometimes a single column of processors. 4.) This code does not currently block the initial transforms so that none of the rows or columns for any bulge are completed until all are started. To offset pipeline start-up it is recommended that at least 2*LCM(NPROW,NPCOL) bulges are used (if possible) 5.) The maximum number of bulges currently supported is fixed at 32. In future versions this will be limited only by the incoming WORK and IWORK array. 6.) The matrix A must be in upper Hessenberg form. If elements below the subdiagonal are nonzero, the resulting transforms may be nonsimilar. This is also true with the LAPACK routine SLAHQR. 7.) For this release, this code has only been tested for RSRC_=CSRC_=0, but it has been written for the general case. 8.) Currently, all the eigenvalues are distributed to all the nodes. Future releases will probably distribute the eigenvalues by the column partitioning. 9.) The internals of this routine are subject to change. 10.) To optimize this for your architecture, try tuning SLAREF. 11.) This code has only been tested for WANTZ = .TRUE. and may behave unpredictably for WANTZ set to .FALSE. Implemented by: G. Henry, May 1, 1997 scalapack-doc-1.5/man/manl/pslahrd.l0100644000056400000620000001050006335610644017054 0ustar pfrauenfstaff.TH PSLAHRD l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLAHRD - reduce the first NB columns of a real general N-by-(N-K+1) distributed matrix A(IA:IA+N-1,JA:JA+N-K) so that elements below the k-th subdiagonal are zero .SH SYNOPSIS .TP 20 SUBROUTINE PSLAHRD( N, K, NB, A, IA, JA, DESCA, TAU, T, Y, IY, JY, DESCY, WORK ) .TP 20 .ti +4 INTEGER IA, IY, JA, JY, K, N, NB .TP 20 .ti +4 INTEGER DESCA( * ), DESCY( * ) .TP 20 .ti +4 REAL A( * ), T( * ), TAU( * ), WORK( * ), Y( * ) .SH PURPOSE PSLAHRD reduces the first NB columns of a real general N-by-(N-K+1) distributed matrix A(IA:IA+N-1,JA:JA+N-K) so that elements below the k-th subdiagonal are zero. The reduction is performed by an orthogo- nal similarity transformation Q' * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T. .br This is an auxiliary routine called by PSGEHRD. In the following comments sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1). .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 K (global input) INTEGER The offset for the reduction. Elements below the k-th subdiagonal in the first NB columns are reduced to zero. .TP 8 NB (global input) INTEGER The number of columns to be reduced. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-K)). On entry, this array contains the the local pieces of the N-by-(N-K+1) general distributed matrix A(IA:IA+N-1,JA:JA+N-K). On exit, the elements on and above the k-th subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced distributed matrix; the elements below the k-th subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors. The other columns of A(IA:IA+N-1,JA:JA+N-K) are unchanged. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) REAL array, dimension LOCc(JA+N-2) The scalar factors of the elementary reflectors (see Further Details). TAU is tied to the distributed matrix A. .TP 8 T (local output) REAL array, dimension (NB_A,NB_A) The upper triangular matrix T. .TP 8 Y (local output) REAL pointer into the local memory to an array of dimension (LLD_Y,NB_A). On exit, this array contains the local pieces of the N-by-NB distributed matrix Y. LLD_Y >= LOCr(IA+N-1). .TP 8 IY (global input) INTEGER The row index in the global array Y indicating the first row of sub( Y ). .TP 8 JY (global input) INTEGER The column index in the global array Y indicating the first column of sub( Y ). .TP 8 DESCY (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix Y. .TP 8 WORK (local workspace) REAL array, dimension (NB) .SH FURTHER DETAILS The matrix Q is represented as a product of nb elementary reflectors Q = H(1) H(2) . . . H(nb). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(ia+i+k:ia+n-1,ja+i-1), and tau in TAU(ja+i-1). .br The elements of the vectors v together form the (n-k+1)-by-nb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form: A(ia:ia+n-1,ja:ja+n-k) := (I-V*T*V')*(A(ia:ia+n-1,ja:ja+n-k)-Y*V'). The contents of A(ia:ia+n-1,ja:ja+n-k) on exit are illustrated by the following example with n = 7, k = 3 and nb = 2: .br ( a h a a a ) .br ( a h a a a ) .br ( a h a a a ) .br ( h h a a a ) .br ( v1 h a a a ) .br ( v1 v2 a a a ) .br ( v1 v2 a a a ) .br where a denotes an element of the original matrix .br A(ia:ia+n-1,ja:ja+n-k), h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i). .br scalapack-doc-1.5/man/manl/pslamch.l0100644000056400000620000000250606335610644017055 0ustar pfrauenfstaff.TH PSLAMCH l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLAMCH - determine single precision machine parameters .SH SYNOPSIS .TP 14 REAL FUNCTION PSLAMCH( ICTXT, CMACH ) .TP 14 .ti +4 CHARACTER CMACH .TP 14 .ti +4 INTEGER ICTXT .SH PURPOSE PSLAMCH determines single precision machine parameters. .SH ARGUMENTS .TP 8 ICTXT (global input) INTEGER The BLACS context handle in which the computation takes place. .TP 8 CMACH (global input) CHARACTER*1 Specifies the value to be returned by PSLAMCH: .br = 'E' or 'e', PSLAMCH := eps .br = 'S' or 's , PSLAMCH := sfmin .br = 'B' or 'b', PSLAMCH := base .br = 'P' or 'p', PSLAMCH := eps*base .br = 'N' or 'n', PSLAMCH := t .br = 'R' or 'r', PSLAMCH := rnd .br = 'M' or 'm', PSLAMCH := emin .br = 'U' or 'u', PSLAMCH := rmin .br = 'L' or 'l', PSLAMCH := emax .br = 'O' or 'o', PSLAMCH := rmax where .TP 6 eps = relative machine precision sfmin = safe minimum, such that 1/sfmin does not overflow base = base of the machine prec = eps*base t = number of (base) digits in the mantissa rnd = 1.0 when rounding occurs in addition, 0.0 otherwise emin = minimum exponent before (gradual) underflow rmin = underflow threshold - base**(emin-1) emax = largest exponent before overflow rmax = overflow threshold - (base**emax)*(1-eps) scalapack-doc-1.5/man/manl/pslange.l0100644000056400000620000001274706335610644017067 0ustar pfrauenfstaff.TH PSLANGE l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLANGE - return the value of the one norm, or the Frobenius norm, .SH SYNOPSIS .TP 14 REAL FUNCTION PSLANGE( NORM, M, N, A, IA, JA, DESCA, WORK ) .TP 14 .ti +4 CHARACTER NORM .TP 14 .ti +4 INTEGER IA, JA, M, N .TP 14 .ti +4 INTEGER DESCA( * ) .TP 14 .ti +4 REAL A( * ), WORK( * ) .SH PURPOSE PSLANGE returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a distributed matrix sub( A ) = A(IA:IA+M-1, JA:JA+N-1). .br PSLANGE returns the value .br ( max(abs(A(i,j))), NORM = 'M' or 'm' with IA <= i <= IA+M-1, ( and JA <= j <= JA+N-1, ( .br ( norm1( sub( A ) ), NORM = '1', 'O' or 'o' .br ( .br ( normI( sub( A ) ), NORM = 'I' or 'i' .br ( .br ( normF( sub( A ) ), NORM = 'F', 'f', 'E' or 'e' .br where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a matrix norm. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies the value to be returned in PSLANGE as described above. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). When M = 0, PSLANGE is set to zero. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). When N = 0, PSLANGE is set to zero. N >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)) containing the local pieces of the distributed matrix sub( A ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 WORK (local workspace) REAL array dimension (LWORK) LWORK >= 0 if NORM = 'M' or 'm' (not referenced), Nq0 if NORM = '1', 'O' or 'o', Mp0 if NORM = 'I' or 'i', 0 if NORM = 'F', 'f', 'E' or 'e' (not referenced), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. scalapack-doc-1.5/man/manl/pslanhs.l0100644000056400000620000001245706335610644017104 0ustar pfrauenfstaff.TH PSLANHS l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLANHS - return the value of the one norm, or the Frobenius norm, .SH SYNOPSIS .TP 14 REAL FUNCTION PSLANHS( NORM, N, A, IA, JA, DESCA, WORK ) .TP 14 .ti +4 CHARACTER NORM .TP 14 .ti +4 INTEGER IA, JA, N .TP 14 .ti +4 INTEGER DESCA( * ) .TP 14 .ti +4 REAL A( * ), WORK( * ) .SH PURPOSE PSLANHS returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hessenberg distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). PSLANHS returns the value .br ( max(abs(A(i,j))), NORM = 'M' or 'm' with IA <= i <= IA+N-1, ( and JA <= j <= JA+N-1, ( .br ( norm1( sub( A ) ), NORM = '1', 'O' or 'o' .br ( .br ( normI( sub( A ) ), NORM = 'I' or 'i' .br ( .br ( normF( sub( A ) ), NORM = 'F', 'f', 'E' or 'e' .br where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a matrix norm. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies the value to be returned in PSLANHS as described above. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on i.e the number of rows and columns of the distributed submatrix sub( A ). When N = 0, PSLANHS is set to zero. N >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1) ) containing the local pieces of sub( A ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 WORK (local workspace) REAL array dimension (LWORK) LWORK >= 0 if NORM = 'M' or 'm' (not referenced), Nq0 if NORM = '1', 'O' or 'o', Mp0 if NORM = 'I' or 'i', 0 if NORM = 'F', 'f', 'E' or 'e' (not referenced), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Np0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. scalapack-doc-1.5/man/manl/pslansy.l0100644000056400000620000001437506335610644017126 0ustar pfrauenfstaff.TH PSLANSY l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLANSY - return the value of the one norm, or the Frobenius norm, .SH SYNOPSIS .TP 14 REAL FUNCTION PSLANSY( NORM, UPLO, N, A, IA, JA, DESCA, WORK ) .TP 14 .ti +4 CHARACTER NORM, UPLO .TP 14 .ti +4 INTEGER IA, JA, N .TP 14 .ti +4 INTEGER DESCA( * ) .TP 14 .ti +4 REAL A( * ), WORK( * ) .SH PURPOSE PSLANSY returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). PSLANSY returns the value .br ( max(abs(A(i,j))), NORM = 'M' or 'm' with IA <= i <= IA+N-1, ( and JA <= j <= JA+N-1, ( .br ( norm1( sub( A ) ), NORM = '1', 'O' or 'o' .br ( .br ( normI( sub( A ) ), NORM = 'I' or 'i' .br ( .br ( normF( sub( A ) ), NORM = 'F', 'f', 'E' or 'e' .br where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a matrix norm. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies the value to be returned in PSLANSY as described above. .TP 8 UPLO (global input) CHARACTER Specifies whether the upper or lower triangular part of the symmetric matrix sub( A ) is to be referenced. = 'U': Upper triangular part of sub( A ) is referenced, .br = 'L': Lower triangular part of sub( A ) is referenced. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on i.e the number of rows and columns of the distributed submatrix sub( A ). When N = 0, PSLANSY is set to zero. N >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)) containing the local pieces of the symmetric distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular matrix which norm is to be computed, and the strictly lower triangular part of this matrix is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular matrix which norm is to be computed, and the strictly upper triangular part of sub( A ) is not referenced. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 WORK (local workspace) REAL array dimension (LWORK) LWORK >= 0 if NORM = 'M' or 'm' (not referenced), 2*Nq0+Np0+LDW if NORM = '1', 'O', 'o', 'I' or 'i', where LDW is given by: IF( NPROW.NE.NPCOL ) THEN LDW = MB_A*CEIL(CEIL(Np0/MB_A)/(LCM/NPROW)) ELSE LDW = 0 END IF 0 if NORM = 'F', 'f', 'E' or 'e' (not referenced), where LCM is the least common multiple of NPROW and NPCOL LCM = ILCM( NPROW, NPCOL ) and CEIL denotes the ceiling operation (ICEIL). IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Np0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), ICEIL, ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. scalapack-doc-1.5/man/manl/pslantr.l0100644000056400000620000001362706335610644017117 0ustar pfrauenfstaff.TH PSLANTR l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLANTR - return the value of the one norm, or the Frobenius norm, .SH SYNOPSIS .TP 14 REAL FUNCTION PSLANTR( NORM, UPLO, DIAG, M, N, A, IA, JA, DESCA, WORK ) .TP 14 .ti +4 CHARACTER DIAG, NORM, UPLO .TP 14 .ti +4 INTEGER IA, JA, M, N .TP 14 .ti +4 INTEGER DESCA( * ) .TP 14 .ti +4 REAL A( * ), WORK( * ) .SH PURPOSE PSLANTR returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular distributed matrix sub( A ) denoting A(IA:IA+M-1, JA:JA+N-1). .br PSLANTR returns the value .br ( max(abs(A(i,j))), NORM = 'M' or 'm' with ia <= i <= ia+m-1, ( and ja <= j <= ja+n-1, ( .br ( norm1( sub( A ) ), NORM = '1', 'O' or 'o' .br ( .br ( normI( sub( A ) ), NORM = 'I' or 'i' .br ( .br ( normF( sub( A ) ), NORM = 'F', 'f', 'E' or 'e' .br where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a matrix norm. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies the value to be returned in PSLANTR as described above. .TP 8 UPLO (global input) CHARACTER Specifies whether the matrix sub( A ) is upper or lower trapezoidal. = 'U': Upper trapezoidal .br = 'L': Lower trapezoidal Note that sub( A ) is triangular instead of trapezoidal if M = N. .TP 8 DIAG (global input) CHARACTER Specifies whether or not the distributed matrix sub( A ) has unit diagonal. = 'N': Non-unit diagonal .br = 'U': Unit diagonal .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). When M = 0, PSLANTR is set to zero. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). When N = 0, PSLANTR is set to zero. N >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1) ) containing the local pieces of sub( A ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 WORK (local workspace) REAL array dimension (LWORK) LWORK >= 0 if NORM = 'M' or 'm' (not referenced), Nq0 if NORM = '1', 'O' or 'o', Mp0 if NORM = 'I' or 'i', 0 if NORM = 'F', 'f', 'E' or 'e' (not referenced), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. scalapack-doc-1.5/man/manl/pslapiv.l0100644000056400000620000001547006335610644017110 0ustar pfrauenfstaff.TH PSLAPIV l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLAPIV - applie either P (permutation matrix indicated by IPIV) or inv( P ) to a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1), resulting in row or column pivoting .SH SYNOPSIS .TP 20 SUBROUTINE PSLAPIV( DIREC, ROWCOL, PIVROC, M, N, A, IA, JA, DESCA, IPIV, IP, JP, DESCIP, IWORK ) .TP 20 .ti +4 CHARACTER*1 DIREC, PIVROC, ROWCOL .TP 20 .ti +4 INTEGER IA, IP, JA, JP, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCIP( * ), IPIV( * ), IWORK( * ) .TP 20 .ti +4 REAL A( * ) .SH PURPOSE PSLAPIV applies either P (permutation matrix indicated by IPIV) or inv( P ) to a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1), resulting in row or column pivoting. The pivot vector may be distributed across a process row or a column. The pivot vector should be aligned with the distributed matrix A. This routine will transpose the pivot vector if necessary. For example if the row pivots should be applied to the columns of sub( A ), pass ROWCOL='C' and PIVROC='C'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Restrictions .br ============ .br IPIV must always be a distributed vector (not a matrix). Thus: IF( ROWPIV .EQ. 'C' ) THEN .br JP must be 1 .br ELSE .br IP must be 1 .br END IF .br The following restrictions apply when IPIV must be transposed: IF( ROWPIV.EQ.'C' .AND. PIVROC.EQ.'C') THEN .br DESCIP(MB_) must equal DESCA(NB_) .br ELSE IF( ROWPIV.EQ.'R" .AND. PIVROC.EQ.'R') THEN .br DESCIP(NB_) must equal DESCA(MB_) .br END IF .br .SH ARGUMENTS .TP 8 DIREC (global input) CHARACTER*1 Specifies in which order the permutation is applied: = 'F' (Forward) Applies pivots Forward from top of matrix. Computes P*sub( A ). = 'B' (Backward) Applies pivots Backward from bottom of matrix. Computes inv( P )*sub( A ). .TP 8 ROWCOL (global input) CHARACTER*1 Specifies if the rows or columns are to be permuted: = 'R' Rows will be permuted, = 'C' Columns will be permuted. .TP 8 PIVROC (global input) CHARACTER*1 Specifies whether IPIV is distributed over a process row or column: = 'R' IPIV distributed over a process row = 'C' IPIV distributed over a process column .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the distributed submatrix sub( A ) to which the row or column interchanges will be applied. On exit, the local pieces of the permuted distributed submatrix. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local input) INTEGER array, dimension >= LOCr(M_A)+MB_A if ROWCOL='R', otherwise LOCc(N_A)+NB_A. It contains the pivoting information. IPIV(i) is the global row (column), local row (column) i was swapped with. The last piece of the array of size MB_A (resp. NB_A) is used as workspace. This array is tied to the distributed matrix A. .TP 8 IWORK (local workspace) INTEGER array, dimension (LDW) where LDW is equal to the workspace necessary for transposition, and the storage of the tranposed IPIV: Let LCM be the least common multiple of NPROW and NPCOL. IF( ROWCOL.EQ.'R' .AND. PIVROC.EQ.'R' ) THEN IF( NPROW.EQ.NPCOL ) THEN LDW = LOCr( N_P + MOD(JP-1, NB_P) ) + NB_P ELSE LDW = LOCr( N_P + MOD(JP-1, NB_P) ) + NB_P * CEIL( CEIL(LOCc(N_P)/NB_P) / (LCM/NPCOL) ) END IF ELSE IF( ROWCOL.EQ.'C' .AND. PIVROC.EQ.'C' ) THEN IF( NPROW.EQ.NPCOL ) THEN LDW = LOCc( M_P + MOD(IP-1, MB_P) ) + MB_P ELSE LDW = LOCc( M_P + MOD(IP-1, MB_P) ) + MB_P * CEIL( CEIL(LOCr(M_P)/MB_P) / (LCM/NPROW) ) END IF ELSE IWORK is not referenced. END IF scalapack-doc-1.5/man/manl/pslapv2.l0100644000056400000620000001356506335610644017024 0ustar pfrauenfstaff.TH PSLAPV2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLAPV2 - applie either P (permutation matrix indicated by IPIV) or inv( P ) to a M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1), resulting in row or column pivoting .SH SYNOPSIS .TP 20 SUBROUTINE PSLAPV2( DIREC, ROWCOL, M, N, A, IA, JA, DESCA, IPIV, IP, JP, DESCIP ) .TP 20 .ti +4 CHARACTER DIREC, ROWCOL .TP 20 .ti +4 INTEGER IA, IP, JA, JP, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCIP( * ), IPIV( * ) .TP 20 .ti +4 REAL A( * ) .SH PURPOSE PSLAPV2 applies either P (permutation matrix indicated by IPIV) or inv( P ) to a M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1), resulting in row or column pivoting. The pivot vector should be aligned with the distributed matrix A. For pivoting the rows of sub( A ), IPIV should be distributed along a process column and replicated over all process rows. Similarly, IPIV should be distributed along a process row and replicated over all process columns for column pivoting. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 DIREC (global input) CHARACTER Specifies in which order the permutation is applied: = 'F' (Forward) Applies pivots Forward from top of matrix. Computes P * sub( A ); = 'B' (Backward) Applies pivots Backward from bottom of matrix. Computes inv( P ) * sub( A ). .TP 8 ROWCOL (global input) CHARACTER Specifies if the rows or columns are to be permuted: = 'R' Rows will be permuted, = 'C' Columns will be permuted. .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this local array contains the local pieces of the distributed matrix sub( A ) to which the row or columns interchanges will be applied. On exit, this array contains the local pieces of the permuted distributed matrix. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (input) INTEGER array, dimension >= LOCr(M_A)+MB_A if ROWCOL = 'R', LOCc(N_A)+NB_A otherwise. It contains the pivoting information. IPIV(i) is the global row (column), local row (column) i was swapped with. The last piece of the array of size MB_A (resp. NB_A) is used as workspace. IPIV is tied to the distributed matrix A. .TP 8 IP (global input) INTEGER IPIV's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JP (global input) INTEGER IPIV's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCIP (global and local input) INTEGER array of dimension 8 The array descriptor for the distributed matrix IPIV. scalapack-doc-1.5/man/manl/pslaqge.l0100644000056400000620000001363606335610644017070 0ustar pfrauenfstaff.TH PSLAQGE l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLAQGE - equilibrate a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) using the row and scaling factors in the vectors R and C .SH SYNOPSIS .TP 20 SUBROUTINE PSLAQGE( M, N, A, IA, JA, DESCA, R, C, ROWCND, COLCND, AMAX, EQUED ) .TP 20 .ti +4 CHARACTER EQUED .TP 20 .ti +4 INTEGER IA, JA, M, N .TP 20 .ti +4 REAL AMAX, COLCND, ROWCND .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), C( * ), R( * ) .SH PURPOSE PSLAQGE equilibrates a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) using the row and scaling factors in the vectors R and C. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)) containing on entry the M-by-N matrix sub( A ). On exit, the equilibrated distributed matrix. See EQUED for the form of the equilibrated distributed submatrix. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 R (local input) REAL array, dimension LOCr(M_A) The row scale factors for sub( A ). R is aligned with the distributed matrix A, and replicated across every process column. R is tied to the distributed matrix A. .TP 8 C (local input) REAL array, dimension LOCc(N_A) The column scale factors of sub( A ). C is aligned with the distributed matrix A, and replicated down every process row. C is tied to the distributed matrix A. .TP 8 ROWCND (global input) REAL The global ratio of the smallest R(i) to the largest R(i), IA <= i <= IA+M-1. .TP 8 COLCND (global input) REAL The global ratio of the smallest C(i) to the largest C(i), JA <= j <= JA+N-1. .TP 8 AMAX (global input) REAL Absolute value of largest distributed submatrix entry. .TP 8 EQUED (global output) CHARACTER Specifies the form of equilibration that was done. = 'N': No equilibration .br = 'R': Row equilibration, i.e., sub( A ) has been pre- .br multiplied by diag(R(IA:IA+M-1)), .br = 'C': Column equilibration, i.e., sub( A ) has been post- .br multiplied by diag(C(JA:JA+N-1)), .br = 'B': Both row and column equilibration, i.e., sub( A ) has been replaced by diag(R(IA:IA+M-1)) * sub( A ) * diag(C(JA:JA+N-1)). .SH PARAMETERS THRESH is a threshold value used to decide if row or column scaling should be done based on the ratio of the row or column scaling factors. If ROWCND < THRESH, row scaling is done, and if COLCND < THRESH, column scaling is done. LARGE and SMALL are threshold values used to decide if row scaling should be done based on the absolute size of the largest matrix element. If AMAX > LARGE or AMAX < SMALL, row scaling is done. scalapack-doc-1.5/man/manl/pslaqsy.l0100644000056400000620000001400506335610644017117 0ustar pfrauenfstaff.TH PSLAQSY l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLAQSY - equilibrate a symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) using the scaling factors in the vectors SR and SC .SH SYNOPSIS .TP 20 SUBROUTINE PSLAQSY( UPLO, N, A, IA, JA, DESCA, SR, SC, SCOND, AMAX, EQUED ) .TP 20 .ti +4 CHARACTER EQUED, UPLO .TP 20 .ti +4 INTEGER IA, JA, N .TP 20 .ti +4 REAL AMAX, SCOND .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), SC( * ), SR( * ) .SH PURPOSE PSLAQSY equilibrates a symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) using the scaling factors in the vectors SR and SC. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the upper or lower triangular part of the symmetric distributed matrix sub( A ) is to be referenced: .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (input/output) REAL pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1)). On entry, the local pieces of the distributed symmetric matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and the strictly lower triangular part of sub( A ) is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and the strictly upper trian- gular part of sub( A ) is not referenced. On exit, if EQUED = 'Y', the equilibrated matrix: .br diag(SR(IA:IA+N-1)) * sub( A ) * diag(SC(JA:JA+N-1)). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 SR (local input) REAL array, dimension LOCr(M_A) The scale factors for A(IA:IA+M-1,JA:JA+N-1). SR is aligned with the distributed matrix A, and replicated across every process column. SR is tied to the distributed matrix A. .TP 8 SC (local input) REAL array, dimension LOCc(N_A) The scale factors for sub( A ). SC is aligned with the dis- tributed matrix A, and replicated down every process row. SC is tied to the distributed matrix A. .TP 8 SCOND (global input) REAL Ratio of the smallest SR(i) (respectively SC(j)) to the largest SR(i) (respectively SC(j)), with IA <= i <= IA+N-1 and JA <= j <= JA+N-1. .TP 8 AMAX (global input) REAL Absolute value of the largest distributed submatrix entry. .TP 8 EQUED (output) CHARACTER*1 Specifies whether or not equilibration was done. = 'N': No equilibration. .br = 'Y': Equilibration was done, i.e., sub( A ) has been re- .br placed by: .br diag(SR(IA:IA+N-1)) * sub( A ) * diag(SC(JA:JA+N-1)). .SH PARAMETERS THRESH is a threshold value used to decide if scaling should be done based on the ratio of the scaling factors. If SCOND < THRESH, scaling is done. LARGE and SMALL are threshold values used to decide if scaling should be done based on the absolute size of the largest matrix element. If AMAX > LARGE or AMAX < SMALL, scaling is done. scalapack-doc-1.5/man/manl/pslared1d.l0100644000056400000620000001052506335610645017306 0ustar pfrauenfstaff.TH PSLARED1D l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSLARED1D - redistribute a 1D array It assumes that the input array, BYCOL, is distributed across rows and that all process column contain the same copy of BYCOL .SH SYNOPSIS .TP 22 SUBROUTINE PSLARED1D( N, IA, JA, DESC, BYCOL, BYALL, WORK, LWORK ) .TP 22 .ti +4 INTEGER IA, JA, LWORK, N .TP 22 .ti +4 INTEGER DESC( * ) .TP 22 .ti +4 REAL BYALL( * ), BYCOL( * ), WORK( LWORK ) .SH PURPOSE PSLARED1D redistributes a 1D array and will contain the entire array. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS NP = Number of local rows in BYCOL() .TP 8 N (global input) INTEGER The number of elements to be redistributed. N >= 0. .TP 8 IA (global input) INTEGER IA must be equal to 1 .TP 8 JA (global input) INTEGER JA must be equal to 1 .TP 8 DESC (global/local input) INTEGER Array of dimension 8 A 2D array descirptor, which describes BYCOL .TP 8 BYCOL (local input) distributed block cyclic REAL array global dimension (N), local dimension NP BYCOL is distributed across the process rows All process columns are assumed to contain the same value .TP 8 BYALL (global output) REAL global dimension( N ) local dimension (N) BYALL is exactly duplicated on all processes It contains the same values as BYCOL, but it is replicated across all processes rather than being distributed BYALL(i) = BYCOL( NUMROC(i,NB,MYROW,0,NPROW ) on the procs whose MYROW == mod((i-1)/NB,NPROW) .TP 8 WORK (local workspace) REAL dimension (LWORK) Used to hold the buffers sent from one process to another .TP 8 LWORK (local input) INTEGER size of WORK array LWORK >= NUMROC(N, DESC( NB_ ), 0, 0, NPCOL) scalapack-doc-1.5/man/manl/pslared2d.l0100644000056400000620000001060406335610645017305 0ustar pfrauenfstaff.TH PSLARED2D l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSLARED2D - redistribute a 1D array It assumes that the input array, BYROW, is distributed across columns and that all process rows contain the same copy of BYROW .SH SYNOPSIS .TP 22 SUBROUTINE PSLARED2D( N, IA, JA, DESC, BYROW, BYALL, WORK, LWORK ) .TP 22 .ti +4 INTEGER IA, JA, LWORK, N .TP 22 .ti +4 INTEGER DESC( * ) .TP 22 .ti +4 REAL BYALL( * ), BYROW( * ), WORK( LWORK ) .SH PURPOSE PSLARED2D redistributes a 1D array and will contain the entire array. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS NP = Number of local rows in BYROW() .TP 8 N (global input) INTEGER The number of elements to be redistributed. N >= 0. .TP 8 IA (global input) INTEGER IA must be equal to 1 .TP 8 JA (global input) INTEGER JA must be equal to 1 .TP 8 DESC (global/local input) INTEGER Array of dimension DLEN_ A 2D array descriptor, which describes BYROW .TP 8 BYROW (local input) distributed block cyclic REAL array global dimension (N), local dimension NP BYCOL is distributed across the process columns All process rows are assumed to contain the same value .TP 8 BYALL (global output) REAL global dimension( N ) local dimension (N) BYALL is exactly duplicated on all processes It contains the same values as BYCOL, but it is replicated across all processes rather than being distributed BYALL(i) = BYCOL( NUMROC(i,NB,MYROW,0,NPROW ) on the procs whose MYROW == mod((i-1)/NB,NPROW) .TP 8 WORK (local workspace) REAL dimension (LWORK) Used to hold the buffers sent from one process to another .TP 8 LWORK (local input) INTEGER size of WORK array LWORK >= LWORK >= NUMROC(N, DESC( NB_ ), 0, 0, NPCOL) scalapack-doc-1.5/man/manl/pslarf.l0100644000056400000620000001777306335610645016732 0ustar pfrauenfstaff.TH PSLARF l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLARF - applie a real elementary reflector Q (or Q**T) to a real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the left or the right .SH SYNOPSIS .TP 19 SUBROUTINE PSLARF( SIDE, M, N, V, IV, JV, DESCV, INCV, TAU, C, IC, JC, DESCC, WORK ) .TP 19 .ti +4 CHARACTER SIDE .TP 19 .ti +4 INTEGER IC, INCV, IV, JC, JV, M, N .TP 19 .ti +4 INTEGER DESCC( * ), DESCV( * ) .TP 19 .ti +4 REAL C( * ), TAU( * ), V( * ), WORK( * ) .SH PURPOSE PSLARF applies a real elementary reflector Q (or Q**T) to a real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the left or the right. Q is represented in the form Q = I - tau * v * v' .br where tau is a real scalar and v is a real vector. .br If tau = 0, then Q is taken to be the unit matrix. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. Restrictions .br ============ .br If SIDE = 'Left' and INCV = 1, then the row process having the first entry V(IV,JV) must also have the first row of sub( C ). Moreover, MOD(IV-1,MB_V) must be equal to MOD(IC-1,MB_C), if INCV=M_V, only the last equality must be satisfied. .br If SIDE = 'Right' and INCV = M_V then the column process having the first entry V(IV,JV) must also have the first column of sub( C ) and MOD(JV-1,NB_V) must be equal to MOD(JC-1,NB_C), if INCV = 1 only the last equality must be satisfied. .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': form Q * sub( C ), .br = 'R': form sub( C ) * Q, Q = Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 V (local input) REAL pointer into the local memory to an array of dimension (LLD_V,*) containing the local pieces of the distributed vectors V representing the Householder transformation Q, V(IV:IV+M-1,JV) if SIDE = 'L' and INCV = 1, .br V(IV,JV:JV+M-1) if SIDE = 'L' and INCV = M_V, .br V(IV:IV+N-1,JV) if SIDE = 'R' and INCV = 1, .br V(IV,JV:JV+N-1) if SIDE = 'R' and INCV = M_V, The vector v in the representation of Q. V is not used if TAU = 0. .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 INCV (global input) INTEGER The global increment for the elements of V. Only two values of INCV are supported in this version, namely 1 and M_V. INCV must not be zero. .TP 8 TAU (local input) REAL, array, dimension LOCc(JV) if INCV = 1, and LOCr(IV) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix V. .TP 8 C (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_C, LOCc(JC+N-1) ), containing the local pieces of sub( C ). On exit, sub( C ) is overwritten by the Q * sub( C ) if SIDE = 'L', or sub( C ) * Q if SIDE = 'R'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace) REAL array, dimension (LWORK) If INCV = 1, if SIDE = 'L', if IVCOL = ICCOL, LWORK >= NqC0 else LWORK >= MpC0 + MAX( 1, NqC0 ) end if else if SIDE = 'R', LWORK >= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC( N+ICOFFC,NB_V,0,0,NPCOL ),NB_V,0,0,LCMQ ) ) end if else if INCV = M_V, if SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_V,0,0,NPROW ),MB_V,0,0,LCMP ) ) else if SIDE = 'R', if IVROW = ICROW, LWORK >= MpC0 else LWORK >= NqC0 + MAX( 1, MpC0 ) end if end if end if where LCM is the least common multiple of NPROW and NPCOL and LCM = ILCM( NPROW, NPCOL ), LCMP = LCM / NPROW, LCMQ = LCM / NPCOL, IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. Alignment requirements ====================== The distributed submatrices V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: MB_V = NB_V, If INCV = 1, If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', ( MB_V.EQ.NB_A .AND. MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if INCV = M_V, If SIDE = 'Left', ( MB_V.EQ.NB_V .AND. MB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND. IVCOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/pslarfb.l0100644000056400000620000001751706335610645017070 0ustar pfrauenfstaff.TH PSLARFB l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLARFB - applie a real block reflector Q or its transpose Q**T to a real distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PSLARFB( SIDE, TRANS, DIRECT, STOREV, M, N, K, V, IV, JV, DESCV, T, C, IC, JC, DESCC, WORK ) .TP 20 .ti +4 CHARACTER SIDE, TRANS, DIRECT, STOREV .TP 20 .ti +4 INTEGER IC, IV, JC, JV, K, M, N .TP 20 .ti +4 INTEGER DESCC( * ), DESCV( * ) .TP 20 .ti +4 REAL C( * ), T( * ), V( * ), WORK( * ) .SH PURPOSE PSLARFB applies a real block reflector Q or its transpose Q**T to a real distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) from the left or the right. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 DIRECT (global input) CHARACTER Indicates how Q is formed from a product of elementary reflectors = 'F': Q = H(1) H(2) . . . H(k) (Forward) .br = 'B': Q = H(k) . . . H(2) H(1) (Backward) .TP 8 STOREV (global input) CHARACTER Indicates how the vectors which define the elementary reflectors are stored: .br = 'C': Columnwise .br = 'R': Rowwise .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The order of the matrix T (= the number of elementary reflectors whose product defines the block reflector). .TP 8 V (local input) REAL pointer into the local memory to an array of dimension ( LLD_V, LOCc(JV+K-1) ) if STOREV = 'C', ( LLD_V, LOCc(JV+M-1)) if STOREV = 'R' and SIDE = 'L', ( LLD_V, LOCc(JV+N-1) ) if STOREV = 'R' and SIDE = 'R'. It contains the local pieces of the distributed vectors V representing the Householder transformation. See further details. If STOREV = 'C' and SIDE = 'L', LLD_V >= MAX(1,LOCr(IV+M-1)); if STOREV = 'C' and SIDE = 'R', LLD_V >= MAX(1,LOCr(IV+N-1)); if STOREV = 'R', LLD_V >= LOCr(IV+K-1). .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 T (local input) REAL array, dimension MB_V by MB_V if STOREV = 'R' and NB_V by NB_V if STOREV = 'C'. The trian- gular matrix T in the representation of the block reflector. .TP 8 C (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the M-by-N distributed matrix sub( C ). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q or sub( C )*Q'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace) REAL array, dimension (LWORK) If STOREV = 'C', if SIDE = 'L', LWORK >= ( NqC0 + MpC0 ) * K else if SIDE = 'R', LWORK >= ( NqC0 + MAX( NpV0 + NUMROC( NUMROC( N+ICOFFC, NB_V, 0, 0, NPCOL ), NB_V, 0, 0, LCMQ ), MpC0 ) ) * K end if else if STOREV = 'R', if SIDE = 'L', LWORK >= ( MpC0 + MAX( MqV0 + NUMROC( NUMROC( M+IROFFC, MB_V, 0, 0, NPROW ), MB_V, 0, 0, LCMP ), NqC0 ) ) * K else if SIDE = 'R', LWORK >= ( MpC0 + NqC0 ) * K end if end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFV = MOD( IV-1, MB_V ), ICOFFV = MOD( JV-1, NB_V ), IVROW = INDXG2P( IV, MB_V, MYROW, RSRC_V, NPROW ), IVCOL = INDXG2P( JV, NB_V, MYCOL, CSRC_V, NPCOL ), MqV0 = NUMROC( M+ICOFFV, NB_V, MYCOL, IVCOL, NPCOL ), NpV0 = NUMROC( N+IROFFV, MB_V, MYROW, IVROW, NPROW ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NpC0 = NUMROC( N+ICOFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. Alignment requirements ====================== The distributed submatrices V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If STOREV = 'Columnwise' If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', ( MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if STOREV = 'Rowwise' If SIDE = 'Left', ( NB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND. IVCOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/pslarfg.l0100644000056400000620000001237506335610645017072 0ustar pfrauenfstaff.TH PSLARFG l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLARFG - generate a real elementary reflector H of order n, such that H * sub( X ) = H * ( x(iax,jax) ) = ( alpha ), H' * H = I .SH SYNOPSIS .TP 20 SUBROUTINE PSLARFG( N, ALPHA, IAX, JAX, X, IX, JX, DESCX, INCX, TAU ) .TP 20 .ti +4 INTEGER IAX, INCX, IX, JAX, JX, N .TP 20 .ti +4 REAL ALPHA .TP 20 .ti +4 INTEGER DESCX( * ) .TP 20 .ti +4 REAL TAU( * ), X( * ) .SH PURPOSE PSLARFG generates a real elementary reflector H of order n, such that ( x ) ( 0 ) .br where alpha is a scalar, and sub( X ) is an (N-1)-element real distributed vector X(IX:IX+N-2,JX) if INCX = 1 and X(IX,JX:JX+N-2) if INCX = DESCX(M_). H is represented in the form .br H = I - tau * ( 1 ) * ( 1 v' ) , .br ( v ) .br where tau is a real scalar and v is a real (N-1)-element .br vector. .br If the elements of sub( X ) are all zero, then tau = 0 and H is taken to be the unit matrix. .br Otherwise 1 <= tau <= 2. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. .SH ARGUMENTS .TP 8 N (global input) INTEGER The global order of the elementary reflector. N >= 0. .TP 8 ALPHA (local output) REAL On exit, alpha is computed in the process scope having the vector sub( X ). .TP 8 IAX (global input) INTEGER The global row index in X of X(IAX,JAX). .TP 8 JAX (global input) INTEGER The global column index in X of X(IAX,JAX). .TP 8 X (local input/local output) REAL, pointer into the local memory to an array of dimension (LLD_X,*). This array contains the local pieces of the distributed vector sub( X ). Before entry, the incremented array sub( X ) must contain the vector x. On exit, it is overwritten with the vector v. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 INCX (global input) INTEGER The global increment for the elements of X. Only two values of INCX are supported in this version, namely 1 and M_X. INCX must not be zero. .TP 8 TAU (local output) REAL, array, dimension LOCc(JX) if INCX = 1, and LOCr(IX) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix X. scalapack-doc-1.5/man/manl/pslarft.l0100644000056400000620000001504006335610645017077 0ustar pfrauenfstaff.TH PSLARFT l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLARFT - form the triangular factor T of a real block reflector H of order n, which is defined as a product of k elementary reflectors .SH SYNOPSIS .TP 20 SUBROUTINE PSLARFT( DIRECT, STOREV, N, K, V, IV, JV, DESCV, TAU, T, WORK ) .TP 20 .ti +4 CHARACTER DIRECT, STOREV .TP 20 .ti +4 INTEGER IV, JV, K, N .TP 20 .ti +4 INTEGER DESCV( * ) .TP 20 .ti +4 REAL TAU( * ), T( * ), V( * ), WORK( * ) .SH PURPOSE PSLARFT forms the triangular factor T of a real block reflector H of order n, which is defined as a product of k elementary reflectors. If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular; If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular. If STOREV = 'C', the vector which defines the elementary reflector H(i) is stored in the i-th column of the distributed matrix V, and H = I - V * T * V' .br If STOREV = 'R', the vector which defines the elementary reflector H(i) is stored in the i-th row of the distributed matrix V, and H = I - V' * T * V .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 DIRECT (global input) CHARACTER*1 Specifies the order in which the elementary reflectors are multiplied to form the block reflector: .br = 'F': H = H(1) H(2) . . . H(k) (Forward) .br = 'B': H = H(k) . . . H(2) H(1) (Backward) .TP 8 STOREV (global input) CHARACTER*1 Specifies how the vectors which define the elementary reflectors are stored (see also Further Details): .br = 'R': rowwise .TP 8 N (global input) INTEGER The order of the block reflector H. N >= 0. .TP 8 K (global input) INTEGER The order of the triangular factor T (= the number of elementary reflectors). 1 <= K <= MB_V (= NB_V). .TP 8 V (input/output) REAL pointer into the local memory to an array of local dimension (LOCr(IV+N-1),LOCc(JV+K-1)) if STOREV = 'C', and (LOCr(IV+K-1),LOCc(JV+N-1)) if STOREV = 'R'. The distributed matrix V contains the Householder vectors. See further details. .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 TAU (local input) REAL, array, dimension LOCr(IV+K-1) if INCV = M_V, and LOCc(JV+K-1) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix V. .TP 8 T (local output) REAL array, dimension (NB_V,NB_V) if STOREV = 'Col', and (MB_V,MB_V) otherwise. It contains the k-by-k triangular factor of the block reflector asso- ciated with V. If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is lower triangular. .TP 8 WORK (local workspace) REAL array, dimension (K*(K-1)/2) .SH FURTHER DETAILS The shape of the matrix V and the storage of the vectors which define the H(i) is best illustrated by the following example with n = 5 and k = 3. The elements equal to 1 are not stored; the corresponding array elements are modified but restored on exit. The rest of the array is not used. .br DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R': V( IV:IV+N-1, ( 1 ) V( IV:IV+K-1, ( 1 v1 v1 v1 v1 ) JV:JV+K-1 ) = ( v1 1 ) JV:JV+N-1 ) = ( 1 v2 v2 v2 ) ( v1 v2 1 ) ( 1 v3 v3 ) ( v1 v2 v3 ) .br ( v1 v2 v3 ) .br DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R': V( IV:IV+N-1, ( v1 v2 v3 ) V( IV:IV+K-1, ( v1 v1 1 ) JV:JV+K-1 ) = ( v1 v2 v3 ) JV:JV+N-1 ) = ( v2 v2 v2 1 ) ( 1 v2 v3 ) ( v3 v3 v3 v3 1 ) ( 1 v3 ) .br ( 1 ) .br scalapack-doc-1.5/man/manl/pslarz.l0100644000056400000620000002041006335610645016734 0ustar pfrauenfstaff.TH PSLARZ l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLARZ - applie a real elementary reflector Q (or Q**T) to a real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the left or the right .SH SYNOPSIS .TP 19 SUBROUTINE PSLARZ( SIDE, M, N, L, V, IV, JV, DESCV, INCV, TAU, C, IC, JC, DESCC, WORK ) .TP 19 .ti +4 CHARACTER SIDE .TP 19 .ti +4 INTEGER IC, INCV, IV, JC, JV, L, M, N .TP 19 .ti +4 INTEGER DESCC( * ), DESCV( * ) .TP 19 .ti +4 REAL C( * ), TAU( * ), V( * ), WORK( * ) .SH PURPOSE PSLARZ applies a real elementary reflector Q (or Q**T) to a real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the left or the right. Q is represented in the form Q = I - tau * v * v' .br where tau is a real scalar and v is a real vector. .br If tau = 0, then Q is taken to be the unit matrix. .br Q is a product of k elementary reflectors as returned by PSTZRZF. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. Restrictions .br ============ .br If SIDE = 'Left' and INCV = 1, then the row process having the first entry V(IV,JV) must also own C(IC+M-L,JC:JC+N-1). Moreover, MOD(IV-1,MB_V) must be equal to MOD(IC+N-L-1,MB_C), if INCV=M_V, only the last equality must be satisfied. .br If SIDE = 'Right' and INCV = M_V then the column process having the first entry V(IV,JV) must also own C(IC:IC+M-1,JC+N-L) and MOD(JV-1,NB_V) must be equal to MOD(JC+N-L-1,NB_C), if INCV = 1 only the last equality must be satisfied. .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': form Q * sub( C ), .br = 'R': form sub( C ) * Q, Q = Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 L (global input) INTEGER The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. .TP 8 V (local input) REAL pointer into the local memory to an array of dimension (LLD_V,*) containing the local pieces of the distributed vectors V representing the Householder transformation Q, V(IV:IV+L-1,JV) if SIDE = 'L' and INCV = 1, .br V(IV,JV:JV+L-1) if SIDE = 'L' and INCV = M_V, .br V(IV:IV+L-1,JV) if SIDE = 'R' and INCV = 1, .br V(IV,JV:JV+L-1) if SIDE = 'R' and INCV = M_V, The vector v in the representation of Q. V is not used if TAU = 0. .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 INCV (global input) INTEGER The global increment for the elements of V. Only two values of INCV are supported in this version, namely 1 and M_V. INCV must not be zero. .TP 8 TAU (local input) REAL, array, dimension LOCc(JV) if INCV = 1, and LOCr(IV) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix V. .TP 8 C (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_C, LOCc(JC+N-1) ), containing the local pieces of sub( C ). On exit, sub( C ) is overwritten by the Q * sub( C ) if SIDE = 'L', or sub( C ) * Q if SIDE = 'R'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace) REAL array, dimension (LWORK) If INCV = 1, if SIDE = 'L', if IVCOL = ICCOL, LWORK >= NqC0 else LWORK >= MpC0 + MAX( 1, NqC0 ) end if else if SIDE = 'R', LWORK >= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC( N+ICOFFC,NB_V,0,0,NPCOL ),NB_V,0,0,LCMQ ) ) end if else if INCV = M_V, if SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_V,0,0,NPROW ),MB_V,0,0,LCMP ) ) else if SIDE = 'R', if IVROW = ICROW, LWORK >= MpC0 else LWORK >= NqC0 + MAX( 1, MpC0 ) end if end if end if where LCM is the least common multiple of NPROW and NPCOL and LCM = ILCM( NPROW, NPCOL ), LCMP = LCM / NPROW, LCMQ = LCM / NPCOL, IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. Alignment requirements ====================== The distributed submatrices V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: MB_V = NB_V, If INCV = 1, If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', ( MB_V.EQ.NB_A .AND. MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if INCV = M_V, If SIDE = 'Left', ( MB_V.EQ.NB_V .AND. MB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND. IVCOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/pslarzb.l0100644000056400000620000001773206335610645017113 0ustar pfrauenfstaff.TH PSLARZB l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLARZB - applie a real block reflector Q or its transpose Q**T to a real distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PSLARZB( SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V, IV, JV, DESCV, T, C, IC, JC, DESCC, WORK ) .TP 20 .ti +4 CHARACTER DIRECT, SIDE, STOREV, TRANS .TP 20 .ti +4 INTEGER IC, IV, JC, JV, K, L, M, N .TP 20 .ti +4 INTEGER DESCC( * ), DESCV( * ) .TP 20 .ti +4 REAL C( * ), T( * ), V( * ), WORK( * ) .SH PURPOSE PSLARZB applies a real block reflector Q or its transpose Q**T to a real distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) from the left or the right. .br Q is a product of k elementary reflectors as returned by PSTZRZF. Currently, only STOREV = 'R' and DIRECT = 'B' are supported. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 DIRECT (global input) CHARACTER Indicates how H is formed from a product of elementary reflectors = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet) .br = 'B': H = H(k) . . . H(2) H(1) (Backward) .TP 8 STOREV (global input) CHARACTER Indicates how the vectors which define the elementary reflectors are stored: .br = 'C': Columnwise (not supported yet) .br = 'R': Rowwise .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The order of the matrix T (= the number of elementary reflectors whose product defines the block reflector). .TP 8 L (global input) INTEGER The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. .TP 8 V (local input) REAL pointer into the local memory to an array of dimension (LLD_V, LOCc(JV+M-1)) if SIDE = 'L', (LLD_V, LOCc(JV+N-1)) if SIDE = 'R'. It contains the local pieces of the distributed vectors V representing the Householder transformation as returned by PSTZRZF. LLD_V >= LOCr(IV+K-1). .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 T (local input) REAL array, dimension MB_V by MB_V The lower triangular matrix T in the representation of the block reflector. .TP 8 C (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the M-by-N distributed matrix sub( C ). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q or sub( C )*Q'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace) REAL array, dimension (LWORK) If STOREV = 'C', if SIDE = 'L', LWORK >= ( NqC0 + MpC0 ) * K else if SIDE = 'R', LWORK >= ( NqC0 + MAX( NpV0 + NUMROC( NUMROC( N+ICOFFC, NB_V, 0, 0, NPCOL ), NB_V, 0, 0, LCMQ ), MpC0 ) ) * K end if else if STOREV = 'R', if SIDE = 'L', LWORK >= ( MpC0 + MAX( MqV0 + NUMROC( NUMROC( M+IROFFC, MB_V, 0, 0, NPROW ), MB_V, 0, 0, LCMP ), NqC0 ) ) * K else if SIDE = 'R', LWORK >= ( MpC0 + NqC0 ) * K end if end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFV = MOD( IV-1, MB_V ), ICOFFV = MOD( JV-1, NB_V ), IVROW = INDXG2P( IV, MB_V, MYROW, RSRC_V, NPROW ), IVCOL = INDXG2P( JV, NB_V, MYCOL, CSRC_V, NPCOL ), MqV0 = NUMROC( M+ICOFFV, NB_V, MYCOL, IVCOL, NPCOL ), NpV0 = NUMROC( N+IROFFV, MB_V, MYROW, IVROW, NPROW ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NpC0 = NUMROC( N+ICOFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. Alignment requirements ====================== The distributed submatrices V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If STOREV = 'Columnwise' If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', ( MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if STOREV = 'Rowwise' If SIDE = 'Left', ( NB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND. IVCOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/pslarzt.l0100644000056400000620000001562606335610645017135 0ustar pfrauenfstaff.TH PSLARZT l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLARZT - form the triangular factor T of a real block reflector H of order > n, which is defined as a product of k elementary reflectors as returned by PSTZRZF .SH SYNOPSIS .TP 20 SUBROUTINE PSLARZT( DIRECT, STOREV, N, K, V, IV, JV, DESCV, TAU, T, WORK ) .TP 20 .ti +4 CHARACTER DIRECT, STOREV .TP 20 .ti +4 INTEGER IV, JV, K, N .TP 20 .ti +4 INTEGER DESCV( * ) .TP 20 .ti +4 REAL TAU( * ), T( * ), V( * ), WORK( * ) .SH PURPOSE PSLARZT forms the triangular factor T of a real block reflector H of order > n, which is defined as a product of k elementary reflectors as returned by PSTZRZF. If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular; If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular. If STOREV = 'C', the vector which defines the elementary reflector H(i) is stored in the i-th column of the array V, and .br H = I - V * T * V' .br If STOREV = 'R', the vector which defines the elementary reflector H(i) is stored in the i-th row of the array V, and .br H = I - V' * T * V .br Currently, only STOREV = 'R' and DIRECT = 'B' are supported. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 DIRECT (global input) CHARACTER Specifies the order in which the elementary reflectors are multiplied to form the block reflector: .br = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet) .br = 'B': H = H(k) . . . H(2) H(1) (Backward) .TP 8 STOREV (global input) CHARACTER Specifies how the vectors which define the elementary reflectors are stored (see also Further Details): .br = 'R': rowwise .TP 8 N (global input) INTEGER The number of meaningful entries of the block reflector H. N >= 0. .TP 8 K (global input) INTEGER The order of the triangular factor T (= the number of elementary reflectors). 1 <= K <= MB_V (= NB_V). .TP 8 V (input/output) REAL pointer into the local memory to an array of local dimension (LOCr(IV+K-1),LOCc(JV+N-1)). The distributed matrix V contains the Householder vectors. See further details. .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 TAU (local input) REAL, array, dimension LOCr(IV+K-1) if INCV = M_V, and LOCc(JV+K-1) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix V. .TP 8 T (local output) REAL array, dimension (MB_V,MB_V) It contains the k-by-k triangular factor of the block reflector associated with V. T is lower triangular. .TP 8 WORK (local workspace) REAL array, dimension (K*(K-1)/2) .SH FURTHER DETAILS The shape of the matrix V and the storage of the vectors which define the H(i) is best illustrated by the following example with n = 5 and k = 3. The elements equal to 1 are not stored; the corresponding array elements are modified but restored on exit. The rest of the array is not used. .br DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R': ______V_____ .br ( v1 v2 v3 ) / \ ( v1 v2 v3 ) ( v1 v1 v1 v1 v1 . . . . 1 ) V = ( v1 v2 v3 ) ( v2 v2 v2 v2 v2 . . . 1 ) ( v1 v2 v3 ) ( v3 v3 v3 v3 v3 . . 1 ) ( v1 v2 v3 ) .br . . . .br . . . .br 1 . . .br 1 . .br 1 .br DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R': ______V_____ 1 / \ . 1 ( 1 . . . . v1 v1 v1 v1 v1 ) . . 1 ( . 1 . . . v2 v2 v2 v2 v2 ) . . . ( . . 1 . . v3 v3 v3 v3 v3 ) . . . .br ( v1 v2 v3 ) .br ( v1 v2 v3 ) .br V = ( v1 v2 v3 ) .br ( v1 v2 v3 ) .br ( v1 v2 v3 ) .br scalapack-doc-1.5/man/manl/pslascl.l0100644000056400000620000001233706335610645017073 0ustar pfrauenfstaff.TH PSLASCL l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLASCL - multiplie the M-by-N real distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) by the real scalar CTO/CFROM .SH SYNOPSIS .TP 20 SUBROUTINE PSLASCL( TYPE, CFROM, CTO, M, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER TYPE .TP 20 .ti +4 INTEGER IA, INFO, JA, M, N .TP 20 .ti +4 REAL CFROM, CTO .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ) .SH PURPOSE PSLASCL multiplies the M-by-N real distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) 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 sub( A ) may be full, upper triangular, lower triangular or upper Hessenberg. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 TYPE (global input) CHARACTER TYPE indices the storage type of the input distributed matrix. = 'G': sub( A ) is a full matrix, .br = 'L': sub( A ) is a lower triangular matrix, .br = 'U': sub( A ) is an upper triangular matrix, .br = 'H': sub( A ) is an upper Hessenberg matrix. .TP 8 CFROM (global input) REAL CTO (global input) REAL The distributed matrix sub( 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. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). This array contains the local pieces of the distributed matrix sub( A ). On exit, this array contains the local pieces of the distributed matrix multiplied by CTO/CFROM. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pslase2.l0100644000056400000620000001214006335610645016773 0ustar pfrauenfstaff.TH PSLASE2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLASE2 - initialize an M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) to BETA on the diagonal and ALPHA on the offdiagonals .SH SYNOPSIS .TP 20 SUBROUTINE PSLASE2( UPLO, M, N, ALPHA, BETA, A, IA, JA, DESCA ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, JA, M, N .TP 20 .ti +4 REAL ALPHA, BETA .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ) .SH PURPOSE PSLASE2 initializes an M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) to BETA on the diagonal and ALPHA on the offdiagonals. PSLASE2 requires that only dimension of the matrix operand is distributed. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies the part of the distributed matrix sub( A ) to be set: .br = 'U': Upper triangular part is set; the strictly lower triangular part of sub( A ) is not changed; = 'L': Lower triangular part is set; the strictly upper triangular part of sub( A ) is not changed; Otherwise: All of the matrix sub( A ) is set. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 ALPHA (global input) REAL The constant to which the offdiagonal elements are to be set. .TP 8 BETA (global input) REAL The constant to which the diagonal elements are to be set. .TP 8 A (local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). This array contains the local pieces of the distributed matrix sub( A ) to be set. On exit, the leading M-by-N submatrix sub( A ) is set as follows: if UPLO = 'U', A(IA+i-1,JA+j-1) = ALPHA, 1<=i<=j-1, 1<=j<=N, if UPLO = 'L', A(IA+i-1,JA+j-1) = ALPHA, j+1<=i<=M, 1<=j<=N, otherwise, A(IA+i-1,JA+j-1) = ALPHA, 1<=i<=M, 1<=j<=N, IA+i.NE.JA+j, and, for all UPLO, A(IA+i-1,JA+i-1) = BETA, 1<=i<=min(M,N). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. scalapack-doc-1.5/man/manl/pslaset.l0100644000056400000620000001202006335610645017072 0ustar pfrauenfstaff.TH PSLASET l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLASET - initialize an M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) to BETA on the diagonal and ALPHA on the offdiagonals .SH SYNOPSIS .TP 20 SUBROUTINE PSLASET( UPLO, M, N, ALPHA, BETA, A, IA, JA, DESCA ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, JA, M, N .TP 20 .ti +4 REAL ALPHA, BETA .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ) .SH PURPOSE PSLASET initializes an M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) to BETA on the diagonal and ALPHA on the offdiagonals. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies the part of the distributed matrix sub( A ) to be set: .br = 'U': Upper triangular part is set; the strictly lower triangular part of sub( A ) is not changed; = 'L': Lower triangular part is set; the strictly upper triangular part of sub( A ) is not changed; Otherwise: All of the matrix sub( A ) is set. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 ALPHA (global input) REAL The constant to which the offdiagonal elements are to be set. .TP 8 BETA (global input) REAL The constant to which the diagonal elements are to be set. .TP 8 A (local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). This array contains the local pieces of the distributed matrix sub( A ) to be set. On exit, the leading M-by-N submatrix sub( A ) is set as follows: if UPLO = 'U', A(IA+i-1,JA+j-1) = ALPHA, 1<=i<=j-1, 1<=j<=N, if UPLO = 'L', A(IA+i-1,JA+j-1) = ALPHA, j+1<=i<=M, 1<=j<=N, otherwise, A(IA+i-1,JA+j-1) = ALPHA, 1<=i<=M, 1<=j<=N, IA+i.NE.JA+j, and, for all UPLO, A(IA+i-1,JA+i-1) = BETA, 1<=i<=min(M,N). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. scalapack-doc-1.5/man/manl/pslasmsub.l0100644000056400000620000001143506335610646017442 0ustar pfrauenfstaff.TH PSLASMSUB l "12 May 1997" "LAPACK version 1.5 " "LAPACK routine (version 1.5 )" .SH NAME PSLASMSUB - look for a small subdiagonal element from the bottom of the matrix that it can safely set to zero .SH SYNOPSIS .TP 22 SUBROUTINE PSLASMSUB( A, DESCA, I, L, K, SMLNUM, BUF, LWORK ) .TP 22 .ti +4 INTEGER I, K, L, LWORK .TP 22 .ti +4 REAL SMLNUM .TP 22 .ti +4 INTEGER DESCA( * ) .TP 22 .ti +4 REAL A( * ), BUF( * ) .SH PURPOSE PSLASMSUB looks for a small subdiagonal element from the bottom of the matrix that it can safely set to zero. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 A (global input) REAL array, dimension (DESCA(LLD_),*) On entry, the Hessenberg matrix whose tridiagonal part is being scanned. Unchanged on exit. .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 I (global input) INTEGER The global location of the bottom of the unreduced submatrix of A. Unchanged on exit. .TP 8 L (global input) INTEGER The global location of the top of the unreduced submatrix of A. Unchanged on exit. .TP 8 K (global output) INTEGER On exit, this yields the bottom portion of the unreduced submatrix. This will satisfy: L <= M <= I-1. .TP 8 SMLNUM (global input) REAL On entry, a "small number" for the given matrix. Unchanged on exit. .TP 8 BUF (local output) REAL array of size LWORK. .TP 8 LWORK (global input) INTEGER On exit, LWORK is the size of the work buffer. This must be at least 2*Ceil( Ceil( (I-L)/HBL ) / LCM(NPROW,NPCOL) ) Here LCM is least common multiple, and NPROWxNPCOL is the logical grid size. Notes: This routine does a global maximum and must be called by all processes. This code is basically a parallelization of the following snip of LAPACK code from SLAHQR: Look for a single small subdiagonal element. DO 20 K = I, L + 1, -1 TST1 = ABS( H( K-1, K-1 ) ) + ABS( H( K, K ) ) IF( TST1.EQ.ZERO ) $ TST1 = SLANHS( '1', I-L+1, H( L, L ), LDH, WORK ) IF( ABS( H( K, K-1 ) ).LE.MAX( ULP*TST1, SMLNUM ) ) $ GO TO 30 20 CONTINUE 30 CONTINUE Implemented by: G. Henry, November 17, 1996 scalapack-doc-1.5/man/manl/pslassq.l0100644000056400000620000001216106335610646017114 0ustar pfrauenfstaff.TH PSLASSQ l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLASSQ - return the values scl and smsq such that ( scl**2 )*smsq = x( 1 )**2 +...+ x( n )**2 + ( scale**2 )*sumsq, .SH SYNOPSIS .TP 20 SUBROUTINE PSLASSQ( N, X, IX, JX, DESCX, INCX, SCALE, SUMSQ ) .TP 20 .ti +4 INTEGER IX, INCX, JX, N .TP 20 .ti +4 REAL SCALE, SUMSQ .TP 20 .ti +4 INTEGER DESCX( * ) .TP 20 .ti +4 REAL X( * ) .SH PURPOSE PSLASSQ returns the values scl and smsq such that where x( i ) = sub( X ) = X( IX+(JX-1)*DESCX(M_)+(i-1)*INCX ). The value of sumsq is assumed to be non-negative and scl returns the value .br scl = max( scale, abs( x( i ) ) ). .br scale and sumsq must be supplied in SCALE and SUMSQ respectively. SCALE and SUMSQ are overwritten by scl and ssq respectively. The routine makes only one pass through the vector sub( X ). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. The result are only available in the scope of sub( X ), i.e if sub( X ) is distributed along a process row, the correct results are only available in this process row of the grid. Similarly if sub( X ) is distributed along a process column, the correct results are only available in this process column of the grid. .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The length of the distributed vector sub( X ). .TP 8 X (input) REAL The vector for which a scaled sum of squares is computed. x( i ) = X(IX+(JX-1)*M_X +(i-1)*INCX ), 1 <= i <= n. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 INCX (global input) INTEGER The global increment for the elements of X. Only two values of INCX are supported in this version, namely 1 and M_X. INCX must not be zero. .TP 8 SCALE (local input/local output) REAL On entry, the value scale in the equation above. On exit, SCALE is overwritten with scl , the scaling factor for the sum of squares. .TP 8 SUMSQ (local input/local output) REAL On entry, the value sumsq in the equation above. On exit, SUMSQ is overwritten with smsq , the basic sum of squares from which scl has been factored out. scalapack-doc-1.5/man/manl/pslaswp.l0100644000056400000620000001224106335610646017116 0ustar pfrauenfstaff.TH PSLASWP l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLASWP - perform a series of row or column interchanges on the distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PSLASWP( DIREC, ROWCOL, N, A, IA, JA, DESCA, K1, K2, IPIV ) .TP 20 .ti +4 CHARACTER DIREC, ROWCOL .TP 20 .ti +4 INTEGER IA, JA, K1, K2, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ) .TP 20 .ti +4 REAL A( * ) .SH PURPOSE PSLASWP performs a series of row or column interchanges on the distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1). One interchange is initiated for each of rows or columns K1 trough K2 of sub( A ). This routine assumes that the pivoting information has already been broadcast along the process row or column. .br Also note that this routine will only work for K1-K2 being in the same MB (or NB) block. If you want to pivot a full matrix, use PSLAPIV. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 DIREC (global input) CHARACTER Specifies in which order the permutation is applied: = 'F' (Forward) = 'B' (Backward) .TP 8 ROWCOL (global input) CHARACTER Specifies if the rows or columns are permuted: = 'R' (Rows) = 'C' (Columns) .TP 8 N (global input) INTEGER If ROWCOL = 'R', the length of the rows of the distributed matrix A(*,JA:JA+N-1) to be permuted; If ROWCOL = 'C', the length of the columns of the distributed matrix A(IA:IA+N-1,*) to be permuted. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, * ). On entry, this array contains the local pieces of the distri- buted matrix to which the row/columns interchanges will be applied. On exit the permuted distributed matrix. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 K1 (global input) INTEGER The first element of IPIV for which a row or column inter- change will be done. .TP 8 K2 (global input) INTEGER The last element of IPIV for which a row or column inter- change will be done. .TP 8 IPIV (local input) INTEGER array, dimension LOCr(M_A)+MB_A for row pivoting and LOCc(N_A)+NB_A for column pivoting. This array is tied to the matrix A, IPIV(K) = L implies rows (or columns) K and L are to be interchanged. scalapack-doc-1.5/man/manl/pslatra.l0100644000056400000620000000765406335610646017107 0ustar pfrauenfstaff.TH PSLATRA l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLATRA - compute the trace of an N-by-N distributed matrix sub( A ) denoting A( IA:IA+N-1, JA:JA+N-1 ) .SH SYNOPSIS .TP 14 REAL FUNCTION PSLATRA( N, A, IA, JA, DESCA ) .TP 14 .ti +4 INTEGER IA, JA, N .TP 14 .ti +4 INTEGER DESCA( * ) .TP 14 .ti +4 REAL A( * ) .SH PURPOSE PSLATRA computes the trace of an N-by-N distributed matrix sub( A ) denoting A( IA:IA+N-1, JA:JA+N-1 ). The result is left on every process of the grid. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on i.e the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension ( LLD_A, LOCc(JA+N-1) ). This array contains the local pieces of the distributed matrix the trace is to be computed. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. scalapack-doc-1.5/man/manl/pslatrd.l0100644000056400000620000002110706335610646017077 0ustar pfrauenfstaff.TH PSLATRD l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLATRD - reduce NB rows and columns of a real symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) to symmetric tridiagonal form by an orthogonal similarity transformation Q' * sub( A ) * Q, .SH SYNOPSIS .TP 20 SUBROUTINE PSLATRD( UPLO, N, NB, A, IA, JA, DESCA, D, E, TAU, W, IW, JW, DESCW, WORK ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IW, JA, JW, N, NB .TP 20 .ti +4 INTEGER DESCA( * ), DESCW( * ) .TP 20 .ti +4 REAL A( * ), D( * ), E( * ), TAU( * ), W( * ), WORK( * ) .SH PURPOSE PSLATRD reduces NB rows and columns of a real symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) to symmetric tridiagonal form by an orthogonal similarity transformation Q' * sub( A ) * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of sub( A ). .br If UPLO = 'U', PSLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; .br if UPLO = 'L', PSLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied. .br This is an auxiliary routine called by PSSYTRD. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the upper or lower triangular part of the symmetric matrix sub( A ) is stored: .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NB (global input) INTEGER The number of rows and columns to be reduced. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the symmetric distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced. On exit, if UPLO = 'U', the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of sub( A ); the elements above the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. If UPLO = 'L', the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of sub( A ); the elements below the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) REAL array, dimension LOCc(JA+N-1) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i). D is tied to the distributed matrix A. .TP 8 E (local output) REAL array, dimension LOCc(JA+N-1) if UPLO = 'U', LOCc(JA+N-2) otherwise. The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. E is tied to the distributed matrix A. .TP 8 TAU (local output) REAL, array, dimension LOCc(JA+N-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 W (local output) REAL pointer into the local memory to an array of dimension (LLD_W,NB_W), This array contains the local pieces of the N-by-NB_W matrix W required to update the unreduced part of sub( A ). .TP 8 IW (global input) INTEGER The row index in the global array W indicating the first row of sub( W ). .TP 8 JW (global input) INTEGER The column index in the global array W indicating the first column of sub( W ). .TP 8 DESCW (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix W. .TP 8 WORK (local workspace) REAL array, dimension (NB_A) .SH FURTHER DETAILS If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors .br Q = H(n) H(n-1) . . . H(n-nb+1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in .br A(ia:ia+i-2,ja+i), and tau in TAU(ja+i-1). .br If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors .br Q = H(1) H(2) . . . H(nb). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in .br A(ia+i+1:ia+n-1,ja+i-1), and tau in TAU(ja+i-1). .br The elements of the vectors v together form the N-by-NB matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a symmetric rank-2k update of the form: sub( A ) := sub( A ) - V*W' - W*V'. .br The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2: .br if UPLO = 'U': if UPLO = 'L': .br ( a a a v4 v5 ) ( d ) ( a a v4 v5 ) ( 1 d ) ( a 1 v5 ) ( v1 1 a ) ( d 1 ) ( v1 v2 a a ) ( d ) ( v1 v2 a a a ) where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i). .br scalapack-doc-1.5/man/manl/pslatrs.l0100644000056400000620000000114306335610646017114 0ustar pfrauenfstaff.TH PSLATRS l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLATRS - solve a triangular system .SH SYNOPSIS .TP 20 SUBROUTINE PSLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, IA, JA, DESCA, X, IX, JX, DESCX, SCALE, CNORM, WORK ) .TP 20 .ti +4 CHARACTER DIAG, NORMIN, TRANS, UPLO .TP 20 .ti +4 INTEGER IA, IX, JA, JX, N .TP 20 .ti +4 REAL SCALE .TP 20 .ti +4 INTEGER DESCA( * ), DESCX( * ) .TP 20 .ti +4 REAL A( * ), CNORM( * ), X( * ), WORK( * ) .SH PURPOSE PSLATRS solves a triangular system. This routine in unfinished at this time, but will be part of the next release. .br scalapack-doc-1.5/man/manl/pslatrz.l0100644000056400000620000001460506335610646017132 0ustar pfrauenfstaff.TH PSLATRZ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSLATRZ - reduce the M-by-N ( M<=N ) real upper trapezoidal matrix sub( A ) = [ A(IA:IA+M-1,JA:JA+M-1) A(IA:IA+M-1,JA+N-L:JA+N-1) ] to upper triangular form by means of orthogonal transformations .SH SYNOPSIS .TP 20 SUBROUTINE PSLATRZ( M, N, L, A, IA, JA, DESCA, TAU, WORK ) .TP 20 .ti +4 INTEGER IA, JA, L, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), TAU( * ), WORK( * ) .SH PURPOSE PSLATRZ reduces the M-by-N ( M<=N ) real upper trapezoidal matrix sub( A ) = [ A(IA:IA+M-1,JA:JA+M-1) A(IA:IA+M-1,JA+N-L:JA+N-1) ] to upper triangular form by means of orthogonal transformations. The upper trapezoidal matrix sub( A ) is factored as .br sub( A ) = ( R 0 ) * Z, .br where Z is an N-by-N orthogonal matrix and R is an M-by-M upper triangular matrix. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 L (global input) INTEGER The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. L > 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the leading M-by-M upper triangular part of sub( A ) contains the upper trian- gular matrix R, and elements N-L+1 to N of the first M rows of sub( A ), with the array TAU, represent the orthogonal matrix Z as a product of M elementary reflectors. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) REAL, array, dimension LOCr(IA+M-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace) REAL array, dimension (LWORK) LWORK >= Nq0 + MAX( 1, Mp0 ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. .SH FURTHER DETAILS The factorization is obtained by Householder's method. The kth transformation matrix, Z( k ), which is used to introduce zeros into the (m - k + 1)th row of sub( A ), is given in the form .br Z( k ) = ( I 0 ), .br ( 0 T( k ) ) .br where .br T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ), ( 0 ) ( z( k ) ) tau is a scalar and z( k ) is an ( n - m ) element vector. tau and z( k ) are chosen to annihilate the elements of the kth row of sub( A ). .br The scalar tau is returned in the kth element of TAU and the vector u( k ) in the kth row of sub( A ), such that the elements of z( k ) are in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in the upper triangular part of sub( A ). .br Z is given by .br Z = Z( 1 ) * Z( 2 ) * ... * Z( m ). .br scalapack-doc-1.5/man/manl/pslauu2.l0100644000056400000620000001157406335610646017030 0ustar pfrauenfstaff.TH PSLAUU2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLAUU2 - compute the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PSLAUU2( UPLO, N, A, IA, JA, DESCA ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ) .SH PURPOSE PSLAUU2 computes the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). If UPLO = 'U' or 'u' then the upper triangle of the result is stored, overwriting the factor U in sub( A ). .br If UPLO = 'L' or 'l' then the lower triangle of the result is stored, overwriting the factor L in sub( A ). .br This is the unblocked form of the algorithm, calling Level 2 BLAS. No communication is performed by this routine, the matrix to operate on should be strictly local to one process. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 Specifies whether the triangular factor stored in the matrix sub( A ) is upper or lower triangular: .br = 'U': Upper triangular, .br = 'L': Lower triangular. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the order of the triangular factor U or L. N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the triangular factor L or U. On exit, if UPLO = 'U', the upper triangle of the distributed matrix sub( A ) is overwritten with the upper triangle of the product U * U'; if UPLO = 'L', the lower triangle of sub( A ) is overwritten with the lower triangle of the product L' * L. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. scalapack-doc-1.5/man/manl/pslauum.l0100644000056400000620000001143206335610646017114 0ustar pfrauenfstaff.TH PSLAUUM l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSLAUUM - compute the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PSLAUUM( UPLO, N, A, IA, JA, DESCA ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ) .SH PURPOSE PSLAUUM computes the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). If UPLO = 'U' or 'u' then the upper triangle of the result is stored, overwriting the factor U in sub( A ). .br If UPLO = 'L' or 'l' then the lower triangle of the result is stored, overwriting the factor L in sub( A ). .br This is the blocked form of the algorithm, calling Level 3 PBLAS. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 Specifies whether the triangular factor stored in the distributed matrix sub( A ) is upper or lower triangular: .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the triangular factor U or L. N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the triangular factor L or U. On exit, if UPLO = 'U', the upper triangle of the distributed matrix sub( A ) is overwritten with the upper triangle of the product U * U'; if UPLO = 'L', the lower triangle of sub( A ) is overwritten with the lower triangle of the product L' * L. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. scalapack-doc-1.5/man/manl/pslawil.l0100644000056400000620000000764706335610646017116 0ustar pfrauenfstaff.TH PSLAWIL l "12 May 1997" "LAPACK version 1.5 " "LAPACK routine (version 1.5 )" .SH NAME PSLAWIL - get the transform given by H44,H33, & H43H34 into V starting at row M .SH SYNOPSIS .TP 20 SUBROUTINE PSLAWIL( II, JJ, M, A, DESCA, H44, H33, H43H34, V ) .TP 20 .ti +4 INTEGER II, JJ, M .TP 20 .ti +4 REAL H33, H43H34, H44 .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), V( * ) .SH PURPOSE PSLAWIL gets the transform given by H44,H33, & H43H34 into V starting at row M. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 II (global input) INTEGER Row owner of H(M+2,M+2) .TP 8 JJ (global input) INTEGER Column owner of H(M+2,M+2) .TP 8 M (global input) INTEGER On entry, this is where the transform starts (row M.) Unchanged on exit. .TP 8 A (global input) REAL array, dimension (DESCA(LLD_),*) On entry, the Hessenberg matrix. Unchanged on exit. .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. Unchanged on exit. H44 H33 H43H34 (global input) REAL These three values are for the double shift QR iteration. Unchanged on exit. .TP 8 V (global output) REAL array of size 3. Contains the transform on output. Implemented by: G. Henry, November 17, 1996 scalapack-doc-1.5/man/manl/psorg2l.l0100644000056400000620000001401506335610646017016 0ustar pfrauenfstaff.TH PSORG2L l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSORG2L - generate an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the last N columns of a product of K elementary reflectors of order M Q = H(k) .SH SYNOPSIS .TP 20 SUBROUTINE PSORG2L( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), TAU( * ), WORK( * ) .SH PURPOSE PSORG2L generates an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the last N columns of a product of K elementary reflectors of order M as returned by PSGEQLF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. M >= N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the j-th column must contain the vector which defines the elementary reflector H(j), JA+N-K <= j <= JA+N-1, as returned by PSGEQLF in the K columns of its distributed matrix argument A(IA:*,JA+N-K:JA+N-1). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) REAL, array, dimension LOCc(JA+N-1) This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PSGEQLF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MpA0 + MAX( 1, NqA0 ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/psorg2r.l0100644000056400000620000001400006335610646017016 0ustar pfrauenfstaff.TH PSORG2R l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSORG2R - generate an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M Q = H(1) H(2) .SH SYNOPSIS .TP 20 SUBROUTINE PSORG2R( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), TAU( * ), WORK( * ) .SH PURPOSE PSORG2R generates an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M as returned by PSGEQRF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. M >= N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the j-th column must contain the vector which defines the elementary reflector H(j), JA <= j <= JA+K-1, as returned by PSGEQRF in the K columns of its array argument A(IA:*,JA:JA+K-1). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) REAL, array, dimension LOCc(JA+K-1). This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PSGEQRF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MpA0 + MAX( 1, NqA0 ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/psorgl2.l0100644000056400000620000001376606335610647017033 0ustar pfrauenfstaff.TH PSORGL2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSORGL2 - generate an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N Q = H(k) .SH SYNOPSIS .TP 20 SUBROUTINE PSORGL2( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), TAU( * ), WORK( * ) .SH PURPOSE PSORGL2 generates an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N as returned by PSGELQF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. N >= M >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PSGELQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) REAL, array, dimension LOCr(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PSGELQF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NqA0 + MAX( 1, MpA0 ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/psorglq.l0100644000056400000620000001377706335610647017134 0ustar pfrauenfstaff.TH PSORGLQ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSORGLQ - generate an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N Q = H(k) .SH SYNOPSIS .TP 20 SUBROUTINE PSORGLQ( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), TAU( * ), WORK( * ) .SH PURPOSE PSORGLQ generates an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N as returned by PSGELQF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. N >= M >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PSGELQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) REAL, array, dimension LOCr(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PSGELQF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MB_A * ( MpA0 + NqA0 + MB_A ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/psorgql.l0100644000056400000620000001402606335610647017120 0ustar pfrauenfstaff.TH PSORGQL l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSORGQL - generate an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the last N columns of a product of K elementary reflectors of order M Q = H(k) .SH SYNOPSIS .TP 20 SUBROUTINE PSORGQL( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), TAU( * ), WORK( * ) .SH PURPOSE PSORGQL generates an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the last N columns of a product of K elementary reflectors of order M as returned by PSGEQLF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. M >= N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the j-th column must contain the vector which defines the elementary reflector H(j), JA+N-K <= j <= JA+N-1, as returned by PSGEQLF in the K columns of its distributed matrix argument A(IA:*,JA+N-K:JA+N-1). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) REAL, array, dimension LOCc(JA+N-1) This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PSGEQLF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB_A * ( NqA0 + MpA0 + NB_A ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/psorgqr.l0100644000056400000620000001402506335610647017125 0ustar pfrauenfstaff.TH PSORGQR l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSORGQR - generate an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M Q = H(1) H(2) .SH SYNOPSIS .TP 20 SUBROUTINE PSORGQR( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), TAU( * ), WORK( * ) .SH PURPOSE PSORGQR generates an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M as returned by PSGEQRF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. M >= N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the j-th column must contain the vector which defines the elementary reflector H(j), JA <= j <= JA+K-1, as returned by PSGEQRF in the K columns of its distributed matrix argument A(IA:*,JA:JA+K-1). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) REAL, array, dimension LOCc(JA+K-1) This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PSGEQRF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB_A * ( NqA0 + MpA0 + NB_A ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/psorgr2.l0100644000056400000620000001400006335610647017017 0ustar pfrauenfstaff.TH PSORGR2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSORGR2 - generate an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N Q = H(1) H(2) .SH SYNOPSIS .TP 20 SUBROUTINE PSORGR2( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), TAU( * ), WORK( * ) .SH PURPOSE PSORGR2 generates an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N as returned by PSGERQF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. N >= M >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA+M-K <= i <= IA+M-1, as returned by PSGERQF in the K rows of its distributed matrix argument A(IA+M-K:IA+M-1,JA:*). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) REAL, array, dimension LOCr(IA+M-1) This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PSGERQF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NqA0 + MAX( 1, MpA0 ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/psorgrq.l0100644000056400000620000001401106335610647017120 0ustar pfrauenfstaff.TH PSORGRQ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSORGRQ - generate an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N Q = H(1) H(2) .SH SYNOPSIS .TP 20 SUBROUTINE PSORGRQ( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), TAU( * ), WORK( * ) .SH PURPOSE PSORGRQ generates an M-by-N real distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N as returned by PSGERQF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. N >= M >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA+M-K <= i <= IA+M-1, as returned by PSGERQF in the K rows of its distributed matrix argument A(IA+M-K:IA+M-1,JA:*). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) REAL, array, dimension LOCr(IA+M-1) This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PSGERQF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MB_A * ( MpA0 + NqA0 + MB_A ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/psorm2l.l0100644000056400000620000001722506335610647017033 0ustar pfrauenfstaff.TH PSORM2L l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSORM2L - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PSORM2L( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 REAL A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PSORM2L overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix defined as the product of K elementary reflectors .br Q = H(k) . . . H(2) H(1) .br as returned by PSGEQLF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+K-1)). On entry, the j-th column must contain the vector which defines the elemen- tary reflector H(j), JA <= j <= JA+K-1, as returned by PSGEQLF in the K columns of its distributed matrix argument A(IA:*,JA:JA+K-1). A(IA:*,JA:JA+K-1) is modified by the routine but restored on exit. If SIDE = 'L', LLD_A >= MAX( 1, LOCr(IA+M-1) ), if SIDE = 'R', LLD_A >= MAX( 1, LOCr(IA+N-1) ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) REAL, array, dimension LOCc(JA+N-1) This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PSGEQLF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MpC0 + MAX( 1, NqC0 ); if SIDE = 'R', LWORK >= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC( N+ICOFFC,NB_A,0,0,NPCOL ),NB_A,0,0,LCMQ ) ); where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/psorm2r.l0100644000056400000620000001722606335610647017042 0ustar pfrauenfstaff.TH PSORM2R l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSORM2R - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PSORM2R( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 REAL A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PSORM2R overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix defined as the product of k elementary reflectors .br Q = H(1) H(2) . . . H(k) .br as returned by PSGEQRF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+K-1)). On entry, the j-th column must contain the vector which defines the elemen- tary reflector H(j), JA <= j <= JA+K-1, as returned by PSGEQRF in the K columns of its distributed matrix argument A(IA:*,JA:JA+K-1). A(IA:*,JA:JA+K-1) is modified by the routine but restored on exit. If SIDE = 'L', LLD_A >= MAX( 1, LOCr(IA+M-1) ); if SIDE = 'R', LLD_A >= MAX( 1, LOCr(IA+N-1) ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) REAL, array, dimension LOCc(JA+K-1). This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PSGEQRF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MpC0 + MAX( 1, NqC0 ); if SIDE = 'R', LWORK >= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC( N+ICOFFC,NB_A,0,0,NPCOL ),NB_A,0,0,LCMQ ) ); where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/psormbr.l0100644000056400000620000002376206335610647017124 0ustar pfrauenfstaff.TH PSORMBR l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSORMBR - VECT = 'Q', PSORMBR overwrites the general real distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PSORMBR( VECT, SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS, VECT .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 REAL A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE If VECT = 'Q', PSORMBR overwrites the general real distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br If VECT = 'P', PSORMBR overwrites sub( C ) with .br SIDE = 'L' SIDE = 'R' .br TRANS = 'N': P * sub( C ) sub( C ) * P .br TRANS = 'T': P**T * sub( C ) sub( C ) * P**T .br Here Q and P**T are the orthogonal distributed matrices determined by PSGEBRD when reducing a real distributed matrix A(IA:*,JA:*) to bidiagonal form: A(IA:*,JA:*) = Q * B * P**T. Q and P**T are defined as products of elementary reflectors H(i) and G(i) respectively. Let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the order of the orthogonal matrix Q or P**T that is applied. If VECT = 'Q', A(IA:*,JA:*) is assumed to have been an NQ-by-K matrix: .br if nq >= k, Q = H(1) H(2) . . . H(k); .br if nq < k, Q = H(1) H(2) . . . H(nq-1). .br If VECT = 'P', A(IA:*,JA:*) is assumed to have been a K-by-NQ matrix: .br if k < nq, P = G(1) G(2) . . . G(k); .br if k >= nq, P = G(1) G(2) . . . G(nq-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 VECT (global input) CHARACTER = 'Q': apply Q or Q**T; .br = 'P': apply P or P**T. .TP 8 SIDE (global input) CHARACTER .br = 'L': apply Q, Q**T, P or P**T from the Left; .br = 'R': apply Q, Q**T, P or P**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q or P; .br = 'T': Transpose, apply Q**T or P**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER If VECT = 'Q', the number of columns in the original distributed matrix reduced by PSGEBRD. If VECT = 'P', the number of rows in the original distributed matrix reduced by PSGEBRD. K >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+MIN(NQ,K)-1)) if VECT='Q', and (LLD_A,LOCc(JA+NQ-1)) if VECT = 'P'. NQ = M if SIDE = 'L', and NQ = N otherwise. The vectors which define the elementary reflectors H(i) and G(i), whose products determine the matrices Q and P, as returned by PSGEBRD. If VECT = 'Q', LLD_A >= max(1,LOCr(IA+NQ-1)); if VECT = 'P', LLD_A >= max(1,LOCr(IA+MIN(NQ,K)-1)). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) REAL array, dimension LOCc(JA+MIN(NQ,K)-1) if VECT = 'Q', LOCr(IA+MIN(NQ,K)-1) if VECT = 'P', TAU(i) must contain the scalar factor of the elementary reflector H(i) or G(i), which determines Q or P, as returned by PDGEBRD in its array argument TAUQ or TAUP. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, if VECT='Q', sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q; if VECT='P, sub( C ) is overwritten by P*sub( C ) or P'*sub( C ) or sub( C )*P or sub( C )*P'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', NQ = M; if( (VECT = 'Q' and NQ >= K) or (VECT <> 'Q' and NQ > K) ), IAA=IA; JAA=JA; MI=M; NI=N; ICC=IC; JCC=JC; else IAA=IA+1; JAA=JA; MI=M-1; NI=N; ICC=IC+1; JCC=JC; end if else if SIDE = 'R', NQ = N; if( (VECT = 'Q' and NQ >= K) or (VECT <> 'Q' and NQ > K) ), IAA=IA; JAA=JA; MI=M; NI=N; ICC=IC; JCC=JC; else IAA=IA; JAA=JA+1; MI=M; NI=N-1; ICC=IC; JCC=JC+1; end if end if If VECT = 'Q', If SIDE = 'L', LWORK >= MAX( (NB_A*(NB_A-1))/2, (NqC0 + MpC0)*NB_A ) + NB_A * NB_A else if SIDE = 'R', LWORK >= MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX( NpA0 + NUMROC( NUMROC( NI+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0 ) )*NB_A ) + NB_A * NB_A end if else if VECT <> 'Q', if SIDE = 'L', LWORK >= MAX( (MB_A*(MB_A-1))/2, ( MpC0 + MAX( MqA0 + NUMROC( NUMROC( MI+IROFFC, MB_A, 0, 0, NPROW ), MB_A, 0, 0, LCMP ), NqC0 ) )*MB_A ) + MB_A * MB_A else if SIDE = 'R', LWORK >= MAX( (MB_A*(MB_A-1))/2, (MpC0 + NqC0)*MB_A ) + MB_A * MB_A end if end if where LCMP = LCM / NPROW, LCMQ = LCM / NPCOL, with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IAA-1, MB_A ), ICOFFA = MOD( JAA-1, NB_A ), IAROW = INDXG2P( IAA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JAA, NB_A, MYCOL, CSRC_A, NPCOL ), MqA0 = NUMROC( MI+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), NpA0 = NUMROC( NI+IROFFA, MB_A, MYROW, IAROW, NPROW ), IROFFC = MOD( ICC-1, MB_C ), ICOFFC = MOD( JCC-1, NB_C ), ICROW = INDXG2P( ICC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JCC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( MI+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( NI+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If VECT = 'Q', If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) else If SIDE = 'L', ( MB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/psormhr.l0100644000056400000620000002010606335610647017117 0ustar pfrauenfstaff.TH PSORMHR l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSORMHR - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PSORMHR( SIDE, TRANS, M, N, ILO, IHI, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, IHI, ILO, INFO, JA, JC, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 REAL A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PSORMHR overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix of order nq, with nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of IHI-ILO elementary reflectors, as returned by PSGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 ILO (global input) INTEGER IHI (global input) INTEGER ILO and IHI must have the same values as in the previous call of PSGEHRD. Q is equal to the unit matrix except in the distributed submatrix Q(ia+ilo:ia+ihi-1,ia+ilo:ja+ihi-1). If SIDE = 'L', 1 <= ILO <= IHI <= max(1,M); if SIDE = 'R', 1 <= ILO <= IHI <= max(1,N); ILO and IHI are relative indexes. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE = 'R'. The vectors which define the elementary reflectors, as returned by PSGEHRD. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) REAL, array, dimension LOCc(JA+M-2) if SIDE = 'L', and LOCc(JA+N-2) if SIDE = 'R'. This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PSGEHRD. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least IAA = IA + ILO; JAA = JA+ILO-1; If SIDE = 'L', MI = IHI-ILO; NI = N; ICC = IC + ILO; JCC = JC; LWORK >= MAX( (NB_A*(NB_A-1))/2, (NqC0 + MpC0)*NB_A ) + NB_A * NB_A else if SIDE = 'R', MI = M; NI = IHI-ILO; ICC = IC; JCC = JC + ILO; LWORK >= MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX( NpA0 + NUMROC( NUMROC( NI+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0 ) )*NB_A ) + NB_A * NB_A end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IAA-1, MB_A ), ICOFFA = MOD( JAA-1, NB_A ), IAROW = INDXG2P( IAA, MB_A, MYROW, RSRC_A, NPROW ), NpA0 = NUMROC( NI+IROFFA, MB_A, MYROW, IAROW, NPROW ), IROFFC = MOD( ICC-1, MB_C ), ICOFFC = MOD( JCC-1, NB_C ), ICROW = INDXG2P( ICC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JCC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( MI+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( NI+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/psorml2.l0100644000056400000620000001720706335610647017033 0ustar pfrauenfstaff.TH PSORML2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSORML2 - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PSORML2( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 REAL A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PSORML2 overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix defined as the product of K elementary reflectors .br Q = H(k) . . . H(2) H(1) .br as returned by PSGELQF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= max(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PSGELQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) REAL, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PSGELQF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_A,0,0,NPROW ),MB_A,0,0,LCMP ) ); if SIDE = 'R', LWORK >= NqC0 + MAX( 1, MpC0 ); where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/psormlq.l0100644000056400000620000001761706335610650017131 0ustar pfrauenfstaff.TH PSORMLQ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSORMLQ - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PSORMLQ( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 REAL A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PSORMLQ overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix defined as the product of K elementary reflectors .br Q = H(k) . . . H(2) H(1) .br as returned by PSGELQF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= max(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PSGELQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) REAL, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PSGELQF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least if SIDE = 'L', LWORK >= MAX( (MB_A*(MB_A-1))/2, ( MpC0 + MAX( MqA0 + NUMROC( NUMROC( M+IROFFC, MB_A, 0, 0, NPROW ), MB_A, 0, 0, LCMP ), NqC0 ) )*MB_A ) + MB_A * MB_A else if SIDE = 'R', LWORK >= MAX( (MB_A*(MB_A-1))/2, (MpC0 + NqC0)*MB_A ) + MB_A * MB_A end if where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MqA0 = NUMROC( M+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/psormql.l0100644000056400000620000001763506335610650017131 0ustar pfrauenfstaff.TH PSORMQL l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSORMQL - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PSORMQL( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 REAL A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PSORMQL overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix defined as the product of K elementary reflectors .br Q = H(k) . . . H(2) H(1) .br as returned by PSGEQLF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+K-1)). On entry, the j-th column must contain the vector which defines the elemen- tary reflector H(j), JA <= j <= JA+K-1, as returned by PSGEQLF in the K columns of its distributed matrix argument A(IA:*,JA:JA+K-1). A(IA:*,JA:JA+K-1) is modified by the routine but restored on exit. If SIDE = 'L', LLD_A >= MAX( 1, LOCr(IA+M-1) ), if SIDE = 'R', LLD_A >= MAX( 1, LOCr(IA+N-1) ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) REAL, array, dimension LOCc(JA+N-1) This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PSGEQLF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MAX( (NB_A*(NB_A-1))/2, (NqC0 + MpC0)*NB_A ) + NB_A * NB_A else if SIDE = 'R', LWORK >= MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX( NpA0 + NUMROC( NUMROC( N+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0 ) )*NB_A ) + NB_A * NB_A end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), NpA0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/psormqr.l0100644000056400000620000001763606335610650017140 0ustar pfrauenfstaff.TH PSORMQR l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSORMQR - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PSORMQR( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 REAL A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PSORMQR overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix defined as the product of k elementary reflectors .br Q = H(1) H(2) . . . H(k) .br as returned by PSGEQRF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+K-1)). On entry, the j-th column must contain the vector which defines the elemen- tary reflector H(j), JA <= j <= JA+K-1, as returned by PSGEQRF in the K columns of its distributed matrix argument A(IA:*,JA:JA+K-1). A(IA:*,JA:JA+K-1) is modified by the routine but restored on exit. If SIDE = 'L', LLD_A >= MAX( 1, LOCr(IA+M-1) ); if SIDE = 'R', LLD_A >= MAX( 1, LOCr(IA+N-1) ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) REAL, array, dimension LOCc(JA+K-1). This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PSGEQRF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MAX( (NB_A*(NB_A-1))/2, (NqC0 + MpC0)*NB_A ) + NB_A * NB_A else if SIDE = 'R', LWORK >= MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX( NpA0 + NUMROC( NUMROC( N+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0 ) )*NB_A ) + NB_A * NB_A end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), NpA0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/psormr2.l0100644000056400000620000001720706335610650017033 0ustar pfrauenfstaff.TH PSORMR2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSORMR2 - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PSORMR2( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 REAL A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PSORMR2 overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix defined as the product of K elementary reflectors .br Q = H(1) H(2) . . . H(k) .br as returned by PSGERQF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= MAX(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PSGERQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) REAL, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PSGERQF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_A,0,0,NPROW ),MB_A,0,0,LCMP ) ); if SIDE = 'R', LWORK >= NqC0 + MAX( 1, MpC0 ); where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/psormr3.l0100644000056400000620000001753206335610650017035 0ustar pfrauenfstaff.TH PSORMR3 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSORMR3 - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PSORMR3( SIDE, TRANS, M, N, K, L, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, L, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 REAL A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PSORMR3 overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix defined as the product of K elementary reflectors .br Q = H(1) H(2) . . . H(k) .br as returned by PSTZRZF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 L (global input) INTEGER The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= MAX(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PSTZRZF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) REAL, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PSTZRZF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_A,0,0,NPROW ),MB_A,0,0,LCMP ) ); if SIDE = 'R', LWORK >= NqC0 + MAX( 1, MpC0 ); where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/psormrq.l0100644000056400000620000001762006335610650017131 0ustar pfrauenfstaff.TH PSORMRQ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSORMRQ - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PSORMRQ( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 REAL A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PSORMRQ overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix defined as the product of K elementary reflectors .br Q = H(1) H(2) . . . H(k) .br as returned by PSGERQF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= MAX(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PSGERQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) REAL, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PSGERQF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least if SIDE = 'L', LWORK >= MAX( (MB_A*(MB_A-1))/2, ( MpC0 + MAX( MqA0 + NUMROC( NUMROC( M+IROFFC, MB_A, 0, 0, NPROW ), MB_A, 0, 0, LCMP ), NqC0 ) )*MB_A ) + MB_A * MB_A else if SIDE = 'R', LWORK >= MAX( (MB_A*(MB_A-1))/2, (MpC0 + NqC0)*MB_A ) + MB_A * MB_A end if where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MqA0 = NUMROC( M+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/psormrz.l0100644000056400000620000002014306335610650017134 0ustar pfrauenfstaff.TH PSORMRZ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSORMRZ - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PSORMRZ( SIDE, TRANS, M, N, K, L, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, L, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 REAL A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PSORMRZ overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix defined as the product of K elementary reflectors .br Q = H(1) H(2) . . . H(k) .br as returned by PSTZRZF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 L (global input) INTEGER The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= MAX(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PSTZRZF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) REAL, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PSTZRZF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least if SIDE = 'L', LWORK >= MAX( (MB_A*(MB_A-1))/2, ( MpC0 + MAX( MqA0 + NUMROC( NUMROC( M+IROFFC, MB_A, 0, 0, NPROW ), MB_A, 0, 0, LCMP ), NqC0 ) )*MB_A ) + MB_A * MB_A else if SIDE = 'R', LWORK >= MAX( (MB_A*(MB_A-1))/2, (MpC0 + NqC0)*MB_A ) + MB_A * MB_A end if where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MqA0 = NUMROC( M+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/psormtr.l0100644000056400000620000002041506335610650017130 0ustar pfrauenfstaff.TH PSORMTR l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSORMTR - overwrite the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PSORMTR( SIDE, UPLO, TRANS, M, N, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS, UPLO .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 REAL A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PSORMTR overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T .br where Q is a real orthogonal distributed matrix of order nq, with nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of nq-1 elementary reflectors, as returned by PSSYTRD: if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1); .br if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**T from the Left; .br = 'R': apply Q or Q**T from the Right. .TP 8 UPLO (global input) CHARACTER .br = 'U': Upper triangle of A(IA:*,JA:*) contains elementary reflectors from PSSYTRD; = 'L': Lower triangle of A(IA:*,JA:*) contains elementary reflectors from PSSYTRD. .TP 8 TRANS (global input) CHARACTER = 'N': No transpose, apply Q; .br = 'T': Transpose, apply Q**T. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', or (LLD_A,LOCc(JA+N-1)) if SIDE = 'R'. The vectors which define the elementary reflectors, as returned by PSSYTRD. If SIDE = 'L', LLD_A >= max(1,LOCr(IA+M-1)); if SIDE = 'R', LLD_A >= max(1,LOCr(IA+N-1)). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) REAL array, dimension LTAU, where if SIDE = 'L' and UPLO = 'U', LTAU = LOCc(M_A), if SIDE = 'L' and UPLO = 'L', LTAU = LOCc(JA+M-2), if SIDE = 'R' and UPLO = 'U', LTAU = LOCc(N_A), if SIDE = 'R' and UPLO = 'L', LTAU = LOCc(JA+N-2). TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by PSSYTRD. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If UPLO = 'U', IAA = IA, JAA = JA+1, ICC = IC, JCC = JC; else UPLO = 'L', IAA = IA+1, JAA = JA; if SIDE = 'L', ICC = IC+1; JCC = JC; else ICC = IC; JCC = JC+1; end if end if If SIDE = 'L', MI = M-1; NI = N; LWORK >= MAX( (NB_A*(NB_A-1))/2, (NqC0 + MpC0)*NB_A ) + NB_A * NB_A else if SIDE = 'R', MI = M; MI = N-1; LWORK >= MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX( NpA0 + NUMROC( NUMROC( NI+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0 ) )*NB_A ) + NB_A * NB_A end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IAA-1, MB_A ), ICOFFA = MOD( JAA-1, NB_A ), IAROW = INDXG2P( IAA, MB_A, MYROW, RSRC_A, NPROW ), NpA0 = NUMROC( NI+IROFFA, MB_A, MYROW, IAROW, NPROW ), IROFFC = MOD( ICC-1, MB_C ), ICOFFC = MOD( JCC-1, NB_C ), ICROW = INDXG2P( ICC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JCC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( MI+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( NI+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/pspbsv.l0100644000056400000620000000140506335610650016735 0ustar pfrauenfstaff.TH PSPBSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSPBSV - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 19 SUBROUTINE PSPBSV( UPLO, N, BW, NRHS, A, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 CHARACTER UPLO .TP 19 .ti +4 INTEGER BW, IB, INFO, JA, LWORK, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 REAL A( * ), B( * ), WORK( * ) .SH PURPOSE PSPBSV solves a system of linear equations where A(1:N, JA:JA+N-1) is an N-by-N real .br banded symmetric positive definite distributed .br matrix with bandwidth BW. .br Cholesky factorization is used to factor a reordering of .br the matrix into L L'. .br See PSPBTRF and PSPBTRS for details. .br scalapack-doc-1.5/man/manl/pspbtrf.l0100644000056400000620000000230406335610650017077 0ustar pfrauenfstaff.TH PSPBTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSPBTRF - compute a Cholesky factorization of an N-by-N real banded symmetric positive definite distributed matrix with bandwidth BW .SH SYNOPSIS .TP 20 SUBROUTINE PSPBTRF( UPLO, N, BW, A, JA, DESCA, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER BW, INFO, JA, LAF, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), AF( * ), WORK( * ) .SH PURPOSE PSPBTRF computes a Cholesky factorization of an N-by-N real banded symmetric positive definite distributed matrix with bandwidth BW: A(1:N, JA:JA+N-1). Reordering is used to increase parallelism in the factorization. This reordering results in factors that are DIFFERENT from those produced by equivalent sequential codes. These factors cannot be used directly by users; however, they can be used in .br subsequent calls to PSPBTRS to solve linear systems. .br The factorization has the form .br P A(1:N, JA:JA+N-1) P^T = U' U , if UPLO = 'U', or P A(1:N, JA:JA+N-1) P^T = L L', if UPLO = 'L' .br where U is a banded upper triangular matrix and L is banded lower triangular, and P is a permutation matrix. .br scalapack-doc-1.5/man/manl/pspbtrs.l0100644000056400000620000000166706335610650017127 0ustar pfrauenfstaff.TH PSPBTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSPBTRS - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 20 SUBROUTINE PSPBTRS( UPLO, N, BW, NRHS, A, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER BW, IB, INFO, JA, LAF, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 REAL A( * ), AF( * ), B( * ), WORK( * ) .SH PURPOSE PSPBTRS solves a system of linear equations where A(1:N, JA:JA+N-1) is the matrix used to produce the factors stored in A(1:N,JA:JA+N-1) and AF by PSPBTRF. .br A(1:N, JA:JA+N-1) is an N-by-N real .br banded symmetric positive definite distributed .br matrix with bandwidth BW. .br Depending on the value of UPLO, A stores either U or L in the equn A(1:N, JA:JA+N-1) = U'*U or L*L' as computed by PSPBTRF. .br Routine PSPBTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pspbtrsv.l0100644000056400000620000000216106335610650017303 0ustar pfrauenfstaff.TH PSPBTRSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSPBTRSV - solve a banded triangular system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 21 SUBROUTINE PSPBTRSV( UPLO, TRANS, N, BW, NRHS, A, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 21 .ti +4 CHARACTER TRANS, UPLO .TP 21 .ti +4 INTEGER BW, IB, INFO, JA, LAF, LWORK, N, NRHS .TP 21 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 21 .ti +4 REAL A( * ), AF( * ), B( * ), WORK( * ) .SH PURPOSE PSPBTRSV solves a banded triangular system of linear equations or .br A(1:N, JA:JA+N-1)^T * X = B(IB:IB+N-1, 1:NRHS) where A(1:N, JA:JA+N-1) is a banded .br triangular matrix factor produced by the .br Cholesky factorization code PSPBTRF .br and is stored in A(1:N,JA:JA+N-1) and AF. .br The matrix stored in A(1:N, JA:JA+N-1) is either .br upper or lower triangular according to UPLO, .br and the choice of solving A(1:N, JA:JA+N-1) or A(1:N, JA:JA+N-1)^T is dictated by the user by the parameter TRANS. .br Routine PSPBTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pspocon.l0100644000056400000620000001500106335610650017076 0ustar pfrauenfstaff.TH PSPOCON l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSPOCON - estimate the reciprocal of the condition number (in the 1-norm) of a real symmetric positive definite distributed matrix using the Cholesky factorization A = U**T*U or A = L*L**T computed by PSPOTRF .SH SYNOPSIS .TP 20 SUBROUTINE PSPOCON( UPLO, N, A, IA, JA, DESCA, ANORM, RCOND, WORK, LWORK, IWORK, LIWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, LIWORK, LWORK, N .TP 20 .ti +4 REAL ANORM, RCOND .TP 20 .ti +4 INTEGER DESCA( * ), IWORK( * ) .TP 20 .ti +4 REAL A( * ), WORK( * ) .SH PURPOSE PSPOCON estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric positive definite distributed matrix using the Cholesky factorization A = U**T*U or A = L*L**T computed by PSPOTRF. An estimate is obtained for norm(inv(A(IA:IA+N-1,JA:JA+N-1))), and the reciprocal of the condition number is computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the factor stored in A(IA:IA+N-1,JA:JA+N-1) is upper or lower triangular. .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER .br The order of the distributed matrix A(IA:IA+N-1,JA:JA+N-1). N >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension ( LLD_A, LOCc(JA+N-1) ). On entry, this array contains the local pieces of the factors L or U from the Cholesky factorization A(IA:IA+N-1,JA:JA+N-1) = U'*U or L*L', as computed by PSPOTRF. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 ANORM (global input) REAL The 1-norm (or infinity-norm) of the symmetric distributed matrix A(IA:IA+N-1,JA:JA+N-1). .TP 8 RCOND (global output) REAL The reciprocal of the condition number of the distributed matrix A(IA:IA+N-1,JA:JA+N-1), computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * .br norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 2*LOCr(N+MOD(IA-1,MB_A)) + 2*LOCc(N+MOD(JA-1,NB_A)) + MAX( 2, MAX(NB_A*CEIL(NPROW-1,NPCOL),LOCc(N+MOD(JA-1,NB_A)) + NB_A*CEIL(NPCOL-1,NPROW)) ). If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/local output) INTEGER array, dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK. .TP 8 LIWORK (local or global input) INTEGER The dimension of the array IWORK. LIWORK is local input and must be at least LIWORK >= LOCr(N+MOD(IA-1,MB_A)). If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pspoequ.l0100644000056400000620000001377306335610651017130 0ustar pfrauenfstaff.TH PSPOEQU l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSPOEQU - compute row and column scalings intended to equilibrate a distributed symmetric positive definite matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) and reduce its condition number (with respect to the two-norm) .SH SYNOPSIS .TP 20 SUBROUTINE PSPOEQU( N, A, IA, JA, DESCA, SR, SC, SCOND, AMAX, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 REAL AMAX, SCOND .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), SC( * ), SR( * ) .SH PURPOSE PSPOEQU computes row and column scalings intended to equilibrate a distributed symmetric positive definite matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) and reduce its condition number (with respect to the two-norm). SR and SC contain the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled distri- buted matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of SR and SC puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings. .br The scaling factor are stored along process rows in SR and along process columns in SC. The duplication of information simplifies greatly the application of the factors. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on i.e the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of local dimension ( LLD_A, LOCc(JA+N-1) ), the N-by-N symmetric positive definite distributed matrix sub( A ) whose scaling factors are to be computed. Only the diagonal elements of sub( A ) are referenced. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 SR (local output) REAL array, dimension LOCr(M_A) If INFO = 0, SR(IA:IA+N-1) contains the row scale factors for sub( A ). SR is aligned with the distributed matrix A, and replicated across every process column. SR is tied to the distributed matrix A. .TP 8 SC (local output) REAL array, dimension LOCc(N_A) If INFO = 0, SC(JA:JA+N-1) contains the column scale factors .br for A(IA:IA+M-1,JA:JA+N-1). SC is aligned with the distribu- ted matrix A, and replicated down every process row. SC is tied to the distributed matrix A. .TP 8 SCOND (global output) REAL If INFO = 0, SCOND contains the ratio of the smallest SR(i) (or SC(j)) to the largest SR(i) (or SC(j)), with IA <= i <= IA+N-1 and JA <= j <= JA+N-1. If SCOND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by SR (or SC). .TP 8 AMAX (global output) REAL Absolute value of largest matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, the K-th diagonal entry of sub( A ) is nonpositive. scalapack-doc-1.5/man/manl/psporfs.l0100644000056400000620000002351306335610651017121 0ustar pfrauenfstaff.TH PSPORFS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSPORFS - improve the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite and provides error bounds and backward error estimates for the solutions .SH SYNOPSIS .TP 20 SUBROUTINE PSPORFS( UPLO, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, B, IB, JB, DESCB, X, IX, JX, DESCX, FERR, BERR, WORK, LWORK, IWORK, LIWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IAF, IB, INFO, IX, JA, JAF, JB, JX, LIWORK, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * ), IWORK( * ) .TP 20 .ti +4 REAL A( * ), AF( * ), B( * ), BERR( * ), FERR( * ), WORK( * ), X( * ) .SH PURPOSE PSPORFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite and provides error bounds and backward error estimates for the solutions. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br In the following comments, sub( A ), sub( X ) and sub( B ) denote respectively A(IA:IA+N-1,JA:JA+N-1), X(IX:IX+N-1,JX:JX+NRHS-1) and B(IB:IB+N-1,JB:JB+NRHS-1). .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix sub( A ) is stored. = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The order of the matrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the matrices sub( B ) and sub( X ). NRHS >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1) ). This array contains the local pieces of the N-by-N symmetric distributed matrix sub( A ) to be factored. If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the distribu- ted matrix, and its strictly upper triangular part is not referenced. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 AF (local input) REAL pointer into the local memory to an array of local dimension (LLD_AF,LOCc(JA+N-1)). On entry, this array contains the factors L or U from the Cholesky factorization sub( A ) = L*L**T or U**T*U, as computed by PSPOTRF. .TP 8 IAF (global input) INTEGER The row index in the global array AF indicating the first row of sub( AF ). .TP 8 JAF (global input) INTEGER The column index in the global array AF indicating the first column of sub( AF ). .TP 8 DESCAF (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix AF. .TP 8 B (local input) REAL pointer into the local memory to an array of local dimension (LLD_B, LOCc(JB+NRHS-1) ). On entry, this array contains the the local pieces of the right hand sides sub( B ). .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 X (local input) REAL pointer into the local memory to an array of local dimension (LLD_X, LOCc(JX+NRHS-1) ). On entry, this array contains the the local pieces of the solution vectors sub( X ). On exit, it contains the improved solution vectors. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 FERR (local output) REAL array of local dimension LOCc(JB+NRHS-1). The estimated forward error bound for each solution vector of sub( X ). If XTRUE is the true solution corresponding to sub( X ), FERR is an estimated upper bound for the magnitude of the largest element in (sub( X ) - XTRUE) divided by the magnitude of the largest element in sub( X ). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. This array is tied to the distributed matrix X. .TP 8 BERR (local output) REAL array of local dimension LOCc(JB+NRHS-1). The componentwise relative backward error of each solution vector (i.e., the smallest re- lative change in any entry of sub( A ) or sub( B ) that makes sub( X ) an exact solution). This array is tied to the distributed matrix X. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 3*LOCr( N + MOD( IA-1, MB_A ) ) If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/local output) INTEGER array, dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK. .TP 8 LIWORK (local or global input) INTEGER The dimension of the array IWORK. LIWORK is local input and must be at least LIWORK >= LOCr( N + MOD( IB-1, MB_B ) ). If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH PARAMETERS ITMAX is the maximum number of steps of iterative refinement. Notes ===== This routine temporarily returns when N <= 1. The distributed submatrices op( A ) and op( AF ) (respectively sub( X ) and sub( B ) ) should be distributed the same way on the same processes. These conditions ensure that sub( A ) and sub( AF ) (resp. sub( X ) and sub( B ) ) are "perfectly" aligned. Moreover, this routine requires the distributed submatrices sub( A ), sub( AF ), sub( X ), and sub( B ) to be aligned on a block boundary, i.e., if f(x,y) = MOD( x-1, y ): f( IA, DESCA( MB_ ) ) = f( JA, DESCA( NB_ ) ) = 0, f( IAF, DESCAF( MB_ ) ) = f( JAF, DESCAF( NB_ ) ) = 0, f( IB, DESCB( MB_ ) ) = f( JB, DESCB( NB_ ) ) = 0, and f( IX, DESCX( MB_ ) ) = f( JX, DESCX( NB_ ) ) = 0. scalapack-doc-1.5/man/manl/psposv.l0100644000056400000620000001456206335610651016763 0ustar pfrauenfstaff.TH PSPOSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSPOSV - compute the solution to a real system of linear equations sub( A ) * X = sub( B ), .SH SYNOPSIS .TP 19 SUBROUTINE PSPOSV( UPLO, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, INFO ) .TP 19 .ti +4 CHARACTER UPLO .TP 19 .ti +4 INTEGER IA, IB, INFO, JA, JB, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 REAL A( * ), B( * ) .SH PURPOSE PSPOSV computes the solution to a real system of linear equations where sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1) and is an N-by-N symmetric distributed positive definite matrix and X and sub( B ) denoting B(IB:IB+N-1,JB:JB+NRHS-1) are N-by-NRHS distributed matrices. .br The Cholesky decomposition is used to factor sub( A ) as .br sub( A ) = U**T * U, if UPLO = 'U', or sub( A ) = L * L**T, if UPLO = 'L', .br where U is an upper triangular matrix and L is a lower triangular matrix. The factored form of sub( A ) is then used to solve the system of equations. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block decomposition ( MB_A = NB_A ). .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER = 'U': Upper triangle of sub( A ) is stored; .br = 'L': Lower triangle of sub( A ) is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed submatrix sub( B ). NRHS >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N symmetric distributed matrix sub( A ) to be factored. If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the distribu- ted matrix, and its strictly upper triangular part is not referenced. On exit, if INFO = 0, this array contains the local pieces of the factor U or L from the Cholesky factori- zation sub( A ) = U**T*U or L*L**T. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_B,LOC(JB+NRHS-1)). On entry, the local pieces of the right hand sides distribu- ted matrix sub( B ). On exit, if INFO = 0, sub( B ) is over- written with the solution distributed matrix X. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, the leading minor of order K, .br A(IA:IA+K-1,JA:JA+K-1) is not positive definite, and the factorization could not be completed, and the solution has not been computed. scalapack-doc-1.5/man/manl/psposvx.l0100644000056400000620000003307106335610651017147 0ustar pfrauenfstaff.TH PSPOSVX l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSPOSVX - use the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system of linear equations A(IA:IA+N-1,JA:JA+N-1) * X = B(IB:IB+N-1,JB:JB+NRHS-1), .SH SYNOPSIS .TP 20 SUBROUTINE PSPOSVX( FACT, UPLO, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, EQUED, SR, SC, B, IB, JB, DESCB, X, IX, JX, DESCX, RCOND, FERR, BERR, WORK, LWORK, IWORK, LIWORK, INFO ) .TP 20 .ti +4 CHARACTER EQUED, FACT, UPLO .TP 20 .ti +4 INTEGER IA, IAF, IB, INFO, IX, JA, JAF, JB, JX, LIWORK, LWORK, N, NRHS .TP 20 .ti +4 REAL RCOND .TP 20 .ti +4 INTEGER DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * ), IWORK( * ) .TP 20 .ti +4 REAL A( * ), AF( * ), B( * ), BERR( * ), FERR( * ), SC( * ), SR( * ), WORK( * ), X( * ) .SH PURPOSE PSPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system of linear equations where A(IA:IA+N-1,JA:JA+N-1) is an N-by-N matrix and X and B(IB:IB+N-1,JB:JB+NRHS-1) are N-by-NRHS matrices. .br Error bounds on the solution and a condition estimate are also provided. In the following comments Y denotes Y(IY:IY+M-1,JY:JY+K-1) a M-by-K matrix where Y can be A, AF, B and X. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH DESCRIPTION The following steps are performed: .br 1. If FACT = 'E', real scaling factors are computed to equilibrate the system: .br diag(SR) * A * diag(SC) * inv(diag(SC)) * X = diag(SR) * 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(SR)*A*diag(SC) and B by diag(SR)*B. 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as A = U**T* U, if UPLO = 'U', or .br A = L * L**T, if UPLO = 'L', .br where U is an upper triangular matrix and L is a lower triangular matrix. .br 3. 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, steps 4-6 are skipped. .br 4. The system of equations is solved for X using the factored form of A. .br 5. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. .br 6. If equilibration was used, the matrix X is premultiplied by diag(SR) so that it solves the original system before .br equilibration. .br .SH ARGUMENTS .TP 8 FACT (global input) CHARACTER 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 contains the factored form of A. If EQUED = 'Y', the matrix A has been equilibrated with scaling factors given by S. A and AF will not be modified. = 'N': The matrix A will be copied to AF and factored. .br = 'E': The matrix A will be equilibrated if necessary, then copied to AF and factored. .TP 8 UPLO (global input) CHARACTER = 'U': Upper triangle of A is stored; .br = 'L': Lower triangle of A is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix A(IA:IA+N-1,JA:JA+N-1). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed submatrices B and X. NRHS >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of local dimension ( LLD_A, LOCc(JA+N-1) ). On entry, the symmetric matrix A, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(SR)*A*diag(SC). 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. A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(SR)*A*diag(SC). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 AF (local input or local output) REAL pointer into the local memory to an array of local dimension ( LLD_AF, LOCc(JA+N-1)). If FACT = 'F', then AF is an input argument and on entry contains the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, in the same storage format as A. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix diag(SR)*A*diag(SC). If FACT = 'N', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the original matrix A. If FACT = 'E', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix). .TP 8 IAF (global input) INTEGER The row index in the global array AF indicating the first row of sub( AF ). .TP 8 JAF (global input) INTEGER The column index in the global array AF indicating the first column of sub( AF ). .TP 8 DESCAF (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix AF. .TP 8 EQUED (global input/global output) CHARACTER Specifies the form of equilibration that was done. = 'N': No equilibration (always true if FACT = 'N'). .br = 'Y': Equilibration was done, i.e., A has been replaced by diag(SR) * A * diag(SC). EQUED is an input variable if FACT = 'F'; otherwise, it is an output variable. .TP 8 SR (local input/local output) REAL array, dimension (LLD_A) The scale factors for A distributed across process rows; not accessed if EQUED = 'N'. SR is an input variable if FACT = 'F'; otherwise, SR is an output variable. If FACT = 'F' and EQUED = 'Y', each element of SR must be positive. .TP 8 SC (local input/local output) REAL array, dimension (LOC(N_A)) The scale factors for A distributed across process columns; not accessed if EQUED = 'N'. SC is an input variable if FACT = 'F'; otherwise, SC is an output variable. If FACT = 'F' and EQUED = 'Y', each element of SC must be positive. .TP 8 B (local input/local output) REAL pointer into the local memory to an array of local dimension ( LLD_B, LOCc(JB+NRHS-1) ). On entry, the N-by-NRHS right-hand side matrix B. On exit, if EQUED = 'N', B is not modified; if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B; if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is overwritten by diag(C)*B. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 X (local input/local output) REAL pointer into the local memory to an array of local dimension ( LLD_X, LOCc(JX+NRHS-1) ). If INFO = 0, the N-by-NRHS 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(SC))*X if TRANS = 'N' and EQUED = 'C' or 'B', or inv(diag(SR))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 RCOND (global output) REAL 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, and the solution and error bounds are not computed. .TP 8 FERR (local output) REAL array, dimension (LOC(N_B)) The estimated forward error bounds for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution, FERR(j) bounds the magnitude of the largest entry in (X(j) - XTRUE) divided by the magnitude of the largest entry in X(j). The quality of the error bound depends on the quality of the estimate of norm(inv(A)) computed in the code; if the estimate of norm(inv(A)) is accurate, the error bound is guaranteed. .TP 8 BERR (local output) REAL array, dimension (LOC(N_B)) The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any entry of A or B that makes X(j) an exact solution). .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK = MAX( PSPOCON( LWORK ), PSPORFS( LWORK ) ) + LOCr( N_A ). LWORK = 3*DESCA( LLD_ ) If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/local output) INTEGER array, dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK. .TP 8 LIWORK (local or global input) INTEGER The dimension of the array IWORK. LIWORK is local input and must be at least LIWORK = DESCA( LLD_ ) LIWORK = LOCr(N_A). If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 0: if INFO = i, and i is .br <= N: if INFO = i, the leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution and error bounds could not be computed. = N+1: RCOND is less than machine precision. The factorization has been completed, but the matrix is singular to working precision, and the solution and error bounds have not been computed. scalapack-doc-1.5/man/manl/pspotf2.l0100644000056400000620000001256606335610651017030 0ustar pfrauenfstaff.TH PSPOTF2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSPOTF2 - compute the Cholesky factorization of a real symmetric positive definite distributed matrix sub( A )=A(IA:IA+N-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PSPOTF2( UPLO, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ) .SH PURPOSE PSPOTF2 computes the Cholesky factorization of a real symmetric positive definite distributed matrix sub( A )=A(IA:IA+N-1,JA:JA+N-1). The factorization has the form .br sub( A ) = U' * U , if UPLO = 'U', or .br sub( A ) = L * L', if UPLO = 'L', .br where U is an upper triangular matrix and L is lower triangular. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires N <= NB_A-MOD(JA-1, NB_A) and square block decomposition ( MB_A = NB_A ). .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER = 'U': Upper triangle of sub( A ) is stored; .br = 'L': Lower triangle of sub( A ) is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N symmetric distributed matrix sub( A ) to be factored. If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the distribu- ted matrix, and its strictly upper triangular part is not referenced. On exit, if UPLO = 'U', the upper triangular part of the distributed matrix contains the Cholesky factor U, if UPLO = 'L', the lower triangular part of the distribu- ted matrix contains the Cholesky factor L. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, the leading minor of order K, .br A(IA:IA+K-1,JA:JA+K-1) is not positive definite, and the factorization could not be completed. scalapack-doc-1.5/man/manl/pspotrf.l0100644000056400000620000001257006335610651017123 0ustar pfrauenfstaff.TH PSPOTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSPOTRF - compute the Cholesky factorization of an N-by-N real symmetric positive definite distributed matrix sub( A ) denoting A(IA:IA+N-1, JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PSPOTRF( UPLO, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ) .SH PURPOSE PSPOTRF computes the Cholesky factorization of an N-by-N real symmetric positive definite distributed matrix sub( A ) denoting A(IA:IA+N-1, JA:JA+N-1). The factorization has the form .br sub( A ) = U' * U , if UPLO = 'U', or .br sub( A ) = L * L', if UPLO = 'L', .br where U is an upper triangular matrix and L is lower triangular. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block decomposition ( MB_A = NB_A ). .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER = 'U': Upper triangle of sub( A ) is stored; .br = 'L': Lower triangle of sub( A ) is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N symmetric distributed matrix sub( A ) to be factored. If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the distribu- ted matrix, and its strictly upper triangular part is not referenced. On exit, if UPLO = 'U', the upper triangular part of the distributed matrix contains the Cholesky factor U, if UPLO = 'L', the lower triangular part of the distribu- ted matrix contains the Cholesky factor L. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, the leading minor of order K, .br A(IA:IA+K-1,JA:JA+K-1) is not positive definite, and the factorization could not be completed. scalapack-doc-1.5/man/manl/pspotri.l0100644000056400000620000001144106335610651017122 0ustar pfrauenfstaff.TH PSPOTRI l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSPOTRI - compute the inverse of a real symmetric positive definite distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) using the Cholesky factorization sub( A ) = U**T*U or L*L**T computed by PSPOTRF .SH SYNOPSIS .TP 20 SUBROUTINE PSPOTRI( UPLO, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ) .SH PURPOSE PSPOTRI computes the inverse of a real symmetric positive definite distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) using the Cholesky factorization sub( A ) = U**T*U or L*L**T computed by PSPOTRF. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 = 'U': Upper triangle of sub( A ) is stored; .br = 'L': Lower triangle of sub( A ) is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the triangular factor U or L from the Cholesky factorization of the distributed matrix sub( A ) = U**T*U or L*L**T, as computed by PSPOTRF. On exit, the local pieces of the upper or lower triangle of the (symmetric) inverse of sub( A ), overwriting the input factor U or L. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = i, the (i,i) element of the factor U or L is zero, and the inverse could not be computed. scalapack-doc-1.5/man/manl/pspotrs.l0100644000056400000620000001266206335610651017142 0ustar pfrauenfstaff.TH PSPOTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSPOTRS - solve a system of linear equations sub( A ) * X = sub( B ) A(IA:IA+N-1,JA:JA+N-1)*X = B(IB:IB+N-1,JB:JB+NRHS-1) .SH SYNOPSIS .TP 20 SUBROUTINE PSPOTRS( UPLO, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IB, INFO, JA, JB, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 REAL A( * ), B( * ) .SH PURPOSE PSPOTRS solves a system of linear equations where sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1) and is a N-by-N symmetric positive definite distributed matrix using the Cholesky factorization sub( A ) = U**T*U or L*L**T computed by PSPOTRF. sub( B ) denotes the distributed matrix B(IB:IB+N-1,JB:JB+NRHS-1). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block decomposition ( MB_A = NB_A ). .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER = 'U': Upper triangle of sub( A ) is stored; .br = 'L': Lower triangle of sub( A ) is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed submatrix sub( B ). NRHS >= 0. .TP 8 A (local input) REAL pointer into local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the factors L or U from the Cholesky facto- rization sub( A ) = L*L**T or U**T*U, as computed by PSPOTRF. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input/local output) REAL pointer into the local memory to an array of local dimension (LLD_B,LOCc(JB+NRHS-1)). On entry, this array contains the the local pieces of the right hand sides sub( B ). On exit, this array contains the local pieces of the solution distributed matrix X. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/psptsv.l0100644000056400000620000000133006335610651016755 0ustar pfrauenfstaff.TH PSPTSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSPTSV - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 19 SUBROUTINE PSPTSV( N, NRHS, D, E, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 INTEGER IB, INFO, JA, LWORK, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 REAL B( * ), D( * ), E( * ), WORK( * ) .SH PURPOSE PSPTSV solves a system of linear equations where A(1:N, JA:JA+N-1) is an N-by-N real .br tridiagonal symmetric positive definite distributed .br matrix. .br Cholesky factorization is used to factor a reordering of .br the matrix into L L'. .br See PSPTTRF and PSPTTRS for details. .br scalapack-doc-1.5/man/manl/pspttrf.l0100644000056400000620000000221606335610651017124 0ustar pfrauenfstaff.TH PSPTTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSPTTRF - compute a Cholesky factorization of an N-by-N real tridiagonal symmetric positive definite distributed matrix A(1:N, JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PSPTTRF( N, D, E, JA, DESCA, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER INFO, JA, LAF, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL AF( * ), D( * ), E( * ), WORK( * ) .SH PURPOSE PSPTTRF computes a Cholesky factorization of an N-by-N real tridiagonal symmetric positive definite distributed matrix A(1:N, JA:JA+N-1). Reordering is used to increase parallelism in the factorization. This reordering results in factors that are DIFFERENT from those produced by equivalent sequential codes. These factors cannot be used directly by users; however, they can be used in .br subsequent calls to PSPTTRS to solve linear systems. .br The factorization has the form .br P A(1:N, JA:JA+N-1) P^T = U' D U or .br P A(1:N, JA:JA+N-1) P^T = L D L', .br where U is a tridiagonal upper triangular matrix and L is tridiagonal lower triangular, and P is a permutation matrix. .br scalapack-doc-1.5/man/manl/pspttrs.l0100644000056400000620000000141206335610651017136 0ustar pfrauenfstaff.TH PSPTTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSPTTRS - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 20 SUBROUTINE PSPTTRS( N, NRHS, D, E, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IB, INFO, JA, LAF, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 REAL AF( * ), B( * ), D( * ), E( * ), WORK( * ) .SH PURPOSE PSPTTRS solves a system of linear equations where A(1:N, JA:JA+N-1) is the matrix used to produce the factors stored in A(1:N,JA:JA+N-1) and AF by PSPTTRF. .br A(1:N, JA:JA+N-1) is an N-by-N real .br tridiagonal symmetric positive definite distributed .br matrix. .br Routine PSPTTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pspttrsv.l0100644000056400000620000000216506335610652017333 0ustar pfrauenfstaff.TH PSPTTRSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSPTTRSV - solve a tridiagonal triangular system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 21 SUBROUTINE PSPTTRSV( UPLO, N, NRHS, D, E, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 21 .ti +4 CHARACTER UPLO .TP 21 .ti +4 INTEGER IB, INFO, JA, LAF, LWORK, N, NRHS .TP 21 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 21 .ti +4 REAL AF( * ), B( * ), D( * ), E( * ), WORK( * ) .SH PURPOSE PSPTTRSV solves a tridiagonal triangular system of linear equations or .br A(1:N, JA:JA+N-1)^T * X = B(IB:IB+N-1, 1:NRHS) where A(1:N, JA:JA+N-1) is a tridiagonal .br triangular matrix factor produced by the .br Cholesky factorization code PSPTTRF .br and is stored in A(1:N,JA:JA+N-1) and AF. .br The matrix stored in A(1:N, JA:JA+N-1) is either .br upper or lower triangular according to UPLO, .br and the choice of solving A(1:N, JA:JA+N-1) or A(1:N, JA:JA+N-1)^T is dictated by the user by the parameter TRANS. .br Routine PSPTTRF MUST be called first. .br scalapack-doc-1.5/man/manl/psrscl.l0100644000056400000620000001107206335610652016731 0ustar pfrauenfstaff.TH PSRSCL l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSRSCL - multiplie an N-element real distributed vector sub( X ) by the real scalar 1/a .SH SYNOPSIS .TP 19 SUBROUTINE PSRSCL( N, SA, SX, IX, JX, DESCX, INCX ) .TP 19 .ti +4 INTEGER IX, INCX, JX, N .TP 19 .ti +4 REAL SA .TP 19 .ti +4 INTEGER DESCX( * ) .TP 19 .ti +4 REAL SX( * ) .SH PURPOSE PSRSCL multiplies an N-element real distributed vector sub( X ) by the real scalar 1/a. This is done without overflow or underflow as long as the final result sub( X )/a does not overflow or underflow. where sub( X ) denotes X(IX:IX+N-1,JX:JX), if INCX = 1, .br X(IX:IX,JX:JX+N-1), if INCX = M_X. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector descA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DT_A (global) descA[ DT_ ] The descriptor type. In this case, DT_A = 1. .br CTXT_A (global) descA[ CTXT_ ] The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) descA[ M_ ] The number of rows in the global array A. .br N_A (global) descA[ N_ ] The number of columns in the global array A. .br MB_A (global) descA[ MB_ ] The blocking factor used to distribu- te the rows of the array. .br NB_A (global) descA[ NB_ ] The blocking factor used to distribu- te the columns of the array. RSRC_A (global) descA[ RSRC_ ] The process row over which the first row of the array A is distributed. CSRC_A (global) descA[ CSRC_ ] The process column over which the first column of the array A is distributed. .br LLD_A (local) descA[ LLD_ ] The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be seen as particular matrices, a distributed vector is considered to be a distributed matrix. .br .SH ARGUMENTS .TP 8 N (global input) pointer to INTEGER The number of components of the distributed vector sub( X ). N >= 0. .TP 8 SA (global input) REAL The scalar a which is used to divide each component of sub( X ). SA must be >= 0, or the subroutine will divide by zero. .TP 8 SX (local input/local output) REAL array containing the local pieces of a distributed matrix of dimension of at least ( (JX-1)*M_X + IX + ( N - 1 )*abs( INCX ) ) This array contains the entries of the distributed vector sub( X ). .TP 8 IX (global input) pointer to INTEGER The global row index of the submatrix of the distributed matrix X to operate on. .TP 8 JX (global input) pointer to INTEGER The global column index of the submatrix of the distributed matrix X to operate on. .TP 8 DESCX (global and local input) INTEGER array of dimension 8. The array descriptor of the distributed matrix X. .TP 8 INCX (global input) pointer to INTEGER The global increment for the elements of X. Only two values of INCX are supported in this version, namely 1 and M_X. scalapack-doc-1.5/man/manl/psstebz.l0100644000056400000620000001763706335610652017132 0ustar pfrauenfstaff.TH PSSTEBZ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSSTEBZ - compute the eigenvalues of a symmetric tridiagonal matrix in parallel .SH SYNOPSIS .TP 20 SUBROUTINE PSSTEBZ( ICTXT, RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E, M, NSPLIT, W, IBLOCK, ISPLIT, WORK, LWORK, IWORK, LIWORK, INFO ) .TP 20 .ti +4 CHARACTER ORDER, RANGE .TP 20 .ti +4 INTEGER ICTXT, IL, INFO, IU, LIWORK, LWORK, M, N, NSPLIT .TP 20 .ti +4 REAL ABSTOL, VL, VU .TP 20 .ti +4 INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * ) .TP 20 .ti +4 REAL D( * ), E( * ), W( * ), WORK( * ) .SH PURPOSE PSSTEBZ computes the eigenvalues of a symmetric tridiagonal matrix in parallel. The user may ask for all eigenvalues, all eigenvalues in the interval [VL, VU], or the eigenvalues indexed IL through IU. A static partitioning of work is done at the beginning of PSSTEBZ which results in all processes finding an (almost) equal number of eigenvalues. .br NOTE : It is assumed that the user is on an IEEE machine. If the user is not on an IEEE mchine, set the compile time flag NO_IEEE to 1 (in SLmake.inc). The features of IEEE arithmetic that are needed for the "fast" Sturm Count are : (a) infinity arithmetic (b) the sign bit of a double precision floating point number is assumed be in the 32nd or 64th bit position (c) the sign of negative zero. .br See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix", Report CS41, Computer Science Dept., Stanford .br University, July 21, 1966. .br .SH ARGUMENTS .TP 8 ICTXT (global input) INTEGER The BLACS context handle. .TP 8 RANGE (global input) CHARACTER Specifies which eigenvalues are to be found. = 'A': ("All") all eigenvalues will be found. .br = 'V': ("Value") all eigenvalues in the interval [VL, VU] will be found. = 'I': ("Index") the IL-th through IU-th eigenvalues (of the entire matrix) will be found. .TP 8 ORDER (global input) CHARACTER Specifies the order in which the eigenvalues and their block numbers are stored in W and IBLOCK. = 'B': ("By Block") the eigenvalues will be grouped by split-off block (see IBLOCK, ISPLIT) and ordered from smallest to largest within the block. = 'E': ("Entire matrix") the eigenvalues for the entire matrix will be ordered from smallest to largest. .TP 8 N (global input) INTEGER The order of the tridiagonal matrix T. N >= 0. .TP 8 VL (global input) REAL If RANGE='V', the lower bound of the interval to be searched for eigenvalues. Eigenvalues less than VL will not be returned. Not referenced if RANGE='A' or 'I'. .TP 8 VU (global input) REAL If RANGE='V', the upper bound of the interval to be searched for eigenvalues. Eigenvalues greater than VU will not be returned. VU must be greater than VL. Not referenced if RANGE='A' or 'I'. .TP 8 IL (global input) INTEGER If RANGE='I', the index (from smallest to largest) of the smallest eigenvalue to be returned. IL must be at least 1. Not referenced if RANGE='A' or 'V'. .TP 8 IU (global input) INTEGER If RANGE='I', the index (from smallest to largest) of the largest eigenvalue to be returned. IU must be at least IL and no greater than N. Not referenced if RANGE='A' or 'V'. .TP 8 ABSTOL (global input) REAL The absolute tolerance for the eigenvalues. An eigenvalue (or cluster) is considered to be located if it has been determined to lie in an interval whose width is ABSTOL or less. If ABSTOL is less than or equal to zero, then ULP*|T| will be used, where |T| means the 1-norm of T. Eigenvalues will be computed most accurately when ABSTOL is set to the underflow threshold SLAMCH('U'), not zero. Note : If eigenvectors are desired later by inverse iteration ( PSSTEIN ), ABSTOL should be set to 2*PSLAMCH('S'). .TP 8 D (global input) REAL array, dimension (N) The n diagonal elements of the tridiagonal matrix T. To avoid overflow, the matrix must be scaled so that its largest entry is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. .TP 8 E (global input) REAL array, dimension (N-1) The (n-1) off-diagonal elements of the tridiagonal matrix T. To avoid overflow, the matrix must be scaled so that its largest entry is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. .TP 8 M (global output) INTEGER The actual number of eigenvalues found. 0 <= M <= N. (See also the description of INFO=2) .TP 8 NSPLIT (global output) INTEGER The number of diagonal blocks in the matrix T. 1 <= NSPLIT <= N. .TP 8 W (global output) REAL array, dimension (N) On exit, the first M elements of W contain the eigenvalues on all processes. .TP 8 IBLOCK (global output) INTEGER array, dimension (N) At each row/column j where E(j) is zero or small, the matrix T is considered to split into a block diagonal matrix. On exit IBLOCK(i) specifies which block (from 1 to the number of blocks) the eigenvalue W(i) belongs to. NOTE: in the (theoretically impossible) event that bisection does not converge for some or all eigenvalues, INFO is set to 1 and the ones for which it did not are identified by a negative block number. .TP 8 ISPLIT (global output) INTEGER array, dimension (N) The splitting points, at which T breaks up into submatrices. The first submatrix consists of rows/columns 1 to ISPLIT(1), the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N. (Only the first NSPLIT elements will actually be used, but since the user cannot know a priori what value NSPLIT will have, N words must be reserved for ISPLIT.) .TP 8 WORK (local workspace) REAL array, dimension ( MAX( 5*N, 7 ) ) .TP 8 LWORK (local input) INTEGER size of array WORK must be >= MAX( 5*N, 7 ) If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace) INTEGER array, dimension ( MAX( 4*N, 14 ) ) .TP 8 LIWORK (local input) INTEGER size of array IWORK must be >= MAX( 4*N, 14, NPROCS ) If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0 : successful exit .br < 0 : if INFO = -i, the i-th argument had an illegal value .br > 0 : some or all of the eigenvalues failed to converge or .br were not computed: .br = 1 : Bisection failed to converge for some eigenvalues; these eigenvalues are flagged by a negative block number. The effect is that the eigenvalues may not be as accurate as the absolute and relative tolerances. This is generally caused by arithmetic which is less accurate than PSLAMCH says. = 2 : There is a mismatch between the number of eigenvalues output and the number desired. = 3 : RANGE='i', and the Gershgorin interval initially used was incorrect. No eigenvalues were computed. Probable cause: your machine has sloppy floating point arithmetic. Cure: Increase the PARAMETER "FUDGE", recompile, and try again. .SH PARAMETERS .TP 8 RELFAC REAL, default = 2.0 The relative tolerance. An interval [a,b] lies within "relative tolerance" if b-a < RELFAC*ulp*max(|a|,|b|), where "ulp" is the machine precision (distance from 1 to the next larger floating point number.) .TP 8 FUDGE REAL, default = 2.0 A "fudge factor" to widen the Gershgorin intervals. Ideally, a value of 1 should work, but on machines with sloppy arithmetic, this needs to be larger. The default for publicly released versions should be large enough to handle the worst machine around. Note that this has no effect on the accuracy of the solution. scalapack-doc-1.5/man/manl/psstein.l0100644000056400000620000002560606335610652017120 0ustar pfrauenfstaff.TH PSSTEIN l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSSTEIN - compute the eigenvectors of a symmetric tridiagonal matrix in parallel, using inverse iteration .SH SYNOPSIS .TP 20 SUBROUTINE PSSTEIN( N, D, E, M, W, IBLOCK, ISPLIT, ORFAC, Z, IZ, JZ, DESCZ, WORK, LWORK, IWORK, LIWORK, IFAIL, ICLUSTR, GAP, INFO ) .TP 20 .ti +4 INTEGER INFO, IZ, JZ, LIWORK, LWORK, M, N .TP 20 .ti +4 REAL ORFAC .TP 20 .ti +4 INTEGER DESCZ( * ), IBLOCK( * ), ICLUSTR( * ), IFAIL( * ), ISPLIT( * ), IWORK( * ) .TP 20 .ti +4 REAL D( * ), E( * ), GAP( * ), W( * ), WORK( * ), Z( * ) .SH PURPOSE PSSTEIN computes the eigenvectors of a symmetric tridiagonal matrix in parallel, using inverse iteration. The eigenvectors found correspond to user specified eigenvalues. PSSTEIN does not orthogonalize vectors that are on different processes. The extent of orthogonalization is controlled by the input parameter LWORK. Eigenvectors that are to be orthogonalized are computed by the same process. PSSTEIN decides on the allocation of work among the processes and then calls SSTEIN2 (modified LAPACK routine) on each individual process. If insufficient workspace is allocated, the expected orthogonalization may not be done. .br Note : If the eigenvectors obtained are not orthogonal, increase LWORK and run the code again. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS P = NPROW * NPCOL is the total number of processes .TP 8 N (global input) INTEGER The order of the tridiagonal matrix T. N >= 0. .TP 8 D (global input) REAL array, dimension (N) The n diagonal elements of the tridiagonal matrix T. .TP 8 E (global input) REAL array, dimension (N-1) The (n-1) off-diagonal elements of the tridiagonal matrix T. .TP 8 M (global input) INTEGER The total number of eigenvectors to be found. 0 <= M <= N. .TP 8 W (global input/global output) REAL array, dim (M) On input, the first M elements of W contain all the eigenvalues for which eigenvectors are to be computed. The eigenvalues should be grouped by split-off block and ordered from smallest to largest within the block (The output array W from PSSTEBZ with ORDER='b' is expected here). This array should be replicated on all processes. On output, the first M elements contain the input eigenvalues in ascending order. Note : To obtain orthogonal vectors, it is best if eigenvalues are computed to highest accuracy ( this can be done by setting ABSTOL to the underflow threshold = SLAMCH('U') --- ABSTOL is an input parameter to PSSTEBZ ) .TP 8 IBLOCK (global input) INTEGER array, dimension (N) The submatrix indices associated with the corresponding eigenvalues in W -- 1 for eigenvalues belonging to the first submatrix from the top, 2 for those belonging to the second submatrix, etc. (The output array IBLOCK from PSSTEBZ is expected here). .TP 8 ISPLIT (global input) INTEGER array, dimension (N) The splitting points, at which T breaks up into submatrices. The first submatrix consists of rows/columns 1 to ISPLIT(1), the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N (The output array ISPLIT from PSSTEBZ is expected here.) .TP 8 ORFAC (global input) REAL ORFAC specifies which eigenvectors should be orthogonalized. Eigenvectors that correspond to eigenvalues which are within ORFAC*||T|| of each other are to be orthogonalized. However, if the workspace is insufficient (see LWORK), this tolerance may be decreased until all eigenvectors to be orthogonalized can be stored in one process. No orthogonalization will be done if ORFAC equals zero. A default value of 10^-3 is used if ORFAC is negative. ORFAC should be identical on all processes. .TP 8 Z (local output) REAL array, dimension (DESCZ(DLEN_), N/npcol + NB) Z contains the computed eigenvectors associated with the specified eigenvalues. Any vector which fails to converge is set to its current iterate after MAXITS iterations ( See SSTEIN2 ). On output, Z is distributed across the P processes in block cyclic format. .TP 8 IZ (global input) INTEGER Z's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JZ (global input) INTEGER Z's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCZ (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix Z. .TP 8 WORK (local workspace/global output) REAL array, dimension ( LWORK ) On output, WORK(1) gives a lower bound on the workspace ( LWORK ) that guarantees the user desired orthogonalization (see ORFAC). Note that this may overestimate the minimum workspace needed. .TP 8 LWORK (local input) integer LWORK controls the extent of orthogonalization which can be done. The number of eigenvectors for which storage is allocated on each process is NVEC = floor(( LWORK- max(5*N,NP00*MQ00) )/N). Eigenvectors corresponding to eigenvalue clusters of size NVEC - ceil(M/P) + 1 are guaranteed to be orthogonal ( the orthogonality is similar to that obtained from SSTEIN2). Note : LWORK must be no smaller than: max(5*N,NP00*MQ00) + ceil(M/P)*N, and should have the same input value on all processes. It is the minimum value of LWORK input on different processes that is significant. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/global output) INTEGER array, dimension ( 3*N+P+1 ) On return, IWORK(1) contains the amount of integer workspace required. On return, the IWORK(2) through IWORK(P+2) indicate the eigenvectors computed by each process. Process I computes eigenvectors indexed IWORK(I+2)+1 thru' IWORK(I+3). .TP 8 LIWORK (local input) INTEGER Size of array IWORK. Must be >= 3*N + P + 1 If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IFAIL (global output) integer array, dimension (M) On normal exit, all elements of IFAIL are zero. If one or more eigenvectors fail to converge after MAXITS iterations (as in SSTEIN), then INFO > 0 is returned. If mod(INFO,M+1)>0, then for I=1 to mod(INFO,M+1), the eigenvector corresponding to the eigenvalue W(IFAIL(I)) failed to converge ( W refers to the array of eigenvalues on output ). ICLUSTR (global output) integer array, dimension (2*P) This output array contains indices of eigenvectors corresponding to a cluster of eigenvalues that could not be orthogonalized due to insufficient workspace (see LWORK, ORFAC and INFO). Eigenvectors corresponding to clusters of eigenvalues indexed ICLUSTR(2*I-1) to ICLUSTR(2*I), I = 1 to INFO/(M+1), could not be orthogonalized due to lack of workspace. Hence the eigenvectors corresponding to these clusters may not be orthogonal. ICLUSTR is a zero terminated array --- ( ICLUSTR(2*K).NE.0 .AND. ICLUSTR(2*K+1).EQ.0 ) if and only if K is the number of clusters. .TP 8 GAP (global output) REAL array, dimension (P) This output array contains the gap between eigenvalues whose eigenvectors could not be orthogonalized. The INFO/M output values in this array correspond to the INFO/(M+1) clusters indicated by the array ICLUSTR. As a result, the dot product between eigenvectors corresponding to the I^th cluster may be as high as ( O(n)*macheps ) / GAP(I). .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. < 0 : if INFO = -I, the I-th argument had an illegal value .br > 0 : if mod(INFO,M+1) = I, then I eigenvectors failed to converge in MAXITS iterations. Their indices are stored in the array IFAIL. if INFO/(M+1) = I, then eigenvectors corresponding to I clusters of eigenvalues could not be orthogonalized due to insufficient workspace. The indices of the clusters are stored in the array ICLUSTR. scalapack-doc-1.5/man/manl/pssyev.l0100644000056400000620000002313406335610652016756 0ustar pfrauenfstaff.TH PSSYEV l "12 May 1997" "LAPACK version 1.3" "LAPACK routine (version 1.3)" .SH NAME .SH SYNOPSIS .TP 19 SUBROUTINE PSSYEV( JOBZ, UPLO, N, A, IA, JA, DESCA, W, Z, IZ, JZ, DESCZ, WORK, LWORK, INFO ) .TP 19 .ti +4 CHARACTER JOBZ, UPLO .TP 19 .ti +4 INTEGER IA, INFO, IZ, JA, JZ, LWORK, N .TP 19 .ti +4 INTEGER DESCA( * ), DESCZ( * ) .TP 19 .ti +4 REAL A( * ), W( * ), WORK( * ), Z( * ) .TP 19 .ti +4 INTEGER BLOCK_CYCLIC_2D, DLEN_, DTYPE_, CTXT_, M_, N_, MB_, NB_, RSRC_, CSRC_, LLD_ .TP 19 .ti +4 PARAMETER ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1, CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6, RSRC_ = 7, CSRC_ = 8, LLD_ = 9 ) .TP 19 .ti +4 REAL ZERO, ONE .TP 19 .ti +4 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) .TP 19 .ti +4 INTEGER ITHVAL .TP 19 .ti +4 PARAMETER ( ITHVAL = 10 ) .TP 19 .ti +4 LOGICAL LOWER, WANTZ .TP 19 .ti +4 INTEGER CONTEXTC, CSRC_A, I, IACOL, IAROW, ICOFFA, IINFO, INDD, INDD2, INDE, INDE2, INDTAU, INDWORK, INDWORK2, IROFFA, IROFFZ, ISCALE, IZROW, J, K, LCM, LCMQ, LDC, LLWORK, LWMIN, MB_A, MB_Z, MYCOL, MYPCOLC, MYPROWC, MYROW, NB, NB_A, NB_Z, NN, NP, NPCOL, NPCOLC, NPROCS, NPROW, NPROWC, NQ, NRC, QRMEM, RSRC_A, RSRC_Z, SIZEMQRLEFT, SIZEMQRRIGHT .TP 19 .ti +4 REAL ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM .TP 19 .ti +4 INTEGER DESCQR( 10 ), IDUM1( 3 ), IDUM2( 3 ) .TP 19 .ti +4 LOGICAL LSAME .TP 19 .ti +4 INTEGER ILCM, INDXG2P, NUMROC, SL_GRIDRESHAPE .TP 19 .ti +4 REAL PSLAMCH, PSLANSY .TP 19 .ti +4 EXTERNAL LSAME, ILCM, INDXG2P, NUMROC, SL_GRIDRESHAPE, PSLAMCH, PSLANSY .TP 19 .ti +4 EXTERNAL BLACS_GRIDEXIT, BLACS_GRIDINFO, CHK1MAT, DESCINIT, PCHK2MAT, PSELGET, PSGEMR2D, PSLASCL, PSLASET, PSORMTR, PSSYTRD, PXERBLA, SCOPY, SGAMN2D, SGAMX2D, SSCAL, SSTEQR2 .TP 19 .ti +4 INTRINSIC ICHAR, MAX, MIN, MOD, REAL, SQRT .TP 19 .ti +4 IF( BLOCK_CYCLIC_2D*CSRC_*CTXT_*DLEN_*DTYPE_*LLD_*MB_*M_*NB_*N_* RSRC_.LT.0 )RETURN .TP 19 .ti +4 IF( N.EQ.0 ) RETURN .TP 19 .ti +4 CALL BLACS_GRIDINFO( DESCA( CTXT_ ), NPROW, NPCOL, MYROW, MYCOL ) .TP 19 .ti +4 INFO = 0 .TP 19 .ti +4 IF( NPROW.EQ.-1 ) THEN .TP 19 .ti +4 INFO = -( 700+CTXT_ ) .TP 19 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCZ( CTXT_ ) ) THEN .TP 19 .ti +4 INFO = -( 1200+CTXT_ ) .TP 19 .ti +4 ELSE .TP 19 .ti +4 CALL CHK1MAT( N, 3, N, 3, IA, JA, DESCA, 7, INFO ) .TP 19 .ti +4 CALL CHK1MAT( N, 3, N, 3, IZ, JZ, DESCZ, 12, INFO ) .TP 19 .ti +4 IF( INFO.EQ.0 ) THEN .TP 19 .ti +4 SAFMIN = PSLAMCH( DESCA( CTXT_ ), 'Safe minimum' ) .TP 19 .ti +4 EPS = PSLAMCH( DESCA( CTXT_ ), 'Precision' ) .TP 19 .ti +4 SMLNUM = SAFMIN / EPS .TP 19 .ti +4 BIGNUM = ONE / SMLNUM .TP 19 .ti +4 RMIN = SQRT( SMLNUM ) .TP 19 .ti +4 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) .TP 19 .ti +4 NPROCS = NPROW*NPCOL .TP 19 .ti +4 NB_A = DESCA( NB_ ) .TP 19 .ti +4 MB_A = DESCA( MB_ ) .TP 19 .ti +4 NB_Z = DESCZ( NB_ ) .TP 19 .ti +4 MB_Z = DESCZ( MB_ ) .TP 19 .ti +4 NB = NB_A .TP 19 .ti +4 LOWER = LSAME( UPLO, 'L' ) .TP 19 .ti +4 WANTZ = LSAME( JOBZ, 'V' ) .TP 19 .ti +4 RSRC_A = DESCA( RSRC_ ) .TP 19 .ti +4 CSRC_A = DESCA( CSRC_ ) .TP 19 .ti +4 RSRC_Z = DESCZ( RSRC_ ) .TP 19 .ti +4 LCM = ILCM( NPROW, NPCOL ) .TP 19 .ti +4 LCMQ = LCM / NPCOL .TP 19 .ti +4 IROFFA = MOD( IA-1, MB_A ) .TP 19 .ti +4 ICOFFA = MOD( JA-1, NB_A ) .TP 19 .ti +4 IROFFZ = MOD( IZ-1, MB_A ) .TP 19 .ti +4 IAROW = INDXG2P( 1, NB_A, MYROW, RSRC_A, NPROW ) .TP 19 .ti +4 IACOL = INDXG2P( 1, MB_A, MYCOL, CSRC_A, NPCOL ) .TP 19 .ti +4 IZROW = INDXG2P( 1, NB_A, MYROW, RSRC_Z, NPROW ) .TP 19 .ti +4 NP = NUMROC( N+IROFFA, NB_Z, MYROW, IAROW, NPROW ) .TP 19 .ti +4 NQ = NUMROC( N+ICOFFA, NB_Z, MYCOL, IACOL, NPCOL ) .TP 19 .ti +4 SIZEMQRLEFT = MAX( ( NB_A*( NB_A-1 ) ) / 2, ( NP+NQ )*NB_A ) + NB_A*NB_A .TP 19 .ti +4 SIZEMQRRIGHT = MAX( ( NB_A*( NB_A-1 ) ) / 2, ( NQ+MAX( NP+NUMROC( NUMROC( N+ICOFFA, NB_A, 0, 0, NPCOL ), NB, 0, 0, LCMQ ), NP ) )* NB_A ) + NB_A*NB_A .TP 19 .ti +4 LDC = 0 .TP 19 .ti +4 IF( WANTZ ) THEN .TP 19 .ti +4 CONTEXTC = SL_GRIDRESHAPE( DESCA( CTXT_ ), 0, 1, 1, NPROCS, 1 ) .TP 19 .ti +4 CALL BLACS_GRIDINFO( CONTEXTC, NPROWC, NPCOLC, MYPROWC, MYPCOLC ) .TP 19 .ti +4 NRC = NUMROC( N, NB_A, MYPROWC, 0, NPROCS ) .TP 19 .ti +4 LDC = MAX( 1, NRC ) .TP 19 .ti +4 CALL DESCINIT( DESCQR, N, N, NB, NB, 0, 0, CONTEXTC, LDC, INFO ) .TP 19 .ti +4 END IF .TP 19 .ti +4 INDTAU = 1 .TP 19 .ti +4 INDE = INDTAU + N .TP 19 .ti +4 INDD = INDE + N .TP 19 .ti +4 INDD2 = INDD + N .TP 19 .ti +4 INDE2 = INDD2 + N .TP 19 .ti +4 INDWORK = INDE2 + N .TP 19 .ti +4 INDWORK2 = INDWORK + N*LDC .TP 19 .ti +4 LLWORK = LWORK - INDWORK + 1 .TP 19 .ti +4 NN = MAX( N, NB, 2 ) .TP 19 .ti +4 IF( WANTZ ) THEN .TP 19 .ti +4 QRMEM = 5*N + MAX( 2*NP+NQ+NB*NN, 2*NN-2 ) + N*LDC .TP 19 .ti +4 LWMIN = MAX( SIZEMQRLEFT, SIZEMQRRIGHT, QRMEM ) .TP 19 .ti +4 ELSE .TP 19 .ti +4 LWMIN = 5*N + 2*NP + NQ + NB*NN .TP 19 .ti +4 END IF .TP 19 .ti +4 END IF .TP 19 .ti +4 IF( INFO.EQ.0 ) THEN .TP 19 .ti +4 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN .TP 19 .ti +4 INFO = -1 .TP 19 .ti +4 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN .TP 19 .ti +4 INFO = -2 .TP 19 .ti +4 ELSE IF( LWORK.LT.LWMIN .AND. LWORK.NE.-1 ) THEN .TP 19 .ti +4 INFO = -14 .TP 19 .ti +4 ELSE IF( IROFFA.NE.IROFFZ ) THEN .TP 19 .ti +4 INFO = -10 .TP 19 .ti +4 ELSE IF( IROFFA.NE.0 ) THEN .TP 19 .ti +4 INFO = -5 .TP 19 .ti +4 ELSE IF( IAROW.NE.IZROW ) THEN .TP 19 .ti +4 INFO = -10 .TP 19 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCA( NB_ ) ) THEN .TP 19 .ti +4 INFO = -( 700+NB_ ) .TP 19 .ti +4 ELSE IF( DESCA( M_ ).NE.DESCZ( M_ ) ) THEN .TP 19 .ti +4 INFO = -( 1200+M_ ) .TP 19 .ti +4 ELSE IF( DESCA( N_ ).NE.DESCZ( N_ ) ) THEN .TP 19 .ti +4 INFO = -( 1200+N_ ) .TP 19 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCZ( MB_ ) ) THEN .TP 19 .ti +4 INFO = -( 1200+MB_ ) .TP 19 .ti +4 ELSE IF( DESCA( NB_ ).NE.DESCZ( NB_ ) ) THEN .TP 19 .ti +4 INFO = -( 1200+NB_ ) .TP 19 .ti +4 ELSE IF( DESCA( RSRC_ ).NE.DESCZ( RSRC_ ) ) THEN .TP 19 .ti +4 INFO = -( 1200+RSRC_ ) .TP 19 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCZ( CTXT_ ) ) THEN .TP 19 .ti +4 INFO = -( 1200+CTXT_ ) .TP 19 .ti +4 END IF .TP 19 .ti +4 END IF .TP 19 .ti +4 IF( WANTZ ) THEN .TP 19 .ti +4 IDUM1( 1 ) = ICHAR( 'V' ) .TP 19 .ti +4 ELSE .TP 19 .ti +4 IDUM1( 1 ) = ICHAR( 'N' ) .TP 19 .ti +4 END IF .TP 19 .ti +4 IDUM2( 1 ) = 1 .TP 19 .ti +4 IF( LOWER ) THEN .TP 19 .ti +4 IDUM1( 2 ) = ICHAR( 'L' ) .TP 19 .ti +4 ELSE .TP 19 .ti +4 IDUM1( 2 ) = ICHAR( 'U' ) .TP 19 .ti +4 END IF .TP 19 .ti +4 IDUM2( 2 ) = 2 .TP 19 .ti +4 IF( LWORK.EQ.-1 ) THEN .TP 19 .ti +4 IDUM1( 3 ) = -1 .TP 19 .ti +4 ELSE .TP 19 .ti +4 IDUM1( 3 ) = 1 .TP 19 .ti +4 END IF .TP 19 .ti +4 IDUM2( 3 ) = 3 .TP 19 .ti +4 CALL PCHK2MAT( N, 3, N, 3, IA, JA, DESCA, 7, N, 3, N, 3, IZ, JZ, DESCZ, 12, 3, IDUM1, IDUM2, INFO ) .TP 19 .ti +4 WORK( 1 ) = REAL( LWMIN ) .TP 19 .ti +4 END IF .TP 19 .ti +4 IF( INFO.NE.0 ) THEN .TP 19 .ti +4 CALL PXERBLA( DESCA( CTXT_ ), 'PSSYEV', -INFO ) .TP 19 .ti +4 IF( WANTZ ) CALL BLACS_GRIDEXIT( CONTEXTC ) .TP 19 .ti +4 RETURN .TP 19 .ti +4 ELSE IF( LWORK.EQ.-1 ) THEN .TP 19 .ti +4 IF( WANTZ ) CALL BLACS_GRIDEXIT( CONTEXTC ) .TP 19 .ti +4 RETURN .TP 19 .ti +4 END IF .TP 19 .ti +4 ISCALE = 0 .TP 19 .ti +4 ANRM = PSLANSY( '1', UPLO, N, A, IA, JA, DESCA, WORK( INDWORK ) ) .TP 19 .ti +4 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN .TP 19 .ti +4 ISCALE = 1 .TP 19 .ti +4 SIGMA = RMIN / ANRM .TP 19 .ti +4 ELSE IF( ANRM.GT.RMAX ) THEN .TP 19 .ti +4 ISCALE = 1 .TP 19 .ti +4 SIGMA = RMAX / ANRM .TP 19 .ti +4 END IF .TP 19 .ti +4 IF( ISCALE.EQ.1 ) THEN .TP 19 .ti +4 CALL PSLASCL( UPLO, ONE, SIGMA, N, N, A, IA, JA, DESCA, IINFO ) .TP 19 .ti +4 END IF .TP 19 .ti +4 CALL PSSYTRD( UPLO, N, A, IA, JA, DESCA, WORK( INDD ), WORK( INDE ), WORK( INDTAU ), WORK( INDWORK ), LLWORK, IINFO ) .TP 19 .ti +4 DO 10 I = 1, N .TP 19 .ti +4 CALL PSELGET( 'A', ' ', WORK( INDD2+I-1 ), A, I+IA-1, I+JA-1, DESCA ) .TP 19 .ti +4 10 CONTINUE .TP 19 .ti +4 IF( LSAME( UPLO, 'U' ) ) THEN .TP 19 .ti +4 DO 20 I = 1, N - 1 .TP 19 .ti +4 CALL PSELGET( 'A', ' ', WORK( INDE2+I-1 ), A, I+IA-1, I+JA, DESCA ) .TP 19 .ti +4 20 CONTINUE .TP 19 .ti +4 ELSE .TP 19 .ti +4 DO 30 I = 1, N - 1 .TP 19 .ti +4 CALL PSELGET( 'A', ' ', WORK( INDE2+I-1 ), A, I+IA, I+JA-1, DESCA ) .TP 19 .ti +4 30 CONTINUE .TP 19 .ti +4 END IF .TP 19 .ti +4 IF( WANTZ ) THEN .TP 19 .ti +4 CALL PSLASET( 'Full', N, N, ZERO, ONE, WORK( INDWORK ), 1, 1, DESCQR ) .TP 19 .ti +4 CALL SSTEQR2( 'I', N, WORK( INDD2 ), WORK( INDE2 ), WORK( INDWORK ), LDC, NRC, WORK( INDWORK2 ), INFO ) .TP 19 .ti +4 CALL PSGEMR2D( N, N, WORK( INDWORK ), 1, 1, DESCQR, Z, 1, 1, DESCZ, CONTEXTC ) .TP 19 .ti +4 CALL PSORMTR( 'L', UPLO, 'N', N, N, A, IA, JA, DESCA, WORK( INDTAU ), Z, IZ, JZ, DESCZ, WORK( INDWORK ), LLWORK, IINFO ) .TP 19 .ti +4 ELSE .TP 19 .ti +4 CALL SSTEQR2( 'N', N, WORK( INDD2 ), WORK( INDE2 ), WORK( INDWORK ), 1, 1, WORK( INDWORK2 ), INFO ) .TP 19 .ti +4 END IF .TP 19 .ti +4 CALL SCOPY( N, WORK( INDD2 ), 1, W, 1 ) .TP 19 .ti +4 IF( ISCALE.EQ.1 ) THEN .TP 19 .ti +4 CALL SSCAL( N, ONE / SIGMA, W, 1 ) .TP 19 .ti +4 END IF .TP 19 .ti +4 WORK( 1 ) = REAL( LWMIN ) .TP 19 .ti +4 IF( WANTZ ) THEN .TP 19 .ti +4 CALL BLACS_GRIDEXIT( CONTEXTC ) .TP 19 .ti +4 END IF .TP 19 .ti +4 IF( N.LE.ITHVAL ) THEN .TP 19 .ti +4 J = N .TP 19 .ti +4 K = 1 .TP 19 .ti +4 ELSE .TP 19 .ti +4 J = N / ITHVAL .TP 19 .ti +4 K = ITHVAL .TP 19 .ti +4 END IF .TP 19 .ti +4 DO 40 I = 1, J .TP 19 .ti +4 WORK( I+INDTAU ) = W( ( I-1 )*K+1 ) .TP 19 .ti +4 WORK( I+INDE ) = W( ( I-1 )*K+1 ) .TP 19 .ti +4 40 CONTINUE .TP 19 .ti +4 CALL SGAMN2D( DESCA( CTXT_ ), 'a', ' ', J, 1, WORK( 1+INDTAU ), J, 1, 1, -1, -1, 0 ) .TP 19 .ti +4 CALL SGAMX2D( DESCA( CTXT_ ), 'a', ' ', J, 1, WORK( 1+INDE ), J, 1, 1, -1, -1, 0 ) .TP 19 .ti +4 DO 50 I = 1, J .TP 19 .ti +4 IF( INFO.EQ.0 .AND. ( WORK( I+INDTAU )-WORK( I+INDE ).NE. ZERO ) ) THEN .TP 19 .ti +4 INFO = N + 1 .TP 19 .ti +4 END IF .TP 19 .ti +4 50 CONTINUE .TP 19 .ti +4 RETURN .TP 19 .ti +4 END .SH PURPOSE scalapack-doc-1.5/man/manl/pssyevx.l0100644000056400000620000003374106335610652017153 0ustar pfrauenfstaff.TH PSSYEVX l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME .SH SYNOPSIS .TP 20 SUBROUTINE PSSYEVX( JOBZ, RANGE, UPLO, N, A, IA, JA, DESCA, VL, VU, IL, IU, ABSTOL, M, NZ, W, ORFAC, Z, IZ, JZ, DESCZ, WORK, LWORK, IWORK, LIWORK, IFAIL, ICLUSTR, GAP, INFO ) .TP 20 .ti +4 CHARACTER JOBZ, RANGE, UPLO .TP 20 .ti +4 INTEGER IA, IL, INFO, IU, IZ, JA, JZ, LIWORK, LWORK, M, N, NZ .TP 20 .ti +4 REAL ABSTOL, ORFAC, VL, VU .TP 20 .ti +4 INTEGER DESCA( * ), DESCZ( * ), ICLUSTR( * ), IFAIL( * ), IWORK( * ) .TP 20 .ti +4 REAL A( * ), GAP( * ), W( * ), WORK( * ), Z( * ) .TP 20 .ti +4 INTEGER BLOCK_CYCLIC_2D, DLEN_, DTYPE_, CTXT_, M_, N_, MB_, NB_, RSRC_, CSRC_, LLD_ .TP 20 .ti +4 PARAMETER ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1, CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6, RSRC_ = 7, CSRC_ = 8, LLD_ = 9 ) .TP 20 .ti +4 REAL ZERO, ONE, TEN, FIVE .TP 20 .ti +4 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TEN = 10.0E+0, FIVE = 5.0E+0 ) .TP 20 .ti +4 INTEGER IERREIN, IERRCLS, IERRSPC, IERREBZ .TP 20 .ti +4 PARAMETER ( IERREIN = 1, IERRCLS = 2, IERRSPC = 4, IERREBZ = 8 ) .TP 20 .ti +4 LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, QUICKRETURN, VALEIG, WANTZ .TP 20 .ti +4 CHARACTER ORDER .TP 20 .ti +4 INTEGER CSRC_A, I, IACOL, IAROW, ICOFFA, IINFO, INDD, INDD2, INDE, INDE2, INDIBL, INDISP, INDTAU, INDWORK, IROFFA, IROFFZ, ISCALE, ISIZESTEBZ, ISIZESTEIN, IZROW, LALLWORK, LIWMIN, LLWORK, LWMIN, MAXEIGS, MB_A, MB_Z, MQ0, MYCOL, MYROW, NB, NB_A, NB_Z, NEIG, NN, NNP, NP0, NPCOL, NPROCS, NPROW, NSPLIT, NZZ, OFFSET, RSRC_A, RSRC_Z, SIZEORMTR, SIZESTEIN, SIZESYEVX .TP 20 .ti +4 REAL ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM, VLL, VUU .TP 20 .ti +4 INTEGER IDUM1( 4 ), IDUM2( 4 ) .TP 20 .ti +4 LOGICAL LSAME .TP 20 .ti +4 INTEGER ICEIL, INDXG2P, NUMROC .TP 20 .ti +4 REAL PSLAMCH, PSLANSY .TP 20 .ti +4 EXTERNAL LSAME, ICEIL, INDXG2P, NUMROC, PSLAMCH, PSLANSY .TP 20 .ti +4 EXTERNAL BLACS_GRIDINFO, CHK1MAT, IGAMN2D, PCHK2MAT, PSELGET, PSLARED1D, PSLASCL, PSORMTR, PSSTEBZ, PSSTEIN, PSSYTRD, PXERBLA, SGEBR2D, SGEBS2D, SLASRT, SSCAL .TP 20 .ti +4 INTRINSIC ABS, ICHAR, MAX, MIN, MOD, REAL, SQRT .TP 20 .ti +4 IF( BLOCK_CYCLIC_2D*CSRC_*CTXT_*DLEN_*DTYPE_*LLD_*MB_*M_*NB_*N_* RSRC_.LT.0 )RETURN .TP 20 .ti +4 QUICKRETURN = ( N.EQ.0 ) .TP 20 .ti +4 CALL BLACS_GRIDINFO( DESCA( CTXT_ ), NPROW, NPCOL, MYROW, MYCOL ) .TP 20 .ti +4 INFO = 0 .TP 20 .ti +4 IF( NPROW.EQ.-1 ) THEN .TP 20 .ti +4 INFO = -( 800+CTXT_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCZ( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+CTXT_ ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 CALL CHK1MAT( N, 4, N, 4, IA, JA, DESCA, 8, INFO ) .TP 20 .ti +4 CALL CHK1MAT( N, 4, N, 4, IZ, JZ, DESCZ, 21, INFO ) .TP 20 .ti +4 IF( INFO.EQ.0 ) THEN .TP 20 .ti +4 SAFMIN = PSLAMCH( DESCA( CTXT_ ), 'Safe minimum' ) .TP 20 .ti +4 EPS = PSLAMCH( DESCA( CTXT_ ), 'Precision' ) .TP 20 .ti +4 SMLNUM = SAFMIN / EPS .TP 20 .ti +4 BIGNUM = ONE / SMLNUM .TP 20 .ti +4 RMIN = SQRT( SMLNUM ) .TP 20 .ti +4 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) .TP 20 .ti +4 NPROCS = NPROW*NPCOL .TP 20 .ti +4 LOWER = LSAME( UPLO, 'L' ) .TP 20 .ti +4 WANTZ = LSAME( JOBZ, 'V' ) .TP 20 .ti +4 ALLEIG = LSAME( RANGE, 'A' ) .TP 20 .ti +4 VALEIG = LSAME( RANGE, 'V' ) .TP 20 .ti +4 INDEIG = LSAME( RANGE, 'I' ) .TP 20 .ti +4 INDTAU = 1 .TP 20 .ti +4 INDE = INDTAU + N .TP 20 .ti +4 INDD = INDE + N .TP 20 .ti +4 INDD2 = INDD + N .TP 20 .ti +4 INDE2 = INDD2 + N .TP 20 .ti +4 INDWORK = INDE2 + N .TP 20 .ti +4 LLWORK = LWORK - INDWORK + 1 .TP 20 .ti +4 ISIZESTEIN = 3*N + NPROCS + 1 .TP 20 .ti +4 ISIZESTEBZ = MAX( 4*N, 14, NPROCS ) .TP 20 .ti +4 INDIBL = ( MAX( ISIZESTEIN, ISIZESTEBZ ) ) + 1 .TP 20 .ti +4 INDISP = INDIBL + N .TP 20 .ti +4 LQUERY = .FALSE. .TP 20 .ti +4 IF( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) LQUERY = .TRUE. .TP 20 .ti +4 NNP = MAX( N, NPROCS+1, 4 ) .TP 20 .ti +4 LIWMIN = 6*NNP .TP 20 .ti +4 NPROCS = NPROW*NPCOL .TP 20 .ti +4 NB_A = DESCA( NB_ ) .TP 20 .ti +4 MB_A = DESCA( MB_ ) .TP 20 .ti +4 NB_Z = DESCZ( NB_ ) .TP 20 .ti +4 MB_Z = DESCZ( MB_ ) .TP 20 .ti +4 NB = NB_A .TP 20 .ti +4 NN = MAX( N, NB, 2 ) .TP 20 .ti +4 RSRC_A = DESCA( RSRC_ ) .TP 20 .ti +4 CSRC_A = DESCA( CSRC_ ) .TP 20 .ti +4 RSRC_Z = DESCZ( RSRC_ ) .TP 20 .ti +4 IROFFA = MOD( IA-1, MB_A ) .TP 20 .ti +4 ICOFFA = MOD( JA-1, NB_A ) .TP 20 .ti +4 IROFFZ = MOD( IZ-1, MB_A ) .TP 20 .ti +4 IAROW = INDXG2P( 1, NB_A, MYROW, RSRC_A, NPROW ) .TP 20 .ti +4 IACOL = INDXG2P( 1, MB_A, MYCOL, CSRC_A, NPCOL ) .TP 20 .ti +4 IZROW = INDXG2P( 1, NB_A, MYROW, RSRC_Z, NPROW ) .TP 20 .ti +4 NP0 = NUMROC( N+IROFFA, NB_Z, MYROW, IAROW, NPROW ) .TP 20 .ti +4 MQ0 = NUMROC( N+ICOFFA, NB_Z, MYCOL, IACOL, NPCOL ) .TP 20 .ti +4 IF( ( .NOT.WANTZ ) .OR. ( VALEIG .AND. ( .NOT.LQUERY ) ) ) THEN .TP 20 .ti +4 LWMIN = 5*N + MAX( 5*NN, NB*( NP0+1 ) ) .TP 20 .ti +4 NEIG = 0 .TP 20 .ti +4 ELSE .TP 20 .ti +4 IF( ALLEIG .OR. VALEIG ) THEN .TP 20 .ti +4 NEIG = N .TP 20 .ti +4 ELSE IF( INDEIG ) THEN .TP 20 .ti +4 NEIG = IU - IL + 1 .TP 20 .ti +4 END IF .TP 20 .ti +4 MQ0 = NUMROC( MAX( NEIG, NB, 2 ), NB, MYCOL, IACOL, NPCOL ) .TP 20 .ti +4 LWMIN = 5*N + MAX( 5*NN, NP0*MQ0+2*NB*NB ) + ICEIL( NEIG, NPROW*NPCOL )*NN .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( INFO.EQ.0 ) THEN .TP 20 .ti +4 IF( MYROW.EQ.0 .AND. MYCOL.EQ.0 ) THEN .TP 20 .ti +4 WORK( 1 ) = ABSTOL .TP 20 .ti +4 IF( VALEIG ) THEN .TP 20 .ti +4 WORK( 2 ) = VL .TP 20 .ti +4 WORK( 3 ) = VU .TP 20 .ti +4 ELSE .TP 20 .ti +4 WORK( 2 ) = ZERO .TP 20 .ti +4 WORK( 3 ) = ZERO .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL SGEBS2D( DESCA( CTXT_ ), 'ALL', ' ', 3, 1, WORK, 3 ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 CALL SGEBR2D( DESCA( CTXT_ ), 'ALL', ' ', 3, 1, WORK, 3, 0, 0 ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN .TP 20 .ti +4 INFO = -1 .TP 20 .ti +4 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN .TP 20 .ti +4 INFO = -2 .TP 20 .ti +4 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN .TP 20 .ti +4 INFO = -3 .TP 20 .ti +4 ELSE IF( VALEIG .AND. N.GT.0 .AND. VU.LE.VL ) THEN .TP 20 .ti +4 INFO = -10 .TP 20 .ti +4 ELSE IF( INDEIG .AND. ( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) ) THEN .TP 20 .ti +4 INFO = -11 .TP 20 .ti +4 ELSE IF( INDEIG .AND. ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) ) THEN .TP 20 .ti +4 INFO = -12 .TP 20 .ti +4 ELSE IF( LWORK.LT.LWMIN .AND. LWORK.NE.-1 ) THEN .TP 20 .ti +4 INFO = -23 .TP 20 .ti +4 ELSE IF( LIWORK.LT.LIWMIN .AND. LIWORK.NE.-1 ) THEN .TP 20 .ti +4 INFO = -25 .TP 20 .ti +4 ELSE IF( VALEIG .AND. ( ABS( WORK( 2 )-VL ).GT.FIVE*EPS* ABS( VL ) ) ) THEN .TP 20 .ti +4 INFO = -9 .TP 20 .ti +4 ELSE IF( VALEIG .AND. ( ABS( WORK( 3 )-VU ).GT.FIVE*EPS* ABS( VU ) ) ) THEN .TP 20 .ti +4 INFO = -10 .TP 20 .ti +4 ELSE IF( ABS( WORK( 1 )-ABSTOL ).GT.FIVE*EPS*ABS( ABSTOL ) ) THEN .TP 20 .ti +4 INFO = -13 .TP 20 .ti +4 ELSE IF( IROFFA.NE.IROFFZ ) THEN .TP 20 .ti +4 INFO = -19 .TP 20 .ti +4 ELSE IF( IROFFA.NE.0 ) THEN .TP 20 .ti +4 INFO = -6 .TP 20 .ti +4 ELSE IF( IAROW.NE.IZROW ) THEN .TP 20 .ti +4 INFO = -19 .TP 20 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCA( NB_ ) ) THEN .TP 20 .ti +4 INFO = -( 800+NB_ ) .TP 20 .ti +4 ELSE IF( DESCA( M_ ).NE.DESCZ( M_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+M_ ) .TP 20 .ti +4 ELSE IF( DESCA( N_ ).NE.DESCZ( N_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+N_ ) .TP 20 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCZ( MB_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+MB_ ) .TP 20 .ti +4 ELSE IF( DESCA( NB_ ).NE.DESCZ( NB_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+NB_ ) .TP 20 .ti +4 ELSE IF( DESCA( RSRC_ ).NE.DESCZ( RSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+RSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CSRC_ ).NE.DESCZ( CSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+CSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCZ( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+CTXT_ ) .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 IDUM1( 1 ) = ICHAR( 'V' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 1 ) = ICHAR( 'N' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 1 ) = 1 .TP 20 .ti +4 IF( LOWER ) THEN .TP 20 .ti +4 IDUM1( 2 ) = ICHAR( 'L' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 2 ) = ICHAR( 'U' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 2 ) = 2 .TP 20 .ti +4 IF( ALLEIG ) THEN .TP 20 .ti +4 IDUM1( 3 ) = ICHAR( 'A' ) .TP 20 .ti +4 ELSE IF( INDEIG ) THEN .TP 20 .ti +4 IDUM1( 3 ) = ICHAR( 'I' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 3 ) = ICHAR( 'V' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 3 ) = 3 .TP 20 .ti +4 IF( LQUERY ) THEN .TP 20 .ti +4 IDUM1( 4 ) = -1 .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 4 ) = 1 .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 4 ) = 4 .TP 20 .ti +4 CALL PCHK2MAT( N, 4, N, 4, IA, JA, DESCA, 8, N, 4, N, 4, IZ, JZ, DESCZ, 21, 4, IDUM1, IDUM2, INFO ) .TP 20 .ti +4 WORK( 1 ) = REAL( LWMIN ) .TP 20 .ti +4 IWORK( 1 ) = LIWMIN .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( INFO.NE.0 ) THEN .TP 20 .ti +4 CALL PXERBLA( DESCA( CTXT_ ), 'PSSYEVX', -INFO ) .TP 20 .ti +4 RETURN .TP 20 .ti +4 ELSE IF( LQUERY ) THEN .TP 20 .ti +4 RETURN .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( QUICKRETURN ) THEN .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 NZ = 0 .TP 20 .ti +4 ICLUSTR( 1 ) = 0 .TP 20 .ti +4 END IF .TP 20 .ti +4 M = 0 .TP 20 .ti +4 WORK( 1 ) = REAL( LWMIN ) .TP 20 .ti +4 IWORK( 1 ) = LIWMIN .TP 20 .ti +4 RETURN .TP 20 .ti +4 END IF .TP 20 .ti +4 ABSTLL = ABSTOL .TP 20 .ti +4 ISCALE = 0 .TP 20 .ti +4 IF( VALEIG ) THEN .TP 20 .ti +4 VLL = VL .TP 20 .ti +4 VUU = VU .TP 20 .ti +4 ELSE .TP 20 .ti +4 VLL = ZERO .TP 20 .ti +4 VUU = ZERO .TP 20 .ti +4 END IF .TP 20 .ti +4 ANRM = PSLANSY( '1', UPLO, N, A, IA, JA, DESCA, WORK( INDWORK ) ) .TP 20 .ti +4 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN .TP 20 .ti +4 ISCALE = 1 .TP 20 .ti +4 SIGMA = RMIN / ANRM .TP 20 .ti +4 ANRM = ANRM*SIGMA .TP 20 .ti +4 ELSE IF( ANRM.GT.RMAX ) THEN .TP 20 .ti +4 ISCALE = 1 .TP 20 .ti +4 SIGMA = RMAX / ANRM .TP 20 .ti +4 ANRM = ANRM*SIGMA .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( ISCALE.EQ.1 ) THEN .TP 20 .ti +4 CALL PSLASCL( UPLO, ONE, SIGMA, N, N, A, IA, JA, DESCA, IINFO ) .TP 20 .ti +4 IF( ABSTOL.GT.0 ) ABSTLL = ABSTOL*SIGMA .TP 20 .ti +4 IF( VALEIG ) THEN .TP 20 .ti +4 VLL = VL*SIGMA .TP 20 .ti +4 VUU = VU*SIGMA .TP 20 .ti +4 IF( VUU.EQ.VLL ) THEN .TP 20 .ti +4 VUU = VUU + 2*MAX( ABS( VUU )*EPS, SAFMIN ) .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 LALLWORK = LLWORK .TP 20 .ti +4 CALL PSSYTRD( UPLO, N, A, IA, JA, DESCA, WORK( INDD ), WORK( INDE ), WORK( INDTAU ), WORK( INDWORK ), LLWORK, IINFO ) .TP 20 .ti +4 OFFSET = 0 .TP 20 .ti +4 IF( IA.EQ.1 .AND. JA.EQ.1 .AND. RSRC_A.EQ.0 .AND. CSRC_A.EQ.0 ) THEN .TP 20 .ti +4 CALL PSLARED1D( N, IA, JA, DESCA, WORK( INDD ), WORK( INDD2 ), WORK( INDWORK ), LLWORK ) .TP 20 .ti +4 CALL PSLARED1D( N, IA, JA, DESCA, WORK( INDE ), WORK( INDE2 ), WORK( INDWORK ), LLWORK ) .TP 20 .ti +4 IF( .NOT.LOWER ) OFFSET = 1 .TP 20 .ti +4 ELSE .TP 20 .ti +4 DO 10 I = 1, N .TP 20 .ti +4 CALL PSELGET( 'A', ' ', WORK( INDD2+I-1 ), A, I+IA-1, I+JA-1, DESCA ) .TP 20 .ti +4 10 CONTINUE .TP 20 .ti +4 IF( LSAME( UPLO, 'U' ) ) THEN .TP 20 .ti +4 DO 20 I = 1, N - 1 .TP 20 .ti +4 CALL PSELGET( 'A', ' ', WORK( INDE2+I-1 ), A, I+IA-1, I+JA, DESCA ) .TP 20 .ti +4 20 CONTINUE .TP 20 .ti +4 ELSE .TP 20 .ti +4 DO 30 I = 1, N - 1 .TP 20 .ti +4 CALL PSELGET( 'A', ' ', WORK( INDE2+I-1 ), A, I+IA, I+JA-1, DESCA ) .TP 20 .ti +4 30 CONTINUE .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 ORDER = 'b' .TP 20 .ti +4 ELSE .TP 20 .ti +4 ORDER = 'e' .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PSSTEBZ( DESCA( CTXT_ ), RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, WORK( INDD2 ), WORK( INDE2+OFFSET ), M, NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWORK ), LLWORK, IWORK( 1 ), ISIZESTEBZ, IINFO ) .TP 20 .ti +4 IF( IINFO.NE.0 ) THEN .TP 20 .ti +4 INFO = INFO + IERREBZ .TP 20 .ti +4 DO 40 I = 1, M .TP 20 .ti +4 IWORK( INDIBL+I-1 ) = ABS( IWORK( INDIBL+I-1 ) ) .TP 20 .ti +4 40 CONTINUE .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 IF( VALEIG ) THEN .TP 20 .ti +4 CALL IGAMN2D( DESCA( CTXT_ ), 'A', ' ', 1, 1, LALLWORK, 1, 1, 1, -1, -1, -1 ) .TP 20 .ti +4 MAXEIGS = DESCZ( N_ ) .TP 20 .ti +4 DO 50 NZ = MIN( MAXEIGS, M ), 0, -1 .TP 20 .ti +4 MQ0 = NUMROC( NZ, NB, 0, 0, NPCOL ) .TP 20 .ti +4 SIZESTEIN = ICEIL( NZ, NPROCS )*N + MAX( 5*N, NP0*MQ0 ) .TP 20 .ti +4 SIZEORMTR = MAX( ( NB*( NB-1 ) ) / 2, ( MQ0+NP0 )*NB ) + NB*NB .TP 20 .ti +4 SIZESYEVX = MAX( SIZESTEIN, SIZEORMTR ) .TP 20 .ti +4 IF( SIZESYEVX.LE.LALLWORK ) GO TO 60 .TP 20 .ti +4 50 CONTINUE .TP 20 .ti +4 60 CONTINUE .TP 20 .ti +4 ELSE .TP 20 .ti +4 NZ = M .TP 20 .ti +4 END IF .TP 20 .ti +4 NZ = MAX( NZ, 0 ) .TP 20 .ti +4 IF( NZ.NE.M ) THEN .TP 20 .ti +4 INFO = INFO + IERRSPC .TP 20 .ti +4 DO 70 I = 1, M .TP 20 .ti +4 IFAIL( I ) = 0 .TP 20 .ti +4 70 CONTINUE .TP 20 .ti +4 IF( NSPLIT.GT.1 ) THEN .TP 20 .ti +4 CALL SLASRT( 'I', M, W, IINFO ) .TP 20 .ti +4 IF( NZ.GT.0 ) THEN .TP 20 .ti +4 VUU = W( NZ ) - TEN*( EPS*ANRM+SAFMIN ) .TP 20 .ti +4 IF( VLL.GE.VUU ) THEN .TP 20 .ti +4 NZZ = 0 .TP 20 .ti +4 ELSE .TP 20 .ti +4 CALL PSSTEBZ( DESCA( CTXT_ ), RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, WORK( INDD2 ), WORK( INDE2+OFFSET ), NZZ, NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWORK ), LLWORK, IWORK( 1 ), ISIZESTEBZ, IINFO ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( MOD( INFO / IERREBZ, 1 ).EQ.0 ) THEN .TP 20 .ti +4 IF( NZZ.GT.NZ .OR. IINFO.NE.0 ) THEN .TP 20 .ti +4 INFO = INFO + IERREBZ .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 NZ = MIN( NZ, NZZ ) .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PSSTEIN( N, WORK( INDD2 ), WORK( INDE2+OFFSET ), NZ, W, IWORK( INDIBL ), IWORK( INDISP ), ORFAC, Z, IZ, JZ, DESCZ, WORK( INDWORK ), LALLWORK, IWORK( 1 ), ISIZESTEIN, IFAIL, ICLUSTR, GAP, IINFO ) .TP 20 .ti +4 IF( IINFO.GE.NZ+1 ) INFO = INFO + IERRCLS .TP 20 .ti +4 IF( MOD( IINFO, NZ+1 ).NE.0 ) INFO = INFO + IERREIN .TP 20 .ti +4 IF( NZ.GT.0 ) THEN .TP 20 .ti +4 CALL PSORMTR( 'L', UPLO, 'N', N, NZ, A, IA, JA, DESCA, WORK( INDTAU ), Z, IZ, JZ, DESCZ, WORK( INDWORK ), LLWORK, IINFO ) .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( ISCALE.EQ.1 ) THEN .TP 20 .ti +4 CALL SSCAL( M, ONE / SIGMA, W, 1 ) .TP 20 .ti +4 END IF .TP 20 .ti +4 WORK( 1 ) = REAL( LWMIN ) .TP 20 .ti +4 IWORK( 1 ) = LIWMIN .TP 20 .ti +4 RETURN .TP 20 .ti +4 END .SH PURPOSE scalapack-doc-1.5/man/manl/pssygs2.l0100644000056400000620000001414006335610652017034 0ustar pfrauenfstaff.TH PSSYGS2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSSYGS2 - reduce a real symmetric-definite generalized eigenproblem to standard form .SH SYNOPSIS .TP 20 SUBROUTINE PSSYGS2( IBTYPE, UPLO, N, A, IA, JA, DESCA, B, IB, JB, DESCB, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IB, IBTYPE, INFO, JA, JB, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 REAL A( * ), B( * ) .SH PURPOSE PSSYGS2 reduces a real symmetric-definite generalized eigenproblem to standard form. In the following sub( A ) denotes A( IA:IA+N-1, JA:JA+N-1 ) and sub( B ) denotes B( IB:IB+N-1, JB:JB+N-1 ). .br If IBTYPE = 1, the problem is sub( A )*x = lambda*sub( B )*x, and sub( A ) is overwritten by inv(U**T)*sub( A )*inv(U) or inv(L)*sub( A )*inv(L**T) .br If IBTYPE = 2 or 3, the problem is sub( A )*sub( B )*x = lambda*x or sub( B )*sub( A )*x = lambda*x, and sub( A ) is overwritten by U*sub( A )*U**T or L**T*sub( A )*L. .br sub( B ) must have been previously factorized as U**T*U or L*L**T by PSPOTRF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 9 IBTYPE (global input) INTEGER = 1: compute inv(U**T)*sub( A )*inv(U) or inv(L)*sub( A )*inv(L**T); = 2 or 3: compute U*sub( A )*U**T or L**T*sub( A )*L. .TP 8 UPLO (global input) CHARACTER .br = 'U': Upper triangle of sub( A ) is stored and sub( B ) is factored as U**T*U; = 'L': Lower triangle of sub( A ) is stored and sub( B ) is factored as L*L**T. .TP 8 N (global input) INTEGER The order of the matrices sub( A ) and sub( B ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N symmetric distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced. On exit, if INFO = 0, the transformed matrix, stored in the same format as sub( A ). .TP 8 IA (global input) INTEGER A's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JA (global input) INTEGER A's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input) REAL pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+N-1)). On entry, this array contains the local pieces of the triangular factor from the Cholesky factorization of sub( B ), as returned by PSPOTRF. .TP 8 IB (global input) INTEGER B's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JB (global input) INTEGER B's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pssygst.l0100644000056400000620000001455706335610652017152 0ustar pfrauenfstaff.TH PSSYGST l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSSYGST - reduce a real symmetric-definite generalized eigenproblem to standard form .SH SYNOPSIS .TP 20 SUBROUTINE PSSYGST( IBTYPE, UPLO, N, A, IA, JA, DESCA, B, IB, JB, DESCB, SCALE, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IB, IBTYPE, INFO, JA, JB, N .TP 20 .ti +4 REAL SCALE .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 REAL A( * ), B( * ) .SH PURPOSE PSSYGST reduces a real symmetric-definite generalized eigenproblem to standard form. In the following sub( A ) denotes A( IA:IA+N-1, JA:JA+N-1 ) and sub( B ) denotes B( IB:IB+N-1, JB:JB+N-1 ). .br If IBTYPE = 1, the problem is sub( A )*x = lambda*sub( B )*x, and sub( A ) is overwritten by inv(U**T)*sub( A )*inv(U) or inv(L)*sub( A )*inv(L**T) .br If IBTYPE = 2 or 3, the problem is sub( A )*sub( B )*x = lambda*x or sub( B )*sub( A )*x = lambda*x, and sub( A ) is overwritten by U*sub( A )*U**T or L**T*sub( A )*L. .br sub( B ) must have been previously factorized as U**T*U or L*L**T by PSPOTRF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 9 IBTYPE (global input) INTEGER = 1: compute inv(U**T)*sub( A )*inv(U) or inv(L)*sub( A )*inv(L**T); = 2 or 3: compute U*sub( A )*U**T or L**T*sub( A )*L. .TP 8 UPLO (global input) CHARACTER .br = 'U': Upper triangle of sub( A ) is stored and sub( B ) is factored as U**T*U; = 'L': Lower triangle of sub( A ) is stored and sub( B ) is factored as L*L**T. .TP 8 N (global input) INTEGER The order of the matrices sub( A ) and sub( B ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N symmetric distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced. On exit, if INFO = 0, the transformed matrix, stored in the same format as sub( A ). .TP 8 IA (global input) INTEGER A's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JA (global input) INTEGER A's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input) REAL pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+N-1)). On entry, this array contains the local pieces of the triangular factor from the Cholesky factorization of sub( B ), as returned by PSPOTRF. .TP 8 IB (global input) INTEGER B's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JB (global input) INTEGER B's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 SCALE (global output) REAL Amount by which the eigenvalues should be scaled to compensate for the scaling performed in this routine. At present, SCALE is always returned as 1.0, it is returned here to allow for future enhancement. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pssygvx.l0100644000056400000620000002375606335610652017162 0ustar pfrauenfstaff.TH PSSYGVX l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME .SH SYNOPSIS .TP 20 SUBROUTINE PSSYGVX( IBTYPE, JOBZ, RANGE, UPLO, N, A, IA, JA, DESCA, B, IB, JB, DESCB, VL, VU, IL, IU, ABSTOL, M, NZ, W, ORFAC, Z, IZ, JZ, DESCZ, WORK, LWORK, IWORK, LIWORK, IFAIL, ICLUSTR, GAP, INFO ) .TP 20 .ti +4 CHARACTER JOBZ, RANGE, UPLO .TP 20 .ti +4 INTEGER IA, IB, IBTYPE, IL, INFO, IU, IZ, JA, JB, JZ, LIWORK, LWORK, M, N, NZ .TP 20 .ti +4 REAL ABSTOL, ORFAC, VL, VU .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ), DESCZ( * ), ICLUSTR( * ), IFAIL( * ), IWORK( * ) .TP 20 .ti +4 REAL A( * ), B( * ), GAP( * ), W( * ), WORK( * ), Z( * ) .TP 20 .ti +4 INTEGER BLOCK_CYCLIC_2D, DLEN_, DTYPE_, CTXT_, M_, N_, MB_, NB_, RSRC_, CSRC_, LLD_ .TP 20 .ti +4 PARAMETER ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1, CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6, RSRC_ = 7, CSRC_ = 8, LLD_ = 9 ) .TP 20 .ti +4 REAL ONE .TP 20 .ti +4 PARAMETER ( ONE = 1.0E+0 ) .TP 20 .ti +4 REAL FIVE, ZERO .TP 20 .ti +4 PARAMETER ( FIVE = 5.0E+0, ZERO = 0.0E+0 ) .TP 20 .ti +4 INTEGER IERRNPD .TP 20 .ti +4 PARAMETER ( IERRNPD = 16 ) .TP 20 .ti +4 LOGICAL ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER IACOL, IAROW, IBCOL, IBROW, ICOFFA, ICOFFB, ICTXT, IROFFA, IROFFB, LIWMIN, LWMIN, MQ0, MYCOL, MYROW, NB, NEIG, NN, NP0, NPCOL, NPROW .TP 20 .ti +4 REAL EPS, SCALE .TP 20 .ti +4 INTEGER IDUM1( 5 ), IDUM2( 5 ) .TP 20 .ti +4 LOGICAL LSAME .TP 20 .ti +4 INTEGER ICEIL, INDXG2P, NUMROC .TP 20 .ti +4 REAL PSLAMCH .TP 20 .ti +4 EXTERNAL LSAME, ICEIL, INDXG2P, NUMROC, PSLAMCH .TP 20 .ti +4 EXTERNAL BLACS_GRIDINFO, CHK1MAT, PCHK1MAT, PCHK2MAT, PSPOTRF, PSSYEVX, PSSYGST, PSTRMM, PSTRSM, PXERBLA, SGEBR2D, SGEBS2D, SSCAL .TP 20 .ti +4 INTRINSIC ABS, ICHAR, MAX, MIN, MOD, REAL .TP 20 .ti +4 IF( BLOCK_CYCLIC_2D*CSRC_*CTXT_*DLEN_*DTYPE_*LLD_*MB_*M_*NB_*N_* RSRC_.LT.0 )RETURN .TP 20 .ti +4 ICTXT = DESCA( CTXT_ ) .TP 20 .ti +4 CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL ) .TP 20 .ti +4 INFO = 0 .TP 20 .ti +4 IF( NPROW.EQ.-1 ) THEN .TP 20 .ti +4 INFO = -( 900+CTXT_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCB( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+CTXT_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCZ( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 2600+CTXT_ ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 EPS = PSLAMCH( DESCA( CTXT_ ), 'Precision' ) .TP 20 .ti +4 WANTZ = LSAME( JOBZ, 'V' ) .TP 20 .ti +4 UPPER = LSAME( UPLO, 'U' ) .TP 20 .ti +4 ALLEIG = LSAME( RANGE, 'A' ) .TP 20 .ti +4 VALEIG = LSAME( RANGE, 'V' ) .TP 20 .ti +4 INDEIG = LSAME( RANGE, 'I' ) .TP 20 .ti +4 CALL CHK1MAT( N, 4, N, 4, IA, JA, DESCA, 9, INFO ) .TP 20 .ti +4 CALL CHK1MAT( N, 4, N, 4, IB, JB, DESCB, 13, INFO ) .TP 20 .ti +4 CALL CHK1MAT( N, 4, N, 4, IZ, JZ, DESCZ, 26, INFO ) .TP 20 .ti +4 IF( INFO.EQ.0 ) THEN .TP 20 .ti +4 IF( MYROW.EQ.0 .AND. MYCOL.EQ.0 ) THEN .TP 20 .ti +4 WORK( 1 ) = ABSTOL .TP 20 .ti +4 IF( VALEIG ) THEN .TP 20 .ti +4 WORK( 2 ) = VL .TP 20 .ti +4 WORK( 3 ) = VU .TP 20 .ti +4 ELSE .TP 20 .ti +4 WORK( 2 ) = ZERO .TP 20 .ti +4 WORK( 3 ) = ZERO .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL SGEBS2D( DESCA( CTXT_ ), 'ALL', ' ', 3, 1, WORK, 3 ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 CALL SGEBR2D( DESCA( CTXT_ ), 'ALL', ' ', 3, 1, WORK, 3, 0, 0 ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IAROW = INDXG2P( IA, DESCA( MB_ ), MYROW, DESCA( RSRC_ ), NPROW ) .TP 20 .ti +4 IBROW = INDXG2P( IB, DESCB( MB_ ), MYROW, DESCB( RSRC_ ), NPROW ) .TP 20 .ti +4 IACOL = INDXG2P( JA, DESCA( NB_ ), MYCOL, DESCA( CSRC_ ), NPCOL ) .TP 20 .ti +4 IBCOL = INDXG2P( JB, DESCB( NB_ ), MYCOL, DESCB( CSRC_ ), NPCOL ) .TP 20 .ti +4 IROFFA = MOD( IA-1, DESCA( MB_ ) ) .TP 20 .ti +4 ICOFFA = MOD( JA-1, DESCA( NB_ ) ) .TP 20 .ti +4 IROFFB = MOD( IB-1, DESCB( MB_ ) ) .TP 20 .ti +4 ICOFFB = MOD( JB-1, DESCB( NB_ ) ) .TP 20 .ti +4 LQUERY = .FALSE. .TP 20 .ti +4 IF( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) LQUERY = .TRUE. .TP 20 .ti +4 LIWMIN = 6*MAX( N, ( NPROW*NPCOL )+1, 4 ) .TP 20 .ti +4 NB = DESCA( MB_ ) .TP 20 .ti +4 NN = MAX( N, NB, 2 ) .TP 20 .ti +4 NP0 = NUMROC( NN, NB, 0, 0, NPROW ) .TP 20 .ti +4 IF( ( .NOT.WANTZ ) .OR. ( VALEIG .AND. ( .NOT.LQUERY ) ) ) THEN .TP 20 .ti +4 LWMIN = 5*N + MAX( 5*NN, NB*( NP0+1 ) ) .TP 20 .ti +4 NEIG = 0 .TP 20 .ti +4 ELSE .TP 20 .ti +4 IF( ALLEIG .OR. VALEIG ) THEN .TP 20 .ti +4 NEIG = N .TP 20 .ti +4 ELSE IF( INDEIG ) THEN .TP 20 .ti +4 NEIG = IU - IL + 1 .TP 20 .ti +4 END IF .TP 20 .ti +4 MQ0 = NUMROC( MAX( NEIG, NB, 2 ), NB, 0, 0, NPCOL ) .TP 20 .ti +4 LWMIN = 5*N + MAX( 5*NN, NP0*MQ0+2*NB*NB ) + ICEIL( NEIG, NPROW*NPCOL )*NN .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( IBTYPE.LT.1 .OR. IBTYPE.GT.3 ) THEN .TP 20 .ti +4 INFO = -1 .TP 20 .ti +4 ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN .TP 20 .ti +4 INFO = -2 .TP 20 .ti +4 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN .TP 20 .ti +4 INFO = -3 .TP 20 .ti +4 ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN .TP 20 .ti +4 INFO = -4 .TP 20 .ti +4 ELSE IF( N.LT.0 ) THEN .TP 20 .ti +4 INFO = -5 .TP 20 .ti +4 ELSE IF( IROFFA.NE.0 ) THEN .TP 20 .ti +4 INFO = -7 .TP 20 .ti +4 ELSE IF( ICOFFA.NE.0 ) THEN .TP 20 .ti +4 INFO = -8 .TP 20 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCA( NB_ ) ) THEN .TP 20 .ti +4 INFO = -( 900+NB_ ) .TP 20 .ti +4 ELSE IF( DESCA( M_ ).NE.DESCB( M_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+M_ ) .TP 20 .ti +4 ELSE IF( DESCA( N_ ).NE.DESCB( N_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+N_ ) .TP 20 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCB( MB_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+MB_ ) .TP 20 .ti +4 ELSE IF( DESCA( NB_ ).NE.DESCB( NB_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+NB_ ) .TP 20 .ti +4 ELSE IF( DESCA( RSRC_ ).NE.DESCB( RSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+RSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CSRC_ ).NE.DESCB( CSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+CSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCB( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+CTXT_ ) .TP 20 .ti +4 ELSE IF( DESCA( M_ ).NE.DESCZ( M_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+M_ ) .TP 20 .ti +4 ELSE IF( DESCA( N_ ).NE.DESCZ( N_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+N_ ) .TP 20 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCZ( MB_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+MB_ ) .TP 20 .ti +4 ELSE IF( DESCA( NB_ ).NE.DESCZ( NB_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+NB_ ) .TP 20 .ti +4 ELSE IF( DESCA( RSRC_ ).NE.DESCZ( RSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+RSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CSRC_ ).NE.DESCZ( CSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+CSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCZ( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+CTXT_ ) .TP 20 .ti +4 ELSE IF( IROFFB.NE.0 .OR. IBROW.NE.IAROW ) THEN .TP 20 .ti +4 INFO = -11 .TP 20 .ti +4 ELSE IF( ICOFFB.NE.0 .OR. IBCOL.NE.IACOL ) THEN .TP 20 .ti +4 INFO = -12 .TP 20 .ti +4 ELSE IF( VALEIG .AND. N.GT.0 .AND. VU.LE.VL ) THEN .TP 20 .ti +4 INFO = -15 .TP 20 .ti +4 ELSE IF( INDEIG .AND. ( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) ) THEN .TP 20 .ti +4 INFO = -16 .TP 20 .ti +4 ELSE IF( INDEIG .AND. ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) ) THEN .TP 20 .ti +4 INFO = -17 .TP 20 .ti +4 ELSE IF( VALEIG .AND. ( ABS( WORK( 2 )-VL ).GT.FIVE*EPS* ABS( VL ) ) ) THEN .TP 20 .ti +4 INFO = -14 .TP 20 .ti +4 ELSE IF( VALEIG .AND. ( ABS( WORK( 3 )-VU ).GT.FIVE*EPS* ABS( VU ) ) ) THEN .TP 20 .ti +4 INFO = -15 .TP 20 .ti +4 ELSE IF( ABS( WORK( 1 )-ABSTOL ).GT.FIVE*EPS*ABS( ABSTOL ) ) THEN .TP 20 .ti +4 INFO = -18 .TP 20 .ti +4 ELSE IF( LWORK.LT.LWMIN .AND. LWORK.NE.-1 ) THEN .TP 20 .ti +4 INFO = -28 .TP 20 .ti +4 ELSE IF( LIWORK.LT.LIWMIN .AND. LIWORK.NE.-1 ) THEN .TP 20 .ti +4 INFO = -30 .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM1( 1 ) = IBTYPE .TP 20 .ti +4 IDUM2( 1 ) = 1 .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 IDUM1( 2 ) = ICHAR( 'V' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 2 ) = ICHAR( 'N' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 2 ) = 2 .TP 20 .ti +4 IF( UPPER ) THEN .TP 20 .ti +4 IDUM1( 3 ) = ICHAR( 'U' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 3 ) = ICHAR( 'L' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 3 ) = 3 .TP 20 .ti +4 IF( ALLEIG ) THEN .TP 20 .ti +4 IDUM1( 4 ) = ICHAR( 'A' ) .TP 20 .ti +4 ELSE IF( INDEIG ) THEN .TP 20 .ti +4 IDUM1( 4 ) = ICHAR( 'I' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 4 ) = ICHAR( 'V' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 4 ) = 4 .TP 20 .ti +4 IF( LQUERY ) THEN .TP 20 .ti +4 IDUM1( 5 ) = -1 .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 5 ) = 1 .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 5 ) = 5 .TP 20 .ti +4 CALL PCHK2MAT( N, 4, N, 4, IA, JA, DESCA, 9, N, 4, N, 4, IB, JB, DESCB, 13, 5, IDUM1, IDUM2, INFO ) .TP 20 .ti +4 CALL PCHK1MAT( N, 4, N, 4, IZ, JZ, DESCZ, 26, 0, IDUM1, IDUM2, INFO ) .TP 20 .ti +4 END IF .TP 20 .ti +4 WORK( 1 ) = REAL( LWMIN ) .TP 20 .ti +4 IWORK( 1 ) = LIWMIN .TP 20 .ti +4 IF( INFO.NE.0 ) THEN .TP 20 .ti +4 CALL PXERBLA( ICTXT, 'PSSYGVX ', -INFO ) .TP 20 .ti +4 RETURN .TP 20 .ti +4 ELSE IF( LQUERY ) THEN .TP 20 .ti +4 RETURN .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PSPOTRF( UPLO, N, B, IB, JB, DESCB, INFO ) .TP 20 .ti +4 IF( INFO.NE.0 ) THEN .TP 20 .ti +4 IFAIL( 1 ) = INFO .TP 20 .ti +4 INFO = IERRNPD .TP 20 .ti +4 RETURN .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PSSYGST( IBTYPE, UPLO, N, A, IA, JA, DESCA, B, IB, JB, DESCB, SCALE, INFO ) .TP 20 .ti +4 CALL PSSYEVX( JOBZ, RANGE, UPLO, N, A, IA, JA, DESCA, VL, VU, IL, IU, ABSTOL, M, NZ, W, ORFAC, Z, IZ, JZ, DESCZ, WORK, LWORK, IWORK, LIWORK, IFAIL, ICLUSTR, GAP, INFO ) .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 NEIG = M .TP 20 .ti +4 IF( IBTYPE.EQ.1 .OR. IBTYPE.EQ.2 ) THEN .TP 20 .ti +4 IF( UPPER ) THEN .TP 20 .ti +4 TRANS = 'N' .TP 20 .ti +4 ELSE .TP 20 .ti +4 TRANS = 'T' .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PSTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, NEIG, ONE, B, IB, JB, DESCB, Z, IZ, JZ, DESCZ ) .TP 20 .ti +4 ELSE IF( IBTYPE.EQ.3 ) THEN .TP 20 .ti +4 IF( UPPER ) THEN .TP 20 .ti +4 TRANS = 'T' .TP 20 .ti +4 ELSE .TP 20 .ti +4 TRANS = 'N' .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PSTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, NEIG, ONE, B, IB, JB, DESCB, Z, IZ, JZ, DESCZ ) .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( SCALE.NE.ONE ) THEN .TP 20 .ti +4 CALL SSCAL( N, SCALE, W, 1 ) .TP 20 .ti +4 END IF .TP 20 .ti +4 RETURN .TP 20 .ti +4 END .SH PURPOSE scalapack-doc-1.5/man/manl/pssytd2.l0100644000056400000620000002020306335610652017027 0ustar pfrauenfstaff.TH PSSYTD2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSSYTD2 - reduce a real symmetric matrix sub( A ) to symmetric tridiagonal form T by an orthogonal similarity transformation .SH SYNOPSIS .TP 20 SUBROUTINE PSSYTD2( UPLO, N, A, IA, JA, DESCA, D, E, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), D( * ), E( * ), TAU( * ), WORK( * ) .SH PURPOSE PSSYTD2 reduces a real symmetric matrix sub( A ) to symmetric tridiagonal form T by an orthogonal similarity transformation: Q' * sub( A ) * Q = T, where sub( A ) = A(IA:IA+N-1,JA:JA+N-1). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the upper or lower triangular part of the symmetric matrix sub( A ) is stored: .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the symmetric distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced. On exit, if UPLO = 'U', the diagonal and first superdiagonal of sub( A ) are over- written by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of sub( A ) are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) REAL array, dimension LOCc(JA+N-1) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i). D is tied to the distributed matrix A. .TP 8 E (local output) REAL array, dimension LOCc(JA+N-1) if UPLO = 'U', LOCc(JA+N-2) otherwise. The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. E is tied to the distributed matrix A. .TP 8 TAU (local output) REAL, array, dimension LOCc(JA+N-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK( 1 ) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 3*N. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors .br Q = H(n-1) . . . H(2) H(1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in .br A(ia:ia+i-2,ja+i), and tau in TAU(ja+i-1). .br If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors .br Q = H(1) H(2) . . . H(n-1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in .br A(ia+i+1:ia+n-1,ja+i-1), and tau in TAU(ja+i-1). .br The contents of sub( A ) on exit are illustrated by the following examples with n = 5: .br if UPLO = 'U': if UPLO = 'L': .br ( d e v2 v3 v4 ) ( d ) ( d e v3 v4 ) ( e d ) ( d e v4 ) ( v1 e d ) ( d e ) ( v1 v2 e d ) ( d ) ( v1 v2 v3 e d ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expression should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) with .br IROFFA = MOD( IA-1, MB_A ) and ICOFFA = MOD( JA-1, NB_A ). scalapack-doc-1.5/man/manl/pssytrd.l0100644000056400000620000002064206335610652017136 0ustar pfrauenfstaff.TH PSSYTRD l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSSYTRD - reduce a real symmetric matrix sub( A ) to symmetric tridiagonal form T by an orthogonal similarity transformation .SH SYNOPSIS .TP 20 SUBROUTINE PSSYTRD( UPLO, N, A, IA, JA, DESCA, D, E, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), D( * ), E( * ), TAU( * ), WORK( * ) .SH PURPOSE PSSYTRD reduces a real symmetric matrix sub( A ) to symmetric tridiagonal form T by an orthogonal similarity transformation: Q' * sub( A ) * Q = T, where sub( A ) = A(IA:IA+N-1,JA:JA+N-1). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the upper or lower triangular part of the symmetric matrix sub( A ) is stored: .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the symmetric distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced. On exit, if UPLO = 'U', the diagonal and first superdiagonal of sub( A ) are over- written by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of sub( A ) are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) REAL array, dimension LOCc(JA+N-1) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i). D is tied to the distributed matrix A. .TP 8 E (local output) REAL array, dimension LOCc(JA+N-1) if UPLO = 'U', LOCc(JA+N-2) otherwise. The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. E is tied to the distributed matrix A. .TP 8 TAU (local output) REAL, array, dimension LOCc(JA+N-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK( 1 ) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MAX( NB * ( NP +1 ), 3 * NB ) where NB = MB_A = NB_A, NP = NUMROC( N, NB, MYROW, IAROW, NPROW ), IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ). INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors .br Q = H(n-1) . . . H(2) H(1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in .br A(ia:ia+i-2,ja+i), and tau in TAU(ja+i-1). .br If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors .br Q = H(1) H(2) . . . H(n-1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a real scalar, and v is a real vector with .br v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in .br A(ia+i+1:ia+n-1,ja+i-1), and tau in TAU(ja+i-1). .br The contents of sub( A ) on exit are illustrated by the following examples with n = 5: .br if UPLO = 'U': if UPLO = 'L': .br ( d e v2 v3 v4 ) ( d ) ( d e v3 v4 ) ( e d ) ( d e v4 ) ( v1 e d ) ( d e ) ( v1 v2 e d ) ( d ) ( v1 v2 v3 e d ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expression should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA .AND. IROFFA.EQ.0 ) with IROFFA = MOD( IA-1, MB_A ) and ICOFFA = MOD( JA-1, NB_A ). scalapack-doc-1.5/man/manl/pstrcon.l0100644000056400000620000001610006335610653017111 0ustar pfrauenfstaff.TH PSTRCON l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSTRCON - estimate the reciprocal of the condition number of a triangular distributed matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm .SH SYNOPSIS .TP 20 SUBROUTINE PSTRCON( NORM, UPLO, DIAG, N, A, IA, JA, DESCA, RCOND, WORK, LWORK, IWORK, LIWORK, INFO ) .TP 20 .ti +4 CHARACTER DIAG, NORM, UPLO .TP 20 .ti +4 INTEGER IA, JA, INFO, LIWORK, LWORK, N .TP 20 .ti +4 REAL RCOND .TP 20 .ti +4 INTEGER DESCA( * ), IWORK( * ) .TP 20 .ti +4 REAL A( * ), WORK( * ) .SH PURPOSE PSTRCON estimates the reciprocal of the condition number of a triangular distributed matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm. The norm of A(IA:IA+N-1,JA:JA+N-1) is computed and an estimate is obtained for norm(inv(A(IA:IA+N-1,JA:JA+N-1))), then the reciprocal of the condition number is computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies whether the 1-norm condition number or the infinity-norm condition number is required: .br = '1' or 'O': 1-norm; .br = 'I': Infinity-norm. .TP 8 UPLO (global input) CHARACTER .br = 'U': A(IA:IA+N-1,JA:JA+N-1) is upper triangular; .br = 'L': A(IA:IA+N-1,JA:JA+N-1) is lower triangular. .TP 8 DIAG (global input) CHARACTER .br = 'N': A(IA:IA+N-1,JA:JA+N-1) is non-unit triangular; .br = 'U': A(IA:IA+N-1,JA:JA+N-1) is unit triangular. .TP 8 N (global input) INTEGER .br The order of the distributed matrix A(IA:IA+N-1,JA:JA+N-1). N >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension ( LLD_A, LOCc(JA+N-1) ). This array contains the local pieces of the triangular distributed matrix A(IA:IA+N-1,JA:JA+N-1). If UPLO = 'U', the leading N-by-N upper triangular part of this distributed matrix con- tains the upper triangular matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of this ditributed matrix contains the lower triangular matrix, and the strictly upper triangular part is not referenced. If DIAG = 'U', the diagonal elements of A(IA:IA+N-1,JA:JA+N-1) are also not referenced and are assumed to be 1. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 RCOND (global output) REAL The reciprocal of the condition number of the distributed matrix A(IA:IA+N-1,JA:JA+N-1), computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * .br norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 2*LOCr(N+MOD(IA-1,MB_A)) + LOCc(N+MOD(JA-1,NB_A)) + MAX( 2, MAX( NB_A*MAX( 1, CEIL(NPROW-1,NPCOL) ), LOCc(N+MOD(JA-1,NB_A)) + NB_A*MAX( 1, CEIL(NPCOL-1,NPROW) ) ). If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/local output) INTEGER array, dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK. .TP 8 LIWORK (local or global input) INTEGER The dimension of the array IWORK. LIWORK is local input and must be at least LIWORK >= LOCr(N+MOD(IA-1,MB_A)). If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pstrrfs.l0100644000056400000620000002262306335610653017133 0ustar pfrauenfstaff.TH PSTRRFS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSTRRFS - provide error bounds and backward error estimates for the solution to a system of linear equations with a triangular coefficient matrix .SH SYNOPSIS .TP 20 SUBROUTINE PSTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, X, IX, JX, DESCX, FERR, BERR, WORK, LWORK, IWORK, LIWORK, INFO ) .TP 20 .ti +4 CHARACTER DIAG, TRANS, UPLO .TP 20 .ti +4 INTEGER INFO, IA, IB, IX, JA, JB, JX, LIWORK, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ), DESCX( * ), IWORK( * ) .TP 20 .ti +4 REAL A( * ), B( * ), BERR( * ), FERR( * ), WORK( * ), X( * ) .SH PURPOSE PSTRRFS provides error bounds and backward error estimates for the solution to a system of linear equations with a triangular coefficient matrix. The solution matrix X must be computed by PSTRTRS or some other means before entering this routine. PSTRRFS does not do iterative refinement because doing so cannot improve the backward error. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br In the following comments, sub( A ), sub( X ) and sub( B ) denote respectively A(IA:IA+N-1,JA:JA+N-1), X(IX:IX+N-1,JX:JX+NRHS-1) and B(IB:IB+N-1,JB:JB+NRHS-1). .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 = 'U': sub( A ) is upper triangular; .br = 'L': sub( A ) is lower triangular. .TP 8 TRANS (global input) CHARACTER*1 Specifies the form of the system of equations. = 'N': sub( A ) * sub( X ) = sub( B ) (No transpose) .br = 'T': sub( A )**T * sub( X ) = sub( B ) (Transpose) .br = 'C': sub( A )**T * sub( X ) = sub( B ) (Conjugate transpose = Transpose) .TP 8 DIAG (global input) CHARACTER*1 = 'N': sub( A ) is non-unit triangular; .br = 'U': sub( A ) is unit triangular. .TP 8 N (global input) INTEGER The order of the matrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the matrices sub( B ) and sub( X ). NRHS >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1) ). This array contains the local pieces of the original triangular distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the distribu- ted matrix, and its strictly upper triangular part is not referenced. If DIAG = 'U', the diagonal elements of sub( A ) are also not referenced and are assumed to be 1. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input) REAL pointer into the local memory to an array of local dimension (LLD_B, LOCc(JB+NRHS-1) ). On entry, this array contains the the local pieces of the right hand sides sub( B ). .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 X (local input) REAL pointer into the local memory to an array of local dimension (LLD_X, LOCc(JX+NRHS-1) ). On entry, this array contains the the local pieces of the solution vectors sub( X ). .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 FERR (local output) REAL array of local dimension LOCc(JB+NRHS-1). The estimated forward error bounds for each solution vector of sub( X ). If XTRUE is the true solution, FERR bounds the magnitude of the largest entry in (sub( X ) - XTRUE) divided by the magnitude of the largest entry in sub( X ). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. This array is tied to the distributed matrix X. .TP 8 BERR (local output) REAL array of local dimension LOCc(JB+NRHS-1). The componentwise relative backward error of each solution vector (i.e., the smallest re- lative change in any entry of sub( A ) or sub( B ) that makes sub( X ) an exact solution). This array is tied to the distributed matrix X. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 3*LOCr( N + MOD( IA-1, MB_A ) ). If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/local output) INTEGER array, dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK. .TP 8 LIWORK (local or global input) INTEGER The dimension of the array IWORK. LIWORK is local input and must be at least LIWORK >= LOCr( N + MOD( IB-1, MB_B ) ). If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Notes ===== This routine temporarily returns when N <= 1. The distributed submatrices sub( X ) and sub( B ) should be distributed the same way on the same processes. These conditions ensure that sub( X ) and sub( B ) are "perfectly" aligned. Moreover, this routine requires the distributed submatrices sub( A ), sub( X ), and sub( B ) to be aligned on a block boundary, i.e., if f(x,y) = MOD( x-1, y ): f( IA, DESCA( MB_ ) ) = f( JA, DESCA( NB_ ) ) = 0, f( IB, DESCB( MB_ ) ) = f( JB, DESCB( NB_ ) ) = 0, and f( IX, DESCX( MB_ ) ) = f( JX, DESCX( NB_ ) ) = 0. scalapack-doc-1.5/man/manl/pstrti2.l0100644000056400000620000001203706335610653017035 0ustar pfrauenfstaff.TH PSTRTI2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSTRTI2 - compute the inverse of a real upper or lower triangular block matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PSTRTI2( UPLO, DIAG, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER DIAG, UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ) .SH PURPOSE PSTRTI2 computes the inverse of a real upper or lower triangular block matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). This matrix should be contained in one and only one process memory space (local operation). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 = 'U': sub( A ) is upper triangular; .br = 'L': sub( A ) is lower triangular. .TP 8 DIAG (global input) CHARACTER*1 .br = 'N': sub( A ) is non-unit triangular .br = 'U': sub( A ) is unit triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)), this array contains the local pieces of the triangular matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of the matrix sub( A ) contains the upper triangular matrix, and the strictly lower triangular part of sub( A ) is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of the matrix sub( A ) contains the lower triangular matrix, and the strictly upper triangular part of sub( A ) is not referenced. If DIAG = 'U', the diagonal elements of sub( A ) are also not referenced and are assumed to be 1. On exit, the (triangular) inverse of the original matrix, in the same storage format. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pstrtri.l0100644000056400000620000001212506335610653017133 0ustar pfrauenfstaff.TH PSTRTRI l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSTRTRI - compute the inverse of a upper or lower triangular distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PSTRTRI( UPLO, DIAG, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER DIAG, UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ) .SH PURPOSE PSTRTRI computes the inverse of a upper or lower triangular distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the distributed matrix sub( A ) is upper or lower triangular: .br = 'U': Upper triangular, .br = 'L': Lower triangular. .TP 8 DIAG (global input) CHARACTER Specifies whether or not the distributed matrix sub( A ) is unit triangular: .br = 'N': Non-unit triangular, .br = 'U': Unit triangular. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the triangular matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of the matrix sub( A ) contains the upper triangular matrix to be inverted, and the strictly lower triangular part of sub( A ) is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of the matrix sub( A ) contains the lower triangular matrix, and the strictly upper triangular part of sub( A ) is not referenced. On exit, the (triangular) inverse of the original matrix. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, A(IA+K-1,JA+K-1) is exactly zero. The triangular matrix sub( A ) is singular and its inverse can not be computed. scalapack-doc-1.5/man/manl/pstrtrs.l0100644000056400000620000001440106335610653017144 0ustar pfrauenfstaff.TH PSTRTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PSTRTRS - solve a triangular system of the form sub( A ) * X = sub( B ) or sub( A )**T * X = sub( B ), .SH SYNOPSIS .TP 20 SUBROUTINE PSTRTRS( UPLO, TRANS, DIAG, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, INFO ) .TP 20 .ti +4 CHARACTER DIAG, TRANS, UPLO .TP 20 .ti +4 INTEGER IA, IB, INFO, JA, JB, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 REAL A( * ), B( * ) .SH PURPOSE PSTRTRS solves a triangular system of the form where sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1) and is a triangular distributed matrix of order N, and B(IB:IB+N-1,JB:JB+NRHS-1) is an N-by-NRHS distributed matrix denoted by sub( B ). A check is made to verify that sub( A ) is nonsingular. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER = 'U': sub( A ) is upper triangular; .br = 'L': sub( A ) is lower triangular. .TP 8 TRANS (global input) CHARACTER .br Specifies the form of the system of equations: .br = 'N': Solve sub( A ) * X = sub( B ) (No transpose) .br = 'T': Solve sub( A )**T * X = sub( B ) (Transpose) .br = 'C': Solve sub( A )**T * X = sub( B ) (Transpose) .TP 8 DIAG (global input) CHARACTER .br = 'N': sub( A ) is non-unit triangular; .br = 'U': sub( A ) is unit triangular. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on i.e the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed matrix sub( B ). NRHS >= 0. .TP 8 A (local input) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1) ). This array contains the local pieces of the distributed triangular matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular matrix, and the strictly lower triangular part of sub( A ) is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular matrix, and the strictly upper triangular part of sub( A ) is not referenced. If DIAG = 'U', the diagonal elements of sub( A ) are also not referenced and are assumed to be 1. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_B,LOCc(JB+NRHS-1)). On entry, this array contains the local pieces of the right hand side distributed matrix sub( B ). On exit, if INFO = 0, sub( B ) is overwritten by the solution matrix X. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = i, the i-th diagonal element of sub( A ) is zero, indicating that the submatrix is singular and the solutions X have not been computed. scalapack-doc-1.5/man/manl/pstzrzf.l0100644000056400000620000001561606335610653017156 0ustar pfrauenfstaff.TH PSTZRZF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PSTZRZF - reduce the M-by-N ( M<=N ) real upper trapezoidal matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper triangular form by means of orthogonal transformations .SH SYNOPSIS .TP 20 SUBROUTINE PSTZRZF( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 REAL A( * ), TAU( * ), WORK( * ) .SH PURPOSE PSTZRZF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper triangular form by means of orthogonal transformations. The upper trapezoidal matrix sub( A ) is factored as .br sub( A ) = ( R 0 ) * Z, .br where Z is an N-by-N orthogonal matrix and R is an M-by-M upper triangular matrix. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the leading M-by-M upper triangular part of sub( A ) contains the upper trian- gular matrix R, and elements M+1 to N of the first M rows of sub( A ), with the array TAU, represent the orthogonal matrix Z as a product of M elementary reflectors. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) REAL, array, dimension LOCr(IA+M-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) REAL array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MB_A * ( Mp0 + Nq0 + MB_A ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The factorization is obtained by Householder's method. The kth transformation matrix, Z( k ), which is used to introduce zeros into the (m - k + 1)th row of sub( A ), is given in the form .br Z( k ) = ( I 0 ), .br ( 0 T( k ) ) .br where .br T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ), ( 0 ) ( z( k ) ) tau is a scalar and z( k ) is an ( n - m ) element vector. tau and z( k ) are chosen to annihilate the elements of the kth row of sub( A ). .br The scalar tau is returned in the kth element of TAU and the vector u( k ) in the kth row of sub( A ), such that the elements of z( k ) are in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in the upper triangular part of sub( A ). .br Z is given by .br Z = Z( 1 ) * Z( 2 ) * ... * Z( m ). .br scalapack-doc-1.5/man/manl/pzdbsv.l0100644000056400000620000000141406335610653016733 0ustar pfrauenfstaff.TH PZDBSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZDBSV - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 19 SUBROUTINE PZDBSV( N, BWL, BWU, NRHS, A, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 INTEGER BWL, BWU, IB, INFO, JA, LWORK, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 COMPLEX*16 A( * ), B( * ), WORK( * ) .SH PURPOSE PZDBSV solves a system of linear equations where A(1:N, JA:JA+N-1) is an N-by-N complex .br banded diagonally dominant-like distributed .br matrix with bandwidth BWL, BWU. .br Gaussian elimination without pivoting .br is used to factor a reordering .br of the matrix into L U. .br See PZDBTRF and PZDBTRS for details. .br scalapack-doc-1.5/man/manl/pzdbtrf.l0100644000056400000620000000214606335610653017101 0ustar pfrauenfstaff.TH PZDBTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZDBTRF - compute a LU factorization of an N-by-N complex banded diagonally dominant-like distributed matrix with bandwidth BWL, BWU .SH SYNOPSIS .TP 20 SUBROUTINE PZDBTRF( N, BWL, BWU, A, JA, DESCA, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER BWL, BWU, INFO, JA, LAF, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), AF( * ), WORK( * ) .SH PURPOSE PZDBTRF computes a LU factorization of an N-by-N complex banded diagonally dominant-like distributed matrix with bandwidth BWL, BWU: A(1:N, JA:JA+N-1). Reordering is used to increase parallelism in the factorization. This reordering results in factors that are DIFFERENT from those produced by equivalent sequential codes. These factors cannot be used directly by users; however, they can be used in .br subsequent calls to PZDBTRS to solve linear systems. .br The factorization has the form .br P A(1:N, JA:JA+N-1) P^T = L U .br where U is a banded upper triangular matrix and L is banded lower triangular, and P is a permutation matrix. .br scalapack-doc-1.5/man/manl/pzdbtrs.l0100644000056400000620000000166606335610653017124 0ustar pfrauenfstaff.TH PZDBTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZDBTRS - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 20 SUBROUTINE PZDBTRS( TRANS, N, BWL, BWU, NRHS, A, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER BWL, BWU, IB, INFO, JA, LAF, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), AF( * ), B( * ), WORK( * ) .SH PURPOSE PZDBTRS solves a system of linear equations or .br A(1:N, JA:JA+N-1)' * X = B(IB:IB+N-1, 1:NRHS) .br where A(1:N, JA:JA+N-1) is the matrix used to produce the factors stored in A(1:N,JA:JA+N-1) and AF by PZDBTRF. .br A(1:N, JA:JA+N-1) is an N-by-N complex .br banded diagonally dominant-like distributed .br matrix with bandwidth BWL, BWU. .br Routine PZDBTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pzdbtrsv.l0100644000056400000620000000221206335610653017276 0ustar pfrauenfstaff.TH PZDBTRSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZDBTRSV - solve a banded triangular system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 21 SUBROUTINE PZDBTRSV( UPLO, TRANS, N, BWL, BWU, NRHS, A, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 21 .ti +4 CHARACTER TRANS, UPLO .TP 21 .ti +4 INTEGER BWL, BWU, IB, INFO, JA, LAF, LWORK, N, NRHS .TP 21 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 21 .ti +4 COMPLEX*16 A( * ), AF( * ), B( * ), WORK( * ) .SH PURPOSE PZDBTRSV solves a banded triangular system of linear equations or .br A(1:N, JA:JA+N-1)^H * X = B(IB:IB+N-1, 1:NRHS) where A(1:N, JA:JA+N-1) is a banded .br triangular matrix factor produced by the .br Gaussian elimination code PZ@(dom_pre)BTRF .br and is stored in A(1:N,JA:JA+N-1) and AF. .br The matrix stored in A(1:N, JA:JA+N-1) is either .br upper or lower triangular according to UPLO, .br and the choice of solving A(1:N, JA:JA+N-1) or A(1:N, JA:JA+N-1)^H is dictated by the user by the parameter TRANS. .br Routine PZDBTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pzdrscl.l0100644000056400000620000001114506335610653017106 0ustar pfrauenfstaff.TH PZDRSCL l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZDRSCL - multiplie an N-element complex distributed vector sub( X ) by the real scalar 1/a .SH SYNOPSIS .TP 20 SUBROUTINE PZDRSCL( N, SA, SX, IX, JX, DESCX, INCX ) .TP 20 .ti +4 INTEGER IX, INCX, JX, N .TP 20 .ti +4 DOUBLE PRECISION SA .TP 20 .ti +4 INTEGER DESCX( * ) .TP 20 .ti +4 COMPLEX*16 SX( * ) .SH PURPOSE PZDRSCL multiplies an N-element complex distributed vector sub( X ) by the real scalar 1/a. This is done without overflow or underflow as long as the final sub( X )/a does not overflow or underflow. .br where sub( X ) denotes X(IX:IX+N-1,JX:JX), if INCX = 1, .br X(IX:IX,JX:JX+N-1), if INCX = M_X. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector descA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DT_A (global) descA[ DT_ ] The descriptor type. In this case, DT_A = 1. .br CTXT_A (global) descA[ CTXT_ ] The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) descA[ M_ ] The number of rows in the global array A. .br N_A (global) descA[ N_ ] The number of columns in the global array A. .br MB_A (global) descA[ MB_ ] The blocking factor used to distribu- te the rows of the array. .br NB_A (global) descA[ NB_ ] The blocking factor used to distribu- te the columns of the array. RSRC_A (global) descA[ RSRC_ ] The process row over which the first row of the array A is distributed. CSRC_A (global) descA[ CSRC_ ] The process column over which the first column of the array A is distributed. .br LLD_A (local) descA[ LLD_ ] The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be seen as particular matrices, a distributed vector is considered to be a distributed matrix. .br .SH ARGUMENTS .TP 8 N (global input) pointer to INTEGER The number of components of the distributed vector sub( X ). N >= 0. .TP 8 SA (global input) DOUBLE PRECISION The scalar a which is used to divide each component of sub( X ). SA must be >= 0, or the subroutine will divide by zero. .TP 8 SX (local input/local output) COMPLEX*16 array containing the local pieces of a distributed matrix of dimension of at least ( (JX-1)*M_X + IX + ( N - 1 )*abs( INCX ) ) This array contains the entries of the distributed vector sub( X ). .TP 8 IX (global input) pointer to INTEGER The global row index of the submatrix of the distributed matrix X to operate on. .TP 8 JX (global input) pointer to INTEGER The global column index of the submatrix of the distributed matrix X to operate on. .TP 8 DESCX (global and local input) INTEGER array of dimension 8. The array descriptor of the distributed matrix X. .TP 8 INCX (global input) pointer to INTEGER The global increment for the elements of X. Only two values of INCX are supported in this version, namely 1 and M_X. scalapack-doc-1.5/man/manl/pzdtsv.l0100644000056400000620000000137706335610653016765 0ustar pfrauenfstaff.TH PZDTSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZDTSV - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 19 SUBROUTINE PZDTSV( N, NRHS, DL, D, DU, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 INTEGER IB, INFO, JA, LWORK, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 COMPLEX*16 B( * ), D( * ), DL( * ), DU( * ), WORK( * ) .SH PURPOSE PZDTSV solves a system of linear equations where A(1:N, JA:JA+N-1) is an N-by-N complex .br tridiagonal diagonally dominant-like distributed .br matrix. .br Gaussian elimination without pivoting .br is used to factor a reordering .br of the matrix into L U. .br See PZDTTRF and PZDTTRS for details. .br scalapack-doc-1.5/man/manl/pzdttrf.l0100644000056400000620000000214106335610653017116 0ustar pfrauenfstaff.TH PZDTTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZDTTRF - compute a LU factorization of an N-by-N complex tridiagonal diagonally dominant-like distributed matrix A(1:N, JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PZDTTRF( N, DL, D, DU, JA, DESCA, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER INFO, JA, LAF, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 AF( * ), D( * ), DL( * ), DU( * ), WORK( * ) .SH PURPOSE PZDTTRF computes a LU factorization of an N-by-N complex tridiagonal diagonally dominant-like distributed matrix A(1:N, JA:JA+N-1). Reordering is used to increase parallelism in the factorization. This reordering results in factors that are DIFFERENT from those produced by equivalent sequential codes. These factors cannot be used directly by users; however, they can be used in .br subsequent calls to PZDTTRS to solve linear systems. .br The factorization has the form .br P A(1:N, JA:JA+N-1) P^T = L U .br where U is a tridiagonal upper triangular matrix and L is tridiagonal lower triangular, and P is a permutation matrix. .br scalapack-doc-1.5/man/manl/pzdttrs.l0100644000056400000620000000165106335610653017140 0ustar pfrauenfstaff.TH PZDTTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZDTTRS - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 20 SUBROUTINE PZDTTRS( TRANS, N, NRHS, DL, D, DU, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER IB, INFO, JA, LAF, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX*16 AF( * ), B( * ), D( * ), DL( * ), DU( * ), WORK( * ) .SH PURPOSE PZDTTRS solves a system of linear equations or .br A(1:N, JA:JA+N-1)' * X = B(IB:IB+N-1, 1:NRHS) .br where A(1:N, JA:JA+N-1) is the matrix used to produce the factors stored in A(1:N,JA:JA+N-1) and AF by PZDTTRF. .br A(1:N, JA:JA+N-1) is an N-by-N complex .br tridiagonal diagonally dominant-like distributed .br matrix. .br Routine PZDTTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pzdttrsv.l0100644000056400000620000000223706335610653017327 0ustar pfrauenfstaff.TH PZDTTRSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZDTTRSV - solve a tridiagonal triangular system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 21 SUBROUTINE PZDTTRSV( UPLO, TRANS, N, NRHS, DL, D, DU, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 21 .ti +4 CHARACTER TRANS, UPLO .TP 21 .ti +4 INTEGER IB, INFO, JA, LAF, LWORK, N, NRHS .TP 21 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 21 .ti +4 COMPLEX*16 AF( * ), B( * ), D( * ), DL( * ), DU( * ), WORK( * ) .SH PURPOSE PZDTTRSV solves a tridiagonal triangular system of linear equations or .br A(1:N, JA:JA+N-1)^H * X = B(IB:IB+N-1, 1:NRHS) where A(1:N, JA:JA+N-1) is a tridiagonal .br triangular matrix factor produced by the .br Gaussian elimination code PZ@(dom_pre)TTRF .br and is stored in A(1:N,JA:JA+N-1) and AF. .br The matrix stored in A(1:N, JA:JA+N-1) is either .br upper or lower triangular according to UPLO, .br and the choice of solving A(1:N, JA:JA+N-1) or A(1:N, JA:JA+N-1)^H is dictated by the user by the parameter TRANS. .br Routine PZDTTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pzgbsv.l0100644000056400000620000000140306335610654016735 0ustar pfrauenfstaff.TH PZGBSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGBSV - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 19 SUBROUTINE PZGBSV( N, BWL, BWU, NRHS, A, JA, DESCA, IPIV, B, IB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 INTEGER BWL, BWU, IB, INFO, JA, LWORK, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ), IPIV( * ) .TP 19 .ti +4 COMPLEX*16 A( * ), B( * ), WORK( * ) .SH PURPOSE PZGBSV solves a system of linear equations where A(1:N, JA:JA+N-1) is an N-by-N complex .br banded distributed .br matrix with bandwidth BWL, BWU. .br Gaussian elimination with pivoting .br is used to factor a reordering .br of the matrix into P L U. .br See PZGBTRF and PZGBTRS for details. .br scalapack-doc-1.5/man/manl/pzgbtrf.l0100644000056400000620000000237606335610654017112 0ustar pfrauenfstaff.TH PZGBTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGBTRF - compute a LU factorization of an N-by-N complex banded distributed matrix with bandwidth BWL, BWU .SH SYNOPSIS .TP 20 SUBROUTINE PZGBTRF( N, BWL, BWU, A, JA, DESCA, IPIV, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER BWL, BWU, INFO, JA, LAF, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), AF( * ), WORK( * ) .SH PURPOSE PZGBTRF computes a LU factorization of an N-by-N complex banded distributed matrix with bandwidth BWL, BWU: A(1:N, JA:JA+N-1). Reordering is used to increase parallelism in the factorization. This reordering results in factors that are DIFFERENT from those produced by equivalent sequential codes. These factors cannot be used directly by users; however, they can be used in .br subsequent calls to PZGBTRS to solve linear systems. .br The factorization has the form .br P A(1:N, JA:JA+N-1) Q = L U .br where U is a banded upper triangular matrix and L is banded lower triangular, and P and Q are permutation matrices. .br The matrix Q represents reordering of columns .br for parallelism's sake, while P represents .br reordering of rows for numerical stability using .br classic partial pivoting. .br scalapack-doc-1.5/man/manl/pzgbtrs.l0100644000056400000620000000165406335610654017125 0ustar pfrauenfstaff.TH PZGBTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGBTRS - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 20 SUBROUTINE PZGBTRS( TRANS, N, BWL, BWU, NRHS, A, JA, DESCA, IPIV, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER BWU, BWL, IB, INFO, JA, LAF, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ), IPIV(*) .TP 20 .ti +4 COMPLEX*16 A( * ), AF( * ), B( * ), WORK( * ) .SH PURPOSE PZGBTRS solves a system of linear equations or .br A(1:N, JA:JA+N-1)' * X = B(IB:IB+N-1, 1:NRHS) .br where A(1:N, JA:JA+N-1) is the matrix used to produce the factors stored in A(1:N,JA:JA+N-1) and AF by PZGBTRF. .br A(1:N, JA:JA+N-1) is an N-by-N complex .br banded distributed .br matrix with bandwidth BWL, BWU. .br Routine PZGBTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pzgebd2.l0100644000056400000620000002232506335610654016765 0ustar pfrauenfstaff.TH PZGEBD2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZGEBD2 - reduce a complex general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form B by an unitary transformation .SH SYNOPSIS .TP 20 SUBROUTINE PZGEBD2( M, N, A, IA, JA, DESCA, D, E, TAUQ, TAUP, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION D( * ), E( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAUP( * ), TAUQ( * ), WORK( * ) .SH PURPOSE PZGEBD2 reduces a complex general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form B by an unitary transformation: Q' * sub( A ) * P = B. If M >= N, B is upper bidiagonal; if M < N, B is lower bidiagonal. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the general distributed matrix sub( A ). On exit, if M >= N, the diagonal and the first superdiagonal of sub( A ) are overwritten with the upper bidiagonal matrix B; the elements below the diagonal, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. If M < N, the diagonal and the first subdiagonal are overwritten with the lower bidiagonal matrix B; the elements below the first subdiagonal, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) DOUBLE PRECISION array, dimension LOCc(JA+MIN(M,N)-1) if M >= N; LOCr(IA+MIN(M,N)-1) otherwise. The distributed diagonal elements of the bidiagonal matrix B: D(i) = A(i,i). D is tied to the distributed matrix A. .TP 8 E (local output) DOUBLE PRECISION array, dimension LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-2) otherwise. The distributed off-diagonal elements of the bidiagonal distributed matrix B: if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. E is tied to the distributed matrix A. .TP 8 TAUQ (local output) COMPLEX*16 array dimension LOCc(JA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the unitary matrix Q. TAUQ is tied to the distributed matrix A. See Further Details. TAUP (local output) COMPLEX*16 array, dimension LOCr(IA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the unitary matrix P. TAUP is tied to the distributed matrix A. See Further Details. WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MAX( MpA0, NqA0 ) where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ) IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, NB, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+IROFFA, NB, MYCOL, IACOL, NPCOL ). INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrices Q and P are represented as products of elementary reflectors: .br If m >= n, .br Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) Each H(i) and G(i) has the form: .br H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are complex scalars, and v and u are complex vectors; .br v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(ia+i:ia+m-1,ja+i-1); .br u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(ia+i-1,ja+i+1:ja+n-1); .br tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br If m < n, .br Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) Each H(i) and G(i) has the form: .br H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are complex scalars, and v and u are complex vectors; .br v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(ia+i+1:ia+m-1,ja+i-1); .br u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(ia+i-1,ja+i:ja+n-1); .br tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br The contents of sub( A ) on exit are illustrated by the following examples: .br m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) ( v1 v2 v3 v4 v5 ) .br where d and e denote diagonal and off-diagonal elements of B, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expressions should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) .br scalapack-doc-1.5/man/manl/pzgebrd.l0100644000056400000620000002234506335610654017067 0ustar pfrauenfstaff.TH PZGEBRD l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGEBRD - reduce a complex general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form B by an unitary transformation .SH SYNOPSIS .TP 20 SUBROUTINE PZGEBRD( M, N, A, IA, JA, DESCA, D, E, TAUQ, TAUP, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION D( * ), E( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAUP( * ), TAUQ( * ), WORK( * ) .SH PURPOSE PZGEBRD reduces a complex general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form B by an unitary transformation: Q' * sub( A ) * P = B. If M >= N, B is upper bidiagonal; if M < N, B is lower bidiagonal. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the general distributed matrix sub( A ). On exit, if M >= N, the diagonal and the first superdiagonal of sub( A ) are overwritten with the upper bidiagonal matrix B; the elements below the diagonal, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. If M < N, the diagonal and the first subdiagonal are overwritten with the lower bidiagonal matrix B; the elements below the first subdiagonal, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) DOUBLE PRECISION array, dimension LOCc(JA+MIN(M,N)-1) if M >= N; LOCr(IA+MIN(M,N)-1) otherwise. The distributed diagonal elements of the bidiagonal matrix B: D(i) = A(i,i). D is tied to the distributed matrix A. .TP 8 E (local output) DOUBLE PRECISION array, dimension LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-2) otherwise. The distributed off-diagonal elements of the bidiagonal distributed matrix B: if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. E is tied to the distributed matrix A. .TP 8 TAUQ (local output) COMPLEX*16 array dimension LOCc(JA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the unitary matrix Q. TAUQ is tied to the distributed matrix A. See Further Details. TAUP (local output) COMPLEX*16 array, dimension LOCr(IA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the unitary matrix P. TAUP is tied to the distributed matrix A. See Further Details. WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK( 1 ) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB*( MpA0 + NqA0 + 1 ) + NqA0 where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ), ICOFFA = MOD( JA-1, NB ), IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, NB, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB, MYCOL, IACOL, NPCOL ). INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrices Q and P are represented as products of elementary reflectors: .br If m >= n, .br Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) Each H(i) and G(i) has the form: .br H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are complex scalars, and v and u are complex vectors; .br v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(ia+i:ia+m-1,ja+i-1); .br u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(ia+i-1,ja+i+1:ja+n-1); .br tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br If m < n, .br Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) Each H(i) and G(i) has the form: .br H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are complex scalars, and v and u are complex vectors; .br v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(ia+i+1:ia+m-1,ja+i-1); .br u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(ia+i-1,ja+i:ja+n-1); .br tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br The contents of sub( A ) on exit are illustrated by the following examples: .br m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) ( v1 v2 v3 v4 v5 ) .br where d and e denote diagonal and off-diagonal elements of B, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expressions should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) .br scalapack-doc-1.5/man/manl/pzgecon.l0100644000056400000620000001552406335610654017100 0ustar pfrauenfstaff.TH PZGECON l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGECON - estimate the reciprocal of the condition number of a general distributed complex matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm, using the LU factorization computed by PZGETRF .SH SYNOPSIS .TP 20 SUBROUTINE PZGECON( NORM, N, A, IA, JA, DESCA, ANORM, RCOND, WORK, LWORK, RWORK, LRWORK, INFO ) .TP 20 .ti +4 CHARACTER NORM .TP 20 .ti +4 INTEGER IA, INFO, JA, LRWORK, LWORK, N .TP 20 .ti +4 DOUBLE PRECISION ANORM, RCOND .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION RWORK( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), WORK( * ) .SH PURPOSE PZGECON estimates the reciprocal of the condition number of a general distributed complex matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm, using the LU factorization computed by PZGETRF. An estimate is obtained for norm(inv(A(IA:IA+N-1,JA:JA+N-1))), and the reciprocal of the condition number is computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies whether the 1-norm condition number or the infinity-norm condition number is required: .br = '1' or 'O': 1-norm .br = 'I': Infinity-norm .TP 8 N (global input) INTEGER .br The order of the distributed matrix A(IA:IA+N-1,JA:JA+N-1). N >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension ( LLD_A, LOCc(JA+N-1) ). On entry, this array contains the local pieces of the factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U; the unit diagonal elements of L are not stored. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 ANORM (global input) DOUBLE PRECISION If NORM = '1' or 'O', the 1-norm of the original distributed matrix A(IA:IA+N-1,JA:JA+N-1). If NORM = 'I', the infinity-norm of the original distributed matrix A(IA:IA+N-1,JA:JA+N-1). .TP 8 RCOND (global output) DOUBLE PRECISION The reciprocal of the condition number of the distributed matrix A(IA:IA+N-1,JA:JA+N-1), computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * .br norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 2*LOCr(N+MOD(IA-1,MB_A)) + MAX( 2, MAX(NB_A*CEIL(NPROW-1,NPCOL),LOCc(N+MOD(JA-1,NB_A)) + NB_A*CEIL(NPCOL-1,NPROW)) ). LOCr and LOCc values can be computed using the ScaLAPACK tool function NUMROC; NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 RWORK (local workspace/local output) DOUBLE PRECISION array, dimension (LRWORK) On exit, RWORK(1) returns the minimal and optimal LRWORK. .TP 8 LRWORK (local or global input) INTEGER The dimension of the array RWORK. LRWORK is local input and must be at least LRWORK >= 2*LOCc(N+MOD(JA-1,NB_A)). If LRWORK = -1, then LRWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pzgeequ.l0100644000056400000620000001452406335610654017112 0ustar pfrauenfstaff.TH PZGEEQU l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGEEQU - compute row and column scalings intended to equilibrate an M-by-N distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA:JA+N-1) and reduce its condition number .SH SYNOPSIS .TP 20 SUBROUTINE PZGEEQU( M, N, A, IA, JA, DESCA, R, C, ROWCND, COLCND, AMAX, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, M, N .TP 20 .ti +4 DOUBLE PRECISION AMAX, COLCND, ROWCND .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION C( * ), R( * ) .TP 20 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZGEEQU computes row and column scalings intended to equilibrate an M-by-N distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA:JA+N-1) and reduce its condition number. R returns the row scale factors and C the column scale factors, chosen to try to make the largest entry in each row and column of the distributed matrix B with elements B(i,j) = R(i) * A(i,j) * C(j) have absolute value 1. .br R(i) and C(j) are restricted to be between SMLNUM = smallest safe number and BIGNUM = largest safe number. Use of these scaling factors is not guaranteed to reduce the condition number of sub( A ) but works well in practice. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension ( LLD_A, LOCc(JA+N-1) ), the local pieces of the M-by-N distributed matrix whose equilibration factors are to be computed. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 R (local output) DOUBLE PRECISION array, dimension LOCr(M_A) If INFO = 0 or INFO > IA+M-1, R(IA:IA+M-1) contains the row scale factors for sub( A ). R is aligned with the distributed matrix A, and replicated across every process column. R is tied to the distributed matrix A. .TP 8 C (local output) DOUBLE PRECISION array, dimension LOCc(N_A) If INFO = 0, C(JA:JA+N-1) contains the column scale factors for sub( A ). C is aligned with the distributed matrix A, and replicated down every process row. C is tied to the distri- buted matrix A. .TP 8 ROWCND (global output) DOUBLE PRECISION If INFO = 0 or INFO > IA+M-1, ROWCND contains the ratio of the smallest R(i) to the largest R(i) (IA <= i <= IA+M-1). If ROWCND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by R(IA:IA+M-1). .TP 8 COLCND (global output) DOUBLE PRECISION If INFO = 0, COLCND contains the ratio of the smallest C(j) to the largest C(j) (JA <= j <= JA+N-1). If COLCND >= 0.1, it is not worth scaling by C(JA:JA+N-1). .TP 8 AMAX (global output) DOUBLE PRECISION Absolute value of largest distributed matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = i, and i is .br <= M: the i-th row of the distributed matrix sub( A ) is exactly zero, > M: the (i-M)-th column of the distributed matrix sub( A ) is exactly zero. scalapack-doc-1.5/man/manl/pzgehd2.l0100644000056400000620000001652506335610654017000 0ustar pfrauenfstaff.TH PZGEHD2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZGEHD2 - reduce a complex general distributed matrix sub( A ) to upper Hessenberg form H by an unitary similarity transformation .SH SYNOPSIS .TP 20 SUBROUTINE PZGEHD2( N, ILO, IHI, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, IHI, ILO, INFO, JA, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), WORK( * ) .SH PURPOSE PZGEHD2 reduces a complex general distributed matrix sub( A ) to upper Hessenberg form H by an unitary similarity transformation: Q' * sub( A ) * Q = H, where .br sub( A ) = A(IA+N-1:IA+N-1,JA+N-1:JA+N-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 ILO (global input) INTEGER IHI (global input) INTEGER It is assumed that sub( A ) is already upper triangular in rows IA:IA+ILO-2 and IA+IHI:IA+N-1 and columns JA:JA+JLO-2 and JA+JHI:JA+N-1. See Further Details. If N > 0, .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N general distributed matrix sub( A ) to be reduced. On exit, the upper triangle and the first subdiagonal of sub( A ) are overwritten with the upper Hessenberg matrix H, and the ele- ments below the first subdiagonal, with the array TAU, repre- sent the unitary matrix Q as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX*16 array, dimension LOCc(JA+N-2) The scalar factors of the elementary reflectors (see Further Details). Elements JA:JA+ILO-2 and JA+IHI:JA+N-2 of TAU are set to zero. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK( 1 ) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB + MAX( NpA0, NB ) where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ), IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ), NpA0 = NUMROC( IHI+IROFFA, NB, MYROW, IAROW, NPROW ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of (ihi-ilo) elementary reflectors .br Q = H(ilo) H(ilo+1) . . . H(ihi-1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on exit in A(ia+ilo+i:ia+ihi-1,ja+ilo+i-2), and tau in TAU(ja+ilo+i-2). The contents of A(IA:IA+N-1,JA:JA+N-1) are illustrated by the follo- wing example, with n = 7, ilo = 2 and ihi = 6: .br on entry on exit .br ( a a a a a a a ) ( a a h h h h a ) ( a a a a a a ) ( a h h h h a ) ( a a a a a a ) ( h h h h h h ) ( a a a a a a ) ( v2 h h h h h ) ( a a a a a a ) ( v2 v3 h h h h ) ( a a a a a a ) ( v2 v3 v4 h h h ) ( a ) ( a ) where a denotes an element of the original matrix sub( A ), h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(ja+ilo+i-2). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expression should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) .br scalapack-doc-1.5/man/manl/pzgehrd.l0100644000056400000620000001716106335610654017075 0ustar pfrauenfstaff.TH PZGEHRD l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGEHRD - reduce a complex general distributed matrix sub( A ) to upper Hessenberg form H by an unitary similarity transformation .SH SYNOPSIS .TP 20 SUBROUTINE PZGEHRD( N, ILO, IHI, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, IHI, ILO, INFO, JA, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), WORK( * ) .SH PURPOSE PZGEHRD reduces a complex general distributed matrix sub( A ) to upper Hessenberg form H by an unitary similarity transformation: Q' * sub( A ) * Q = H, where .br sub( A ) = A(IA+N-1:IA+N-1,JA+N-1:JA+N-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 ILO (global input) INTEGER IHI (global input) INTEGER It is assumed that sub( A ) is already upper triangular in rows IA:IA+ILO-2 and IA+IHI:IA+N-1 and columns JA:JA+ILO-2 and JA+IHI:JA+N-1. See Further Details. If N > 0, .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N general distributed matrix sub( A ) to be reduced. On exit, the upper triangle and the first subdiagonal of sub( A ) are overwritten with the upper Hessenberg matrix H, and the ele- ments below the first subdiagonal, with the array TAU, repre- sent the unitary matrix Q as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX*16 array, dimension LOCc(JA+N-2) The scalar factors of the elementary reflectors (see Further Details). Elements JA:JA+ILO-2 and JA+IHI:JA+N-2 of TAU are set to zero. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK( 1 ) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB*NB + NB*MAX( IHIP+1, IHLP+INLQ ) where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ), ICOFFA = MOD( JA-1, NB ), IOFF = MOD( IA+ILO-2, NB ), IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ), IHIP = NUMROC( IHI+IROFFA, NB, MYROW, IAROW, NPROW ), ILROW = INDXG2P( IA+ILO-1, NB, MYROW, RSRC_A, NPROW ), IHLP = NUMROC( IHI-ILO+IOFF+1, NB, MYROW, ILROW, NPROW ), ILCOL = INDXG2P( JA+ILO-1, NB, MYCOL, CSRC_A, NPCOL ), INLQ = NUMROC( N-ILO+IOFF+1, NB, MYCOL, ILCOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of (ihi-ilo) elementary reflectors .br Q = H(ilo) H(ilo+1) . . . H(ihi-1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(1:I) = 0, v(I+1) = 1 and v(IHI+1:N) = 0; v(I+2:IHI) is stored on exit in A(IA+ILO+I:IA+IHI-1,JA+ILO+I-2), and tau in TAU(JA+ILO+I-2). The contents of A(IA:IA+N-1,JA:JA+N-1) are illustrated by the follow- ing example, with N = 7, ILO = 2 and IHI = 6: .br on entry on exit .br ( a a a a a a a ) ( a a h h h h a ) ( a a a a a a ) ( a h h h h a ) ( a a a a a a ) ( h h h h h h ) ( a a a a a a ) ( v2 h h h h h ) ( a a a a a a ) ( v2 v3 h h h h ) ( a a a a a a ) ( v2 v3 v4 h h h ) ( a ) ( a ) where a denotes an element of the original matrix sub( A ), H denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(JA+ILO+I-2). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expression should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) .br scalapack-doc-1.5/man/manl/pzgelq2.l0100644000056400000620000001423706335610654017017 0ustar pfrauenfstaff.TH PZGELQ2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGELQ2 - compute a LQ factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = L * Q .SH SYNOPSIS .TP 20 SUBROUTINE PZGELQ2( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), WORK( * ) .SH PURPOSE PZGELQ2 computes a LQ factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = L * Q. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the elements on and below the diagonal of sub( A ) contain the M by min(M,N) lower trapezoidal matrix L (L is lower triangular if M <= N); the elements above the diagonal, with the array TAU, repre- sent the unitary matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX*16, array, dimension LOCr(IA+MIN(M,N)-1). This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= Nq0 + MAX( 1, Mp0 ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ia+k-1)' H(ia+k-2)' . . . H(ia)', where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; conjg(v(i+1:n)) is stored on exit in A(ia+i-1,ja+i:ja+n-1), and tau in TAU(ia+i-1). .br scalapack-doc-1.5/man/manl/pzgelqf.l0100644000056400000620000001425006335610654017076 0ustar pfrauenfstaff.TH PZGELQF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGELQF - compute a LQ factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = L * Q .SH SYNOPSIS .TP 20 SUBROUTINE PZGELQF( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), WORK( * ) .SH PURPOSE PZGELQF computes a LQ factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = L * Q. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the elements on and below the diagonal of sub( A ) contain the M by min(M,N) lower trapezoidal matrix L (L is lower triangular if M <= N); the elements above the diagonal, with the array TAU, repre- sent the unitary matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX*16, array, dimension LOCr(IA+MIN(M,N)-1). This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MB_A * ( Mp0 + Nq0 + MB_A ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ia+k-1)' H(ia+k-2)' . . . H(ia)', where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; conjg(v(i+1:n)) is stored on exit in A(ia+i-1,ja+i:ja+n-1), and tau in TAU(ia+i-1). .br scalapack-doc-1.5/man/manl/pzgels.l0100644000056400000620000002222606335610655016735 0ustar pfrauenfstaff.TH PZGELS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGELS - solve overdetermined or underdetermined complex linear systems involving an M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1), .SH SYNOPSIS .TP 19 SUBROUTINE PZGELS( TRANS, M, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 CHARACTER TRANS .TP 19 .ti +4 INTEGER IA, IB, INFO, JA, JB, LWORK, M, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 COMPLEX*16 A( * ), B( * ), WORK( * ) .SH PURPOSE PZGELS solves overdetermined or underdetermined complex linear systems involving an M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1), or its conjugate-transpose, using a QR or LQ factorization of sub( A ). It is assumed that sub( A ) has full rank. .br The following options are provided: .br 1. If TRANS = 'N' and m >= n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem minimize || sub( B ) - sub( A )*X ||. .br 2. If TRANS = 'N' and m < n: find the minimum norm solution of an underdetermined system sub( A ) * X = sub( B ). .br 3. If TRANS = 'C' and m >= n: find the minimum norm solution of an undetermined system sub( A )**H * X = sub( B ). .br 4. If TRANS = 'C' and m < n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem minimize || sub( B ) - sub( A )**H * X ||. where sub( B ) denotes B( IB:IB+M-1, JB:JB+NRHS-1 ) when TRANS = 'N' and B( IB:IB+N-1, JB:JB+NRHS-1 ) otherwise. Several right hand side vectors b and solution vectors x can be handled in a single call; When TRANS = 'N', the solution vectors are stored as the columns of the N-by-NRHS right hand side matrix sub( B ) and the M-by-NRHS right hand side matrix sub( B ) otherwise. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 TRANS (global input) CHARACTER = 'N': the linear system involves sub( A ); .br = 'C': the linear system involves sub( A )**H. .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e. the number of columns of the distributed submatrices sub( B ) and X. NRHS >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of local dimension ( LLD_A, LOCc(JA+N-1) ). On entry, the M-by-N matrix A. if M >= N, sub( A ) is overwritten by details of its QR factorization as returned by PZGEQRF; if M < N, sub( A ) is overwritten by details of its LQ factorization as returned by PZGELQF. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input/local output) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_B, LOCc(JB+NRHS-1)). On entry, this array contains the local pieces of the distributed matrix B of right hand side vectors, stored columnwise; sub( B ) is M-by-NRHS if TRANS='N', and N-by-NRHS otherwise. On exit, sub( B ) is overwritten by the solution vectors, stored columnwise: if TRANS = 'N' and M >= N, rows 1 to N of sub( B ) contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements N+1 to M in that column; if TRANS = 'N' and M < N, rows 1 to N of sub( B ) contain the minimum norm solution vectors; if TRANS = 'C' and M >= N, rows 1 to M of sub( B ) contain the minimum norm solution vectors; if TRANS = 'C' and M < N, rows 1 to M of sub( B ) contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements M+1 to N in that column. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= LTAU + MAX( LWF, LWS ) where If M >= N, then LTAU = NUMROC( JA+MIN(M,N)-1, NB_A, MYCOL, CSRC_A, NPCOL ), LWF = NB_A * ( MpA0 + NqA0 + NB_A ) LWS = MAX( (NB_A*(NB_A-1))/2, (NRHSqB0 + MpB0)*NB_A ) + NB_A * NB_A Else LTAU = NUMROC( IA+MIN(M,N)-1, MB_A, MYROW, RSRC_A, NPROW ), LWF = MB_A * ( MpA0 + NqA0 + MB_A ) LWS = MAX( (MB_A*(MB_A-1))/2, ( NpB0 + MAX( NqA0 + NUMROC( NUMROC( N+IROFFB, MB_A, 0, 0, NPROW ), MB_A, 0, 0, LCMP ), NRHSqB0 ) )*MB_A ) + MB_A * MB_A End if where LCMP = LCM / NPROW with LCM = ILCM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFB = MOD( IB-1, MB_B ), ICOFFB = MOD( JB-1, NB_B ), IBROW = INDXG2P( IB, MB_B, MYROW, RSRC_B, NPROW ), IBCOL = INDXG2P( JB, NB_B, MYCOL, CSRC_B, NPCOL ), MpB0 = NUMROC( M+IROFFB, MB_B, MYROW, IBROW, NPROW ), NpB0 = NUMROC( N+IROFFB, MB_B, MYROW, IBROW, NPROW ), NRHSqB0 = NUMROC( NRHS+ICOFFB, NB_B, MYCOL, IBCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pzgeql2.l0100644000056400000620000001440106335610655017011 0ustar pfrauenfstaff.TH PZGEQL2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGEQL2 - compute a QL factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * L .SH SYNOPSIS .TP 20 SUBROUTINE PZGEQL2( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), WORK( * ) .SH PURPOSE PZGEQL2 computes a QL factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * L. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, if M >= N, the lower triangle of the distributed submatrix A( IA+M-N:IA+M-1, JA:JA+N-1 ) contains the N-by-N lower triangular matrix L; if M <= N, the elements on and below the (N-M)-th superdiagonal contain the M by N lower trapezoidal matrix L; the remaining elements, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX*16, array, dimension LOCc(JA+N-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= Mp0 + MAX( 1, Nq0 ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ja+k-1) . . . H(ja+1) H(ja), where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in A(ia:ia+m-k+i-2,ja+n-k+i-1), and tau in TAU(ja+n-k+i-1). .br scalapack-doc-1.5/man/manl/pzgeqlf.l0100644000056400000620000001441206335610655017077 0ustar pfrauenfstaff.TH PZGEQLF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGEQLF - compute a QL factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * L .SH SYNOPSIS .TP 20 SUBROUTINE PZGEQLF( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), WORK( * ) .SH PURPOSE PZGEQLF computes a QL factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * L. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, if M >= N, the lower triangle of the distributed submatrix A( IA+M-N:IA+M-1, JA:JA+N-1 ) contains the N-by-N lower triangular matrix L; if M <= N, the elements on and below the (N-M)-th superdiagonal contain the M by N lower trapezoidal matrix L; the remaining elements, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX*16, array, dimension LOCc(JA+N-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB_A * ( Mp0 + Nq0 + NB_A ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ja+k-1) . . . H(ja+1) H(ja), where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in A(ia:ia+m-k+i-2,ja+n-k+i-1), and tau in TAU(ja+n-k+i-1). .br scalapack-doc-1.5/man/manl/pzgeqpf.l0100644000056400000620000001631006335610655017102 0ustar pfrauenfstaff.TH PZGEQPF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGEQPF - compute a QR factorization with column pivoting of a M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PZGEQPF( M, N, A, IA, JA, DESCA, IPIV, TAU, WORK, LWORK, RWORK, LRWORK, INFO ) .TP 20 .ti +4 INTEGER IA, JA, INFO, LRWORK, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ) .TP 20 .ti +4 DOUBLE PRECISION RWORK( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), WORK( * ) .SH PURPOSE PZGEQPF computes a QR factorization with column pivoting of a M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1): sub( A ) * P = Q * R. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the elements on and above the diagonal of sub( A ) contain the min(M,N) by N upper trapezoidal matrix R (R is upper triangular if M >= N); the elements below the diagonal, with the array TAU, repre- sent the unitary matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local output) INTEGER array, dimension LOCc(JA+N-1). On exit, if IPIV(I) = K, the local i-th column of sub( A )*P was the global K-th column of sub( A ). IPIV is tied to the distributed matrix A. .TP 8 TAU (local output) COMPLEX*16, array, dimension LOCc(JA+MIN(M,N)-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MAX(3,Mp0 + Nq0). If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 RWORK (local workspace/local output) DOUBLE PRECISION array, dimension (LRWORK) On exit, RWORK(1) returns the minimal and optimal LRWORK. .TP 8 LRWORK (local or global input) INTEGER The dimension of the array RWORK. LRWORK is local input and must be at least LRWORK >= LOCc(JA+N-1)+Nq0. IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), LOCc(JA+N-1) = NUMROC( JA+N-1, NB_A, MYCOL, CSRC_A, NPCOL ) and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LRWORK = -1, then LRWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) . . . H(n) .br Each H(i) has the form .br H = I - tau * v * v' .br 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 .br A(ia+i-1:ia+m-1,ja+i-1). .br The matrix P is represented in jpvt as follows: If .br jpvt(j) = i .br then the jth column of P is the ith canonical unit vector. scalapack-doc-1.5/man/manl/pzgeqr2.l0100644000056400000620000001423106335610655017020 0ustar pfrauenfstaff.TH PZGEQR2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGEQR2 - compute a QR factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * R .SH SYNOPSIS .TP 20 SUBROUTINE PZGEQR2( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), WORK( * ) .SH PURPOSE PZGEQR2 computes a QR factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * R. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the elements on and above the diagonal of sub( A ) contain the min(M,N) by N upper trapezoidal matrix R (R is upper triangular if M >= N); the elements below the diagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX*16, array, dimension LOCc(JA+MIN(M,N)-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= Mp0 + MAX( 1, Nq0 ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ja) H(ja+1) . . . H(ja+k-1), where k = min(m,n). Each H(i) has the form .br H(j) = I - tau * v * v' .br 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 .br A(ia+i:ia+m-1,ja+i-1), and tau in TAU(ja+i-1). .br scalapack-doc-1.5/man/manl/pzgeqrf.l0100644000056400000620000001424206335610655017106 0ustar pfrauenfstaff.TH PZGEQRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGEQRF - compute a QR factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * R .SH SYNOPSIS .TP 20 SUBROUTINE PZGEQRF( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), WORK( * ) .SH PURPOSE PZGEQRF computes a QR factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * R. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the elements on and above the diagonal of sub( A ) contain the min(M,N) by N upper trapezoidal matrix R (R is upper triangular if M >= N); the elements below the diagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX*16, array, dimension LOCc(JA+MIN(M,N)-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB_A * ( Mp0 + Nq0 + NB_A ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ja) H(ja+1) . . . H(ja+k-1), where k = min(m,n). Each H(i) has the form .br H(j) = I - tau * v * v' .br 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 .br A(ia+i:ia+m-1,ja+i-1), and tau in TAU(ja+i-1). .br scalapack-doc-1.5/man/manl/pzgerfs.l0100644000056400000620000002340506335610655017111 0ustar pfrauenfstaff.TH PZGERFS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGERFS - improve the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solutions .SH SYNOPSIS .TP 20 SUBROUTINE PZGERFS( TRANS, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, IPIV, B, IB, JB, DESCB, X, IX, JX, DESCX, FERR, BERR, WORK, LWORK, RWORK, LRWORK, INFO ) .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER IA, IAF, IB, IX, INFO, JA, JAF, JB, JX, LRWORK, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * ), IPIV( * ) .TP 20 .ti +4 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), AF( * ), B( * ), WORK( * ), X( * ) .SH PURPOSE PZGERFS improves the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solutions. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br In the following comments, sub( A ), sub( X ) and sub( B ) denote respectively A(IA:IA+N-1,JA:JA+N-1), X(IX:IX+N-1,JX:JX+NRHS-1) and B(IB:IB+N-1,JB:JB+NRHS-1). .br .SH ARGUMENTS .TP 8 TRANS (global input) CHARACTER*1 Specifies the form of the system of equations. = 'N': sub( A ) * sub( X ) = sub( B ) (No transpose) .br = 'T': sub( A )**T * sub( X ) = sub( B ) (Transpose) .br = 'C': sub( A )**H * sub( X ) = sub( B ) (Conjugate transpose) .TP 8 N (global input) INTEGER The order of the matrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the matrices sub( B ) and sub( X ). NRHS >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1)). This array contains the local pieces of the distributed matrix sub( A ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 AF (local input) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_AF,LOCc(JA+N-1)). This array contains the local pieces of the distributed factors of the matrix sub( A ) = P * L * U as computed by PZGETRF. .TP 8 IAF (global input) INTEGER The row index in the global array AF indicating the first row of sub( AF ). .TP 8 JAF (global input) INTEGER The column index in the global array AF indicating the first column of sub( AF ). .TP 8 DESCAF (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix AF. .TP 8 IPIV (local input) INTEGER array of dimension LOCr(M_AF)+MB_AF. This array contains the pivoting information as computed by PZGETRF. IPIV(i) -> The global row local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 B (local input) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_B,LOCc(JB+NRHS-1)). This array contains the local pieces of the distributed matrix of right hand sides sub( B ). .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 X (local input and output) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_X,LOCc(JX+NRHS-1)). On entry, this array contains the local pieces of the distributed matrix solution sub( X ). On exit, the improved solution vectors. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 FERR (local output) DOUBLE PRECISION array of local dimension LOCc(JB+NRHS-1). The estimated forward error bound for each solution vector of sub( X ). If XTRUE is the true solution corresponding to sub( X ), FERR is an estimated upper bound for the magnitude of the largest element in (sub( X ) - XTRUE) divided by the magnitude of the largest element in sub( X ). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. This array is tied to the distributed matrix X. .TP 8 BERR (local output) DOUBLE PRECISION array of local dimension LOCc(JB+NRHS-1). The componentwise relative backward error of each solution vector (i.e., the smallest re- lative change in any entry of sub( A ) or sub( B ) that makes sub( X ) an exact solution). This array is tied to the distributed matrix X. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 2*LOCr( N + MOD(IA-1,MB_A) ) If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 RWORK (local workspace/local output) DOUBLE PRECISION array, dimension (LRWORK) On exit, RWORK(1) returns the minimal and optimal LRWORK. .TP 8 LRWORK (local or global input) INTEGER The dimension of the array RWORK. LRWORK is local input and must be at least LRWORK >= LOCr( N + MOD(IB-1,MB_B) ). If LRWORK = -1, then LRWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH PARAMETERS ITMAX is the maximum number of steps of iterative refinement. Notes ===== This routine temporarily returns when N <= 1. The distributed submatrices op( A ) and op( AF ) (respectively sub( X ) and sub( B ) ) should be distributed the same way on the same processes. These conditions ensure that sub( A ) and sub( AF ) (resp. sub( X ) and sub( B ) ) are "perfectly" aligned. Moreover, this routine requires the distributed submatrices sub( A ), sub( AF ), sub( X ), and sub( B ) to be aligned on a block boundary, i.e., if f(x,y) = MOD( x-1, y ): f( IA, DESCA( MB_ ) ) = f( JA, DESCA( NB_ ) ) = 0, f( IAF, DESCAF( MB_ ) ) = f( JAF, DESCAF( NB_ ) ) = 0, f( IB, DESCB( MB_ ) ) = f( JB, DESCB( NB_ ) ) = 0, and f( IX, DESCX( MB_ ) ) = f( JX, DESCX( NB_ ) ) = 0. scalapack-doc-1.5/man/manl/pzgerq2.l0100644000056400000620000001435306335610655017025 0ustar pfrauenfstaff.TH PZGERQ2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGERQ2 - compute a RQ factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = R * Q .SH SYNOPSIS .TP 20 SUBROUTINE PZGERQ2( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), WORK( * ) .SH PURPOSE PZGERQ2 computes a RQ factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = R * Q. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, if M <= N, the upper triangle of A( IA:IA+M-1, JA+N-M:JA+N-1 ) 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 unitary matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX*16, array, dimension LOCr(IA+M-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= Nq0 + MAX( 1, Mp0 ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ia)' H(ia+1)' . . . H(ia+k-1)', where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br 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(ia+m-k+i-1,ja:ja+n-k+i-2), and tau in TAU(ia+m-k+i-1). scalapack-doc-1.5/man/manl/pzgerqf.l0100644000056400000620000001436406335610655017113 0ustar pfrauenfstaff.TH PZGERQF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGERQF - compute a RQ factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = R * Q .SH SYNOPSIS .TP 20 SUBROUTINE PZGERQF( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), WORK( * ) .SH PURPOSE PZGERQF computes a RQ factorization of a complex distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = R * Q. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, if M <= N, the upper triangle of A( IA:IA+M-1, JA+N-M:JA+N-1 ) 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 unitary matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX*16, array, dimension LOCr(IA+M-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MB_A * ( Mp0 + Nq0 + MB_A ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ia)' H(ia+1)' . . . H(ia+k-1)', where k = min(m,n). Each H(i) has the form .br H(i) = I - tau * v * v' .br 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(ia+m-k+i-1,ja:ja+n-k+i-2), and tau in TAU(ia+m-k+i-1). scalapack-doc-1.5/man/manl/pzgesv.l0100644000056400000620000001377006335610655016753 0ustar pfrauenfstaff.TH PZGESV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGESV - compute the solution to a complex system of linear equations sub( A ) * X = sub( B ), .SH SYNOPSIS .TP 19 SUBROUTINE PZGESV( N, NRHS, A, IA, JA, DESCA, IPIV, B, IB, JB, DESCB, INFO ) .TP 19 .ti +4 INTEGER IA, IB, INFO, JA, JB, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ), IPIV( * ) .TP 19 .ti +4 COMPLEX*16 A( * ), B( * ) .SH PURPOSE PZGESV computes the solution to a complex system of linear equations where sub( A ) = A(IA:IA+N-1,JA:JA+N-1) is an N-by-N distributed matrix and X and sub( B ) = B(IB:IB+N-1,JB:JB+NRHS-1) are N-by-NRHS distributed matrices. .br The LU decomposition with partial pivoting and row interchanges is used to factor sub( A ) as sub( A ) = P * L * U, where P is a permu- tation matrix, L is unit lower triangular, and U is upper triangular. L and U are stored in sub( A ). The factored form of sub( A ) is then used to solve the system of equations sub( A ) * X = sub( B ). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block decomposition ( MB_A = NB_A ). .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed submatrix sub( A ). NRHS >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the local pieces of the N-by-N distributed matrix sub( A ) to be factored. On exit, this array contains the local pieces of the factors L and U from the factorization sub( A ) = P*L*U; the unit diagonal elements of L are not stored. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local output) INTEGER array, dimension ( LOCr(M_A)+MB_A ) This array contains the pivoting information. IPIV(i) -> The global row local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 B (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_B,LOCc(JB+NRHS-1)). On entry, the right hand side distributed matrix sub( B ). On exit, if INFO = 0, sub( B ) is overwritten by the solution distributed matrix X. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, U(IA+K-1,JA+K-1) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution could not be computed. scalapack-doc-1.5/man/manl/pzgesvx.l0100644000056400000620000004050106335610655017133 0ustar pfrauenfstaff.TH PZGESVX l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGESVX - use the LU factorization to compute the solution to a complex system of linear equations A(IA:IA+N-1,JA:JA+N-1) * X = B(IB:IB+N-1,JB:JB+NRHS-1), .SH SYNOPSIS .TP 20 SUBROUTINE PZGESVX( FACT, TRANS, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, IPIV, EQUED, R, C, B, IB, JB, DESCB, X, IX, JX, DESCX, RCOND, FERR, BERR, WORK, LWORK, RWORK, LRWORK, INFO ) .TP 20 .ti +4 CHARACTER EQUED, FACT, TRANS .TP 20 .ti +4 INTEGER IA, IAF, IB, INFO, IX, JA, JAF, JB, JX, LRWORK, LWORK, N, NRHS .TP 20 .ti +4 DOUBLE PRECISION RCOND .TP 20 .ti +4 INTEGER DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * ), IPIV( * ) .TP 20 .ti +4 DOUBLE PRECISION BERR( * ), C( * ), FERR( * ), R( * ), RWORK( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), AF( * ), B( * ), WORK( * ), X( * ) .SH PURPOSE PZGESVX uses the LU factorization to compute the solution to a complex system of linear equations where A(IA:IA+N-1,JA:JA+N-1) is an N-by-N matrix and X and B(IB:IB+N-1,JB:JB+NRHS-1) are N-by-NRHS matrices. .br Error bounds on the solution and a condition estimate are also provided. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH DESCRIPTION In the following description, A denotes A(IA:IA+N-1,JA:JA+N-1), B denotes B(IB:IB+N-1,JB:JB+NRHS-1) and X denotes .br X(IX:IX+N-1,JX:JX+NRHS-1). .br The following steps are performed: .br 1. If FACT = 'E', real scaling factors are computed to equilibrate the system: .br 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'). .br 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as .br A = P * L * U, .br where P is a permutation matrix, L is a unit lower triangular matrix, and U is upper triangular. .br 3. 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, steps 4-6 are skipped. .br 4. The system of equations is solved for X using the factored form of A. .br 5. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. .br 6. If FACT = 'E' and 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. .br .SH ARGUMENTS .TP 8 FACT (global input) CHARACTER Specifies whether or not the factored form of the matrix A(IA:IA+N-1,JA:JA+N-1) is supplied on entry, and if not, .br whether the matrix A(IA:IA+N-1,JA:JA+N-1) should be equilibrated before it is factored. = 'F': On entry, AF(IAF:IAF+N-1,JAF:JAF+N-1) and IPIV con- .br tain the factored form of A(IA:IA+N-1,JA:JA+N-1). If EQUED is not 'N', the matrix A(IA:IA+N-1,JA:JA+N-1) has been equilibrated with scaling factors given by R and C. A(IA:IA+N-1,JA:JA+N-1), AF(IAF:IAF+N-1,JAF:JAF+N-1), and IPIV are not modified. = 'N': The matrix A(IA:IA+N-1,JA:JA+N-1) will be copied to .br AF(IAF:IAF+N-1,JAF:JAF+N-1) and factored. .br = 'E': The matrix A(IA:IA+N-1,JA:JA+N-1) will be equili- brated if necessary, then copied to AF(IAF:IAF+N-1,JAF:JAF+N-1) and factored. .TP 8 TRANS (global input) CHARACTER .br Specifies the form of the system of equations: .br = 'N': A(IA:IA+N-1,JA:JA+N-1) * X(IX:IX+N-1,JX:JX+NRHS-1) .br = B(IB:IB+N-1,JB:JB+NRHS-1) (No transpose) .br = 'T': A(IA:IA+N-1,JA:JA+N-1)**T * X(IX:IX+N-1,JX:JX+NRHS-1) .br = B(IB:IB+N-1,JB:JB+NRHS-1) (Transpose) .br = 'C': A(IA:IA+N-1,JA:JA+N-1)**H * X(IX:IX+N-1,JX:JX+NRHS-1) .br = B(IB:IB+N-1,JB:JB+NRHS-1) (Conjugate transpose) .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix A(IA:IA+N-1,JA:JA+N-1). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right-hand sides, i.e., the number of columns of the distributed submatrices B(IB:IB+N-1,JB:JB+NRHS-1) and .br X(IX:IX+N-1,JX:JX+NRHS-1). NRHS >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1)). On entry, the N-by-N matrix A(IA:IA+N-1,JA:JA+N-1). If FACT = 'F' and EQUED is not 'N', .br then A(IA:IA+N-1,JA:JA+N-1) must have been equilibrated by .br the scaling factors in R and/or C. A(IA:IA+N-1,JA:JA+N-1) is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. On exit, if EQUED .ne. 'N', A(IA:IA+N-1,JA:JA+N-1) is scaled as follows: .br EQUED = 'R': A(IA:IA+N-1,JA:JA+N-1) := .br diag(R) * A(IA:IA+N-1,JA:JA+N-1) .br EQUED = 'C': A(IA:IA+N-1,JA:JA+N-1) := .br A(IA:IA+N-1,JA:JA+N-1) * diag(C) .br EQUED = 'B': A(IA:IA+N-1,JA:JA+N-1) := .br diag(R) * A(IA:IA+N-1,JA:JA+N-1) * diag(C). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 AF (local input or local output) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_AF,LOCc(JA+N-1)). If FACT = 'F', then AF(IAF:IAF+N-1,JAF:JAF+N-1) is an input argument and on entry contains the factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U as computed by PZGETRF. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix A(IA:IA+N-1,JA:JA+N-1). If FACT = 'N', then AF(IAF:IAF+N-1,JAF:JAF+N-1) is an output argument and on exit returns the factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the original .br matrix A(IA:IA+N-1,JA:JA+N-1). If FACT = 'E', then AF(IAF:IAF+N-1,JAF:JAF+N-1) is an output argument and on exit returns the factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the equili- .br brated matrix A(IA:IA+N-1,JA:JA+N-1) (see the description of .br A(IA:IA+N-1,JA:JA+N-1) for the form of the equilibrated matrix). .TP 8 IAF (global input) INTEGER The row index in the global array AF indicating the first row of sub( AF ). .TP 8 JAF (global input) INTEGER The column index in the global array AF indicating the first column of sub( AF ). .TP 8 DESCAF (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix AF. .TP 8 IPIV (local input or local output) INTEGER array, dimension LOCr(M_A)+MB_A. If FACT = 'F', then IPIV is an input argu- ment and on entry contains the pivot indices from the fac- torization A(IA:IA+N-1,JA:JA+N-1) = P*L*U as computed by PZGETRF; IPIV(i) -> The global row local row i was swapped with. This array must be aligned with A( IA:IA+N-1, * ). If FACT = 'N', then IPIV is an output argument and on exit contains the pivot indices from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the original matrix .br A(IA:IA+N-1,JA:JA+N-1). If FACT = 'E', then IPIV is an output argument and on exit contains the pivot indices from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the equilibrated matrix .br A(IA:IA+N-1,JA:JA+N-1). .TP 8 EQUED (global input or global output) CHARACTER Specifies the form of equilibration that was done. = 'N': No equilibration (always true if FACT = 'N'). .br = 'R': Row equilibration, i.e., A(IA:IA+N-1,JA:JA+N-1) has been premultiplied by diag(R). = 'C': Column equilibration, i.e., A(IA:IA+N-1,JA:JA+N-1) has been postmultiplied by diag(C). = 'B': Both row and column equilibration, i.e., .br A(IA:IA+N-1,JA:JA+N-1) has been replaced by .br diag(R) * A(IA:IA+N-1,JA:JA+N-1) * diag(C). EQUED is an input variable if FACT = 'F'; otherwise, it is an output variable. .TP 8 R (local input or local output) DOUBLE PRECISION array, dimension LOCr(M_A). The row scale factors for A(IA:IA+N-1,JA:JA+N-1). .br If EQUED = 'R' or 'B', A(IA:IA+N-1,JA:JA+N-1) is multiplied on the left by diag(R); if EQUED='N' or 'C', R is not acces- sed. R is an input variable if FACT = 'F'; otherwise, R is an output variable. If FACT = 'F' and EQUED = 'R' or 'B', each element of R must be positive. R is replicated in every process column, and is aligned with the distributed matrix A. .TP 8 C (local input or local output) DOUBLE PRECISION array, dimension LOCc(N_A). The column scale factors for A(IA:IA+N-1,JA:JA+N-1). .br If EQUED = 'C' or 'B', A(IA:IA+N-1,JA:JA+N-1) is multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is not accessed. C is an input variable if FACT = 'F'; otherwise, C is an output variable. If FACT = 'F' and EQUED = 'C' or 'B', each element of C must be positive. C is replicated in every process row, and is aligned with the distributed matrix A. .TP 8 B (local input/local output) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_B,LOCc(JB+NRHS-1) ). On entry, the N-by-NRHS right-hand side matrix B(IB:IB+N-1,JB:JB+NRHS-1). On exit, if .br EQUED = 'N', B(IB:IB+N-1,JB:JB+NRHS-1) is not modified; if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B(IB:IB+N-1,JB:JB+NRHS-1); if TRANS = 'T' or 'C' .br and EQUED = 'C' or 'B', B(IB:IB+N-1,JB:JB+NRHS-1) is over- .br written by diag(C)*B(IB:IB+N-1,JB:JB+NRHS-1). .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 X (local input/local output) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_X, LOCc(JX+NRHS-1)). If INFO = 0, the N-by-NRHS solution matrix X(IX:IX+N-1,JX:JX+NRHS-1) to the original .br system of equations. Note that A(IA:IA+N-1,JA:JA+N-1) and .br B(IB:IB+N-1,JB:JB+NRHS-1) are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(C))*X(IX:IX+N-1,JX:JX+NRHS-1) if TRANS = 'N' and EQUED = 'C' or 'B', or inv(diag(R))*X(IX:IX+N-1,JX:JX+NRHS-1) if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 RCOND (global output) DOUBLE PRECISION The estimate of the reciprocal condition number of the matrix A(IA:IA+N-1,JA:JA+N-1) 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. .TP 8 FERR (local output) DOUBLE PRECISION array, dimension LOCc(N_B) The estimated forward error bounds for each solution vector X(j) (the j-th column of the solution matrix X(IX:IX+N-1,JX:JX+NRHS-1). If XTRUE is the true solution, FERR(j) bounds the magnitude of the largest entry in (X(j) - XTRUE) divided by the magnitude of the largest entry in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. FERR is replicated in every process row, and is aligned with the matrices B and X. .TP 8 BERR (local output) DOUBLE PRECISION array, dimension LOCc(N_B). The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any entry of A(IA:IA+N-1,JA:JA+N-1) or .br B(IB:IB+N-1,JB:JB+NRHS-1) that makes X(j) an exact solution). BERR is replicated in every process row, and is aligned with the matrices B and X. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK = MAX( PZGECON( LWORK ), PZGERFS( LWORK ) ) + LOCr( N_A ). If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 RWORK (local workspace/local output) DOUBLE PRECISION array, dimension (LRWORK) On exit, RWORK(1) returns the minimal and optimal LRWORK. .TP 8 LRWORK (local or global input) INTEGER The dimension of the array RWORK. LRWORK is local input and must be at least LRWORK = 2*LOCc(N_A). If LRWORK = -1, then LRWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 0: if INFO = i, and i is .br <= N: U(IA+I-1,IA+I-1) 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. = N+1: RCOND is less than machine precision. The factorization has been completed, but the matrix is singular to working precision, and the solution and error bounds have not been computed. scalapack-doc-1.5/man/manl/pzgetf2.l0100644000056400000620000001261406335610656017013 0ustar pfrauenfstaff.TH PZGETF2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGETF2 - compute an LU factorization of a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) using partial pivoting with row interchanges .SH SYNOPSIS .TP 20 SUBROUTINE PZGETF2( M, N, A, IA, JA, DESCA, IPIV, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, M, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ) .TP 20 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZGETF2 computes an LU factorization of a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) using partial pivoting with row interchanges. The factorization has the form sub( A ) = P * L * U, where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n). .br This is the right-looking Parallel Level 2 BLAS version of the algorithm. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires N <= NB_A-MOD(JA-1, NB_A) and square block decomposition ( MB_A = NB_A ). .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). NB_A-MOD(JA-1, NB_A) >= N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the M-by-N distributed matrix sub( A ). On exit, this array contains the local pieces of the factors L and U from the factoriza- tion sub( A ) = P*L*U; the unit diagonal elements of L are not stored. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local output) INTEGER array, dimension ( LOCr(M_A)+MB_A ) This array contains the pivoting information. IPIV(i) -> The global row local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, U(IA+K-1,JA+K-1) 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. scalapack-doc-1.5/man/manl/pzgetrf.l0100644000056400000620000001257606335610656017122 0ustar pfrauenfstaff.TH PZGETRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGETRF - compute an LU factorization of a general M-by-N distributed matrix sub( A ) = (IA:IA+M-1,JA:JA+N-1) using partial pivoting with row interchanges .SH SYNOPSIS .TP 20 SUBROUTINE PZGETRF( M, N, A, IA, JA, DESCA, IPIV, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, M, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ) .TP 20 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZGETRF computes an LU factorization of a general M-by-N distributed matrix sub( A ) = (IA:IA+M-1,JA:JA+N-1) using partial pivoting with row interchanges. The factorization has the form sub( A ) = P * L * U, where P is a permutation matrix, L is lower triangular with unit diagonal ele- ments (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n). L and U are stored in sub( A ). This is the right-looking Parallel Level 3 BLAS version of the algorithm. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block decomposition ( MB_A = NB_A ). .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the M-by-N distributed matrix sub( A ) to be factored. On exit, this array contains the local pieces of the factors L and U from the factorization sub( A ) = P*L*U; the unit diagonal ele- ments of L are not stored. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local output) INTEGER array, dimension ( LOCr(M_A)+MB_A ) This array contains the pivoting information. IPIV(i) -> The global row local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, U(IA+K-1,JA+K-1) 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. scalapack-doc-1.5/man/manl/pzgetri.l0100644000056400000620000001442606335610656017121 0ustar pfrauenfstaff.TH PZGETRI l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGETRI - compute the inverse of a distributed matrix using the LU factorization computed by PZGETRF .SH SYNOPSIS .TP 20 SUBROUTINE PZGETRI( N, A, IA, JA, DESCA, IPIV, WORK, LWORK, IWORK, LIWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LIWORK, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ), IWORK( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), WORK( * ) .SH PURPOSE PZGETRI computes the inverse of a distributed matrix using the LU factorization computed by PZGETRF. This method inverts U and then computes the inverse of sub( A ) = A(IA:IA+N-1,JA:JA+N-1) denoted InvA by solving the system InvA*L = inv(U) for InvA. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the local pieces of the L and U obtained by the factorization sub( A ) = P*L*U computed by PZGETRF. On exit, if INFO = 0, sub( A ) contains the inverse of the original distributed matrix sub( A ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local input) INTEGER array, dimension LOCr(M_A)+MB_A keeps track of the pivoting information. IPIV(i) is the global row index the local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK = LOCr(N+MOD(IA-1,MB_A))*NB_A. WORK is used to keep a copy of at most an entire column block of sub( A ). If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/local output) INTEGER array, dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK. .TP 8 LIWORK (local or global input) INTEGER The dimension of the array IWORK used as workspace for physically transposing the pivots. LIWORK is local input and must be at least if NPROW == NPCOL then LIWORK = LOCc( N_A + MOD(JA-1, NB_A) ) + NB_A, else LIWORK = LOCc( N_A + MOD(JA-1, NB_A) ) + MAX( CEIL(CEIL(LOCr(M_A)/MB_A)/(LCM/NPROW)), NB_A ) where LCM is the least common multiple of process rows and columns (NPROW and NPCOL). end if If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, U(IA+K-1,IA+K-1) is exactly zero; the matrix is singular and its inverse could not be computed. scalapack-doc-1.5/man/manl/pzgetrs.l0100644000056400000620000001331106335610656017123 0ustar pfrauenfstaff.TH PZGETRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGETRS - solve a system of distributed linear equations op( sub( A ) ) * X = sub( B ) with a general N-by-N distributed matrix sub( A ) using the LU factorization computed by PZGETRF .SH SYNOPSIS .TP 20 SUBROUTINE PZGETRS( TRANS, N, NRHS, A, IA, JA, DESCA, IPIV, B, IB, JB, DESCB, INFO ) .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER IA, IB, INFO, JA, JB, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ), IPIV( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), B( * ) .SH PURPOSE PZGETRS solves a system of distributed linear equations sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1), op( A ) = A, A**T or A**H and sub( B ) denotes B(IB:IB+N-1,JB:JB+NRHS-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block data decomposition ( MB_A=NB_A ). .SH ARGUMENTS .TP 8 TRANS (global input) CHARACTER Specifies the form of the system of equations: .br = 'N': sub( A ) * X = sub( B ) (No transpose) .br = 'T': sub( A )**T * X = sub( B ) (Transpose) .br = 'C': sub( A )**H * X = sub( B ) (Conjugate transpose) .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed submatrix sub( B ). NRHS >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the factors L and U from the factorization sub( A ) = P*L*U; the unit diagonal elements of L are not stored. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local input) INTEGER array, dimension ( LOCr(M_A)+MB_A ) This array contains the pivoting information. IPIV(i) -> The global row local row i was swapped with. This array is tied to the distributed matrix A. .TP 8 B (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_B,LOCc(JB+NRHS-1)). On entry, the right hand sides sub( B ). On exit, sub( B ) is overwritten by the solution distributed matrix X. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pzggqrf.l0100644000056400000620000002353606335610656017117 0ustar pfrauenfstaff.TH PZGGQRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGGQRF - compute a generalized QR factorization of an N-by-M matrix sub( A ) = A(IA:IA+N-1,JA:JA+M-1) and an N-by-P matrix sub( B ) = B(IB:IB+N-1,JB:JB+P-1) .SH SYNOPSIS .TP 20 SUBROUTINE PZGGQRF( N, M, P, A, IA, JA, DESCA, TAUA, B, IB, JB, DESCB, TAUB, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, IB, INFO, JA, JB, LWORK, M, N, P .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), B( * ), TAUA( * ), TAUB( * ), WORK( * ) .SH PURPOSE PZGGQRF computes a generalized QR factorization of an N-by-M matrix sub( A ) = A(IA:IA+N-1,JA:JA+M-1) and an N-by-P matrix sub( B ) = B(IB:IB+N-1,JB:JB+P-1): sub( A ) = Q*R, sub( B ) = Q*T*Z, .br where Q is an N-by-N unitary matrix, Z is a P-by-P unitary matrix, and R and T assume one of the forms: .br if N >= M, R = ( R11 ) M , or if N < M, R = ( R11 R12 ) N, ( 0 ) N-M N M-N M .br where R11 is upper triangular, and .br if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) N-P, P-N N ( T21 ) P P .br where T12 or T21 is upper triangular. .br In particular, if sub( B ) is square and nonsingular, the GQR factorization of sub( A ) and sub( B ) implicitly gives the QR factorization of inv( sub( B ) )* sub( A ): .br inv( sub( B ) )*sub( A )= Z'*(inv(T)*R) .br where inv( sub( B ) ) denotes the inverse of the matrix sub( B ), and Z' denotes the conjugate transpose of matrix Z. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrices sub( A ) and sub( B ). N >= 0. .TP 8 M (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). M >= 0. .TP 8 P (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( B ). P >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+M-1)). On entry, the local pieces of the N-by-M distributed matrix sub( A ) which is to be factored. On exit, the elements on and above the diagonal of sub( A ) contain the min(N,M) by M upper trapezoidal matrix R (R is upper triangular if N >= M); the elements below the diagonal, with the array TAUA, represent the unitary matrix Q as a product of min(N,M) elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAUA (local output) COMPLEX*16, array, dimension LOCc(JA+MIN(N,M)-1). This array contains the scalar factors TAUA of the elementary reflectors which represent the unitary matrix Q. TAUA is tied to the distributed matrix A. (see Further Details). B (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+P-1)). On entry, the local pieces of the N-by-P distributed matrix sub( B ) which is to be factored. On exit, if N <= P, the upper triangle of B(IB:IB+N-1,JB+P-N:JB+P-1) contains the N by N upper triangular matrix T; if N > P, the elements on and above the (N-P)-th subdiagonal contain the N by P upper trapezoidal matrix T; the remaining elements, with the array TAUB, represent the unitary matrix Z as a product of elementary reflectors (see Further Details). IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 TAUB (local output) COMPLEX*16, array, dimension LOCr(IB+N-1) This array contains the scalar factors of the elementary reflectors which represent the unitary matrix Z. TAUB is tied to the distributed matrix B (see Further Details). WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MAX( NB_A * ( NpA0 + MqA0 + NB_A ), MAX( (NB_A*(NB_A-1))/2, (PqB0 + NpB0)*NB_A ) + NB_A * NB_A, MB_B * ( NpB0 + PqB0 + MB_B ) ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), NpA0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), MqA0 = NUMROC( M+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFB = MOD( IB-1, MB_B ), ICOFFB = MOD( JB-1, NB_B ), IBROW = INDXG2P( IB, MB_B, MYROW, RSRC_B, NPROW ), IBCOL = INDXG2P( JB, NB_B, MYCOL, CSRC_B, NPCOL ), NpB0 = NUMROC( N+IROFFB, MB_B, MYROW, IBROW, NPROW ), PqB0 = NUMROC( P+ICOFFB, NB_B, MYCOL, IBCOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ja) H(ja+1) . . . H(ja+k-1), where k = min(n,m). Each H(i) has the form .br H(i) = I - taua * v * v' .br where taua is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in .br A(ia+i:ia+n-1,ja+i-1), and taua in TAUA(ja+i-1). .br To form Q explicitly, use ScaLAPACK subroutine PZUNGQR. .br To use Q to update another matrix, use ScaLAPACK subroutine PZUNMQR. The matrix Z is represented as a product of elementary reflectors Z = H(ib)' H(ib+1)' . . . H(ib+k-1)', where k = min(n,p). Each H(i) has the form .br H(i) = I - taub * v * v' .br where taub is a complex scalar, and v is a complex vector with v(p-k+i+1:p) = 0 and v(p-k+i) = 1; conjg(v(1:p-k+i-1)) is stored on exit in B(ib+n-k+i-1,jb:jb+p-k+i-2), and taub in TAUB(ib+n-k+i-1). To form Z explicitly, use ScaLAPACK subroutine PZUNGRQ. .br To use Z to update another matrix, use ScaLAPACK subroutine PZUNMRQ. Alignment requirements .br ====================== .br The distributed submatrices sub( A ) and sub( B ) must verify some alignment properties, namely the following expression should be true: ( MB_A.EQ.MB_B .AND. IROFFA.EQ.IROFFB .AND. IAROW.EQ.IBROW ) scalapack-doc-1.5/man/manl/pzggrqf.l0100644000056400000620000002343606335610656017116 0ustar pfrauenfstaff.TH PZGGRQF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZGGRQF - compute a generalized RQ factorization of an M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PZGGRQF( M, P, N, A, IA, JA, DESCA, TAUA, B, IB, JB, DESCB, TAUB, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, IB, INFO, JA, JB, LWORK, M, N, P .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), B( * ), TAUA( * ), TAUB( * ), WORK( * ) .SH PURPOSE PZGGRQF computes a generalized RQ factorization of an M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) and a P-by-N matrix sub( B ) = B(IB:IB+P-1,JB:JB+N-1): .br sub( A ) = R*Q, sub( B ) = Z*T*Q, .br where Q is an N-by-N unitary matrix, Z is a P-by-P unitary matrix, and R and T assume one of the forms: .br if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N, N-M M ( R21 ) N N .br where R12 or R21 is upper triangular, and .br if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P, ( 0 ) P-N P N-P N .br where T11 is upper triangular. .br In particular, if sub( B ) is square and nonsingular, the GRQ factorization of sub( A ) and sub( B ) implicitly gives the RQ factorization of sub( A )*inv( sub( B ) ): .br sub( A )*inv( sub( B ) ) = (R*inv(T))*Z' .br where inv( sub( B ) ) denotes the inverse of the matrix sub( B ), and Z' denotes the conjugate transpose of matrix Z. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 P (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( B ). P >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrices sub( A ) and sub( B ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, if M <= N, the upper triangle of A( IA:IA+M-1, JA+N-M:JA+N-1 ) 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 TAUA, represent the unitary matrix Q as a product of elementary reflectors (see Further Details). IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAUA (local output) COMPLEX*16, array, dimension LOCr(IA+M-1) This array contains the scalar factors of the elementary reflectors which represent the unitary matrix Q. TAUA is tied to the distributed matrix A (see Further Details). B (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+N-1)). On entry, the local pieces of the P-by-N distributed matrix sub( B ) which is to be factored. On exit, the elements on and above the diagonal of sub( B ) contain the min(P,N) by N upper trapezoidal matrix T (T is upper triangular if P >= N); the elements below the diagonal, with the array TAUB, represent the unitary matrix Z as a product of elementary reflectors (see Further Details). IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 TAUB (local output) COMPLEX*16, array, dimension LOCc(JB+MIN(P,N)-1). This array contains the scalar factors TAUB of the elementary reflectors which represent the unitary matrix Z. TAUB is tied to the distributed matrix B (see Further Details). WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MAX( MB_A * ( MpA0 + NqA0 + MB_A ), MAX( (MB_A*(MB_A-1))/2, (PpB0 + NqB0)*MB_A ) + MB_A * MB_A, NB_B * ( PpB0 + NqB0 + NB_B ) ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFB = MOD( IB-1, MB_B ), ICOFFB = MOD( JB-1, NB_B ), IBROW = INDXG2P( IB, MB_B, MYROW, RSRC_B, NPROW ), IBCOL = INDXG2P( JB, NB_B, MYCOL, CSRC_B, NPCOL ), PpB0 = NUMROC( P+IROFFB, MB_B, MYROW, IBROW, NPROW ), NqB0 = NUMROC( N+ICOFFB, NB_B, MYCOL, IBCOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The matrix Q is represented as a product of elementary reflectors Q = H(ia)' H(ia+1)' . . . H(ia+k-1)', where k = min(m,n). Each H(i) has the form .br H(i) = I - taua * v * v' .br where taua 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(ia+m-k+i-1,ja:ja+n-k+i-2), and taua in TAUA(ia+m-k+i-1). To form Q explicitly, use ScaLAPACK subroutine PZUNGRQ. .br To use Q to update another matrix, use ScaLAPACK subroutine PZUNMRQ. The matrix Z is represented as a product of elementary reflectors Z = H(jb) H(jb+1) . . . H(jb+k-1), where k = min(p,n). Each H(i) has the form .br H(i) = I - taub * v * v' .br where taub is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in .br B(ib+i:ib+p-1,jb+i-1), and taub in TAUB(jb+i-1). .br To form Z explicitly, use ScaLAPACK subroutine PZUNGQR. .br To use Z to update another matrix, use ScaLAPACK subroutine PZUNMQR. Alignment requirements .br ====================== .br The distributed submatrices sub( A ) and sub( B ) must verify some alignment properties, namely the following expression should be true: ( NB_A.EQ.NB_B .AND. ICOFFA.EQ.ICOFFB .AND. IACOL.EQ.IBCOL ) scalapack-doc-1.5/man/manl/pzheevx.l0100644000056400000620000003541606335610656017130 0ustar pfrauenfstaff.TH PZHEEVX l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME .SH SYNOPSIS .TP 20 SUBROUTINE PZHEEVX( JOBZ, RANGE, UPLO, N, A, IA, JA, DESCA, VL, VU, IL, IU, ABSTOL, M, NZ, W, ORFAC, Z, IZ, JZ, DESCZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, IFAIL, ICLUSTR, GAP, INFO ) .TP 20 .ti +4 CHARACTER JOBZ, RANGE, UPLO .TP 20 .ti +4 INTEGER IA, IL, INFO, IU, IZ, JA, JZ, LIWORK, LRWORK, LWORK, M, N, NZ .TP 20 .ti +4 DOUBLE PRECISION ABSTOL, ORFAC, VL, VU .TP 20 .ti +4 INTEGER DESCA( * ), DESCZ( * ), ICLUSTR( * ), IFAIL( * ), IWORK( * ) .TP 20 .ti +4 DOUBLE PRECISION GAP( * ), RWORK( * ), W( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), WORK( * ), Z( * ) .TP 20 .ti +4 INTEGER BLOCK_CYCLIC_2D, DLEN_, DTYPE_, CTXT_, M_, N_, MB_, NB_, RSRC_, CSRC_, LLD_ .TP 20 .ti +4 PARAMETER ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1, CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6, RSRC_ = 7, CSRC_ = 8, LLD_ = 9 ) .TP 20 .ti +4 DOUBLE PRECISION ZERO, ONE, TEN, FIVE .TP 20 .ti +4 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TEN = 10.0D+0, FIVE = 5.0D+0 ) .TP 20 .ti +4 INTEGER IERREIN, IERRCLS, IERRSPC, IERREBZ .TP 20 .ti +4 PARAMETER ( IERREIN = 1, IERRCLS = 2, IERRSPC = 4, IERREBZ = 8 ) .TP 20 .ti +4 LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, QUICKRETURN, VALEIG, WANTZ .TP 20 .ti +4 CHARACTER ORDER .TP 20 .ti +4 INTEGER CSRC_A, I, IACOL, IAROW, ICOFFA, IINFO, INDD, INDD2, INDE, INDE2, INDIBL, INDISP, INDRWORK, INDTAU, INDWORK, IROFFA, IROFFZ, ISCALE, ISIZESTEBZ, ISIZESTEIN, IZROW, LALLWORK, LIWMIN, LLRWORK, LLWORK, LRWMIN, LWMIN, MAXEIGS, MB_A, MB_Z, MQ0, MYCOL, MYROW, NB, NB_A, NB_Z, NEIG, NN, NNP, NP0, NPCOL, NPROCS, NPROW, NQ0, NSPLIT, NZZ, OFFSET, RSRC_A, RSRC_Z, SIZEHEEVX, SIZEORMTR, SIZESTEIN .TP 20 .ti +4 DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM, VLL, VUU .TP 20 .ti +4 INTEGER IDUM1( 4 ), IDUM2( 4 ) .TP 20 .ti +4 LOGICAL LSAME .TP 20 .ti +4 INTEGER ICEIL, INDXG2P, NUMROC .TP 20 .ti +4 DOUBLE PRECISION PDLAMCH, PZLANHE .TP 20 .ti +4 EXTERNAL LSAME, ICEIL, INDXG2P, NUMROC, PDLAMCH, PZLANHE .TP 20 .ti +4 EXTERNAL BLACS_GRIDINFO, CHK1MAT, DGEBR2D, DGEBS2D, DLASRT, DSCAL, IGAMN2D, PCHK2MAT, PDLARED1D, PDSTEBZ, PXERBLA, PZELGET, PZHETRD, PZLASCL, PZSTEIN, PZUNMTR .TP 20 .ti +4 INTRINSIC ABS, DBLE, DCMPLX, ICHAR, MAX, MIN, MOD, SQRT .TP 20 .ti +4 IF( BLOCK_CYCLIC_2D*CSRC_*CTXT_*DLEN_*DTYPE_*LLD_*MB_*M_*NB_*N_* RSRC_.LT.0 )RETURN .TP 20 .ti +4 QUICKRETURN = ( N.EQ.0 ) .TP 20 .ti +4 CALL BLACS_GRIDINFO( DESCA( CTXT_ ), NPROW, NPCOL, MYROW, MYCOL ) .TP 20 .ti +4 INFO = 0 .TP 20 .ti +4 IF( NPROW.EQ.-1 ) THEN .TP 20 .ti +4 INFO = -( 800+CTXT_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCZ( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+CTXT_ ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 CALL CHK1MAT( N, 4, N, 4, IA, JA, DESCA, 8, INFO ) .TP 20 .ti +4 CALL CHK1MAT( N, 4, N, 4, IZ, JZ, DESCZ, 21, INFO ) .TP 20 .ti +4 IF( INFO.EQ.0 ) THEN .TP 20 .ti +4 SAFMIN = PDLAMCH( DESCA( CTXT_ ), 'Safe minimum' ) .TP 20 .ti +4 EPS = PDLAMCH( DESCA( CTXT_ ), 'Precision' ) .TP 20 .ti +4 SMLNUM = SAFMIN / EPS .TP 20 .ti +4 BIGNUM = ONE / SMLNUM .TP 20 .ti +4 RMIN = SQRT( SMLNUM ) .TP 20 .ti +4 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) .TP 20 .ti +4 NPROCS = NPROW*NPCOL .TP 20 .ti +4 LOWER = LSAME( UPLO, 'L' ) .TP 20 .ti +4 WANTZ = LSAME( JOBZ, 'V' ) .TP 20 .ti +4 ALLEIG = LSAME( RANGE, 'A' ) .TP 20 .ti +4 VALEIG = LSAME( RANGE, 'V' ) .TP 20 .ti +4 INDEIG = LSAME( RANGE, 'I' ) .TP 20 .ti +4 INDTAU = 1 .TP 20 .ti +4 INDWORK = INDTAU + N .TP 20 .ti +4 LLWORK = LWORK - INDWORK + 1 .TP 20 .ti +4 INDE = 1 .TP 20 .ti +4 INDD = INDE + N .TP 20 .ti +4 INDD2 = INDD + N .TP 20 .ti +4 INDE2 = INDD2 + N .TP 20 .ti +4 INDRWORK = INDE2 + N .TP 20 .ti +4 LLRWORK = LRWORK - INDRWORK + 1 .TP 20 .ti +4 ISIZESTEIN = 3*N + NPROCS + 1 .TP 20 .ti +4 ISIZESTEBZ = MAX( 4*N, 14, NPROCS ) .TP 20 .ti +4 INDIBL = ( MAX( ISIZESTEIN, ISIZESTEBZ ) ) + 1 .TP 20 .ti +4 INDISP = INDIBL + N .TP 20 .ti +4 LQUERY = .FALSE. .TP 20 .ti +4 IF( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 .OR. LRWORK.EQ.-1 ) LQUERY = .TRUE. .TP 20 .ti +4 NNP = MAX( N, NPROCS+1, 4 ) .TP 20 .ti +4 LIWMIN = 6*NNP .TP 20 .ti +4 NPROCS = NPROW*NPCOL .TP 20 .ti +4 NB_A = DESCA( NB_ ) .TP 20 .ti +4 MB_A = DESCA( MB_ ) .TP 20 .ti +4 NB_Z = DESCZ( NB_ ) .TP 20 .ti +4 MB_Z = DESCZ( MB_ ) .TP 20 .ti +4 NB = NB_A .TP 20 .ti +4 NN = MAX( N, NB, 2 ) .TP 20 .ti +4 RSRC_A = DESCA( RSRC_ ) .TP 20 .ti +4 CSRC_A = DESCA( CSRC_ ) .TP 20 .ti +4 RSRC_Z = DESCZ( RSRC_ ) .TP 20 .ti +4 IROFFA = MOD( IA-1, MB_A ) .TP 20 .ti +4 ICOFFA = MOD( JA-1, NB_A ) .TP 20 .ti +4 IROFFZ = MOD( IZ-1, MB_A ) .TP 20 .ti +4 IAROW = INDXG2P( 1, NB_A, MYROW, RSRC_A, NPROW ) .TP 20 .ti +4 IACOL = INDXG2P( 1, MB_A, MYCOL, CSRC_A, NPCOL ) .TP 20 .ti +4 IZROW = INDXG2P( 1, NB_A, MYROW, RSRC_Z, NPROW ) .TP 20 .ti +4 NP0 = NUMROC( N+IROFFA, NB_Z, MYROW, IAROW, NPROW ) .TP 20 .ti +4 MQ0 = NUMROC( N+ICOFFA, NB_Z, MYCOL, IACOL, NPCOL ) .TP 20 .ti +4 IF( ( .NOT.WANTZ ) .OR. ( VALEIG .AND. ( .NOT.LQUERY ) ) ) THEN .TP 20 .ti +4 LWMIN = N + MAX( NB*( NP0+1 ), 3 ) .TP 20 .ti +4 LRWMIN = 5*NN + 4*N .TP 20 .ti +4 NEIG = 0 .TP 20 .ti +4 ELSE .TP 20 .ti +4 IF( ALLEIG .OR. VALEIG ) THEN .TP 20 .ti +4 NEIG = N .TP 20 .ti +4 ELSE IF( INDEIG ) THEN .TP 20 .ti +4 NEIG = IU - IL + 1 .TP 20 .ti +4 END IF .TP 20 .ti +4 MQ0 = NUMROC( MAX( NEIG, NB, 2 ), NB, MYCOL, IACOL, NPCOL ) .TP 20 .ti +4 NQ0 = NUMROC( NN, NB, 0, 0, NPCOL ) .TP 20 .ti +4 LWMIN = N + ( NP0+NQ0+NB )*NB .TP 20 .ti +4 LRWMIN = 4*N + MAX( 5*NN, NP0*MQ0 ) + ICEIL( NEIG, NPROW*NPCOL )*NN .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( INFO.EQ.0 ) THEN .TP 20 .ti +4 IF( MYROW.EQ.0 .AND. MYCOL.EQ.0 ) THEN .TP 20 .ti +4 RWORK( 1 ) = ABSTOL .TP 20 .ti +4 IF( VALEIG ) THEN .TP 20 .ti +4 RWORK( 2 ) = VL .TP 20 .ti +4 RWORK( 3 ) = VU .TP 20 .ti +4 ELSE .TP 20 .ti +4 RWORK( 2 ) = ZERO .TP 20 .ti +4 RWORK( 3 ) = ZERO .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL DGEBS2D( DESCA( CTXT_ ), 'ALL', ' ', 3, 1, RWORK, 3 ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 CALL DGEBR2D( DESCA( CTXT_ ), 'ALL', ' ', 3, 1, RWORK, 3, 0, 0 ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN .TP 20 .ti +4 INFO = -1 .TP 20 .ti +4 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN .TP 20 .ti +4 INFO = -2 .TP 20 .ti +4 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN .TP 20 .ti +4 INFO = -3 .TP 20 .ti +4 ELSE IF( VALEIG .AND. N.GT.0 .AND. VU.LE.VL ) THEN .TP 20 .ti +4 INFO = -10 .TP 20 .ti +4 ELSE IF( INDEIG .AND. ( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) ) THEN .TP 20 .ti +4 INFO = -11 .TP 20 .ti +4 ELSE IF( INDEIG .AND. ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) ) THEN .TP 20 .ti +4 INFO = -12 .TP 20 .ti +4 ELSE IF( LWORK.LT.LWMIN .AND. LWORK.NE.-1 ) THEN .TP 20 .ti +4 INFO = -23 .TP 20 .ti +4 ELSE IF( LRWORK.LT.LRWMIN .AND. LRWORK.NE.-1 ) THEN .TP 20 .ti +4 INFO = -25 .TP 20 .ti +4 ELSE IF( LIWORK.LT.LIWMIN .AND. LIWORK.NE.-1 ) THEN .TP 20 .ti +4 INFO = -27 .TP 20 .ti +4 ELSE IF( VALEIG .AND. ( ABS( RWORK( 2 )-VL ).GT.FIVE*EPS* ABS( VL ) ) ) THEN .TP 20 .ti +4 INFO = -9 .TP 20 .ti +4 ELSE IF( VALEIG .AND. ( ABS( RWORK( 3 )-VU ).GT.FIVE*EPS* ABS( VU ) ) ) THEN .TP 20 .ti +4 INFO = -10 .TP 20 .ti +4 ELSE IF( ABS( RWORK( 1 )-ABSTOL ).GT.FIVE*EPS* ABS( ABSTOL ) ) THEN .TP 20 .ti +4 INFO = -13 .TP 20 .ti +4 ELSE IF( IROFFA.NE.IROFFZ ) THEN .TP 20 .ti +4 INFO = -19 .TP 20 .ti +4 ELSE IF( IROFFA.NE.0 ) THEN .TP 20 .ti +4 INFO = -6 .TP 20 .ti +4 ELSE IF( IAROW.NE.IZROW ) THEN .TP 20 .ti +4 INFO = -19 .TP 20 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCA( NB_ ) ) THEN .TP 20 .ti +4 INFO = -( 800+NB_ ) .TP 20 .ti +4 ELSE IF( DESCA( M_ ).NE.DESCZ( M_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+M_ ) .TP 20 .ti +4 ELSE IF( DESCA( N_ ).NE.DESCZ( N_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+N_ ) .TP 20 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCZ( MB_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+MB_ ) .TP 20 .ti +4 ELSE IF( DESCA( NB_ ).NE.DESCZ( NB_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+NB_ ) .TP 20 .ti +4 ELSE IF( DESCA( RSRC_ ).NE.DESCZ( RSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+RSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CSRC_ ).NE.DESCZ( CSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+CSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCZ( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 2100+CTXT_ ) .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 IDUM1( 1 ) = ICHAR( 'V' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 1 ) = ICHAR( 'N' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 1 ) = 1 .TP 20 .ti +4 IF( LOWER ) THEN .TP 20 .ti +4 IDUM1( 2 ) = ICHAR( 'L' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 2 ) = ICHAR( 'U' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 2 ) = 2 .TP 20 .ti +4 IF( ALLEIG ) THEN .TP 20 .ti +4 IDUM1( 3 ) = ICHAR( 'A' ) .TP 20 .ti +4 ELSE IF( INDEIG ) THEN .TP 20 .ti +4 IDUM1( 3 ) = ICHAR( 'I' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 3 ) = ICHAR( 'V' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 3 ) = 3 .TP 20 .ti +4 IF( LQUERY ) THEN .TP 20 .ti +4 IDUM1( 4 ) = -1 .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 4 ) = 1 .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 4 ) = 4 .TP 20 .ti +4 CALL PCHK2MAT( N, 4, N, 4, IA, JA, DESCA, 8, N, 4, N, 4, IZ, JZ, DESCZ, 21, 4, IDUM1, IDUM2, INFO ) .TP 20 .ti +4 WORK( 1 ) = DCMPLX( LWMIN ) .TP 20 .ti +4 RWORK( 1 ) = DBLE( LRWMIN ) .TP 20 .ti +4 IWORK( 1 ) = LIWMIN .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( INFO.NE.0 ) THEN .TP 20 .ti +4 CALL PXERBLA( DESCA( CTXT_ ), 'PZHEEVX', -INFO ) .TP 20 .ti +4 RETURN .TP 20 .ti +4 ELSE IF( LQUERY ) THEN .TP 20 .ti +4 RETURN .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( QUICKRETURN ) THEN .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 NZ = 0 .TP 20 .ti +4 ICLUSTR( 1 ) = 0 .TP 20 .ti +4 END IF .TP 20 .ti +4 M = 0 .TP 20 .ti +4 WORK( 1 ) = DCMPLX( LWMIN ) .TP 20 .ti +4 RWORK( 1 ) = DBLE( LRWMIN ) .TP 20 .ti +4 IWORK( 1 ) = LIWMIN .TP 20 .ti +4 RETURN .TP 20 .ti +4 END IF .TP 20 .ti +4 ABSTLL = ABSTOL .TP 20 .ti +4 ISCALE = 0 .TP 20 .ti +4 IF( VALEIG ) THEN .TP 20 .ti +4 VLL = VL .TP 20 .ti +4 VUU = VU .TP 20 .ti +4 ELSE .TP 20 .ti +4 VLL = ZERO .TP 20 .ti +4 VUU = ZERO .TP 20 .ti +4 END IF .TP 20 .ti +4 ANRM = PZLANHE( '1', UPLO, N, A, IA, JA, DESCA, RWORK( INDRWORK ) ) .TP 20 .ti +4 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN .TP 20 .ti +4 ISCALE = 1 .TP 20 .ti +4 SIGMA = RMIN / ANRM .TP 20 .ti +4 ANRM = ANRM*SIGMA .TP 20 .ti +4 ELSE IF( ANRM.GT.RMAX ) THEN .TP 20 .ti +4 ISCALE = 1 .TP 20 .ti +4 SIGMA = RMAX / ANRM .TP 20 .ti +4 ANRM = ANRM*SIGMA .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( ISCALE.EQ.1 ) THEN .TP 20 .ti +4 CALL PZLASCL( UPLO, ONE, SIGMA, N, N, A, IA, JA, DESCA, IINFO ) .TP 20 .ti +4 IF( ABSTOL.GT.0 ) ABSTLL = ABSTOL*SIGMA .TP 20 .ti +4 IF( VALEIG ) THEN .TP 20 .ti +4 VLL = VL*SIGMA .TP 20 .ti +4 VUU = VU*SIGMA .TP 20 .ti +4 IF( VUU.EQ.VLL ) THEN .TP 20 .ti +4 VUU = VUU + 2*MAX( ABS( VUU )*EPS, SAFMIN ) .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 LALLWORK = LLRWORK .TP 20 .ti +4 CALL PZHETRD( UPLO, N, A, IA, JA, DESCA, RWORK( INDD ), RWORK( INDE ), WORK( INDTAU ), WORK( INDWORK ), LLWORK, IINFO ) .TP 20 .ti +4 OFFSET = 0 .TP 20 .ti +4 IF( IA.EQ.1 .AND. JA.EQ.1 .AND. RSRC_A.EQ.0 .AND. CSRC_A.EQ.0 ) THEN .TP 20 .ti +4 CALL PDLARED1D( N, IA, JA, DESCA, RWORK( INDD ), RWORK( INDD2 ), RWORK( INDRWORK ), LLRWORK ) .TP 20 .ti +4 CALL PDLARED1D( N, IA, JA, DESCA, RWORK( INDE ), RWORK( INDE2 ), RWORK( INDRWORK ), LLRWORK ) .TP 20 .ti +4 IF( .NOT.LOWER ) OFFSET = 1 .TP 20 .ti +4 ELSE .TP 20 .ti +4 DO 10 I = 1, N .TP 20 .ti +4 CALL PZELGET( 'A', ' ', WORK( INDD2+I-1 ), A, I+IA-1, I+JA-1, DESCA ) .TP 20 .ti +4 RWORK( INDD2+I-1 ) = DBLE( WORK( INDD2+I-1 ) ) .TP 20 .ti +4 10 CONTINUE .TP 20 .ti +4 IF( LSAME( UPLO, 'U' ) ) THEN .TP 20 .ti +4 DO 20 I = 1, N - 1 .TP 20 .ti +4 CALL PZELGET( 'A', ' ', WORK( INDE2+I-1 ), A, I+IA-1, I+JA, DESCA ) .TP 20 .ti +4 RWORK( INDE2+I-1 ) = DBLE( WORK( INDE2+I-1 ) ) .TP 20 .ti +4 20 CONTINUE .TP 20 .ti +4 ELSE .TP 20 .ti +4 DO 30 I = 1, N - 1 .TP 20 .ti +4 CALL PZELGET( 'A', ' ', WORK( INDE2+I-1 ), A, I+IA, I+JA-1, DESCA ) .TP 20 .ti +4 RWORK( INDE2+I-1 ) = DBLE( WORK( INDE2+I-1 ) ) .TP 20 .ti +4 30 CONTINUE .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 ORDER = 'b' .TP 20 .ti +4 ELSE .TP 20 .ti +4 ORDER = 'e' .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PDSTEBZ( DESCA( CTXT_ ), RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, RWORK( INDD2 ), RWORK( INDE2+OFFSET ), M, NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWORK ), LLRWORK, IWORK( 1 ), ISIZESTEBZ, IINFO ) .TP 20 .ti +4 IF( IINFO.NE.0 ) THEN .TP 20 .ti +4 INFO = INFO + IERREBZ .TP 20 .ti +4 DO 40 I = 1, M .TP 20 .ti +4 IWORK( INDIBL+I-1 ) = ABS( IWORK( INDIBL+I-1 ) ) .TP 20 .ti +4 40 CONTINUE .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 IF( VALEIG ) THEN .TP 20 .ti +4 CALL IGAMN2D( DESCA( CTXT_ ), 'A', ' ', 1, 1, LALLWORK, 1, 1, 1, -1, -1, -1 ) .TP 20 .ti +4 MAXEIGS = DESCZ( N_ ) .TP 20 .ti +4 DO 50 NZ = MIN( MAXEIGS, M ), 0, -1 .TP 20 .ti +4 MQ0 = NUMROC( NZ, NB, 0, 0, NPCOL ) .TP 20 .ti +4 SIZESTEIN = ICEIL( NZ, NPROCS )*N + MAX( 5*N, NP0*MQ0 ) .TP 20 .ti +4 SIZEORMTR = MAX( ( NB*( NB-1 ) ) / 2, ( MQ0+NP0 )*NB ) + NB*NB .TP 20 .ti +4 SIZEHEEVX = MAX( SIZESTEIN, SIZEORMTR ) .TP 20 .ti +4 IF( SIZEHEEVX.LE.LALLWORK ) GO TO 60 .TP 20 .ti +4 50 CONTINUE .TP 20 .ti +4 60 CONTINUE .TP 20 .ti +4 ELSE .TP 20 .ti +4 NZ = M .TP 20 .ti +4 END IF .TP 20 .ti +4 NZ = MAX( NZ, 0 ) .TP 20 .ti +4 IF( NZ.NE.M ) THEN .TP 20 .ti +4 INFO = INFO + IERRSPC .TP 20 .ti +4 DO 70 I = 1, M .TP 20 .ti +4 IFAIL( I ) = 0 .TP 20 .ti +4 70 CONTINUE .TP 20 .ti +4 IF( NSPLIT.GT.1 ) THEN .TP 20 .ti +4 CALL DLASRT( 'I', M, W, IINFO ) .TP 20 .ti +4 IF( NZ.GT.0 ) THEN .TP 20 .ti +4 VUU = W( NZ ) - TEN*( EPS*ANRM+SAFMIN ) .TP 20 .ti +4 IF( VLL.GE.VUU ) THEN .TP 20 .ti +4 NZZ = 0 .TP 20 .ti +4 ELSE .TP 20 .ti +4 CALL PDSTEBZ( DESCA( CTXT_ ), RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, RWORK( INDD2 ), RWORK( INDE2+ OFFSET ), NZZ, NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWORK ), LLRWORK, IWORK( 1 ), ISIZESTEBZ, IINFO ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( MOD( INFO / IERREBZ, 1 ).EQ.0 ) THEN .TP 20 .ti +4 IF( NZZ.GT.NZ .OR. IINFO.NE.0 ) THEN .TP 20 .ti +4 INFO = INFO + IERREBZ .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 NZ = MIN( NZ, NZZ ) .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PZSTEIN( N, RWORK( INDD2 ), RWORK( INDE2+OFFSET ), NZ, W, IWORK( INDIBL ), IWORK( INDISP ), ORFAC, Z, IZ, JZ, DESCZ, RWORK( INDRWORK ), LALLWORK, IWORK( 1 ), ISIZESTEIN, IFAIL, ICLUSTR, GAP, IINFO ) .TP 20 .ti +4 IF( IINFO.GE.NZ+1 ) INFO = INFO + IERRCLS .TP 20 .ti +4 IF( MOD( IINFO, NZ+1 ).NE.0 ) INFO = INFO + IERREIN .TP 20 .ti +4 IF( NZ.GT.0 ) THEN .TP 20 .ti +4 CALL PZUNMTR( 'L', UPLO, 'N', N, NZ, A, IA, JA, DESCA, WORK( INDTAU ), Z, IZ, JZ, DESCZ, WORK( INDWORK ), LLWORK, IINFO ) .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( ISCALE.EQ.1 ) THEN .TP 20 .ti +4 CALL DSCAL( M, ONE / SIGMA, W, 1 ) .TP 20 .ti +4 END IF .TP 20 .ti +4 WORK( 1 ) = DCMPLX( LWMIN ) .TP 20 .ti +4 RWORK( 1 ) = DBLE( LRWMIN ) .TP 20 .ti +4 IWORK( 1 ) = LIWMIN .TP 20 .ti +4 RETURN .TP 20 .ti +4 END .SH PURPOSE scalapack-doc-1.5/man/manl/pzhegs2.l0100644000056400000620000001417006335610656017013 0ustar pfrauenfstaff.TH PZHEGS2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZHEGS2 - reduce a complex Hermitian-definite generalized eigenproblem to standard form .SH SYNOPSIS .TP 20 SUBROUTINE PZHEGS2( IBTYPE, UPLO, N, A, IA, JA, DESCA, B, IB, JB, DESCB, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IB, IBTYPE, INFO, JA, JB, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), B( * ) .SH PURPOSE PZHEGS2 reduces a complex Hermitian-definite generalized eigenproblem to standard form. In the following sub( A ) denotes A( IA:IA+N-1, JA:JA+N-1 ) and sub( B ) denotes B( IB:IB+N-1, JB:JB+N-1 ). .br If IBTYPE = 1, the problem is sub( A )*x = lambda*sub( B )*x, and sub( A ) is overwritten by inv(U**H)*sub( A )*inv(U) or inv(L)*sub( A )*inv(L**H) .br If IBTYPE = 2 or 3, the problem is sub( A )*sub( B )*x = lambda*x or sub( B )*sub( A )*x = lambda*x, and sub( A ) is overwritten by U*sub( A )*U**H or L**H*sub( A )*L. .br sub( B ) must have been previously factorized as U**H*U or L*L**H by PZPOTRF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 9 IBTYPE (global input) INTEGER = 1: compute inv(U**H)*sub( A )*inv(U) or inv(L)*sub( A )*inv(L**H); = 2 or 3: compute U*sub( A )*U**H or L**H*sub( A )*L. .TP 8 UPLO (global input) CHARACTER .br = 'U': Upper triangle of sub( A ) is stored and sub( B ) is factored as U**H*U; = 'L': Lower triangle of sub( A ) is stored and sub( B ) is factored as L*L**H. .TP 8 N (global input) INTEGER The order of the matrices sub( A ) and sub( B ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N Hermitian distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced. On exit, if INFO = 0, the transformed matrix, stored in the same format as sub( A ). .TP 8 IA (global input) INTEGER A's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JA (global input) INTEGER A's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+N-1)). On entry, this array contains the local pieces of the triangular factor from the Cholesky factorization of sub( B ), as returned by PZPOTRF. .TP 8 IB (global input) INTEGER B's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JB (global input) INTEGER B's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pzhegst.l0100644000056400000620000001463706335610656017125 0ustar pfrauenfstaff.TH PZHEGST l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZHEGST - reduce a complex Hermitian-definite generalized eigenproblem to standard form .SH SYNOPSIS .TP 20 SUBROUTINE PZHEGST( IBTYPE, UPLO, N, A, IA, JA, DESCA, B, IB, JB, DESCB, SCALE, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IB, IBTYPE, INFO, JA, JB, N .TP 20 .ti +4 DOUBLE PRECISION SCALE .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), B( * ) .SH PURPOSE PZHEGST reduces a complex Hermitian-definite generalized eigenproblem to standard form. In the following sub( A ) denotes A( IA:IA+N-1, JA:JA+N-1 ) and sub( B ) denotes B( IB:IB+N-1, JB:JB+N-1 ). .br If IBTYPE = 1, the problem is sub( A )*x = lambda*sub( B )*x, and sub( A ) is overwritten by inv(U**H)*sub( A )*inv(U) or inv(L)*sub( A )*inv(L**H) .br If IBTYPE = 2 or 3, the problem is sub( A )*sub( B )*x = lambda*x or sub( B )*sub( A )*x = lambda*x, and sub( A ) is overwritten by U*sub( A )*U**H or L**H*sub( A )*L. .br sub( B ) must have been previously factorized as U**H*U or L*L**H by PZPOTRF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 9 IBTYPE (global input) INTEGER = 1: compute inv(U**H)*sub( A )*inv(U) or inv(L)*sub( A )*inv(L**H); = 2 or 3: compute U*sub( A )*U**H or L**H*sub( A )*L. .TP 8 UPLO (global input) CHARACTER .br = 'U': Upper triangle of sub( A ) is stored and sub( B ) is factored as U**H*U; = 'L': Lower triangle of sub( A ) is stored and sub( B ) is factored as L*L**H. .TP 8 N (global input) INTEGER The order of the matrices sub( A ) and sub( B ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N Hermitian distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced. On exit, if INFO = 0, the transformed matrix, stored in the same format as sub( A ). .TP 8 IA (global input) INTEGER A's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JA (global input) INTEGER A's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+N-1)). On entry, this array contains the local pieces of the triangular factor from the Cholesky factorization of sub( B ), as returned by PZPOTRF. .TP 8 IB (global input) INTEGER B's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JB (global input) INTEGER B's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 SCALE (global output) DOUBLE PRECISION Amount by which the eigenvalues should be scaled to compensate for the scaling performed in this routine. At present, SCALE is always returned as 1.0, it is returned here to allow for future enhancement. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pzhegvx.l0100644000056400000620000002457006335610657017132 0ustar pfrauenfstaff.TH PZHEGVX l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME .SH SYNOPSIS .TP 20 SUBROUTINE PZHEGVX( IBTYPE, JOBZ, RANGE, UPLO, N, A, IA, JA, DESCA, B, IB, JB, DESCB, VL, VU, IL, IU, ABSTOL, M, NZ, W, ORFAC, Z, IZ, JZ, DESCZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, IFAIL, ICLUSTR, GAP, INFO ) .TP 20 .ti +4 CHARACTER JOBZ, RANGE, UPLO .TP 20 .ti +4 INTEGER IA, IB, IBTYPE, IL, INFO, IU, IZ, JA, JB, JZ, LIWORK, LRWORK, LWORK, M, N, NZ .TP 20 .ti +4 DOUBLE PRECISION ABSTOL, ORFAC, VL, VU .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ), DESCZ( * ), ICLUSTR( * ), IFAIL( * ), IWORK( * ) .TP 20 .ti +4 DOUBLE PRECISION GAP( * ), RWORK( * ), W( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), B( * ), WORK( * ), Z( * ) .TP 20 .ti +4 INTEGER BLOCK_CYCLIC_2D, DLEN_, DTYPE_, CTXT_, M_, N_, MB_, NB_, RSRC_, CSRC_, LLD_ .TP 20 .ti +4 PARAMETER ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1, CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6, RSRC_ = 7, CSRC_ = 8, LLD_ = 9 ) .TP 20 .ti +4 COMPLEX*16 ONE .TP 20 .ti +4 PARAMETER ( ONE = 1.0D+0 ) .TP 20 .ti +4 DOUBLE PRECISION FIVE, ZERO .TP 20 .ti +4 PARAMETER ( FIVE = 5.0D+0, ZERO = 0.0D+0 ) .TP 20 .ti +4 INTEGER IERRNPD .TP 20 .ti +4 PARAMETER ( IERRNPD = 16 ) .TP 20 .ti +4 LOGICAL ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER IACOL, IAROW, IBCOL, IBROW, ICOFFA, ICOFFB, ICTXT, IROFFA, IROFFB, LIWMIN, LRWMIN, LWMIN, MQ0, MYCOL, MYROW, NB, NEIG, NN, NP0, NPCOL, NPROW .TP 20 .ti +4 DOUBLE PRECISION EPS, SCALE .TP 20 .ti +4 INTEGER IDUM1( 5 ), IDUM2( 5 ) .TP 20 .ti +4 LOGICAL LSAME .TP 20 .ti +4 INTEGER ICEIL, INDXG2P, NUMROC .TP 20 .ti +4 DOUBLE PRECISION PDLAMCH .TP 20 .ti +4 EXTERNAL LSAME, ICEIL, INDXG2P, NUMROC, PDLAMCH .TP 20 .ti +4 EXTERNAL BLACS_GRIDINFO, CHK1MAT, DGEBR2D, DGEBS2D, DSCAL, PCHK1MAT, PCHK2MAT, PXERBLA, PZHEEVX, PZHEGST, PZPOTRF, PZTRMM, PZTRSM .TP 20 .ti +4 INTRINSIC ABS, DBLE, DCMPLX, ICHAR, MAX, MIN, MOD .TP 20 .ti +4 IF( BLOCK_CYCLIC_2D*CSRC_*CTXT_*DLEN_*DTYPE_*LLD_*MB_*M_*NB_*N_* RSRC_.LT.0 )RETURN .TP 20 .ti +4 ICTXT = DESCA( CTXT_ ) .TP 20 .ti +4 CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL ) .TP 20 .ti +4 INFO = 0 .TP 20 .ti +4 IF( NPROW.EQ.-1 ) THEN .TP 20 .ti +4 INFO = -( 900+CTXT_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCB( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+CTXT_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCZ( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 2600+CTXT_ ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 EPS = PDLAMCH( DESCA( CTXT_ ), 'Precision' ) .TP 20 .ti +4 WANTZ = LSAME( JOBZ, 'V' ) .TP 20 .ti +4 UPPER = LSAME( UPLO, 'U' ) .TP 20 .ti +4 ALLEIG = LSAME( RANGE, 'A' ) .TP 20 .ti +4 VALEIG = LSAME( RANGE, 'V' ) .TP 20 .ti +4 INDEIG = LSAME( RANGE, 'I' ) .TP 20 .ti +4 CALL CHK1MAT( N, 4, N, 4, IA, JA, DESCA, 9, INFO ) .TP 20 .ti +4 CALL CHK1MAT( N, 4, N, 4, IB, JB, DESCB, 13, INFO ) .TP 20 .ti +4 CALL CHK1MAT( N, 4, N, 4, IZ, JZ, DESCZ, 26, INFO ) .TP 20 .ti +4 IF( INFO.EQ.0 ) THEN .TP 20 .ti +4 IF( MYROW.EQ.0 .AND. MYCOL.EQ.0 ) THEN .TP 20 .ti +4 RWORK( 1 ) = ABSTOL .TP 20 .ti +4 IF( VALEIG ) THEN .TP 20 .ti +4 RWORK( 2 ) = VL .TP 20 .ti +4 RWORK( 3 ) = VU .TP 20 .ti +4 ELSE .TP 20 .ti +4 RWORK( 2 ) = ZERO .TP 20 .ti +4 RWORK( 3 ) = ZERO .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL DGEBS2D( DESCA( CTXT_ ), 'ALL', ' ', 3, 1, RWORK, 3 ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 CALL DGEBR2D( DESCA( CTXT_ ), 'ALL', ' ', 3, 1, RWORK, 3, 0, 0 ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IAROW = INDXG2P( IA, DESCA( MB_ ), MYROW, DESCA( RSRC_ ), NPROW ) .TP 20 .ti +4 IBROW = INDXG2P( IB, DESCB( MB_ ), MYROW, DESCB( RSRC_ ), NPROW ) .TP 20 .ti +4 IACOL = INDXG2P( JA, DESCA( NB_ ), MYCOL, DESCA( CSRC_ ), NPCOL ) .TP 20 .ti +4 IBCOL = INDXG2P( JB, DESCB( NB_ ), MYCOL, DESCB( CSRC_ ), NPCOL ) .TP 20 .ti +4 IROFFA = MOD( IA-1, DESCA( MB_ ) ) .TP 20 .ti +4 ICOFFA = MOD( JA-1, DESCA( NB_ ) ) .TP 20 .ti +4 IROFFB = MOD( IB-1, DESCB( MB_ ) ) .TP 20 .ti +4 ICOFFB = MOD( JB-1, DESCB( NB_ ) ) .TP 20 .ti +4 LQUERY = .FALSE. .TP 20 .ti +4 IF( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 .OR. LRWORK.EQ.-1 ) LQUERY = .TRUE. .TP 20 .ti +4 LIWMIN = 6*MAX( N, ( NPROW*NPCOL )+1, 4 ) .TP 20 .ti +4 NB = DESCA( MB_ ) .TP 20 .ti +4 NN = MAX( N, NB, 2 ) .TP 20 .ti +4 NP0 = NUMROC( NN, NB, 0, 0, NPROW ) .TP 20 .ti +4 IF( ( .NOT.WANTZ ) .OR. ( VALEIG .AND. ( .NOT.LQUERY ) ) ) THEN .TP 20 .ti +4 LWMIN = N + MAX( NB*( NP0+1 ), 3 ) .TP 20 .ti +4 LRWMIN = 5*NN + 4*N .TP 20 .ti +4 NEIG = 0 .TP 20 .ti +4 ELSE .TP 20 .ti +4 IF( ALLEIG .OR. VALEIG ) THEN .TP 20 .ti +4 NEIG = N .TP 20 .ti +4 ELSE IF( INDEIG ) THEN .TP 20 .ti +4 NEIG = IU - IL + 1 .TP 20 .ti +4 END IF .TP 20 .ti +4 MQ0 = NUMROC( MAX( NEIG, NB, 2 ), NB, 0, 0, NPCOL ) .TP 20 .ti +4 LWMIN = N + ( NP0+MQ0+NB )*NB .TP 20 .ti +4 LRWMIN = 4*N + MAX( 5*NN, NP0*MQ0 ) + ICEIL( NEIG, NPROW*NPCOL )*NN .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( IBTYPE.LT.1 .OR. IBTYPE.GT.3 ) THEN .TP 20 .ti +4 INFO = -1 .TP 20 .ti +4 ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN .TP 20 .ti +4 INFO = -2 .TP 20 .ti +4 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN .TP 20 .ti +4 INFO = -3 .TP 20 .ti +4 ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN .TP 20 .ti +4 INFO = -4 .TP 20 .ti +4 ELSE IF( N.LT.0 ) THEN .TP 20 .ti +4 INFO = -5 .TP 20 .ti +4 ELSE IF( IROFFA.NE.0 ) THEN .TP 20 .ti +4 INFO = -7 .TP 20 .ti +4 ELSE IF( ICOFFA.NE.0 ) THEN .TP 20 .ti +4 INFO = -8 .TP 20 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCA( NB_ ) ) THEN .TP 20 .ti +4 INFO = -( 900+NB_ ) .TP 20 .ti +4 ELSE IF( DESCA( M_ ).NE.DESCB( M_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+M_ ) .TP 20 .ti +4 ELSE IF( DESCA( N_ ).NE.DESCB( N_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+N_ ) .TP 20 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCB( MB_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+MB_ ) .TP 20 .ti +4 ELSE IF( DESCA( NB_ ).NE.DESCB( NB_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+NB_ ) .TP 20 .ti +4 ELSE IF( DESCA( RSRC_ ).NE.DESCB( RSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+RSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CSRC_ ).NE.DESCB( CSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+CSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCB( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 1300+CTXT_ ) .TP 20 .ti +4 ELSE IF( DESCA( M_ ).NE.DESCZ( M_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+M_ ) .TP 20 .ti +4 ELSE IF( DESCA( N_ ).NE.DESCZ( N_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+N_ ) .TP 20 .ti +4 ELSE IF( DESCA( MB_ ).NE.DESCZ( MB_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+MB_ ) .TP 20 .ti +4 ELSE IF( DESCA( NB_ ).NE.DESCZ( NB_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+NB_ ) .TP 20 .ti +4 ELSE IF( DESCA( RSRC_ ).NE.DESCZ( RSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+RSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CSRC_ ).NE.DESCZ( CSRC_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+CSRC_ ) .TP 20 .ti +4 ELSE IF( DESCA( CTXT_ ).NE.DESCZ( CTXT_ ) ) THEN .TP 20 .ti +4 INFO = -( 2200+CTXT_ ) .TP 20 .ti +4 ELSE IF( IROFFB.NE.0 .OR. IBROW.NE.IAROW ) THEN .TP 20 .ti +4 INFO = -11 .TP 20 .ti +4 ELSE IF( ICOFFB.NE.0 .OR. IBCOL.NE.IACOL ) THEN .TP 20 .ti +4 INFO = -12 .TP 20 .ti +4 ELSE IF( VALEIG .AND. N.GT.0 .AND. VU.LE.VL ) THEN .TP 20 .ti +4 INFO = -15 .TP 20 .ti +4 ELSE IF( INDEIG .AND. ( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) ) THEN .TP 20 .ti +4 INFO = -16 .TP 20 .ti +4 ELSE IF( INDEIG .AND. ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) ) THEN .TP 20 .ti +4 INFO = -17 .TP 20 .ti +4 ELSE IF( VALEIG .AND. ( ABS( RWORK( 2 )-VL ).GT.FIVE*EPS* ABS( VL ) ) ) THEN .TP 20 .ti +4 INFO = -14 .TP 20 .ti +4 ELSE IF( VALEIG .AND. ( ABS( RWORK( 3 )-VU ).GT.FIVE*EPS* ABS( VU ) ) ) THEN .TP 20 .ti +4 INFO = -15 .TP 20 .ti +4 ELSE IF( ABS( RWORK( 1 )-ABSTOL ).GT.FIVE*EPS* ABS( ABSTOL ) ) THEN .TP 20 .ti +4 INFO = -18 .TP 20 .ti +4 ELSE IF( LWORK.LT.LWMIN .AND. LWORK.NE.-1 ) THEN .TP 20 .ti +4 INFO = -28 .TP 20 .ti +4 ELSE IF( LRWORK.LT.LRWMIN .AND. LRWORK.NE.-1 ) THEN .TP 20 .ti +4 INFO = -30 .TP 20 .ti +4 ELSE IF( LIWORK.LT.LIWMIN .AND. LIWORK.NE.-1 ) THEN .TP 20 .ti +4 INFO = -32 .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM1( 1 ) = IBTYPE .TP 20 .ti +4 IDUM2( 1 ) = 1 .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 IDUM1( 2 ) = ICHAR( 'V' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 2 ) = ICHAR( 'N' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 2 ) = 2 .TP 20 .ti +4 IF( UPPER ) THEN .TP 20 .ti +4 IDUM1( 3 ) = ICHAR( 'U' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 3 ) = ICHAR( 'L' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 3 ) = 3 .TP 20 .ti +4 IF( ALLEIG ) THEN .TP 20 .ti +4 IDUM1( 4 ) = ICHAR( 'A' ) .TP 20 .ti +4 ELSE IF( INDEIG ) THEN .TP 20 .ti +4 IDUM1( 4 ) = ICHAR( 'I' ) .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 4 ) = ICHAR( 'V' ) .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 4 ) = 4 .TP 20 .ti +4 IF( LQUERY ) THEN .TP 20 .ti +4 IDUM1( 5 ) = -1 .TP 20 .ti +4 ELSE .TP 20 .ti +4 IDUM1( 5 ) = 1 .TP 20 .ti +4 END IF .TP 20 .ti +4 IDUM2( 5 ) = 5 .TP 20 .ti +4 CALL PCHK2MAT( N, 4, N, 4, IA, JA, DESCA, 9, N, 4, N, 4, IB, JB, DESCB, 13, 5, IDUM1, IDUM2, INFO ) .TP 20 .ti +4 CALL PCHK1MAT( N, 4, N, 4, IZ, JZ, DESCZ, 26, 0, IDUM1, IDUM2, INFO ) .TP 20 .ti +4 END IF .TP 20 .ti +4 WORK( 1 ) = DCMPLX( DBLE( LWMIN ) ) .TP 20 .ti +4 RWORK( 1 ) = DBLE( LRWMIN ) .TP 20 .ti +4 IWORK( 1 ) = LIWMIN .TP 20 .ti +4 IF( INFO.NE.0 ) THEN .TP 20 .ti +4 CALL PXERBLA( ICTXT, 'PZHEGVX ', -INFO ) .TP 20 .ti +4 RETURN .TP 20 .ti +4 ELSE IF( LQUERY ) THEN .TP 20 .ti +4 RETURN .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PZPOTRF( UPLO, N, B, IB, JB, DESCB, INFO ) .TP 20 .ti +4 IF( INFO.NE.0 ) THEN .TP 20 .ti +4 IFAIL( 1 ) = INFO .TP 20 .ti +4 INFO = IERRNPD .TP 20 .ti +4 RETURN .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PZHEGST( IBTYPE, UPLO, N, A, IA, JA, DESCA, B, IB, JB, DESCB, SCALE, INFO ) .TP 20 .ti +4 CALL PZHEEVX( JOBZ, RANGE, UPLO, N, A, IA, JA, DESCA, VL, VU, IL, IU, ABSTOL, M, NZ, W, ORFAC, Z, IZ, JZ, DESCZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, IFAIL, ICLUSTR, GAP, INFO ) .TP 20 .ti +4 IF( WANTZ ) THEN .TP 20 .ti +4 NEIG = M .TP 20 .ti +4 IF( IBTYPE.EQ.1 .OR. IBTYPE.EQ.2 ) THEN .TP 20 .ti +4 IF( UPPER ) THEN .TP 20 .ti +4 TRANS = 'N' .TP 20 .ti +4 ELSE .TP 20 .ti +4 TRANS = 'C' .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PZTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, NEIG, ONE, B, IB, JB, DESCB, Z, IZ, JZ, DESCZ ) .TP 20 .ti +4 ELSE IF( IBTYPE.EQ.3 ) THEN .TP 20 .ti +4 IF( UPPER ) THEN .TP 20 .ti +4 TRANS = 'C' .TP 20 .ti +4 ELSE .TP 20 .ti +4 TRANS = 'N' .TP 20 .ti +4 END IF .TP 20 .ti +4 CALL PZTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, NEIG, ONE, B, IB, JB, DESCB, Z, IZ, JZ, DESCZ ) .TP 20 .ti +4 END IF .TP 20 .ti +4 END IF .TP 20 .ti +4 IF( SCALE.NE.ONE ) THEN .TP 20 .ti +4 CALL DSCAL( N, SCALE, W, 1 ) .TP 20 .ti +4 END IF .TP 20 .ti +4 RETURN .TP 20 .ti +4 END .SH PURPOSE scalapack-doc-1.5/man/manl/pzhetd2.l0100644000056400000620000002031706335610657017012 0ustar pfrauenfstaff.TH PZHETD2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZHETD2 - reduce a complex Hermitian matrix sub( A ) to Hermitian tridiagonal form T by an unitary similarity transformation .SH SYNOPSIS .TP 20 SUBROUTINE PZHETD2( UPLO, N, A, IA, JA, DESCA, D, E, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION D( * ), E( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), WORK( * ) .SH PURPOSE PZHETD2 reduces a complex Hermitian matrix sub( A ) to Hermitian tridiagonal form T by an unitary similarity transformation: Q' * sub( A ) * Q = T, where sub( A ) = A(IA:IA+N-1,JA:JA+N-1). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the upper or lower triangular part of the Hermitian matrix sub( A ) is stored: .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the Hermitian distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced. On exit, if UPLO = 'U', the diagonal and first superdiagonal of sub( A ) are over- written by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of sub( A ) are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-1) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i). D is tied to the distributed matrix A. .TP 8 E (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-1) if UPLO = 'U', LOCc(JA+N-2) otherwise. The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. E is tied to the distributed matrix A. .TP 8 TAU (local output) COMPLEX*16, array, dimension LOCc(JA+N-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK( 1 ) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 3*N. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors .br Q = H(n-1) . . . H(2) H(1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in .br A(ia:ia+i-2,ja+i), and tau in TAU(ja+i-1). .br If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors .br Q = H(1) H(2) . . . H(n-1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in .br A(ia+i+1:ia+n-1,ja+i-1), and tau in TAU(ja+i-1). .br The contents of sub( A ) on exit are illustrated by the following examples with n = 5: .br if UPLO = 'U': if UPLO = 'L': .br ( d e v2 v3 v4 ) ( d ) ( d e v3 v4 ) ( e d ) ( d e v4 ) ( v1 e d ) ( d e ) ( v1 v2 e d ) ( d ) ( v1 v2 v3 e d ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expression should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) with .br IROFFA = MOD( IA-1, MB_A ) and ICOFFA = MOD( JA-1, NB_A ). scalapack-doc-1.5/man/manl/pzhetrd.l0100644000056400000620000002075606335610657017121 0ustar pfrauenfstaff.TH PZHETRD l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZHETRD - reduce a complex Hermitian matrix sub( A ) to Hermitian tridiagonal form T by an unitary similarity transformation .SH SYNOPSIS .TP 20 SUBROUTINE PZHETRD( UPLO, N, A, IA, JA, DESCA, D, E, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION D( * ), E( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), WORK( * ) .SH PURPOSE PZHETRD reduces a complex Hermitian matrix sub( A ) to Hermitian tridiagonal form T by an unitary similarity transformation: Q' * sub( A ) * Q = T, where sub( A ) = A(IA:IA+N-1,JA:JA+N-1). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the upper or lower triangular part of the Hermitian matrix sub( A ) is stored: .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the Hermitian distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced. On exit, if UPLO = 'U', the diagonal and first superdiagonal of sub( A ) are over- written by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of sub( A ) are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-1) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i). D is tied to the distributed matrix A. .TP 8 E (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-1) if UPLO = 'U', LOCc(JA+N-2) otherwise. The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. E is tied to the distributed matrix A. .TP 8 TAU (local output) COMPLEX*16, array, dimension LOCc(JA+N-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK( 1 ) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MAX( NB * ( NP +1 ), 3 * NB ) where NB = MB_A = NB_A, NP = NUMROC( N, NB, MYROW, IAROW, NPROW ), IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ). INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors .br Q = H(n-1) . . . H(2) H(1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in .br A(ia:ia+i-2,ja+i), and tau in TAU(ja+i-1). .br If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors .br Q = H(1) H(2) . . . H(n-1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in .br A(ia+i+1:ia+n-1,ja+i-1), and tau in TAU(ja+i-1). .br The contents of sub( A ) on exit are illustrated by the following examples with n = 5: .br if UPLO = 'U': if UPLO = 'L': .br ( d e v2 v3 v4 ) ( d ) ( d e v3 v4 ) ( e d ) ( d e v4 ) ( v1 e d ) ( d e ) ( v1 v2 e d ) ( d ) ( v1 v2 v3 e d ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i). .br Alignment requirements .br ====================== .br The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the following expression should be true: .br ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA .AND. IROFFA.EQ.0 ) with IROFFA = MOD( IA-1, MB_A ) and ICOFFA = MOD( JA-1, NB_A ). scalapack-doc-1.5/man/manl/pzlabrd.l0100644000056400000620000002360406335610657017072 0ustar pfrauenfstaff.TH PZLABRD l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLABRD - reduce the first NB rows and columns of a complex general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form by an unitary transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transfor- mation to the unreduced part of sub( A ) .SH SYNOPSIS .TP 20 SUBROUTINE PZLABRD( M, N, NB, A, IA, JA, DESCA, D, E, TAUQ, TAUP, X, IX, JX, DESCX, Y, IY, JY, DESCY, WORK ) .TP 20 .ti +4 INTEGER IA, IX, IY, JA, JX, JY, M, N, NB .TP 20 .ti +4 INTEGER DESCA( * ), DESCX( * ), DESCY( * ) .TP 20 .ti +4 DOUBLE PRECISION D( * ), E( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAUP( * ), TAUQ( * ), X( * ), Y( * ), WORK( * ) .SH PURPOSE PZLABRD reduces the first NB rows and columns of a complex general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form by an unitary transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transfor- mation to the unreduced part of sub( A ). If M >= N, sub( A ) is reduced to upper bidiagonal form; if M < N, to lower bidiagonal form. .br This is an auxiliary routine called by PZGEBRD. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 NB (global input) INTEGER The number of leading rows and columns of sub( A ) to be reduced. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the general distributed matrix sub( A ) to be reduced. On exit, the first NB rows and columns of the matrix are overwritten; the rest of the distributed matrix sub( A ) is unchanged. If m >= n, elements on and below the diagonal in the first NB columns, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors. If m < n, elements below the diagonal in the first NB columns, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and elements on and above the diagonal in the first NB rows, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) DOUBLE PRECISION array, dimension LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-1) otherwise. The distributed diagonal elements of the bidiagonal matrix B: D(i) = A(ia+i-1,ja+i-1). D is tied to the distributed matrix A. .TP 8 E (local output) DOUBLE PRECISION array, dimension LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-2) otherwise. The distributed off-diagonal elements of the bidiagonal distributed matrix B: if m >= n, E(i) = A(ia+i-1,ja+i) for i = 1,2,...,n-1; if m < n, E(i) = A(ia+i,ja+i-1) for i = 1,2,...,m-1. E is tied to the distributed matrix A. .TP 8 TAUQ (local output) COMPLEX*16 array dimension LOCc(JA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the unitary matrix Q. TAUQ is tied to the distributed matrix A. See Further Details. TAUP (local output) COMPLEX*16 array, dimension LOCr(IA+MIN(M,N)-1). The scalar factors of the elementary reflectors which represent the unitary matrix P. TAUP is tied to the distributed matrix A. See Further Details. X (local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_X,NB). On exit, the local pieces of the distributed M-by-NB matrix X(IX:IX+M-1,JX:JX+NB-1) required to update the unreduced part of sub( A ). .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 Y (local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_Y,NB). On exit, the local pieces of the distributed N-by-NB matrix Y(IY:IY+N-1,JY:JY+NB-1) required to update the unreduced part of sub( A ). .TP 8 IY (global input) INTEGER The row index in the global array Y indicating the first row of sub( Y ). .TP 8 JY (global input) INTEGER The column index in the global array Y indicating the first column of sub( Y ). .TP 8 DESCY (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix Y. .TP 8 WORK (local workspace) COMPLEX*16 array, dimension (LWORK) LWORK >= NB_A + NQ, with NQ = NUMROC( N+MOD( IA-1, NB_Y ), NB_Y, MYCOL, IACOL, NPCOL ) IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ) INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. .SH FURTHER DETAILS The matrices Q and P are represented as products of elementary reflectors: .br Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb) Each H(i) and G(i) has the form: .br H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are complex scalars, and v and u are complex vectors. .br If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in A(ia+i-1:ia+m-1,ja+i-1); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in A(ia+i-1,ja+i:ja+n-1); tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in A(ia+i+1:ia+m-1,ja+i-1); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in A(ia+i-1,ja+i:ja+n-1); tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). .br The elements of the vectors v and u together form the m-by-nb matrix V and the nb-by-n matrix U' which are needed, with X and Y, to apply the transformation to the unreduced part of the matrix, using a block update of the form: sub( A ) := sub( A ) - V*Y' - X*U'. .br The contents of sub( A ) on exit are illustrated by the following examples with nb = 2: .br m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 ) ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 ) ( v1 v2 a a a ) ( v1 1 a a a a ) ( v1 v2 a a a ) ( v1 v2 a a a a ) ( v1 v2 a a a ) ( v1 v2 a a a a ) ( v1 v2 a a a ) .br where a denotes an element of the original matrix which is unchanged, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i). .br scalapack-doc-1.5/man/manl/pzlacgv.l0100644000056400000620000001027306335610657017100 0ustar pfrauenfstaff.TH PZLACGV l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLACGV - conjugate a complex vector of length N, sub( X ), where sub( X ) denotes X(IX,JX:JX+N-1) if INCX = DESCX( M_ ) and X(IX:IX+N-1,JX) if INCX = 1, and Notes ===== Each global data object is described by an associated description vector .SH SYNOPSIS .TP 20 SUBROUTINE PZLACGV( N, X, IX, JX, DESCX, INCX ) .TP 20 .ti +4 INTEGER INCX, IX, JX, N .TP 20 .ti +4 INTEGER DESCX( * ) .TP 20 .ti +4 COMPLEX*16 X( * ) .SH PURPOSE PZLACGV conjugates a complex vector of length N, sub( X ), where sub( X ) denotes X(IX,JX:JX+N-1) if INCX = DESCX( M_ ) and X(IX:IX+N-1,JX) if INCX = 1, and the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. .SH ARGUMENTS .TP 8 N (global input) INTEGER The length of the distributed vector sub( X ). .TP 8 X (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_X,*). On entry the vector to be conjugated x( i ) = X(IX+(JX-1)*M_X +(i-1)*INCX ), 1 <= i <= N. On exit the conjugated vector. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 INCX (global input) INTEGER The global increment for the elements of X. Only two values of INCX are supported in this version, namely 1 and M_X. INCX must not be zero. scalapack-doc-1.5/man/manl/pzlacon.l0100644000056400000620000001266106335610657017103 0ustar pfrauenfstaff.TH PZLACON l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLACON - estimate the 1-norm of a square, complex distributed matrix A .SH SYNOPSIS .TP 20 SUBROUTINE PZLACON( N, V, IV, JV, DESCV, X, IX, JX, DESCX, EST, KASE ) .TP 20 .ti +4 INTEGER IV, IX, JV, JX, KASE, N .TP 20 .ti +4 DOUBLE PRECISION EST .TP 20 .ti +4 INTEGER DESCV( * ), DESCX( * ) .TP 20 .ti +4 COMPLEX*16 V( * ), X( * ) .SH PURPOSE PZLACON estimates the 1-norm of a square, complex distributed matrix A. Reverse communication is used for evaluating matrix-vector products. X and V are aligned with the distributed matrix A, this information is implicitly contained within IV, IX, DESCV, and DESCX. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The length of the distributed vectors V and X. N >= 0. .TP 8 V (local workspace) COMPLEX*16 pointer into the local memory to an array of dimension LOCr(N+MOD(IV-1,MB_V)). On the final return, V = A*W, where EST = norm(V)/norm(W) (W is not returned). .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 X (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension LOCr(N+MOD(IX-1,MB_X)). On an intermediate return, X should be overwritten by A * X, if KASE=1, A' * X, if KASE=2, where A' is the conjugate transpose of A, and PZLACON must be re-called with all the other parameters unchanged. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 EST (global output) DOUBLE PRECISION An estimate (a lower bound) for norm(A). .TP 8 KASE (local input/local output) INTEGER On the initial call to PZLACON, KASE should be 0. On an intermediate return, KASE will be 1 or 2, indicating whether X should be overwritten by A * X or A' * X. On the final return from PZLACON, KASE will again be 0. .SH FURTHER DETAILS The serial version ZLACON has been contributed by Nick Higham, University of Manchester. It was originally named SONEST, dated March 16, 1988. .br Reference: N.J. Higham, "FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation", ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988. scalapack-doc-1.5/man/manl/pzlacp2.l0100644000056400000620000001270506335610657017007 0ustar pfrauenfstaff.TH PZLACP2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLACP2 - copie all or part of a distributed matrix A to another distributed matrix B .SH SYNOPSIS .TP 20 SUBROUTINE PZLACP2( UPLO, M, N, A, IA, JA, DESCA, B, IB, JB, DESCB ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IB, JA, JB, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), B( * ) .SH PURPOSE PZLACP2 copies all or part of a distributed matrix A to another distributed matrix B. No communication is performed, PZLACP2 performs a local copy sub( A ) := sub( B ), where sub( A ) denotes A(IA:IA+M-1,JA:JA+N-1) and sub( B ) denotes B(IB:IB+M-1,JB:JB+N-1). PZLACP2 requires that only dimension of the matrix operands is distributed. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies the part of the distributed matrix sub( A ) to be copied: .br = 'U': Upper triangular part is copied; the strictly lower triangular part of sub( A ) is not referenced; = 'L': Lower triangular part is copied; the strictly upper triangular part of sub( A ) is not referenced; Otherwise: All of the matrix sub( A ) is copied. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1) ). This array contains the local pieces of the distributed matrix sub( A ) to be copied from. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+N-1) ). This array contains on exit the local pieces of the distributed matrix sub( B ) set as follows: if UPLO = 'U', B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), 1<=i<=j, 1<=j<=N; if UPLO = 'L', B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), j<=i<=M, 1<=j<=N; otherwise, B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), 1<=i<=M, 1<=j<=N. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. scalapack-doc-1.5/man/manl/pzlacpy.l0100644000056400000620000001256506335610657017122 0ustar pfrauenfstaff.TH PZLACPY l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLACPY - copie all or part of a distributed matrix A to another distributed matrix B .SH SYNOPSIS .TP 20 SUBROUTINE PZLACPY( UPLO, M, N, A, IA, JA, DESCA, B, IB, JB, DESCB ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IB, JA, JB, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), B( * ) .SH PURPOSE PZLACPY copies all or part of a distributed matrix A to another distributed matrix B. No communication is performed, PZLACPY performs a local copy sub( A ) := sub( B ), where sub( A ) denotes A(IA:IA+M-1,JA:JA+N-1) and sub( B ) denotes B(IB:IB+M-1,JB:JB+N-1). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies the part of the distributed matrix sub( A ) to be copied: .br = 'U': Upper triangular part is copied; the strictly lower triangular part of sub( A ) is not referenced; = 'L': Lower triangular part is copied; the strictly upper triangular part of sub( A ) is not referenced; Otherwise: All of the matrix sub( A ) is copied. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1) ). This array contains the local pieces of the distributed matrix sub( A ) to be copied from. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+N-1) ). This array contains on exit the local pieces of the distributed matrix sub( B ) set as follows: if UPLO = 'U', B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), 1<=i<=j, 1<=j<=N; if UPLO = 'L', B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), j<=i<=M, 1<=j<=N; otherwise, B(IB+i-1,JB+j-1) = A(IA+i-1,JA+j-1), 1<=i<=M, 1<=j<=N. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. scalapack-doc-1.5/man/manl/pzlaevswp.l0100644000056400000620000001224406335610657017465 0ustar pfrauenfstaff.TH PZLAEVSWP l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZLAEVSWP - move the eigenvectors (potentially unsorted) from where they are computed, to a ScaLAPACK standard block cyclic array, sorted so that the corresponding eigenvalues are sorted .SH SYNOPSIS .TP 22 SUBROUTINE PZLAEVSWP( N, ZIN, LDZI, Z, IZ, JZ, DESCZ, NVS, KEY, RWORK, LRWORK ) .TP 22 .ti +4 INTEGER IZ, JZ, LDZI, LRWORK, N .TP 22 .ti +4 INTEGER DESCZ( * ), KEY( * ), NVS( * ) .TP 22 .ti +4 DOUBLE PRECISION RWORK( * ), ZIN( LDZI, * ) .TP 22 .ti +4 COMPLEX*16 Z( * ) .SH PURPOSE PZLAEVSWP moves the eigenvectors (potentially unsorted) from where they are computed, to a ScaLAPACK standard block cyclic array, sorted so that the corresponding eigenvalues are sorted. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS NP = the number of rows local to a given process. NQ = the number of columns local to a given process. .TP 8 N (global input) INTEGER The order of the matrix A. N >= 0. .TP 8 ZIN (local input) DOUBLE PRECISION array, dimension ( LDZI, NVS(iam) ) The eigenvectors on input. Each eigenvector resides entirely in one process. Each process holds a contiguous set of NVS(iam) eigenvectors. The first eigenvector which the process holds is: sum for i=[0,iam-1) of NVS(i) .TP 8 LDZI (locl input) INTEGER leading dimension of the ZIN array .TP 8 Z (local output) COMPLEX*16 array global dimension (N, N), local dimension (DESCZ(DLEN_), NQ) The eigenvectors on output. The eigenvectors are distributed in a block cyclic manner in both dimensions, with a block size of NB. .TP 8 IZ (global input) INTEGER Z's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JZ (global input) INTEGER Z's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCZ (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix Z. .TP 8 NVS (global input) INTEGER array, dimension( nprocs+1 ) nvs(i) = number of processes number of eigenvectors held by processes [0,i-1) nvs(1) = number of eigen vectors held by [0,1-1) == 0 nvs(nprocs+1) = number of eigen vectors held by [0,nprocs) == total number of eigenvectors .TP 8 KEY (global input) INTEGER array, dimension( N ) Indicates the actual index (after sorting) for each of the eigenvectors. .TP 9 RWORK (local workspace) DOUBLE PRECISION array, dimension (LRWORK) .TP 9 LRWORK (local input) INTEGER dimension of RWORK scalapack-doc-1.5/man/manl/pzlahrd.l0100644000056400000620000001054706335610657017102 0ustar pfrauenfstaff.TH PZLAHRD l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLAHRD - reduce the first NB columns of a complex general N-by-(N-K+1) distributed matrix A(IA:IA+N-1,JA:JA+N-K) so that elements below the k-th subdiagonal are zero .SH SYNOPSIS .TP 20 SUBROUTINE PZLAHRD( N, K, NB, A, IA, JA, DESCA, TAU, T, Y, IY, JY, DESCY, WORK ) .TP 20 .ti +4 INTEGER IA, IY, JA, JY, K, N, NB .TP 20 .ti +4 INTEGER DESCA( * ), DESCY( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), T( * ), TAU( * ), WORK( * ), Y( * ) .SH PURPOSE PZLAHRD reduces the first NB columns of a complex general N-by-(N-K+1) distributed matrix A(IA:IA+N-1,JA:JA+N-K) so that elements below the k-th subdiagonal are zero. The reduction is performed by an unitary similarity transformation Q' * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T. .br This is an auxiliary routine called by PZGEHRD. In the following comments sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1). .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 K (global input) INTEGER The offset for the reduction. Elements below the k-th subdiagonal in the first NB columns are reduced to zero. .TP 8 NB (global input) INTEGER The number of columns to be reduced. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-K)). On entry, this array contains the the local pieces of the N-by-(N-K+1) general distributed matrix A(IA:IA+N-1,JA:JA+N-K). On exit, the elements on and above the k-th subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced distributed matrix; the elements below the k-th subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors. The other columns of A(IA:IA+N-1,JA:JA+N-K) are unchanged. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX*16 array, dimension LOCc(JA+N-2) The scalar factors of the elementary reflectors (see Further Details). TAU is tied to the distributed matrix A. .TP 8 T (local output) COMPLEX*16 array, dimension (NB_A,NB_A) The upper triangular matrix T. .TP 8 Y (local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_Y,NB_A). On exit, this array contains the local pieces of the N-by-NB distributed matrix Y. LLD_Y >= LOCr(IA+N-1). .TP 8 IY (global input) INTEGER The row index in the global array Y indicating the first row of sub( Y ). .TP 8 JY (global input) INTEGER The column index in the global array Y indicating the first column of sub( Y ). .TP 8 DESCY (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix Y. .TP 8 WORK (local workspace) COMPLEX*16 array, dimension (NB) .SH FURTHER DETAILS The matrix Q is represented as a product of nb elementary reflectors Q = H(1) H(2) . . . H(nb). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(ia+i+k:ia+n-1,ja+i-1), and tau in TAU(ja+i-1). .br The elements of the vectors v together form the (n-k+1)-by-nb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form: A(ia:ia+n-1,ja:ja+n-k) := (I-V*T*V')*(A(ia:ia+n-1,ja:ja+n-k)-Y*V'). The contents of A(ia:ia+n-1,ja:ja+n-k) on exit are illustrated by the following example with n = 7, k = 3 and nb = 2: .br ( a h a a a ) .br ( a h a a a ) .br ( a h a a a ) .br ( h h a a a ) .br ( v1 h a a a ) .br ( v1 v2 a a a ) .br ( v1 v2 a a a ) .br where a denotes an element of the original matrix .br A(ia:ia+n-1,ja:ja+n-k), h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i). .br scalapack-doc-1.5/man/manl/pzlange.l0100644000056400000620000001305106335610657017067 0ustar pfrauenfstaff.TH PZLANGE l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLANGE - return the value of the one norm, or the Frobenius norm, .SH SYNOPSIS .TP 17 DOUBLE PRECISION FUNCTION PZLANGE( NORM, M, N, A, IA, JA, DESCA, WORK ) .TP 17 .ti +4 CHARACTER NORM .TP 17 .ti +4 INTEGER IA, JA, M, N .TP 17 .ti +4 INTEGER DESCA( * ) .TP 17 .ti +4 DOUBLE PRECISION WORK( * ) .TP 17 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZLANGE returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a distributed matrix sub( A ) = A(IA:IA+M-1, JA:JA+N-1). .br PZLANGE returns the value .br ( max(abs(A(i,j))), NORM = 'M' or 'm' with IA <= i <= IA+M-1, ( and JA <= j <= JA+N-1, ( .br ( norm1( sub( A ) ), NORM = '1', 'O' or 'o' .br ( .br ( normI( sub( A ) ), NORM = 'I' or 'i' .br ( .br ( normF( sub( A ) ), NORM = 'F', 'f', 'E' or 'e' .br where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a matrix norm. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies the value to be returned in PZLANGE as described above. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). When M = 0, PZLANGE is set to zero. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). When N = 0, PZLANGE is set to zero. N >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)) containing the local pieces of the distributed matrix sub( A ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 WORK (local workspace) DOUBLE PRECISION array dimension (LWORK) LWORK >= 0 if NORM = 'M' or 'm' (not referenced), Nq0 if NORM = '1', 'O' or 'o', Mp0 if NORM = 'I' or 'i', 0 if NORM = 'F', 'f', 'E' or 'e' (not referenced), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. scalapack-doc-1.5/man/manl/pzlanhe.l0100644000056400000620000001447706335610657017105 0ustar pfrauenfstaff.TH PZLANHE l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLANHE - return the value of the one norm, or the Frobenius norm, .SH SYNOPSIS .TP 17 DOUBLE PRECISION FUNCTION PZLANHE( NORM, UPLO, N, A, IA, JA, DESCA, WORK ) .TP 17 .ti +4 CHARACTER NORM, UPLO .TP 17 .ti +4 INTEGER IA, JA, N .TP 17 .ti +4 INTEGER DESCA( * ) .TP 17 .ti +4 DOUBLE PRECISION WORK( * ) .TP 17 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZLANHE returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex hermitian distributed matrix sub(A) = A(IA:IA+N-1,JA:JA+N-1). PZLANHE returns the value .br ( max(abs(A(i,j))), NORM = 'M' or 'm' with IA <= i <= IA+N-1, ( and JA <= j <= JA+N-1, ( .br ( norm1( sub( A ) ), NORM = '1', 'O' or 'o' .br ( .br ( normI( sub( A ) ), NORM = 'I' or 'i' .br ( .br ( normF( sub( A ) ), NORM = 'F', 'f', 'E' or 'e' .br where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a matrix norm. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies the value to be returned in PZLANHE as described above. .TP 8 UPLO (global input) CHARACTER Specifies whether the upper or lower triangular part of the hermitian matrix sub( A ) is to be referenced. = 'U': Upper triangular part of sub( A ) is referenced, .br = 'L': Lower triangular part of sub( A ) is referenced. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on i.e the number of rows and columns of the distributed submatrix sub( A ). When N = 0, PZLANHE is set to zero. N >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)) containing the local pieces of the hermitian distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular matrix which norm is to be computed, and the strictly lower triangular part of this matrix is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular matrix which norm is to be computed, and the strictly upper triangular part of sub( A ) is not referenced. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 WORK (local workspace) DOUBLE PRECISION array dimension (LWORK) LWORK >= 0 if NORM = 'M' or 'm' (not referenced), 2*Nq0+Np0+LDW if NORM = '1', 'O', 'o', 'I' or 'i', where LDW is given by: IF( NPROW.NE.NPCOL ) THEN LDW = MB_A*CEIL(CEIL(Np0/MB_A)/(LCM/NPROW)) ELSE LDW = 0 END IF 0 if NORM = 'F', 'f', 'E' or 'e' (not referenced), where LCM is the least common multiple of NPROW and NPCOL LCM = ILCM( NPROW, NPCOL ) and CEIL denotes the ceiling operation (ICEIL). IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Np0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), ICEIL, ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. scalapack-doc-1.5/man/manl/pzlanhs.l0100644000056400000620000001256106335610657017113 0ustar pfrauenfstaff.TH PZLANHS l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLANHS - return the value of the one norm, or the Frobenius norm, .SH SYNOPSIS .TP 17 DOUBLE PRECISION FUNCTION PZLANHS( NORM, N, A, IA, JA, DESCA, WORK ) .TP 17 .ti +4 CHARACTER NORM .TP 17 .ti +4 INTEGER IA, JA, N .TP 17 .ti +4 INTEGER DESCA( * ) .TP 17 .ti +4 DOUBLE PRECISION WORK( * ) .TP 17 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZLANHS returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hessenberg distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). PZLANHS returns the value .br ( max(abs(A(i,j))), NORM = 'M' or 'm' with IA <= i <= IA+N-1, ( and JA <= j <= JA+N-1, ( .br ( norm1( sub( A ) ), NORM = '1', 'O' or 'o' .br ( .br ( normI( sub( A ) ), NORM = 'I' or 'i' .br ( .br ( normF( sub( A ) ), NORM = 'F', 'f', 'E' or 'e' .br where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a matrix norm. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies the value to be returned in PZLANHS as described above. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on i.e the number of rows and columns of the distributed submatrix sub( A ). When N = 0, PZLANHS is set to zero. N >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1) ) containing the local pieces of sub( A ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 WORK (local workspace) DOUBLE PRECISION array dimension (LWORK) LWORK >= 0 if NORM = 'M' or 'm' (not referenced), Nq0 if NORM = '1', 'O' or 'o', Mp0 if NORM = 'I' or 'i', 0 if NORM = 'F', 'f', 'E' or 'e' (not referenced), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Np0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. scalapack-doc-1.5/man/manl/pzlansy.l0100644000056400000620000001447706335610660017136 0ustar pfrauenfstaff.TH PZLANSY l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLANSY - return the value of the one norm, or the Frobenius norm, .SH SYNOPSIS .TP 17 DOUBLE PRECISION FUNCTION PZLANSY( NORM, UPLO, N, A, IA, JA, DESCA, WORK ) .TP 17 .ti +4 CHARACTER NORM, UPLO .TP 17 .ti +4 INTEGER IA, JA, N .TP 17 .ti +4 INTEGER DESCA( * ) .TP 17 .ti +4 DOUBLE PRECISION WORK( * ) .TP 17 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZLANSY returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). PZLANSY returns the value .br ( max(abs(A(i,j))), NORM = 'M' or 'm' with IA <= i <= IA+N-1, ( and JA <= j <= JA+N-1, ( .br ( norm1( sub( A ) ), NORM = '1', 'O' or 'o' .br ( .br ( normI( sub( A ) ), NORM = 'I' or 'i' .br ( .br ( normF( sub( A ) ), NORM = 'F', 'f', 'E' or 'e' .br where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a matrix norm. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies the value to be returned in PZLANSY as described above. .TP 8 UPLO (global input) CHARACTER Specifies whether the upper or lower triangular part of the symmetric matrix sub( A ) is to be referenced. = 'U': Upper triangular part of sub( A ) is referenced, .br = 'L': Lower triangular part of sub( A ) is referenced. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on i.e the number of rows and columns of the distributed submatrix sub( A ). When N = 0, PZLANSY is set to zero. N >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)) containing the local pieces of the symmetric distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular matrix which norm is to be computed, and the strictly lower triangular part of this matrix is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular matrix which norm is to be computed, and the strictly upper triangular part of sub( A ) is not referenced. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 WORK (local workspace) DOUBLE PRECISION array dimension (LWORK) LWORK >= 0 if NORM = 'M' or 'm' (not referenced), 2*Nq0+Np0+LDW if NORM = '1', 'O', 'o', 'I' or 'i', where LDW is given by: IF( NPROW.NE.NPCOL ) THEN LDW = MB_A*CEIL(CEIL(Np0/MB_A)/(LCM/NPROW)) ELSE LDW = 0 END IF 0 if NORM = 'F', 'f', 'E' or 'e' (not referenced), where LCM is the least common multiple of NPROW and NPCOL LCM = ILCM( NPROW, NPCOL ) and CEIL denotes the ceiling operation (ICEIL). IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Np0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), ICEIL, ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. scalapack-doc-1.5/man/manl/pzlantr.l0100644000056400000620000001373106335610660017120 0ustar pfrauenfstaff.TH PZLANTR l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLANTR - return the value of the one norm, or the Frobenius norm, .SH SYNOPSIS .TP 17 DOUBLE PRECISION FUNCTION PZLANTR( NORM, UPLO, DIAG, M, N, A, IA, JA, DESCA, WORK ) .TP 17 .ti +4 CHARACTER DIAG, NORM, UPLO .TP 17 .ti +4 INTEGER IA, JA, M, N .TP 17 .ti +4 INTEGER DESCA( * ) .TP 17 .ti +4 DOUBLE PRECISION WORK( * ) .TP 17 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZLANTR returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular distributed matrix sub( A ) denoting A(IA:IA+M-1, JA:JA+N-1). .br PZLANTR returns the value .br ( max(abs(A(i,j))), NORM = 'M' or 'm' with ia <= i <= ia+m-1, ( and ja <= j <= ja+n-1, ( .br ( norm1( sub( A ) ), NORM = '1', 'O' or 'o' .br ( .br ( normI( sub( A ) ), NORM = 'I' or 'i' .br ( .br ( normF( sub( A ) ), NORM = 'F', 'f', 'E' or 'e' .br where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a matrix norm. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies the value to be returned in PZLANTR as described above. .TP 8 UPLO (global input) CHARACTER Specifies whether the matrix sub( A ) is upper or lower trapezoidal. = 'U': Upper trapezoidal .br = 'L': Lower trapezoidal Note that sub( A ) is triangular instead of trapezoidal if M = N. .TP 8 DIAG (global input) CHARACTER Specifies whether or not the distributed matrix sub( A ) has unit diagonal. = 'N': Non-unit diagonal .br = 'U': Unit diagonal .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). When M = 0, PZLANTR is set to zero. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). When N = 0, PZLANTR is set to zero. N >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1) ) containing the local pieces of sub( A ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 WORK (local workspace) DOUBLE PRECISION array dimension (LWORK) LWORK >= 0 if NORM = 'M' or 'm' (not referenced), Nq0 if NORM = '1', 'O' or 'o', Mp0 if NORM = 'I' or 'i', 0 if NORM = 'F', 'f', 'E' or 'e' (not referenced), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. scalapack-doc-1.5/man/manl/pzlapiv.l0100644000056400000620000001550406335610660017113 0ustar pfrauenfstaff.TH PZLAPIV l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLAPIV - applie either P (permutation matrix indicated by IPIV) or inv( P ) to a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1), resulting in row or column pivoting .SH SYNOPSIS .TP 20 SUBROUTINE PZLAPIV( DIREC, ROWCOL, PIVROC, M, N, A, IA, JA, DESCA, IPIV, IP, JP, DESCIP, IWORK ) .TP 20 .ti +4 CHARACTER*1 DIREC, PIVROC, ROWCOL .TP 20 .ti +4 INTEGER IA, IP, JA, JP, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCIP( * ), IPIV( * ), IWORK( * ) .TP 20 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZLAPIV applies either P (permutation matrix indicated by IPIV) or inv( P ) to a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1), resulting in row or column pivoting. The pivot vector may be distributed across a process row or a column. The pivot vector should be aligned with the distributed matrix A. This routine will transpose the pivot vector if necessary. For example if the row pivots should be applied to the columns of sub( A ), pass ROWCOL='C' and PIVROC='C'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Restrictions .br ============ .br IPIV must always be a distributed vector (not a matrix). Thus: IF( ROWPIV .EQ. 'C' ) THEN .br JP must be 1 .br ELSE .br IP must be 1 .br END IF .br The following restrictions apply when IPIV must be transposed: IF( ROWPIV.EQ.'C' .AND. PIVROC.EQ.'C') THEN .br DESCIP(MB_) must equal DESCA(NB_) .br ELSE IF( ROWPIV.EQ.'R" .AND. PIVROC.EQ.'R') THEN .br DESCIP(NB_) must equal DESCA(MB_) .br END IF .br .SH ARGUMENTS .TP 8 DIREC (global input) CHARACTER*1 Specifies in which order the permutation is applied: = 'F' (Forward) Applies pivots Forward from top of matrix. Computes P*sub( A ). = 'B' (Backward) Applies pivots Backward from bottom of matrix. Computes inv( P )*sub( A ). .TP 8 ROWCOL (global input) CHARACTER*1 Specifies if the rows or columns are to be permuted: = 'R' Rows will be permuted, = 'C' Columns will be permuted. .TP 8 PIVROC (global input) CHARACTER*1 Specifies whether IPIV is distributed over a process row or column: = 'R' IPIV distributed over a process row = 'C' IPIV distributed over a process column .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the distributed submatrix sub( A ) to which the row or column interchanges will be applied. On exit, the local pieces of the permuted distributed submatrix. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (local input) INTEGER array, dimension >= LOCr(M_A)+MB_A if ROWCOL='R', otherwise LOCc(N_A)+NB_A. It contains the pivoting information. IPIV(i) is the global row (column), local row (column) i was swapped with. The last piece of the array of size MB_A (resp. NB_A) is used as workspace. This array is tied to the distributed matrix A. .TP 8 IWORK (local workspace) INTEGER array, dimension (LDW) where LDW is equal to the workspace necessary for transposition, and the storage of the tranposed IPIV: Let LCM be the least common multiple of NPROW and NPCOL. IF( ROWCOL.EQ.'R' .AND. PIVROC.EQ.'R' ) THEN IF( NPROW.EQ.NPCOL ) THEN LDW = LOCr( N_P + MOD(JP-1, NB_P) ) + NB_P ELSE LDW = LOCr( N_P + MOD(JP-1, NB_P) ) + NB_P * CEIL( CEIL(LOCc(N_P)/NB_P) / (LCM/NPCOL) ) END IF ELSE IF( ROWCOL.EQ.'C' .AND. PIVROC.EQ.'C' ) THEN IF( NPROW.EQ.NPCOL ) THEN LDW = LOCc( M_P + MOD(IP-1, MB_P) ) + MB_P ELSE LDW = LOCc( M_P + MOD(IP-1, MB_P) ) + MB_P * CEIL( CEIL(LOCr(M_P)/MB_P) / (LCM/NPROW) ) END IF ELSE IWORK is not referenced. END IF scalapack-doc-1.5/man/manl/pzlapv2.l0100644000056400000620000001360106335610660017020 0ustar pfrauenfstaff.TH PZLAPV2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLAPV2 - applie either P (permutation matrix indicated by IPIV) or inv( P ) to a M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1), resulting in row or column pivoting .SH SYNOPSIS .TP 20 SUBROUTINE PZLAPV2( DIREC, ROWCOL, M, N, A, IA, JA, DESCA, IPIV, IP, JP, DESCIP ) .TP 20 .ti +4 CHARACTER DIREC, ROWCOL .TP 20 .ti +4 INTEGER IA, IP, JA, JP, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCIP( * ), IPIV( * ) .TP 20 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZLAPV2 applies either P (permutation matrix indicated by IPIV) or inv( P ) to a M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1), resulting in row or column pivoting. The pivot vector should be aligned with the distributed matrix A. For pivoting the rows of sub( A ), IPIV should be distributed along a process column and replicated over all process rows. Similarly, IPIV should be distributed along a process row and replicated over all process columns for column pivoting. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 DIREC (global input) CHARACTER Specifies in which order the permutation is applied: = 'F' (Forward) Applies pivots Forward from top of matrix. Computes P * sub( A ); = 'B' (Backward) Applies pivots Backward from bottom of matrix. Computes inv( P ) * sub( A ). .TP 8 ROWCOL (global input) CHARACTER Specifies if the rows or columns are to be permuted: = 'R' Rows will be permuted, = 'C' Columns will be permuted. .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this local array contains the local pieces of the distributed matrix sub( A ) to which the row or columns interchanges will be applied. On exit, this array contains the local pieces of the permuted distributed matrix. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 IPIV (input) INTEGER array, dimension >= LOCr(M_A)+MB_A if ROWCOL = 'R', LOCc(N_A)+NB_A otherwise. It contains the pivoting information. IPIV(i) is the global row (column), local row (column) i was swapped with. The last piece of the array of size MB_A (resp. NB_A) is used as workspace. IPIV is tied to the distributed matrix A. .TP 8 IP (global input) INTEGER IPIV's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JP (global input) INTEGER IPIV's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCIP (global and local input) INTEGER array of dimension 8 The array descriptor for the distributed matrix IPIV. scalapack-doc-1.5/man/manl/pzlaqge.l0100644000056400000620000001402006335610660017061 0ustar pfrauenfstaff.TH PZLAQGE l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLAQGE - equilibrate a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) using the row and scaling factors in the vectors R and C .SH SYNOPSIS .TP 20 SUBROUTINE PZLAQGE( M, N, A, IA, JA, DESCA, R, C, ROWCND, COLCND, AMAX, EQUED ) .TP 20 .ti +4 CHARACTER EQUED .TP 20 .ti +4 INTEGER IA, JA, M, N .TP 20 .ti +4 DOUBLE PRECISION AMAX, COLCND, ROWCND .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION C( * ), R( * ) .TP 20 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZLAQGE equilibrates a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) using the row and scaling factors in the vectors R and C. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)) containing on entry the M-by-N matrix sub( A ). On exit, the equilibrated distributed matrix. See EQUED for the form of the equilibrated distributed submatrix. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 R (local input) DOUBLE PRECISION array, dimension LOCr(M_A) The row scale factors for sub( A ). R is aligned with the distributed matrix A, and replicated across every process column. R is tied to the distributed matrix A. .TP 8 C (local input) DOUBLE PRECISION array, dimension LOCc(N_A) The column scale factors of sub( A ). C is aligned with the distributed matrix A, and replicated down every process row. C is tied to the distributed matrix A. .TP 8 ROWCND (global input) DOUBLE PRECISION The global ratio of the smallest R(i) to the largest R(i), IA <= i <= IA+M-1. .TP 8 COLCND (global input) DOUBLE PRECISION The global ratio of the smallest C(i) to the largest C(i), JA <= j <= JA+N-1. .TP 8 AMAX (global input) DOUBLE PRECISION Absolute value of largest distributed submatrix entry. .TP 8 EQUED (global output) CHARACTER Specifies the form of equilibration that was done. = 'N': No equilibration .br = 'R': Row equilibration, i.e., sub( A ) has been pre- .br multiplied by diag(R(IA:IA+M-1)), .br = 'C': Column equilibration, i.e., sub( A ) has been post- .br multiplied by diag(C(JA:JA+N-1)), .br = 'B': Both row and column equilibration, i.e., sub( A ) has been replaced by diag(R(IA:IA+M-1)) * sub( A ) * diag(C(JA:JA+N-1)). .SH PARAMETERS THRESH is a threshold value used to decide if row or column scaling should be done based on the ratio of the row or column scaling factors. If ROWCND < THRESH, row scaling is done, and if COLCND < THRESH, column scaling is done. LARGE and SMALL are threshold values used to decide if row scaling should be done based on the absolute size of the largest matrix element. If AMAX > LARGE or AMAX < SMALL, row scaling is done. scalapack-doc-1.5/man/manl/pzlaqsy.l0100644000056400000620000001415306335610660017130 0ustar pfrauenfstaff.TH PZLAQSY l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLAQSY - equilibrate a symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) using the scaling factors in the vectors SR and SC .SH SYNOPSIS .TP 20 SUBROUTINE PZLAQSY( UPLO, N, A, IA, JA, DESCA, SR, SC, SCOND, AMAX, EQUED ) .TP 20 .ti +4 CHARACTER EQUED, UPLO .TP 20 .ti +4 INTEGER IA, JA, N .TP 20 .ti +4 DOUBLE PRECISION AMAX, SCOND .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION SC( * ), SR( * ) .TP 20 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZLAQSY equilibrates a symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) using the scaling factors in the vectors SR and SC. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the upper or lower triangular part of the symmetric distributed matrix sub( A ) is to be referenced: .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (input/output) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1)). On entry, the local pieces of the distributed symmetric matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and the strictly lower triangular part of sub( A ) is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and the strictly upper trian- gular part of sub( A ) is not referenced. On exit, if EQUED = 'Y', the equilibrated matrix: .br diag(SR(IA:IA+N-1)) * sub( A ) * diag(SC(JA:JA+N-1)). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 SR (local input) DOUBLE PRECISION array, dimension LOCr(M_A) The scale factors for A(IA:IA+M-1,JA:JA+N-1). SR is aligned with the distributed matrix A, and replicated across every process column. SR is tied to the distributed matrix A. .TP 8 SC (local input) DOUBLE PRECISION array, dimension LOCc(N_A) The scale factors for sub( A ). SC is aligned with the dis- tributed matrix A, and replicated down every process row. SC is tied to the distributed matrix A. .TP 8 SCOND (global input) DOUBLE PRECISION Ratio of the smallest SR(i) (respectively SC(j)) to the largest SR(i) (respectively SC(j)), with IA <= i <= IA+N-1 and JA <= j <= JA+N-1. .TP 8 AMAX (global input) DOUBLE PRECISION Absolute value of the largest distributed submatrix entry. .TP 8 EQUED (output) CHARACTER*1 Specifies whether or not equilibration was done. = 'N': No equilibration. .br = 'Y': Equilibration was done, i.e., sub( A ) has been re- .br placed by: .br diag(SR(IA:IA+N-1)) * sub( A ) * diag(SC(JA:JA+N-1)). .SH PARAMETERS THRESH is a threshold value used to decide if scaling should be done based on the ratio of the scaling factors. If SCOND < THRESH, scaling is done. LARGE and SMALL are threshold values used to decide if scaling should be done based on the absolute size of the largest matrix element. If AMAX > LARGE or AMAX < SMALL, scaling is done. scalapack-doc-1.5/man/manl/pzlarf.l0100644000056400000620000002001506335610660016715 0ustar pfrauenfstaff.TH PZLARF l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLARF - applie a complex elementary reflector Q to a complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the left or the right .SH SYNOPSIS .TP 19 SUBROUTINE PZLARF( SIDE, M, N, V, IV, JV, DESCV, INCV, TAU, C, IC, JC, DESCC, WORK ) .TP 19 .ti +4 CHARACTER SIDE .TP 19 .ti +4 INTEGER IC, INCV, IV, JC, JV, M, N .TP 19 .ti +4 INTEGER DESCC( * ), DESCV( * ) .TP 19 .ti +4 COMPLEX*16 C( * ), TAU( * ), V( * ), WORK( * ) .SH PURPOSE PZLARF applies a complex elementary reflector Q to a complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the left or the right. Q is represented in the form Q = I - tau * v * v' .br where tau is a complex scalar and v is a complex vector. .br If tau = 0, then Q is taken to be the unit matrix. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. Restrictions .br ============ .br If SIDE = 'Left' and INCV = 1, then the row process having the first entry V(IV,JV) must also have the first row of sub( C ). Moreover, MOD(IV-1,MB_V) must be equal to MOD(IC-1,MB_C), if INCV=M_V, only the last equality must be satisfied. .br If SIDE = 'Right' and INCV = M_V then the column process having the first entry V(IV,JV) must also have the first column of sub( C ) and MOD(JV-1,NB_V) must be equal to MOD(JC-1,NB_C), if INCV = 1 only the last equality must be satisfied. .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': form Q * sub( C ), .br = 'R': form sub( C ) * Q. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 V (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_V,*) containing the local pieces of the distributed vectors V representing the Householder transformation Q, V(IV:IV+M-1,JV) if SIDE = 'L' and INCV = 1, .br V(IV,JV:JV+M-1) if SIDE = 'L' and INCV = M_V, .br V(IV:IV+N-1,JV) if SIDE = 'R' and INCV = 1, .br V(IV,JV:JV+N-1) if SIDE = 'R' and INCV = M_V, The vector v in the representation of Q. V is not used if TAU = 0. .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 INCV (global input) INTEGER The global increment for the elements of V. Only two values of INCV are supported in this version, namely 1 and M_V. INCV must not be zero. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCc(JV) if INCV = 1, and LOCr(IV) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix V. .TP 8 C (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_C, LOCc(JC+N-1) ), containing the local pieces of sub( C ). On exit, sub( C ) is overwritten by the Q * sub( C ) if SIDE = 'L', or sub( C ) * Q if SIDE = 'R'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace) COMPLEX*16 array, dimension (LWORK) If INCV = 1, if SIDE = 'L', if IVCOL = ICCOL, LWORK >= NqC0 else LWORK >= MpC0 + MAX( 1, NqC0 ) end if else if SIDE = 'R', LWORK >= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC( N+ICOFFC,NB_V,0,0,NPCOL ),NB_V,0,0,LCMQ ) ) end if else if INCV = M_V, if SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_V,0,0,NPROW ),MB_V,0,0,LCMP ) ) else if SIDE = 'R', if IVROW = ICROW, LWORK >= MpC0 else LWORK >= NqC0 + MAX( 1, MpC0 ) end if end if end if where LCM is the least common multiple of NPROW and NPCOL and LCM = ILCM( NPROW, NPCOL ), LCMP = LCM / NPROW, LCMQ = LCM / NPCOL, IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. Alignment requirements ====================== The distributed submatrices V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: MB_V = NB_V, If INCV = 1, If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', ( MB_V.EQ.NB_A .AND. MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if INCV = M_V, If SIDE = 'Left', ( MB_V.EQ.NB_V .AND. MB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND. IVCOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/pzlarfb.l0100644000056400000620000001767606335610660017102 0ustar pfrauenfstaff.TH PZLARFB l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLARFB - applie a complex block reflector Q or its conjugate transpose Q**H to a complex M-by-N distributed matrix sub( C ) denoting C(IC:IC+M-1,JC:JC+N-1), from the left or the right .SH SYNOPSIS .TP 20 SUBROUTINE PZLARFB( SIDE, TRANS, DIRECT, STOREV, M, N, K, V, IV, JV, DESCV, T, C, IC, JC, DESCC, WORK ) .TP 20 .ti +4 CHARACTER SIDE, TRANS, DIRECT, STOREV .TP 20 .ti +4 INTEGER IC, IV, JC, JV, K, M, N .TP 20 .ti +4 INTEGER DESCC( * ), DESCV( * ) .TP 20 .ti +4 COMPLEX*16 C( * ), T( * ), V( * ), WORK( * ) .SH PURPOSE PZLARFB applies a complex block reflector Q or its conjugate transpose Q**H to a complex M-by-N distributed matrix sub( C ) denoting C(IC:IC+M-1,JC:JC+N-1), from the left or the right. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 DIRECT (global input) CHARACTER Indicates how Q is formed from a product of elementary reflectors = 'F': Q = H(1) H(2) . . . H(k) (Forward) .br = 'B': Q = H(k) . . . H(2) H(1) (Backward) .TP 8 STOREV (global input) CHARACTER Indicates how the vectors which define the elementary reflectors are stored: .br = 'C': Columnwise .br = 'R': Rowwise .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The order of the matrix T (= the number of elementary reflectors whose product defines the block reflector). .TP 8 V (local input) COMPLEX*16 pointer into the local memory to an array of dimension ( LLD_V, LOCc(JV+K-1) ) if STOREV = 'C', ( LLD_V, LOCc(JV+M-1)) if STOREV = 'R' and SIDE = 'L', ( LLD_V, LOCc(JV+N-1) ) if STOREV = 'R' and SIDE = 'R'. It contains the local pieces of the distributed vectors V representing the Householder transformation. See further details. If STOREV = 'C' and SIDE = 'L', LLD_V >= MAX(1,LOCr(IV+M-1)); if STOREV = 'C' and SIDE = 'R', LLD_V >= MAX(1,LOCr(IV+N-1)); if STOREV = 'R', LLD_V >= LOCr(IV+K-1). .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 T (local input) COMPLEX*16 array, dimension MB_V by MB_V if STOREV = 'R' and NB_V by NB_V if STOREV = 'C'. The trian- gular matrix T in the representation of the block reflector. .TP 8 C (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the M-by-N distributed matrix sub( C ). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q or sub( C )*Q'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace) COMPLEX*16 array, dimension (LWORK) If STOREV = 'C', if SIDE = 'L', LWORK >= ( NqC0 + MpC0 ) * K else if SIDE = 'R', LWORK >= ( NqC0 + MAX( NpV0 + NUMROC( NUMROC( N+ICOFFC, NB_V, 0, 0, NPCOL ), NB_V, 0, 0, LCMQ ), MpC0 ) ) * K end if else if STOREV = 'R', if SIDE = 'L', LWORK >= ( MpC0 + MAX( MqV0 + NUMROC( NUMROC( M+IROFFC, MB_V, 0, 0, NPROW ), MB_V, 0, 0, LCMP ), NqC0 ) ) * K else if SIDE = 'R', LWORK >= ( MpC0 + NqC0 ) * K end if end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFV = MOD( IV-1, MB_V ), ICOFFV = MOD( JV-1, NB_V ), IVROW = INDXG2P( IV, MB_V, MYROW, RSRC_V, NPROW ), IVCOL = INDXG2P( JV, NB_V, MYCOL, CSRC_V, NPCOL ), MqV0 = NUMROC( M+ICOFFV, NB_V, MYCOL, IVCOL, NPCOL ), NpV0 = NUMROC( N+IROFFV, MB_V, MYROW, IVROW, NPROW ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NpC0 = NUMROC( N+ICOFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. Alignment requirements ====================== The distributed submatrices V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If STOREV = 'Columnwise' If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', ( MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if STOREV = 'Rowwise' If SIDE = 'Left', ( NB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND. IVCOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/pzlarfc.l0100644000056400000620000002000106335610660017053 0ustar pfrauenfstaff.TH PZLARFC l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLARFC - applie a complex elementary reflector Q**H to a complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), .SH SYNOPSIS .TP 20 SUBROUTINE PZLARFC( SIDE, M, N, V, IV, JV, DESCV, INCV, TAU, C, IC, JC, DESCC, WORK ) .TP 20 .ti +4 CHARACTER SIDE .TP 20 .ti +4 INTEGER IC, INCV, IV, JC, JV, M, N .TP 20 .ti +4 INTEGER DESCC( * ), DESCV( * ) .TP 20 .ti +4 COMPLEX*16 C( * ), TAU( * ), V( * ), WORK( * ) .SH PURPOSE PZLARFC applies a complex elementary reflector Q**H to a complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the left or the right. Q is represented in the form Q = I - tau * v * v' .br where tau is a complex scalar and v is a complex vector. .br If tau = 0, then Q is taken to be the unit matrix. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. Restrictions .br ============ .br If SIDE = 'Left' and INCV = 1, then the row process having the first entry V(IV,JV) must also have the first row of sub( C ). Moreover, MOD(IV-1,MB_V) must be equal to MOD(IC-1,MB_C), if INCV=M_V, only the last equality must be satisfied. .br If SIDE = 'Right' and INCV = M_V then the column process having the first entry V(IV,JV) must also have the first column of sub( C ) and MOD(JV-1,NB_V) must be equal to MOD(JC-1,NB_C), if INCV = 1 only the last equality must be satisfied. .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': form Q**H * sub( C ), .br = 'R': form sub( C ) * Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 V (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_V,*) containing the local pieces of the distributed vectors V representing the Householder transformation Q, V(IV:IV+M-1,JV) if SIDE = 'L' and INCV = 1, .br V(IV,JV:JV+M-1) if SIDE = 'L' and INCV = M_V, .br V(IV:IV+N-1,JV) if SIDE = 'R' and INCV = 1, .br V(IV,JV:JV+N-1) if SIDE = 'R' and INCV = M_V, The vector v in the representation of Q. V is not used if TAU = 0. .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 INCV (global input) INTEGER The global increment for the elements of V. Only two values of INCV are supported in this version, namely 1 and M_V. INCV must not be zero. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCc(JV) if INCV = 1, and LOCr(IV) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix V. .TP 8 C (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_C, LOCc(JC+N-1) ), containing the local pieces of sub( C ). On exit, sub( C ) is overwritten by the Q**H * sub( C ) if SIDE = 'L', or sub( C ) * Q**H if SIDE = 'R'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace) COMPLEX*16 array, dimension (LWORK) If INCV = 1, if SIDE = 'L', if IVCOL = ICCOL, LWORK >= NqC0 else LWORK >= MpC0 + MAX( 1, NqC0 ) end if else if SIDE = 'R', LWORK >= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC( N+ICOFFC,NB_V,0,0,NPCOL ),NB_V,0,0,LCMQ ) ) end if else if INCV = M_V, if SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_V,0,0,NPROW ),MB_V,0,0,LCMP ) ) else if SIDE = 'R', if IVROW = ICROW, LWORK >= MpC0 else LWORK >= NqC0 + MAX( 1, MpC0 ) end if end if end if where LCM is the least common multiple of NPROW and NPCOL and LCM = ILCM( NPROW, NPCOL ), LCMP = LCM / NPROW, LCMQ = LCM / NPCOL, IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. Alignment requirements ====================== The distributed submatrices V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: MB_V = NB_V, If INCV = 1, If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', ( MB_V.EQ.NB_A .AND. MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if INCV = M_V, If SIDE = 'Left', ( MB_V.EQ.NB_V .AND. MB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND. IVCOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/pzlarfg.l0100644000056400000620000001256606335610660017100 0ustar pfrauenfstaff.TH PZLARFG l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLARFG - generate a complex elementary reflector H of order n, such that H * sub( X ) = H * ( x(iax,jax) ) = ( alpha ), H' * H = I .SH SYNOPSIS .TP 20 SUBROUTINE PZLARFG( N, ALPHA, IAX, JAX, X, IX, JX, DESCX, INCX, TAU ) .TP 20 .ti +4 INTEGER IAX, INCX, IX, JAX, JX, N .TP 20 .ti +4 COMPLEX*16 ALPHA .TP 20 .ti +4 INTEGER DESCX( * ) .TP 20 .ti +4 COMPLEX*16 TAU( * ), X( * ) .SH PURPOSE PZLARFG generates a complex elementary reflector H of order n, such that ( x ) ( 0 ) .br where alpha is a real scalar, and sub( X ) is an (N-1)-element complex distributed vector X(IX:IX+N-2,JX) if INCX = 1 and X(IX,JX:JX+N-2) if INCX = DESCX(M_). H is represented in the form H = I - tau * ( 1 ) * ( 1 v' ) , .br ( v ) .br where tau is a complex scalar and v is a complex (N-1)-element vector. Note that H is not Hermitian. .br If the elements of sub( X ) are all zero and X(IAX,JAX) is real, then tau = 0 and H is taken to be the unit matrix. .br Otherwise 1 <= real(tau) <= 2 and abs(tau-1) <= 1. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. .SH ARGUMENTS .TP 8 N (global input) INTEGER The global order of the elementary reflector. N >= 0. .TP 8 ALPHA (local output) COMPLEX*16 On exit, alpha is computed in the process scope having the vector sub( X ). .TP 8 IAX (global input) INTEGER The global row index in X of X(IAX,JAX). .TP 8 JAX (global input) INTEGER The global column index in X of X(IAX,JAX). .TP 8 X (local input/local output) COMPLEX*16, pointer into the local memory to an array of dimension (LLD_X,*). This array contains the local pieces of the distributed vector sub( X ). Before entry, the incremented array sub( X ) must contain the vector x. On exit, it is overwritten with the vector v. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 INCX (global input) INTEGER The global increment for the elements of X. Only two values of INCX are supported in this version, namely 1 and M_X. INCX must not be zero. .TP 8 TAU (local output) COMPLEX*16, array, dimension LOCc(JX) if INCX = 1, and LOCr(IX) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix X. scalapack-doc-1.5/man/manl/pzlarft.l0100644000056400000620000001510406335610660017104 0ustar pfrauenfstaff.TH PZLARFT l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLARFT - form the triangular factor T of a complex block reflector H of order n, which is defined as a product of k elementary reflectors .SH SYNOPSIS .TP 20 SUBROUTINE PZLARFT( DIRECT, STOREV, N, K, V, IV, JV, DESCV, TAU, T, WORK ) .TP 20 .ti +4 CHARACTER DIRECT, STOREV .TP 20 .ti +4 INTEGER IV, JV, K, N .TP 20 .ti +4 INTEGER DESCV( * ) .TP 20 .ti +4 COMPLEX*16 TAU( * ), T( * ), V( * ), WORK( * ) .SH PURPOSE PZLARFT forms the triangular factor T of a complex block reflector H of order n, which is defined as a product of k elementary reflectors. If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular; If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular. If STOREV = 'C', the vector which defines the elementary reflector H(i) is stored in the i-th column of the distributed matrix V, and H = I - V * T * V' .br If STOREV = 'R', the vector which defines the elementary reflector H(i) is stored in the i-th row of the distributed matrix V, and H = I - V' * T * V .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 DIRECT (global input) CHARACTER*1 Specifies the order in which the elementary reflectors are multiplied to form the block reflector: .br = 'F': H = H(1) H(2) . . . H(k) (Forward) .br = 'B': H = H(k) . . . H(2) H(1) (Backward) .TP 8 STOREV (global input) CHARACTER*1 Specifies how the vectors which define the elementary reflectors are stored (see also Further Details): .br = 'R': rowwise .TP 8 N (global input) INTEGER The order of the block reflector H. N >= 0. .TP 8 K (global input) INTEGER The order of the triangular factor T (= the number of elementary reflectors). 1 <= K <= MB_V (= NB_V). .TP 8 V (input/output) COMPLEX*16 pointer into the local memory to an array of local dimension (LOCr(IV+N-1),LOCc(JV+K-1)) if STOREV = 'C', and (LOCr(IV+K-1),LOCc(JV+N-1)) if STOREV = 'R'. The distributed matrix V contains the Householder vectors. See further details. .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCr(IV+K-1) if INCV = M_V, and LOCc(JV+K-1) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix V. .TP 8 T (local output) COMPLEX*16 array, dimension (NB_V,NB_V) if STOREV = 'Col', and (MB_V,MB_V) otherwise. It contains the k-by-k triangular factor of the block reflector asso- ciated with V. If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is lower triangular. .TP 8 WORK (local workspace) COMPLEX*16 array, dimension (K*(K-1)/2) .SH FURTHER DETAILS The shape of the matrix V and the storage of the vectors which define the H(i) is best illustrated by the following example with n = 5 and k = 3. The elements equal to 1 are not stored; the corresponding array elements are modified but restored on exit. The rest of the array is not used. .br DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R': V( IV:IV+N-1, ( 1 ) V( IV:IV+K-1, ( 1 v1 v1 v1 v1 ) JV:JV+K-1 ) = ( v1 1 ) JV:JV+N-1 ) = ( 1 v2 v2 v2 ) ( v1 v2 1 ) ( 1 v3 v3 ) ( v1 v2 v3 ) .br ( v1 v2 v3 ) .br DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R': V( IV:IV+N-1, ( v1 v2 v3 ) V( IV:IV+K-1, ( v1 v1 1 ) JV:JV+K-1 ) = ( v1 v2 v3 ) JV:JV+N-1 ) = ( v2 v2 v2 1 ) ( 1 v2 v3 ) ( v3 v3 v3 v3 1 ) ( 1 v3 ) .br ( 1 ) .br scalapack-doc-1.5/man/manl/pzlarz.l0100644000056400000620000002043206335610660016744 0ustar pfrauenfstaff.TH PZLARZ l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLARZ - applie a complex elementary reflector Q to a complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the left or the right .SH SYNOPSIS .TP 19 SUBROUTINE PZLARZ( SIDE, M, N, L, V, IV, JV, DESCV, INCV, TAU, C, IC, JC, DESCC, WORK ) .TP 19 .ti +4 CHARACTER SIDE .TP 19 .ti +4 INTEGER IC, INCV, IV, JC, JV, L, M, N .TP 19 .ti +4 INTEGER DESCC( * ), DESCV( * ) .TP 19 .ti +4 COMPLEX*16 C( * ), TAU( * ), V( * ), WORK( * ) .SH PURPOSE PZLARZ applies a complex elementary reflector Q to a complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the left or the right. Q is represented in the form Q = I - tau * v * v' .br where tau is a complex scalar and v is a complex vector. .br If tau = 0, then Q is taken to be the unit matrix. .br Q is a product of k elementary reflectors as returned by PZTZRZF. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. Restrictions .br ============ .br If SIDE = 'Left' and INCV = 1, then the row process having the first entry V(IV,JV) must also own C(IC+M-L,JC:JC+N-1). Moreover, MOD(IV-1,MB_V) must be equal to MOD(IC+N-L-1,MB_C), if INCV=M_V, only the last equality must be satisfied. .br If SIDE = 'Right' and INCV = M_V then the column process having the first entry V(IV,JV) must also own C(IC:IC+M-1,JC+N-L) and MOD(JV-1,NB_V) must be equal to MOD(JC+N-L-1,NB_C), if INCV = 1 only the last equality must be satisfied. .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': form Q * sub( C ), .br = 'R': form sub( C ) * Q. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 L (global input) INTEGER The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. .TP 8 V (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_V,*) containing the local pieces of the distributed vectors V representing the Householder transformation Q, V(IV:IV+L-1,JV) if SIDE = 'L' and INCV = 1, .br V(IV,JV:JV+L-1) if SIDE = 'L' and INCV = M_V, .br V(IV:IV+L-1,JV) if SIDE = 'R' and INCV = 1, .br V(IV,JV:JV+L-1) if SIDE = 'R' and INCV = M_V, The vector v in the representation of Q. V is not used if TAU = 0. .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 INCV (global input) INTEGER The global increment for the elements of V. Only two values of INCV are supported in this version, namely 1 and M_V. INCV must not be zero. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCc(JV) if INCV = 1, and LOCr(IV) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix V. .TP 8 C (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_C, LOCc(JC+N-1) ), containing the local pieces of sub( C ). On exit, sub( C ) is overwritten by the Q * sub( C ) if SIDE = 'L', or sub( C ) * Q if SIDE = 'R'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace) COMPLEX*16 array, dimension (LWORK) If INCV = 1, if SIDE = 'L', if IVCOL = ICCOL, LWORK >= NqC0 else LWORK >= MpC0 + MAX( 1, NqC0 ) end if else if SIDE = 'R', LWORK >= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC( N+ICOFFC,NB_V,0,0,NPCOL ),NB_V,0,0,LCMQ ) ) end if else if INCV = M_V, if SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_V,0,0,NPROW ),MB_V,0,0,LCMP ) ) else if SIDE = 'R', if IVROW = ICROW, LWORK >= MpC0 else LWORK >= NqC0 + MAX( 1, MpC0 ) end if end if end if where LCM is the least common multiple of NPROW and NPCOL and LCM = ILCM( NPROW, NPCOL ), LCMP = LCM / NPROW, LCMQ = LCM / NPCOL, IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. Alignment requirements ====================== The distributed submatrices V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: MB_V = NB_V, If INCV = 1, If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', ( MB_V.EQ.NB_A .AND. MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if INCV = M_V, If SIDE = 'Left', ( MB_V.EQ.NB_V .AND. MB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND. IVCOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/pzlarzb.l0100644000056400000620000002011106335610661017101 0ustar pfrauenfstaff.TH PZLARZB l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLARZB - applie a complex block reflector Q or its conjugate transpose Q**H to a complex M-by-N distributed matrix sub( C ) denoting C(IC:IC+M-1,JC:JC+N-1), from the left or the right .SH SYNOPSIS .TP 20 SUBROUTINE PZLARZB( SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V, IV, JV, DESCV, T, C, IC, JC, DESCC, WORK ) .TP 20 .ti +4 CHARACTER DIRECT, SIDE, STOREV, TRANS .TP 20 .ti +4 INTEGER IC, IV, JC, JV, K, L, M, N .TP 20 .ti +4 INTEGER DESCC( * ), DESCV( * ) .TP 20 .ti +4 COMPLEX*16 C( * ), T( * ), V( * ), WORK( * ) .SH PURPOSE PZLARZB applies a complex block reflector Q or its conjugate transpose Q**H to a complex M-by-N distributed matrix sub( C ) denoting C(IC:IC+M-1,JC:JC+N-1), from the left or the right. Q is a product of k elementary reflectors as returned by PZTZRZF. Currently, only STOREV = 'R' and DIRECT = 'B' are supported. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 DIRECT (global input) CHARACTER Indicates how H is formed from a product of elementary reflectors = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet) .br = 'B': H = H(k) . . . H(2) H(1) (Backward) .TP 8 STOREV (global input) CHARACTER Indicates how the vectors which define the elementary reflectors are stored: .br = 'C': Columnwise (not supported yet) .br = 'R': Rowwise .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The order of the matrix T (= the number of elementary reflectors whose product defines the block reflector). .TP 8 L (global input) INTEGER The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. .TP 8 V (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_V, LOCc(JV+M-1)) if SIDE = 'L', (LLD_V, LOCc(JV+N-1)) if SIDE = 'R'. It contains the local pieces of the distributed vectors V representing the Householder transformation as returned by PZTZRZF. LLD_V >= LOCr(IV+K-1). .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 T (local input) COMPLEX*16 array, dimension MB_V by MB_V The lower triangular matrix T in the representation of the block reflector. .TP 8 C (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the M-by-N distributed matrix sub( C ). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q or sub( C )*Q'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace) COMPLEX*16 array, dimension (LWORK) If STOREV = 'C', if SIDE = 'L', LWORK >= ( NqC0 + MpC0 ) * K else if SIDE = 'R', LWORK >= ( NqC0 + MAX( NpV0 + NUMROC( NUMROC( N+ICOFFC, NB_V, 0, 0, NPCOL ), NB_V, 0, 0, LCMQ ), MpC0 ) ) * K end if else if STOREV = 'R', if SIDE = 'L', LWORK >= ( MpC0 + MAX( MqV0 + NUMROC( NUMROC( M+IROFFC, MB_V, 0, 0, NPROW ), MB_V, 0, 0, LCMP ), NqC0 ) ) * K else if SIDE = 'R', LWORK >= ( MpC0 + NqC0 ) * K end if end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFV = MOD( IV-1, MB_V ), ICOFFV = MOD( JV-1, NB_V ), IVROW = INDXG2P( IV, MB_V, MYROW, RSRC_V, NPROW ), IVCOL = INDXG2P( JV, NB_V, MYCOL, CSRC_V, NPCOL ), MqV0 = NUMROC( M+ICOFFV, NB_V, MYCOL, IVCOL, NPCOL ), NpV0 = NUMROC( N+IROFFV, MB_V, MYROW, IVROW, NPROW ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NpC0 = NUMROC( N+ICOFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. Alignment requirements ====================== The distributed submatrices V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If STOREV = 'Columnwise' If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', ( MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if STOREV = 'Rowwise' If SIDE = 'Left', ( NB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND. IVCOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/pzlarzc.l0100644000056400000620000002041606335610661017112 0ustar pfrauenfstaff.TH PZLARZC l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLARZC - applie a complex elementary reflector Q**H to a complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), .SH SYNOPSIS .TP 20 SUBROUTINE PZLARZC( SIDE, M, N, L, V, IV, JV, DESCV, INCV, TAU, C, IC, JC, DESCC, WORK ) .TP 20 .ti +4 CHARACTER SIDE .TP 20 .ti +4 INTEGER IC, INCV, IV, JC, JV, L, M, N .TP 20 .ti +4 INTEGER DESCC( * ), DESCV( * ) .TP 20 .ti +4 COMPLEX*16 C( * ), TAU( * ), V( * ), WORK( * ) .SH PURPOSE PZLARZC applies a complex elementary reflector Q**H to a complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the left or the right. Q is represented in the form Q = I - tau * v * v' .br where tau is a complex scalar and v is a complex vector. .br If tau = 0, then Q is taken to be the unit matrix. .br Q is a product of k elementary reflectors as returned by PZTZRZF. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. Restrictions .br ============ .br If SIDE = 'Left' and INCV = 1, then the row process having the first entry V(IV,JV) must also own C(IC+M-L,JC:JC+N-1). Moreover, MOD(IV-1,MB_V) must be equal to MOD(IC+N-L-1,MB_C), if INCV=M_V, only the last equality must be satisfied. .br If SIDE = 'Right' and INCV = M_V then the column process having the first entry V(IV,JV) must also own C(IC:IC+M-1,JC+N-L) and MOD(JV-1,NB_V) must be equal to MOD(JC+N-L-1,NB_C), if INCV = 1 only the last equality must be satisfied. .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': form Q**H * sub( C ), .br = 'R': form sub( C ) * Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 L (global input) INTEGER The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. .TP 8 V (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_V,*) containing the local pieces of the distributed vectors V representing the Householder transformation Q, V(IV:IV+L-1,JV) if SIDE = 'L' and INCV = 1, .br V(IV,JV:JV+L-1) if SIDE = 'L' and INCV = M_V, .br V(IV:IV+L-1,JV) if SIDE = 'R' and INCV = 1, .br V(IV,JV:JV+L-1) if SIDE = 'R' and INCV = M_V, The vector v in the representation of Q. V is not used if TAU = 0. .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 INCV (global input) INTEGER The global increment for the elements of V. Only two values of INCV are supported in this version, namely 1 and M_V. INCV must not be zero. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCc(JV) if INCV = 1, and LOCr(IV) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix V. .TP 8 C (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_C, LOCc(JC+N-1) ), containing the local pieces of sub( C ). On exit, sub( C ) is overwritten by the Q**H * sub( C ) if SIDE = 'L', or sub( C ) * Q**H if SIDE = 'R'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace) COMPLEX*16 array, dimension (LWORK) If INCV = 1, if SIDE = 'L', if IVCOL = ICCOL, LWORK >= NqC0 else LWORK >= MpC0 + MAX( 1, NqC0 ) end if else if SIDE = 'R', LWORK >= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC( N+ICOFFC,NB_V,0,0,NPCOL ),NB_V,0,0,LCMQ ) ) end if else if INCV = M_V, if SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_V,0,0,NPROW ),MB_V,0,0,LCMP ) ) else if SIDE = 'R', if IVROW = ICROW, LWORK >= MpC0 else LWORK >= NqC0 + MAX( 1, MpC0 ) end if end if end if where LCM is the least common multiple of NPROW and NPCOL and LCM = ILCM( NPROW, NPCOL ), LCMP = LCM / NPROW, LCMQ = LCM / NPCOL, IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. Alignment requirements ====================== The distributed submatrices V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: MB_V = NB_V, If INCV = 1, If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', ( MB_V.EQ.NB_A .AND. MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if INCV = M_V, If SIDE = 'Left', ( MB_V.EQ.NB_V .AND. MB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND. IVCOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/pzlarzt.l0100644000056400000620000001567206335610661017143 0ustar pfrauenfstaff.TH PZLARZT l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLARZT - form the triangular factor T of a complex block reflector H of order > n, which is defined as a product of k elementary reflectors as returned by PZTZRZF .SH SYNOPSIS .TP 20 SUBROUTINE PZLARZT( DIRECT, STOREV, N, K, V, IV, JV, DESCV, TAU, T, WORK ) .TP 20 .ti +4 CHARACTER DIRECT, STOREV .TP 20 .ti +4 INTEGER IV, JV, K, N .TP 20 .ti +4 INTEGER DESCV( * ) .TP 20 .ti +4 COMPLEX*16 TAU( * ), T( * ), V( * ), WORK( * ) .SH PURPOSE PZLARZT forms the triangular factor T of a complex block reflector H of order > n, which is defined as a product of k elementary reflectors as returned by PZTZRZF. If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular; If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular. If STOREV = 'C', the vector which defines the elementary reflector H(i) is stored in the i-th column of the array V, and .br H = I - V * T * V' .br If STOREV = 'R', the vector which defines the elementary reflector H(i) is stored in the i-th row of the array V, and .br H = I - V' * T * V .br Currently, only STOREV = 'R' and DIRECT = 'B' are supported. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 DIRECT (global input) CHARACTER Specifies the order in which the elementary reflectors are multiplied to form the block reflector: .br = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet) .br = 'B': H = H(k) . . . H(2) H(1) (Backward) .TP 8 STOREV (global input) CHARACTER Specifies how the vectors which define the elementary reflectors are stored (see also Further Details): .br = 'R': rowwise .TP 8 N (global input) INTEGER The number of meaningful entries of the block reflector H. N >= 0. .TP 8 K (global input) INTEGER The order of the triangular factor T (= the number of elementary reflectors). 1 <= K <= MB_V (= NB_V). .TP 8 V (input/output) COMPLEX*16 pointer into the local memory to an array of local dimension (LOCr(IV+K-1),LOCc(JV+N-1)). The distributed matrix V contains the Householder vectors. See further details. .TP 8 IV (global input) INTEGER The row index in the global array V indicating the first row of sub( V ). .TP 8 JV (global input) INTEGER The column index in the global array V indicating the first column of sub( V ). .TP 8 DESCV (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCr(IV+K-1) if INCV = M_V, and LOCc(JV+K-1) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix V. .TP 8 T (local output) COMPLEX*16 array, dimension (MB_V,MB_V) It contains the k-by-k triangular factor of the block reflector associated with V. T is lower triangular. .TP 8 WORK (local workspace) COMPLEX*16 array, dimension (K*(K-1)/2) .SH FURTHER DETAILS The shape of the matrix V and the storage of the vectors which define the H(i) is best illustrated by the following example with n = 5 and k = 3. The elements equal to 1 are not stored; the corresponding array elements are modified but restored on exit. The rest of the array is not used. .br DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R': ______V_____ .br ( v1 v2 v3 ) / \ ( v1 v2 v3 ) ( v1 v1 v1 v1 v1 . . . . 1 ) V = ( v1 v2 v3 ) ( v2 v2 v2 v2 v2 . . . 1 ) ( v1 v2 v3 ) ( v3 v3 v3 v3 v3 . . 1 ) ( v1 v2 v3 ) .br . . . .br . . . .br 1 . . .br 1 . .br 1 .br DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R': ______V_____ 1 / \ . 1 ( 1 . . . . v1 v1 v1 v1 v1 ) . . 1 ( . 1 . . . v2 v2 v2 v2 v2 ) . . . ( . . 1 . . v3 v3 v3 v3 v3 ) . . . .br ( v1 v2 v3 ) .br ( v1 v2 v3 ) .br V = ( v1 v2 v3 ) .br ( v1 v2 v3 ) .br ( v1 v2 v3 ) .br scalapack-doc-1.5/man/manl/pzlascl.l0100644000056400000620000001242506335610661017076 0ustar pfrauenfstaff.TH PZLASCL l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLASCL - multiplie the M-by-N complex distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) by the real scalar CTO/CFROM .SH SYNOPSIS .TP 20 SUBROUTINE PZLASCL( TYPE, CFROM, CTO, M, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER TYPE .TP 20 .ti +4 INTEGER IA, INFO, JA, M, N .TP 20 .ti +4 DOUBLE PRECISION CFROM, CTO .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZLASCL multiplies the M-by-N complex distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) 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 sub( A ) may be full, upper triangular, lower triangular or upper Hessenberg. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 TYPE (global input) CHARACTER TYPE indices the storage type of the input distributed matrix. = 'G': sub( A ) is a full matrix, .br = 'L': sub( A ) is a lower triangular matrix, .br = 'U': sub( A ) is an upper triangular matrix, .br = 'H': sub( A ) is an upper Hessenberg matrix. .TP 8 CFROM (global input) DOUBLE PRECISION CTO (global input) DOUBLE PRECISION The distributed matrix sub( 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. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). This array contains the local pieces of the distributed matrix sub( A ). On exit, this array contains the local pieces of the distributed matrix multiplied by CTO/CFROM. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pzlase2.l0100644000056400000620000001217606335610661017011 0ustar pfrauenfstaff.TH PZLASE2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLASE2 - initialize an M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) to BETA on the diagonal and ALPHA on the offdiagonals .SH SYNOPSIS .TP 20 SUBROUTINE PZLASE2( UPLO, M, N, ALPHA, BETA, A, IA, JA, DESCA ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, JA, M, N .TP 20 .ti +4 COMPLEX*16 ALPHA, BETA .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZLASE2 initializes an M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) to BETA on the diagonal and ALPHA on the offdiagonals. PZLASE2 requires that only dimension of the matrix operand is distributed. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies the part of the distributed matrix sub( A ) to be set: .br = 'U': Upper triangular part is set; the strictly lower triangular part of sub( A ) is not changed; = 'L': Lower triangular part is set; the strictly upper triangular part of sub( A ) is not changed; Otherwise: All of the matrix sub( A ) is set. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 ALPHA (global input) COMPLEX*16 The constant to which the offdiagonal elements are to be set. .TP 8 BETA (global input) COMPLEX*16 The constant to which the diagonal elements are to be set. .TP 8 A (local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). This array contains the local pieces of the distributed matrix sub( A ) to be set. On exit, the leading M-by-N submatrix sub( A ) is set as follows: if UPLO = 'U', A(IA+i-1,JA+j-1) = ALPHA, 1<=i<=j-1, 1<=j<=N, if UPLO = 'L', A(IA+i-1,JA+j-1) = ALPHA, j+1<=i<=M, 1<=j<=N, otherwise, A(IA+i-1,JA+j-1) = ALPHA, 1<=i<=M, 1<=j<=N, IA+i.NE.JA+j, and, for all UPLO, A(IA+i-1,JA+i-1) = BETA, 1<=i<=min(M,N). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. scalapack-doc-1.5/man/manl/pzlaset.l0100644000056400000620000001205606335610661017110 0ustar pfrauenfstaff.TH PZLASET l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLASET - initialize an M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) to BETA on the diagonal and ALPHA on the offdiagonals .SH SYNOPSIS .TP 20 SUBROUTINE PZLASET( UPLO, M, N, ALPHA, BETA, A, IA, JA, DESCA ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, JA, M, N .TP 20 .ti +4 COMPLEX*16 ALPHA, BETA .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZLASET initializes an M-by-N distributed matrix sub( A ) denoting A(IA:IA+M-1,JA:JA+N-1) to BETA on the diagonal and ALPHA on the offdiagonals. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies the part of the distributed matrix sub( A ) to be set: .br = 'U': Upper triangular part is set; the strictly lower triangular part of sub( A ) is not changed; = 'L': Lower triangular part is set; the strictly upper triangular part of sub( A ) is not changed; Otherwise: All of the matrix sub( A ) is set. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 ALPHA (global input) COMPLEX*16 The constant to which the offdiagonal elements are to be set. .TP 8 BETA (global input) COMPLEX*16 The constant to which the diagonal elements are to be set. .TP 8 A (local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). This array contains the local pieces of the distributed matrix sub( A ) to be set. On exit, the leading M-by-N submatrix sub( A ) is set as follows: if UPLO = 'U', A(IA+i-1,JA+j-1) = ALPHA, 1<=i<=j-1, 1<=j<=N, if UPLO = 'L', A(IA+i-1,JA+j-1) = ALPHA, j+1<=i<=M, 1<=j<=N, otherwise, A(IA+i-1,JA+j-1) = ALPHA, 1<=i<=M, 1<=j<=N, IA+i.NE.JA+j, and, for all UPLO, A(IA+i-1,JA+i-1) = BETA, 1<=i<=min(M,N). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. scalapack-doc-1.5/man/manl/pzlassq.l0100644000056400000620000001251406335610661017122 0ustar pfrauenfstaff.TH PZLASSQ l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLASSQ - return the values scl and smsq such that ( scl**2 )*smsq = x( 1 )**2 +...+ x( n )**2 + ( scale**2 )*sumsq, .SH SYNOPSIS .TP 20 SUBROUTINE PZLASSQ( N, X, IX, JX, DESCX, INCX, SCALE, SUMSQ ) .TP 20 .ti +4 INTEGER IX, INCX, JX, N .TP 20 .ti +4 DOUBLE PRECISION SCALE, SUMSQ .TP 20 .ti +4 INTEGER DESCX( * ) .TP 20 .ti +4 COMPLEX*16 X( * ) .SH PURPOSE PZLASSQ returns the values scl and smsq such that where x( i ) = sub( X ) = abs( X( IX+(JX-1)*DESCX(M_)+(i-1)*INCX ) ). The value of sumsq is assumed to be at least unity and the value of ssq will then satisfy .br 1.0 .le. ssq .le. ( sumsq + 2*n ). .br scale is assumed to be non-negative and scl returns the value scl = max( scale, abs( real( x( i ) ) ), abs( aimag( x( i ) ) ) ), i .br scale and sumsq must be supplied in SCALE and SUMSQ respectively. SCALE and SUMSQ are overwritten by scl and ssq respectively. The routine makes only one pass through the vector sub( X ). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. The result are only available in the scope of sub( X ), i.e if sub( X ) is distributed along a process row, the correct results are only available in this process row of the grid. Similarly if sub( X ) is distributed along a process column, the correct results are only available in this process column of the grid. .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The length of the distributed vector sub( X ). .TP 8 X (input) COMPLEX*16 The vector for which a scaled sum of squares is computed. x( i ) = X(IX+(JX-1)*M_X +(i-1)*INCX ), 1 <= i <= n. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 INCX (global input) INTEGER The global increment for the elements of X. Only two values of INCX are supported in this version, namely 1 and M_X. INCX must not be zero. .TP 8 SCALE (local input/local output) DOUBLE PRECISION On entry, the value scale in the equation above. On exit, SCALE is overwritten with scl , the scaling factor for the sum of squares. .TP 8 SUMSQ (local input/local output) DOUBLE PRECISION On entry, the value sumsq in the equation above. On exit, SUMSQ is overwritten with smsq , the basic sum of squares from which scl has been factored out. scalapack-doc-1.5/man/manl/pzlaswp.l0100644000056400000620000001225506335610661017127 0ustar pfrauenfstaff.TH PZLASWP l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLASWP - perform a series of row or column interchanges on the distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PZLASWP( DIREC, ROWCOL, N, A, IA, JA, DESCA, K1, K2, IPIV ) .TP 20 .ti +4 CHARACTER DIREC, ROWCOL .TP 20 .ti +4 INTEGER IA, JA, K1, K2, N .TP 20 .ti +4 INTEGER DESCA( * ), IPIV( * ) .TP 20 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZLASWP performs a series of row or column interchanges on the distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1). One interchange is initiated for each of rows or columns K1 trough K2 of sub( A ). This routine assumes that the pivoting information has already been broadcast along the process row or column. .br Also note that this routine will only work for K1-K2 being in the same MB (or NB) block. If you want to pivot a full matrix, use PZLAPIV. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 DIREC (global input) CHARACTER Specifies in which order the permutation is applied: = 'F' (Forward) = 'B' (Backward) .TP 8 ROWCOL (global input) CHARACTER Specifies if the rows or columns are permuted: = 'R' (Rows) = 'C' (Columns) .TP 8 N (global input) INTEGER If ROWCOL = 'R', the length of the rows of the distributed matrix A(*,JA:JA+N-1) to be permuted; If ROWCOL = 'C', the length of the columns of the distributed matrix A(IA:IA+N-1,*) to be permuted. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, * ). On entry, this array contains the local pieces of the distri- buted matrix to which the row/columns interchanges will be applied. On exit the permuted distributed matrix. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 K1 (global input) INTEGER The first element of IPIV for which a row or column inter- change will be done. .TP 8 K2 (global input) INTEGER The last element of IPIV for which a row or column inter- change will be done. .TP 8 IPIV (local input) INTEGER array, dimension LOCr(M_A)+MB_A for row pivoting and LOCc(N_A)+NB_A for column pivoting. This array is tied to the matrix A, IPIV(K) = L implies rows (or columns) K and L are to be interchanged. scalapack-doc-1.5/man/manl/pzlatra.l0100644000056400000620000000767606335610661017117 0ustar pfrauenfstaff.TH PZLATRA l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLATRA - compute the trace of an N-by-N distributed matrix sub( A ) denoting A( IA:IA+N-1, JA:JA+N-1 ) .SH SYNOPSIS .TP 20 COMPLEX*16 FUNCTION PZLATRA( N, A, IA, JA, DESCA ) .TP 20 .ti +4 INTEGER IA, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZLATRA computes the trace of an N-by-N distributed matrix sub( A ) denoting A( IA:IA+N-1, JA:JA+N-1 ). The result is left on every process of the grid. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on i.e the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension ( LLD_A, LOCc(JA+N-1) ). This array contains the local pieces of the distributed matrix the trace is to be computed. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. scalapack-doc-1.5/man/manl/pzlatrd.l0100644000056400000620000002140106335610661017100 0ustar pfrauenfstaff.TH PZLATRD l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLATRD - reduce NB rows and columns of a complex Hermitian distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) to complex tridiagonal form by an unitary similarity transformation Q' * sub( A ) * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of sub( A ) .SH SYNOPSIS .TP 20 SUBROUTINE PZLATRD( UPLO, N, NB, A, IA, JA, DESCA, D, E, TAU, W, IW, JW, DESCW, WORK ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IW, JA, JW, N, NB .TP 20 .ti +4 INTEGER DESCA( * ), DESCW( * ) .TP 20 .ti +4 DOUBLE PRECISION D( * ), E( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), W( * ), WORK( * ) .SH PURPOSE PZLATRD reduces NB rows and columns of a complex Hermitian distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) to complex tridiagonal form by an unitary similarity transformation Q' * sub( A ) * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of sub( A ). If UPLO = 'U', PZLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; .br if UPLO = 'L', PZLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied. .br This is an auxiliary routine called by PZHETRD. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the upper or lower triangular part of the Hermitian matrix sub( A ) is stored: .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NB (global input) INTEGER The number of rows and columns to be reduced. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the Hermitian distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced. On exit, if UPLO = 'U', the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of sub( A ); the elements above the diagonal with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. If UPLO = 'L', the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of sub( A ); the elements below the diagonal with the array TAU, represent the unitary matrix Q as a product of elementary reflectors; See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 D (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-1) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i). D is tied to the distributed matrix A. .TP 8 E (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-1) if UPLO = 'U', LOCc(JA+N-2) otherwise. The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. E is tied to the distributed matrix A. .TP 8 TAU (local output) COMPLEX*16, array, dimension LOCc(JA+N-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 W (local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_W,NB_W), This array contains the local pieces of the N-by-NB_W matrix W required to update the unreduced part of sub( A ). .TP 8 IW (global input) INTEGER The row index in the global array W indicating the first row of sub( W ). .TP 8 JW (global input) INTEGER The column index in the global array W indicating the first column of sub( W ). .TP 8 DESCW (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix W. .TP 8 WORK (local workspace) COMPLEX*16 array, dimension (NB_A) .SH FURTHER DETAILS If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors .br Q = H(n) H(n-1) . . . H(n-nb+1). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in .br A(ia:ia+i-2,ja+i), and tau in TAU(ja+i-1). .br If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors .br Q = H(1) H(2) . . . H(nb). .br Each H(i) has the form .br H(i) = I - tau * v * v' .br where tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in .br A(ia+i+1:ia+n-1,ja+i-1), and tau in TAU(ja+i-1). .br The elements of the vectors v together form the N-by-NB matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a Hermitian rank-2k update of the form: sub( A ) := sub( A ) - V*W' - W*V'. .br The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2: .br if UPLO = 'U': if UPLO = 'L': .br ( a a a v4 v5 ) ( d ) ( a a v4 v5 ) ( 1 d ) ( a 1 v5 ) ( v1 1 a ) ( d 1 ) ( v1 v2 a a ) ( d ) ( v1 v2 a a a ) where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i). .br scalapack-doc-1.5/man/manl/pzlatrs.l0100644000056400000620000000122306335610661017117 0ustar pfrauenfstaff.TH PZLATRS l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLATRS - solve a triangular system .SH SYNOPSIS .TP 20 SUBROUTINE PZLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, IA, JA, DESCA, X, IX, JX, DESCX, SCALE, CNORM, WORK ) .TP 20 .ti +4 CHARACTER DIAG, NORMIN, TRANS, UPLO .TP 20 .ti +4 INTEGER IA, IX, JA, JX, N .TP 20 .ti +4 DOUBLE PRECISION SCALE .TP 20 .ti +4 INTEGER DESCA( * ), DESCX( * ) .TP 20 .ti +4 DOUBLE PRECISION CNORM( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), X( * ), WORK( * ) .SH PURPOSE PZLATRS solves a triangular system. This routine in unfinished at this time, but will be part of the next release. .br scalapack-doc-1.5/man/manl/pzlatrz.l0100644000056400000620000001455206335610661017137 0ustar pfrauenfstaff.TH PZLATRZ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZLATRZ - reduce the M-by-N ( M<=N ) complex upper trapezoidal matrix sub( A ) = [A(IA:IA+M-1,JA:JA+M-1) A(IA:IA+M-1,JA+N-L:JA+N-1)] .SH SYNOPSIS .TP 20 SUBROUTINE PZLATRZ( M, N, L, A, IA, JA, DESCA, TAU, WORK ) .TP 20 .ti +4 INTEGER IA, JA, L, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), WORK( * ) .SH PURPOSE PZLATRZ reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix sub( A ) = [A(IA:IA+M-1,JA:JA+M-1) A(IA:IA+M-1,JA+N-L:JA+N-1)] to upper triangular form by means of unitary transformations. The upper trapezoidal matrix sub( A ) is factored as .br sub( A ) = ( R 0 ) * Z, .br where Z is an N-by-N unitary matrix and R is an M-by-M upper triangular matrix. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 L (global input) INTEGER The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. L > 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the leading M-by-M upper triangular part of sub( A ) contains the upper trian- gular matrix R, and elements N-L+1 to N of the first M rows of sub( A ), with the array TAU, represent the unitary matrix Z as a product of M elementary reflectors. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX*16, array, dimension LOCr(IA+M-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace) COMPLEX*16 array, dimension (LWORK) LWORK >= Nq0 + MAX( 1, Mp0 ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. .SH FURTHER DETAILS The factorization is obtained by Householder's method. The kth transformation matrix, Z( k ), whose conjugate transpose is used to introduce zeros into the (m - k + 1)th row of sub( A ), is given in the form .br Z( k ) = ( I 0 ), .br ( 0 T( k ) ) .br where .br T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ), ( 0 ) ( z( k ) ) tau is a scalar and z( k ) is an ( n - m ) element vector. tau and z( k ) are chosen to annihilate the elements of the kth row of sub( A ). .br The scalar tau is returned in the kth element of TAU and the vector u( k ) in the kth row of sub( A ), such that the elements of z( k ) are in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in the upper triangular part of sub( A ). .br Z is given by .br Z = Z( 1 ) * Z( 2 ) * ... * Z( m ). .br scalapack-doc-1.5/man/manl/pzlauu2.l0100644000056400000620000001161006335610662017024 0ustar pfrauenfstaff.TH PZLAUU2 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLAUU2 - compute the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PZLAUU2( UPLO, N, A, IA, JA, DESCA ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZLAUU2 computes the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). If UPLO = 'U' or 'u' then the upper triangle of the result is stored, overwriting the factor U in sub( A ). .br If UPLO = 'L' or 'l' then the lower triangle of the result is stored, overwriting the factor L in sub( A ). .br This is the unblocked form of the algorithm, calling Level 2 BLAS. No communication is performed by this routine, the matrix to operate on should be strictly local to one process. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 Specifies whether the triangular factor stored in the matrix sub( A ) is upper or lower triangular: .br = 'U': Upper triangular, .br = 'L': Lower triangular. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the order of the triangular factor U or L. N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the triangular factor L or U. On exit, if UPLO = 'U', the upper triangle of the distributed matrix sub( A ) is overwritten with the upper triangle of the product U * U'; if UPLO = 'L', the lower triangle of sub( A ) is overwritten with the lower triangle of the product L' * L. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. scalapack-doc-1.5/man/manl/pzlauum.l0100644000056400000620000001144606335610662017126 0ustar pfrauenfstaff.TH PZLAUUM l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZLAUUM - compute the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PZLAUUM( UPLO, N, A, IA, JA, DESCA ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZLAUUM computes the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). If UPLO = 'U' or 'u' then the upper triangle of the result is stored, overwriting the factor U in sub( A ). .br If UPLO = 'L' or 'l' then the lower triangle of the result is stored, overwriting the factor L in sub( A ). .br This is the blocked form of the algorithm, calling Level 3 PBLAS. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 Specifies whether the triangular factor stored in the distributed matrix sub( A ) is upper or lower triangular: .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the triangular factor U or L. N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the triangular factor L or U. On exit, if UPLO = 'U', the upper triangle of the distributed matrix sub( A ) is overwritten with the upper triangle of the product U * U'; if UPLO = 'L', the lower triangle of sub( A ) is overwritten with the lower triangle of the product L' * L. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. scalapack-doc-1.5/man/manl/pzmax1.l0100644000056400000620000001341106335610662016643 0ustar pfrauenfstaff.TH PZMAX1 l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZMAX1 - compute the global index of the maximum element in absolute value of a distributed vector sub( X ) .SH SYNOPSIS .TP 19 SUBROUTINE PZMAX1( N, AMAX, INDX, X, IX, JX, DESCX, INCX ) .TP 19 .ti +4 INTEGER INDX, INCX, IX, JX, N .TP 19 .ti +4 COMPLEX*16 AMAX .TP 19 .ti +4 INTEGER DESCX( * ) .TP 19 .ti +4 COMPLEX*16 X( * ) .SH PURPOSE PZMAX1 computes the global index of the maximum element in absolute value of a distributed vector sub( X ). The global index is returned in INDX and the value is returned in AMAX, .br where sub( X ) denotes X(IX:IX+N-1,JX) if INCX = 1, .br X(IX,JX:JX+N-1) if INCX = M_X. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. When the result of a vector-oriented PBLAS call is a scalar, it will be made available only within the scope which owns the vector(s) being operated on. Let X be a generic term for the input vector(s). Then, the processes which receive the answer will be (note that if an operation involves more than one vector, the processes which re- ceive the result will be the union of the following calculation for each vector): .br If N = 1, M_X = 1 and INCX = 1, then one can't determine if a process row or process column owns the vector operand, therefore only the process of coordinate {RSRC_X, CSRC_X} receives the result; If INCX = M_X, then sub( X ) is a vector distributed over a process row. Each process part of this row receives the result; .br If INCX = 1, then sub( X ) is a vector distributed over a process column. Each process part of this column receives the result; Based on PZAMAX from Level 1 PBLAS. The change is to use the 'genuine' absolute value. .br The serial version was contributed to LAPACK by Nick Higham for use with ZLACON. .br .SH ARGUMENTS .TP 8 N (global input) pointer to INTEGER The number of components of the distributed vector sub( X ). N >= 0. .TP 8 AMAX (global output) pointer to DOUBLE PRECISION The absolute value of the largest entry of the distributed vector sub( X ) only in the scope of sub( X ). .TP 8 INDX (global output) pointer to INTEGER The global index of the element of the distributed vector sub( X ) whose real part has maximum absolute value. .TP 8 X (local input) COMPLEX*16 array containing the local pieces of a distributed matrix of dimension of at least ( (JX-1)*M_X + IX + ( N - 1 )*abs( INCX ) ) This array contains the entries of the distributed vector sub( X ). .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 INCX (global input) INTEGER The global increment for the elements of X. Only two values of INCX are supported in this version, namely 1 and M_X. INCX must not be zero. scalapack-doc-1.5/man/manl/pzpbsv.l0100644000056400000620000000141606335610662016751 0ustar pfrauenfstaff.TH PZPBSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZPBSV - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 19 SUBROUTINE PZPBSV( UPLO, N, BW, NRHS, A, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 CHARACTER UPLO .TP 19 .ti +4 INTEGER BW, IB, INFO, JA, LWORK, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 COMPLEX*16 A( * ), B( * ), WORK( * ) .SH PURPOSE PZPBSV solves a system of linear equations where A(1:N, JA:JA+N-1) is an N-by-N complex .br banded symmetric positive definite distributed .br matrix with bandwidth BW. .br Cholesky factorization is used to factor a reordering of .br the matrix into L L'. .br See PZPBTRF and PZPBTRS for details. .br scalapack-doc-1.5/man/manl/pzpbtrf.l0100644000056400000620000000232006335610662017107 0ustar pfrauenfstaff.TH PZPBTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZPBTRF - compute a Cholesky factorization of an N-by-N complex banded symmetric positive definite distributed matrix with bandwidth BW .SH SYNOPSIS .TP 20 SUBROUTINE PZPBTRF( UPLO, N, BW, A, JA, DESCA, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER BW, INFO, JA, LAF, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), AF( * ), WORK( * ) .SH PURPOSE PZPBTRF computes a Cholesky factorization of an N-by-N complex banded symmetric positive definite distributed matrix with bandwidth BW: A(1:N, JA:JA+N-1). Reordering is used to increase parallelism in the factorization. This reordering results in factors that are DIFFERENT from those produced by equivalent sequential codes. These factors cannot be used directly by users; however, they can be used in .br subsequent calls to PZPBTRS to solve linear systems. .br The factorization has the form .br P A(1:N, JA:JA+N-1) P^T = U' U , if UPLO = 'U', or P A(1:N, JA:JA+N-1) P^T = L L', if UPLO = 'L' .br where U is a banded upper triangular matrix and L is banded lower triangular, and P is a permutation matrix. .br scalapack-doc-1.5/man/manl/pzpbtrs.l0100644000056400000620000000170006335610662017125 0ustar pfrauenfstaff.TH PZPBTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZPBTRS - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 20 SUBROUTINE PZPBTRS( UPLO, N, BW, NRHS, A, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER BW, IB, INFO, JA, LAF, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), AF( * ), B( * ), WORK( * ) .SH PURPOSE PZPBTRS solves a system of linear equations where A(1:N, JA:JA+N-1) is the matrix used to produce the factors stored in A(1:N,JA:JA+N-1) and AF by PZPBTRF. .br A(1:N, JA:JA+N-1) is an N-by-N complex .br banded symmetric positive definite distributed .br matrix with bandwidth BW. .br Depending on the value of UPLO, A stores either U or L in the equn A(1:N, JA:JA+N-1) = U'*U or L*L' as computed by PZPBTRF. .br Routine PZPBTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pzpbtrsv.l0100644000056400000620000000216706335610662017323 0ustar pfrauenfstaff.TH PZPBTRSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZPBTRSV - solve a banded triangular system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 21 SUBROUTINE PZPBTRSV( UPLO, TRANS, N, BW, NRHS, A, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 21 .ti +4 CHARACTER TRANS, UPLO .TP 21 .ti +4 INTEGER BW, IB, INFO, JA, LAF, LWORK, N, NRHS .TP 21 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 21 .ti +4 COMPLEX*16 A( * ), AF( * ), B( * ), WORK( * ) .SH PURPOSE PZPBTRSV solves a banded triangular system of linear equations or .br A(1:N, JA:JA+N-1)^H * X = B(IB:IB+N-1, 1:NRHS) where A(1:N, JA:JA+N-1) is a banded .br triangular matrix factor produced by the .br Cholesky factorization code PZPBTRF .br and is stored in A(1:N,JA:JA+N-1) and AF. .br The matrix stored in A(1:N, JA:JA+N-1) is either .br upper or lower triangular according to UPLO, .br and the choice of solving A(1:N, JA:JA+N-1) or A(1:N, JA:JA+N-1)^H is dictated by the user by the parameter TRANS. .br Routine PZPBTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pzpocon.l0100644000056400000620000001511106335610662017112 0ustar pfrauenfstaff.TH PZPOCON l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZPOCON - estimate the reciprocal of the condition number (in the 1-norm) of a complex Hermitian positive definite distributed matrix using the Cholesky factorization A = U**H*U or A = L*L**H computed by PZPOTRF .SH SYNOPSIS .TP 20 SUBROUTINE PZPOCON( UPLO, N, A, IA, JA, DESCA, ANORM, RCOND, WORK, LWORK, RWORK, LRWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, LRWORK, LWORK, N .TP 20 .ti +4 DOUBLE PRECISION ANORM, RCOND .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION RWORK( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), WORK( * ) .SH PURPOSE PZPOCON estimates the reciprocal of the condition number (in the 1-norm) of a complex Hermitian positive definite distributed matrix using the Cholesky factorization A = U**H*U or A = L*L**H computed by PZPOTRF. An estimate is obtained for norm(inv(A(IA:IA+N-1,JA:JA+N-1))), and the reciprocal of the condition number is computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the factor stored in A(IA:IA+N-1,JA:JA+N-1) is upper or lower triangular. .br = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER .br The order of the distributed matrix A(IA:IA+N-1,JA:JA+N-1). N >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension ( LLD_A, LOCc(JA+N-1) ). On entry, this array contains the local pieces of the factors L or U from the Cholesky factorization A(IA:IA+N-1,JA:JA+N-1) = U'*U or L*L', as computed by PZPOTRF. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 ANORM (global input) DOUBLE PRECISION The 1-norm (or infinity-norm) of the hermitian distributed matrix A(IA:IA+N-1,JA:JA+N-1). .TP 8 RCOND (global output) DOUBLE PRECISION The reciprocal of the condition number of the distributed matrix A(IA:IA+N-1,JA:JA+N-1), computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * .br norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 2*LOCr(N+MOD(IA-1,MB_A)) + MAX( 2, MAX(NB_A*MAX(1,CEIL(P-1,Q)),LOCc(N+MOD(JA-1,NB_A)) + NB_A*MAX(1,CEIL(Q-1,P))) ). If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 RWORK (local workspace/local output) DOUBLE PRECISION array, dimension (LRWORK) On exit, RWORK(1) returns the minimal and optimal LRWORK. .TP 8 LRWORK (local or global input) INTEGER The dimension of the array RWORK. LRWORK is local input and must be at least LRWORK >= 2*LOCc(N+MOD(JA-1,NB_A)). If LRWORK = -1, then LRWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pzpoequ.l0100644000056400000620000001414106335610662017127 0ustar pfrauenfstaff.TH PZPOEQU l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZPOEQU - compute row and column scalings intended to equilibrate a distributed Hermitian positive definite matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) and reduce its condition number (with respect to the two-norm) .SH SYNOPSIS .TP 20 SUBROUTINE PZPOEQU( N, A, IA, JA, DESCA, SR, SC, SCOND, AMAX, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 DOUBLE PRECISION AMAX, SCOND .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION SC( * ), SR( * ) .TP 20 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZPOEQU computes row and column scalings intended to equilibrate a distributed Hermitian positive definite matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) and reduce its condition number (with respect to the two-norm). SR and SC contain the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled distri- buted matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of SR and SC puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings. .br The scaling factor are stored along process rows in SR and along process columns in SC. The duplication of information simplifies greatly the application of the factors. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 N (global input) INTEGER The number of rows and columns to be operated on i.e the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of local dimension ( LLD_A, LOCc(JA+N-1) ), the N-by-N Hermitian positive definite distributed matrix sub( A ) whose scaling factors are to be computed. Only the diagonal elements of sub( A ) are referenced. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 SR (local output) DOUBLE PRECISION array, dimension LOCr(M_A) If INFO = 0, SR(IA:IA+N-1) contains the row scale factors for sub( A ). SR is aligned with the distributed matrix A, and replicated across every process column. SR is tied to the distributed matrix A. .TP 8 SC (local output) DOUBLE PRECISION array, dimension LOCc(N_A) If INFO = 0, SC(JA:JA+N-1) contains the column scale factors .br for A(IA:IA+M-1,JA:JA+N-1). SC is aligned with the distribu- ted matrix A, and replicated down every process row. SC is tied to the distributed matrix A. .TP 8 SCOND (global output) DOUBLE PRECISION If INFO = 0, SCOND contains the ratio of the smallest SR(i) (or SC(j)) to the largest SR(i) (or SC(j)), with IA <= i <= IA+N-1 and JA <= j <= JA+N-1. If SCOND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by SR (or SC). .TP 8 AMAX (global output) DOUBLE PRECISION Absolute value of largest matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, the K-th diagonal entry of sub( A ) is nonpositive. scalapack-doc-1.5/man/manl/pzporfs.l0100644000056400000620000002365606335610662017142 0ustar pfrauenfstaff.TH PZPORFS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZPORFS - improve the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite and provides error bounds and backward error estimates for the solutions .SH SYNOPSIS .TP 20 SUBROUTINE PZPORFS( UPLO, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, B, IB, JB, DESCB, X, IX, JX, DESCX, FERR, BERR, WORK, LWORK, RWORK, LRWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IAF, IB, INFO, IX, JA, JAF, JB, JX, LRWORK, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), AF( * ), B( * ), BERR( * ), FERR( * ), WORK( * ), X( * ) .TP 20 .ti +4 DOUBLE PRECISION RWORK( * ) .SH PURPOSE PZPORFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite and provides error bounds and backward error estimates for the solutions. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br In the following comments, sub( A ), sub( X ) and sub( B ) denote respectively A(IA:IA+N-1,JA:JA+N-1), X(IX:IX+N-1,JX:JX+NRHS-1) and B(IB:IB+N-1,JB:JB+NRHS-1). .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix sub( A ) is stored. = 'U': Upper triangular .br = 'L': Lower triangular .TP 8 N (global input) INTEGER The order of the matrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the matrices sub( B ) and sub( X ). NRHS >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1) ). This array contains the local pieces of the N-by-N Hermitian distributed matrix sub( A ) to be factored. If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the distribu- ted matrix, and its strictly upper triangular part is not referenced. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 AF (local input) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_AF,LOCc(JA+N-1)). On entry, this array contains the factors L or U from the Cholesky factorization sub( A ) = L*L**H or U**H*U, as computed by PZPOTRF. .TP 8 IAF (global input) INTEGER The row index in the global array AF indicating the first row of sub( AF ). .TP 8 JAF (global input) INTEGER The column index in the global array AF indicating the first column of sub( AF ). .TP 8 DESCAF (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix AF. .TP 8 B (local input) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_B, LOCc(JB+NRHS-1) ). On entry, this array contains the the local pieces of the right hand sides sub( B ). .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 X (local input) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_X, LOCc(JX+NRHS-1) ). On entry, this array contains the the local pieces of the solution vectors sub( X ). On exit, it contains the improved solution vectors. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 FERR (local output) DOUBLE PRECISION array of local dimension LOCc(JB+NRHS-1). The estimated forward error bound for each solution vector of sub( X ). If XTRUE is the true solution corresponding to sub( X ), FERR is an estimated upper bound for the magnitude of the largest element in (sub( X ) - XTRUE) divided by the magnitude of the largest element in sub( X ). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. This array is tied to the distributed matrix X. .TP 8 BERR (local output) DOUBLE PRECISION array of local dimension LOCc(JB+NRHS-1). The componentwise relative backward error of each solution vector (i.e., the smallest re- lative change in any entry of sub( A ) or sub( B ) that makes sub( X ) an exact solution). This array is tied to the distributed matrix X. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 2*LOCr( N + MOD( IA-1, MB_A ) ) If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 RWORK (local workspace/local output) DOUBLE PRECISION array, dimension (LRWORK) On exit, RWORK(1) returns the minimal and optimal LRWORK. .TP 8 LRWORK (local or global input) INTEGER The dimension of the array RWORK. LRWORK is local input and must be at least LRWORK >= LOCr( N + MOD( IB-1, MB_B ) ). If LRWORK = -1, then LRWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH PARAMETERS ITMAX is the maximum number of steps of iterative refinement. Notes ===== This routine temporarily returns when N <= 1. The distributed submatrices op( A ) and op( AF ) (respectively sub( X ) and sub( B ) ) should be distributed the same way on the same processes. These conditions ensure that sub( A ) and sub( AF ) (resp. sub( X ) and sub( B ) ) are "perfectly" aligned. Moreover, this routine requires the distributed submatrices sub( A ), sub( AF ), sub( X ), and sub( B ) to be aligned on a block boundary, i.e., if f(x,y) = MOD( x-1, y ): f( IA, DESCA( MB_ ) ) = f( JA, DESCA( NB_ ) ) = 0, f( IAF, DESCAF( MB_ ) ) = f( JAF, DESCAF( NB_ ) ) = 0, f( IB, DESCB( MB_ ) ) = f( JB, DESCB( NB_ ) ) = 0, and f( IX, DESCX( MB_ ) ) = f( JX, DESCX( NB_ ) ) = 0. scalapack-doc-1.5/man/manl/pzposv.l0100644000056400000620000001461206335610662016770 0ustar pfrauenfstaff.TH PZPOSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZPOSV - compute the solution to a complex system of linear equations sub( A ) * X = sub( B ), .SH SYNOPSIS .TP 19 SUBROUTINE PZPOSV( UPLO, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, INFO ) .TP 19 .ti +4 CHARACTER UPLO .TP 19 .ti +4 INTEGER IA, IB, INFO, JA, JB, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 COMPLEX*16 A( * ), B( * ) .SH PURPOSE PZPOSV computes the solution to a complex system of linear equations where sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1) and is an N-by-N hermitian distributed positive definite matrix and X and sub( B ) denoting B(IB:IB+N-1,JB:JB+NRHS-1) are N-by-NRHS distributed matrices. .br The Cholesky decomposition is used to factor sub( A ) as .br sub( A ) = U**H * U, if UPLO = 'U', or sub( A ) = L * L**H, if UPLO = 'L', .br where U is an upper triangular matrix and L is a lower triangular matrix. The factored form of sub( A ) is then used to solve the system of equations. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block decomposition ( MB_A = NB_A ). .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER = 'U': Upper triangle of sub( A ) is stored; .br = 'L': Lower triangle of sub( A ) is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed submatrix sub( B ). NRHS >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N symmetric distributed matrix sub( A ) to be factored. If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the distribu- ted matrix, and its strictly upper triangular part is not referenced. On exit, if INFO = 0, this array contains the local pieces of the factor U or L from the Cholesky factori- zation sub( A ) = U**H*U or L*L**H. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_B,LOC(JB+NRHS-1)). On entry, the local pieces of the right hand sides distribu- ted matrix sub( B ). On exit, if INFO = 0, sub( B ) is over- written with the solution distributed matrix X. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, the leading minor of order K, .br A(IA:IA+K-1,JA:JA+K-1) is not positive definite, and the factorization could not be completed, and the solution has not been computed. scalapack-doc-1.5/man/manl/pzposvx.l0100644000056400000620000003326106335610662017161 0ustar pfrauenfstaff.TH PZPOSVX l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZPOSVX - use the Cholesky factorization A = U**H*U or A = L*L**H to compute the solution to a complex system of linear equations A(IA:IA+N-1,JA:JA+N-1) * X = B(IB:IB+N-1,JB:JB+NRHS-1), .SH SYNOPSIS .TP 20 SUBROUTINE PZPOSVX( FACT, UPLO, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, EQUED, SR, SC, B, IB, JB, DESCB, X, IX, JX, DESCX, RCOND, FERR, BERR, WORK, LWORK, RWORK, LRWORK, INFO ) .TP 20 .ti +4 CHARACTER EQUED, FACT, UPLO .TP 20 .ti +4 INTEGER IA, IAF, IB, INFO, IX, JA, JAF, JB, JX, LRWORK, LWORK, N, NRHS .TP 20 .ti +4 DOUBLE PRECISION RCOND .TP 20 .ti +4 INTEGER DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * ) .TP 20 .ti +4 DOUBLE PRECISION BERR( * ), FERR( * ), SC( * ), SR( * ), RWORK( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), AF( * ), B( * ), WORK( * ), X( * ) .SH PURPOSE PZPOSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to compute the solution to a complex system of linear equations where A(IA:IA+N-1,JA:JA+N-1) is an N-by-N matrix and X and B(IB:IB+N-1,JB:JB+NRHS-1) are N-by-NRHS matrices. .br Error bounds on the solution and a condition estimate are also provided. In the following comments Y denotes Y(IY:IY+M-1,JY:JY+K-1) a M-by-K matrix where Y can be A, AF, B and X. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH DESCRIPTION The following steps are performed: .br 1. If FACT = 'E', real scaling factors are computed to equilibrate the system: .br diag(SR) * A * diag(SC) * inv(diag(SC)) * X = diag(SR) * 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(SR)*A*diag(SC) and B by diag(SR)*B. 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as A = U**T* U, if UPLO = 'U', or .br A = L * L**T, if UPLO = 'L', .br where U is an upper triangular matrix and L is a lower triangular matrix. .br 3. 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, steps 4-6 are skipped. .br 4. The system of equations is solved for X using the factored form of A. .br 5. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. .br 6. If equilibration was used, the matrix X is premultiplied by diag(SR) so that it solves the original system before .br equilibration. .br .SH ARGUMENTS .TP 8 FACT (global input) CHARACTER 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 contains the factored form of A. If EQUED = 'Y', the matrix A has been equilibrated with scaling factors given by S. A and AF will not be modified. = 'N': The matrix A will be copied to AF and factored. .br = 'E': The matrix A will be equilibrated if necessary, then copied to AF and factored. .TP 8 UPLO (global input) CHARACTER = 'U': Upper triangle of A is stored; .br = 'L': Lower triangle of A is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix A(IA:IA+N-1,JA:JA+N-1). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed submatrices B and X. NRHS >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of local dimension ( LLD_A, LOCc(JA+N-1) ). On entry, the Hermitian matrix A, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(SR)*A*diag(SC). 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. A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(SR)*A*diag(SC). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 AF (local input or local output) COMPLEX*16 pointer into the local memory to an array of local dimension ( LLD_AF, LOCc(JA+N-1)). If FACT = 'F', then AF is an input argument and on entry contains the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, in the same storage format as A. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix diag(SR)*A*diag(SC). If FACT = 'N', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the original matrix A. If FACT = 'E', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix). .TP 8 IAF (global input) INTEGER The row index in the global array AF indicating the first row of sub( AF ). .TP 8 JAF (global input) INTEGER The column index in the global array AF indicating the first column of sub( AF ). .TP 8 DESCAF (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix AF. .TP 8 EQUED (global input/global output) CHARACTER Specifies the form of equilibration that was done. = 'N': No equilibration (always true if FACT = 'N'). .br = 'Y': Equilibration was done, i.e., A has been replaced by diag(SR) * A * diag(SC). EQUED is an input variable if FACT = 'F'; otherwise, it is an output variable. .TP 8 SR (local input/local output) COMPLEX*16 array, dimension (LLD_A) The scale factors for A distributed across process rows; not accessed if EQUED = 'N'. SR is an input variable if FACT = 'F'; otherwise, SR is an output variable. If FACT = 'F' and EQUED = 'Y', each element of SR must be positive. .TP 8 SC (local input/local output) COMPLEX*16 array, dimension (LOC(N_A)) The scale factors for A distributed across process columns; not accessed if EQUED = 'N'. SC is an input variable if FACT = 'F'; otherwise, SC is an output variable. If FACT = 'F' and EQUED = 'Y', each element of SC must be positive. .TP 8 B (local input/local output) COMPLEX*16 pointer into the local memory to an array of local dimension ( LLD_B, LOCc(JB+NRHS-1) ). On entry, the N-by-NRHS right-hand side matrix B. On exit, if EQUED = 'N', B is not modified; if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B; if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is overwritten by diag(C)*B. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 X (local input/local output) COMPLEX*16 pointer into the local memory to an array of local dimension ( LLD_X, LOCc(JX+NRHS-1) ). If INFO = 0, the N-by-NRHS 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(SC))*X if TRANS = 'N' and EQUED = 'C' or 'B', or inv(diag(SR))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'. .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 RCOND (global 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, and the solution and error bounds are not computed. .TP 8 FERR (local output) DOUBLE PRECISION array, dimension (LOC(N_B)) The estimated forward error bounds for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution, FERR(j) bounds the magnitude of the largest entry in (X(j) - XTRUE) divided by the magnitude of the largest entry in X(j). The quality of the error bound depends on the quality of the estimate of norm(inv(A)) computed in the code; if the estimate of norm(inv(A)) is accurate, the error bound is guaranteed. .TP 8 BERR (local output) DOUBLE PRECISION array, dimension (LOC(N_B)) The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any entry of A or B that makes X(j) an exact solution). .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK = MAX( PZPOCON( LWORK ), PZPORFS( LWORK ) ) + LOCr( N_A ). LWORK = 3*DESCA( LLD_ ) If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 RWORK (local workspace/local output) DOUBLE PRECISION array, dimension (LRWORK) On exit, RWORK(1) returns the minimal and optimal LRWORK. .TP 8 LRWORK (local or global input) INTEGER The dimension of the array RWORK. LRWORK is local input and must be at least LRWORK = 2*LOCc(N_A). If LRWORK = -1, then LRWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 0: if INFO = i, and i is .br <= N: if INFO = i, the leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution and error bounds could not be computed. = N+1: RCOND is less than machine precision. The factorization has been completed, but the matrix is singular to working precision, and the solution and error bounds have not been computed. scalapack-doc-1.5/man/manl/pzpotf2.l0100644000056400000620000001261006335610662017027 0ustar pfrauenfstaff.TH PZPOTF2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZPOTF2 - compute the Cholesky factorization of a complex hermitian positive definite distributed matrix sub( A )=A(IA:IA+N-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PZPOTF2( UPLO, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZPOTF2 computes the Cholesky factorization of a complex hermitian positive definite distributed matrix sub( A )=A(IA:IA+N-1,JA:JA+N-1). The factorization has the form .br sub( A ) = U' * U , if UPLO = 'U', or .br sub( A ) = L * L', if UPLO = 'L', .br where U is an upper triangular matrix and L is lower triangular. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires N <= NB_A-MOD(JA-1, NB_A) and square block decomposition ( MB_A = NB_A ). .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER = 'U': Upper triangle of sub( A ) is stored; .br = 'L': Lower triangle of sub( A ) is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N symmetric distributed matrix sub( A ) to be factored. If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the distribu- ted matrix, and its strictly upper triangular part is not referenced. On exit, if UPLO = 'U', the upper triangular part of the distributed matrix contains the Cholesky factor U, if UPLO = 'L', the lower triangular part of the distribu- ted matrix contains the Cholesky factor L. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, the leading minor of order K, .br A(IA:IA+K-1,JA:JA+K-1) is not positive definite, and the factorization could not be completed. scalapack-doc-1.5/man/manl/pzpotrf.l0100644000056400000620000001261206335610662017131 0ustar pfrauenfstaff.TH PZPOTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZPOTRF - compute the Cholesky factorization of an N-by-N complex hermitian positive definite distributed matrix sub( A ) denoting A(IA:IA+N-1, JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PZPOTRF( UPLO, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZPOTRF computes the Cholesky factorization of an N-by-N complex hermitian positive definite distributed matrix sub( A ) denoting A(IA:IA+N-1, JA:JA+N-1). The factorization has the form .br sub( A ) = U' * U , if UPLO = 'U', or .br sub( A ) = L * L', if UPLO = 'L', .br where U is an upper triangular matrix and L is lower triangular. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block decomposition ( MB_A = NB_A ). .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER = 'U': Upper triangle of sub( A ) is stored; .br = 'L': Lower triangle of sub( A ) is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N Hermitian distributed matrix sub( A ) to be factored. If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the distribu- ted matrix, and its strictly upper triangular part is not referenced. On exit, if UPLO = 'U', the upper triangular part of the distributed matrix contains the Cholesky factor U, if UPLO = 'L', the lower triangular part of the distribu- ted matrix contains the Cholesky factor L. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, the leading minor of order K, .br A(IA:IA+K-1,JA:JA+K-1) is not positive definite, and the factorization could not be completed. scalapack-doc-1.5/man/manl/pzpotri.l0100644000056400000620000001146306335610663017140 0ustar pfrauenfstaff.TH PZPOTRI l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZPOTRI - compute the inverse of a complex Hermitian positive definite distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) using the Cholesky factorization sub( A ) = U**H*U or L*L**H computed by PZPOTRF .SH SYNOPSIS .TP 20 SUBROUTINE PZPOTRI( UPLO, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZPOTRI computes the inverse of a complex Hermitian positive definite distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) using the Cholesky factorization sub( A ) = U**H*U or L*L**H computed by PZPOTRF. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 = 'U': Upper triangle of sub( A ) is stored; .br = 'L': Lower triangle of sub( A ) is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the triangular factor U or L from the Cholesky factorization of the distributed matrix sub( A ) = U**H*U or L*L**H, as computed by PZPOTRF. On exit, the local pieces of the upper or lower triangle of the (Hermitian) inverse of sub( A ), overwriting the input factor U or L. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = i, the (i,i) element of the factor U or L is zero, and the inverse could not be computed. scalapack-doc-1.5/man/manl/pzpotrs.l0100644000056400000620000001270406335610663017151 0ustar pfrauenfstaff.TH PZPOTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZPOTRS - solve a system of linear equations sub( A ) * X = sub( B ) A(IA:IA+N-1,JA:JA+N-1)*X = B(IB:IB+N-1,JB:JB+NRHS-1) .SH SYNOPSIS .TP 20 SUBROUTINE PZPOTRS( UPLO, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IA, IB, INFO, JA, JB, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), B( * ) .SH PURPOSE PZPOTRS solves a system of linear equations where sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1) and is a N-by-N hermitian positive definite distributed matrix using the Cholesky factorization sub( A ) = U**H*U or L*L**H computed by PZPOTRF. sub( B ) denotes the distributed matrix B(IB:IB+N-1,JB:JB+NRHS-1). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br This routine requires square block decomposition ( MB_A = NB_A ). .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER = 'U': Upper triangle of sub( A ) is stored; .br = 'L': Lower triangle of sub( A ) is stored. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed submatrix sub( B ). NRHS >= 0. .TP 8 A (local input) COMPLEX*16 pointer into local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the factors L or U from the Cholesky facto- rization sub( A ) = L*L**H or U**H*U, as computed by PZPOTRF. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input/local output) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_B,LOCc(JB+NRHS-1)). On entry, this array contains the the local pieces of the right hand sides sub( B ). On exit, this array contains the local pieces of the solution distributed matrix X. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pzptsv.l0100644000056400000620000000144206335610663016773 0ustar pfrauenfstaff.TH PZPTSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZPTSV - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 19 SUBROUTINE PZPTSV( UPLO, N, NRHS, D, E, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO ) .TP 19 .ti +4 CHARACTER UPLO .TP 19 .ti +4 INTEGER IB, INFO, JA, LWORK, N, NRHS .TP 19 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 19 .ti +4 COMPLEX*16 B( * ), E( * ), WORK( * ) .TP 19 .ti +4 DOUBLE PRECISION D( * ) .SH PURPOSE PZPTSV solves a system of linear equations where A(1:N, JA:JA+N-1) is an N-by-N complex .br tridiagonal symmetric positive definite distributed .br matrix. .br Cholesky factorization is used to factor a reordering of .br the matrix into L L'. .br See PZPTTRF and PZPTTRS for details. .br scalapack-doc-1.5/man/manl/pzpttrf.l0100644000056400000620000000227006335610663017136 0ustar pfrauenfstaff.TH PZPTTRF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZPTTRF - compute a Cholesky factorization of an N-by-N complex tridiagonal symmetric positive definite distributed matrix A(1:N, JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PZPTTRF( N, D, E, JA, DESCA, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER INFO, JA, LAF, LWORK, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 AF( * ), E( * ), WORK( * ) .TP 20 .ti +4 DOUBLE PRECISION D( * ) .SH PURPOSE PZPTTRF computes a Cholesky factorization of an N-by-N complex tridiagonal symmetric positive definite distributed matrix A(1:N, JA:JA+N-1). Reordering is used to increase parallelism in the factorization. This reordering results in factors that are DIFFERENT from those produced by equivalent sequential codes. These factors cannot be used directly by users; however, they can be used in .br subsequent calls to PZPTTRS to solve linear systems. .br The factorization has the form .br P A(1:N, JA:JA+N-1) P^T = U' D U or .br P A(1:N, JA:JA+N-1) P^T = L D L', .br where U is a tridiagonal upper triangular matrix and L is tridiagonal lower triangular, and P is a permutation matrix. .br scalapack-doc-1.5/man/manl/pzpttrs.l0100644000056400000620000000172406335610663017156 0ustar pfrauenfstaff.TH PZPTTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZPTTRS - solve a system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 20 SUBROUTINE PZPTTRS( UPLO, N, NRHS, D, E, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER UPLO .TP 20 .ti +4 INTEGER IB, INFO, JA, LAF, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX*16 AF( * ), B( * ), E( * ), WORK( * ) .TP 20 .ti +4 DOUBLE PRECISION D( * ) .SH PURPOSE PZPTTRS solves a system of linear equations where A(1:N, JA:JA+N-1) is the matrix used to produce the factors stored in A(1:N,JA:JA+N-1) and AF by PZPTTRF. .br A(1:N, JA:JA+N-1) is an N-by-N complex .br tridiagonal symmetric positive definite distributed .br matrix. .br Depending on the value of UPLO, A stores either U or L in the equn A(1:N, JA:JA+N-1) = U'D *U or L*D L' as computed by PZPTTRF. Routine PZPTTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pzpttrsv.l0100644000056400000620000000224706335610663017345 0ustar pfrauenfstaff.TH PZPTTRSV l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZPTTRSV - solve a tridiagonal triangular system of linear equations A(1:N, JA:JA+N-1) * X = B(IB:IB+N-1, 1:NRHS) .SH SYNOPSIS .TP 21 SUBROUTINE PZPTTRSV( UPLO, TRANS, N, NRHS, D, E, JA, DESCA, B, IB, DESCB, AF, LAF, WORK, LWORK, INFO ) .TP 21 .ti +4 CHARACTER TRANS, UPLO .TP 21 .ti +4 INTEGER IB, INFO, JA, LAF, LWORK, N, NRHS .TP 21 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 21 .ti +4 COMPLEX*16 AF( * ), B( * ), E( * ), WORK( * ) .TP 21 .ti +4 DOUBLE PRECISION D( * ) .SH PURPOSE PZPTTRSV solves a tridiagonal triangular system of linear equations or .br A(1:N, JA:JA+N-1)^H * X = B(IB:IB+N-1, 1:NRHS) where A(1:N, JA:JA+N-1) is a tridiagonal .br triangular matrix factor produced by the .br Cholesky factorization code PZPTTRF .br and is stored in A(1:N,JA:JA+N-1) and AF. .br The matrix stored in A(1:N, JA:JA+N-1) is either .br upper or lower triangular according to UPLO, .br and the choice of solving A(1:N, JA:JA+N-1) or A(1:N, JA:JA+N-1)^H is dictated by the user by the parameter TRANS. .br Routine PZPTTRF MUST be called first. .br scalapack-doc-1.5/man/manl/pzstein.l0100644000056400000620000002600406335610663017122 0ustar pfrauenfstaff.TH PZSTEIN l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZSTEIN - compute the eigenvectors of a symmetric tridiagonal matrix in parallel, using inverse iteration .SH SYNOPSIS .TP 20 SUBROUTINE PZSTEIN( N, D, E, M, W, IBLOCK, ISPLIT, ORFAC, Z, IZ, JZ, DESCZ, WORK, LWORK, IWORK, LIWORK, IFAIL, ICLUSTR, GAP, INFO ) .TP 20 .ti +4 INTEGER INFO, IZ, JZ, LIWORK, LWORK, M, N .TP 20 .ti +4 DOUBLE PRECISION ORFAC .TP 20 .ti +4 INTEGER DESCZ( * ), IBLOCK( * ), ICLUSTR( * ), IFAIL( * ), ISPLIT( * ), IWORK( * ) .TP 20 .ti +4 DOUBLE PRECISION D( * ), E( * ), GAP( * ), W( * ), WORK( * ) .TP 20 .ti +4 COMPLEX*16 Z( * ) .SH PURPOSE PZSTEIN computes the eigenvectors of a symmetric tridiagonal matrix in parallel, using inverse iteration. The eigenvectors found correspond to user specified eigenvalues. PZSTEIN does not orthogonalize vectors that are on different processes. The extent of orthogonalization is controlled by the input parameter LWORK. Eigenvectors that are to be orthogonalized are computed by the same process. PZSTEIN decides on the allocation of work among the processes and then calls DSTEIN2 (modified LAPACK routine) on each individual process. If insufficient workspace is allocated, the expected orthogonalization may not be done. .br Note : If the eigenvectors obtained are not orthogonal, increase LWORK and run the code again. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS P = NPROW * NPCOL is the total number of processes .TP 8 N (global input) INTEGER The order of the tridiagonal matrix T. N >= 0. .TP 8 D (global input) DOUBLE PRECISION array, dimension (N) The n diagonal elements of the tridiagonal matrix T. .TP 8 E (global input) DOUBLE PRECISION array, dimension (N-1) The (n-1) off-diagonal elements of the tridiagonal matrix T. .TP 8 M (global input) INTEGER The total number of eigenvectors to be found. 0 <= M <= N. .TP 8 W (global input/global output) DOUBLE PRECISION array, dim (M) On input, the first M elements of W contain all the eigenvalues for which eigenvectors are to be computed. The eigenvalues should be grouped by split-off block and ordered from smallest to largest within the block (The output array W from PDSTEBZ with ORDER='b' is expected here). This array should be replicated on all processes. On output, the first M elements contain the input eigenvalues in ascending order. Note : To obtain orthogonal vectors, it is best if eigenvalues are computed to highest accuracy ( this can be done by setting ABSTOL to the underflow threshold = DLAMCH('U') --- ABSTOL is an input parameter to PDSTEBZ ) .TP 8 IBLOCK (global input) INTEGER array, dimension (N) The submatrix indices associated with the corresponding eigenvalues in W -- 1 for eigenvalues belonging to the first submatrix from the top, 2 for those belonging to the second submatrix, etc. (The output array IBLOCK from PDSTEBZ is expected here). .TP 8 ISPLIT (global input) INTEGER array, dimension (N) The splitting points, at which T breaks up into submatrices. The first submatrix consists of rows/columns 1 to ISPLIT(1), the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N (The output array ISPLIT from PDSTEBZ is expected here.) .TP 8 ORFAC (global input) DOUBLE PRECISION ORFAC specifies which eigenvectors should be orthogonalized. Eigenvectors that correspond to eigenvalues which are within ORFAC*||T|| of each other are to be orthogonalized. However, if the workspace is insufficient (see LWORK), this tolerance may be decreased until all eigenvectors to be orthogonalized can be stored in one process. No orthogonalization will be done if ORFAC equals zero. A default value of 10^-3 is used if ORFAC is negative. ORFAC should be identical on all processes. .TP 8 Z (local output) COMPLEX*16 array, dimension (DESCZ(DLEN_), N/npcol + NB) Z contains the computed eigenvectors associated with the specified eigenvalues. Any vector which fails to converge is set to its current iterate after MAXITS iterations ( See DSTEIN2 ). On output, Z is distributed across the P processes in block cyclic format. .TP 8 IZ (global input) INTEGER Z's global row index, which points to the beginning of the submatrix which is to be operated on. .TP 8 JZ (global input) INTEGER Z's global column index, which points to the beginning of the submatrix which is to be operated on. .TP 8 DESCZ (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix Z. .TP 8 WORK (local workspace/global output) DOUBLE PRECISION array, dimension ( LWORK ) On output, WORK(1) gives a lower bound on the workspace ( LWORK ) that guarantees the user desired orthogonalization (see ORFAC). Note that this may overestimate the minimum workspace needed. .TP 8 LWORK (local input) integer LWORK controls the extent of orthogonalization which can be done. The number of eigenvectors for which storage is allocated on each process is NVEC = floor(( LWORK- max(5*N,NP00*MQ00) )/N). Eigenvectors corresponding to eigenvalue clusters of size NVEC - ceil(M/P) + 1 are guaranteed to be orthogonal ( the orthogonality is similar to that obtained from ZSTEIN2). Note : LWORK must be no smaller than: max(5*N,NP00*MQ00) + ceil(M/P)*N, and should have the same input value on all processes. It is the minimum value of LWORK input on different processes that is significant. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IWORK (local workspace/global output) INTEGER array, dimension ( 3*N+P+1 ) On return, IWORK(1) contains the amount of integer workspace required. On return, the IWORK(2) through IWORK(P+2) indicate the eigenvectors computed by each process. Process I computes eigenvectors indexed IWORK(I+2)+1 thru' IWORK(I+3). .TP 8 LIWORK (local input) INTEGER Size of array IWORK. Must be >= 3*N + P + 1 If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 IFAIL (global output) integer array, dimension (M) On normal exit, all elements of IFAIL are zero. If one or more eigenvectors fail to converge after MAXITS iterations (as in ZSTEIN), then INFO > 0 is returned. If mod(INFO,M+1)>0, then for I=1 to mod(INFO,M+1), the eigenvector corresponding to the eigenvalue W(IFAIL(I)) failed to converge ( W refers to the array of eigenvalues on output ). ICLUSTR (global output) integer array, dimension (2*P) This output array contains indices of eigenvectors corresponding to a cluster of eigenvalues that could not be orthogonalized due to insufficient workspace (see LWORK, ORFAC and INFO). Eigenvectors corresponding to clusters of eigenvalues indexed ICLUSTR(2*I-1) to ICLUSTR(2*I), I = 1 to INFO/(M+1), could not be orthogonalized due to lack of workspace. Hence the eigenvectors corresponding to these clusters may not be orthogonal. ICLUSTR is a zero terminated array --- ( ICLUSTR(2*K).NE.0 .AND. ICLUSTR(2*K+1).EQ.0 ) if and only if K is the number of clusters. .TP 8 GAP (global output) DOUBLE PRECISION array, dimension (P) This output array contains the gap between eigenvalues whose eigenvectors could not be orthogonalized. The INFO/M output values in this array correspond to the INFO/(M+1) clusters indicated by the array ICLUSTR. As a result, the dot product between eigenvectors corresponding to the I^th cluster may be as high as ( O(n)*macheps ) / GAP(I). .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. < 0 : if INFO = -I, the I-th argument had an illegal value .br > 0 : if mod(INFO,M+1) = I, then I eigenvectors failed to converge in MAXITS iterations. Their indices are stored in the array IFAIL. if INFO/(M+1) = I, then eigenvectors corresponding to I clusters of eigenvalues could not be orthogonalized due to insufficient workspace. The indices of the clusters are stored in the array ICLUSTR. scalapack-doc-1.5/man/manl/pztrcon.l0100644000056400000620000001612306335610663017126 0ustar pfrauenfstaff.TH PZTRCON l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZTRCON - estimate the reciprocal of the condition number of a triangular distributed matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm .SH SYNOPSIS .TP 20 SUBROUTINE PZTRCON( NORM, UPLO, DIAG, N, A, IA, JA, DESCA, RCOND, WORK, LWORK, RWORK, LRWORK, INFO ) .TP 20 .ti +4 CHARACTER DIAG, NORM, UPLO .TP 20 .ti +4 INTEGER IA, JA, INFO, LRWORK, LWORK, N .TP 20 .ti +4 DOUBLE PRECISION RCOND .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 DOUBLE PRECISION RWORK( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), WORK( * ) .SH PURPOSE PZTRCON estimates the reciprocal of the condition number of a triangular distributed matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm. The norm of A(IA:IA+N-1,JA:JA+N-1) is computed and an estimate is obtained for norm(inv(A(IA:IA+N-1,JA:JA+N-1))), then the reciprocal of the condition number is computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 NORM (global input) CHARACTER Specifies whether the 1-norm condition number or the infinity-norm condition number is required: .br = '1' or 'O': 1-norm; .br = 'I': Infinity-norm. .TP 8 UPLO (global input) CHARACTER .br = 'U': A(IA:IA+N-1,JA:JA+N-1) is upper triangular; .br = 'L': A(IA:IA+N-1,JA:JA+N-1) is lower triangular. .TP 8 DIAG (global input) CHARACTER .br = 'N': A(IA:IA+N-1,JA:JA+N-1) is non-unit triangular; .br = 'U': A(IA:IA+N-1,JA:JA+N-1) is unit triangular. .TP 8 N (global input) INTEGER .br The order of the distributed matrix A(IA:IA+N-1,JA:JA+N-1). N >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension ( LLD_A, LOCc(JA+N-1) ). This array contains the local pieces of the triangular distributed matrix A(IA:IA+N-1,JA:JA+N-1). If UPLO = 'U', the leading N-by-N upper triangular part of this distributed matrix con- tains the upper triangular matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of this ditributed matrix contains the lower triangular matrix, and the strictly upper triangular part is not referenced. If DIAG = 'U', the diagonal elements of A(IA:IA+N-1,JA:JA+N-1) are also not referenced and are assumed to be 1. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 RCOND (global output) DOUBLE PRECISION The reciprocal of the condition number of the distributed matrix A(IA:IA+N-1,JA:JA+N-1), computed as .br RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) * .br norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ). .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 2*LOCr(N+MOD(IA-1,MB_A)) + MAX( 2, MAX(NB_A*CEIL(P-1,Q),LOCc(N+MOD(JA-1,NB_A)) + NB_A*CEIL(Q-1,P)) ). If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 RWORK (local workspace/local output) DOUBLE PRECISION array, dimension (LRWORK) On exit, RWORK(1) returns the minimal and optimal LRWORK. .TP 8 LRWORK (local or global input) INTEGER The dimension of the array RWORK. LRWORK is local input and must be at least LRWORK >= LOCc(N+MOD(JA-1,NB_A)). If LRWORK = -1, then LRWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pztrrfs.l0100644000056400000620000002274406335610663017147 0ustar pfrauenfstaff.TH PZTRRFS l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZTRRFS - provide error bounds and backward error estimates for the solution to a system of linear equations with a triangular coefficient matrix .SH SYNOPSIS .TP 20 SUBROUTINE PZTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, X, IX, JX, DESCX, FERR, BERR, WORK, LWORK, RWORK, LRWORK, INFO ) .TP 20 .ti +4 CHARACTER DIAG, TRANS, UPLO .TP 20 .ti +4 INTEGER INFO, IA, IB, IX, JA, JB, JX, LRWORK, LWORK, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ), DESCX( * ) .TP 20 .ti +4 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), B( * ), WORK( * ), X( * ) .SH PURPOSE PZTRRFS provides error bounds and backward error estimates for the solution to a system of linear equations with a triangular coefficient matrix. The solution matrix X must be computed by PZTRTRS or some other means before entering this routine. PZTRRFS does not do iterative refinement because doing so cannot improve the backward error. Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br In the following comments, sub( A ), sub( X ) and sub( B ) denote respectively A(IA:IA+N-1,JA:JA+N-1), X(IX:IX+N-1,JX:JX+NRHS-1) and B(IB:IB+N-1,JB:JB+NRHS-1). .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 = 'U': sub( A ) is upper triangular; .br = 'L': sub( A ) is lower triangular. .TP 8 TRANS (global input) CHARACTER*1 Specifies the form of the system of equations. = 'N': sub( A ) * sub( X ) = sub( B ) (No transpose) .br = 'T': sub( A )**T * sub( X ) = sub( B ) (Transpose) .br = 'C': sub( A )**H * sub( X ) = sub( B ) (Conjugate transpose) .TP 8 DIAG (global input) CHARACTER*1 = 'N': sub( A ) is non-unit triangular; .br = 'U': sub( A ) is unit triangular. .TP 8 N (global input) INTEGER The order of the matrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the matrices sub( B ) and sub( X ). NRHS >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1) ). This array contains the local pieces of the original triangular distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the distribu- ted matrix, and its strictly upper triangular part is not referenced. If DIAG = 'U', the diagonal elements of sub( A ) are also not referenced and are assumed to be 1. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_B, LOCc(JB+NRHS-1) ). On entry, this array contains the the local pieces of the right hand sides sub( B ). .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 X (local input) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_X, LOCc(JX+NRHS-1) ). On entry, this array contains the the local pieces of the solution vectors sub( X ). .TP 8 IX (global input) INTEGER The row index in the global array X indicating the first row of sub( X ). .TP 8 JX (global input) INTEGER The column index in the global array X indicating the first column of sub( X ). .TP 8 DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. .TP 8 FERR (local output) DOUBLE PRECISION array of local dimension LOCc(JB+NRHS-1). The estimated forward error bounds for each solution vector of sub( X ). If XTRUE is the true solution, FERR bounds the magnitude of the largest entry in (sub( X ) - XTRUE) divided by the magnitude of the largest entry in sub( X ). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. This array is tied to the distributed matrix X. .TP 8 BERR (local output) DOUBLE PRECISION array of local dimension LOCc(JB+NRHS-1). The componentwise relative backward error of each solution vector (i.e., the smallest re- lative change in any entry of sub( A ) or sub( B ) that makes sub( X ) an exact solution). This array is tied to the distributed matrix X. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 2*LOCr( N + MOD( IA-1, MB_A ) ). If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 RWORK (local workspace/local output) DOUBLE PRECISION array, dimension (LRWORK) On exit, RWORK(1) returns the minimal and optimal LRWORK. .TP 8 LRWORK (local or global input) INTEGER The dimension of the array RWORK. LRWORK is local input and must be at least LRWORK >= LOCr( N + MOD( IB-1, MB_B ) ). If LRWORK = -1, then LRWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Notes ===== This routine temporarily returns when N <= 1. The distributed submatrices sub( X ) and sub( B ) should be distributed the same way on the same processes. These conditions ensure that sub( X ) and sub( B ) are "perfectly" aligned. Moreover, this routine requires the distributed submatrices sub( A ), sub( X ), and sub( B ) to be aligned on a block boundary, i.e., if f(x,y) = MOD( x-1, y ): f( IA, DESCA( MB_ ) ) = f( JA, DESCA( NB_ ) ) = 0, f( IB, DESCB( MB_ ) ) = f( JB, DESCB( NB_ ) ) = 0, and f( IX, DESCX( MB_ ) ) = f( JX, DESCX( NB_ ) ) = 0. scalapack-doc-1.5/man/manl/pztrti2.l0100644000056400000620000001206106335610663017042 0ustar pfrauenfstaff.TH PZTRTI2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZTRTI2 - compute the inverse of a complex upper or lower triangular block matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PZTRTI2( UPLO, DIAG, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER DIAG, UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZTRTI2 computes the inverse of a complex upper or lower triangular block matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). This matrix should be contained in one and only one process memory space (local operation). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER*1 = 'U': sub( A ) is upper triangular; .br = 'L': sub( A ) is lower triangular. .TP 8 DIAG (global input) CHARACTER*1 .br = 'N': sub( A ) is non-unit triangular .br = 'U': sub( A ) is unit triangular .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)), this array contains the local pieces of the triangular matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of the matrix sub( A ) contains the upper triangular matrix, and the strictly lower triangular part of sub( A ) is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of the matrix sub( A ) contains the lower triangular matrix, and the strictly upper triangular part of sub( A ) is not referenced. If DIAG = 'U', the diagonal elements of sub( A ) are also not referenced and are assumed to be 1. On exit, the (triangular) inverse of the original matrix, in the same storage format. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pztrtri.l0100644000056400000620000001214106335610663017141 0ustar pfrauenfstaff.TH PZTRTRI l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZTRTRI - compute the inverse of a upper or lower triangular distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) .SH SYNOPSIS .TP 20 SUBROUTINE PZTRTRI( UPLO, DIAG, N, A, IA, JA, DESCA, INFO ) .TP 20 .ti +4 CHARACTER DIAG, UPLO .TP 20 .ti +4 INTEGER IA, INFO, JA, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ) .SH PURPOSE PZTRTRI computes the inverse of a upper or lower triangular distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER Specifies whether the distributed matrix sub( A ) is upper or lower triangular: .br = 'U': Upper triangular, .br = 'L': Lower triangular. .TP 8 DIAG (global input) CHARACTER Specifies whether or not the distributed matrix sub( A ) is unit triangular: .br = 'N': Non-unit triangular, .br = 'U': Unit triangular. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the triangular matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of the matrix sub( A ) contains the upper triangular matrix to be inverted, and the strictly lower triangular part of sub( A ) is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of the matrix sub( A ) contains the lower triangular matrix, and the strictly upper triangular part of sub( A ) is not referenced. On exit, the (triangular) inverse of the original matrix. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = K, A(IA+K-1,JA+K-1) is exactly zero. The triangular matrix sub( A ) is singular and its inverse can not be computed. scalapack-doc-1.5/man/manl/pztrtrs.l0100644000056400000620000001447506335610663017167 0ustar pfrauenfstaff.TH PZTRTRS l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME PZTRTRS - solve a triangular system of the form sub( A ) * X = sub( B ) or sub( A )**T * X = sub( B ) or sub( A )**H * X = sub( B ), .SH SYNOPSIS .TP 20 SUBROUTINE PZTRTRS( UPLO, TRANS, DIAG, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, INFO ) .TP 20 .ti +4 CHARACTER DIAG, TRANS, UPLO .TP 20 .ti +4 INTEGER IA, IB, INFO, JA, JB, N, NRHS .TP 20 .ti +4 INTEGER DESCA( * ), DESCB( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), B( * ) .SH PURPOSE PZTRTRS solves a triangular system of the form where sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1) and is a triangular distributed matrix of order N, and B(IB:IB+N-1,JB:JB+NRHS-1) is an N-by-NRHS distributed matrix denoted by sub( B ). A check is made to verify that sub( A ) is nonsingular. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 UPLO (global input) CHARACTER = 'U': sub( A ) is upper triangular; .br = 'L': sub( A ) is lower triangular. .TP 8 TRANS (global input) CHARACTER .br Specifies the form of the system of equations: .br = 'N': Solve sub( A ) * X = sub( B ) (No transpose) .br = 'T': Solve sub( A )**T * X = sub( B ) (Transpose) .br = 'C': Solve sub( A )**H * X = sub( B ) (Conjugate transpose) .TP 8 DIAG (global input) CHARACTER .br = 'N': sub( A ) is non-unit triangular; .br = 'U': sub( A ) is unit triangular. .TP 8 N (global input) INTEGER The number of rows and columns to be operated on i.e the order of the distributed submatrix sub( A ). N >= 0. .TP 8 NRHS (global input) INTEGER The number of right hand sides, i.e., the number of columns of the distributed matrix sub( B ). NRHS >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1) ). This array contains the local pieces of the distributed triangular matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular matrix, and the strictly lower triangular part of sub( A ) is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular matrix, and the strictly upper triangular part of sub( A ) is not referenced. If DIAG = 'U', the diagonal elements of sub( A ) are also not referenced and are assumed to be 1. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 B (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_B,LOCc(JB+NRHS-1)). On entry, this array contains the local pieces of the right hand side distributed matrix sub( B ). On exit, if INFO = 0, sub( B ) is overwritten by the solution matrix X. .TP 8 IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ). .TP 8 JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ). .TP 8 DESCB (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: If INFO = i, the i-th diagonal element of sub( A ) is zero, indicating that the submatrix is singular and the solutions X have not been computed. scalapack-doc-1.5/man/manl/pztzrzf.l0100644000056400000620000001566406335610664017172 0ustar pfrauenfstaff.TH PZTZRZF l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZTZRZF - reduce the M-by-N ( M<=N ) complex upper trapezoidal matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper triangular form by means of unitary transformations .SH SYNOPSIS .TP 20 SUBROUTINE PZTZRZF( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), WORK( * ) .SH PURPOSE PZTZRZF reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper triangular form by means of unitary transformations. The upper trapezoidal matrix sub( A ) is factored as .br sub( A ) = ( R 0 ) * Z, .br where Z is an N-by-N unitary matrix and R is an M-by-M upper triangular matrix. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the leading M-by-M upper triangular part of sub( A ) contains the upper trian- gular matrix R, and elements M+1 to N of the first M rows of sub( A ), with the array TAU, represent the unitary matrix Z as a product of M elementary reflectors. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local output) COMPLEX*16, array, dimension LOCr(IA+M-1) This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MB_A * ( Mp0 + Nq0 + MB_A ), where IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ), and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. .SH FURTHER DETAILS The factorization is obtained by Householder's method. The kth transformation matrix, Z( k ), whose conjugate transpose is used to introduce zeros into the (m - k + 1)th row of sub( A ), is given in the form .br Z( k ) = ( I 0 ), .br ( 0 T( k ) ) .br where .br T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ), ( 0 ) ( z( k ) ) tau is a scalar and z( k ) is an ( n - m ) element vector. tau and z( k ) are chosen to annihilate the elements of the kth row of sub( A ). .br The scalar tau is returned in the kth element of TAU and the vector u( k ) in the kth row of sub( A ), such that the elements of z( k ) are in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in the upper triangular part of sub( A ). .br Z is given by .br Z = Z( 1 ) * Z( 2 ) * ... * Z( m ). .br scalapack-doc-1.5/man/manl/pzung2l.l0100644000056400000620000001405306335610664017031 0ustar pfrauenfstaff.TH PZUNG2L l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZUNG2L - generate an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the last N columns of a product of K elementary reflectors of order M Q = H(k) .SH SYNOPSIS .TP 20 SUBROUTINE PZUNG2L( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), WORK( * ) .SH PURPOSE PZUNG2L generates an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the last N columns of a product of K elementary reflectors of order M as returned by PZGEQLF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. M >= N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the j-th column must contain the vector which defines the elementary reflector H(j), JA+N-K <= j <= JA+N-1, as returned by PZGEQLF in the K columns of its distributed matrix argument A(IA:*,JA+N-K:JA+N-1). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCc(JA+N-1) This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PZGEQLF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MpA0 + MAX( 1, NqA0 ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pzung2r.l0100644000056400000620000001403606335610664017040 0ustar pfrauenfstaff.TH PZUNG2R l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZUNG2R - generate an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M Q = H(1) H(2) .SH SYNOPSIS .TP 20 SUBROUTINE PZUNG2R( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), WORK( * ) .SH PURPOSE PZUNG2R generates an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M as returned by PZGEQRF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. M >= N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the j-th column must contain the vector which defines the elementary reflector H(j), JA <= j <= JA+K-1, as returned by PZGEQRF in the K columns of its array argument A(IA:*,JA:JA+K-1). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCc(JA+K-1). This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PZGEQRF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MpA0 + MAX( 1, NqA0 ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pzungl2.l0100644000056400000620000001402506335610664017030 0ustar pfrauenfstaff.TH PZUNGL2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZUNGL2 - generate an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N Q = H(k)' .SH SYNOPSIS .TP 20 SUBROUTINE PZUNGL2( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), WORK( * ) .SH PURPOSE PZUNGL2 generates an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N as returned by PZGELQF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. N >= M >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PZGELQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCr(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PZGELQF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NqA0 + MAX( 1, MpA0 ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pzunglq.l0100644000056400000620000001403606335610664017131 0ustar pfrauenfstaff.TH PZUNGLQ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZUNGLQ - generate an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N Q = H(k)' .SH SYNOPSIS .TP 20 SUBROUTINE PZUNGLQ( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), WORK( * ) .SH PURPOSE PZUNGLQ generates an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N as returned by PZGELQF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. N >= M >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PZGELQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCr(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PZGELQF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MB_A * ( MpA0 + NqA0 + MB_A ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pzungql.l0100644000056400000620000001406406335610664017132 0ustar pfrauenfstaff.TH PZUNGQL l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZUNGQL - generate an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the last N columns of a product of K elementary reflectors of order M Q = H(k) .SH SYNOPSIS .TP 20 SUBROUTINE PZUNGQL( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), WORK( * ) .SH PURPOSE PZUNGQL generates an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the last N columns of a product of K elementary reflectors of order M as returned by PZGEQLF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. M >= N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the j-th column must contain the vector which defines the elementary reflector H(j), JA+N-K <= j <= JA+N-1, as returned by PZGEQLF in the K columns of its distributed matrix argument A(IA:*,JA+N-K:JA+N-1). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCc(JA+N-1) This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PZGEQLF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB_A * ( NqA0 + MpA0 + NB_A ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pzungqr.l0100644000056400000620000001406306335610664017137 0ustar pfrauenfstaff.TH PZUNGQR l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZUNGQR - generate an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M Q = H(1) H(2) .SH SYNOPSIS .TP 20 SUBROUTINE PZUNGQR( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), WORK( * ) .SH PURPOSE PZUNGQR generates an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M as returned by PZGEQRF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. M >= N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the j-th column must contain the vector which defines the elementary reflector H(j), JA <= j <= JA+K-1, as returned by PZGEQRF in the K columns of its distributed matrix argument A(IA:*,JA:JA+K-1). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCc(JA+K-1) This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PZGEQRF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NB_A * ( NqA0 + MpA0 + NB_A ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pzungr2.l0100644000056400000620000001404006335610664017033 0ustar pfrauenfstaff.TH PZUNGR2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZUNGR2 - generate an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N Q = H(1)' H(2)' .SH SYNOPSIS .TP 20 SUBROUTINE PZUNGR2( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), WORK( * ) .SH PURPOSE PZUNGR2 generates an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N as returned by PZGERQF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. N >= M >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA+M-K <= i <= IA+M-1, as returned by PZGERQF in the K rows of its distributed matrix argument A(IA+M-K:IA+M-1,JA:*). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCr(IA+M-1) This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PZGERQF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= NqA0 + MAX( 1, MpA0 ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pzungrq.l0100644000056400000620000001405106335610664017134 0ustar pfrauenfstaff.TH PZUNGRQ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZUNGRQ - generate an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N Q = H(1)' H(2)' .SH SYNOPSIS .TP 20 SUBROUTINE PZUNGRQ( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO ) .TP 20 .ti +4 INTEGER IA, INFO, JA, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), TAU( * ), WORK( * ) .SH PURPOSE PZUNGRQ generates an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N as returned by PZGERQF. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix Q. N >= M >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0. .TP 8 A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA+M-K <= i <= IA+M-1, as returned by PZGERQF in the K rows of its distributed matrix argument A(IA+M-K:IA+M-1,JA:*). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCr(IA+M-1) This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PZGERQF. TAU is tied to the distributed matrix A. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MB_A * ( MpA0 + NqA0 + MB_A ), where IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. scalapack-doc-1.5/man/manl/pzunm2l.l0100644000056400000620000001730206335610664017037 0ustar pfrauenfstaff.TH PZUNM2L l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZUNM2L - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PZUNM2L( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PZUNM2L overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C ) sub( C ) * Q**H .br where Q is a complex unitary distributed matrix defined as the product of K elementary reflectors .br Q = H(k) . . . H(2) H(1) .br as returned by PZGEQLF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+K-1)). On entry, the j-th column must contain the vector which defines the elemen- tary reflector H(j), JA <= j <= JA+K-1, as returned by PZGEQLF in the K columns of its distributed matrix argument A(IA:*,JA:JA+K-1). A(IA:*,JA:JA+K-1) is modified by the routine but restored on exit. If SIDE = 'L', LLD_A >= MAX( 1, LOCr(IA+M-1) ), if SIDE = 'R', LLD_A >= MAX( 1, LOCr(IA+N-1) ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCc(JA+N-1) This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PZGEQLF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MpC0 + MAX( 1, NqC0 ); if SIDE = 'R', LWORK >= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC( N+ICOFFC,NB_A,0,0,NPCOL ),NB_A,0,0,LCMQ ) ); where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/pzunm2r.l0100644000056400000620000001730406335610664017047 0ustar pfrauenfstaff.TH PZUNM2R l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZUNM2R - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PZUNM2R( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PZUNM2R overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C ) sub( C ) * Q**H .br where Q is a complex unitary distributed matrix defined as the product of k elementary reflectors .br Q = H(1) H(2) . . . H(k) .br as returned by PZGEQRF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+K-1)). On entry, the j-th column must contain the vector which defines the elemen- tary reflector H(j), JA <= j <= JA+K-1, as returned by PZGEQRF in the K columns of its distributed matrix argument A(IA:*,JA:JA+K-1). A(IA:*,JA:JA+K-1) is modified by the routine but restored on exit. If SIDE = 'L', LLD_A >= MAX( 1, LOCr(IA+M-1) ); if SIDE = 'R', LLD_A >= MAX( 1, LOCr(IA+N-1) ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCc(JA+K-1). This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PZGEQRF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MpC0 + MAX( 1, NqC0 ); if SIDE = 'R', LWORK >= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC( N+ICOFFC,NB_A,0,0,NPCOL ),NB_A,0,0,LCMQ ) ); where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/pzunmbr.l0100644000056400000620000002404106335610664017123 0ustar pfrauenfstaff.TH PZUNMBR l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZUNMBR - VECT = 'Q', PZUNMBR overwrites the general complex distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PZUNMBR( VECT, SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS, VECT .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE If VECT = 'Q', PZUNMBR overwrites the general complex distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C ) sub( C ) * Q**H .br If VECT = 'P', PZUNMBR overwrites sub( C ) with .br SIDE = 'L' SIDE = 'R' .br TRANS = 'N': P * sub( C ) sub( C ) * P .br TRANS = 'C': P**H * sub( C ) sub( C ) * P**H .br Here Q and P**H are the unitary distributed matrices determined by PZGEBRD when reducing a complex distributed matrix A(IA:*,JA:*) to bidiagonal form: A(IA:*,JA:*) = Q * B * P**H. Q and P**H are defined as products of elementary reflectors H(i) and G(i) respectively. Let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the order of the unitary matrix Q or P**H that is applied. .br If VECT = 'Q', A(IA:*,JA:*) is assumed to have been an NQ-by-K matrix: .br if nq >= k, Q = H(1) H(2) . . . H(k); .br if nq < k, Q = H(1) H(2) . . . H(nq-1). .br If VECT = 'P', A(IA:*,JA:*) is assumed to have been a K-by-NQ matrix: .br if k < nq, P = G(1) G(2) . . . G(k); .br if k >= nq, P = G(1) G(2) . . . G(nq-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 VECT (global input) CHARACTER = 'Q': apply Q or Q**H; .br = 'P': apply P or P**H. .TP 8 SIDE (global input) CHARACTER .br = 'L': apply Q, Q**H, P or P**H from the Left; .br = 'R': apply Q, Q**H, P or P**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q or P; .br = 'C': Conjugate transpose, apply Q**H or P**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER If VECT = 'Q', the number of columns in the original distributed matrix reduced by PZGEBRD. If VECT = 'P', the number of rows in the original distributed matrix reduced by PZGEBRD. K >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+MIN(NQ,K)-1)) if VECT='Q', and (LLD_A,LOCc(JA+NQ-1)) if VECT = 'P'. NQ = M if SIDE = 'L', and NQ = N otherwise. The vectors which define the elementary reflectors H(i) and G(i), whose products determine the matrices Q and P, as returned by PZGEBRD. If VECT = 'Q', LLD_A >= max(1,LOCr(IA+NQ-1)); if VECT = 'P', LLD_A >= max(1,LOCr(IA+MIN(NQ,K)-1)). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX*16 array, dimension LOCc(JA+MIN(NQ,K)-1) if VECT = 'Q', LOCr(IA+MIN(NQ,K)-1) if VECT = 'P', TAU(i) must contain the scalar factor of the elementary reflector H(i) or G(i), which determines Q or P, as returned by PDGEBRD in its array argument TAUQ or TAUP. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, if VECT='Q', sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q; if VECT='P, sub( C ) is overwritten by P*sub( C ) or P'*sub( C ) or sub( C )*P or sub( C )*P'. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', NQ = M; if( (VECT = 'Q' and NQ >= K) or (VECT <> 'Q' and NQ > K) ), IAA=IA; JAA=JA; MI=M; NI=N; ICC=IC; JCC=JC; else IAA=IA+1; JAA=JA; MI=M-1; NI=N; ICC=IC+1; JCC=JC; end if else if SIDE = 'R', NQ = N; if( (VECT = 'Q' and NQ >= K) or (VECT <> 'Q' and NQ > K) ), IAA=IA; JAA=JA; MI=M; NI=N; ICC=IC; JCC=JC; else IAA=IA; JAA=JA+1; MI=M; NI=N-1; ICC=IC; JCC=JC+1; end if end if If VECT = 'Q', If SIDE = 'L', LWORK >= MAX( (NB_A*(NB_A-1))/2, (NqC0 + MpC0)*NB_A ) + NB_A * NB_A else if SIDE = 'R', LWORK >= MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX( NpA0 + NUMROC( NUMROC( NI+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0 ) )*NB_A ) + NB_A * NB_A end if else if VECT <> 'Q', if SIDE = 'L', LWORK >= MAX( (MB_A*(MB_A-1))/2, ( MpC0 + MAX( MqA0 + NUMROC( NUMROC( MI+IROFFC, MB_A, 0, 0, NPROW ), MB_A, 0, 0, LCMP ), NqC0 ) )*MB_A ) + MB_A * MB_A else if SIDE = 'R', LWORK >= MAX( (MB_A*(MB_A-1))/2, (MpC0 + NqC0)*MB_A ) + MB_A * MB_A end if end if where LCMP = LCM / NPROW, LCMQ = LCM / NPCOL, with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IAA-1, MB_A ), ICOFFA = MOD( JAA-1, NB_A ), IAROW = INDXG2P( IAA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JAA, NB_A, MYCOL, CSRC_A, NPCOL ), MqA0 = NUMROC( MI+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), NpA0 = NUMROC( NI+IROFFA, MB_A, MYROW, IAROW, NPROW ), IROFFC = MOD( ICC-1, MB_C ), ICOFFC = MOD( JCC-1, NB_C ), ICROW = INDXG2P( ICC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JCC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( MI+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( NI+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If VECT = 'Q', If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) else If SIDE = 'L', ( MB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) end if scalapack-doc-1.5/man/manl/pzunmhr.l0100644000056400000620000002016406335610665017134 0ustar pfrauenfstaff.TH PZUNMHR l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZUNMHR - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PZUNMHR( SIDE, TRANS, M, N, ILO, IHI, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, IHI, ILO, INFO, JA, JC, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PZUNMHR overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C ) sub( C ) * Q**H .br where Q is a complex unitary distributed matrix of order nq, with nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of IHI-ILO elementary reflectors, as returned by PZGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 ILO (global input) INTEGER IHI (global input) INTEGER ILO and IHI must have the same values as in the previous call of PZGEHRD. Q is equal to the unit matrix except in the distributed submatrix Q(ia+ilo:ia+ihi-1,ia+ilo:ja+ihi-1). If SIDE = 'L', 1 <= ILO <= IHI <= max(1,M); if SIDE = 'R', 1 <= ILO <= IHI <= max(1,N); ILO and IHI are relative indexes. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE = 'R'. The vectors which define the elementary reflectors, as returned by PZGEHRD. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCc(JA+M-2) if SIDE = 'L', and LOCc(JA+N-2) if SIDE = 'R'. This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PZGEHRD. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least IAA = IA + ILO; JAA = JA+ILO-1; If SIDE = 'L', MI = IHI-ILO; NI = N; ICC = IC + ILO; JCC = JC; LWORK >= MAX( (NB_A*(NB_A-1))/2, (NqC0 + MpC0)*NB_A ) + NB_A * NB_A else if SIDE = 'R', MI = M; NI = IHI-ILO; ICC = IC; JCC = JC + ILO; LWORK >= MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX( NpA0 + NUMROC( NUMROC( NI+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0 ) )*NB_A ) + NB_A * NB_A end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IAA-1, MB_A ), ICOFFA = MOD( JAA-1, NB_A ), IAROW = INDXG2P( IAA, MB_A, MYROW, RSRC_A, NPROW ), NpA0 = NUMROC( NI+IROFFA, MB_A, MYROW, IAROW, NPROW ), IROFFC = MOD( ICC-1, MB_C ), ICOFFC = MOD( JCC-1, NB_C ), ICROW = INDXG2P( ICC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JCC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( MI+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( NI+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/pzunml2.l0100644000056400000620000001727006335610665017044 0ustar pfrauenfstaff.TH PZUNML2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZUNML2 - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PZUNML2( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PZUNML2 overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C sub( C ) * Q**H .br where Q is a complex unitary distributed matrix defined as the product of K elementary reflectors .br Q = H(k)' . . . H(2)' H(1)' .br as returned by PZGELQF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= max(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PZGELQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PZGELQF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_A,0,0,NPROW ),MB_A,0,0,LCMP ) ); if SIDE = 'R', LWORK >= NqC0 + MAX( 1, MpC0 ); where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/pzunmlq.l0100644000056400000620000001770006335610665017141 0ustar pfrauenfstaff.TH PZUNMLQ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZUNMLQ - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PZUNMLQ( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PZUNMLQ overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C sub( C ) * Q**H .br where Q is a complex unitary distributed matrix defined as the product of K elementary reflectors .br Q = H(k)' . . . H(2)' H(1)' .br as returned by PZGELQF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= max(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PZGELQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PZGELQF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least if SIDE = 'L', LWORK >= MAX( (MB_A*(MB_A-1))/2, ( MpC0 + MAX( MqA0 + NUMROC( NUMROC( M+IROFFC, MB_A, 0, 0, NPROW ), MB_A, 0, 0, LCMP ), NqC0 ) )*MB_A ) + MB_A * MB_A else if SIDE = 'R', LWORK >= MAX( (MB_A*(MB_A-1))/2, (MpC0 + NqC0)*MB_A ) + MB_A * MB_A end if where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MqA0 = NUMROC( M+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/pzunmql.l0100644000056400000620000001771206335610665017144 0ustar pfrauenfstaff.TH PZUNMQL l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZUNMQL - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PZUNMQL( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PZUNMQL overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C ) sub( C ) * Q**H .br where Q is a complex unitary distributed matrix defined as the product of K elementary reflectors .br Q = H(k) . . . H(2) H(1) .br as returned by PZGEQLF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+K-1)). On entry, the j-th column must contain the vector which defines the elemen- tary reflector H(j), JA <= j <= JA+K-1, as returned by PZGEQLF in the K columns of its distributed matrix argument A(IA:*,JA:JA+K-1). A(IA:*,JA:JA+K-1) is modified by the routine but restored on exit. If SIDE = 'L', LLD_A >= MAX( 1, LOCr(IA+M-1) ), if SIDE = 'R', LLD_A >= MAX( 1, LOCr(IA+N-1) ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCc(JA+N-1) This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PZGEQLF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MAX( (NB_A*(NB_A-1))/2, (NqC0 + MpC0)*NB_A ) + NB_A * NB_A else if SIDE = 'R', LWORK >= MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX( NpA0 + NUMROC( NUMROC( N+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0 ) )*NB_A ) + NB_A * NB_A end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), NpA0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/pzunmqr.l0100644000056400000620000001771406335610665017154 0ustar pfrauenfstaff.TH PZUNMQR l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZUNMQR - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PZUNMQR( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PZUNMQR overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C ) sub( C ) * Q**H .br where Q is a complex unitary distributed matrix defined as the product of k elementary reflectors .br Q = H(1) H(2) . . . H(k) .br as returned by PZGEQRF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+K-1)). On entry, the j-th column must contain the vector which defines the elemen- tary reflector H(j), JA <= j <= JA+K-1, as returned by PZGEQRF in the K columns of its distributed matrix argument A(IA:*,JA:JA+K-1). A(IA:*,JA:JA+K-1) is modified by the routine but restored on exit. If SIDE = 'L', LLD_A >= MAX( 1, LOCr(IA+M-1) ); if SIDE = 'R', LLD_A >= MAX( 1, LOCr(IA+N-1) ). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCc(JA+K-1). This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PZGEQRF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MAX( (NB_A*(NB_A-1))/2, (NqC0 + MpC0)*NB_A ) + NB_A * NB_A else if SIDE = 'R', LWORK >= MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX( NpA0 + NUMROC( NUMROC( N+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0 ) )*NB_A ) + NB_A * NB_A end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), NpA0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/pzunmr2.l0100644000056400000620000001727006335610665017052 0ustar pfrauenfstaff.TH PZUNMR2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZUNMR2 - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PZUNMR2( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PZUNMR2 overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C ) sub( C ) * Q**H .br where Q is a complex unitary distributed matrix defined as the product of K elementary reflectors .br Q = H(1)' H(2)' . . . H(k)' .br as returned by PZGERQF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= MAX(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PZGERQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PZGERQF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_A,0,0,NPROW ),MB_A,0,0,LCMP ) ); if SIDE = 'R', LWORK >= NqC0 + MAX( 1, MpC0 ); where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/pzunmr3.l0100644000056400000620000001761306335610665017054 0ustar pfrauenfstaff.TH PZUNMR3 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZUNMR3 - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PZUNMR3( SIDE, TRANS, M, N, K, L, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, L, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PZUNMR3 overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C ) sub( C ) * Q**H .br where Q is a complex unitary distributed matrix defined as the product of K elementary reflectors .br Q = H(1)' H(2)' . . . H(k)' .br as returned by PZTZRZF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 L (global input) INTEGER The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= MAX(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PZTZRZF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PZTZRZF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_A,0,0,NPROW ),MB_A,0,0,LCMP ) ); if SIDE = 'R', LWORK >= NqC0 + MAX( 1, MpC0 ); where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (local output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/pzunmrq.l0100644000056400000620000001770106335610665017150 0ustar pfrauenfstaff.TH PZUNMRQ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZUNMRQ - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PZUNMRQ( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PZUNMRQ overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C ) sub( C ) * Q**H .br where Q is a complex unitary distributed matrix defined as the product of K elementary reflectors .br Q = H(1)' H(2)' . . . H(k)' .br as returned by PZGERQF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= MAX(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PZGERQF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PZGERQF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least if SIDE = 'L', LWORK >= MAX( (MB_A*(MB_A-1))/2, ( MpC0 + MAX( MqA0 + NUMROC( NUMROC( M+IROFFC, MB_A, 0, 0, NPROW ), MB_A, 0, 0, LCMP ), NqC0 ) )*MB_A ) + MB_A * MB_A else if SIDE = 'R', LWORK >= MAX( (MB_A*(MB_A-1))/2, (MpC0 + NqC0)*MB_A ) + MB_A * MB_A end if where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MqA0 = NUMROC( M+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/pzunmrz.l0100644000056400000620000002022406335610665017153 0ustar pfrauenfstaff.TH PZUNMRZ l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZUNMRZ - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PZUNMRZ( SIDE, TRANS, M, N, K, L, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, K, L, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PZUNMRZ overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C ) sub( C ) * Q**H .br where Q is a complex unitary distributed matrix defined as the product of K elementary reflectors .br Q = H(1)' H(2)' . . . H(k)' .br as returned by PZTZRZF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 TRANS (global input) CHARACTER .br = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 K (global input) INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0, if SIDE = 'R', N >= K >= 0. .TP 8 L (global input) INTEGER The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= MAX(1,LOCr(IA+K-1)); On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PZTZRZF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). .br A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit. .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX*16, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PZTZRZF. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least if SIDE = 'L', LWORK >= MAX( (MB_A*(MB_A-1))/2, ( MpC0 + MAX( MqA0 + NUMROC( NUMROC( M+IROFFC, MB_A, 0, 0, NPROW ), MB_A, 0, 0, LCMP ), NqC0 ) )*MB_A ) + MB_A * MB_A else if SIDE = 'R', LWORK >= MAX( (MB_A*(MB_A-1))/2, (MpC0 + NqC0)*MB_A ) + MB_A * MB_A end if where LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), MqA0 = NUMROC( M+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) scalapack-doc-1.5/man/manl/pzunmtr.l0100644000056400000620000002047306335610665017153 0ustar pfrauenfstaff.TH PZUNMTR l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME PZUNMTR - overwrite the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N' .SH SYNOPSIS .TP 20 SUBROUTINE PZUNMTR( SIDE, UPLO, TRANS, M, N, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO ) .TP 20 .ti +4 CHARACTER SIDE, TRANS, UPLO .TP 20 .ti +4 INTEGER IA, IC, INFO, JA, JC, LWORK, M, N .TP 20 .ti +4 INTEGER DESCA( * ), DESCC( * ) .TP 20 .ti +4 COMPLEX*16 A( * ), C( * ), TAU( * ), WORK( * ) .SH PURPOSE PZUNMTR overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'C': Q**H * sub( C ) sub( C ) * Q**H .br where Q is a complex unitary distributed matrix of order nq, with nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of nq-1 elementary reflectors, as returned by PZHETRD: if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1); .br if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1). .br Notes .br ===== .br Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. .br Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array". .br NOTATION STORED IN EXPLANATION .br --------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1. .br CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary. .br M_A (global) DESCA( M_ ) The number of rows in the global array A. .br N_A (global) DESCA( N_ ) The number of columns in the global array A. .br MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array. .br NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array. .br RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed. .br LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)). Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. .br LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column. .br Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row. .br The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC: .br LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by: .br LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A .br LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A .br .SH ARGUMENTS .TP 8 SIDE (global input) CHARACTER = 'L': apply Q or Q**H from the Left; .br = 'R': apply Q or Q**H from the Right. .TP 8 UPLO (global input) CHARACTER .br = 'U': Upper triangle of A(IA:*,JA:*) contains elementary reflectors from PZHETRD; = 'L': Lower triangle of A(IA:*,JA:*) contains elementary reflectors from PZHETRD. .TP 8 TRANS (global input) CHARACTER = 'N': No transpose, apply Q; .br = 'C': Conjugate transpose, apply Q**H. .TP 8 M (global input) INTEGER The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0. .TP 8 N (global input) INTEGER The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0. .TP 8 A (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', or (LLD_A,LOCc(JA+N-1)) if SIDE = 'R'. The vectors which define the elementary reflectors, as returned by PZHETRD. If SIDE = 'L', LLD_A >= max(1,LOCr(IA+M-1)); if SIDE = 'R', LLD_A >= max(1,LOCr(IA+N-1)). .TP 8 IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). .TP 8 JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). .TP 8 DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. .TP 8 TAU (local input) COMPLEX*16 array, dimension LTAU, where if SIDE = 'L' and UPLO = 'U', LTAU = LOCc(M_A), if SIDE = 'L' and UPLO = 'L', LTAU = LOCc(JA+M-2), if SIDE = 'R' and UPLO = 'U', LTAU = LOCc(N_A), if SIDE = 'R' and UPLO = 'L', LTAU = LOCc(JA+N-2). TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by PZHETRD. TAU is tied to the distributed matrix A. .TP 8 C (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q. .TP 8 IC (global input) INTEGER The row index in the global array C indicating the first row of sub( C ). .TP 8 JC (global input) INTEGER The column index in the global array C indicating the first column of sub( C ). .TP 8 DESCC (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C. .TP 8 WORK (local workspace/local output) COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK. .TP 8 LWORK (local or global input) INTEGER The dimension of the array WORK. LWORK is local input and must be at least If UPLO = 'U', IAA = IA, JAA = JA+1, ICC = IC, JCC = JC; else UPLO = 'L', IAA = IA+1, JAA = JA; if SIDE = 'L', ICC = IC+1; JCC = JC; else ICC = IC; JCC = JC+1; end if end if If SIDE = 'L', MI = M-1; NI = N; LWORK >= MAX( (NB_A*(NB_A-1))/2, (NqC0 + MpC0)*NB_A ) + NB_A * NB_A else if SIDE = 'R', MI = M; MI = N-1; LWORK >= MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX( NpA0 + NUMROC( NUMROC( NI+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0 ) )*NB_A ) + NB_A * NB_A end if where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ), IROFFA = MOD( IAA-1, MB_A ), ICOFFA = MOD( JAA-1, NB_A ), IAROW = INDXG2P( IAA, MB_A, MYROW, RSRC_A, NPROW ), NpA0 = NUMROC( NI+IROFFA, MB_A, MYROW, IAROW, NPROW ), IROFFC = MOD( ICC-1, MB_C ), ICOFFC = MOD( JCC-1, NB_C ), ICROW = INDXG2P( ICC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JCC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( MI+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( NI+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ), ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. .TP 8 INFO (global output) INTEGER = 0: successful exit .br < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. Alignment requirements ====================== The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true: If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) scalapack-doc-1.5/man/manl/sdbtf2.l0100644000056400000620000000463306335610665016620 0ustar pfrauenfstaff.SH NAME SDBTF2 - compute an LU factorization of a real m-by-n band matrix A without using partial pivoting with row interchanges .SH SYNOPSIS .TP 19 SUBROUTINE SDBTF2( M, N, KL, KU, AB, LDAB, INFO ) .TP 19 .ti +4 INTEGER INFO, KL, KU, LDAB, M, N .TP 19 .ti +4 REAL AB( LDAB, * ) .SH PURPOSE Sdbtrf computes an LU factorization of a real m-by-n band matrix A without using partial pivoting with row interchanges. This is the unblocked version of the algorithm, calling Level 2 BLAS. .SH ARGUMENTS .TP 8 M (input) INTEGER The number of rows of the matrix A. M >= 0. .TP 8 N (input) INTEGER The number of columns of the matrix A. N >= 0. .TP 8 KL (input) INTEGER The number of subdiagonals within the band of A. KL >= 0. .TP 8 KU (input) INTEGER The number of superdiagonals within the band of A. KU >= 0. .TP 8 AB (input/output) REAL array, dimension (LDAB,N) On entry, the matrix A in band storage, in rows KL+1 to 2*KL+KU+1; rows 1 to KL of the array need not be set. The j-th column of A is stored in the j-th column of the array AB as follows: AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl) On exit, details of the factorization: U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. See below for further details. .TP 8 LDAB (input) INTEGER The leading dimension of the array AB. LDAB >= 2*KL+KU+1. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 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. .SH FURTHER DETAILS The band storage scheme is illustrated by the following example, when M = N = 6, KL = 2, KU = 1: .br On entry: On exit: .br * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 * a31 a42 a53 a64 * * m31 m42 m53 m64 * * Array elements marked * are not used by the routine; elements marked + need not be set on entry, but are required by the routine to store elements of U, because of fill-in resulting from the row .br interchanges. .br scalapack-doc-1.5/man/manl/sdbtrf.l0100644000056400000620000000434706335610665016722 0ustar pfrauenfstaff.SH NAME SDBTRF - compute an LU factorization of a real m-by-n band matrix A without using partial pivoting or row interchanges .SH SYNOPSIS .TP 19 SUBROUTINE SDBTRF( M, N, KL, KU, AB, LDAB, INFO ) .TP 19 .ti +4 INTEGER INFO, KL, KU, LDAB, M, N .TP 19 .ti +4 REAL AB( LDAB, * ) .SH PURPOSE Sdbtrf computes an LU factorization of a real m-by-n band matrix A without using partial pivoting or row interchanges. This is the blocked version of the algorithm, calling Level 3 BLAS. .SH ARGUMENTS .TP 8 M (input) INTEGER The number of rows of the matrix A. M >= 0. .TP 8 N (input) INTEGER The number of columns of the matrix A. N >= 0. .TP 8 KL (input) INTEGER The number of subdiagonals within the band of A. KL >= 0. .TP 8 KU (input) INTEGER The number of superdiagonals within the band of A. KU >= 0. .TP 8 AB (input/output) REAL array, dimension (LDAB,N) On entry, the matrix A in band storage, in rows KL+1 to 2*KL+KU+1; rows 1 to KL of the array need not be set. The j-th column of A is stored in the j-th column of the array AB as follows: AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl) On exit, details of the factorization: U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. See below for further details. .TP 8 LDAB (input) INTEGER The leading dimension of the array AB. LDAB >= 2*KL+KU+1. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 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. .SH FURTHER DETAILS The band storage scheme is illustrated by the following example, when M = N = 6, KL = 2, KU = 1: .br On entry: On exit: .br * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 * a31 a42 a53 a64 * * m31 m42 m53 m64 * * Array elements marked * are not used by the routine. .br scalapack-doc-1.5/man/manl/sdttrf.l0100644000056400000620000000331406335610665016735 0ustar pfrauenfstaff.TH SDTTRF l "12 May 1997" "modified LAPACK routine" "LAPACK routine (version 2.0)" .SH NAME SDTTRF - compute an LU factorization of a complex tridiagonal matrix A using elimination without partial pivoting .SH SYNOPSIS .TP 19 SUBROUTINE SDTTRF( N, DL, D, DU, INFO ) .TP 19 .ti +4 INTEGER INFO, N .TP 19 .ti +4 REAL D( * ), DL( * ), DU( * ) .SH PURPOSE SDTTRF computes an LU factorization of a complex tridiagonal matrix A using elimination without partial pivoting. The factorization has the form .br A = L * U .br where L is a product of unit lower bidiagonal .br matrices and U is upper triangular with nonzeros in only the main diagonal and first superdiagonal. .br .SH ARGUMENTS .TP 8 N (input) INTEGER The order of the matrix A. N >= 0. .TP 8 DL (input/output) COMPLEX array, dimension (N-1) On entry, DL must contain the (n-1) subdiagonal elements of A. On exit, DL is overwritten by the (n-1) multipliers that define the matrix L from the LU factorization of A. .TP 8 D (input/output) COMPLEX array, dimension (N) On entry, D must contain the diagonal elements of A. On exit, D is overwritten by the n diagonal elements of the upper triangular matrix U from the LU factorization of A. .TP 8 DU (input/output) COMPLEX array, dimension (N-1) On entry, DU must contain the (n-1) superdiagonal elements of A. On exit, DU is overwritten by the (n-1) elements of the first superdiagonal of U. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 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. scalapack-doc-1.5/man/manl/sdttrsv.l0100644000056400000620000000352106335610666017141 0ustar pfrauenfstaff.TH SDTTRSV l "12 May 1997" "modified LAPACK routine" "LAPACK routine (version 2.0)" .SH NAME SDTTRSV - solve one of the systems of equations L * X = B, L**T * X = B, or L**H * X = B, .SH SYNOPSIS .TP 20 SUBROUTINE SDTTRSV( UPLO, TRANS, N, NRHS, DL, D, DU, B, LDB, INFO ) .TP 20 .ti +4 CHARACTER UPLO, TRANS .TP 20 .ti +4 INTEGER INFO, LDB, N, NRHS .TP 20 .ti +4 REAL B( LDB, * ), D( * ), DL( * ), DU( * ) .SH PURPOSE SDTTRSV solves one of the systems of equations L * X = B, L**T * X = B, or L**H * X = B, U * X = B, U**T * X = B, or U**H * X = B, .br with factors of the tridiagonal matrix A from the LU factorization computed by SDTTRF. .br .SH ARGUMENTS .TP 8 UPLO (input) CHARACTER*1 Specifies whether to solve with L or U. .TP 8 TRANS (input) CHARACTER Specifies the form of the system of equations: .br = 'N': A * X = B (No transpose) .br = 'T': A**T * X = B (Transpose) .br = 'C': A**H * X = B (Conjugate transpose) .TP 8 N (input) INTEGER The order of the matrix A. N >= 0. .TP 8 NRHS (input) INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. .TP 8 DL (input) COMPLEX array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A. .TP 8 D (input) COMPLEX array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A. .TP 8 DU (input) COMPLEX array, dimension (N-1) The (n-1) elements of the first superdiagonal of U. .TP 8 B (input/output) COMPLEX array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, B is overwritten by the solution matrix X. .TP 8 LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value scalapack-doc-1.5/man/manl/slamsh.l0100644000056400000620000000444406335610666016724 0ustar pfrauenfstaff.TH SLAMSH l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME SLAMSH - send multiple shifts through a small (single node) matrix to see how consecutive small subdiagonal elements are modified by subsequent shifts in an effort to maximize the number of bulges that can be sent through .SH SYNOPSIS .TP 18 SUBROUTINE SLAMSH ( S, LDS, NBULGE, JBLK, H, LDH, N, ULP ) .TP 18 .ti +4 INTEGER LDS, NBULGE, JBLK, LDH, N .TP 18 .ti +4 REAL ULP .TP 18 .ti +4 REAL S(LDS,*), H(LDH,*) .SH PURPOSE SLAMSH sends multiple shifts through a small (single node) matrix to see how consecutive small subdiagonal elements are modified by subsequent shifts in an effort to maximize the number of bulges that can be sent through. SLAMSH should only be called when there are multiple shifts/bulges (NBULGE > 1) and the first shift is starting in the middle of an unreduced Hessenberg matrix because of two or more consecutive small subdiagonal elements. .br .SH ARGUMENTS .TP 8 S (local input/output) REAL array, (LDS,*) On entry, the matrix of shifts. Only the 2x2 diagonal of S is referenced. It is assumed that S has JBLK double shifts (size 2). On exit, the data is rearranged in the best order for applying. .TP 8 LDS (local input) INTEGER On entry, the leading dimension of S. Unchanged on exit. 1 < NBULGE <= JBLK <= LDS/2 .TP 8 NBULGE (local input/output) INTEGER On entry, the number of bulges to send through H ( >1 ). NBULGE should be less than the maximum determined (JBLK). 1 < NBULGE <= JBLK <= LDS/2 On exit, the maximum number of bulges that can be sent through. .TP 8 JBLK (local input) INTEGER On entry, the number of shifts determined for S. Unchanged on exit. .TP 8 H (local input/output) REAL array (LDH,N) On entry, the local matrix to apply the shifts on. H should be aligned so that the starting row is 2. On exit, the data is destroyed. .TP 8 LDS (local input) INTEGER On entry, the leading dimension of S. Unchanged on exit. .TP 8 N (local input) INTEGER On entry, the size of H. If all the bulges are expected to go through, N should be at least 4*NBULGE+2. Otherwise, NBULGE may be reduced by this routine. .TP 8 ULP (local input) REAL On entry, machine precision Unchanged on exit. Implemented by: G. Henry, May 1, 1997 scalapack-doc-1.5/man/manl/slaref.l0100644000056400000620000000615306335610666016710 0ustar pfrauenfstaff.TH SLAREF l "12 May 1997" "LAPACK version 1.5" "LAPACK auxiliary routine (version 1.5)" .SH NAME SLAREF - applie one or several Householder reflectors of size 3 to one or two matrices (if column is specified) on either their rows or columns .SH SYNOPSIS .TP 19 SUBROUTINE SLAREF( TYPE, A, LDA, WANTZ, Z, LDZ, BLOCK, IROW1, ICOL1, ISTART, ISTOP, ITMP1, ITMP2, LILOZ, LIHIZ, VECS, V2, V3, T1, T2, T3 ) .TP 19 .ti +4 LOGICAL BLOCK, WANTZ .TP 19 .ti +4 CHARACTER TYPE .TP 19 .ti +4 INTEGER ICOL1, IROW1, ISTART, ISTOP, ITMP1, ITMP2, LDA, LDZ, LIHIZ, LILOZ .TP 19 .ti +4 REAL T1, T2, T3, V2, V3 .TP 19 .ti +4 REAL A( LDA, * ), VECS( * ), Z( LDZ, * ) .SH PURPOSE SLAREF applies one or several Householder reflectors of size 3 to one or two matrices (if column is specified) on either their rows or columns. .SH ARGUMENTS .TP 8 TYPE (global input) CHARACTER*1 If 'R': Apply reflectors to the rows of the matrix (apply from left) Otherwise: Apply reflectors to the columns of the matrix Unchanged on exit. .TP 8 A (global input/output) REAL array, (LDA,*) On entry, the matrix to receive the reflections. The updated matrix on exit. .TP 8 LDA (local input) INTEGER On entry, the leading dimension of A. Unchanged on exit. .TP 8 WANTZ (global input) LOGICAL If .TRUE., then apply any column reflections to Z as well. If .FALSE., then do no additional work on Z. .TP 8 Z (global input/output) REAL array, (LDZ,*) On entry, the second matrix to receive column reflections. This is changed only if WANTZ is set. .TP 8 LDZ (local input) INTEGER On entry, the leading dimension of Z. Unchanged on exit. .TP 8 BLOCK (global input) LOGICAL If .TRUE., then apply several reflectors at once and read their data from the VECS array. If .FALSE., apply the single reflector given by V2, V3, T1, T2, and T3. .TP 8 IROW1 (local input/output) INTEGER On entry, the local row element of A. Undefined on output. .TP 8 ICOL1 (local input/output) INTEGER On entry, the local column element of A. Undefined on output. .TP 8 ISTART (global input) INTEGER Specifies the "number" of the first reflector. This is used as an index into VECS if BLOCK is set. ISTART is ignored if BLOCK is .FALSE.. .TP 8 ISTOP (global input) INTEGER Specifies the "number" of the last reflector. This is used as an index into VECS if BLOCK is set. ISTOP is ignored if BLOCK is .FALSE.. .TP 8 ITMP1 (local input) INTEGER Starting range into A. For rows, this is the local first column. For columns, this is the local first row. .TP 8 ITMP2 (local input) INTEGER Ending range into A. For rows, this is the local last column. For columns, this is the local last row. LILOZ LIHIZ (local input) INTEGER These serve the same purpose as ITMP1,ITMP2 but for Z when WANTZ is set. .TP 8 VECS (global input) REAL array of size 3*N (matrix size) This holds the size 3 reflectors one after another and this is only accessed when BLOCK is .TRUE. V2 V3 T1 T2 T3 (global input/output) REAL This holds information on a single size 3 Householder reflector and is read when BLOCK is .FALSE., and overwritten when BLOCK is .TRUE. Implemented by: G. Henry, May 1, 1997 scalapack-doc-1.5/man/manl/slasorte.l0100644000056400000620000000262106335610666017264 0ustar pfrauenfstaff.TH SLASORTE l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME SLASORTE - sort eigenpairs so that real eigenpairs are together and complex are together .SH SYNOPSIS .TP 20 SUBROUTINE SLASORTE ( S, LDS, J, OUT, INFO ) .TP 20 .ti +4 INTEGER INFO, J, LDS .TP 20 .ti +4 REAL OUT( J, * ), S( LDS, * ) .SH PURPOSE SLASORTE sorts eigenpairs so that real eigenpairs are together and complex are together. This way one can employ 2x2 shifts easily since every 2nd subdiagonal is guaranteed to be zero. .br This routine does no parallel work and makes no calls. .br .SH ARGUMENTS .TP 8 S (local input/output) REAL array, dimension LDS On entry, a matrix already in Schur form. On exit, the diagonal blocks of S have been rewritten to pair the eigenvalues. The resulting matrix is no longer similar to the input. .TP 8 LDS (local input) INTEGER On entry, the leading dimension of the local array S. Unchanged on exit. .TP 8 J (local input) INTEGER On entry, the order of the matrix S. Unchanged on exit. .TP 8 OUT (local input/output) REAL array, dimension Jx2 This is the work buffer required by this routine. .TP 8 INFO (local input) INTEGER This is set if the input matrix had an odd number of real eigenvalues and things couldn't be paired or if the input matrix S was not originally in Schur form. 0 indicates successful completion. Implemented by: G. Henry, May 1, 1997 scalapack-doc-1.5/man/manl/slasrt2.l0100644000056400000620000000263206335610666017024 0ustar pfrauenfstaff.TH SLASRT2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME SLASRT2 - the numbers in D in increasing order (if ID = 'I') or in decreasing order (if ID = 'D' ) .SH SYNOPSIS .TP 20 SUBROUTINE SLASRT2( ID, N, D, KEY, INFO ) .TP 20 .ti +4 CHARACTER ID .TP 20 .ti +4 INTEGER INFO, N .TP 20 .ti +4 INTEGER KEY( * ) .TP 20 .ti +4 REAL D( * ) .SH PURPOSE Sort the numbers in D in increasing order (if ID = 'I') or in decreasing order (if ID = 'D' ). Use Quick Sort, reverting to Insertion sort on arrays of .br size <= 20. Dimension of STACK limits N to about 2**32. .br .SH ARGUMENTS .TP 8 ID (input) CHARACTER*1 = 'I': sort D in increasing order; .br = 'D': sort D in decreasing order. .TP 8 N (input) INTEGER The length of the array D. .TP 8 D (input/output) REAL array, dimension (N) On entry, the array to be sorted. On exit, D has been sorted into increasing order (D(1) <= ... <= D(N) ) or into decreasing order (D(1) >= ... >= D(N) ), depending on ID. .TP 8 KEY (input/output) INTEGER array, dimension (N) On entry, KEY contains a key to each of the entries in D() Typically, KEY(I) = I for all I On exit, KEY is permuted in exactly the same manner as D() was permuted from input to output Therefore, if KEY(I) = I for all I upon input, then D_out(I) = D_in(KEY(I)) .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value scalapack-doc-1.5/man/manl/spttrsv.l0100644000056400000620000000314406335610666017156 0ustar pfrauenfstaff.TH SPTTRSV l "12 May 1997" "modified LAPACK routine" "LAPACK routine (version 2.0)" .SH NAME SPTTRSV - solve one of the triangular systems L**T* X = B, or L * X = B, .SH SYNOPSIS .TP 20 SUBROUTINE SPTTRSV( TRANS, N, NRHS, D, E, B, LDB, INFO ) .TP 20 .ti +4 CHARACTER TRANS .TP 20 .ti +4 INTEGER INFO, LDB, N, NRHS .TP 20 .ti +4 REAL D( * ) .TP 20 .ti +4 REAL B( LDB, * ), E( * ) .SH PURPOSE SPTTRSV solves one of the triangular systems L**T* X = B, or L * X = B, where L is the Cholesky factor of a Hermitian positive .br definite tridiagonal matrix A such that .br A = L*D*L**H (computed by SPTTRF). .br .SH ARGUMENTS .TP 8 TRANS (input) CHARACTER Specifies the form of the system of equations: .br = 'N': L * X = B (No transpose) .br = 'T': L**T * X = B (Transpose) .TP 8 N (input) INTEGER The order of the tridiagonal matrix A. N >= 0. .TP 8 NRHS (input) INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. .TP 8 D (input) REAL array, dimension (N) The n diagonal elements of the diagonal matrix D from the factorization computed by SPTTRF. .TP 8 E (input) COMPLEX array, dimension (N-1) The (n-1) off-diagonal elements of the unit bidiagonal factor U or L from the factorization computed by SPTTRF (see UPLO). .TP 8 B (input/output) COMPLEX array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X. .TP 8 LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value scalapack-doc-1.5/man/manl/sstein2.l0100644000056400000620000000666706335610666017035 0ustar pfrauenfstaff.TH SSTEIN2 l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)" .SH NAME SSTEIN2 - compute the eigenvectors of a real symmetric tridiagonal matrix T corresponding to specified eigenvalues, using inverse iteration .SH SYNOPSIS .TP 20 SUBROUTINE SSTEIN2( N, D, E, M, W, IBLOCK, ISPLIT, ORFAC, Z, LDZ, WORK, IWORK, IFAIL, INFO ) .TP 20 .ti +4 INTEGER INFO, LDZ, M, N .TP 20 .ti +4 REAL ORFAC .TP 20 .ti +4 INTEGER IBLOCK( * ), IFAIL( * ), ISPLIT( * ), IWORK( * ) .TP 20 .ti +4 REAL D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * ) .SH PURPOSE SSTEIN2 computes the eigenvectors of a real symmetric tridiagonal matrix T corresponding to specified eigenvalues, using inverse iteration. The maximum number of iterations allowed for each eigenvector is specified by an internal parameter MAXITS (currently set to 5). .SH ARGUMENTS .TP 8 N (input) INTEGER The order of the matrix. N >= 0. .TP 8 D (input) REAL array, dimension (N) The n diagonal elements of the tridiagonal matrix T. .TP 8 E (input) REAL array, dimension (N) The (n-1) subdiagonal elements of the tridiagonal matrix T, in elements 1 to N-1. E(N) need not be set. .TP 8 M (input) INTEGER The number of eigenvectors to be found. 0 <= M <= N. .TP 8 W (input) REAL array, dimension (N) The first M elements of W contain the eigenvalues for which eigenvectors are to be computed. The eigenvalues should be grouped by split-off block and ordered from smallest to largest within the block. ( The output array W from SSTEBZ with ORDER = 'B' is expected here. ) .TP 8 IBLOCK (input) INTEGER array, dimension (N) The submatrix indices associated with the corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to the first submatrix from the top, =2 if W(i) belongs to the second submatrix, etc. ( The output array IBLOCK from SSTEBZ is expected here. ) .TP 8 ISPLIT (input) INTEGER array, dimension (N) The splitting points, at which T breaks up into submatrices. The first submatrix consists of rows/columns 1 to ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1 through ISPLIT( 2 ), etc. ( The output array ISPLIT from SSTEBZ is expected here. ) .TP 8 ORFAC (input) REAL ORFAC specifies which eigenvectors should be orthogonalized. Eigenvectors that correspond to eigenvalues which are within ORFAC*||T|| of each other are to be orthogonalized. .TP 8 Z (output) REAL array, dimension (LDZ, M) The computed eigenvectors. The eigenvector associated with the eigenvalue W(i) is stored in the i-th column of Z. Any vector which fails to converge is set to its current iterate after MAXITS iterations. .TP 8 LDZ (input) INTEGER The leading dimension of the array Z. LDZ >= max(1,N). .TP 8 WORK (workspace) REAL array, dimension (5*N) .TP 8 IWORK (workspace) INTEGER array, dimension (N) .TP 8 IFAIL (output) INTEGER array, dimension (M) On normal exit, all elements of IFAIL are zero. If one or more eigenvectors fail to converge after MAXITS iterations, then their indices are stored in array IFAIL. .TP 8 INFO (output) INTEGER = 0: successful exit. .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 0: if INFO = i, then i eigenvectors failed to converge in MAXITS iterations. Their indices are stored in array IFAIL. .SH PARAMETERS .TP 8 MAXITS INTEGER, default = 5 The maximum number of iterations performed. .TP 8 EXTRA INTEGER, default = 2 The number of iterations performed after norm growth criterion is satisfied, should be at least 1. scalapack-doc-1.5/man/manl/ssteqr2.l0100644000056400000620000000503706335610666017037 0ustar pfrauenfstaff.TH SSTEQR2 l "12 May 1997" "modified LAPACK routine" "LAPACK routine (version 2.0)" .SH NAME SSTEQR2 - i a modified version of LAPACK routine SSTEQR .SH SYNOPSIS .TP 20 SUBROUTINE SSTEQR2( COMPZ, N, D, E, Z, LDZ, NR, WORK, INFO ) .TP 20 .ti +4 CHARACTER COMPZ .TP 20 .ti +4 INTEGER INFO, LDZ, N, NR .TP 20 .ti +4 REAL D( * ), E( * ), WORK( * ), Z( LDZ, * ) .SH PURPOSE SSTEQR2 is a modified version of LAPACK routine SSTEQR. SSTEQR2 computes all eigenvalues and, optionally, eigenvectors of a symmetric tridiagonal matrix using the implicit QL or QR method. running SSTEQR2 to perform updates on a distributed matrix Q. Proper usage of SSTEQR2 can be gleaned from examination of ScaLAPACK's PSSYEV. .br .SH ARGUMENTS .TP 8 COMPZ (input) CHARACTER*1 = 'N': Compute eigenvalues only. .br = 'I': Compute eigenvalues and eigenvectors of the tridiagonal matrix. Z must be initialized to the identity matrix by PDLASET or DLASET prior to entering this subroutine. .TP 8 N (input) INTEGER The order of the matrix. N >= 0. .TP 8 D (input/output) REAL array, dimension (N) On entry, the diagonal elements of the tridiagonal matrix. On exit, if INFO = 0, the eigenvalues in ascending order. .TP 8 E (input/output) REAL array, dimension (N-1) On entry, the (n-1) subdiagonal elements of the tridiagonal matrix. On exit, E has been destroyed. .TP 8 Z (local input/local output) REAL array, global dimension (N, N), local dimension (LDZ, NR). On entry, if COMPZ = 'V', then Z contains the orthogonal matrix used in the reduction to tridiagonal form. On exit, if INFO = 0, then if COMPZ = 'V', Z contains the orthonormal eigenvectors of the original symmetric matrix, and if COMPZ = 'I', Z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix. If COMPZ = 'N', then Z is not referenced. .TP 8 LDZ (input) INTEGER The leading dimension of the array Z. LDZ >= 1, and if eigenvectors are desired, then LDZ >= max(1,N). .TP 8 NR (input) INTEGER NR = MAX(1, NUMROC( N, NB, MYPROW, 0, NPROCS ) ). If COMPZ = 'N', then NR is not referenced. .TP 8 WORK (workspace) REAL array, dimension (max(1,2*N-2)) If COMPZ = 'N', then WORK is not referenced. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 0: the algorithm has failed to find all the eigenvalues in a total of 30*N iterations; if INFO = i, then i elements of E have not converged to zero; on exit, D and E contain the elements of a symmetric tridiagonal matrix which is orthogonally similar to the original matrix. scalapack-doc-1.5/man/manl/zdbtf2.l0100644000056400000620000000464106335610666016627 0ustar pfrauenfstaff.SH NAME ZDBTF2 - compute an LU factorization of a real m-by-n band matrix A without using partial pivoting with row interchanges .SH SYNOPSIS .TP 19 SUBROUTINE ZDBTF2( M, N, KL, KU, AB, LDAB, INFO ) .TP 19 .ti +4 INTEGER INFO, KL, KU, LDAB, M, N .TP 19 .ti +4 COMPLEX*16 AB( LDAB, * ) .SH PURPOSE Zdbtrf computes an LU factorization of a real m-by-n band matrix A without using partial pivoting with row interchanges. This is the unblocked version of the algorithm, calling Level 2 BLAS. .SH ARGUMENTS .TP 8 M (input) INTEGER The number of rows of the matrix A. M >= 0. .TP 8 N (input) INTEGER The number of columns of the matrix A. N >= 0. .TP 8 KL (input) INTEGER The number of subdiagonals within the band of A. KL >= 0. .TP 8 KU (input) INTEGER The number of superdiagonals within the band of A. KU >= 0. .TP 8 AB (input/output) COMPLEX*16 array, dimension (LDAB,N) On entry, the matrix A in band storage, in rows KL+1 to 2*KL+KU+1; rows 1 to KL of the array need not be set. The j-th column of A is stored in the j-th column of the array AB as follows: AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl) On exit, details of the factorization: U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. See below for further details. .TP 8 LDAB (input) INTEGER The leading dimension of the array AB. LDAB >= 2*KL+KU+1. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 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. .SH FURTHER DETAILS The band storage scheme is illustrated by the following example, when M = N = 6, KL = 2, KU = 1: .br On entry: On exit: .br * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 * a31 a42 a53 a64 * * m31 m42 m53 m64 * * Array elements marked * are not used by the routine; elements marked + need not be set on entry, but are required by the routine to store elements of U, because of fill-in resulting from the row .br interchanges. .br scalapack-doc-1.5/man/manl/zdbtrf.l0100644000056400000620000000435506335610666016731 0ustar pfrauenfstaff.SH NAME ZDBTRF - compute an LU factorization of a real m-by-n band matrix A without using partial pivoting or row interchanges .SH SYNOPSIS .TP 19 SUBROUTINE ZDBTRF( M, N, KL, KU, AB, LDAB, INFO ) .TP 19 .ti +4 INTEGER INFO, KL, KU, LDAB, M, N .TP 19 .ti +4 COMPLEX*16 AB( LDAB, * ) .SH PURPOSE Zdbtrf computes an LU factorization of a real m-by-n band matrix A without using partial pivoting or row interchanges. This is the blocked version of the algorithm, calling Level 3 BLAS. .SH ARGUMENTS .TP 8 M (input) INTEGER The number of rows of the matrix A. M >= 0. .TP 8 N (input) INTEGER The number of columns of the matrix A. N >= 0. .TP 8 KL (input) INTEGER The number of subdiagonals within the band of A. KL >= 0. .TP 8 KU (input) INTEGER The number of superdiagonals within the band of A. KU >= 0. .TP 8 AB (input/output) REAL array, dimension (LDAB,N) On entry, the matrix A in band storage, in rows KL+1 to 2*KL+KU+1; rows 1 to KL of the array need not be set. The j-th column of A is stored in the j-th column of the array AB as follows: AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl) On exit, details of the factorization: U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. See below for further details. .TP 8 LDAB (input) INTEGER The leading dimension of the array AB. LDAB >= 2*KL+KU+1. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 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. .SH FURTHER DETAILS The band storage scheme is illustrated by the following example, when M = N = 6, KL = 2, KU = 1: .br On entry: On exit: .br * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 * a31 a42 a53 a64 * * m31 m42 m53 m64 * * Array elements marked * are not used by the routine. .br scalapack-doc-1.5/man/manl/zdttrf.l0100644000056400000620000000332206335610666016744 0ustar pfrauenfstaff.TH ZDTTRF l "12 May 1997" "modified LAPACK routine" "LAPACK routine (version 2.0)" .SH NAME ZDTTRF - compute an LU factorization of a complex tridiagonal matrix A using elimination without partial pivoting .SH SYNOPSIS .TP 19 SUBROUTINE ZDTTRF( N, DL, D, DU, INFO ) .TP 19 .ti +4 INTEGER INFO, N .TP 19 .ti +4 COMPLEX*16 D( * ), DL( * ), DU( * ) .SH PURPOSE ZDTTRF computes an LU factorization of a complex tridiagonal matrix A using elimination without partial pivoting. The factorization has the form .br A = L * U .br where L is a product of unit lower bidiagonal .br matrices and U is upper triangular with nonzeros in only the main diagonal and first superdiagonal. .br .SH ARGUMENTS .TP 8 N (input) INTEGER The order of the matrix A. N >= 0. .TP 8 DL (input/output) COMPLEX array, dimension (N-1) On entry, DL must contain the (n-1) subdiagonal elements of A. On exit, DL is overwritten by the (n-1) multipliers that define the matrix L from the LU factorization of A. .TP 8 D (input/output) COMPLEX array, dimension (N) On entry, D must contain the diagonal elements of A. On exit, D is overwritten by the n diagonal elements of the upper triangular matrix U from the LU factorization of A. .TP 8 DU (input/output) COMPLEX array, dimension (N-1) On entry, DU must contain the (n-1) superdiagonal elements of A. On exit, DU is overwritten by the (n-1) elements of the first superdiagonal of U. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 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. scalapack-doc-1.5/man/manl/zdttrsv.l0100644000056400000620000000352706335610666017156 0ustar pfrauenfstaff.TH ZDTTRSV l "12 May 1997" "modified LAPACK routine" "LAPACK routine (version 2.0)" .SH NAME ZDTTRSV - solve one of the systems of equations L * X = B, L**T * X = B, or L**H * X = B, .SH SYNOPSIS .TP 20 SUBROUTINE ZDTTRSV( UPLO, TRANS, N, NRHS, DL, D, DU, B, LDB, INFO ) .TP 20 .ti +4 CHARACTER UPLO, TRANS .TP 20 .ti +4 INTEGER INFO, LDB, N, NRHS .TP 20 .ti +4 COMPLEX*16 B( LDB, * ), D( * ), DL( * ), DU( * ) .SH PURPOSE ZDTTRSV solves one of the systems of equations L * X = B, L**T * X = B, or L**H * X = B, U * X = B, U**T * X = B, or U**H * X = B, .br with factors of the tridiagonal matrix A from the LU factorization computed by ZDTTRF. .br .SH ARGUMENTS .TP 8 UPLO (input) CHARACTER*1 Specifies whether to solve with L or U. .TP 8 TRANS (input) CHARACTER Specifies the form of the system of equations: .br = 'N': A * X = B (No transpose) .br = 'T': A**T * X = B (Transpose) .br = 'C': A**H * X = B (Conjugate transpose) .TP 8 N (input) INTEGER The order of the matrix A. N >= 0. .TP 8 NRHS (input) INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. .TP 8 DL (input) COMPLEX array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A. .TP 8 D (input) COMPLEX array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A. .TP 8 DU (input) COMPLEX array, dimension (N-1) The (n-1) elements of the first superdiagonal of U. .TP 8 B (input/output) COMPLEX array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, B is overwritten by the solution matrix X. .TP 8 LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value scalapack-doc-1.5/man/manl/zpttrsv.l0100644000056400000620000000413306335610666017164 0ustar pfrauenfstaff.TH ZPTTRSV l "12 May 1997" "modified LAPACK routine" "LAPACK routine (version 2.0)" .SH NAME ZPTTRSV - solve one of the triangular systems L * X = B, or L**H * X = B, .SH SYNOPSIS .TP 20 SUBROUTINE ZPTTRSV( UPLO, TRANS, N, NRHS, D, E, B, LDB, INFO ) .TP 20 .ti +4 CHARACTER UPLO, TRANS .TP 20 .ti +4 INTEGER INFO, LDB, N, NRHS .TP 20 .ti +4 DOUBLE PRECISION D( * ) .TP 20 .ti +4 COMPLEX*16 B( LDB, * ), E( * ) .SH PURPOSE ZPTTRSV solves one of the triangular systems L * X = B, or L**H * X = B, U * X = B, or U**H * X = B, .br where L or U is the Cholesky factor of a Hermitian positive definite tridiagonal matrix A such that .br A = U**H*D*U or A = L*D*L**H (computed by ZPTTRF). .br .SH ARGUMENTS .TP 8 UPLO (input) CHARACTER*1 Specifies whether the superdiagonal or the subdiagonal of the tridiagonal matrix A is stored and the form of the factorization: .br = 'U': E is the superdiagonal of U, and A = U'*D*U; .br = 'L': E is the subdiagonal of L, and A = L*D*L'. (The two forms are equivalent if A is real.) .TP 8 TRANS (input) CHARACTER Specifies the form of the system of equations: .br = 'N': L * X = B (No transpose) .br = 'N': L * X = B (No transpose) .br = 'C': U**H * X = B (Conjugate transpose) .br = 'C': L**H * X = B (Conjugate transpose) .TP 8 N (input) INTEGER The order of the tridiagonal matrix A. N >= 0. .TP 8 NRHS (input) INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. .TP 8 D (input) REAL array, dimension (N) The n diagonal elements of the diagonal matrix D from the factorization computed by ZPTTRF. .TP 8 E (input) COMPLEX array, dimension (N-1) The (n-1) off-diagonal elements of the unit bidiagonal factor U or L from the factorization computed by ZPTTRF (see UPLO). .TP 8 B (input/output) COMPLEX array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X. .TP 8 LDB (input) INTEGER The leading dimension of the array B. 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D<00FE0007FF800F87C01E01E03E01F07C00F07C00F8FC00F8FC00F8FFFFF8FFFFF8 FC0000FC0000FC00007C00007C00007E00003E00181F00300FC07003FFC000FF0015167E 951A>101 D108 D 118 D E end TeXDict begin @landscape 1 0 bop -465 8 a Fi(Lev)n(el)17 b(1)h(PBLAS)-249 87 y Fh(dim)f(scalar)103 b(vector)316 b(vector)2295 b Fg(pre\014xes)-465 126 y Fh(P)p Ff(2)p Fh(SWAP)66 b(\()17 b(N,)246 b(X,)17 b(IX,)g(JX,)f(DESCX,)g(INCX,)g(Y,)h(IY,)f(JY,)h(DESCY,)e(INCY)h(\))886 b Fe(x)10 b Fd($)g Fe(y)1002 b Fg(S,)11 b(D,)h(C,)g(Z)-465 166 y Fh(P)p Ff(2)p Fh(SCAL)66 b(\()17 b(N,)g(ALPHA,)121 b(X,)17 b(IX,)g(JX,)f(DESCX,)g(INCX)g(\))1309 b Fe(x)10 b Fd( )g Fe(\013x)976 b Fg(S,)11 b(D,)h(C,)g(Z,)f(CS,)h(ZD)-465 205 y Fh(P)p Ff(2)p Fh(COPY)66 b(\()17 b(N,)246 b(X,)17 b(IX,)g(JX,)f(DESCX,)g(INCX,)g(Y,)h(IY,)f(JY,)h(DESCY,)e(INCY)h(\))886 b Fe(y)12 b Fd( )e Fe(x)1000 b Fg(S,)11 b(D,)h(C,)g(Z)-465 245 y Fh(P)p Ff(2)p Fh(AXPY)66 b(\()17 b(N,)g(ALPHA,)121 b(X,)17 b(IX,)g(JX,)f(DESCX,)g(INCX,)g(Y,)h(IY,)f(JY,)h(DESCY,)e(INCY)h (\))886 b Fe(y)12 b Fd( )e Fe(\013x)d Fg(+)i Fe(y)917 b Fg(S,)11 b(D,)h(C,)g(Z)-465 285 y Fh(P)p Ff(2)p Fh(DOT)84 b(\()17 b(N,)g(DOT,)157 b(X,)17 b(IX,)g(JX,)f(DESCX,)g(INCX,)g(Y,)h (IY,)f(JY,)h(DESCY,)e(INCY)h(\))886 b Fe(dot)10 b Fd( )g Fe(x)1889 274 y Fc(T)1913 285 y Fe(y)930 b Fg(S,)11 b(D)-465 326 y Fh(P)p Ff(2)p Fh(DOTU)66 b(\()17 b(N,)g(DOTU,)139 b(X,)17 b(IX,)g(JX,)f(DESCX,)g(INCX,)g(Y,)h(IY,)f(JY,)h(DESCY,)e(INCY)h (\))886 b Fe(dotu)11 b Fd( )f Fe(x)1910 314 y Fc(T)1934 326 y Fe(y)909 b Fg(C,)12 b(Z)-465 367 y Fh(P)p Ff(2)p Fh(DOTC)66 b(\()17 b(N,)g(DOTC,)139 b(X,)17 b(IX,)g(JX,)f(DESCX,)g (INCX,)g(Y,)h(IY,)f(JY,)h(DESCY,)e(INCY)h(\))886 b Fe(dotc)10 b Fd( )g Fe(x)1904 355 y Fc(H)1933 367 y Fe(y)910 b Fg(C,)12 b(Z)-465 406 y Fh(P)p Ff(2)p Fh(NRM2)66 b(\()17 b(N,)g(NORM2,)121 b(X,)17 b(IX,)g(JX,)f(DESCX,)g(INCX)g(\))1309 b Fe(nor)q(m)p Fg(2)10 b Fd( )g(k)p Fe(x)p Fd(k)1981 411 y Fb(2)2860 406 y Fg(S,)h(D,)h(SC,)g(DZ)-465 445 y Fh(P)p Ff(2)p Fh(ASUM)66 b(\()17 b(N,)g(ASUM,)139 b(X,)17 b(IX,)g(JX,)f(DESCX,)g (INCX)g(\))1309 b Fe(asum)11 b Fd( )f(k)p Fe(r)q(e)p Fg(\()p Fe(x)p Fg(\))p Fd(k)2025 450 y Fb(1)2049 445 y Fg(+)e Fd(k)p Fe(im)p Fg(\()p Fe(x)p Fg(\))p Fd(k)2211 450 y Fb(1)2860 445 y Fg(S,)j(D,)h(SC,)g(DZ)-465 485 y Fh(P)p Ff(2)p Fh(AMAX)66 b(\()17 b(N,)g(AMAX,)f(INDX,)33 b(X,)17 b(IX,)g(JX,)f(DESCX,)g(INCX)g(\))1309 b Fe(indx)11 b Fd( )f Fg(1)1911 473 y Fc(st)1951 485 y Fe(k)h Fd(3)e(j)p Fe(Re)p Fg(\()p Fe(x)2100 491 y Fc(k)2119 485 y Fg(\))p Fd(j)e Fg(+)h Fd(j)p Fe(I)s(m)p Fg(\()p Fe(x)2278 491 y Fc(k)2297 485 y Fg(\))p Fd(j)539 b Fg(S,)11 b(D,)h(C,)g(Z)1990 524 y(=)e Fe(max)p Fg(\()p Fd(j)p Fe(Re)p Fg(\()p Fe(x)2198 529 y Fc(i)2210 524 y Fg(\))p Fd(j)e Fg(+)g Fd(j)p Fe(I)s(m)p Fg(\()p Fe(x)2370 529 y Fc(i)2383 524 y Fg(\))p Fd(j)p Fg(\))h(=)h Fe(amax)-465 610 y Fi(Lev)n(el)17 b(2)h(PBLAS)-249 689 y Fh(options)209 b(dim)52 b(scalar)16 b(matrix)209 b(vector)298 b(scalar)15 b(vector)-465 730 y(P)p Ff(2)p Fh(GEMV)66 b(\()123 b(TRANS,)e(M,)17 b(N,)g(ALPHA,)f(A,)g(IA,)h(JA,)g (DESCA,)e(X,)i(IX,)g(JX,)f(DESCX,)g(INCX,)f(BETA,)h(Y,)h(IY,)g(JY,)f (DESCY,)g(INCY)g(\))180 b Fe(y)12 b Fd( )e Fe(\013op)p Fg(\()p Fe(A)p Fg(\))p Fe(x)d Fg(+)h Fe(\014)r(y)q(;)e(op)p Fg(\()p Fe(A)p Fg(\))k(=)g Fe(A;)c(A)2274 718 y Fc(T)2299 730 y Fe(;)f(A)2340 718 y Fc(H)2370 730 y Fe(;)g(A)k Fd(\000)f Fe(m)g Fd(\002)g Fe(n)310 b Fg(S,)11 b(D,)h(C,)g(Z)-465 769 y Fh(P)p Ff(2)p Fh(HEMV)66 b(\()17 b(UPLO,)298 b(N,)17 b(ALPHA,)f(A,)g(IA,)h(JA,)g(DESCA,)e(X,)i(IX,)g(JX,)f(DESCX,)g(INCX,)f (BETA,)h(Y,)h(IY,)g(JY,)f(DESCY,)g(INCY)g(\))180 b Fe(y)12 b Fd( )e Fe(\013Ax)e Fg(+)g Fe(\014)r(y)869 b Fg(C,)12 b(Z)-465 808 y Fh(P)p Ff(2)p Fh(SYMV)66 b(\()17 b(UPLO,)298 b(N,)17 b(ALPHA,)f(A,)g(IA,)h(JA,)g(DESCA,)e(X,)i(IX,)g(JX,)f(DESCX,)g (INCX,)f(BETA,)h(Y,)h(IY,)g(JY,)f(DESCY,)g(INCY)g(\))180 b Fe(y)12 b Fd( )e Fe(\013Ax)e Fg(+)g Fe(\014)r(y)869 b Fg(S,)11 b(D)-465 849 y Fh(P)p Ff(2)p Fh(TRMV)66 b(\()17 b(UPLO,)f(TRANS,)f(DIAG,)69 b(N,)141 b(A,)16 b(IA,)h(JA,)g(DESCA,)e(X,) i(IX,)g(JX,)f(DESCX,)g(INCX)g(\))709 b Fe(x)10 b Fd( )g Fe(\013Ax;)c(x)j Fd( )h Fe(\013A)2049 837 y Fc(T)2074 849 y Fe(x;)5 b(x)10 b Fd( )g Fe(\013A)2233 837 y Fc(H)2262 849 y Fe(x;)568 b Fg(S,)11 b(D,)h(C,)g(Z)-465 890 y Fh(P)p Ff(2)p Fh(TRSV)66 b(\()17 b(UPLO,)f(TRANS,)f(DIAG,)69 b(N,)141 b(A,)16 b(IA,)h(JA,)g(DESCA,)e(X,)i(IX,)g(JX,)f(DESCX,)g(INCX) g(\))709 b Fe(x)10 b Fd( )g Fe(\013A)1890 878 y Fa(\000)p Fb(1)1931 890 y Fe(x;)c(x)k Fd( )g Fe(\013A)2091 878 y Fa(\000)p Fc(T)2139 890 y Fe(x;)c(x)j Fd( )h Fe(\013A)2298 878 y Fa(\000)p Fc(H)2351 890 y Fe(x;)479 b Fg(S,)11 b(D,)h(C,)g(Z)-249 968 y Fh(options)209 b(dim)52 b(scalar)16 b(vector)315 b(vector)g(matrix)-465 1009 y(P)p Ff(2)p Fh(GER)84 b(\()352 b(M,)17 b(N,)g(ALPHA,)f(X,)g(IX,)h(JX,)g(DESCX,)e (INCX,)h(Y,)h(IY,)f(JY,)h(DESCY,)e(INCY,)h(A,)h(IA,)g(JA,)f(DESCA)g(\)) 286 b Fe(A)11 b Fd( )f Fe(\013xy)1909 997 y Fc(T)1941 1009 y Fg(+)e Fe(A;)e(A)j Fd(\000)f Fe(m)g Fd(\002)g Fe(n)677 b Fg(S,)11 b(D)-465 1050 y Fh(P)p Ff(2)p Fh(GERU)66 b(\()352 b(M,)17 b(N,)g(ALPHA,)f(X,)g(IX,)h(JX,)g(DESCX,)e(INCX,)h(Y,)h (IY,)f(JY,)h(DESCY,)e(INCY,)h(A,)h(IA,)g(JA,)f(DESCA)g(\))286 b Fe(A)11 b Fd( )f Fe(\013xy)1909 1038 y Fc(T)1941 1050 y Fg(+)e Fe(A;)e(A)j Fd(\000)f Fe(m)g Fd(\002)g Fe(n)677 b Fg(C,)12 b(Z)-465 1090 y Fh(P)p Ff(2)p Fh(GERC)66 b(\()352 b(M,)17 b(N,)g(ALPHA,)f(X,)g(IX,)h(JX,)g(DESCX,)e(INCX,)h(Y,)h(IY,)f (JY,)h(DESCY,)e(INCY,)h(A,)h(IA,)g(JA,)f(DESCA)g(\))286 b Fe(A)11 b Fd( )f Fe(\013xy)1909 1078 y Fc(H)1945 1090 y Fg(+)f Fe(A;)d(A)i Fd(\000)g Fe(m)h Fd(\002)f Fe(n)672 b Fg(C,)12 b(Z)-465 1131 y Fh(P)p Ff(2)p Fh(HER)84 b(\()17 b(UPLO,)298 b(N,)17 b(ALPHA,)f(X,)g(IX,)h(JX,)g(DESCX,)e(INCX,)439 b(A,)17 b(IA,)g(JA,)f(DESCA)g(\))286 b Fe(A)11 b Fd( )f Fe(\013xx)1911 1119 y Fc(H)1947 1131 y Fg(+)e Fe(A)852 b Fg(C,)12 b(Z)-465 1171 y Fh(P)p Ff(2)p Fh(HER2)66 b(\()17 b(UPLO,)298 b(N,)17 b(ALPHA,)f(X,)g(IX,)h(JX,)g(DESCX,)e(INCX,)h(Y,)h (IY,)f(JY,)h(DESCY,)e(INCY,)h(A,)h(IA,)g(JA,)f(DESCA)g(\))286 b Fe(A)11 b Fd( )f Fe(\013xy)1909 1160 y Fc(H)1945 1171 y Fg(+)f Fe(y)q Fg(\()p Fe(\013x)p Fg(\))2070 1160 y Fc(H)2106 1171 y Fg(+)f Fe(A)693 b Fg(C,)12 b(Z)-465 1212 y Fh(P)p Ff(2)p Fh(SYR)84 b(\()17 b(UPLO,)298 b(N,)17 b(ALPHA,)f(X,)g(IX,)h(JX,)g(DESCX,)e(INCX,)439 b(A,)17 b(IA,)g(JA,)f(DESCA)g(\))286 b Fe(A)11 b Fd( )f Fe(\013xx)1911 1200 y Fc(T)1942 1212 y Fg(+)e Fe(A)857 b Fg(S,)11 b(D)-465 1252 y Fh(P)p Ff(2)p Fh(SYR2)66 b(\()17 b(UPLO,)298 b(N,)17 b(ALPHA,)f(X,)g(IX,)h(JX,)g(DESCX,)e(INCX,)h(Y,)h(IY,)f(JY,)h(DESCY,)e (INCY,)h(A,)h(IA,)g(JA,)f(DESCA)g(\))286 b Fe(A)11 b Fd( )f Fe(\013xy)1909 1241 y Fc(T)1941 1252 y Fg(+)e Fe(\013y)q(x)2037 1241 y Fc(T)2070 1252 y Fg(+)g Fe(A)729 b Fg(S,)11 b(D)-465 1378 y Fi(Lev)n(el)17 b(3)h(PBLAS)-249 1457 y Fh(options)474 b(dim)105 b(scalar)15 b(matrix)210 b(matrix)f(scalar)16 b(matrix)-465 1497 y(P)p Ff(2)p Fh(GEMM)66 b(\()229 b(TRANSA,)15 b(TRANSB,)121 b(M,)17 b(N,)g(K,)g(ALPHA,)e(A,)i(IA,)g(JA,)f(DESCA,)g(B,)h(IB,)f(JB,)h(DESCB,) e(BETA,)h(C,)h(IC,)g(JC,)f(DESCC)g(\))74 b Fe(C)13 b Fd( )d Fe(\013op)p Fg(\()p Fe(A)p Fg(\))p Fe(op)p Fg(\()p Fe(B)r Fg(\))5 b(+)k Fe(\014)r(C;)c(op)p Fg(\()p Fe(X)s Fg(\))k(=)h Fe(X)q(;)c(X)2375 1485 y Fc(T)2399 1497 y Fe(;)f(X)2446 1485 y Fc(H)2475 1497 y Fe(;)g(C)11 b Fd(\000)d Fe(m)g Fd(\002)g Fe(n)204 b Fg(S,)11 b(D,)h(C,)g(Z)-465 1538 y Fh(P)p Ff(2)p Fh(SYMM)66 b(\()17 b(SIDE,)f(UPLO,)404 b(M,)17 b(N,)70 b(ALPHA,)15 b(A,)i(IA,)g(JA,)f(DESCA,)g(B,)h(IB,)f(JB,) h(DESCB,)e(BETA,)h(C,)h(IC,)g(JC,)f(DESCC)g(\))74 b Fe(C)13 b Fd( )d Fe(\013AB)f Fg(+)f Fe(\014)r(C;)e(C)12 b Fd( )e Fe(\013B)r(A)d Fg(+)i Fe(\014)r(C;)c(C)11 b Fd(\000)d Fe(m)g Fd(\002)g Fe(n;)f(A)j Fg(=)g Fe(A)2579 1526 y Fc(T)2860 1538 y Fg(S,)h(D,)h(C,)g(Z)-465 1578 y Fh(P)p Ff(2)p Fh(HEMM)66 b(\()17 b(SIDE,)f(UPLO,)404 b(M,)17 b(N,)70 b(ALPHA,)15 b(A,)i(IA,)g(JA,)f(DESCA,)g(B,)h(IB,)f(JB,)h (DESCB,)e(BETA,)h(C,)h(IC,)g(JC,)f(DESCC)g(\))74 b Fe(C)13 b Fd( )d Fe(\013AB)f Fg(+)f Fe(\014)r(C;)e(C)12 b Fd( )e Fe(\013B)r(A)d Fg(+)i Fe(\014)r(C;)c(C)11 b Fd(\000)d Fe(m)g Fd(\002)g Fe(n;)f(A)j Fg(=)g Fe(A)2579 1567 y Fc(H)2860 1578 y Fg(C,)i(Z)-465 1619 y Fh(P)p Ff(2)p Fh(SYRK)66 b(\()123 b(UPLO,)16 b(TRANS,)333 b(N,)17 b(K,)g(ALPHA,)e(A,) i(IA,)g(JA,)f(DESCA,)333 b(BETA,)16 b(C,)h(IC,)g(JC,)f(DESCC)g(\))74 b Fe(C)13 b Fd( )d Fe(\013AA)1924 1607 y Fc(T)1956 1619 y Fg(+)f Fe(\014)r(C;)c(C)13 b Fd( )d Fe(\013A)2186 1607 y Fc(T)2210 1619 y Fe(A)e Fg(+)h Fe(\014)r(C;)c(C)11 b Fd(\000)d Fe(n)g Fd(\002)h Fe(n)361 b Fg(S,)11 b(D,)h(C,)g(Z)-465 1659 y Fh(P)p Ff(2)p Fh(HERK)66 b(\()123 b(UPLO,)16 b(TRANS,)333 b(N,)17 b(K,)g(ALPHA,)e(A,)i(IA,)g(JA,)f(DESCA,)333 b(BETA,)16 b(C,)h(IC,)g(JC,)f(DESCC)g(\))74 b Fe(C)13 b Fd( )d Fe(\013AA)1924 1648 y Fc(H)1961 1659 y Fg(+)e Fe(\014)r(C;)e(C)12 b Fd( )e Fe(\013A)2190 1648 y Fc(H)2219 1659 y Fe(A)f Fg(+)f Fe(\014)r(C;)e(C)k Fd(\000)e Fe(n)g Fd(\002)h Fe(n)352 b Fg(C,)12 b(Z)-465 1700 y Fh(P)p Ff(2)p Fh(SYR2K)48 b(\()123 b(UPLO,)16 b(TRANS,)333 b(N,)17 b(K,)g(ALPHA,)e(A,)i(IA,)g (JA,)f(DESCA,)g(B,)h(IB,)f(JB,)h(DESCB,)e(BETA,)h(C,)h(IC,)g(JC,)f (DESCC)g(\))74 b Fe(C)13 b Fd( )d Fe(\013AB)1927 1688 y Fc(T)1958 1700 y Fg(+)f Fe(\013B)r(A)2072 1688 y Fc(T)2103 1700 y Fg(+)f Fe(\014)r(C;)e(C)12 b Fd( )e Fe(\013A)2332 1688 y Fc(T)2357 1700 y Fe(B)f Fg(+)f Fe(\013B)2480 1688 y Fc(T)2503 1700 y Fe(A)h Fg(+)f Fe(\014)r(C;)e(C)k Fd(\000)e Fe(n)h Fd(\002)f Fe(n)68 b Fg(S,)11 b(D,)h(C,)g(Z)-465 1741 y Fh(P)p Ff(2)p Fh(HER2K)48 b(\()123 b(UPLO,)16 b(TRANS,)333 b(N,)17 b(K,)g(ALPHA,)e(A,)i(IA,)g(JA,)f(DESCA,)g(B,)h (IB,)f(JB,)h(DESCB,)e(BETA,)h(C,)h(IC,)g(JC,)f(DESCC)g(\))74 b Fe(C)13 b Fd( )d Fe(\013AB)1927 1729 y Fc(H)1963 1741 y Fg(+)i(\026)-22 b Fe(\013B)r(A)2076 1729 y Fc(H)2112 1741 y Fg(+)8 b Fe(\014)r(C;)e(C)12 b Fd( )e Fe(\013A)2341 1729 y Fc(H)2370 1741 y Fe(B)f Fg(+)j(\026)-21 b Fe(\013)o(B)2493 1729 y Fc(H)2522 1741 y Fe(A)8 b Fg(+)g Fe(\014)r(C;)e(C)k Fd(\000)e Fe(n)h Fd(\002)f Fe(n)50 b Fg(C,)12 b(Z)-465 1781 y Fh(P)p Ff(2)p Fh(TRAN)66 b(\()617 b(M,)17 b(N,)70 b(ALPHA,)15 b(A,)i(IA,)g(JA,)f(DESCA,)333 b(BETA,)16 b(C,)h(IC,)g(JC,)f(DESCC)g(\))74 b Fe(C)13 b Fd( )d Fe(\014)r(C)g Fg(+)e Fe(\013A)1990 1769 y Fc(T)2014 1781 y Fe(;)e(A)i Fd(\000)g Fe(n)h Fd(\002)f Fe(m;)e(C)k Fd(\000)f Fe(m)f Fd(\002)g Fe(n)483 b Fg(S,)11 b(D)-465 1822 y Fh(P)p Ff(2)p Fh(TRANU)48 b(\()617 b(M,)17 b(N,)70 b(ALPHA,)15 b(A,)i(IA,)g(JA,)f(DESCA,)333 b(BETA,)16 b(C,)h(IC,)g(JC,)f(DESCC)g(\)) 74 b Fe(C)13 b Fd( )d Fe(\014)r(C)g Fg(+)e Fe(\013A)1990 1810 y Fc(T)2014 1822 y Fe(;)e(A)i Fd(\000)g Fe(n)h Fd(\002)f Fe(m;)e(C)k Fd(\000)f Fe(m)f Fd(\002)g Fe(n)483 b Fg(C,)12 b(Z)-465 1862 y Fh(P)p Ff(2)p Fh(TRANC)48 b(\()617 b(M,)17 b(N,)70 b(ALPHA,)15 b(A,)i(IA,)g(JA,)f(DESCA,)333 b(BETA,)16 b(C,)h(IC,)g(JC,)f(DESCC)g(\))74 b Fe(C)13 b Fd( )d Fe(\014)r(C)g Fg(+)e Fe(\013A)1990 1851 y Fc(H)2019 1862 y Fe(;)d(A)k Fd(\000)f Fe(n)g Fd(\002)h Fe(m;)c(C)11 b Fd(\000)d Fe(m)g Fd(\002)g Fe(n)479 b Fg(C,)12 b(Z)-465 1903 y Fh(P)p Ff(2)p Fh(TRMM)66 b(\()17 b(SIDE,)f(UPLO,)g(TRANSA,)156 b(DIAG,)16 b(M,)h(N,)70 b(ALPHA,)15 b(A,)i(IA,)g(JA,)f(DESCA,)g(B,)h (IB,)f(JB,)h(DESCB)f(\))497 b Fe(B)11 b Fd( )f Fe(\013op)p Fg(\()p Fe(A)p Fg(\))p Fe(B)r(;)5 b(B)11 b Fd( )f Fe(\013B)r(op)p Fg(\()p Fe(A)p Fg(\))p Fe(;)t(op)p Fg(\()p Fe(A)p Fg(\))g(=)g Fe(A;)c(A)2447 1891 y Fc(T)2472 1903 y Fe(;)f(A)2513 1891 y Fc(H)2543 1903 y Fe(;)g(B)10 b Fd(\000)e Fe(m)g Fd(\002)g Fe(n)135 b Fg(S,)11 b(D,)h(C,)g(Z)-465 1943 y Fh(P)p Ff(2)p Fh(TRSM)66 b(\()17 b(SIDE,)f(UPLO,)g(TRANSA,)156 b(DIAG,)16 b(M,)h(N,)70 b(ALPHA,)15 b(A,)i(IA,)g(JA,)f(DESCA,)g(B,)h (IB,)f(JB,)h(DESCB)f(\))497 b Fe(B)11 b Fd( )f Fe(\013op)p Fg(\()p Fe(A)1947 1932 y Fa(\000)p Fb(1)1988 1943 y Fg(\))p Fe(B)r(;)5 b(B)11 b Fd( )f Fe(\013B)r(op)p Fg(\()p Fe(A)2256 1932 y Fa(\000)p Fb(1)2296 1943 y Fg(\))p Fe(;)c(op)p Fg(\()p Fe(A)p Fg(\))j(=)h Fe(A;)d(A)2530 1932 y Fc(T)2554 1943 y Fe(;)f(A)2596 1932 y Fc(H)2625 1943 y Fe(;)g(B)j Fd(\000)f Fe(m)g Fd(\002)h Fe(n)52 b Fg(S,)11 b(D,)h(C,)g(Z)p eop end userdict /end-hook known{end-hook}if %%EndDocument endTexFig eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF scalapack-doc-1.5/README0100644000056400000620000000245606335611037014430 0ustar pfrauenfstaffScaLAPACK, version 1.5 DATE: May 1, 1997 To install the ScaLAPACK documentation, do the following: 1) Copy the ScaLAPACK manual pages to the desired location. This will probably be /usr/local/man/manl or other.dir/man/manl. % cp man/manl/* /usr/local/man/manl Be aware that if the manpages are installed in some directory other than /usr/local/man/manl you will need to add that directory to your MANPATH environment variable. First type: % echo $MANPATH If it says that MANPATH is undefined, type: % setenv MANPATH /usr/man:/usr/local/man:/user_path/man_dir Otherwise, if MANPATH is defined, type: % setenv MANPATH ${MANPATH}:/user_path/man_dir where man_dir is the directory of manual pages containing /man/manl. ('%' is assumed to be your shell prompt.) 2) Add the LAPACK manual pages to the whatis database for keyword searching via 'man -k': % catman -w -M /usr/local/man or % catman -w -M other.dir/man 3) Test out the installation. First make sure your current shell knows about the installation: % rehash Now test out the manual pages: % man dgetrf You should get the man page for dgetrf. BE AWARE that your request should be in lower case. If you have any comments/suggestions, please direct them to scalapack@cs.utk.edu. scalapack-doc-1.5/scalapackqref.ps0100644000056400000620000052335506710272402016716 0ustar pfrauenfstaff%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: scalapackqref.dvi %%Pages: 2 %%PageOrder: Ascend %%Orientation: Landscape %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips -p 2 -l 3 -t landscape -o scalapackqref.ps %+ scalapackqref.dvi %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 1997.05.11:1342 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if 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b(De\014nite)h(Band)376 b Fa(PCPBSV\()38 b(UPLO,)e(N,)g(BW,)177 b(NRHS,)37 b(A,)f(JA,)g(DESCA,)178 b(B,)36 b(IB,)g(DESCB,)h(WORK,)g (LWORK,)g(INFO)f(\))1236 2442 y Fb(Symmetric/Hermitian)390 b Fa(PSPTSV\()249 b(N,)318 b(NRHS,)37 b(D,)f(E,)g(JA,)g(DESCA,)72 b(B,)36 b(IB,)g(DESCB,)h(WORK,)g(LWORK,)g(INFO)f(\))1236 2521 y Fb(P)n(ositiv)n(e)24 b(De\014nite)h(T)-6 b(ridiagonal)188 b Fa(PCPTSV\()249 b(N,)318 b(NRHS,)37 b(D,)f(E,)g(JA,)g(DESCA,)72 b(B,)36 b(IB,)g(DESCB,)h(WORK,)g(LWORK,)g(INFO)f(\))1794 2679 y Fc(Simple)26 b(Driv)n(er)g(Routines)h(for)f(Standard)h(and)g (Generalized)f(Linear)h(Least)f(Squares)h(Problems)1236 2867 y Fb(Problem)22 b(T)n(yp)r(e)630 b(Routine)p 1186 2894 V 1236 2949 a(Solv)n(e)24 b(Using)g(Orthogonal)g(F)-6 b(actor,)100 b Fa(PSGELS\()38 b(TRANS,)f(M,)e(N,)h(NRHS,)h(A,)f(IA,)g (JA,)g(DESCA,)h(B,)f(IB,)g(JB,)g(DESCB,)h(WORK,)g(LWORK,)g(INFO)f(\)) 1236 3028 y Fb(Assuming)22 b(F)-6 b(ull)23 b(Rank)442 b Fa(PCGELS\()38 b(TRANS,)f(M,)e(N,)h(NRHS,)h(A,)f(IA,)g(JA,)g(DESCA,)h (B,)f(IB,)g(JB,)g(DESCB,)h(WORK,)g(LWORK,)g(INFO)f(\))1928 3186 y Fc(Simple)25 b(Driv)n(er)i(Routines)f(for)h(Standard)f(Eigen)n (v)-5 b(alue)27 b(and)g(Singular)f(V)-7 b(alue)27 b(Problems)1236 3358 y Fb(Matrix/Problem)22 b(T)n(yp)r(e)383 b(Routine)p 1186 3385 V 1236 3441 a(Symmetric/Hermitian)390 b Fa(PSSYEV\()38 b(JOBZ,)e(UPLO,)h(N,)f(A,)g(IA,)g(JA,)g(DESCA,)h(W,)f(Z,)g(IZ,)g(JZ,)g (DESCZ,)h(WORK,)g(LWORK,)g(INFO)f(\))1236 3519 y Fb(Eigen)n(v)l (alues/v)n(ectors)1236 3677 y(General)832 b Fa(PSGESVD\()38 b(JOBU,)f(JOBVT,)g(M,)e(N,)h(A,)g(IA,)g(JA,)g(DESCA,)h(S,)f(U,)g(IU,)g (JU,)g(DESCU,)h(VT,)g(IVT,)f(JVT,)g(DESCVT,)1236 3756 y Fb(Singular)23 b(V)-6 b(alues/V)g(ectors)649 b Fa(WORK,)37 b(LWORK,)g(INFO)f(\))p eop %%Page: 3 2 3 1 bop -459 -201 a Fd(Exp)t(ert)45 b(Driv)l(ers)1717 -55 y Fc(Exp)r(ert)27 b(Driv)n(er)f(Routines)h(for)f(Linear)g (Equations)-409 163 y Fb(Matrix)d(T)n(yp)r(e)385 b(Routine)p -459 190 5959 4 v -409 245 a(General)539 b Fa(PSGESVX\()37 b(FACT,)g(TRANS,)g(N,)f(NRHS,)h(A,)e(IA,)i(JA,)f(DESCA,)h(AF,)f(IAF,)g (JAF,)h(DESCAF,)g(IPIV,)g(EQUED,)g(R,)f(C,)f(B,)h(IB,)g(JB,)h(DESCB,)g (X,)e(IX,)h(JX,)h(DESCX,)g(RCOND,)g(FERR,)f(BERR,)686 324 y(WORK,)h(LWORK,)g(IWORK,)g(LIWORK,)g(INFO)g(\))369 403 y(PCGESVX\()g(FACT,)g(TRANS,)g(N,)f(NRHS,)h(A,)e(IA,)i(JA,)f (DESCA,)h(AF,)f(IAF,)g(JAF,)h(DESCAF,)g(IPIV,)g(EQUED,)g(R,)f(C,)f(B,)h (IB,)g(JB,)h(DESCB,)g(X,)e(IX,)h(JX,)h(DESCX,)g(RCOND,)g(FERR,)f(BERR,) 686 482 y(WORK,)h(LWORK,)g(RWORK,)g(LRWORK,)g(INFO)g(\))-409 640 y Fb(Symmetric/Hermitian)97 b Fa(PSPOSVX\()37 b(FACT,)g(UPLO,)72 b(N,)36 b(NRHS,)h(A,)e(IA,)i(JA,)f(DESCA,)h(AF,)f(IAF,)g(JAF,)h (DESCAF,)249 b(EQUED,)37 b(S,)141 b(B,)36 b(IB,)g(JB,)h(DESCB,)g(X,)e (IX,)h(JX,)h(DESCX,)g(RCOND,)g(FERR,)f(BERR,)-409 718 y Fb(P)n(ositiv)n(e)24 b(De\014nite)588 b Fa(WORK,)37 b(LWORK,)g(IWORK,)g(LIWORK,)g(INFO)g(\))369 797 y(PCPOSVX\()g(FACT,)g (UPLO,)72 b(N,)36 b(NRHS,)h(A,)e(IA,)i(JA,)f(DESCA,)h(AF,)f(IAF,)g (JAF,)h(DESCAF,)249 b(EQUED,)37 b(S,)141 b(B,)36 b(IB,)g(JB,)h(DESCB,)g (X,)e(IX,)h(JX,)h(DESCX,)g(RCOND,)g(FERR,)f(BERR,)686 876 y(WORK,)h(LWORK,)g(RWORK,)g(LRWORK,)g(INFO)g(\))1006 1222 y Fc(Exp)r(ert)27 b(Driv)n(er)f(Routines)h(for)f(Standard)g(and)i (Generalized)e(Symmetric)e(Eigen)n(v)-5 b(alue)27 b(Problems)-409 1440 y Fb(Matrix/Problem)22 b(T)n(yp)r(e)100 b(Routine)p -459 1467 6040 4 v -409 1522 a(Symmetric)455 b Fa(PSSYEVX\()38 b(JOBZ,)f(RANGE,)g(UPLO,)f(N,)g(A,)g(IA,)g(JA,)g(DESCA,)h(VL,)g(VU,)f (IL,)g(IU,)g(ABSTOL,)h(M,)f(NZ,)g(W,)g(ORFAC,)h(Z,)f(IZ,)g(JZ,)g (DESCZ,)h(WORK,)g(LWORK,)g(IWORK,)g(LIWORK,)-409 1601 y Fb(Eigen)n(v)l(alues/v)n(ectors)499 b Fa(IFAIL,)37 b(ICLUSTR,)h(GAP,)e(INFO)g(\))-409 1680 y Fb(Hermitian)475 b Fa(PCHEEVX\()38 b(JOBZ,)f(RANGE,)g(UPLO,)f(N,)g(A,)g(IA,)g(JA,)g (DESCA,)h(VL,)g(VU,)f(IL,)g(IU,)g(ABSTOL,)h(M,)f(NZ,)g(W,)g(ORFAC,)h (Z,)f(IZ,)g(JZ,)g(DESCZ,)h(WORK,)g(LWORK,)g(RWORK,)g(LRWORK,)-409 1759 y Fb(Eigen)n(v)l(alues/v)n(ectors)499 b Fa(IWORK,)37 b(LIWORK,)g(IFAIL,)g(ICLUSTR,)h(GAP,)e(INFO)h(\))-409 1917 y Fb(Symmetric)455 b Fa(PSSYGVX\()38 b(IBTYPE,)f(JOBZ,)g(RANGE,)g (UPLO,)g(N,)f(A,)f(IA,)i(JA,)f(DESCA,)h(B,)f(IB,)g(JB,)g(DESCB,)h(VL,)f (VU,)g(IL,)g(IU,)g(ABSTOL,)i(M,)e(NZ,)g(W,)g(ORFAC,)h(Z,)e(IZ,)i(JZ,)f (DESCZ,)h(WORK,)f(LWORK,)-409 1995 y Fb(Eigen)n(v)l(alues/v)n(ectors) 499 b Fa(IWORK,)37 b(LIWORK,)g(IFAIL,)g(ICLUSTR,)h(GAP,)e(INFO)h(\)) -409 2074 y Fb(Hermitian)475 b Fa(PCHEGVX\()38 b(IBTYPE,)f(JOBZ,)g (RANGE,)g(UPLO,)g(N,)f(A,)f(IA,)i(JA,)f(DESCA,)h(B,)f(IB,)g(JB,)g (DESCB,)h(VL,)f(VU,)g(IL,)g(IU,)g(ABSTOL,)i(M,)e(NZ,)g(W,)g(ORFAC,)h (Z,)e(IZ,)i(JZ,)f(DESCZ,)h(WORK,)f(LWORK,)-409 2153 y Fb(Eigen)n(v)l(alues/v)n(ectors)499 b Fa(RWORK,)37 b(LRWORK,)g(IWORK,)g (LIWORK,)h(IFAIL,)f(ICLUSTR,)g(GAP,)g(INFO)f(\))2191 2470 y Fc(Meaning)26 b(of)h(pre\014xes)1817 2609 y Fb(Routines)e(b)r (eginning)f(with)g(\\PS")g(are)g(a)n(v)l(ailable)g(in:)2382 2747 y Fa(PS)36 b(-)g(REAL)2171 2826 y(PD)f(-)h(DOUBLE)h(PRECISION)1812 2905 y Fb(Routines)24 b(b)r(eginning)g(with)g(\\PC")g(are)g(a)n(v)l (ailable)g(in:)2329 3044 y Fa(PC)36 b(-)g(COMPLEX)2276 3123 y(PZ)g(-)g(COMPLEX*16)1662 3202 y Fb(Note:)c Fa(COMPLEX*16)27 b Fb(ma)n(y)c(not)h(b)r(e)g(supp)r(orted)h(b)n(y)f(all)f(mac)n(hines)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF